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. . 00:! .. 5 . I? .0 o. . Q 00. r . 0 0 . 00.. .0 . .. o . I“ 0 0 0 .. .. 0 o. . . .05....0 .. . .. .u‘ r‘ '00 '1 ‘
0 . 0 _ 0. l . . ‘. 0... . . 3... O. 0 u . I . .0 .... . 0.. . 04 0. '0. I . 00. . . . . . . .' I V t o
. 0M0. I . . . V . . 0 >04 .5: . . 00 .0 n I . . -I. . ... 0 .0. . . V .
n. . . I 0’ ...O. 0.0, ..0 ‘0I .000 -... 0' I.’ 0 0 . .. ~ U
o .530 ’I ...l. 0 ....I .... 0 mi. II . O .. 0.. 5'. 0 . .. V. ... I
t l . I . I ‘7 . . . . u .. 0 . . . . I ..
. 0- . 1 LI ... .. I 0 . .... 0. .. 0'0..- - Mount. 0 4 . .. n H .
I. ...-0 D o 0... .-00. ’ I. .I a .
060 . l .300 .. II I . . . . q . 0 .
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1 t.
. ‘0 \.J. .1530 I 0‘}
9... . ...0. O . .
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.I II‘ .. 0.
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0. O 0 I
. . ......

T'f-J": ? q; 7's
2

 

i ((3 ' LIBRARY
EHNOW ‘\ Michiga Tit-ate
University

 

 

 

This is to certify that the
dissertation entitled
Numerical Simulations of Waves in a
Magnetically Structured Atmosphere
presented by

Thomas Peter Espinola

has been accepted towards fulfillment
of the requirements for

Ph-D. degree in Physics

 

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Major professor

Date I? 4pr:l 'qgfi

MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

PLACE IN RETURN Box to remove this checkout from your record.
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Numerical Simulations of Waves in a

Magnetically Structured Atmosphere
by
Thomas Peter Espinola

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
tor the degree of
DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1988

 

Abstact

Numerical Simulations of Waves in a
Magnetically Structered Atmosphere

by
Thomas Peter Espinola

A physical model for simulating waves in a stellar atmosphere was
developed from a combination of basic fluid mechanics, plasma
physics, and electrodynamics. The model was three dimensional and
included the effects of gravity, magnetic fields, and viscosity. An
algorithm was developed to numerically implement this model. The
resulting program used an explicit time integration scheme based on
Runge-Kutta and a combination of finite difference and spectral
methods to evaluate the spatial derivatives. A number of numerical
boundary conditions were developed— the most successful used a
modified Sommerfeld radiation condition. The program was written
and coded in Fortran on a Vax computer. Additional routines were
written to evaluate the required fast fourier transforms and to graph
and display the data. The program was tested on a large number of
one and two dimensional problems for which the solutions were
known. These problems included acoustic waves. Alfvén waves,
magnetoacoustic waves, shocks, rarefactions, and contact
discontinuities. The numerical results agreed with the analytic -
solutions of the physical problems to within the precision requested of
the simulation. The program proved to be stable and robust for all the
problems attempted.

This program was then used to simulate three problems for which
analytic solutions are not known. All three simulations concerned the
propagation of waves in magnetically structured atmospheres and
may be applied to outstanding problems in solar physics. First, the
interactions of non-linear waves and a flux slab were studied. From
the result it is apparent that sources of shocks and rarefactions, such

as the solar convection zone, do not concentrate the magnetic field in
flux sheaths. Next I used the program to simulate the interaction of
non-linear waves with a flux tube. The results suggest that the
magnetic fields in flux tubes are also not concentrated by pairs of
passing shocks and rarefactions; however, a complete demonstration
of this would require a finer grid than was practical with the available
computing power. The last and most interesting problem simulated
was the interaction of oblique acoustic waves and a plane magnetic
interface. The results show that while a region of the interface does
remove energy from the waves by resonant heating, resonant heating
within a sunspot-photosphere interface cannot account for the
observed absorption of solar p-mode waves. Sufficient transmission
of energy occurs so that the results are consistent with the
observations if the absorption takes place by another mechanism in
the sunspot interior. The simulation suggests that a two dimensional
fine magnetic structure may be capable of absorbing the observed
fraction of wave energy.

 

Contents

Add—Ad
Uta-coma

2.1
2.2
2.3
2.4
2.5
2.6

3.1
3.2
3.3
3.4
3.5
3.6
3.7

4.1
4.2
4.3

Chapter 1. Physical Theory

Electrodynamics

Fluid Equations

Equations in Conservation Form

Addition of Stratification

Equations in Dimensionless Conservation Form

Chapter 2. Numerical Methods

Spatial Derivatives Using Finite Difference Methods
Spatial Derivatives Using Spectral Methods
Coordinate Transformations for Dynamic Grid

Time Integration of the Equations

Numerical Boundary Conditions

Numerical Viscosity

Chapter 3. Test Problems

Acoustic Waves

One Dimensional Shock Waves

One Dimensional Rarefactions

Shock Tubes

Alivén Waves

Magnetoacoustic Waves

Nonlinear Waves in Uniform an Atmosphere

Chapter 4. Waves in Magnetically Structured Media

Nonlinear Waves Incident Normal to a Magnetic Flux Slab
Nonlinear Waves Incident Normal to a Magnetic Flux Tube
Acoustic Waves Incident Obliquely to a Magnetic Interface

Chapter 5. Conclusion and Discussion
General Conclusions

Discussion of Acoustic Absorption by Sunspots
Future Problems

Appendices

Glossary of Symbols

Summary of Equations

Analytic Solutions to Test Problems
Auxiliary Programs

GMHD.FOR

Acknowlegements

Bibliography

iv

GINA

11
14

17
18
21
22
25
27
29

30
31
35
41
45
49
52
58

69
70
74
80

91

92
95
98

101
102
106
112
142

177

178

 

1
Physical Theory

The basic physical theory, developed in the following chapter, is a combination of
fluid mechanics, plasma physics, and electrodynamics. The theory is greatly
simplified by a number of assumptions about the atmosphere. The resulting eight
partial differential equations give a complete local description of the behavior of the

gas and fields. L

 

 

 

1 .1 Electrodynamics

The problems of interest concern the electrodynamics of moving media; therefore,
two frames of reference are of primary importance. The laboratory frame of reference
is at rest relative to the observer while the oo-moving frame is at rest relative to the
moving media. Quantities in the laboratory frame will be represented by unprimed
variables and those in the co-moving frame by primed variables. For a complete
glossary of symbols, see Appendix A.

First, consider the Maxwell equations which hold for either frame:

V0 D = P, (1.1a)
8B

Vx =-5-f (Lib)

V0 B = 0 (1.10)

mm = 433% (1.1a)

where D is the electric flux density in coulombs/meter, 99 is the charge density in
coulombs/m3, E is the electric field in volts/meter, B is the magnetic flux density in
Teslas, H is the magnetic field in amperes/meter, and J is the current density in
amperes/meterz. A well posed problem requires three additional equations. I will be
considering only linear, isotropic media, for which Ohm’s holds in the co-moving
frame

J' = oE' (1.2)
where o is the conductivity of the medium in mho/meter. The constituent equations
also hold only in the co-moving frame,

0' = eE' (1.3a)

B‘ = uH' - (1.3b)
where e is the permitivity in farads/meter and u is the permeability in henrys/meter.

These seven equations interrelate the electromagnetic field quantities and seem to
be uncoupled from the motion of the fluid; however, J may depend explicitly on the
velocity of the medium. The motion of the fluid is coupled to the fields through the
Lorentz force equation, which is covariant,

F = q(E + VXB) (1.4)
For the problems of interest, one may make the following simplifying assumptions,
which are known as the magnetohydrodynamic, or MHD, approximations (Hughes &
Young, 1966, §6.3):

1.

The fluid velocities will remain nonrelativistic. One may then replace ywith 1 in
the Lorentz transforms. Therefore,

 

 

E'= 15+sz (1.5a)
szH

D= D+ (1.5b)

H'=H-vxb (1.5c)

B'=B-"”c‘2IE (1.5a)

J'=J - pev (1.5a)

P: P,--‘%- (1. 5:)

The conductivity is very large so that E's-'0. The Lorentz transform then yields
E a —vxa and De—vxi-I/c2 a o.

The permitivity and permeability equal those of free space. Both these
assumptions are very good for astrophysical plasmas. The assumption that
e=eo sometimes breaks down for other uses of the MHD approximations.

The induced magnetic fields are small compared to the external fields. This
assumption, which follows for (1), yields 8' :8 as follows

8- _-Vx( chB)_ B+0r( _v_:)

 

     

Vc2E -
B'eB

Since u=p°, H = H' follows directly. For another permeability, H=_ H' may still be
a good approximation.

The displacement current may be neglected as compared to the conduction
current. Therefore, high frequency phenomena cannot be considered.

The current densities in the laboratory and co-moving frames are equal since
. 2
Ig _ I. _ I V.J ~ - V
J J pQV J (p: c2)v -J 0r(Jc7)
Also, the charge density in the co-moving frame, p'e, vanishes since the medium
is a conductor. Thus

 

J'=J=0'(E+VXB)
Moreover, o is assumed constant and independent of frequency, 8, and E but not
necessarily independent of temperature or other fluid variables.

3

The Maxwell equations with the MHD approximations thus may be written as:

 

v-D= v31 (1.6a)
an
VxIE-----a—t (1.6b)
V-B=O (1.6c)
VxH=J (1.6d)
and Ohm's law is
J=O’(E+VXB) (1.6s)

 

Using these and the constituent equations allows one to solve for the time evolution
of any of the fields. Of most interest here is the evolution of the magnetic field.
Solving Ohm's law for E yields

 

     

J
E = '6: “ V X B
Substituting VXH for J
V x H
E a G ' v X B
Using this in the curl E equation (1.6b) yields
V x H
T = V x(V xB)- 0'
Therefore, one gets the following for the time evolution of B:
68
—=Vx(VxB) VxMVx B) (1.7a)

where n: 1/ouo is the magnetic diffusivity. This equation may be simplified when
written using the Einstein summation convention.

33—?” ‘ng [B,v,-- Bji+v ”(3%? 35.)]: o (1.710)

If n is independent of position, one may use the ideJntity
VxVxB= V(V-B)-
to get

T'VX(VXB)+71VZB (1.70)

Equation 1.7 fully specifies the electrodynamics under the MHD approximations.

1 .2 Fluid Equations

A complete description of the MHD system requires equations describing the
behavior of the material particles as well as those describing the electromagnetic
fields. The media of primary interest are astrophysical plasmas. The plasma
equations can be quite complicated when the electrons, ions and neutral particles
are treated separately and the binary interactions are considered in detail. For
simplicity, I will consider only ideal plasmas which may be described by one-fluid
equations (Krall & Trivelpiece, 1973, §3.5). The equations describing such a plasma
reduce to those of a fluid with a local charge density.

First consider the continuity equation

a

a—fw-pv = o (1.8)
The validity of this equation may be shown by considering an arbitrary closed, fixed
volume, V, within the viscous, compressible fluid. It has a boundary (surface area)
av. Let M be the total mass, in kilograms, within this volume. Thus

r1 =va dzx

Where 9 is the mass density in kilograms/meter? Taking the time derivative of both
sides and remembering the volume is fixed,

THIS—tpdsx

However, the change of mass in the absence of sources or sinks must result from the

flow across the boundary. Thus

8M

at :16?!) -dA

where the negative comes from the convention that dA is directed outward causing
positive flows to decrease the mass within the volume. Equating these two forms for
the time change in mass yields

J'V-a—pdsx- =I-pv-dA

The integral on the right hand side may be 3converted to a volume integral using the
divergence theorem to yield

IV?“ = )vw pv)dx

LES—Ft) +V-3pv)dx =

Since the volume is arbitrary, the integrand must vanish everywhere; thus the
continuity equation is verified.

To derive the momentum equation, once again consider the closed volume, V. Let m

 

be the total momentum in this volume. Thus
3 I
m=I pV d X
and
3
5L“- -J(-§-{p v)d x (1.9)
a v

The change of momentum within the volume, in the absence of sinks or sources for
mass, results from the following three sources:

1. External body forces acting within the volume. Let f be the sum of all such forces.
I will postpone considering the nature of these forces until later. The time rate of
change of momentum resulting from these forces is given by

3
9.91:1de
at

2. The flow of momentum across the boundary given by
a_r_n2 :I—(pV)V-UA
at W

3. Pressure and viscous forces acting on the boundary;
5.31%; =I-PdA +[1-dA
av av
where ‘l: is the viscous stress tensor discussed later.

Thus the total change in momentum in the volume is given by
3
5L“ :I f d x +I—(pv)v-dA +I-PdA +I1-dA
5‘ v av v a

The three surface integrals may be combined to yield
3
abi? =I f d X +I-(pVV+P;-1)'GA
V av

where 1 is the unit tensor (order two). This surface integral may then be converted to
a volume integral to yield

am _ 3 - - 3
a -Ivf d x +IvV-(pvv+P; I” x
Equating this to the time rate of change of momentum from equation (1.9) yields

3
I [gipv- f+ V-(pvv+P;-1)] d x = 0
v
Once again the integrand must vanish since the volume is arbitrary. This gives the

momentum equation

Ba—{pV+V-(pVV+P,1,-L)=f (1.10)

The viscous stress tensor, t,is defined such that
dfv = jgdA

or, using the Einstein summation convention

where 1v is the viscous force acting on the boundary of the volume. For the form of
Tij- one must rely on a combination of experiment and theory to give (Huang, 1963,
pp 113-116)

_ aVi an 2 aVk aVk
Tij - §l[ m‘kai - 3611.3)?“ ] + {237k

where {1 and {2 are the coefficients of viscosity which may be functions of the fluid
variables.

The external forces of primary interest include gravity,

f9= P9 (1.11)
and the magnetic body force
fm=JxB=-BxJ=-BXVXH =-LBxVxB (1.1213)
2 0
f '(B'V)B _ VIBI
m _—”o —2“° (1.12b)

or in summation notation
. 3.3.

E, Jaxj ax, 211o

 

(1.120)

Combining these two external forces gives

2
(B V__)_B _VIBI
f P9+-—11° 2n,

The last term in equation (1.13) may be grouped with the pressure on the left side of
the momentum equation to yield

(1.13)

7§pv+V- '[pVV+(P+'ZB—u|: )1- 1']: pg +(—B-—- V)B (1.14)

Consideration of the energy equations begins with the first law of thermodynamics
dU = (id + 6w

which holds in the co-moving frame. U is the internal energy contained in dV, a small

volume moving with the fluid. Defining a as the internal energy per unit mass and

using -Pdv for work yields
f’ldez dq- Pdv

d8: 99 - EV
M M
The dq term includes the energy gained from external forces, viscous damping, joule

heating, as well as heat transfer. As the volume is allowed to become smaller and
smaller, one may make the following substitutions

_ r1
dV---p—zdp

run»
V 91-111- 10
mpvp

where Q is the energy input per unit time per unit volume. Combining these gives

de_ _lidP 0
dt pza‘t p (145)

as the time rate of change for e. To find the rate of change of e in the laboratory
frame, one must use

and

in equation (1.15) to yield

be -P Q___P Q
g—t+v-.Ve p2(-pV-v)+5— FV'V-r-fi

I am interested in finding the time rate of change of pa; so,

198 be .93 ._ 9-
g-tpez e —ta +p-a—tze +p pV-V+p v-Ve)

g—tpe= e— i):- -vae- PV- v+Q=e g—pwV-pv- -V-pev- -PV- V+Q
From the continuity equation, the first two terms on the right hand side equal zero so
g—tpez—Vpev -PV-v+Q =-Vpev -V-Pv+ v-VP+Q
Thus
%pe+V-[v(pe+P)]=V-VP+Q (1.16)

This equation determines the time rate of change of the internal energy. Q contains
the external heating as well as heat transport, so

0 : QT+ and... QEM +0vis
where
CIT = V-kTVT (1.17a)
8v
0 = 1,557 (1.17b)
- . 12 m III2 II 2
0: M=J.E=_=_T_ =—IVxB| (1.17c)
Po
Therefore the internal energy equation has the final formV
giPe+V-[V(pe+P)]: v--VP+VkTVT+Tij::‘j+-1|VXB|2 (1.18)

The Grad term has been dropped since radiative transport is not being considered.

An additional equation, which considers the evolution of kinetic energy, can be
obtained by taking the scalar product of the momentum equation and the the velocity.
This equation, although unnecessary to describe the fluid, is useful in verifying the
conservation of energy since it may be added to the internal energy equation to
obtain the total (mechanical) energy equation. The scalar product is easiest to

evaluate in summation notation.
a a
V,- gPVI ” V157} PVIVJ' +P6ij- TIj )= vifi

This may be simplified by using

a 1 2 . = a I 2 a _

to yield

5_1 2.1 222 2.1 2 -1. 25 BE- 3
atzpv 2v t+axj2pv vJ 2v fipvl+iavaifiJ1113va1
and
a 1 2_ 1 2 313 b 1 2 3P 3
__ V — - .—— .— . a . .
5129 v (57" a—x.pvJ)+§- -)Z'J.2pv VJ"'V1 axi viaxJTiJ thi
The termin the pflarentheses vanishes from the continuity equation. Thus
a 2 6P 3
5‘29V215‘xJ2p" VJ' Vja—xj’via‘xJTij ' Vifi
11 2 9:1 2 5P a 3V1
612’” +axJ-zp" Vj” VJa"><J-’ axJ-ViiJ'I TJ‘J'axJ- Viif
a 2 a V2 1 3P
3%!” +59%“ v J—vJ'tJJ.)= vaJ- 115M}- vJS-J-J-j (1.19)

Adding the internal energy equation (1.18) and the kinetic energy equation (1.19)
yields an equation for the total (mechanical energy),

a 2 a 2
3742‘” +pe)+é—§j[(-;—pv +pe+P)Vj_V111j]= VIfI+QT+QEM (1.20s)
or in vector notation
3 1 2
$913)! +Pe)+V-[V(%pv2+pe+P)-V-1-I<JVT]= v 1+ ileBI2
(1.20b)

Notice that the viscous heating no longer appears on the right hand side. Viscosity
only changes kinetic energy into thermal energy so does not appear as a source in
the total energy equation. Similarly, the resistive heating term would not appear as a
source if the “total” energy included the magnetic energy density, 82/2110.

These fluid equations are not complete since there is one more variable than
equation. The equations are closed using an appropriate equation of state. For the
completely ionized astrophysical plasmas being considered, the ideal gas law may
be used. For the variables chosen, the law has the form
P=Ir1lpe (1 21)

where y is now the ratio of specific heats. Use of this equation of state assumes that
all the internal energy is thermal. For other MHD problems, as well as for
astrophysical problems considering ionization, a more complicated equation may be
required.

10

1.3 Equations in Conservation Form

The eight equations which describe the time evolution of the system may be
simplified, both conceptually and computationally, by transforming them into the
conservation form. Each equation then assumes the form

8R
fi+V-F=S (1.22)
where R is the variable (such as por pe), F is the flux, and S is the source term.

The fluid equations assume this form if we make the following identifications:
1. For the continuity equation (1.8),
R: p F: pV S=0
Fi= 9V:

2. For the momentum equations (1.10), consider the x component equation. The
other components have the analogous form.

R: va Fi= pVxVi+P5xi-txi S=fx
3. For the internal energy equation (1.17)
R: pe F=(pe+P)v S:v-VP+Q
bP
FJ-z (pe+P)vJ- Sva + 37:0

The magnetic equations (1.7) also assume the conservation form if we make the
following identifications:

4. Once again consider only the x component.

BB 33-
Rsz FiszVJ-BJ-Vx+fl(a—x':-a‘) 8:0

These eight conservation equations completely specify the time evolution of the
magnetohydrodynamic fluid. They are written out in full, with the addition of
stratification, in appendix B.

Notice that these equations may be written as a single matrix equation with the form
3R
a—t“ +V. F“! S“
The program treats the equations this way with or=1 for p; a=2,3,4 for pvx, pvy, pvz;
or=5 for pe; and a=6,7,8 for Bx, By,Bz.

11

1.4 Addition of Stratification

In the presence of an uniform external gravitational field, the compressible fluid will
exhibit stratification. The equations, as derived to this point, have an equilibrium
solution giving the resulting stratification for any given temperature profile. I have
chosen to eliminate this explicit dependence in the equations, however, by a change
of variables. I believe this is preferable because it emphasizes the variation from
equilibrium, minimizes the range in magnitude of the fluid variables, and, perhaps
most importantly, reduces the dependence of the numerical equilibrium solution on
the numerical boundary conditions. The exact equilibrium solution satisfying the
equations under the approximations introduced by the the numerical scheme
depends on the choice of numerical method. For problems in which the variation
from equilibrium is small compared to the equilibrium values themselves, the errors
introduced by requiring the equilibrium solution to satisfy the numerical scheme may
exceed the numerical errors in the variation from equilibrium by orders of magnitude.
The numerical method at the boundary is often of a lower order than in the interior;
so, this effect is exaggerated near the boundary. This boundary effect on the
equilibrium solution may be removed by eliminating the explicit stratification in the
variables as follows.

First, include the gravity body force as given in equation (1.11) with the restriction the
g be constant in the negative 2 direction. In effect, the direction of gravity defines the
z direction. For an isothermal atmosphere, the equilibrium atmosphere is given by

p(z)=p(0)exp(-%) H=£P_9_ (1.23)

where H is the scale height. If one lets pJJ and Po be defined as follows:

(1.24)

 

where the subscript indicates the fluid variable has the stratification

12

 

divided out as in equation (1.24). The gravitational force term cancels
out of the momentum equations and the calculations are considerably
simplified. A similar technique can be employed for any case in which
the equilibrium temperature profile is explicitly known as a function of
height. The full equations giving the flux and source terms are
summarized in Appendix B.

13

 

1.5 Equations in Dimensionless Conservation Form

There are three primary reasons for stating the equations in dimensionless form.
First, by appropriate scaling of the dimensionless variables, one may avoid the
computational difficulties arising from very small or large numbers. The computer
program is less likely to suffer from rounding errors and the user is less likely to be
buried in numbers which are difficult to interpret. Second, one may discover the
dimensionless physical parameters which determine the problem. Third, the results
of a run may be applied to all problems which have the same dimensionless
parameters. The effects of scaling will be removed from the equations.

To make the equations dimensionless, I chose three scaling parameters. The first, M,
has dimensionality of density and is characteristic of the plasma densities in the
problem. The second, L, is a characteristic length in the problem, while the third, T, is
a characteristic time scale. If one makes the following substitutions:

..1
M
Z
x=iL9 g=—Jg_2 2=T9
2
8-8012
L
T T
V=V°—L- C=C°T

where the variable with the nought subscript are the variables from the previous
section and the unsubscripted variables are dimensionless, the equations are
unchanged in form. The value of these three constant were chosen so that the
density, the cell size, and the speed of sound each equal 1.0 for the equilibrium
solution.

The parameters which specify the problem must also be scaled as follows:

14

With these substitutions, the eight equations retain their form but the variables are all
dimensionless.

gagii'g—JBIVJB BMW-323 3:30] 0
g—f+v.pV:0

g—tpwv- [pVV+(P+ J2—ilj)1'1]‘ pg +(___B V)B

Jitpew-h V(Pe+P)]=V-VP+T ~~3JJ1L~¢|V=KBI2

Ij-ZTV‘:

A number of dimensionless parameters appear in the usual theoretical consideration
of fluid mechanics and magnetohydrodyanmics. These parameters determine the
nature of the problem independent of the physical scaling. A number of these
parameters, their physical interpretation, and their dependence on program variables
are outlined below (Hughes&Young,1966).

1. The Reynolds number (=LV0/fi) is a measure of the ratio of inertial to viscous
forces. It is specified in the choice of g; however, the real viscosity is vanishingly
small. Only the numerical viscosity is of importance and its size is determined by
computational considerations. In this program, the Reynolds number is always
large.

15

 

. The Frounde number (=V02/gL) is a measure of the ratio of inertial forces to
gravitational forces. It and the scale height are determined by the choice of g.

. The Prandtl number (=cp§po/kT) is determined by kT(see 1.18a) and g. It is a
measure of the ratio of viscosity to thermal diffusivity. As with the Reynolds
number, it is not physically significant for the problems considered but may be
used for computational purposes.

. The plasma [3 (=2uoP/BZ) is determined by the choice of B. This parameter,
which is the ratio of gas pressure to magnetic pressure, determines the Alvfén
speed and, therefore, the slow mode and fast mode magnetoacoustic wave
speeds.

The Strouhal number (=voT/L) may be of importance for fluid flows including

oscillations under the action of external driving forces (of period T). It is a
measure of the size of the oscillation to the characteristic length. For this
program, the Strouhal number is generally in the range of 10-100.

16

 

2
Numerical Methods

The eight partial differential equations derived in chapter 1 describe the time
evolution of the fluid in the laboratory frame. Several numerical methods must be
developed to use these equations computationally. Finite difference methods are
used to evaluate the 2 spatial derivatives while pseudospectral methods are used for
the x and y derivatives. The grid points are allowed to move arbitrarily in the z
direction, so a coordinate transformation is required. Several different scheme were
tried for time integration of the equations. l have chosen to use a fourth order Runge
Kutta like scheme which treats the equations as coupled ordinary differential
equations. Each of these numerical techniques is developed in the following
sections.

17

2.1 Spatial Derivatives Using Finite Difference Methods

I wish to approximate the 2 spatial derivative of a function whose value is known at a
finite set of discrete points. Assume the value of the function f 13 =f(z )3) is known for
all{£ : 2 =0,1,2,...,NL}. To find the finite differrence approximation to the
derivative 81/82, define L to be an independent, continuous variable. Also define
z(L)= z); and f(L)= f 3 for L=£. For other values of L, I only require that f(L) and z(L)
be continuous and sufficiently differentiable. Since f(L)=f(x,y,z,t),

o o
ar_araz+_a;a Jaawfla

aL'azaL ax L by L at L

_31
g;_ aL
5L

Therefore the problem reduces to finding the finite difference approximation for af/BL
and az/aL. Values of these functions are known for an evenly spaced (in L) set of
points.

Expand 1 about the point L=2 using a Taylor expansion to find an approximation forf
at the grid points L<£,

3 4

ar a’r 3231 32 51

2'4 f3 4 al- 8 5L2 3 31.3 + 3 3L4
1 4-3—bf 913-11%,221‘1
r -r-2—af+ .33__4.il+£fl
J z _111.1121 .121 ,1 a‘r
1" aL 2 aL 6 5L 24 314

where all the partial derivatives are evaluated at L=J2. Use has been made of the
fact AL=1 between adjacent points. Similarly for points with L>£.

f£+J=f2+ Igf+%-§L7 +%-§%§- +2i4$
f£+2=f£ 2%4-2? +.%.:_:.;. + -§-%:_4[
f2+3=f2+3fi+3§§ +%-§—:§- +%7_§%
fJ+4=fJ+4-§-|‘:—+8.aa_:§ +3336} +§§§g

This expansion can be inverted (neglecting terms of order greater than four) to give
the required relation between anal. and f at adjacent points. This inversion is most
easily done using matrix methods. As an example, consider the fourth order
centered difference. The earlier expansion becomes, in matrix form,

 

 

 

 

 

 

4 2 f

- -— — 2
{in (‘2? 1.31/1)
112-1 1 '1 —2' "3' 2‘2 5'2}
r = l o o o o 2—
i I I I 5L2
9+1 ‘ ‘ 7 ? 2‘4 23.31.
f 3 2 5'-
\ 2+2/ I 2 2 3 3 / \fl/
.aL“.

This has the form f=A-af. The solution is then af=A'1-f. A microcomputer was used
to invert the matrix to yield

 

 

 

 

 

 

f
/ a: \ / o o I o o \ (1J_2\
_ 1 -£ .2 --1_

M 12 3 0 3 I2 1'“
_afi _ -1 .1 -1 _1 -1

3

at __I_ _ _1_

3L 2 I 0 I 2 1M
4
\11/ \ I -4 6 -4 I / \rJJZ/
'aL ‘ 'J I I. .3

Considering only af/BL one finds

19

bf

‘51" = (1 31-2-81,» +82+1-+2)/12 (2.26)
Similarly
.33 - -

The same method is used to find the approximation for ar/aL near the boundary;
however, the presence of the boundary requires an off center scheme. The resulting
fourth order off-center differences are

(%E)=(-251+4812-3613 +1614 ~315 )/12 (22c)
(-§[- )2=(- -3r -Ior2 +1813 -6I'4 +I5 )/I2 (2211)
€-[-)NL_-=1(-1NL_4+61NL_3-Iar L_2+lOfNL_1+31NL)/12 (229)
%[-)NL= (3rNL_4- I fifNL_3+36fNL_2-481NL_1+2SfNL )/ I 2 (2.20

Sometimes a lower order method may be used at the boundary to minimize the
boundary effects.

These finite difference approximations are good to fourth order only for continuous,
four time differentiable functions. This limitation is sometimes important because
several test problems have discontinuous initial conditions.

20

2.2 Spatial Derivatives Using Spectral Methods

Derivatives in the x and y direction are evaluated using a pseudo-spectral method
and a fast Fourier transform (Gottlieb and Orzsag,1977). For simplicity, I will discuss
the x direction only. The y direction is completely analogous.

The function, f, is known at a discrete set of evenly spaced points ({xi} j=0,1,...,NX).
This function can be approximated be a complex Fourier transform of the form

NJ 2ni -
f(x): 2 ’fj e(Ax M)” (2.3)
J=0

If NJ=NX, the Fourier series will give the exact values for x=xj while giving a smooth
approximation to f between the grid points. One may take the derivative of this
function to yield

 

NJ 2711' -
Z x . —— JX
bf: 3:0 ofj( Aiflnixhegx NX) (24)

In my program a fast Fourier transform is used to find the coefficients in
equation (2.3) and the inverse transform is then used to reconstruct the derivatives
in equation (2.4). Since the Fourier approximation to f is continuous, this method
does not work well for some functions. The well known Gibbs phenomena makes
this method especially inappropriate for problems in which the function or its first
derivative is discontinuous. This becomes important for several of the test problems
which have discontinuous initial conditions.

21

2.3 Coordinate Transformations for Dynamic Grid

The equations developed in chapter one are for a system of coordinates at rest
relative to the laboratory. If I wish to evaluate the functions and their derivatives in
another reference frame, I will need to develop the appropriate transformations. I wish
to use the new system of coordinates associated with the moving, or dynamic, grid.
The grid is allowed to move in the z direction to facilitate driving waves from the
boundary as well as to allow the points to be concentrated in regions of interest.
Although there are occasions when it may be useful to allow the grid to move in the x
and y directions as well, the grid may then become twisted, which adds an
unacceptable level of complication. This sort of problem is associated with two and
three dimensional Lagrangian methods, which attempt to use the co-moving
coordinates.

Denote the laboratory coordinates as xi ({xi : x,y,z,t}) and the new frame coordinates as
xi' ({xi' : x',y',z',t'}). The coordinate transform, Fl, which may be nonlinear, is assumed
to be known

K x=x(x',g',z',t') \
l U=U(x',y',z',t')
Z=Z(x',g',z',t')
t=t(xfigfizfit?
\ z

N
I
N -

 

 

To transform the derivatives required by the equation, one needs the relation

a _ 3X- a
a7;_(_1. _

Define 8N as the matrix which transforms the derivatives in one frame to derivatives in
the other. It is then given by

~ aX‘
..= _J.
aN,J (ax;

22

therefore

/ \

(9:. w 9; al

' bx' bx' bx' ' '
9L .92 9; 9.1

... 09' 09' 09' bg'
a”’ 2x. 29. 0.2. _t
02' bz' bz' bz'

ax fl a: at .

I
\bt' bt' bt' bt'll

This matrix is nonsingular if the coordinate transform can be inverted to give xi'(xj). It is
assumed that transforms which are useful here can be inverted. Thus

This may be used to calculate the laboratory frame derivatives from any moving frame
denvafives.

For this program, x, y, and t remain the same as the laboratory frame while the grid is
allowed to move in the z direction. The new coordinates are {x,y,L,t} and the required
coordinate transform is

Z!
\
II II
N c x_
J

(X'.y'. L .t')

F‘NCCX

ll
('0'-

 

 

This yields

23

/ \
II I 0 sf. 0 \|
.. o I 33. o
3N = 02
0 0 3L 0
' 0 o 95. I '
\\ bt //
This matrix is easily inverted to give
/ \
.5;- Q?-
.l 1 0 bx' 0L 0 \-
I J‘- 9-z o

by' DL

0

(3N)-1= Oz
0 O I/ZTL 0
0

 

' -9; 9.2
\ 0 bt' DL 1,,

Therefore, one may evaluate the laboratory frame derivatives required in the
equations with

9_= 9. _ 9.2. 92 9.

bx bx' ax' bL DL (2.58)
9.= 9__9_z. 92 9_

59 09' 09' DL 9L (2%)
L- 9.2 L

bz ’ 1/aL bL (2.5c)
L - L- 0—2 Q’- L

at " bt' bt' bL bL (2.5d)

using only derivatives in the moving frame.

24

 

2.4 Time Integration of the Equations

The equations from chapter 1 and numerical methods from 2.1-2.3 are used together
to calculate the time rate of change of each of the eight gas variables. A numerical
scheme is still required for the time integration. A number of explicit techniques were
tried, including Euler’s, predictor-corrector, and leapfrog methods. The final program
uses the fourth order Runge Kutta like method outlined below. It was chosen
because it permits larger time steps and, more importantly, gives a measure of the
accumulating numerical error resulting from the time integration.

Using the equations of chapter 1 and the earlier derived numerical methods, I have
written a subroutine which returns the value of dR/dt using the variables at the current
time step so
is}? 5 f'(R(t),t)

A second subroutine uses these derivatives to find R at t+At using the following fourth
order Plunge-Kutta formula (Abramowitz and Stegun,1964, §9.2)

k1: At f'(R(t),t)

k2= At f'(R(t)+ k,/2,t+At/2)

k3= At f'(R(t)+ k2/2,t+At/2)

k4= At f'(R(t)+ k3,t+At)

k1 k2 k3 k4
RM”: “‘0 * F * a * ‘3' * '5' (2.6)

The resulting error is of order (At)5. By calculating a new value R1 using one step of
2At and comparing the result with the value, R2, obtain from two steps of At, one gets
an indication of the size of the error, E, as follows:

R, = R(t+2 At) + «9 (2M)5

R2= R(t+2 At) +2¢(At)5

£52¢(At)5=(R1-R2)/15

Where CD is a numerical constant (assumed independent of At) characteristic of the
equations. This can be used to choose the largest time step which produces a
desired accuracy, 60, (Press et al, 1986, §15.2). If the current step size results in an
unacceptably large error, it can be repeated with a smaller time step chosen as
follows:

25

1/4
E
Atnw = 0.9 At“ K12) (2.7)

The factor of 0.9 has been included as a safety factor to reduce the probability of
repeating a step because At is slightly too large. The exponent, 1/4, comes from the
desire to limit the error accumulated over all steps. If the error is acceptable, At can
be increased with the following

1/5
E
Atnew = 0.9Ato1 d(_Esz) (2.8)

This method provides a straightforward way to obtain the maximum At consistent with
a desired accuracy; moreover, it produces large time steps for the slowly changing
parts of problem and concentrates many steps during the rapidly varying portions.
The cost is an extra three evaluations of dR for each two At. The derivative must be
evaluated four times for each At. Thus the cost is about 3/8, which is generally made
up by the larger time step.

26

2.5 Numerical Boundary Conditions

The eight equations describe the behavior of the fluid throughout its volume;
however, the region treated numerically is restricted. Since the problems to be
considered are not strictly outflow, numerical boundary conditions will be required to
maintain numerical stability. Periodic boundary conditions are used for the x and y
boundaries as is implicit in the use of pseudo-spectral methods. The lower 2
boundary is used to drive the waves into the fluid and so explicit conditions will be
specified for most problems. The upper boundary may be open or reflecting. To
produce an open boundary, l have used a modified Sommerfeld radiation condition
similar to the method suggested by Orlanski (Orlanski, 1976, pp251-269).

It is assumed that a disturbance near the boundary obeys the wave equation

.51 fl -
at + C 52 '0 (2.9)

For each variable, t, this equation is used to determine the wave speed. If c>Az/At,
the flow is strongly outward and ft+At(NL) can be extrapolated from ft(NL-1). If c<0
then the wave propagation is into the region and the boundary conditions must be
given explicitly. When 0<c<Az/At, there is a outward moving wave (assumed to be
obeying equation 2.9). Any function of the form

f(Z,t):f(Z"Ct)
identically satisfies this equation so, the new value of ft+At(NL) is given by

t+At _ t+At_ _
f(zNL ,t+At)--f(zNL ct CAt)

=r(z;:“-cAt,t)

t

=1'(zt (2.110)

t+At_ t _ fl
NL-l’t)+(zNL zNL-l CA”(62)

NL-l

The new value of R is calculated and the required value of dR/dt is returned by the
subroutine calculating the time derivative. Note that each variable will have its own
characteristic wave speed and thus its own extrapolation. In the program p, Vx: vy, vz,
e, Bx, By, and B2 are chosen as the {f} rather than p, pvx, pvy, pvz, pe, Bx, By, and
32.

The dynamic grid greatly complicates the implementation of this method. Moreover,
the time derivative used in equation (2.10) is evaluated only to second order, so the
boundary may dominate in determining the time step. This becomes a problem when

27

a large amplitude wave (or shock) passes the boundary. If this occurs and is not the
region of interest, the boundary may be “stepped” over the shock by suddenly
truncating the region to exclude the shock. The number of grid points may be
permanently reduced or extra points may be added between existing points by
interpolation.

Another approach may be used for an open boundary when a single outgoing wave
velocity can be determined. The values of the variables at an interior point are noted
and the position of the point is propagated outward at the wave velocity. When this
point passes the boundary, its position and its values of the variables replace the
boundary values. A new interior point is chosen and the process repeated. This
method worked well for acoustic waves, Alfvén waves, and other cases with a single
well defined wave velocity.

To produce a reflecting boundary, two algorithms were used. The simplest was to
continually set the variables at the boundary to the initial values. A better method
was to set the values at the boundary to those of the next point towards the interior
and to reverse the normal velocity; that is,

ONE PNL-I

(Vx)NL=(Vx)NL-1

(Vz)NL‘-' ‘(Vz)NL-1

eNL=9NL-1

BNL=BNL-1
These conditions were used on both boundaries for some problems in which the
motion of interest was along the x direction.

Periodic boundary conditions were also occasionally used. In this case, I set
fNL=f4
fNL-1=f3
f2=fNL-2

f1=tNL-3
for each of the variables. These conditions were most useful for problems which

were independent of 2, such as the ID (in x) test problems.

The effectiveness of these conditions is discussed in chapter 3 for several test
problems.

28

2.6 Numerical Viscosity

Explicit numerical schemes for nonlinear problems are frequently subject to an
instability arising from a transfer of energy from long wavelength to short wavelength
oscillations. As a result, an oscillation with a wavelength equal to the grid spacing
eventually grows to swamp the motions of interest. A similar difficulty arises from the
presence of shocks. The standard way of handling these problems is to add
artificially large viscosity to the equations. I have already included viscosity in the
equations, so the these instabilities can be controlled by the proper choice of
coefficients in the equation

av av
T" = g {(137,- ex.) 2 3513' ext] C2 aii "al,:

The second coefficient, £2, is sufficient to control the short wavelength oscillation;
however, £1, may be needed in the presence of oblique shocks or transverse waves.

If the oscillations occur in the fourier (x) direction, they may be reduced or eliminated
by removing the highest frequency components. This fourier smoothing has much
the same effect as artificial viscosity.

As a result of the addition of this artificial viscosity, real shock waves are widened in

profile. The solutions in front of and behind the shocks are unchanged but the shock
itself is rounded. This is discussed in greater detail in section 3.2.

29

3

Test Problems

The program was tested on a number of problems for which the analytic solutions
were known. Each of the following sections contains one class of test problem and
the results of the computer simulations of that problem. The first few problems are
one dimensional, the next few are plane-wave, and the last few are two dimensional.

30

3.1 Acoustic Waves

The first and simplest test problem was to drive low amplitude sound waves from the
lower 2 boundary. First, a single sinusoidal (in time) pulse was started and followed
as it moved in the positive 2 direction. A sample graph of the resulting wave is given
figure 3.1. The wave traveled with the expected speed and pulse length. There was
some raggedness at the trailing edge of the pulse resulting from difficulty in stopping
the driving force at the correct time (which generally fell in the middle of a time step).
There was a reflection from the upper boundary with an amplitude of about 5%.

A similar simulation was performed with an acoustic pulse traveling in the x-direction.
Since the boundary conditions are periodic in x, the pulse was driven from an interior
point by sinusoidally varying the density and pressure, but not the velocity. One pulse
traveled in each direction. Numerical viscosity and fourier smoothing were used to
minimize the difficulties arising from the spatial discontinuity at the driving point. A
sample graph is given in figure 3.2. Once again, the wave traveled with the expected
speed and pulse length. The ragged trailing edge is again apparent.

Next, a continuous acoustic wave was driven from the lower boundary. The results,
as shown in figure 3.3, were as predicted by theory. After the initial transient
reflections passed the upper boundary, the reflections decreased in amplitude to less
than 2%. This improvement can be attributed to use of the the Sommerfeld wave
boundary conditions. These boundary conditions should work best when the net
outflow results from a wave with uniform and constant speed. They did not work as
well for the abrupt beginning and ending of the acoustic pulse.

31

 

1 .0090”-
1 .0080-
1 .0070-
1 . 0060*

I

d

Velocitg
QQOQQQQQQQQ

Temperature

‘

.9050»-
.0040»-
ease--
.eezed-
.eetou
.eoeo--
.etee--
.oege--
.0096»-
.0070»-
.9069-
.eese-L
.9949"
.9039“-
.eeze«
.9010-
.0069

. 0060--
. 0050--
. 0040--
.0030"
. 0020'-

.0010“

. 0000--

 

_III

I

-r'

-I.

 

 

J

j

 

0.00

1‘ i i m i i 3
10.00 20.00 30.00 40.00 50.00 60.00 70.00

Figure 3.1: An acoustic pulse traveling in the z direction. T=60.00
Amplitude, vp=0.01.

I
80.00

I
90.90

 

 

1.0070--
1.0060--
1.0050”-
1.0040--

:5

...:

'5 1 . 0030--

C

a)

0

1.0020-

1.0010-

 

 

1.0000-l

 

-L

0.0060-

0.0040-

0.0020-

9

 

 

I

0.0000-

I

Velocit

-0.0020-

-0.0040-

-0.0060-

 

1.0050-

1.0040-

‘

.0030-

.0020-

Temperature

‘

.0010-r

 

 

I

 

1.0000-_ L . . . . 4L

 

 

I I I I I I
0.00 20.00 40.00 60.00 80.00 100.00 120.00

Figure 3.2: Acoustic pulses traveling in the x direction. T=30.00
Amplitude. vp=0.008.

33

Velocity

Temperature

.0150-

.0100“

.0050-

.0000-

.9950d

.9900a

.9850‘

.0150-

.0100'

.0050‘

.0000‘

.0050~

.0100‘

.0150-

.0110-

.0080-

.0050-

.0020-

.9990‘

.99604

.9930-

.9900-

 

 

1

-r

 

dh

 

_I-

_I-
.1-
III)-

j

l

 

i i i u 3
10.00 20.00 30.00 40.00 50.00

i 1 l i
60.00 70.00 80.00 90.00

Figure 3.3: Acoustic wave propagating in the positive 2 direction. T=200.00
Amplitude, vp=0.02.

34

 

3.2 One Dimenslonal Shock Wave

Another simple problem which is analytically soluble is that of a piston moving into or
out of a uniform, non-magnetic fluid. If the piston moves inward, the fluid is divided
into two regions separated by a shock as pictured in figure 3.4.

 

V Regionl Region2
% :1 :2

 

Figure 3.4: Adiabatic shock resulting from an inward moving piston.
To the right of the shock, the fluid parameters are all the same as the initial fluid at

rest. To the left of the shock, the velocity of the fluid is that of the piston. The
solutions for the remaining post—shock quantities are (see Appendix C)

2 2 2 2'
14-74419 +/i +(7+1_Vp
4 2 2 4 2
c2 c c,

 

 

 

 

 

 

 

91 2
(i=5: v2
1+7-l—p-
2
2 (:2
v2
e1_ 7-1 P o(+i
'9‘- 1+ "2'
2 2 c1 0(-1
Ci
”:7:er

The values which result from my choices of the initial fluid parameters and some
values for the piston velocity are given in Table 3.1. When the code was used to
simulate this problem using no viscosity, the post-shock instability alluded to in
section 2.6 dominated. Sample output for such a case (vp=0.1) is given in figure 3.5.
This instability was eliminated by the addition of viscosity.

35

vp pt T1 Vs

0.01 1.01003 1.00667 1.00667
0.05 1 .05082 1 .03363 1.03389
0.10 1.10321 1.06793 1.06889
0.25 1.26868 1.17591 1.18046
0.50 1.56343 1.37914 1.38743
1.00 2.15139 1.91234 1.86852
vp—aao 4.00000 3/4vp2/y 4/3 vp

Table 3.1: Solution for piston problem, y=5/3, p2=1.0 , T2=1.0

Sample output is given for such a case (vp=o.1, §2=0.15) in figure 3.6. The
post-shock oscillations have been eliminated at the cost of widening the shock. For
weak shocks, the shock assumes a profile given by a hyperbolic tangent. In this
case, the thickness of the shock can be described by a single parameter 8 where
tanh(z/8) describes the shape of the profile. For weak shocks in an ideal gas,
(Landau & Lifshitz, 1959, §92)

f -I ge2

13161

The program was run for each of the first five piston speeds in table 3.1 with the
viscosity chosen to give a shock thickness of about 4.0. The program worked well in
each case. The average value of the gas variables behind the shock and the speed
of the shock varied from the theoretical values by less than 1%. The amplitude of the
reflection of weak shocks from the open boundary were of approximately the same
size as those from the acoustic pulses; however, strong shocks were reflected with
an amplitude of 10-15%. For problems in which this presents a problem, the
program would have to step over the shock.

36

 

1. 10*-

Density

.08"

 

 

Velocity

l
j,

 

 

1 . 09-
1 . 08-
1 .07-
1 . 06-
. 05-
1 . 04-

Temperature

1.03-
1.02-
1.01‘
1.00-

 

-b

1

-(r

 

-b

 

0.00

Figure 3.5 : Sample output for piston with vp=0.1 and no viscosity.

t=60.0

Z

37

1 l I 1 I T r r 1
10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00

 

Density

Velocity
PPPPPPPPPPP

Temperature

Figure 3.6: Computational results for shock wave in z direction.

. 06‘-

.05"

.e4~-

.03"

. 02--

.01“

 

 

 

 

uh
db
.1

A

A

 

'-‘n

 

i i I
10.00 20.00 30.00

vp=0.1 , §2=0.15 , and 1=60.0

l i 3
40.00 50.00 60.00

2

38

I
70.00

I
80.00

I
90.00

 

A similar problem was run in the spectral direction. Since the shock could not be
driven from a boundary, an initial discontinuity was used to create the shock. The
numerical domain was initially divided into two regions — one region had v=vp while
the other had v=-vp. Both regions had the same density and temperature. For t>0,
there is a region in which the gas has come to rest as well as the initial two regions.
A shock is driven into each of the initial regions. The solution is the same as the
piston problem given in figure 3.4 except that region 1 is at rest rather than region 2.
Because of the periodic boundary conditions, there is also an interface at which the
gases are moving away from each other. Rarefaction waves are generated at this
junction. These are considered in the next section.

The program was run for several of the piston speeds in table 3.1. Numerical
viscosity and fourier smoothing were used to minimize the effect of the spatial
discontinuity; however, very small time steps were still initially required. These small
steps and the requirement for a large number of cells (only one fourth of the domain
was useful for the shock problem) slowed the program considerably —- each run took
about eight hours on the Vax.

The post shock values again were within 1% of the theoretical values; however, the
program had some difficulty with modeling the shock itself. This is most evident in
the dip just behind the shock in figure 3.7. This was a remnant of the initial
discontinuity and decreased with time.

39

 

 

Density
3?

fl
0
Q
d
l
I

 

 

 

I I
D O
Q Q
N -.

l#

I I

-0 . 031v

Velocity
A
Q
9‘

 

 

. 03"

1.022-

Temperature

 

f

 

 

 

l

 

i i i i r fi‘ 1
8.00 16.00 24.00 32.00 X 40.00 48.00 56.00

Figure 3.7: Computational results for shock wave in x direction.

vp=0.1 , §2=0.15 , and t=45.0.

40

3.3 One Dimensional Rarefaction

If a piston moves oUt of a uniform fluid, a rarefaction wave travels into fluid. For
piston velocities less than the critical value, 2c/(?f -1), one expects the solution shown
in figure 3.8 (Landau &Lifshitz,1959,§92.).

y Region 1 .................................... Region 2

IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII

OOOOOOOOOOOOOOOOOO
IIIIIIIIIIIIIIIIII

IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII

OOOOOOOOOOOOOOOOOO
IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII

 

dUJD_.<_:o
ll
v<
UCD<‘O
N N N”
II II II II
U.(D°°‘o
N)

Figure 3.8: Theoretical solution for outward moving piston.

The fluid in region 1 has the same velocity as the piston. In region 2, the original
conditions persist. The two regions are separated by a rarefaction wave; the leading
edge of which moves with the velocity, va=co (the initial sound speed). The velocity
of the trailing edge, vb, may be either to the right or to the left and depends on v as
follows:

p

_ 7+1
”1: - Co"? ”v'

The value of the fluid variables in region 1 also depend on v and are given by

2 p
_ 7-1 1va 7:1
91-41(1'7'30)
3!
EM IXEI)?"
2 co

Table 3.2 contains the values of the fluid variables which result from my choice of
density and temperature and several piston speeds. Within the rarefaction wave
itself, the speed of the gas is given by

IvI=-.,—i,—(c.- %)
where x is the distance from the front of the rarefaction and t is the time since the
piston began moving outward.

41

The computer program was run for each of the piston speeds in table 3.2. Sample
output is shown in figure 3.9. The average values for the gas variables in region I
agreed with theory to within 1% for vpso.6. For greater piston speeds, the numerical
domain did not contain any of region 1. The rarefaction wave had the expected
profile and traveled with the correct speed. There was a small wave packet which
preceded the rarefaction wave which I believe was caused by the initial temporal
discontinuity. It may be eliminated by increasing the numerical viscosity; however,
this also changes the profile of the rarefaction. For purposes of this test, I felt the
small oscillations were not significant and the viscosity was not increased.

vp p p e vb
0.00000 1.00000 0.60000 0.90000 1 .00000
0.10000 0.90330 0.50645 0.84100 0.86667
0.20000 0.81304 0.42495 0.78400 0.73333
0.30000 0.72900 0.35429 0.72900 0.60000
0.40000 0.65096 0.29337 0.67600 0.46667
0.50000 0.57870 0.241 13 0.62500 0.33333
0.80000 0.51200 0.19661 0.57600 0.20000
0.80000 0.39437 0.12725 0.48400 -0.06667
1.00000 0.29630 0.07901 0.40000 0.33333
1.50000 0.12500 0.01875 0.22500 -1.00000
2.00000 0.03704 0.00247 0.10000 -1 .66667

Table 3.2: Gas variables after the passage of the rarefaction cause by a piston moving
at selected speeds.

The upper boundary performed fairly well in this test. It reflected the waves with an
amplitude of about 5%. The reflected rarefaction eventually passed through the
original rarefaction and continued to propagate through region 1. '

A similar test was run in the spectral direction. Since the rarefaction could not be
driven from the boundary, an initial discontinuity in velocity was used as explained in
section 3.2. Because the numerical domain also included a shock, the viscosity
could not be kept as low. The program was run for the lower piston speeds and the
numerical results for region 1 were again within 1% of the theoretical values.
Sample output is given in figure 3.10.

42

 

 

-0 . 20‘-

. 30--

Velocity
A

-0 . 50--

 

 

0.90"

Temperature

 

 

Figure 3.9:

l j I 5
0.00 20.00 40.00 60.00 80.00

Computational results for rarefaction wave in the z direction.
vp= -0.60 , §2=0.05 , and t=40.0.

43

 

 

‘
O
O
Q
J
1

 

Density
999999999
10
1.»

 

A
4

 

Velocity

Q
N
1
l

.011

I

 

 

 

HQGOOOGOQQQQ
0
(II
I
I

Temperature
.0 P P
to ID ID
7' ‘P '9

to
10
at

*1

 

A

 

1 i i
8.00 32.00 40.00 48.00

1
24.00 X

I
0.00 16.00

Figure 3.10: Computational results for rarefaction wave in the x direction.
vp=0.1 , t 2=0.15 , and t=30.0.

44

1
56.00

 

3.4 Shock tube

Another one dimensional problem which was run in both directions consists of two
semi-infinite regions of gas initially separated by a partition. The two regions are in
thermal equilibrium but have different pressures. At t=0, the partition is broken and
the gas begins to move due to the pressure gradient. For t>0, the gas is divided into
four regions as shown in figure 3.11.

 

 

 

Region I shfioack Region 2 Region 3 iii'giig'figg'ffgfi Region 4
—) 35:35:22::3:3:3:1:3:3:3:3:3:=:——)
Vs V, 533555555533555553555323535555555323 9.
P. P. P. P. ‘2
V1=0 V2 V3 Vt=0
91 e2 e3 535352333355353552535555553335335235 e‘
D, D2 D3 D;
s c b a

Figure 3.11: Theoretical solution for the shock tube.

Regions 1 and 4 are the remaining parts of the original regions and the initial
conditions persist in them. Regions 2 and 3 share the same velocity (toward
regiont) and pressure but have different densities and temperatures. The regions
are separated from each other by a shock wave moving into region 1, a contact
discontinuity moving with the gas in regions 2 and 3, and a rarefaction wave moving
into region 4. The leading edge of the rarefaction moves with initial sound speed.
The speeds of the remaining interfaces, as well as the values of the pressure,
density, velocity, and temperature in regions 2 and 3 depend on the initial pressure
ratio. The theoretical solution for this problem follows directly from the combination

ratio V3 P3 92 T2 VS 93 T3 Vb
1.50000 .12158 1.21972 1.12628 1.08296 1.08433 1.32492 .92059 .83790
2.00000 .20764 1.39727 1.22082 1.14454 1.14796 1.61280 .86636 .72315
4.00000 .41361 1.90515 1.45984 1.30504 1.31306 2.56319 .74327 .44853
5.00000 .47931 2.09391 1.53853 1.36099 1.36935 2.96595 .70599 .36092
6.00000 .53272 2.25753 1.60289 1.40841 1.41634 3.33765 .67638 .28970
8.00000 .61643 2.53295 1.70395 1.48652 1.49210 4.01247 .63127 .17810
10.00000 .68081 2.76108 1.78142 1.54993 1.55205 4.62006 .59763 .09226

Table 3.3: Theoretical values for shock tubes.

45

of the shock and rarefaction wave solutions. The values which result for my choice
of variables and several initial pressure ratios are given in table 3.3.

The program was used to simulate a shock tube with motion in the finite difference
direction for each of the initial pressure ratios in table 3.3. Numerical viscosity was
used because of the presence of the shock. The numerical results agreed with the
theoretical values given in the table to within 1%. The program did have some
difficulty modeling the contact discontinuity as can be seen in the temperature
oscillations near the discontinuity in figure 3.12, sample output for a shock tube with
an initial pressure ratio of 10. I believe this difficulty could be eliminated by using a
non-zero thermal conductivity (kT in eq 1.17a). I did not incorporate this change
since contact discontinuities only result from initial discontinuities, which can be
avoided when necessary.

The program was also used to simulate a shock tube with motion in the spectral
direction for the smaller values of initial pressure ratios (rati055). For the larger
values, the time steps became extremely small for the time just after t=0. Numerical
viscosity and fourier smoothing were both used to minimize the difficulties arising
from all the discontinuities. The numerical results were again within 1% of the
theoretical values except for the value of gas velocity in regions 2 and 3. The
average value there was still close to the theoretical, but there was a spatial variation
in the velocity caused by the contact discontinuity which caused the average value to
be uncertain by about 2%. Figure 3.13, which is sample output for an intitial ratio of
2, shows the difficulty in both the temperature and velocity graphs.

The difficulty experienced by both spatial methods in attempting to model the contact
discontinuity makes it essential to avoid this type of discontinuity in the initial
conditions. This program would need to be improved to give a useful simulation of
problems in which contact discontinuities cannot be avoided.

46

 

 

‘
Q
Q
Q
1
1
1
1
1
1

Density
99.99.9999
0
9

 

Q
Q
1
l
1

 

0.30--

Velocity

0.20-P
0. 10--

0 . 00--l

 

...
..(
u
_.

1.40“
1.30--
1.20--
. 10"

 

Temperature

.90"
.80"
.709-
.60". . . . . . .
I I I I l T
0.00 20.00 40.00 60.00 2 80.00 100.00 120.00

 

 

 

Figure 3.12: Computational results for shock tube in finite difference direction, t=30.
Initial pressure ratio=10:1.

47

 

Velocity
Q

o d
1

Temperature

. 10‘-
.00“
.21--

. 18"

.0691-

.03”-

. 00--

d
I

. 96‘-
. 92‘-

.331-

 

 

 

j

 

j

 

 

1 1 1
72.00 80.00 88.00

i i i 1 1
96.00 X 104.00 112.00 120.00 128.00

Figure 3.13: Computational results for shock tube in spectral direction, t=20.

Initial pressure ratio = 2:1.

3.5 Alfvén Waves

With the addition of a magnetic field, several new wave modes are possible. The
simplest of these are Alfvén waves which propagate along the field lines with a

velocity A, where
2

A‘= BO
61%

These waves are transverse oscillations in the velocity and magnetic field — the
density, temperature, and longitudinal components of the velocity and magnetic field
are undisturbed. In simulating this mode, I used the usual initial conditions and

Bz=Bzo

Bx=on= 0.0
An Alfven wave was driven from the lower boundary by sinusoidally varying Bx and
vx as follows:

 

Vx = Vo Sin(C01)
and Bx = -Bzo vo sin(oot)/A
The program was run for several choices of Va, which effects only the amplitude of
the wave, Bzo,which effects the Alfvén speed, and 0). Sample output showing the
spatial variation in B)( and vx is given in figure 3.14. The remaining variables were
constant (20.05%) throughout the numerical domain and the wave propagated at the
theoretical Alfvén speed to within 1%(the accuracy of determining the group velocity).

The program was then run with propagation in the x direction. The waves were
generated by varying the transverse velocity. Both fourier smoothing and artificially
large viscosity (9, in this case) were required to overcome the spatial discontinuity at
the driving point. I obtained results equal in accuracy to that of the wave propagating
in the z direction.

Next the program was used to simulate an Alfvén wave propagating at an angle to
the magnetic field. The background magnetic field had both x and 2 components and
the wave propagated in the x or z direction. The y components of the velocity and
magnetic field varied while all the other variables remained constant. Sample output
is given in figure 3.15. Once again only By and vy varied - the other variables
remained constant (:l:0.05%). The wave velocity was once again within uncertainty of
the theoretical value(i1%). In this case, the coupling between the driving point and
the wave was not as good as in the case of the propagation along the field lines. The

49

driving point in figure 3.15 oscillated with an amplitude of 0.020 in vy. The wave,
however, had an amplitude of only 0.017 (or 85% of expected). No other modes
were excited and I am not sure of the cause for this decoupling.

 

0 . 0 100-
0 . 0080-
0 . 0060-
0 . 0040-
0
0

I

I

. 0020-

U

 

. 0000'
-0 . 0020-
-0 . 0040-
-0 . 0060‘
-0 . 0080‘

I

I

 

0.03"

0.02"

0.01"

 

X
m 0.00"

-0.01-

-0 . 02-"

-0 . 03“

 

 

 

‘ 1 1 1 1 1 1 1 1 fi'
0.00 10.00 20.00 30.00 40.00250.00 60.00 70.00 80.00 90.00

Figure 3.14: Alfvén wave propagating in the z direction. Bz=1.0 ,Az= 0.282 ,t=150.0.
B: 15.08 A = 0.282

50

 

0.0160--
0.01209-
0. 0080--

0. 0040--

> 0 . 00009-
-0.0040"-
-0 . 0080‘-
-0.0120‘-

-0.0160--

 

1b
1
41-
q
.l

8.06--
0.04--
002-)

3
00 0.00--
—0.02--

-8 . 044'-

 

 

-0.06--, .
I I I I I I I I I
0.88 28.08 40.00 60.00 8800210000 120.80 140.00 160.88

 

Figure 3.15: Alfvén wave pulse traveling at 45° angle to field lines. Bx=3'.0 , Bz=3.0 ,
B= 1.676 ,A= 1.197 ,Az= 0.846 ,t=150.0

51

3.6 Magnetoacoustic Waves

The introduction of a magnetic field causes the acoustic mode of oscillation to split
into two modes. These two modes, called the fast mode and slow mode, consist of a
coupled oscillation in all the variables except those tangential components which
vary in the Alfvén mode of oscillation. The phase velocities of these two modes are
given by (Hughes & Young,1966)

 

 

 

 

 

 

1.
2
A2+A2+c2 2 2 2 2
a =:l: x z 0.. Ax+Az+C° 22
- C A
slow 2 2 o z
1.
2
A2+A2+C2 2 2 2 2
a = X z 0 + AX+AZ+CO _ CZAZ
fast 2 2 o 2

where co is the sonic speed, Az is the Alfvén speed in the direction of propagation,
and A)( is the Alfvén speed in the direction perpendicular to the direction of
propagation. The coordinate axes have been chosen such that By=0.0 (without loss
of generality).

For propagation along the magnetic field (Ax=0.0), there is only one mode of
oscillation, in which a=co. This case was easily simulated by the program. The
waves were driven exactly as they were for the acoustic wave in section 3.1.

For the general case, I tried several different ways of driving the waves. When the
transverse magnetic field (Bx) and velocity were varied and the remaining variables
held constant at the driving point, both modes were generated simultaneously.
Figure 3.16 shows sample output for such a case. The relative strength of the modes
depended on the relative amplitude of the oscillation in velocity and magnetic field.
A considerable amount of “noise” was also generated which was reduced by using
non-zero viscosity and magnetic diffusion.

52

Density

.0000

 

.0050‘
.0040-
.0030‘
.0020-
.0010-
.0000-
.9990-
.9980-

.0030‘

.0020-

.0010‘

.0010'

.0020-

.0000-
.9980'
.9960-
.9940-
.9920‘
.9900‘
.9880~

.98601

I

I

I

 

0.00

l l l j l l l

I I I I I
20.00 40.00 60.00 80.00

i i
2 100.00

Figure 3.16: Simultaneously generated fast and slow mode magnetoacoustic wave pulses.

t=60.0 , «1:.01, §1=.05.

53

Next I solved the linearized equations to find the relations in the variation of p ,vx, vy.

e, and Bx to get
vzzw sin(wt)

p=p,(1+%"-sin(wt))
-A A
v = _I—z— w sin(cot)
X
oz- A2z
e=e°[ l+(7-1)%sin(wt)]
a

2_ A22

 

B=B (1+ wsin(cot))
X X0

0
where a is either afast or aslow. When these relations were used to drive waves from
the lower 2 boundary, only the chosen wave was generated. The program was used
to simulate fast and slow mode magnetoacoustic waves propagating at a variety of
angles to the magnetic field (by vary 8)( and 82). Figures 3.17 and 3.18 show some
typical output (propagation at 45° to the field). A numerical instability developed in
the propagation of the slow mode wave which was eliminated by using a non-zero
magnetic diffusivity, n. The value used (11:0.01) still corresponded to a very low
resistivity.

For propagation in the spectral direction, I used both of the methods outlined above.
When just the tangential velocity and field were varied, the results were much the
same as in the finite difference direction. There were fewer difficulties with the
“noise”, probably resulting from the fourier smoothing; therefore, I did not need to
use a non-zero magnetic diffusivity.

When the driving point was oscillated using the theoretical relations above, both
modes were generated. Since the relations could be correct for only one of the two
directions in which the waves propagated, the waves propagating in the “wrong"
directions were about equal in amplitude. In the forward direction, the chosen mode
dominated. Sample output is shown in figure 3.19, which was driven as a fast wave.
Notice that the slow mode waves appear symmetrically. This suggests to me that it
may be possible to drive only the fast mode if the correct driving mode can be
determined.

54

 

1.0110"
mesa--
331.0050"
3:
g 1.092e--
m
0 3999--
peso--

0
0
0 . 9930--
0 . 9900"
0

. 9870--

 

 

0 . 0090--

0 . 0060--

0 . 0030"

0 . 0000--

 

-0 . 0030-

-0 . 0060-

 

3.06‘
3.04-
3.02-

x
CD 3.00-
2.981

2.96-

F

 

 

2.94-1

l l I l A l l

 

I I I I I I I I I JI
0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00
2

Figure 3.17: Fast mode magnetoacoustic wave. t=50.

 

 

0 . 9990-

0 . 9960-

. 9930-

. 9900-

. 9870-

. 9840‘-

Density

.9810?-

fi

. 9780-

. 9750- .
. 0000

 

-0 . 0020-

I

-0 . 0040'

-0 . 0060'-

-0 . 0080-

1

-0.0100‘

-0.0120-

 

3.044"

3.03--

3.02"-

3.01"

 

 

3.00 . . . . ' “‘—‘-—.‘-—'

l A

I I I I 4I I I
0.00 20.00 40.00 60.00 2 80.00 100.00 120.00 140.00

 

Figure 3.18: Slow mode magnetoacoustic wave pulse. 1:150 , n=.01 .

. 0 toe-r
. 0080-
. 0060-
. 0040u
. 0020j
. 0000-

. 9980‘

. 0020-
. 0000‘
. 0020-
. 0040-
.0060-

. 0080!

 

I

I

I

 

I

I

-i-

cli-

 

3.05"

3.04-

3.03--

3.02--

3.01“-

3.00--

 

 

 

 

 

1
20.00

%
40.00

3
60.00

A:
80.00

i
100.00

E
120.00

 

I1.

X

Figure 3.19: Magnetoacoustic wave pulses generated by driving with +x fast mode
phase relations. t=30 , 11:0.

57

3.7 Nonlinear Waves in Magnetic Atmosphere

The last class of test problems considered was that of shock waves and rarefaction
waves propagating through a uniform magnetic atmosphere. Theoretical solutions
are available for the simpler cases and the results for the cases for which theoretical
solutions are not known will be needed for comparison in chapter 4. For the simplest
case of waves propagating in the direction of the field, the problem may be solved
theoretically. The waves propagate exactly as they would in the absence of the
magnetic field and the the solutions from section 3.1 and 3.2 still pertain.

For the case where the waves propagate perpendicular to the field, theoretical
solutions are also available. The solution to the piston driven shock given in
appendix C, can be extended to include the magnetic field; however, the resulting
equation in a, the ratio of densities, is cubic. Analysis of this equation revealed one
real root which was found numerically for the chosen initial fluid parameters and
several values for the piston speed. These values are listed in table 3.4. The results
obtained from the simulation were in excellent agreement (i0.1°/o) with these values.
Sample output is given in figure 3.20.

vp/c p1 T1 vs B1

0.010 1.00765 1.00507 1.31539 3.02296
0.050 1.03852 1.02572 1.34818 3.1 1555
0.100 1.07799 1.05158 1.38227 3.23396
0.250 1.20061 1.13274 1.49617 3.60182
0.500 1.41553 1.28602 1.70328 4.24658
1.000 1.85403 1.70618 2.17092 5.56209

Table 3.4: Theoretical Solution for Shock— Bz=3.0

58

d
I

eraturg

D-

_Tem

. 07--

. 06‘-

. 05*-

.a4--

.03--

. 02‘

.01n

. 00~

QGGOQOZOQQQ

 

Q
.3
1
I

 

 

 

 

 

I
g.
b l
l:
‘1
‘1

 

 

 

 

 

 

teioa zafao setoa 40190 saiaozsataa 701.00 80.00 90'.ea 100100110108

Figure 3.20a: Shock moving normal to a uniform magnetic field. Vp=0.1, Bz=3.0

59

 

 

3.21--

3.181-

3.15--

On)

G

O
l
I

3.06”-

3.03--

 

 

3 06“ AA.MAAAAAA
. I I I J I I I I I I

I I I I I I I I I I I
10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00100.00110.00
Z

 

 

 

Figure 3.20b: Shock moving normal to a uniform magnetic field. Vp=0.1 B2=-3.0

l was unable to find a solution in the literature for a rarefaction wave in a magnetic
fluid. The solution used in section 3.3 (Landau & Lifshitz, 1959, §92) is a similarity
solution with the acoustic speed as the characteristic velocity. I assumed that the
presence of the magnetic field would not introduce a characteristic length and,
therefore, the similarity solution could still be used. The characteristic velocity, in this
case, would be the magnetoacoustic wave speed (only one mode is possible for
propagation perpendicular to the field). Table 3.5 lists the values given by this
solution for several piston speeds.

vp/c p1 T1 vb B1

0.010 0.99239 0.99492 1.29670 2.97717
0.050 0.96232 0.97472 1.24340 2.88696
0.100 0.92559 0.94976 1.17670 2.77677
0.250 0.82105 0.87682 0.97670 2.46315
0.500 0.66483 0.76174 0.64337 1.99449

Table 3.5: Theoretical values for rarefactions— Bz=3.0

Once again the computer simulation of this problem yielded results in agreement

with the analytic values (10.1%). Sample output is given in figure 3.21.

60

 

Densitg
Q 0 0 O 0 Q Q o
. . . . b» . . .
-?

AAAA

 

 

erature
.0
o
9

Temp
9 .0 P
(D m (n
5 f 1'

Q
«I
“P

 

 

-0 . 20-

I

I

 

AAAA

 

 

A

 

Figure 3.21a: Rarefaction moving normal to a uniform magnetic field. Vp=-0.5 83:30

T

i i u i
0.00 20.00 40.00 2 60.00 80.00 100.00

61

 

NX

NNNN

AA AA
I 06-. vvv

.80-
.70"
.60"
.50"
.40"
.30"
.20"
. 101-
.00"

NNNNNQ

 

 

o
o
f

 

J I l I_

 

 

 

I7 I I I I I
0.00 20.00 40.00 2 60.00 80.00 100.00

Figure 3.21b: Rarefaction moving normal to a uniform magnetic field.

For the most general case of a shock propagating at an angle to the magnetic field,
the equation in a, the ratios of densities, is cubic in a2 and cannot be solved
analytically. An analysis given by Hughes and Young (Hughes & Young, 1966,
§10.4) reveals that there are three real solutions for a and vs. The first solution is
called a slow shock since it reduces to a slow mode magnetoacoustic wave in the
small amplitude limit. The second solution, called a fast shock, reduces to a fast
mode magnetoacoustic wave. The third solution, call an intermediate shock, has a
shock speed which approaches the Alfvén speed in the small amplitude limit.

Regioni Region2 Region3

oooo
0".
c 0'
:-:-°

'0

 

I
I I.
'Sg3- 03::-

 

 

’o::'

 

W

Figure 3.22: Shocks resulting from driving a piston at an angle to the magnetic field.
Figure 3.22 illustrates what happens when a piston is driven into the fluid at an angle

to the magnetic field. The boundary values specified at the piston did not include the
transverse velocity or magnetic field; therefore, a frictionless piston was being

62

modeled. In this case, any or all of these type of shocks may result from the piston.
Output for a sample run is given in figure 4.4. As can be seen, two shocks result and
propagate independently. The first and larger shock is a fast shock and the second
is a slow shock. No intermediate shock was generated. The results for the case

Bx=Bz=3.0 (8:45°) are summarized in the following table.

Vp

0.010
0.050
0.100
0.250
0.500
1.000

P1
1.0081

1.0411
1.0837
1.218
1.460
1.98

in

-0.0025
-0.0112
-0.0208
-0.042
-0.063

-0.06

1}
1.0054
1.0273
1.0555
1.145
1.315
1.77

3.023
3.112
3.221
3.541
4.062
5.03

132
1.0062

1.0309
1.0624
1.161
1.338
1.724

-0.0046
-0.0226
-0.0443
-0.1048
-0.1908

V22
0.0089
0.0443
0.0887
0.224
0.454
0.935
Table 3.6: Shocks caused by driving a piston at 6=4S° to a uniform magnetic field.

T2

1.0041
1.0205
1.0414
1.107
1.235
1.604

Bx2
3.028
3.140
3.281
3.709
4.438
5.89

These results can be used to determine what transverse velocity must be specified to
generate any fast shock. This was done, iteratively if necessary, to obtain or. and vs
as a function of vp for the fast shocks. Since the slow shock in figure 3.22 moves into
a region whose fluid parameters were determined by the fast shock, it was more
difficult to determine what transverse velocity was required to produce the desired
slow shock moving into the undisturbed fluid.

Vp
0.000
0.010
0.050
0.100
0.250

0.500

Table 3.7: Post shock fluid values for a oblique fast shock.

p
1.0000

1.0069
1.0351
1.070
1.180
1.377

Vx

0.0000
-0.0052

-0.025
-0.049
-0.116
-0.206

63

T
1.0000
1.0046
1.0232
1.0470
1.122

1.268

BX
3.000
3.032
3.157
3.317
3.794
4.58

1.453
1.482
1.519
1.634
1.839

[empegaturg

‘

slooaoo

@009

. 0070--
.0060‘
.0050"
.0040--
.0030"
. 0020--
.0010--

.0000-

.0050-~
.0040--
.0030"
.0020“
.0010‘-

.0000-

.0090‘
.0080-
.0070-
.0060‘
.0050-
.0040-
.0030'
.0020J
.0010

.0000"

 

I

 

I
p

 

 

 

‘l‘

 

 

I

I

I

I

I

 

—-)-

 

A

.L

A

 

 

I
0.00

I
20.00

40.00

1 1
60.00 2 80.00

T

100.00

I
120.00

Figure 3.233: Piston driven oblique shocks. Vp=0.01. Bzx=Bzz=3-°

64

I
140.00

 

 

.0000“

.0010-

. 00201

.0030-

.0040‘

I

 

 

 

.0280.

.0240‘

.0200-

.0160-

.0120-

.0080-

.0040‘

I

I

I

I

 

.0000‘

41-

AA

 

'

 

A

 

0.00

%
20.00

1
40.00

4
100.00

T
120.00

I
140.00

60.90 2 80.00
Figure 3.23b: Piston driven oblique shocks.

To isolate the dependence of slow and fast shocks on the angle between the shock
and the magnetic field, I ran the piston at the same speed at different angles. The

results are summarized in table 3.8.

9 Pt Vx1 T1 91 92 Vx2 V22 T2 92
0 1.1032 0.0000 1.0679 0.00

15 1.1013 -0.0073 1.0668 16.30 1.0886 -0.0334 0.0914 1.0585 17.95
30 1.0967 -0.0123 1.0639 32.12 1.0823 -0.0351 0.0918 1.0544 33.28
45 1.0906 -0.0142 1.0600 47.30 1.0805 -0.0293 0.0955 1.0533 47.86
60 1.0849 -0.0126 1.0563 61.86 1.0794 -0.0206 0.0984 1.0525 62.08
75 1.080 -0.0077 1.0537 76.05 1.0784 -0.0105 0.0997 1.0519 76.07
90 1.0779 0.0000 0.1000 1.0517 90.00

Table 3.8: Post shock fluid values for different angles between piston and field.

65

When the piston was withdrawn from the fluid, two rarefactions resulted. This case is
illustrated in figure 3.24. The two waves have leading edge velocities equal to the
slow and fast magnetoacoustic wave speeds for the region, so I have called them fast
and slow rarefactions. Sample output is given in figure 3.25.

IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII

IIIIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIII

 

 

 

, . E E E E 5 SI owgigigigégég - 33535555353 fa st 355335553; -
// Regtonl rarefaction Regtonz rarefaction Reg1on3
/ p Vc 35323353555E;EgEgEgEgEgSgEgEgSgEgSgi slow V5353553555E;Egigigigigfgégigigigégigi fast
V1=vp §s§s§s§2§2:2§2§2§2§5§s§s§2§s§2§2§5§s V2 Vs=0
91 s:33sis§a;a§s§s§s§s§2§s§s§5§s§s§s§2§ e2 es=e°
p1 '=:3:E:E:E:E:E:E:E:Eziziriziriziziti p2 :3:3:5:3:5152:53323355:3383 '33“).
C b a

Figure 3.24: Rarefactions resulting from withdrawing a piston at an angle to the field.

The results of running the simulations for a number of different piston velocities are
given in the following table.

Vp P1 Vx1 T1 8x1 92 Vx2 V22 T2 Bx2
-0.010 0.9921 0.0028 0.9947 2.976 0.9938 0.0047 -0.0090 0.9958 2.972
-0.050 0.9607 0.0139 0.9736 2.883 0.9695 0.0238 -0.0446 0.9795 2.859
-0.100 0.9219 0.0275 0.9473 2.771 0.9399 0.0481 -0.0883 0.9596 2.722
-0.250 0.810 0.068 0.869 2.450 0.858 0.124 -0.213 0.904 2.329
-0.500 0.640 0.141 0.744 1.970 0.747 0.259 -0.395 0.824 1.735
-1.000 0.364 0.316 0.527 1.149 0.625 0.510 -0.617 0.732 0.868

Table 3.9: Flarefactions caused by driving a piston at 8=45° to a uniform magnetic field.

66

.04--

. 03“-

.01“

 

 

10
'1"
l
l
1

ID
to
9.

 

 

.1)-
.11-
-
-1

 

1

 

V"

i i
0.00 30.00 60.00 90.00

I
120.00

1
150.00

1
180.00

Figure 3.248: Sample output for oblique rarefactions. vp=0.1 , B3X=B3z=30

67

 

Temperature
.Q 0 O

. 00"

.99“

. 98"-

.97"

. 95--

. 00"

.96"

.92“

.88"

.84--

. 80"

. '26--

 

A A A AA A A A AA AA A A AAAAA
vv

A

1!

 

 

db

-1-

 

I I I I
0.00 30.00 60.00 90.00 120.00

Figure 3.24b: Sample output for oblique rarefactions.

These results for nonlinear waves in a uniform magnetic field will be used for
comparison when I consider the propagation of nonlinear waves through

magnetically structured media.

68

I
150.00

I
180.00

 

 

4

Waves in Magnetically Structured Media

In the first three chapters, 1 have developed and tested a computer program for
modeling nonlinear disturbances in magnetic fluids. The program was next used to
model several problems for which thetheoretical solution are not known. These
problems all consider waves in magnetically structured media.

 

69

4.1 Nonlinear Waves Incident Normal to 8 Magnetic Flux Slab

The program is able to model the propagation of nonlinear waves through a
magnetically structured atmosphere. Theoretical solutions to problems of this type
are generally unavailable.

The simplest problem of this type is the normal incidence of a nonlinear wave on a
magnetic flux slab. This slab consists of region of high magnetic field embedded in
an environment of weaker magnetic field. The simple slab considered here has a
magnetic field given by

30(2) {50“. |z-z°| < 5

0 , IZ'ZOI >6

The region is initially isothermal and the environment has a higher density to balance
the total internal and external pressures. The wave, propagating in the z direction, is
normal to the surface of the slab.

The computer program was used to model this problem for a variety of shock waves
and magnetic field strengths. From the results of these simulations, I can draw the
following conclusions:

1. Each time a wave crosses an interface, a reflection is possible. The
magnitude of the reflection depends on the strength of the magnetic field
within the slab. As a result of these reflections, the slab sends out wave
pulses in both directions. The amplitude of these waves decline with each
reflection and only the initial reflections were easily followed back through
the environment. The amplitude of the pulses depend on the strength .of the
shock and the magnitude of the magnetic field. The width of the pulse also
depends on the width of the slab.

2. After the shock passes and the slab reaches its final state, the slab is again
in pressure equilibrium with the environment. It is narrower and the
magnetic field strength is increased. The slab is no longer in thermal
equilibrium with the environment. For all cases tried the slab was cooler
than the environment. The density of the slab is increased by a lower ratio
than that of the environment.

3. The slab is still at rest relative to the environment.

4. The final state of the environment is the same as it would have been if the
slab had not been present. The two regions of the environment, one on each

70

side of the slab, are still identical.

5. The final state of the slab is denser, hotter, and has a stronger magnetic field
than if a shock had passed through a uniform region with the same initial
conditions.

The results of the simulations are given on the following table. The ratio of densities,
aim is also the ratio of final magnetic field to initial magnetic field. The last column is
the ratio of final to initial width for the slab.

leab vp “in “out Tslab/Tenv waN i
0.10 0.100 1.1031 1.1032 0.9996 0.905
0.50 0.010 1.010 1.010 0.9999 0.990

0.050 1.051 1.051 0.9998 0.951
0.100 1.102 1.103 0.9993 0.907
0.250 1.266 1.268 0.998 0.789
1.00 0.010 1.0099 1.0099 0.9999 0.990
0.050 1.050 1.050 0.9997 0.953
0.100 1.101 1.102 0.9992 0.908
0.250 1.261 1.268 0.998 0.796
3.00 0.010 1.0093 1.010 0.9995 0.991
0.050 1.047 1.051 0.9976 0.954
0.100 1.094 1.103 0.994 0.912
0.250 1.246 1.269 0.991 0.801
0.500 1.519 1.564 0.987 0.656
10.00 0.010 1.0086 1.010 0.999 0.991
0.050 1.043 1.051 0.995 0.957
0.100 1.087 1.103 0.991 0.917 '
0.250 1.226 1.270 0.981 0.810

Table 4.1: Results of simulating a shock incident normal to a slab.

Next, I used an outward moving piston to generate rarefactions which were normal to
the slab. From the results, lcan make the following conclusions.
1. Wave pulses are generated similar to those produced by the shocks—
though opposite in magnitude.
2. After the rarefaction passes and the slab reaches its final state, the slab is
again in pressure equilibrium with the environment. It is wider and the
magnetic field strength is decreased. The slab is no longer in thermal

71

equilibrium with the environment. For all cases the slab was hotter than the
environment.

3. The slab is still at rest relative to the environment.

4. The final state of the environment is the same as it would have been if the
slab had not been present. The two regions of the environment, one on each
side of the slab, are still identical.

5. The final state of the slab is less dense, cooler, and has a weaker magnetic
field than if a shock had passed through a uniform region with the same
initial conditions.

The results of the simulations are summarized in the following table.

leab Vp “in “out Tslab/Tenv wf/wi

0.10 -0.010 0.9900 0.9900 1.000 1.010

-0.050 0.9508 0.9507 1.000 1.053

-0.100 0.9033 0.9032 1.000 1.108

-0.250 0.7700 0.7698 1.000 1.300

0.50 -0.010 0.9900 0.9900 1.000 1.010

-0.050 0.9510 0.9507 1.000 1.053

-0.100 0.9036 0.9032 1.000 1.109

-0.250 0.7708 0.7701 1.000 1.305

1.00 -0.010 0.990 0.990 1.000 1.010

-0.050 0.951 0.951 1.000 1.052

-0.100 0.904 0.903 1.000 1.107

-0.250 0.772 0.770 1.002 1.297

3.00 -0.010 0.991 0.990 1.000 1.010
-0.050 0.954 0.951 1.002 1.050 .

-0.100 0.915 0.903 1.008 1.093

-0.250 0.779 0.770 1.007 1.284

-0.500 0.600 0.578 1.024 1.667

10.00 -0.010 0.992 0.990 1.001 1.009

-0.050 0.958 0.951 1.005 1.046

-0.100 0.917 0.903 1.009 1.095

-0.250 0.800 0.770 1.027 1.253

 

Table 4.2: Results of simulating a rarefaction normal to a slab.

It is of interest to note that if a shock is followed by a rarefaction driven by the same
piston speed, the slab and environment each return to its initial state. Therefore a

72

source of shocks and rarefactions, such as a convection zone, would not serve to
strengthen or weaken a flux slab provided the source produced identical distributions
of shock and rarefaction strengths.

73

4.2 Nonlinear Waves Incident Normal to a Magnetic Flux Tube

As a second magnetic structure, I considered a magnetic flux tube whose magnetic

field was given by

[Bog , x2+22<r2
Bo=

0 , >12+22>r2
Since the symmetry of the tube is not the same as that of the rectangular grid, I used
a gradual boundary described by a hyperbolic tangent:

Bo=;—Bo‘g[ 1+tanh( r-V x2+gz]]

The numerical region was initially isothermal and the density varied to maintain the
pressure balance.

The shock generated by the moving piston propagated in the +2 direction and was
therefore incident normal to tube boundary. Figure 4.1 shows a series of density
plots as a shock passes the tube. Also shown in the figure, are contour plots of the
temperature and speed after the shock has passed. In addition to noise, one can
clearly see the difference in temperature and speed between the tube and its
environment. The noise pictured is due to fluctuations less than the accuracy
requested for the simulation. They are visible because of the near uniformity of the
atmosphere (note the small contour spacings).

74

 

 

 

 

 

T=15

 

 

 

 

 

 

 

 

T

20

 

 

 

 

 

T=25

 

 

 

 

 

 

 

Figure 4.1a: Contour plot of density as a shock wave passes a flux tube.

B°= 3.0 , vp=0.1

75

 

 

 

T=30 1=40

   

 

 

 

 

 

 

 

 

 

 

 

 

 

Min= 1.0577 Mins- 0.0955

max= 1.0726 m8x= 0.1100

contours= 030149 contours: 0.00145
SPEEDATT= 40.000

TEMPERATURE AT T= 40.000

Figure 4.1b: Contour plots of density as a shock wave passes. Also showns
are contour plots of temperature and speed after the shock.

76

 

Because of the amount of time required for each run, I did not use as large a variety
of shock strengths and field strengths as for the slab. From the results of these
simulations 1 can draw the following conclusions:

1.

After the shock passes and the tube reaches its final state, it is in pressure
equilibrium with the environment. The tube is no longer in thermal
equilibrium with the environment. For all cases, the tube was cooler than the
environment.

. The density of the tube has increased by a lower ratio than the environment.
. The tube is no longer cylindrical in cross section. The 2 dimension has been

decreased by a fraction similar to that of the slab. The x dimension increases
slightly.

. The tube is not at rest relative to the environment but rather has a relative

velocity in the +2 direction. The fluid in the environment moves around the
tube as it passes.

. Far from the tube, the environment is the same as it would have been if the

tube had not been present. Near the tube, there is a “wake” region where
the velocity is higher and the temperature and density are slightly lower. To
the sides (ix) of the tube, the z velocity is slightly slower. This effect
decreases farther from the tube.

. The final state of the tube is denser, hotter, and has a stronger magnetic field

than if the shock had passed through a uniform region with the tube’s
internal conditions. The average final density and temperature of the tube
were slightly less than those of a slab with the same internal initial
conditions.

. There was some evidence that the tube was not as uniform after the shock

as before. The center was less dense, cooler, moving faster. The magnetic
field was greatest near the center of the tube. To quantify these results
would require a larger grid than is practical on the Vax.

The results of the simulations are given on the following table.

77

leab Vp Vin ain aout Tslab/Tenv WH/Wi wL/wi
0.10 0.100 0.1001 1.1030 1.1032 1.000 1.000 0.907
0.30 0.100 0.1002 1.1029 1.1032 0.999 1.000 0.908
1.00 0.100 0.1013 1.1017 1.1032 0.998 1.001 0.908

3.00 0.010 0.012 1.009 1.010 0.999 1.000 0.991
0.050 0.055 1.045 1.051 0.997 1.002 0.954

0.100 0.11 1.09 1.10 0.994 1.004 0.907

0.250 0.264 1.25 1.27 0.989 1.008 0.792

0.500 0.53 1.50 1.56 0.986 1.039 0.637

Table 4.2: Results of simulating a shock incident normally on a flux tube.

Next rarefaction waves were generated normal to the tube. From the results of the
numerical simulation I can make the following conclusions:

1. After the rarefaction passes and the tube reaches its final state, it is in
pressure equilibrium with the environment. The tube is no longer in thermal
equilibrium with the environment. For all cases, the tube was hotter than the
environment.

2. The density of the tube has decreased less than the environment.

3. The tube is no longer cylindrical in cross section. The 2 dimension has been
increased by a fraction similar to that of the slab and the x dimension
decreases slightly.

4. The tube is not at rest relative to the environment but rather has a relative
velocity in the -z direction. The fluid in the environment moves around the
tube as it passes.

5. Far from the tube, the environment is the same as it would have been if the
tube had not been present. Near the tube, there is a “wake” region where
the velocity is higher and the temperature and density are slightly greater.
To the sides (:lzx) of the tube, the z velocity is slightly slower. This effect
decreases farther from the tube.

6. The final state of the tube is less dense, cooler, and has a weaker magnetic
field than if the shock had passed through a uniform region with the tube’s
internal conditions. The average final density and temperature of the tube
were slightly greater than those of a slab with the same internal initial
conditions.

The results of the simulations are given on the following table.

78

leab vp V'n 0‘in “out Tslab/Tenv WHNVI WI/w i
0.10 -0.100 -0.1005 0.9033 0.9033 1.000 1.000 1.107
0.30 -0.100 -0.1006 0.9034 0.9033 1.000 1 .000 1 .107
1.00 -0.100 -0.1014 0.9045 0.9033 1.001 0.999 1.105
3.00 -0.010 -0.01 15 0.991 0.990 1.000 1.000 1.109

-0.050 -0.055 0.956 0.951 1.002 0.998 1 .045
-0.100 -0.108 0.912 0.903 1.005 0.996 1 .095
-0.250 -0.261 0.788 0.770 1.013 0.993 1.27
-0.500 -0.51 0.60 0.579 1.038 0.971 1.66

Table 4.2: Results of simulating a rarefaction incident normally on a flux tube.
Because of the coarse grid and final non-uniformity, I am unable to make a

conclusion regarding the effect of alternating shocks and rarefactions. A larger grid
and greater time would be required than is practical on the Vax.

79

4.3 Acoustic Waves Incident Obliquely to 8 Magnetic Interface

As the last and most interesting problem, I considered acoustic waves incident
obliquely on a plane parallel magnetic interface. The results of this simulation have
immediate application to several outstanding problems in the solar atmosphere. The
most important of these is the reported absorption of solar p-mode oscillations by sun
spots (Braun et al, 1987). The details of this problem and the application of the
results of this simulation to this problem are considered in the discussion section of
chapter 5.

The magnetic structure for this problem was a plane interface between a field free
region and a region of uniform magnetic field. The numerical domain was originally
isothermal and the density in the the field free region was increased to maintain
pressure equilibrium. The number and range of controllable parameters in this
problem, as well as the large amount of CPU time required for each run, made it
impossible to simulate the full range of problems possible. Therefore, the internal
magnetic field strength was chosen to give a magnetic beta similar to that of a sun
spot and the direction of the field was chosen normal to both the interface and the
direction of wave propagation. These conditions are illustrated in figure 4.2. Several
different magnetic field profiles were used for the interface. None of the profiles used
was based on a sun spot model.

..........
IIIIIIIII
..........
OOOOOOOOO
..........
.........
IIIIIIIIII
.........
OOOOOOOOOO
OOOOOOOOO
..........
IIIIIIIII
IIIIIIIIII
.........
IIIIIIIIII

..........
.........
IIIIIIIIII
IIIIIIIII
IIIIIIIIII
.........
..........
IIIIIIIII
0000000000
.........
..........
.........
IIIIIIIIII
IIIIIIIII
IIIIIIIIII
.........
IIIIIIIIII
IIIIIIIII
..........
.........
..........
IIIIIIIII
IIIIIIIIII

Figure 4.2: Acoustic wave incident at angle 0 to the interface

To study the large scale effects and the angular dependences, I chose the following
profile:
B z—z
B -—° 1+tanh(—-9-)]
’- 2 [ s

80

The parameter s determined the width of the interface and B0 was the interior
magnetic field. The cells were uniform in size. A variety of values was tried for 5
without effecting the large scale results. A value s=2L2 was chosen for the remaining
simulations with this profile.

A piston moving with velocity VI) was used to drive an acoustic wave from the lower
boundary at an angle 6 to the interface. The wavelength, 2., was specified and the
width of the cells in the spectral direction was redefined to meet the periodic
boundary conditions. The required angular frequency for the driving force was
calculated from the wavelength and the speed of sound. The simulation was run
long enough for the transient affects to disappear— at least 8 periods for all
frequencies used. The simulation was run for a variety of wavelengths and angles.
The large scale results were the same for all choices of wavelength within the range
10Lzs7ts100Lz. The angle of incidence was central in determining whether the wave
would be transmitted or reflected. The results can summarized as follows:

1. For small angles of incidence, the wave is mostly transmitted. For 851 5°, the
energy flux in the z direction leaving the interface was greater than 95% of the
incident flux. The transmitted wave was refracted away from the normal as
expected.

2. For angles greater than 30°, the wave was mostly reflected. For 30°56$60°,
the energy flux in the z direction leaving the interface was less than 5% of the
incident flux. A standing wave resulted from the interference of the incident
and reflected waves.

These results are illustrated in the following figures. Figure 4.3 shoWs contour plots
of the flow resulting for 6=15° and vp=0.01. The refraction of the wave can be clearly
seen. The noise in the temperature graph is smaller in magnitude than the accuracy
requested of the simulation and is apparent because of the nearly uniform
temperature. Figure 4.4 illustrates the results for 6=4S°. The standing wave pattern
can be clearly seen in the region above the interface. The distance between the
piston and the interface is not an integral number of half-wavelengths and the results
are similar for the range of wavelengths simulated.

81

 

H
H1118 0.9947
m. mfix' 47818
W contours: 037871
DENSITYATTI 200.00

 

 

 

Min= 0.9928

max= 1.0073

contours= 0.00145
TEMPERATUREATT= 200.00

 

 

 

 

Figure 4.3a: Contour plots of density and temperature for 9=15°, and vp=0.01.

82

 

 

 

 

My

 

 

 

Mins 0.9977

max: 4.9097
contours= 0.39119
DENSITYATTs 200.00

Min= 0.9768

max= 1.0264

contours= 0.00496
TEMPERATUREATT= 200.00

Figure 4.4: Contour plots of density and temperature for 0=4S° and vp=0.01.

 

 

o i 1110- 0.0000

max: 0.0275
contours- 0.00275
SPEEDATT- 200.00

 

 

 

 

 

 

 

 

Min= 4.0048

max= 4.5329

contours= 0.05281

TOTAL ENERGY AT T= 200.00

 

 

 

Figure 4.4b: Contour plots of speed and total energy density for 0=45° and vp=0.01.

85

A second interface profile was chosen to study the small scale details. Since the
interior Alfvén speed exceeds talkx for 6230°, a zone of resonant absorption was
expected within the interface (Steinolfson, 1984) (Tataronis, 1975). To concentrate
on this region, I chose the following profile:

'- 0 Z< Zb
00+01(2-Zo)+02(Z-Zo)2+03(2-Z°)3 2155-25-20
59(2)“ 2 3
Co+C1 (Z- Zo)+C2( 2" 2°) +03( 2' 20) 2. f. 2 5 21

 

The coefficients were chosen such that the Alfvén speed equaled colkx at 20 for 6=45°
and both B and 68/82 were continuous everywhere. The value of zt and 2b were
chosen to determine the width of the interface and nonuniform cell spacing was
used. Half of the cells were within the interface while one quarter was above and
one quarter below the interface. The values used for the majority of the simulations
gave an interface width about one quarter the thickness of the tanh profile and a cell
size one tenth as large. This profile is illustrated in figure 4.5. In addition to the usual
output, I had the program calculate and print the average density, velocities, and
temperature for a full cycle as a function of z; therefore, small differences in these
variables could be differentiated from the periodic variation of the wave.

86

 

 

3.501

9

3.003-

Densit

 

 

 

5. 00*

4.00--

3.00--

2.00~

1.00"

 

 

 

0. 00-- . A . . A

A
I I I

I I I I I
0.00 3.00 5.00 9.00 2 12.00 15.00 18.00 21.00

 

Figure 4.5: Profile of density and By. 20:10.00, zb=9.00, and z,=11.00.

The finer grid required a smaller time step and thus considerably more CPU time.
Each simulation required 50-60 hours of CPU time and thus only a few trials were
run. For 8=15°, the results were consistent with the earlier simulations. No resonant
heating was observed. For 8=45°, resonant heating was clearly evident in cells near
20. Figure 4.5 shows contour plots of density and total energy for this simulation. The
region shown is only 40% as large in 2 as figure 4.4. Although the total energy plot is
shifted horizontally because of a difference in phase, the results are clearly
consistent with the earlier simulation. There is only a hint of the resonance zone in
this graph. The heated zone is clearly visible in figure 4.7 which shows temperature
versus 2 for the central (in x) zone.

87

 

Mina 0.9966

max: 4.8002
contours: 0.38035
DENSITYATT- 200.00

 

 

 

 

 

 

 

Min= 4.1367
, 1 max: 4.3750
contours= 0.02383
TOTAL ENERGYAT T= 200.00

 

 

 

 

 

Figure 4.6: Contour plot of density and total energy for the fine grid.

88

 

1 . 0080-
1 . 0070-
1 . 0060-
1 . 0050-
. 0040-

1 . 0030-

temperature

1. 0020-
1 . 0010-

 

1 . 0000

P.

l

.l

I

l

 

I

0.00

3.00

l
A',

6.00

I
15.00

I I
9.00 2 12.00

Figure 4.7: Temperature versus 2 for the central x zone. T=200.

I
18.00

db

The resonant absorption region can be seen even more clearly in figure 4.8 which
shows the temperature average over a cycle versus 2. The region begins at or just
before 20 and continues to zr l-ran shorter simulations with half the cell size and
compared the results. The temperature rise was similar to that observed for larger
cells and the region was same size; therefore, I concluded that smaller cell size

would not improve the resolution of this resonant heating zone.

 

1 . 0024'"-

.I

‘

temperature

d

.8828-
.0016?’
.00121
.00081
.8084-

. 0000--

 

J

 

 

I

2.00

4.00

6.00

8.00

1
10.00
2

1
12.00

1
14.00

1
16.00

Figure 4.8: Temperature averaged over one cycle versus 2. T=200.

89

1
18.00

I
20.00

 

 

I calculated the amount of surplus thermal energy in the resonant zone to be
(8.9:l:1.9)% of energy available from the wave. Since the region remained in
pressure equilibrium, the magnetic field, and thus the energy stored in the field,
decreased. Considering both thermal energy gain and magnetic energy loss, I found
the resonant zone had absorbed (3.010.6)% of the energy available from the wave.
An additional (4.9io.4)% of the energy was transmitted. I could not directly calculate
the fraction of the energy reflected by the interface; however, the standing wave
pattern was consistent with earlier simulations having reflections greater than 90%.

90

5

Conclusion and Discussion

91

 

5.1 General Conclusions

The results of the simulations run in chapter 3 and 4 allow me to make several
conclusions regarding the general approach used. First and foremost, the program
was able to model all of the physical problems attempted. When accepted values
were known, the computational results agreed within the precision requested. The
only exception was in modeling contact discontinuities. I believe this shortcoming
could have been overcome by introducing non-zero thermal conductivity or another
smoothing mechanism. This was not done, however, because contact discontinuities
did not develop in the physical problems.

Several features of the program make it suitable for a wide variety of problems. First,
it is an explicit code. Since no Iinearization or iterative calculations are required, it is
straightforward to include any forces or other extra terms required for a particular
problem. I also believe that explicit methods are easier to understand and
implement.

Second, the dynamic grid allow the program to concentrate on the region of interest.
Small zones, such as the resonant heating zone in section 4.3, could be studied
without increasing the number of grid points beyond reason. Furthermore, fast
moving features, such as shocks, could be followed without increasing the numerical
domain. As another benefit, driving waves from the boundary was straightforward
using the moving grid. For many problems, it would be beneficial to have the grid
dynamic in two or three dimensions; however, this introduces the complication of
grid twisting. More complicated algorithms for managing the grid motion may be able
to overcome this difficulty. I used only the simplest grid motions.

Last, the Runge Kutta like method I used for the time integration proved to be very
stable. It allowed the program to use the largest possible steps consistent with the
requested precision with a minimum of overhead. It is straightforward to understand
and implement. The method was clearly superior to using fixed time steps, either
user chosen or calculated, or to using the Courant condition. Using the Courant
condition becomes complicated with the addition of viscosity, conductivity, and other
dispersive effects. Once implemented, the method was insensitive to other changes
in the program. I would strongly recommend this method to others.

92

A number of the aspects of the program, although satisfactory, leave room for change
and improvement. Foremost among these is the choice of numerical boundary
conditions. From my work with this program, I would have to conclude that finding
appropriate, stable, and easily implemented boundary conditions is the most difficult
aspect of simulating magnetohydrodynamics. The method of following a grid point at
the local wave velocity worked well for simulations of waves and other smooth
variations but was not perfect for shocks and other discontinuities. Problem specific
numerical boundary conditions may be required for these types of flow.

The use of a spectral method for taking spatial derivatives worked well for the
problems on which I concentrated; however, there is some cost is using them. The
accuracy of the derivatives obtained from the spectral method greatly exceeded that
of the finite difference method for a given number of grid points, N. The finite
difference method was faster, however, for a given N. This trade-off favored the
spectral method only for N232. Thus for many problems with a small number of grid
points, it would have been preferable to use finite difference methods in both
directions. When computer power allows a larger grid or when using vector
machines, the spectral method is clearly more powerful. For problems for which
periodic boundary conditions are not appropriate spectral methods should not be
used. The modular structure used in my program allowed me to easily substitute
different routines for the spectral derivative subroutine.

Another aspect of this research which may require further work for future problems is
the presentation and interpretation of the data. This type of program generates large
volumes of data which can be difficult to interpret. The linegraph and contour plotting
routines which I wrote were adequate for displaying the static characteristics of one
and two dimensional problems; however, some form of animation would have been
helpful in understanding some of the dynamic aspects of the problems. Additional
display techniques will be required to adequately interpret three dimensional results.
Both animation and three dimensional problems require more computing power than
I had available.

In general, I would conclude that the algorithms, numerical methods, and the
resulting program were successful in simulating the problems attempted within the

constraints imposed by available computing power.

The conclusions which can be drawn from the simulation of non-linear waves

93

incident normal to a magnetic flux slab are quite straightforward. First the presence
of the slab does not affect the final state of the fluid away from the slab. Second, the
final state of the interior is different from that which would result if there were no
magnetic structure. The details of the differences are given in section 4.1. Lastly, a
source of shocks and rarefactions, such as the solar convection zone, would not
concentrate the field.

Similar conclusions can be drawn for the simulation of non-linear waves incident
normal to a flux tube. Once again, the magnetic structure effects the final state within
the tube but not in the environment far from the slab. It was not as clear that the
internal structure would be unaffected by repeated passes of shocks and rarefactions
of equal amplitudes. A finer grid and more CPU time would be required to make a
definite conclusion.

The conclusions which can be drawn from the last simulation are discussed in the
next section.

94

5.2 Discussion of Acoustic Absorption by Sunspots

The recent observation of absorption of high-degree p-mode solar wave energy by
sunspots (Braun et al, 1987) can be considered in light of the numerical results of
section 4.3. The simulated problem can be interpreted as a simplified simulation of
the interaction between solar p-mode acoustic waves and the sun spot boundary.
From this simple approach, I should be able to address several questions. First, is
the resonant absorption of wave energy of a sufficient magnitude to explain the
observed results? If not, does a sufficient fraction of the incident wave energy pass
through the interface into the sun spot interior where other mechanism could absorb
or redirect the energy? Lastly, is the two dimensional magnetic structure of the sun
spot simulation sufficient to explain the observed absorption or is three dimensional
structure required?

First I must show that the dimensionless parameters chosen are appropriate for the
physical problem. I used the following for the values of the sun spot (Allen,1973):

Rs=15Mm radius of sunspot
B=2750 gauss magnetic field
P=8 x104 dyn/cm2 gas pressure
c=8 km/sec . speed of sound

The units were chosen to be consistent with those used by Braun et al. The choices
for Rs, P, and c yielded the following scaling factors (see section 1.5):

M=2 x 10'7 grams/cm3 density scale factor

L=0.1 Mm length scale factor

T=12.5 sec time scale factor
Using these scaling values and the parameters used in the simulation, I get the
following for the physical parameters:

parameter scaled physical

7. 25 2.5 Mm

Tmax 200 2500 sec

(0 0.25 0.02 rad/sec
B 7.5 2730 gauss

6 interface thickness 2 0.2 Mm

p 1 2 x 10-7 g/cm3

These values are consistent with the sun spot values and the frequencies used by
Braun et al. Note that for the radius and length scaling factor used, the plane

95

 

interface should have been an arc of about 10°. An additional simplification is the
uniform temperature. The above scaling factors give a physical temperature of about
5500K while the actual temperature varies from 4240K on the interior of the sunspot
to 6050K in the surrounding photosphere.

Addressing the first question above, I can consider the energy absorption by the
resonant layer in the interface. Braun et al observed up to 50% absorption. Since
the amount of energy absorbed for a simulation at any angle never exceeded 5%
and was ~0% for angles less than 25°, 1 can conclude that the resonant absorption
can not be the primary mechanism for the observed energy absorption.

To answer the second question, consider the following figure.

 

Incident wave

transmitted

122831119 = «sinee

C

 

 

 

 

Figure 5.1: Wave incident on circular sunspot.

The critical angle, 6c, is the angle at which the majority of the wave energy is
reflected. If 9c=(25°:l;4°) is used as the results of the simulation suggest, the fraction,
f, of the incident energy which passes into the interior is (4215)%. This result is
consistent with observation if a mechanism can be found inside the sunspot which
can absorb nearly all of the acoustic energy which enters.

Lastly, can any two dimensional magnetic structure account for the observed
absorption? The most promising model for such a sunspot consists of a large
number of narrow flux tubes. Incident acoustic waves would be partially transmitted
and partially reflected repeatedly as they pass through this bundle of tubes. Since
the results of the simulations (and observations) suggest that the absorption rate is
largely independent of scaling effects, the results of the simulations may be
applicable to each of these tubes. Using 60:25° and an absorption rate of 3% for the
reflected part of the wave, one gets an average absorption of 1-2% for each tube

96

interaction. The observed absorption would require an average of 40-50
interactions. This value is not completely unreasonable. Since the critical region
occurs where the magnetic field is relatively large, the magnetic field need not vanish
between the narrow flux tubes. Three dimensional structures are even more
promising since all is required is a mechanism for deflecting the energy downward.

In summary, 1 would conclude that the simulation shows that resonant absorption by
a single two dimensional magnetic structure cannot account for the observed
absorption. However, the numerical results are consistent with a mechanism inside
the sunspot which absorbs or deflects the energy downward. Moreover, the results
are consistent with absorption by a two dimension fine magnetic structure in which
the magnetic field varies about a relatively large value.

97

5.3 Future Problems

There are a very large number of interesting problems which can be simulated by
this program, by its three dimensional sibling, and by programs using similar
algorithms written for different geometries. Most of these require more computing
power than I currently have available; however, some of them do not require a
supercomputer.

The current program should be used to simulate the normal incidence of non-linear
waves on flux tubes with a much finer grid than was used here. This should resolve
the question whether a source of small shocks and rarefactions can concentrate the
field in a tube. The interaction of acoustic waves and sunspots could be further
studied using the same circular flux tube. My experiences in section 4.3 suggest that
a grid at least 10 times as fine as that used in section 4.2 would be required to study
the absorption zone. The two dimensional code could also be used to study surface
waves and solitons along flux slabs. I was able to take a preliminary look at these
problems using the Vax but a detailed analysis required more grid points and CPU
time than was practical.

A completely three dimensional Cartesian program would allow for the study of flux
tube models of sunspots. By placing the z direction along the symmetry axis of the
tube, gravity could be included. This would allow for the modeling of tapering flux
tubes, sausage and kink mode tube waves, and a large number of similar problems.
Even using a supercomputer, however, the total number of grid points would have to
be kept fairly small. Thus problems such as the oblique incidence of acoustic waves
on flux tubes could not be done with the resolution necessary to study small
resonance zones.

Perhaps the greatest promise for simulating these flux tube problems lies in rewriting
the program in cylindrical coordinates. Small cells could be used in the interface
while larger cells could be used on both the inside and outside. The two
dimensional program could be used to simulate normal incidence. This problem
may be within the grasp of the Vax and will probably be the next problem I consider
unless a larger computer becomes available. A three dimensional program would
put the modeling of oblique incidence of acoustic waves on flux tubes within the
grasp of supercomputers. Simulations of non-symmetric tube waves would also be

98

possible.

A two or three dimensional spherical algorithm would be possible using methods
similar to those used in my current program. I have no plans at present to attempt
this since no currently accessible astrophysical problems lend themselves to this
symmetry. When sufficient computing power becomes available it may be possible
to model proto-stellar collapse including magnetic fields. A Poisson solving
algorithm would be required to handle the self-gravity. I include this aside because it
was interest in this problem which originally started me down this line of research.

99

Appendix A
Glossary of Symbols

B— magnetic flux density

c— speed of light

D— electric flux density

E— electric field

F— total force

f— body force per unit volume
g— gravitational field

H— magnetic field

J— current density

P— pressure

R— array containing all eight fluid variables
q— electric charge

t— time

v— fluid velocity

oc—ratio of densities

B—ratio of gas pressure to magnetic pressure
y—ratio of specific heats
e—permitivity

C—viscosity

n— magnetic diffusivity
u—permeability

p— mass density

96'— charge density

o— electric conductivity
t—viscous stress tensor

101

 

Appendix B
Summary of Equations

In chapter 2, l have developed the following equations which describe the time
evolution of a magnetohydrodynamic fluid. The variables are defined in the
laboratory frame, are dimensionless, and have the stratification divided out. Each
equation is written below in vector form and summation form. The equations are
presented in complete detail on the pages 104-105.

1. The continuity equation

 

g—p+V- -=pv 0 (1.8)
becomes
_23 -21:
+-V -pv H
with the addition of stratification. In summation form
1_p _a_ -va
at ax,pj= H
2. The momentum equations
Bl2
a (B-V)B
_ v .. = _—
atp +v. [pVV+(P+IB 2110 l); ;] pg+ “a (1.14)
becomes
2
a 181 (____B' V)B vaz
_ v . vv — _—
atp +V[p +0” 211,.)1 3h!“ u, H

In summation form
a a 51.5., a B +PV1Vz
— v.+— v-v. P+ 8---t-- : - B
atP 1 axj[P 1 J{ 2P6] 1) 1)] 991{ kaT k]p—:-+ H

3. The internal energy equations

a by
513‘31V[‘-'(Pe+P)]=v-Mr~1VT11‘fija—v1j11mlvxBl2 (1.170)

 

 

becomes

"1' 11v): BIZ fl—ze"

a
'ane+V'[' (pe+P)]= --1-VVP+VkTVT+Tij:-;j+ H

102

In summation form

a a 3P 6 _3_T
—pe+—lv,-<pe+p)1=v,-—+—[I<— ax
J

av- evz
+ .,-— —+J-:'- va 312+—
at a J a a xJ big 116 H

4. The magnetic equations

 

bB
—=Vx(VxB)-VanxB (1.70)
as, a 3B _aa_j,
becomes

38 By
—=Vx(VxB)-Vx?1VxB+ ‘
at H

6B; 6 BBL 3B.- ]8 B_1_Vz

a—t' “ ‘67J[Bi"i ' 51' ”1 ”‘( axi TJ

103

The continuity equation

bp 8
—+— v+
bt bxp x by

The momentum equations

P 2 2 2
a a B +8 +3 3
b_tp ”1+3; ."_PV1<"1*[P*'L2i—z]"”‘] au‘vav" W“ 2‘9“ "“1

1 68x] 1[ 65.] 1[ 68.] pmz
= +— B — +— B — +— B +
ng Fo[ x 3X “0 9 by 110 z 32 H

 

 

2 2 2
b b b Bx+By +Br. b
-11 171111110945190919“? $999111]

b bB,J bB v v
=P99+‘L[ Bx— B__‘J_]+ I _[Bu_]+l[az__9].j~p_9__z
Po bx Po by bz H

a a a J B2+B2+B2
Sfpvz+a 2)—<—[pvzv -txz]+— gz[pv vJJ —1,J,J] bz szVz-I‘ P+——— ~99

L 211‘0
1[ sz] 1[ b8 ‘11:”
= B —— B— 82
sz 110 x 5X “0 1;— by)“

 

 

Bzz]H PVsz

The energy equation

3—pe+—tv.(pe+p)1 +-:—IV9(P9+D)I+:—ZIVZ(PG+D)I

bt
=Vx3£+V92£+Vz1a—P+L k1 fl +:—[kT%I]+L[kTfl-]+
bx by bz bx

bx by by bz bz
bvx bvx bvx bv bv,J bv bvz
1"“ bx “"9 by +sz bz +19“ bx HWEH‘” bz 1“ bx

 

2
bvz b_vz_ 11 [$338!] as _xzaa] ab,J szn pevz

+ __ _-_
1‘9 by 1“ b2 by bx bx b2 b2 by H

104

 

The magnetic equations

bB a[ [ng bB MD az[ [sz_ bBXJ] vaz
Buv -B,Jv —- Bv Bv + =
bt by Mnax by a x””1”“ bz H

as bB
+ 21.4
gal—37]] a—z{B v‘ B V” "[ by bz “'9
5 +3—

Bz be _b_Bz +_a_ bB,J sz Bzvz
Bzxv -szv+n Bzvg-B9Vg+n—-— =
bt +b_x bz b—x +by bz by H

 

 

 

 

 

ng
at

 

“V

 

b
+bx[B” v“ B xaaBW

  

 

 

105

Appendix C
Analytic Solutions to Test Problems

Some of the analytic solutions used in chapter 3 were developed by me specifically
for this research. None of these represent completely original work since they are
based on well known results or published works. These derivations are include in
the following section. References are included as appropriate.

106

C.1 Piston Driven Shock

The solution of the piston driven shock is shown in figure 3.4. It consists of two fluid
regions separated by a shock. The shock moves at a constant speed and has zero
thickness. In the frame of reference moving with the shock, fluid moves into the
shock from the right with a constant speed of u2 and fluid leaves the shock toward the
left with a constant speed of uJ. Since the problem is steady state, all time
derivatives vanish in the equations. Thus the continuity equation

g¥+V-pv = 0 (1.8)

shows that pu must be conserved across the shock. Thus
p1u1=p2u2 (C. 1)
Similarly, the momentum equation
5619‘” V-(pvv+P;-1)=f (1.10)
with the idealization that viscosity and external forces are zero yields
p, u%+P,=pzu§+P2 (c2)
If the same technique and idealizations are applied to the total energy equation
g—JGPV2+PB)+V- [v(%pv2+pe+P )-v-:l_’, -kTV'r ]= v-f+ E—OIV x BI2
(1.2010)
one gets
u1[;-P1 u11TH 91+P1]=U2[%P2U2+P292+P2] (c3)

Equation (c.1) can be used in equation (0.3) to yield
P P

1 2 1 1 2 2
2 ‘ ' PI 2 2 2 Pz

These three equations may now be transformed, nonrelativistically, back to the
laboratory frame of reference—

P1(V1'Vs)=Pz(V2"Vs)
2 2
P1+P1(V1'Vs) =P2+P2(V1‘V2)

P P
1—(u1-v,)2+ej+—1 . 1(02-vs)2+ez+—2-
2 P1 2 P2

107

 

In region 1, behind the shock, one knows vJ=vp. ln region 2, ahead of the shock one
knows v2=0. The values of p2 and 62 are those of the undisturbed gas and are
specified by the initial conditions. In addition to these three equations, one has the
equations of state
Pt-(rl)piei P2=(\r-l)pz

Therefore P1 and P2 are known in terms of p191 and p292; thus we have three
equations and three unknowns. Using the known values and eliminating the
pressures, one gets

P1Vp
P1'P2

 

91(Vp-Vs)=-F’2vs =9 vs= (C.4)

plop-192+ (y - 09181 = pzv§+ <1 — time:

We can use the first equation, (0.4), to eliminate vs from the other two equations
getting

 

 

 

 

 

P1P2
-1 e - v2 = -I 8
(Y )P1 1 pp1_p2 (Y )P2 2
and
V§[P1+P2] e e e +V§[P1+P2] ( S)
-— = = = — u
1 2 P1'P2 Y2 1 2 ZY P1'P2 c
This can then be used to eliminate 81. Thus
(v-l)[P1+P2] 2 2 P1P2
-1 e + v -v = -l 9

Using the value of the speed of sound in the undisturbed fluid

c§=y<v -1)e2
one may solve this equation for the ratio of densities. Since the equation is
quadratic, there are two solutions. Discarding the negative, non-physical, solution
one gets

108

 

 

1.2.33. fi.[m3§]
,..a. 4 1
pz 1 L113.
2 C3

The ratio of densities, which I have called at here, can then be plugged back into
equations (c.4) and (0.5) to yield

fl.-[1+f_"_l’§.]&t‘.

109

0.2 Weak Shock Profile

The profile of a weak shock may be derived using a method given by Landau and
Lifshitz. They determined the variation of pressure within a shock (Landau & Lifshitz,
1959,§87)as

62v _ _ _v3 4 ]1< bT] av] 22}dp
3[— bp2 —]S(D D1)(P D2)= §{[-3'11+C +27%; 1'5; pC P dx

where VEI/p , hng =0 , C252, and KEKT =0 in the symbols I am using. Substituting in
these symbols and values yields

2V 3
-[ b —:s] (P'D1XD Dz)= "‘V—Ea— dp
2 bp2 c3 dx

One can use pVY = constant for an adiabatic process to evaluate the derivative on the
right side of the equation:

[sz} _ y+21_\_/_
602 12 02

Using this and c2 = ypV, the equation simplifies to

 

 

 

 

 

 

'2652 1 Q
1+1 (D-Pi)(D-Pz) dx
Integrating
1+de [*X-chiz 1 tipdx JIM-20132 1
-x -x y+1 (D‘Df)(D‘Dz)dX pl 1+1 (p-Dlxp-pz)
yields
1
D'-(92+D1)
C -
2x= 6'2 4 tanh1 2

1+1 92-01 15032-91)

110

Therefore, one gets the following variation of pressure

.. 1. 1. _ i]
p - 2(92+Dt) + 2(Dz p1)tanh[5
with

= 20C2 1 2C§2 i

Y“ Dz’Pt 1,2-1 Pzez‘P1e1

6

 

 

 

 

For a shock with this profile, over 90% of the pressure variation occurs with in a
distance of 5.

111

Appendix D
Auxiliary Programs

I have written a number of programs to graph, analyze, or calculate data. In this
section I have included the source codes for these. The FORTRAN routines were
compiled and included in an object library. The Pascal programs were written on
and for the Macintosh computer.

112

D.1 Graph6

program linegraph(input,output);

{This program uses the linegraph routines to read output from GMHD and draw a
graph for each of six variables. It has options of connecting the points with
line segments or using a cubic spline. The spline is slow and not needed for

n2 40.}

Uses MacIntf:
{This program uses the Macintosh interface}

type{global types}
parray= array[0..500] of integer;
glnarray= array[0..256] of real:

var{global variables}

thefile: text:
n,i,ni: integer:
z,rho,vx,vz,bx,bz,t: glnarray:
quit,done,
splineflag,
tickflag,
pointflag: Boolean:
theWindow: Windothr:
SKIP: CHAR:
data: array [l..80] of char:
procedure getn:{reads number of points from file}
begin
readln(thefile,n):
writeln('Found table with ',n,' entries.');
ni:=l:

while ((not eoln(theFile)) and (ni < 80)) do begin
read(theFile, data[ni]):
ni:=ni+l
end;{while}

readln(thefile):

end: {getn}

procedure getxy;{reads the points from file}
var
i,q: integer:
zz,rr,vvx,vvz,tt,bbx,bbz: real;
begin
readln(theFile);
for i:=0 to n do begin
readln(thefile,q,zz,rr,vvx,vvz,tt,bbx,bbz);
z[i]:=zz:
rho[i]:=rr;
vx[i]:=vvx;
vz[i]:=vvz;
t[i]:=tt:
bx[i]:=bbx:
bz[i]:=bbz:
end: {for}
end:{getxy}

procedure graph( x,yz glnarray:

113

n: integer:
tflag,
pflag: boolean);{the actual graphing program}

const
hmin=60:
hmax=505:
numticks=8:
vmin=25:
vmax=225:
grit=3:
type
harray= array{hmin..hmax] of real;

var
h,vo: integer:
ypl,ypn: real:
xmax,xmin: real:
ymax,ymin: real:

y2: glnarray:
xh,v: harray:
xp,yp: parray:
graphrect: rect:
loglO: real:

q: char:

procedure spline(ypl,ypn: real:
var y2: glnarray):{determines the cubic spline}
{This procedure is based on Press et al.,l986}

var ilk: integer;
qun,sig,un: real;
u: glnarray:

begin {spline}

if(ypl>0.99e30) then begin
y2[l]:=0.0:
u[l]:=0.0
end {then}

else begin
y[2]:=-0.5:
u[l]:=(3.O/(x[2]-X[l]))*((Y[2]-Y[1])/(X[2]-X[l])'YP1)
end: {else}

for 1:: 2 to n-l do begin
sig==(x{i]-x{i-1])/(x{i+1]-x{i-1]):
p :=sig*y2[i-l]+2.0:
y2[i]:=(sig-l.0)/p:
Uli]:=(y{i+1]-y[i])/(x{i+1]-x{i])-(y[i]-y[i-l])/(x{i+l]-X[i]):
u[i]:=(6.0*u[i]/(x[i+1]-x[i-l])-sig*u[i-1])/p
end:{for}

if (ypn>0.99e30) then begin
qn := 0.0:
un := 0.0
end {then}

else begin
qn:= 0.5:
un:= (3.0/(xtn1-XIn-11))*(ypn-(y[n]-y[n-1])/(x[n]-x{n-1]))
end:{else}

114

y2[n]:= (un-qn*u[n-l])/(qn*y2[n-l]+l.0):
for k:=n-l downto 1 do begin

y2[k]:= y2{k]*y2[k+l]+u[k}

end:{for}
end:{spline}

procedure splint( xa,ya,y2a: glnarray:

n. integer:
x: real:
var y: real);
var
klo,khi,k: integer:
a,b,h: real:

begin {splint}
klo:=l:
khi:=n:

while (khi-klo>l) do begin
k:=(khi+klo) div 2;
if(xa[k]>x) then khi:=k else kloz=k
end; {while}

h:=xa[khi]-xa[klo]:

if (h = 0.0) then begin
writeln(' Bad xa input'):
readln
end: {if}

a:=(xa[khi]-x)/h;

b:=(x-xa[klo])/h;

y:=a*ya[klo]+b*ya[khi]+((a*a*a-a)*y2a[klo]+(b*b*b-b)*y2a[khi])*(h*h)/6.0
end:{splint}

procedure splint2( xa,ya,y2a: glnarray:

n: integer:
x: real:
var y: real:
var klo,khi: integer):
var
a,b,h: real:

begin {splint2}

while (x>xa[khi]) do begin
khi:=khi+1:
kloz=klo+l

end; {while}

h:=xa[khi]-xa[klo]:

a:=(xa[khi]-x)/h;

b:=(x-xa[klo])/h;

y:=a*ya[klo]+b*ya[khi]+((a*a*a-a)*y2a[klo]+(b*b*b-b)*y2a[khi])*(h*h)/6.0
end:{splint2}

procedure scale( x: glnarray:
n: integer:
var max,min: real);{This procedure scales the data to fit}
var

115

i: integer;

begin{sca1e}
max:=x[0]:
min:=x[0]:
for i:=1 to n do begin
if (x[i] > max) then max:=x[i]:
if (x[i] < min) then min:=x[i};
end:{for}
if (abs(max-min)<0.0005) then max:=min+1.0
end:{scale}

procedure calcxhspline:

var
delta: real:
h,i: integer:
begin{calcxhspline}

delta:=(xmax—xmin)/(hmax-hmin);

for h:=hmin to hmax do xh{h]:=xmin+delta*(h-hmin);

for i:=1 to n do xp[i]:=round(hmin+(x[i]-xmin)/delta);
end:{calcxhspline}

procedure calcxpyp:

var
deltax,
deltay: real:
i: integer:
begin{calcxpvp}

deltax:=(xmax-xmin)/(hmax—hmin);
deltay:=(vmax—vmin)/(ymax—ymin);
for i:=1 to n do begin
xp[i]:=round(hmin+(x[i]-xmin)/deltax);
yp[i]:=round(vmax -deltay*(y[i]-ymin))
end:{for}
end:{calcxpvp}

procedure calcvspline(y2: glnarray);
var

delta,yh: real:

h,i,klo,khi: integer:
begin{calcvspline}

khi:=2:

klo:=l:

delta:=(vmax-vmin)/(ymax-ymin);

h:=hmin;

while h<= hmax do begin
splint2(x,y,y2,n,xh[h],yh,klo,khi);
v[h]:=vmax -delta*(yh-ymin):
h:=h+grit
end:{while}

khi:=2:

klo:=1:

for i:= 1 to n do begin
splintZ(x,y,y2,n,x[i],yh,klo,khi);
yp[i]:=round(vmax -delta*(yh-ymin))
end:{for}

116

end:{calcvspline}

Procedure addline:
var
h: integer:
begin
pensize(l,1);
penmode (PatOr) ,-
vo:=round(v[hmin]):
moveto(hmin,vo);
h:=hmin:
while h<=hmax do begin
lineto(h,round(v[h])):
h:=h+grit
end:{while}
moveto(0,0):
penNormal
end:{addline}

Procedure connect:
var
i: integer:

begin
pensize(l,l);
penmode(PatOr):
moveto(xp[l],yp[l]):
for i:=2 to n do lineto(xp[i],yp[i]);
moveto(0,0):
PenNormal
end:{connect}

function rounder(x: real): real;

var
d: integer:
q,
tens: real:

begin
d:=trunc(ln(x)/log10):
repeat

tens:=exp(d*log10):
q:=round(x/tens)*tens:
d:=d-l:
until (q <> 0.0):
rounder:=q

end:{rounder}

procedure tickmark:{calculate and place the tickmarks}
var

xinc,yinc,xtick,ytick: real;

deltay,deltax: real;

i,top,bot: integer;

vtick,htick: integer;

w,d: integer:
begin

deltay:=(vmax—vmin)/(ymax-ymin);
deltax:=(xmax-xmin)/(hmax-hmin);
xinc:=(xmax—xmin)/numticks;

117

yinc:=(ymax—ymin)/numticks:

xinc:=rounder(xinc):

if xinc = 0.0 then xinc:=0.l;

yinc:=rounder(yinc):

if yinc = 0.0 then yinc := 0.1:

top:=trunc(ymax/yinc):

bot:=trunc(ymin/yinc+0.999);

w:=7: d:=2:

if yinc<0.01 then d:=4:

for i:=bot to top do begin
ytick:=i*yinc:
vtick:=round(vmax -deltay*(ytick—ymin));
moveto(hmin-8,vtick);
lineto(hmin-2,vtick):
moveto(10,vtick+4);
writeln(ytick:w:d)

end:{for}

top:=trunc(xmax/xinc);

bot:=trunc(xmin/xinc+0.999):

w:=7: d:=2:

if xinc<0.01 then d:=4:

for i:=bot to top do begin
xtick:=i*xinc:
htick:=round(hmin+(xtick—xmin)/deltax);
moveto(htick,vmax+8):
lineto(htick,vmax+2):
moveto(htick-25,vmax+20):
writeln(xtick:w:d)

end:{for}

moveto(0,0)
end: {tickmark}

procedure addpoints;{add the points as triangles}
var
i: integer:
begin
for i:= l to n do begin
moveto(xp[i},yp[i]):
lineto(xp[i],yp[i]-2):
lineto(xp[i]-2,yp[i]+2):
lineto(xp[i]+2,yp[i]+2):
lineto(XP[i]rYP[i]-2)
end:{for}
moveto(0,0)
end:{addpoints}

begin {graph}
moveto(0,10);
for i:=1 to ni do write( data[i]):
writeln:
log10:=ln(10.0);
scale(x,n,xmax,xmin):
scale(y,n,ymax,ymin):
SetRect(graphrect,hmin—5,vmin-5,hmax+3,vmax+5):
if splineflag then begin
spline(le50,le50,y2):
calcxhspline:

118

calcvspline(y2);

addline

end{then}
else begin

calcxpyp:

connect:

end:{else}
FrameRect(graphrect):
if tflag then tickmark:
if pflag then addpoints:
readln(q):
if q:'.' then quit:=true
end: {graph}

procedure findnext:{finds the next table to graph}
var
one,s,t,p: char;
begin
repeat
readln(theFile,one,s,t,p):
if eof(theFile) then begin
quit:=true:
exit(findnext):
end:{if eof}
if one ='@' then begin
if s='F' then splineflag:=false else splineflag:=true:
if t=‘F' then tickflag:=false else tickflag:=true:
if p='T' then pointflag:=true else pointflag:=false:
end:{then}
until (one = '@') or quit
end:{findnext}

Procedure OpenFile;{opens a new file or quits if cancel button}
var

reply: SFreply:
typeList: SFTypeList;
where: point:
begin
typeList[0}:='TEXT';
where.h:=80:
where.v:=100:
initCursor:

SFGetFile(where,",NIL,l,TypeList,NIL,reply);
done:= not reply.good:
if reply.good
then
open(theFile,reply.fName);
end:{OpenFile}

begin{main}
theWindow:=FrontWindow;setWTitle(theWindow,'Linegraph');
MoveWindow(theWindow,1,35,true);
SizeWindow(theWindow,510,348,true);
done :=false:

openFile:
while not done do begin
quit:=false:

tickflag:=true:
pointflag:=false:
repeat

119

page(output):
setWTitle(theWindow,'Searching File');
findnext:
if not quit then begin
getn:
if (n>256)then begin
writeln(’n larger than 256, using n=256'):
n:=256
end:{if n>256}
writeln('reading data from file');
getxy:
page(output):
HideCursor:
setWTitle(theWindow,'Density vs 2'):
graph(z,rho,n,tickflag,pointflag);
moveto(0,0):
page(output):
setWTitle(theWindow,'Vx vs 2');
graph(z,vx,n,tickflag,pointflag);
moveto(0,0):
page(output):
setWTitle(theWindow,'Vz vs 2'):
graph(z,vz,n,tickflag,pointflag):
moveto(0,0):
page(output):
setWTitle(theWindow,'Temperature vs 2');
graph(z,t,n,tickflag,pointflag);
moveto(0,0):
page(output):
setWTitle(theWindow,'Bx vs 2');
graph(z,bx,n,tickflag,pointflag);
moveto(0,0):
page(output):
setWTitle(theWindow,'Bz vs 2');
graph(z,bz,n,tickflag,pointflag):
moveto(0,0):
end:{if not quit}
until quit:
close(thefile):
openFile:
end:{while not done}
end.{main}

120

D.2 Contour

PROGRAM Contour:

{This program produces contour plots on the Macintosh using encoded data

send by subroutine MAKCON (in FORTRAN) from the mainframe.

full Macintosh interfacel}
CONST {menu stuff}
AppleID = 1000:

FileID = 1001:

EditID = 1002:

nextID = 1003:
VAR

done, doneWithFile, next : Boolean;
theEvt : eventRecord:

wRecord : windowRecord:

myWindow, TheWindow : windothr;
WindowRect, blankRect : Rect:

appleMenu, fileMenu, editMenu, nextMenu : MenuHandle;

thefile : text:
PROCEDURE ProcessMenu_in (CodeWOrd : longint);
forward:

PROCEDURE plot:

CONST
hmin
vmin
space = 0:
skip = 65:
nextcol = 66:
quit = 67:

VAR
c, t : char:
d, h, i, ni, ii : integer:
thru : boolean:
plotrect : rect:

5:
4

o
I

BEGIN
thru := false:
moveto(hmin, vmin);
h := hmin:
REPEAT
BEGIN
WHILE ((NOT thru) AND (NOT eoln(theFile)))
BEGIN
read(theFile, c);
d := ord(c) — 40:
CASE d OF
space :

skip :
move(0, 64);
nextcol -
BEGIN
h := h + l:
moveto(h, vmin)
END;{nextcol}
quit .
BEGIN
moveto(300, 10);

121

It supports the

ni := l:
readln(theFile):
FOR ii := 1 TO 4 DO
BEGIN
WHILE ((NOT eoln(theFile)) AND (ni < 35)) DO
BEGIN
read(theFile, t):
DrawChar(t):
ni := ni + 1
END;{while}
ni := 1:
readln(theFile):
moveto(300, 10 + 12 * ii):
END:{for}
thru := true:
END:
OTHERWISE
BEGIN {all valid moves}
move(0, d):
1ine(0, 0):
END;{otherwise}
END;{case d}
END;{while not eoln}
readln(thefile):
END:{until}
UNTIL (thru):
SetRect(plotRect, hmin - 3, vmin - 3, h + 3, vmin + 293);
FrameRect(plotRect):
END:{plot}

PROCEDURE miniloop:
VAR
WindowPointedTo : Windothr:
MouseLoc : Point:
WindoLoc : integer:
BEGIN
HiliteMenu(0);
REPEAT
IF GetNextEvent(everyEvent, theEvt) THEN
CASE theEvt.what OF

mousedown :
BEGIN
MouseLoc := theEvt.Where;
WindoLoc := FindWindow(MouseLoc, WindowPointedTo):
CASE WindoLoc OF
inMenuBar :
ProcessMenu;in(MenuSelect(MouseLoc));
inSysWindow :
SystemClick(theEvt, WindowPointedTo):

OTHERWISE
END;{case theEvt.what}
END:{mousedown}
mouseup :

updateEvt, activateEvt :

OTHERWISE

I

122

END:{case what event}
{and of then}
UNTIL done OR doneWithFile OR next:
END:{miniloop}

PROCEDURE findnext:
VAR
one : char:
BEGIN
REPEAT
IF eof(theFile) THEN
DoneWithFile := true
ELSE
readln(theFile, one);
UNTIL (one = '~') OR DoneWithFile
END:{findnext}

PROCEDURE openplot:
VAR

reply : SFreply:

typeList : SFTypeList:

where : point:

BEGIN
EnableItem(nextMenu, O);
DisableItem(fileMenu, 3):
EnableItem<fileMenu, 4);
enableItem(NextMenu, 1):
DrawMenuBar:
initCursor:
where.h := 80:
where.v := 100:
typeListh] := 'TEXT';
SFGetFile(where, ", NIL, l, TypeList, NIL, reply);
IF reply.good THEN
BEGIN
doneWithFile := false:
open(theFile, reply.fName);
findNext:
WHILE NOT doneWithFile DO
BEGIN
plot:
next := false:
miniloop:
eraseRect(blankrect):
findNext:
END:{while}
close(theFile);

END:{then and if}
DisableItem(nextMenu, 0):
DisableItem(fileMenu, 4);
EnableItem(fileMenu, 3):
DrawMenuBar:

DrawMenuBar:
END:{openplot}

PROCEDURE Setup:
BEGIN

123

Initgraf(@thePort);

Initfonts:

FlushEvents(everyEvent, 0):
SetEventMask(everyEvent):

InitCursor:

InitWindows:

SetRect(WindowRect, 1, 20, 510, 350);
setRect(blankRect, 1, 1, 509, 310);

myWindow := NewWindow(@wRecord, WindowRect, 'Contour Plot', false, 4,
pointer(-1), false, 0):
ShowWindow(myWindow);
SetPort(myWindow);
myWindow“.thize := 10:
SelectWindow(myWindow);
END:{Setup}
PROCEDURE SetupMenus:
VAR
MenuTopic : MenuHandle:
apple : char:
BEGIN
apple := '@':
apple := chr(AppleMark);
appleMenu := NewMenu(AppleID, apple); {get the apple desk accessories
menu}
AddResMenu(appleMenu, 'DRVR'); {adds all names into item list}
InsertMenu(appleMenu, 0); {put in list held by menu manager}
fileMenu := NewMenu(FileID, 'File'): {always need this for Quiting}
AppendMenu(FileMenu, 'Quit/Q'):
AppendMenu(fileMenu, '(-'):
AppendMenu(fileMenu, 'Open.../O:(Close ');
InsertMenu(fileMenu, 0):
editMenu := NewMenu(EditID, 'Edit'): {always need for editing Desk
Accessories}
AppendMenu(EditMenu, 'Undo/z');
Appendmenu(editMenu, '(—');
appendMenu(editMenu, 'Cut/x;Copy/C:Paste/V;Clear');
InsertMenu(editMenu, O):
nextMenu := newMenu(NextID, 'Next'):
AppendMenu(nextMenu, '(Next Plot');
InsertMenu(nextMenu, 0);
DrawMenuBar: {all done so show the menu bar}
END;
PROCEDURE ProcessMenu_in; {(CodeWord : longint)}
VAR
Menu_No : integer; {menu number that was selected}
Item;No : integer; {item in menu that was selected}
NameHolder : Str255: {name holder for desk accessory or font}
DNA : integer: {OpenDA will never return 0, so don't care}
BEGIN

IF CodeWord <> 0 THEN
BEGIN {go ahead and process the command}
Menu_No HiWord(CodeWord): {get the Hi word of...}
Item;no : LoWord(CodeWord): {get the L0 word of...}

124

CASE Menu_No OF
AppleID :
BEGIN
GetItem(GetMHandle(AppleID), Item;No, NameHolder);
DNA := OpenDeskAcc(NameHolder);

END;
FileID :
BEGIN
CASE Item_No OF
1 :
BEGIN
Done := True: {quit}
doneWithFile := true:
END:{quit}
{ 2: ------- }
3 :
openplot:
doneWithFile := true; {close}
END:
END:
EditID :
BEGIN
IF NOT SystemEdit(Item_no — 1) THEN {if not for a desk
accessory}
CASE Item4No OF
1 :
BEGIN
END; {undo}
{ 2: line divider}
3 :
BEGIN
END; {cut}
4 :
BEGI
END; {copy}
5 :
BEGIN
END; {paste}
6 :
BEGIN
END; {clear}
END;
END:
NextID :
next := true:
END:
HiliteMenu(O): {unhilite after processing menu}
END:

END; {of ProcessMenu_in procedure}

PROCEDURE initThings:

BEGIN{initThings}
setupmenus:

END:{initThings}

PROCEDURE loop:
VAR

125

WindowPointedTo : Windothr:
MouseLoc : Point:
WindoLoc : integer:

BEGIN

REPEAT
IF GetNextEvent(everyEvent, theEvt) THEN
CASE theEvt.what OF
mousedown :
BEGIN

MouseLoc := theEvt.Where:

WindoLoc := FindWindow(MouseLoc, WindowPointedTo);

CASE WindoLoc OF
inContent

IF WindowPointedTo <> FrontWindow THEN
SelectWindow(WindowPointedTo);

inDrag :

inGrow :

inGkoay :

done := TrackGkoay(WindowPointedTo, MouseLoc);
inMenuBar

ProcessMenu_in(MenuSelect(MouseLoc));
inSysWindow :

SystemClick(theEvt, WindowPointedTo);
OTHERWISE

END:{case theEvt.what}
END:{mousedown}
mouseup

updateEvt, activateEvt
OTHERWISE
END:{case what event}
{and of then}
UNTIL done:
END:{loop}
BEGIN{main program}
Setup:
initThings:
done := false:
loop:
END.{Main program}

END.

126

D.3 Three-D

PROGRAM ThreeD:

{This program produces a projection of a three dimensional graph from
data encoded and sent by subroutine Threed (in FORTRAN) from the Mainframe
It supports the full Macintosh Interface and shares much of its

Macintosh skeleton with contour.pas}

CONST {menu stuff}
AppleID = 1000:

FileID = 1001:

EditID = 1002:

nextID = 1003:
VAR

done, doneWithFile, next : Boolean:
theEvt : eventRecord:
wRecord : windowRecord:
myWindow, TheWindow : windothr;
WindowRect, blankRect : Rect:
appleMenu, fileMenu, editMenu, nextMenu : MenuHandle;
pen : penState:
thefile : text:
PROCEDURE ProcessMenu_in (CodeWord : longint);
forward;

PROCEDURE plot:

CONST
numlines = 16:
dhdi = 16:
dhdj = 0;
dhdk = —16:
dvdi = 2:
dvdj = -2:
dvdk = 12;
h0 = 256:
v0 = 128:

VAR
d : char:

nx, nz, i, j, k : integer:
x, 2, h, v : integer:
dxdi, dzdk : integer:
f : ARRAY[0..63, 0..63] OF integer;
BEGIN
readln(theFile, nx, nz);
FOR x := 0 TO nx - 1 DO
BEGIN
FOR 2 := 0 TO nz - 1 DO
BEGIN
read(theFile, d);
flx, z} := ord(d) - 40;
END:{for z}
readln(thefile);
END:{for x}
dxdi := nx DIV numlines:
dzdk := nz DIV numlines;
FOR 1 := 0 TO numlines — 1 DO
BEGIN
x := i * dxdi;

127

h := hO + dhdi * i +
v = v0 + dvdi * i +
moveto (h, v) ;
FOR 2 = 1 TO nz — 1
BEGIN
h °= hO + dhdi
v := v0 + dvdi
lineto(h, v);
END:{for z}
END:{for 1}
FOR k := 0 TO numlines — 1
BEGIN
z = k * dzdk;
h = hO + dhdk * k +
v °= v0 + dvdk * k +
moveto(h, v);
FOR x °= 1 TO nx - 1
BEGIN
h '= hO + dhdk
v = v0 + dvdk
lineto(h, v):
END:{for x}
END:{for k}
h := hO:
v := v0:

moveto(h, v):

getPenState(pen):
2:
:= 2;
setPenState(pen):
64

pen.anize.v :
pen.pnsize.h

line(64 * dhdj,
moveto(h, v):
line(l? * dhdi,
moveto(h, v):
line(17 * dhdk,

l7

pen.anize.v := 1:
pen.pnsize.h := 1:
setPenState(pen):
END:{Plot}
PROCEDURE miniloop:
VAR
WindowPointedTo
MouseLoc : Point:
WindoLoc : integer:
BEGIN
HiliteMenu(O):
REPEAT

* dvdj);

* dvdi);

dhdj * f[x, 0];
dvdj * f[x, 0]:

DO

(2 * dhdk) DIV dzdk + dhdj * f[x, z]:

* i +
* i + (z * dvdk) DIV dzdk + dvdj * f[x, 2]:

DO
dhdj * f[0, z];
dvdj * f[O, 2]:
DO

(x * dhdi) DIV dxdi + dhdj * f[x, z];

* k +
* k + (x * dvdi) DIV dxdi + dvdj * f[x, z]:

{y axis}

{x axis}

17 * dvdk):{z axis}

: Windothr:

IF GetNextEvent(everyEvent, theEvt) THEN
CASE theEvt.what OF
mousedown :
BEGIN

MouseLoc :
WindoLoc :

theEvt.Where:
FindWindow(MouseLoc, WindowPointedTo);

CASE WindoLoc OF
inMenuBar :
ProcessMenu_in(MenuSelect(MouseLoc));
inSysWindow :
SystemClick(theEvt, WindowPointedTo):

128

OTHERWISE

END:{case theEvt.what}
END:{mousedown}
mouseup :

updateEvt, activateEvt :
OTHERWISE

END:{case what event}
{and of then}
UNTIL done OR doneWithFile OR next:
END:{miniloop}

PROCEDURE findnext (VAR t : char);
VAR
one : char:
ni : integer:
Qstring : str255:
BEGIN
Qstring := 'Type s to skip, <ret> to plot';
REPEAT
IF eof(theFile) THEN
DoneWithFile := true
ELSE
readln(theFile, one);
UNTIL (one = '~') OR DoneWithFile;
moveto(lOO, 10):
hi := 1:
IF (NOT donewithfile) THEN
BEGIN
WHILE ((NOT eoln(theFile)) AND (ni < 72)) DO
BEGIN
read(theFile, t);
DrawChar(t):
ni := ni + 1
END:{while}
ni := 1:
readln(theFile);
moveto(lOO, 30):
drawstring(qstring);
read(t);
END:{if}
eraseRect(blankrect);
END:{findnext}

PROCEDURE openplot:
VAR
reply : SFreply:
typeList : SFTypeList:
where : point:
t : char:
BEGIN
EnableItem(nextMEnu, 0):
DisableItem(fileMenu, 3);
EnableItem(fileMenu, 4):
enableItem(NextMenu, 1):
DrawMenuBar:

129

initCursor:
where.h := 80:
where.v := 100:
typeList[0] := 'TEXT';
SFGetFile(where, ", NIL, 1, TypeList, NIL, reply);
IF reply.good THEN
BEGIN
doneWithFile := false:
open(theFile, reply.fName):
findNext(t);
WHILE NOT doneWithFile DO
BEGIN
CASE t OF
's' :

OTHERWISE
BEGIN
plot:
next := false:
miniloop:
eraseRect(blankrect);
END;{Otherwise}
END:{case}
IF NOT doneWithFile THEN
findNext(t);
END:{while}
close(theFile);

END:{then and if}
DisableItem(nextMenu, 0):
DisableItem(fileMenu, 4):
EnableItem(fileMenu, 3):
DrawMenuBar:

DrawMenuBar:
END:{openplot}

PROCEDURE Setup:
BEGIN
Initgraf(@thePort);
Initfonts:
FlushEvents(everyEvent, 0);
SetEventMask(everyEvent):
InitCursor:
InitWindows:
SetRect(WindowRect, l, 20, 512, 350);
setRect(blankRect, 0, 0, 511, 340);

myWindow := NewWindow(@wRecord, WindowRect, 'Contour Plot',

pointer(—1), false, 0):
ShowWindow(myWindow);
SetPort(myWindow);
myWindow*.thize := 10:
SelectWindow(myWindow);

END:{Setup}
PROCEDURE SetupMenus:
VAR
MenuTopic : MenuHandle:
apple : Char:
BEGIN
apple := '@':

130

false,

4,

apple := chr(AppleMark);
appleMenu := NewMenu(AppleID, apple): {get the apple desk accessories
menu}

AddResMenu(appleMenu, 'DRVR'); {adds all names into item list}
InsertMenu(appleMenu, 0): {put in list held by menu manager}
fileMenu := NewMenu(FileID, 'File'); {always need this for Quiting}

AppendMenu(FileMenu, 'Quit/Q'):
AppendMenu(fileMenu, '(-'):
AppendMenu(fileMenu, 'Open.../O:(Close '):
InsertMenu(fileMenu, 0):

editMenu := NewMenu(EditID, 'Edit'); {always need for editing Desk
Accessories}

AppendMenu(EditMenu, 'UndO/z');

Appendmenu(editMenu, '(-');

appendMenu<editMenu, 'Cut/x:COpy/C;Paste/V;Clear');

InsertMenu(editMenu, 0);

nextMenu := newMenu(NextID, 'Next'):
AppendMenu(nextMenu, '(Next Plot'):
InsertMenu(nextMenu, 0):

DrawMenuBar: {all done so show the menu bar}
END:

PROCEDURE ProcessMenu_in; {(CodeWord : longint)}

VAR
Menu_No : integer; {menu number that was selected}
Item;No : integer; {item in menu that was selected}
NameHolder : Str255; {name holder for desk accessory or font}
DNA : integer: {OpenDA will never return 0, so don't care}
BEGIN

IF CodeWord <> 0 THEN
BEGIN {go ahead and process the command}
Menu_NO := HiWOrd(COdeWord): {get the Hi word of...}
Item_no := LoWord(CodeWord): {get the Lo word of...}

CASE Menu;No OF
AppleID :
BEGIN
GetItem(GetMHandle(AppleID), Item_No, NameHolder);
DNA := OpenDeskAcc(NameHolder);

END,
FileID :
BEGIN
CASE Item;No OF
1 :
BEGIN
Done := True: {quit}
doneWithFile := true:
END:{quit}
1 2: -——----1
3 :
openplot:
4 :
doneWithFile := true; {close}
END:
END:

EditID :

131

BEGIN
IF NOT SystemEdit(Item_no - 1) THEN {if not for a desk

accessory}
CASE Item_No OF

1 :
BEGIN
END; {undo}

{ 2: line divider}

3 :
BEGI
END; {cut}

4 :
BEGI

5 END: {copy}
BEGIN
END; {paste}

6 :
BEGIN
END; {clear}

END:
END:
NextID :
next := true:
END:
HiliteMenu(O): {unhilite after processing menu}
END:

END; {of ProcessMenu_in procedure}

PROCEDURE initThings:

BEGIN{initThings}
setupmenus:

END:{initThings}

PROCEDURE loop:
VAR
WindowPointedTo : Windothr:
MouseLoc : Point:
WindoLoc : integer:
BEGIN

REPEAT
IF GetNextEvent(everyEvent, theEvt) THEN
CASE theEvt.what OF
mousedown :
BEGIN

MouseLoc := theEvt.Where:

WindoLoc := FindWindow(MouseLoc, WindowPointedTo):

CASE WindoLoc OF
inContent

IF WindowPointedTo <> FrontWindow THEN
SelectWindow(WindowPointedTo);

inDrag :

inGrow :
iMmey:
done := TrackGOAway(WindowPointedTo, MouseLoc):

inMenuBar :
ProcessMenu_in(MenuSelect(MouseLoc));

132

inSysWindow :
SystemClick(theEvt, WindowPointedTo):
OTHERWISE

END:{case theEvt.what}
END;{mousedown}
mouseup :

updateEvt, activateEvt
OTHERWISE

END:{case what event}
{and of then}
UNTIL done:
END;{loop}

BEGIN{main program}
Setup:
initThings:
done := false:
loop:

END.{Main program}

END.

133

 

D.4 FFT

* THESE ROUTINES ARE THE PAST FOURIER TRANSFORM ROUTINES

* BASED ON PRESS ET AL
SUBROUTINE FOUR1(DATA,NN,ISIGN)

* THIS SUBROUTINE IS FROM PRESS ET AL
DOUBLE PRECISION WR,WI,WPR,WPI,WTEMP,THETA
REAL DATA(2*NN)

N=2*NN
J=1
DO 11 I=1,N,2
IF(J.GT.I)THEN
TEMPR=DATA(J)
TEMPI=DATA(J+1)
DATA (J) =DATA (I)
DATA(J+1)=DATA(I+1)
DATA(I)=TEMPR
DATA(I+1)=TEMPI
ENDIF
M=N/ 2
1 IF((M.GE.2).AND.(J.GT.M)) THEN
J=J€M
M=M/ 2
GOTO 1
ENDIF
J=J+M
11 CONTINUE
MMAX=2
2 IF(N.GT.MMAX) THEN
ISTEP=2*MMAX
THETA=6.28318530717959D0/(ISIGN*MMAX)
WPR=-2.DO*DSIN(0.5D0*THETA)**2
WPI=DSIN(THETA)
WR=1.D0
WI=0.D0
DO 13 M=1,MMAX,2
DO 12 I=M,N,ISTEP
J=I+MMAX
TEMPR=SNGL(WR)*DATA(J)-SNGL(WI)*DATA(J+1)
TEMPI=SNGL(WR)*DATA(J+1)+SNGL(WI)*DATA(J)
DATA(J)=DATA(I)-TEMPR
DATA(J+1)=DATA(I+1)-TEMPI
DATA (I) =DATA (I) +TEMPR
DATA(I+1)=DATA(I+1)+TEMPI
12 CONTINUE
WTEMP=WR
WR=WR*WPR-WI*WPI+WR
WI= WI*WPR+WTEMP*WPI+WI
13 CONTINUE
MMAX=ISTEP
GOTO 2
ENDIF
RETURN
END

SUBROUTINE TWOFFT(DATA1,DATA2,FFT1,FFT2,N)
* THIS ROUTINE WAS ADAPTED FROM PRESS ET AL
* MODIFIED T.ESPINOLA

REAL DATA1(N),DATA2(N)

COMPLEX FFT1(N+1),FFT2(N+1),H1,H2,C1,C2

134

11

12

* THE

11

12

10

Cl=(0.5,0.0)

C2=(0.0,-0.5)

DO 11 J=1,N
FFT1(J)=CMPLX(DATA1(J),DATA2(J))
CONTINUE

CALL FOUR1(FFT1,N,1)

FFT1(N+1)=FFT1 (l)

N2=N+2

DO 12 J=1,N/2+1

H1=Cl*(FFT1(J)+CONJG(FFT1(N2-J)))
H2=C2*(FFT1(J)—CONJG(FFT1(N2-J)))
FFT1(J)=H1
FFT1(N2-J)=CONJG(H1)
FFT2 (J) =H2
FFT2(N2-J)=CONJG(H2)
CONTINUE

RETURN

END

REMAINING ROUTINES ARE BY T. ESPINOLA
SUBROUTINE FFTR(DATA1,DATA2,FFT1,N)
REAL DATA1(N),DATA2 (N)

COMPLEX FFT1(N+1) , H1, H2, C1, C2

C1=(0.5,0.0)

C2=(0.0,-0.5)

DO 11 J=1,N
FFT1(J)=CMPLX(DATA1(J),DATA2(J))
CONTINUE

CALL FOUR1(FFT1,N,l)

FFT1(N+1)=FFT1(1)

N2=N+2

DO 12 J=1,N/2+1
H1=C1*(FFT1(J)+CONJG(FFT1(N2—J)))
H2=C2* (FFTl (J) -CONJG(FFT1 (NZ-J) ) )
FFT1(J)=H1
FFTl (NZ-J) =H2
CONTINUE

RETURN

END

SUBROUTINE FFTIZ (DATAl , DATA2 , WORK, N)
COMPLEX DATAl (N/Z) , DATA2 (N/2) ,WORK(N+1) , I
I=(0.0,1.0)
=N/2
N2=N+2
DO 10 J:2,NN
WORK (J) =DATA1 (J) +DATA2 (J) *I
WORK (NZ-J) =CONJG (DATAl (J) )+I*CONJG (DATA2 (J) )
CONTINUE
WORK(1)=CMPLX(REAL(DATA1(1) ) ,REAL(DATA2 (1) ) )
WORK (NN+1) =CMPLX (AIMAG (DATA1 (1) ),AIMAG(DATA2(1) ) )
CALL FOURl(WORK,N,-1)
RETURN
END

SUBROUT INE FFTI (DATAl , DATA2 , WORK, N)
REAL DATAl (N) , DATA2 (N)

COMPLEX WORK (N+1)

CALL FFTI2(DATA1, DATA2 , WORK, N)

DO 10 J=1, N

135

fi-X-fi-fl-l-fi-Jl'

DATAl (J) =REAL (WORK (J) ) /N
DATA2 (J) =AIMAG (WORK (J) ) /N
10 CONTINUE
RETURN
END

SUBROUTINE FFT2 (DATAl , DATA2 , WORK, N, IS IGN )
THIS IS THE ROUTINE DIRECTLY CALLED BY GMHD
DATAl AND DATA2 ARE THE TWO ARRAYS TO BE TRANSFORMED
THE TRANSFORMATION IS STORED IN PLACE
WORK IS WORK SPACE AND MUST BE DIMENSIONED AT LEAST 4* (N+1)
N IS THE SIZE OF THE ARRAY AND MUST BE A POWER OF 2
ISIGN DETERMINES WHETHER IT IS A THE TRANSFORM OR AN INVERSE TRANSFORM
ISIGN =1 FOR TRANSFORM ISGN=-1 FOR INVERSE TRANSFORM
REAL DATAl (N) ,DATA2 (N) ,WORK (2*N-i-2, 2)
IF (ISIGN.EQ.1)THEN
CALL TWOFFT (DATA1,DATA2,WORK(1, 1) ,WORK(1,2) ,N)
DO 10 I=1,N
DATA1(I)=WORK(I, 1)
DATA2 (I)=WORK(I, 2)
10 CONTINUE
DATA1 (2) =WORK (N+1, 1)
DATA2 (2) =WORK (N+1, 2)
ELSE
CALL FFTI (DATAl , DATA2 , WORK, N)
ENDIF
RETURN
END

136

D.5 Makcon

*li-ilrllrll-

I»

*

SUBROUTINE MakCon(F,Z,NX,NZ,NUMLIN,TITLE)
THESE ROUTINES MAKE A CONTOUR PLOT OF THE ARRAY F
THE CELLS ARE EVENLY SPACED IN THE X DIRECTION WHILE THE ARRAY
Z CONTAINS THE Z POSITION OF THE GRID POINTS
THE PLOTTING INFORMATION IS ENCODED AS CHARACTERS TO BE DOWNLOADED
TO THE MAC

IMPLICIT NONE
PASSED VARIABLES

INTEGER NX,NZ,NUMLIN

REAL F(NX,NZ),Z(NX,NZ)

CHARACTER*20 TITLE
LOCAL VARIABLES

INTEGER NY, NH, NAX, MAXA, NHMAX, I, J, NYY,MA.XAA

PARAMETER(NY=292,MAXA=1024,NHMAX=292)

REAL A(NY,0:33),FMIN,FMAX,ZMIN,ZMAX,DF,DZ

INTEGER FIRST(NY),LAST(NY)

COMMON /CONZZL/ NH, NYY, MAXAA, Dz, FMIN, ZMIN, DF

IF(NX.LT.8)THEN

PRINT *,‘NX MUST >= 8'

RETURN

ENDIF

NYY=NY

MAXAA=MAXA

CALL FINDRA(F,NX,NZ,FMIN,FMAX)

CALL FINDRA(Z,NX,NZ,ZMIN,ZMAX)

IF(FMIN.EQ.FMAX.OR.ZMIN.EQ.ZMAX)THEN

PRINT*,'Error in array passed to countour.‘
PRINT*,'Function or 2 has no variation.’
ELSE
PRINT *,CHAR(126),' Start of plot. ',TITLE
DF=REAL (MAXA) / {FMAX-FMIN)
DZ=(ZMAX-ZMIN)/NY
J:1
CALL GETCOL(FIRST,F,Z,NX,NZ,J)
NAX=NHMAX/NX
NH=1+(NX-1)*NAX

DO 10 J=2,NX
CALL GETCOL(LAST,F,Z,NX,NZ,J)
CALL FILCOL(A,FIRST,LAST,NAX,NUMLIN)
CALL OUTA(A,NAX)
DO 20 I=1,NY
FIRST(I)=LAST (I)
20 CONTINUE
10 CONTINUE
ENDIF
PRINT *,CHAR(107),'End Of plot.’
PRINT 1000,FMIN,FMAX,(fmax-fmin)/numlin

1000 FORMaT(1X,'Min = ',f9.4,/,1x,'max= ',f9.4,/,1x,'contours= ',f9.5)

RETURN
END

SUBROUTINE GETCOL(COL,F,Z,NX,NZ,J)
IMPLICIT NONE
GLOBAL VARIABLES
INTEGER NH,NY,MAXA
REAL DZ,FMIN,ZMIN,DF

137

COMMON /CONZZL/NH,NY,MAXA,DZ,FMIN,ZMIN,DF
* PASSED VARIABLES

INTEGER NX,NZ,J

REAL F(NX,NZ),Z(NX,NZ)

INTEGER COL(NY)
* LOCAL VARIABLES

INTEGER K,L,KO,KK,K1

REAL DELTA,F0
* BEGIN GETCOL

K=1
L=1
* WHILE
1 IF(Z(J,L).GT.(ZMIN+K*DZ))THEN
COL(K)=-MAXA
K=K+1
GOTOl
ENDIF
K1=K
L=L+1
3 IF (L.LT.NZ)THEN
KO=K
DELTA=DZ*(F(J,L+l)-F(J,L))/(Z(J,L+1)-Z(J,L))
F0=F(J,L)-FMIN
2 IF(Z(J,L).GT.(ZMIN+K*DZ))THEN
COL(K)=(F0+(K-KO)*DELTA)*DF
K=K+1
GOTO 2
ENDIF
=L+1
GOTO 3
ENDIF
COL(K)=(F(J,NZ)-FMIN)*DF
DO 10 KK=K+1,NY
COL(KK)=-MAXA
10 CONTINUE
RETURN
END

SUBROUTINE FILCOL(A,FIRST,LAST,NAX,NUMLIN)
IMPLICIT NONE
* GLOBAL VARIABLES
INTEGER NH,NY,MAXA
REAL DZ,FMIN,ZMIN,DF
COMMON /CONZZL/NH,NY,MAXA,DZ,FMIN,ZMIN,DF
* PASSED VARIABLES
INTEGER A(NY, 0 : 33) ,FIRST (NY) ,LAST (NY) ,NAX,NUMLIN
* LOCAL VARIABLES
INTEGER I,K
REAL DELTA,DC
* BEGIN FILCOL
DO 10 K=1,NY
A(K,0)=FIRST(K)
A(K,NAX)=LAST(K)
DELTA=REAL(LAST(K)-FIRST(K))/NAX
Do 20 I= l,NAX-l
A(K,I)=FIRST(K)+DELTA*I
20 CONTINUE
10 CONTINUE
DC=REAL(NUMLIN)/REAL(MAXA)*.5

138

DO 30 I=0,NAX
DO 30 K=1,NY
A(K,I)=(A(K,I)+ABS(A(K,I)))*DC
30 CONTINUE
RETURN
END

SUBROUTINE OUTA(A,NAX)
IMPLICIT NONE
* GLOBAL VARIABLES
INTEGER NH,NY,MAXA
REAL DZ,FMIN,ZMIN,DF
COMMON /CONZZL/NH,NY,MAXA,DZ,FMIN,ZMIN,DF
* PASSED VARIABLES
INTEGER.A(NY,0:33),NAX
* LOCAL VARIABLES
INTEGER NUMOUT,COUNT,K,I
CHARACTER*1 OUTIT(5000)

* BEGIN OUTA
NUMOUT=1
DO 10 I=0,NAX-l
OUTIT(NUMOUT)=CHAR(106)
NUMOUT=NUMOUT+1

K=1
l IF(K.LT.NY)THEN
COUNT=1
2 IF((COUNT.LE.64).AND.(K.LT.NY))THEN

IF( ( (A(K,I).EQ.A(K,I+1)) .OR. (A(K,I)*A(K,I+1).EQ.O))
$.AND. ((A(K,I).EQ.A(K+1,I)).OR.(A(K,I)*A(K+1,I).EQ.O)))THEN
COUNT=COUNT+1
ELSE
OUTIT(NUMOUT)=CHAR(40+COUNT)
NUMOUT=NUMOUT+1
COUNT=1
ENDIF
K=K+1
GOTO 2
ENDIF
IF(COUNT.GE.65)THEN
OUTIT(NUMOUT)=CHAR(40+COUNT)
NUMOUT=NUMOUT+1
ENDIF
GOTO 1
ENDIF
10 CONTINUE
WRITE(*,1000)(OUTIT(I),I=1,NUMOUT-1)
1000 FORMAT(1X,80A1)
RETURN
END

SUBROUTINE FINDRA(F,NX,NZ,EMIN,FMAX)
* PASSED VARIABLES

INTEGER NX,NZ

REAL F(NX,NZ),FMIN,FMAX
* LOCAL VARIABLES

INTEGER J,K
* BEGIN FINDRA

FMIN=F(1,1)

FMAX=F(1,1)

139

10

DO 10 J=1,NX

DO 10 K=1,NZ
FMIN=AMIN1(FMIN, F (J, K) )
FMAX=AMAX1(FMAX, F (J, K) )
CONTINUE

RETURN

END

140

 

 

 

D.6 Make3D

*ii-ifil-

SUBROUTINE THREED(F,NX,NZ,TITLE)

THESE ROUTINES PRODUCE A 3D PROJECTION GRAPH OF THE DATA IN ARRAY F
IT DOES NOT ACCOUNT FOR THE Z POSITION OF THE GRID POINTS BUT PLOTS
THEM EQUALLY SPACED. THE PLOTTING INFORMATION IS ENCODED AS CHARACTERS

FOR

DOWNLOADING TO THE MAC. THIS IS A BETA
IMPLICIT NONE

PASSED VARIABLES

INTEGER NX,NZ
REAL F(-1:NX,0:NZ-l)
CHARACTER*40 TITLE

LOCAL VARIABLES

2000

20
10

1000
100

REAL MAX,MIN,DF,DX,DZ,D,X,Z
INTEGER J,K,I,NH,NV,NL,KK
CHARACTER*1 A(0:63,0:63)

MAX=F(0,0)
MIN=F(0,0)
DO 5 J=0,NX-l
DO 5 K=0,NZ-1
MIN=AMIN1(MIN,F(J,K))
MAX=AMAX1 (MAX,F (J, K) )
CONTINUE
IF(MAX.EQ.MIN)THEN
PRINT *,‘NO VARIATION IN PLOT ’,TITLE
RETURN
ENDIF
DF=64/(MAX-MIN)
PRINT *,'~'
WRITE( *,2000)TITLE(1:30),MAX,MIN
FORMAT(1X,A30,' MAX=',F10.4,' MIN=',F10.4)
NH=64
NV=64

DX=FLOAT(NX-l)/(NH-l)
DZ=FLOAT(NZ-l)/(NV-l)
DO 10 K=O,NV-1
Z=K*DZ
KK=Z
DO 20 J=0,NH-1
X=J*DX
I=X
D=F(I,KK)+(F(I+1,KK)-F(I,KK))*(X-I)+

$ {F(I,KK+1)-F(I,KK))*(Z-IG<)

A(J,K)=CHAR(IFIX((D-MIN)*DF+40))
CONTINUE
CONTINUE

PRINT *,NH,NV

DO 100 K=0,NV-l
WRITE(*,1000)(A(J,K),J=0,NH-1)
FORMAT(1X,80A1)

CONTINUE

END

141

 

Appendix E
The Program

The main program and its included subroutines and functions are given below. I have
included the Nassi-Shneidarman diagrams, where appropriate to illustrate their
structure.

Main Program

 

Call lnit
T-O
count-0

 

While T<Tmax

 

increment count
call BC for boundary conditions

gall Grid
If

Tviewso

     
   

  

yes

 

 

call output
tview=tview+tprint

 

 

aka 2 small
call Calcnewr to get newr and newz
steps
call BC(newr,newz,t+At)
call Grid(n.ewr,newz,t+At)
call Calcnewr to get newr2
take 1 big call BC(r,z,t)
step call Grid(r,z,t)

call Calcnewr to get newr and z

 

repeatacheckdt(newr,newr2)

comparing newr and newr2

to see if At is small enough
if it is, set next At

 

 

repeat until At small enough

 

set r and 2 using newr, newz,
newr2,newz2 to get 5th order
t=t+2At
tview=tview+2At

 

 

 

142

 

THIS

PROGRAM GMHD
IS THE MAIN PROGRAM FOR DOING GRAVITATIONAL MAGNETOHYDRODYMIC

SIMULATIONS. IT IS THE 2D VERSION USED PRIMARILY ON THE VAX

IT D
IT I

GLOB

0

\JOAUIDWNH

OEs NOT CONTAIN ALL THE VECTORIZING OPTIMIZATION

5 ONLY SPARSELY COMMENTED BECAUSE IT Is DOWNLOADED AT 1200 BAUD

IMPLICIT NONE

AL VARIABLES

REAL F1D15,F16D15,0VER8PI,PI,FOURPI

INTEGER IFAX(13),INC,JUMP,NPTS,NVEC

REAL WORK(0:31,1:50,8),TRIGS(451)

INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF

REAL TMAX,TPRINT,PFLAG,TVIEW

REAL MASS, ENERGY, MAsso , ENERGYO, DENERGY, DMAss, OLDM, OLDE

REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)

COMPLEX DX(O:31)

INTEGER SYMFLAG

REAL C(0:31,l:50,8)

COMMON /FFT/WORK,SYMFLAG
/NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
/CONST/FlDlS,F16D15,0VER8PI,PI,FOURPI
/PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
/PRINTS/TMAX,TPRINT,PFLAG,TVIEW
/CHECK/MASS,ENERGY,MASSO,ENERGYO,DENERGY,DMASS,OLDM,OLDE
/GRID/DZDX,DZDL,DZDT,DX
/SOMMER/C

LOCAL VARIABLES

0
1
2
3

BEGI

THE
2

REAL R(O:3l,l:50,8),NEWR(O:31,1:50,8),QR(0:31,5,8),
Z(0:31,1:50),NEWZ(O:31,1:50),
NEWR2(0:31,1:50,8),NEW22(O:31,1:50),NEWT,T,
DT,DTX2,CURRENT,DT2

INTEGER J,L,M,COUNT

LOGICAL REPEAT,CHECKDT

N MAIN PROGRAM

CALL INIT(R,Z,DT)

T=0.0

COUNT=O

MAIN LOOP-- WHILE T<TMAX
IF(T.LE.(TMAX+.01))THEN

A DO-UNTIL LOOP WHICH REPEATS UNTIL THE ERROR IS SMALL ENOUGH

3

CONTINUE
COUNT=COUNT+1
CALL BC(R,Z,T)
CALL GRID(R,Z,T,DT)
IF(TVIEW.LE.1E-6) THEN
PRINT 1000,T,COUNT,T/COUNT

1000FORMAT(' T=',F9.5,' STEPS=',IS,' AVERAGE DT=',F9.7)

CALL OUTPUT(R,T,Z)
TVIEW=TVIEW+TPRINT
ENDIF
CALL CALCNEWR(R,NEWR,Z,NEWZ,T,DT)
=T+DT
CALL BC(NEWR,NEWZ,NEWT)
CALL GRID(NEWR,NEWZ,NEWT,DT)
CALL CALCNEWR(NEWR,NEWR2,NEWZ,NEWZZ,NEWT,DT)
DTX2=2.*DT
CALL BC(R,Z,T)
CALL GRID(R,Z,T,DT)
CALL CALCNEWR (R, NEWR, Z, NEWZ, T, DTX2)
REPEAT=CHECKDT(R,NEWR,NEWR2,T,DT)

143

 

 

 

* END
* USE

10

20

IF(REPEAT)GOTO 3

OF DO UNTIL ERROR IS SMALL ENOUGH

THE EXTRA INFO TO FIND NEWR TO 5TH ORDER

DO 10 M=1,8

DO 10 J=XLO,XHI
DO 10 L=ZLO,ZHI

R(J,L,M)=NEWR2(J,L,M)

R(J,L,M)=F16D15*NEWR2(J,L,M)~F1D15*NEWR(J,L,M)
R(J,L,M)=NEWR2(J,L,M)+F1D15*(NEWR2(J,L,M)-NEWR(J,L,M))
CONTINUE
DO 20 J=XLO,XHI
DO 20 L=ZLO,ZHI
Z(J,L)=NEWZZ(J,L)

Z(J,L)=F16D15*NEWZZ(J,L)-F1D15*NEWZ(J,L)

Z(J,L)=NEWZ2(J,L)+F1D15*(NEW22(J,L)-NEWZ(J,L))
CONTINUE

T=T+DTX2

TVIEW=TVIEW-DTX2

GOTO 2

OF WHILE

ENDIF

PRINT *,‘REACHED MAX TIME OF ',TMAX

END

144

 

CalcNewR

 

Set A12 =At/2
A13 :At/3

 

Call Cachf'(f',Z,t)

 

For all cells

 

 

Set Ztemp=z+dzdt*At6
Newz=z+dzdt*At2
rtemp=r+drdt*At6
newr=r+drdt*At2

 

 

Call BC(newr,newz,t+At2)
Call Grid(newr,newz,t+At2)
Coll cochr(newr,newz,t+At2)

 

For all cells

 

 

Set Ztemp=ztemp+dzdtffAt3
Newz=z+dzdt*At2
rtemp:rtemp+drdt*At3
newr=r+drdtffAt2

 

 

Call BC(newr,newz,t+At2)
Call Grid(newr,newz,t+At2)
Call cachr(newr,newz,t+At2)

 

For all cells

 

 

Set Ztemp=ztemp+dzdtffAt3
Newz=z+dzdt*At
rtemp:rtemp+drdt*At3
newr=r+drdt*At

 

 

Call BC(newr,newz,t+At)
Call Grid(newr,newz,t+At)
Coll cochr(newr,new2,t+At)

 

For all cells

 

 

 

Set Newz=ztemp+dzdtttAt6
newr=rtemp+drdt"‘At6

 

 

145

 

 

 

*****************‘k*‘k‘k*****'A'***************‘k************************‘k

* THIS SUBROUTINE USES A FOURTH ORDER SCHEME TO FIND THE NEW

* VARIABLES.
********************************************************************

SUBROUTINE CALCNEWR(R,NEWR,Z,NEWZ,T,DT)
IMPLICIT NONE

* PASSED VARIABLES
OREAL R(0:31,1:50,8),NEWR(0:31,1:50,8),
1 Z(0:31,1:50),NEWZ(0:31,1:50),T,DT

* GLOBAL VARIABLES _
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL TMAX,TPRINT,PFLAG,TVIEW
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50) .g
COMPLEX DX(0:31) '

0COMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

 

l /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
2 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW
3 /GRID/DZDX, DZDL, DZDT, DX L

* LOCAL VARIABLES
OREAL DT2,DT3,DT6,DR(O:31,1:50, 8) ,RTEMP (0:31, 1:50, 8),
1 ZTEMP(0:31,1:50),NEWT
INTEGER J,L,M

1(-

BEGIN CALCNEWR

FIND THE DT'S NEEDED
DT2=.5*DT
DT3=DT/3.0
DT6=.5*DT3

1(-

CALL CALCDR(R,Z,DR,T,DT2)
* DR AT T
DO 10 J:XLO,XHI
DO 20 L=ZLO,ZHI
ZTEMP(J,L)=Z(J,L)+DT6*DZDT(J,L)
NEWZ(J,L)=Z(J,L)+DT2*DZDT(J,L)
20 CONTINUE
DO 30 M=1,8
DO 30 L=ZLO,ZHI
NEWR(J,L,M)=R(J,L,M)+DT2*DR(J,L,M)
RTEMP(J,L,M)=R(J,L,M)+DT6*DR(J,L,M)

30 CONTINUE
10 CONTINUE
NEWT=T+DT2

CALL BC(NEWR,NEWZ,NEWT)
CALL GRID(NEWR,NEWZ,NEWT,DT)
CALL CALCDR(NEWR,NEWZ,DR,NEWT,DT2)
* DR AT T+DT/2 FIRST TIME
DO 40 J=XLO,XHI
DO 50 L=ZLO,ZHI
ZTEMP(J,L)=ZTEMP(J,L)+DT3*DZDT(J,L)
50 NEWZ(J,L)=Z(J,L)+DT2*DZDT(J,L)
DO 60 M:1,8
DO 60 L=ZLO,ZHI
NEWR(J,L,M)=R(J,L,M)+DT2*DR(J,L,M)
RTEMP(J,L,M)=RTEMP(J,L,M)+DT3*DR(J,L,M)
6O CONTINUE

146 '

 

4O CONTINUE

CALL BC(NEWR,NEWZ,NEWT)
CALL GRID(NEWR,NEWZ,NEWT,DT)
CALL CALCDR(NEWR,NEWZ,DR,NEWT,DT)
* DR AT T+DT/2 SECOND TIME
DO 70 J=XLO,XHI
DO 80 L=ZLO,ZHI
ZTEMP(J,L)=ZTEMP(J,L)+DT3*DZDT(J,L)
80 NEWZ(J,L)=Z(J,L)+DT*DZDT(J,L)
DO 90 M=1, 8
DO 90 L=ZLO,ZHI
NEWR(J,L,M)=R(J,L,M)+DT*DR(J,L,M)
RTEMP(J,L,M)=RTEMP(J,L,M)+DT3*DR(J,L,M)
90 CONTINUE
7O CONTINUE

NEWT=T+DT

CALL BC(NEWR,NEWZ,NEWT)

CALL GRID(NEWR,NEWZ,NEWT,DT)

CALL CALCDR(NEWR,NEWZ,DR,NEWT,DT)
* DR AT T+DT

 

DO 110 J=XLO,XHI
DO 120 L=ZLO,ZHI
120 NEWZ(J,L)=ZTEMP(J,L)+DT6*DZDT(J,L)
DO 130 M=1,8
DO 130 L=ZLO,ZHI
NEWR(J,L,M)=RTEMP(J,L,M)+DT6*DR(J,L,M)
130 CONTINUE
110 CONTINUE
C CALL SOMMER(R,NEWR,DT,Z,NEWZ)
CALL BC(NEWR,NEWZ,NEWT)
RETURN
END

147

 

cachR

 

 

 

call calcVfl to get velogitigs gng fields
call diffVB to get derivatives 91' v and B

for all values of j (x direction)

 

 

‘ for all values of l (2 direction)

 

Set pressure: P=( Y-l) pe
F"4:(Bo)/2Ll

 

 

 

call calcTij to get viscous stress tensor

 

for all values of l (2 direction)

 

calculate Fx
and advecZ

 

 

 

 

for all values of l (2 direction)

 

calculate FZ using advecZ
and DR (source terms)

 

 

 

 

 

for all cells and gas variables

 

Use ddl to find the z derivatives of F2
and ddx to find the x derivatives of FX
and combine with the source terms in DR

 

 

 

 

********************************************************************

* THIS SUBROUTINE CALCULATES THE DERIVATIVES TO BE USED IN CALCNEWR
********************************************************************
SUBROUTINE CALCDR(R,Z,DR,T,DT)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8),Z(0:31,1:50),DR(0:31,1:50,8),T,DT
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL F1D15,F16D15,0VER8PI,PI,FOURPI
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
REAL TMAX,TPRINT,PFLAG,TVIEW
REAL DZDX(0:31,1:50),DZDL(O:31,1:50),DZDT(0:31,1:50)
REAL MASS,ENERGY,MASSO,ENERGYO,DENERGY,DMASS,OLDM,OLDE
COMPLEX DX(0:31)
INTEGER FXSYM(8)
148

 

'k

'k
'k
”k
'k

*

OCOMMON
/NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
/CONST/F1D15,F16D15,0VER8PI,PI,FOURPI
/PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
/PRINTS/TMAX,TPRINT,PFLAG,TVIEW
/CHECK/MASS,ENERGY,MASSO,ENERGYO,DENERGY,DMASS,OLDM,OLDE
/GRID/DZDX,DZDL,DZDT,DX

/SYMS/FXSYM

\lONU'lubUJNH

LOCAL VARIABLES
INTEGER J,L,M

OREAL ETA(1:50),P(1:50),PM(1:50),
U(0:31,1:50),V(0:31,1:50),W(0:31,1:50),TXX(1:50),
TXY(1:50),TXZ(1:50),TYY(1:50),TYZ(1:50),TZZ(1:50),
DUDX(0:31,1:50),DWDX(0:31,1:50),ADVECZ(1:50),
DUDZ(O:31,1:50),DVDZ(0:31,1:50),DWDZ(0:31,1:50),
DVDX(0:31,1:50),BX(0:31,1:50),BY(0:31,1:50),
BZ(O:3l,1:50),DBXDX(0:31,1:50),DBYDX(O:31,1:50),
DBZDX(0:31,1:50),DBXDZ(0:31,1:50),DBYDZ(0:31,1:50),
DBZDZ(0:31,1:50),CURLB2(1:50),FX(0:31,1:50,8),
FZ(0:31,1:50,8)

\DCDxJONUTiwaf-J

BEGIN CALCDR
GET THE VELOCITIES, B FIELDS, AND THEIR DERIVATIVES
CALL CALCVB(R,U,V,W,BX,BY,BZ)
OCALL DIFFVB(U,DUDX,DUDZ,V,DVDX,DVDZ,W,DWDX,DWDZ,
1 BX,DBXDX,DBXDZ,BY,DBYDX,DBYDZ,BZ,DBZDX,DBZDZ)

 

THE MAIN LOOP OVER J FOR CALCULATING FX,FZ,DR
DO 5 J=XLO,XHI
CALCULATE THE PRESSURES,ETA, AND THE CURL OF B SQUARED
DO 10 L=ZLO,ZHI
P(L)=GM1*R(J,L,5)/DZDL(J,L)
PM(L)=(BX(J,L)*BX(J,L)+BY(J,L)*BY(J,L)+BZ(J,L)*BZ(J,L))*OVER8PI
0 CURLB2(L)=DBYDZ(J,L)*DBYDZ(J,L)+DBYDX(J,L)*DBYDX(J,L)
1 +(DBXDZ(J,L)-DBZDX(J,L))*(DBXDZ(J,L)-DBZDX(J,L))
ETA(L)=ETAO
10 CONTINUE

CALCULATE THE VISCOUS STRESS TENSOR TIJ
OCALL CALCTIJ(R,J,TXX,TXY,TXZ,TYY,TYZ,TZZ,
1 U,DUDX,DUDZ,V,DVDX,DVDZ,W,DWDX,DWDZ)

 

BLOCK:CALCFX
CALCULATE FX AND ADVECZ

 

DO 20 L=ZLO,ZHI
FX(J,L,1)=R(J,L,2)
FX(J,L,2)=R(J,L,2)*U(J,L)+(P(L)+PM(L)-TXX(L))*DZDL(J,L)
FX(J,L,3)=R(J,L,3)*U(J,L)-TXY(L)*DZDL(J,L)
FX(J,L,4)=R(J,L,4)*U(J,L)-TXZ(L)*DZDL(J,L)
FX(J,L,5)=R(J,L,5)*U(J,L)
FX(J,L,6)=0.
FX(J,L,7)=(BY(J,L)*U(J,L)-BX(J,L)*V(J,L)-ETA(L)*DBYDX(J,L))
*DZDL(J,L)
FX(J,L,8)=(BZ(J,L)*U(J,L)-BX(J,L)*W(J,L)+
ETA(L)*(DBXDZ(J,L)-DBZDX(J,L)))*DZDL(J,L)
CALCULATE THE ADVECTION TERM
ADVECZ(L)=(W(J,L)-DZDT(J,L))/DZDL(J,L)
20 CONTINUE

HOl—‘O

 

149

 

* BLOCK: CALCFZ
* CALCULATE FZ, DR

*

 

DO 30 L=ZLO,ZHI
FZ(J,L,1)=ADVECZ(L)*R(J,L,1)-FX(J,L,1)*DZDX(J,L)
FZ(J,L,2)=ADVECZ(L)*R(J,L,2)-TXZ(L)-FX(J,L,2)*DZDX(J,L)
FZ(J,L,3)=ADVECZ(L)*R(J,L,3)-TYZ(L)-FX(J,L,3)*DZDX(J,L)
FZ (J, L, 4)=ADVECZ (L) *R(J, L, 4) +P (L) +PM (L) -TZZ (L)

$ -FX(J,L,4)*DZDX(J,L)
FZ(J,L,5)=ADVECZ(L)*R(J,L,5)-FX(J,L,5)*DZDX(J,L)
FZ(J,L,6)=BX(J,L)*W(J,L)-BZ(J,L)*U(J,L)+ETA(L)*
$ (DBZDX(J,L)-DBXDZ(J,L))-BX(J,L)*DZDT(J,L)-FX(J,L,6)*DZDX(J,L)
FZ(J,L,7)=BY(J,L)*W(J,L)-BZ(J,L)*V(J,L)-
$ ETA(L)*DBYDZ(J,L)-BY(J,L)*DZDT(J,L)
FZ(J,L,8)=-BZ(J,L)*DZDT(J,L)-FX(J,L,8)*DZDX(J,L)
* NOW DR
DR(J,L,1)=QH*R(J,L,4)
DR(J,L,2)=QH*R(J,L,4)*U(J,L)
+2.*(R(J,L,6)*DBXDX(J,L)+R(J,L,8)*DBXDZ(J,L))*OVER8PI
DR(J,L,3)=QH*R(J,L,4)*V(J,L)
+2.*(R(J,L,6)*DBYDX(J,L)+R(J,L,8)*DBYDZ(J,L))*OVER8PI
DR(J,L,4)=QH*R(J,L,4)*W(J,L)
+2.*(R(J,L,6)*DBZDX(J,L)+R(J,L,8)*DBZDZ(J,L))*OVER8PI
DR(J,L,5)=(-P(L)*(DUDX(J,L)+DWDZ(J,L))+
TXX(L)*DUDX(J,L)+TZZ(L)*DWDZ(J,L)+
TXY(L)*(DVDX(J,L))+TYZ(L)*(DVDZ(J,L))
+TXZ(L)*(DUDZ(J,L)))*DZDL(J,L)+QH*R(J,L,4)*R(J,L,5)/R(J,L,l)
+2.*ETA(L)*OVER8PI*CURLBZ(L)*DZDL(J,L)
DR(J,L,6)=0.
DR(J,L,7)=0.
DR(J,L,8)=O.
30 CONTINUE
* END OF LOOP
5 CONTINUE

mmmm U) {D '0')

*

 

*BLOCK: COMBINEFZ

* NOW COMBINE DR WITH THE DERIVATIVE OF F2 AND STORE IN DR
* USE V AND DVDZ AS A TEMP STORAGE FOR F2 AND DFZDZ SINCE
* FZ DOES NOT HAVE THE RIGHT SHAPE

*

 

DO 60 M=1,8
DO 40 J=XLO,XHI
DO 40 L=ZLO,ZHI
V(J,L)=FZ(J,L,M)
40 CONTINUE
CALL DDL(V,DVDZ)
DO 50 J=XLO,XHI
DO 50 L=ZLO,ZHI
DR(J,L,M)=DR(J,L,M)-DVDZ(J,L)
50 CONTINUE
60 CONTINUE

 

BLOCK:COMBINEFX
NOW COMBINE DR WITH X DERIVATIVE OF FX AND STORE IN DR
USE V AS TEMP STORAGE FOR FX AND DFXDX

Sl-Irfl-Irfl-

 

DO 90 M=1,8
DO 70 J:XLO,XHI
DO 70 L=ZLO,ZHI
V(J,L)=FX(J,L,M)

150

 

‘1!

 

70 CONTINUE
CALL DDX(FXSYM(M),V,DVDZ)
DO 80 J=XLO,XHI
DO 80 L=ZLO,ZHI
DR(J,L,M)=DR(J,L,M)-DVDZ(J,L)
80 CONTINUE
90 CONTINUE

 

RETURN
END

*********************************************************************

SUBROUTINE CALCVB(R,U,V,W,BX,BY,BZ)
IMPLICIT NONE
* PASSED VARIABLES
OREAL R(0:31,1:50,8),U(0:31,1:50),V(O:31,1:50),
1 W(0:31,1:50),BX(0:31,1:50),BY(0:31,1:50),BZ(0:3l,1:50)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(O:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1 /GRID/DZDX,DZDL,DZDT,DX
* LOCAL VARIABLES
INTEGER J,L

* BEGIN CALCVB
DO 10 J=XLO,XHI
DO 10 L=ZLO,ZHI
U(JIL)=R(JILI2)/R(JILI1)
V(JIL)=R(JILI 3) /R(JILI1)
W(J,L)=R(J,L, 4)/R(J,L, 1)
BX(J,L)=R(J,L, 6) /DZDL(J,L)
BY(J,L)=R(J,L,7)/DZDL(J,L)
BZ (J,L)=R(J,L, 8) /DZDL(J,L)
10 CONTINUE
RETURN
END

*********************************************************************

OSUBROUTINE DIFFVB(U,DUDX,DUDZ,V,DVDX,DVDZ,W,DWDX,DWDZ,
1 BX,DBXDX,DBXDZ,BY,DBYDX,DBYDZ,BZ,DBZDX,DBZDZ)
IMPLICIT NONE
* PASSED VARIABLES
REAL U(0:31,1:50),DUDX(0:31,1:50),DUDZ(0:31,1:50),
V(O:31,1:50),DVDX(0:31,1:50),DVDZ(0:31,1:50),
W(0:31,1:50),DWDX(0:31,1:50),DWDZ(0:31,1:50),
BX(0:31,1:50),DBXDX(0:31,1:50),DBXDZ(0:31,1:50),
BY(0:31,1:50),DBYDX(0:31,1:50),DBYDZ(0:31,1:50),
BZ(0:31,1:50),DBZDX(0:31,1:50),DBZDZ(0:31,1:50)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1 /GRID/DZDX,DZDL,DZDT,DX
* LOCAL VARIABLES
INTEGER J,L

U1bCQkJH

151

* BEGIN DIFFVB
* CALCULATE X DERIVATIVES USING SPECTRAL METHODS
CALL DDX(-1,U,DUDX)
CALL DDX(1,V,DVDX)
CALL DDX(1,W,DWDX)
CALL DDX(1,BX,DBXDX)
CALL DDX(-1,BY,DBYDX)
CALL DDX(-1,BZ,DBZDX)

* CALCULATE DUDZ, DVDZ, DWDZ USING FINITE DIFFERENCE
CALL DDL(U,DUDZ)
CALL DDL(V,DVDZ)
CALL DDL(W,DWDZ)
CALL DDL(BX,DBXDZ)
CALL DDL(BY,DBYDZ)
CALL DDL(BZ,DBZDZ)
* CORRECT FOR MOVING GRID
DO 10 J=XLO,XHI
DO 10 L=ZLO,ZHI
DUDX(J,L)=DUDX(J,L)-DZDX(J,L)*DUDZ(J,L)
DVDX(J,L)=DVDX(J,L)-DZDX(J,L)*DVDZ(J,L)
DWDX(J,L)=DWDX(J,L)-DZDX(J,L)*DWDZ(J,L)
DBXDX(J,L)=DBXDX(J,L)-DZDX(J,L)*DBXDZ(J,L)
DBYDX(J,L)=DBYDX(J,L)-DZDX(J,L)*DBYDZ(J,L)
DBZDX(J,L)=DBZDX(J,L)-DZDX(J,L)*DBZDZ(J,L)

DUDZ(J,L)=DUDZ(J,L)/DZDL(J,L)
DVDZ(J,L)=DVDZ(J,L)/DZDL(J,L)
DWDZ(J,L)=DWDZ(J,L)/DZDL(J,L)
DBXDZ(J,L)=DBXDZ(J,L)/DZDL(J,L)
DBYDZ(J,L)=DBYDZ(J,L)/DZDL(J,L)
DBZDZ(J,L)=DBZDZ(J,L)/DZDL(J,L)
10 CONTINUE
RETURN
END

****3):****************************************************************

OSUBROUTINE CALCTIJ(R,J,TXX,TXY,TXZ,TYY,TYZ,TZZ,
1 U, DUDX, DUDZ, V, DVDX, DVDZ, W, DWDX, DWDZ)
IMPLICIT NONE
* PASSED VARIABLES

INTEGER J
OREAL R(0:31,1:50,8),TXX(1:50),TXY(1:50),TXZ(1:50),
1 TYY(1:50),TYZ(1:50),TZZ(1:50),
2 U(0:31,1:50),DUDX(0:31,1:50),DUDZ(0:31,1:50),
3 V(0:31,l:50),DVDX(0:31,1:50),DVDZ(0:31,1:50),
4 W(0:31,1:50),DWDX(0:31,1:50),DWDZ(O:31,1:50)

* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF

OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /GRID/DZDX,DZDL,DZDT,DX

2 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF

* LOCAL VARIABLES
REAL DVKDXK(1:50)

* TO SAVE DIVISIONS, DVKDXK(L)=1/3 OF ACTUAL VALUE THIS CHANGES ZETAZ

* to 1/3 OF THE VALUE IN THE PROGRAM
INTEGER L

152

* BEGIN CALCTIJ
DO 10 L=ZLO,ZHI
DVKDXK(L)=(DUDX(J,L)+DWDZ(J,L))/3.
10CONTINUE
* CALCULATE Txx, TXY, sz, TYY, TYZ, Tzz
DO 20 L=ZLO,ZHI
TXX(L)=ZETA1*(DUDX(J,L)-2.*DVKDXK(L))+ZETA2*DVKDXK(L)
TXY(L)=ZETA1*DVDX(J,L)
TXZ(L)=ZETA1*(DUDZ(J,L)+DWDX(J,L))
TYY(L)=DVKDXK(L)*(ZETA2-2.*ZETA1)
TYZ(L)=ZETA1*DVDZ(J,L)
TZZ(L)=ZETA1*(DWDZ(J,L)-2.*DVKDXK(L))+ZETA2*DVKDXK(L)
20CONTINUE
RETURN
END

*********************************************************************

* THIS SUBROUTINE FINDS THE Z DERIVATIVE OF THE ELEMENTS OF THE
* PASSED ARRAY. THE ARRAY MUST BE SHAPED (0:NR,1:NL)

*********************************************************************

SUBROUTINE DDL(F,DFDL)
IMPLICIT NONE
* PASSED VARIABLES
REAL F(0:3l,l:50),DFDL(0:31,1:50)
INTEGER SYM
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
COMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
LOCAL VARIABLES
INTEGER J,L,M
* BEGIN DDL USING FOURTH ORDER EVERYWHERE
DO 20 J:XLO,XHI
DO 10 L=3,LM2
C PRINT *,J,L,DFDL(J,L)
DFDL(J,L)=F(J,L-2)-8.*F(J,L-1)+8.*F(J,L+l)-F(J,L+2)
10 CONTINUE
O DFDL(J,1)=-25.*F(J,1)+48.*F(J,2)-36.*F(J,3)+
116.*F(J,4)-3.*F(J,5)
DFDL(J,2)=-3.*F(J,1)-10.*F(J,2)+18.*F(J,3)-6.*F(J,4)+F(J,5)
0 DFDL(J,LM)=-F(J,LM4)+6.*F(J,LM3)-18.*F(J,LM2)+10.*F(J,LM)
1 +3.*F(J,NL)
0 DFDL(J,NL)=3.*F(J,LM4)-16.*F(J,LM3)+36.*F(J,LM2)-
148.*F(J,LM)+25.*F(J,NL)
20 CONTINUE
120 CONTINUE

)0-

*********************************************************************

* THIS ROUTINE USES FINTIE DIFFERENCE TO CALCULATE THE X DERIVATIVES

* IT IS ONLY USED WHEN PERIODIC BOUNDARY CONDITIONS ARE INAPPROPRIATE
*********************************************************************

SUBROUTINE DDJ(F,DFDL)
IMPLICIT NONE
* PASSED VARIABLES

153

 

 

 

REAL F(0:3l,1:50),DFDL(0:31,1:50)
INTEGER SYM
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
COMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
$ /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
* LOCAL VARIABLES
INTEGER J, L,M, JM, JM2, JM3, JM4
REAL DXDJ
* BEGIN DDL USING FOURTH ORDER EVERYWHERE
JM:NR-l
JM2=NR-2
JM3=NR-3
JM4=NR—4
DXDJ=1.0/(12.0*LX)
DO 20 L=ZLO,ZHI
DO 10 J=2,JM2
DFDL(J,L)=F(J-2,L)-8.*F(J41,L)+8.*F(J+l,L)-F(J+2,L)
10 CONTINUE
O DFDL(0,L)=—25.*F(0,L)+48.*F(1,L)-36.*F(2,L)+
116.*F(3,L)-3.*F(4,L)
DFDL(1,L)=-3.*F(0,L)-10.*F(1,L)+18.*F(2,L)-6.*F(3,L)+F(4,L)
O DFDL(JM,L)=-F(JM4,L)+6.*F(JM3,L)-18.*F(JM2,L)+10.*F(JM,L)
1 +3.*F(NR,L)
O DFDL(NR,L)=3.*F(JM4,L)-16.*F(JM3,L)+36.*F(JM2,L)—
148.*F(JM,L)+25.*F(NR,L)
20 CONTINUE
120 CONTINUE
DO 200 J:0,NR
DO 200 L=ZLO,ZHI
DFDL(J,L)=DFDL(J,L)*DXDJ
200 CONTINUE
END

 

**********************************************************************

* THIS ROUTINE CALCULATES x DERIVATIVES BY SPECTRAL METHODS
**********************************************************************
SUBROUTINE DDX(SYM,A,DADX)
IMPLICIT NONE
* GLOBAL VARIABLES
COMPLEX DX(0:31)
REAL W(0:31,1:50,8),TRIGS(451)
INTEGER IFAX(13),INC,JUMP,NPTS,NVEC
INTEGER SYMFLAG
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1 /FFT/W,SYMFLAG
2 /GRID/DZDX,DZDL,DZDT,DX
* PASSED VARIABLES
COMPLEX DADX(0:NC,NL),A(0:NC,NL)
INTEGER SYM
* LOCAL VARIABLES
INTEGER J,L,M,L1,NX2,NR2
COMPLEX B(0:255),C(0:255),D(0:31,1:50)

GOTO (1,2,3,4, 1)SYMFLAG
* ONE D
3 DO 10 J=0,NC

154

DO 10 L=1,NL
10 DADX(J,L)=0.0
GOTO 999
* NO SYMMETRY
1 Do 110 L=ZLO,ZHI,2
L1=L+1
DO 120 J=0,NC
B(J)=A(J,L)
C(J)=A(J,Ll)
120 CONTINUE
CALL FFT2(B,C,W,NX,1)
DO 130 J=0,NC
B(J)=B(J)*DX(J)
C(J)=C(J)*DX(J)
13o CONTINUE
CALL FFT2(B,C,W,NX,-1)
DO 140 J=0,NC
DADX(J,L)=B(J)
DADX(J,L1)=C(J)
140 CONTINUE
110 CONTINUE
- IF(SYMFLAG.EQ.5)THEN
CALL DDJ(A,D)
DO 150 L=ZLO,ZHI
DO 160 J=O,1
DADX(J,L)=D(J,L)
DADX (NR-J, L) =D (NR-J, L)

160 CONTINUE
150 CONTINUE
ENDIF
GOTO 999

*EVEN SYMMETRY IN X
2 NX2=2*NX
NR2=NX2-1
DO 210 L=ZLO,ZHI,2
L1=L+1
DO 220 J=0,NC
B(J)=A(J,L)
C(J)=A(J,Ll)
220 CONTINUE
CALL EXTEND(B,NR,NR2,SYM)
CALL EXTEND(C,NR,NR2,SYM)
CALL FFT2(B,C,W,NX2,1)
DO 230 J=XLO,XHI
B(J)=B(J)*DX(J)
C(J)=C(J)*DX(J)
230 CONTINUE
CALL FFT2(B,C,W,NX2,-1)
DO 240 J=0,NC
DADX(J,L)=B(J)
DADX(J,L1)=C(J)

240 CONTINUE
210 CONTINUE
GOTO 999

* USE FINITE DIFFERENCE IN X DIRECTION
4 CALL DDJ(A,DADX)

999 RETURN

END

155

10

SUBROUTINE EXTEND(B,N1,N2,SYM)
IMPLICIT NONE
INTEGER N1,N2,SYM,J
REAL B(O:N2)
DO 10 J:O,Nl
B(N2-J)=B(J)*SYM
CONTINUE
RETURN
END

156

 

*********************************************************************

* THIS SUBROUTINE OUTPUTS THE RESULTS
* PFLAG=1 CAUSES 1D IN Z TO BE OUTPUT
* 2 CAUSES 2D TABLE

*3 CAUSES 2D TABLE AND CONTOUR PLOT
*4 CAUSES CONTOUR PLOT ONLY

*5 CAUSE 1D IN X

*****t***************************************************************

 

SUBROUTINE OUTPUT(R,T,Z)

IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8),T,Z(0:31,1:50)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
REAL TMAX,TPRINT,PFLAG,TVIEW
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

 

1 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
2 /GRID/DZDX,DZDL,DZDT,DX
3 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW

* LOCAL VARIABLES
0 REAL D(O:31,l:50),VX(0:31,1:50),VY(0:31,1:50),VZ(0:31,1:50),
1 E(0:31,1:50),P(0:31,1:50),BX(O:31,1:50),BY(O:31,1:50),
2 BZ(0:31,1:50),V(O:31,1:50),A(0:3l,0:31),AZ(0:31,0:31)
INTEGER J,L,K,M,NX2,FLAG3D,Q
CHARACTER*4O TITLE

FLAG3D=2
IF(PFLAG.GE.2.)THEN
DO 10 J=0,NR
DO 10 L=1,NL
D(J,L)=R(J,L,1)/DZDL(J,L)
VX(J,L)=R(J,L,2)/R(J,L,l)
VY(JIL)=R(JILI3)/R(JILI1)
VZ(JIL)=R(JILI4)/R(JILI1)
V(J,L)=SQRT(VX(J,L)*VX(J,L)+VZ(J,L)*VZ(J,L))
B(J,L)=R(J,L, 5)/(0.9D0*R(J,L, 1))
C P(J,L)=R(J,L,5)/DZDL(J,L)
BX(J,L)=R(J,L,6)/DZDL(J,L)
BY(J,L)=R(J,L,7)/DZDL(J,L)
BZ(J,L)=R(J,L,8)/DZDL(J,L)
10 CONTINUE
ENDIF

IF(PFLAG.GE.2.AND.PFLAG.LE.3)THEN
DO 20 L=1,NL

 

WRITE(6,2000)L
2000 FORMAT(' > L=',I4)
WRITE(6,3000)'J','Z','DENSITY','VX','VY','VZ','TEMP',
$ 'BY','BZ'

3000 FORMAT(1X,A2,1X,9(A8,1X))
DO 30 J:XLO,XHI
WRITE(6,4000)J,Z (J,L) ,D(J,L) ,VX(J,L) ,VY(J,L) ,

$ VZlJ,L),E(J,L).BY(J,L),BZ(J.L)
4000 FORMAT(1X,I3,1X,F8.4,1X,9(F8.4,1X))
30 CONTINUE
20 CONTINUE
ENDIF

157

 

00000000

1

2

4O

41

42

43

999

IF ((PFLAG.GE.3..AND.PFLAG.LT.5.).OR.(PFLAG.EQ.9))THEN
GOTO (1,2)FLAG3D

TITLE='DENSITY'

CALL MAKCON(D,Z,NX,NL,10,TITLE)
PRINT *,‘DENSITY AT T= ',T
PRINT*,' '

TITLE='VX'

CALL MAKCON(VX,Z,NX,NL,10,TITLE)
PRINT *,‘PLOT OF VX AT T= ',T
PRINT*,' '

TITLE='VZ'

CALL MAKCON(VZ,Z,NX,NL,10,TITLE)

PRINT *,‘VZ AT T= ',T

PRINT*,' '

PRINT *,‘3D PLOTS AT T= ',T

TITLE='PLOT OF ENERGY'

CALL MAKCON(E,Z,NX,NL,10,TITLE)
PRINT *,‘ENERGY DENSITY AT T= ',T
PRINT*,' '
TITLE='SPEED'
CALL MAKCON(V,Z,NX,NL,10,TITLE)
PRINT *,‘SPEED AT T= ',T
PRINT*,' '
GOTO 999
CONTINUE
TITLE='DENSITY'
CALL THREED(D,NX-2,NL,TITLE)
TITLE='TEMPERATURE'
CALL THREED(E,NX-2,NL,TITLE)
TITLE='SPEED'
CALL THREED(V,NX-2,NL,TITLE)
GOTO 999
CONTINUE
Q=0.4*NL—NC
DO 40 J: 0,NR
DO 40 K= 0,NR
AZ(J,K)=Z(J,K+Q)
DO 41 J= 0,NR
DO 41 K= 0,NR
A(J,K)=D(J,K+Q)
TITLE='DENSITY'
CALL MAKCON(A,AZ,NX,NX,10,TITLE)
PRINT *,‘PLOT OF DENSITY AT T= ',T
PRINT*,' '
DO 42 J"= 0,NR
DO 42 K= 0,NR
A(J.K) =E (J,K+Q)
TITLE='PLOT OF ENERGY'
CALL MAKCON(A,AZ,NX,NX,10,TITLE)
PRINT *,‘TEMPERATURE AT T= ',T
PRINT*,' '
DO 43 J= 0,NR
DO 43 K: 0,NR
A(JIK)=V(J,K+Q)
TITLE='SPEED'
CALL MAKCON(A,AZ,NX,NX,10,TITLE)
PRINT *,‘SPEED AT T= ',T
PRINT*,' '
CONTINUE
ENDIF

IF((PFLAG.EQ.1).OR.(PFLAG.EQ.3.5))CALL OUTZlD(R,Z,T)
158

 

 

 

IF((PFLAG.EQ.5).OR.(PFLAG.EQ.9))CALL OUTX1D(R,Z,T)

IF(PFLAG.EQ.25.)THEN
DO 720 L=10, 30
WRITE(6,2000)L
WRITE(6,3000)'J','Z','DENSITY','VX','VY','VZ','TEMP',
s 'BY"'BZ'
DO 730 J:XLO,XHI
'WRITE(6,4000)J,Z(J,L),D(J,L),VX(J,L),VY(J,L),
$ VZ(J,L),E(J.L),BY(J,L),BZ(J,L)
730 CONTINUE
720 CONTINUE
ENDIF
RETURN
END

159

 

***************************************************************

* THIS IS THE BOUNDARY CONDITION ROUTINE
***************************************************************
SUBROUTINE BC(R,Z,T)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8),Z(O:31,1:50),T
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
REAL BXO,BYO,BZO,E0,R0,VXO,VYO,VZO
REAL ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO
REAL WORK(0:31,1:50,8)
INTEGER SYMFLAG
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /GRID/DZDX,DZDL,DZDT,DX

2 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
3 /ZEROES/BXO,BYO,BZO,E0,RO,VXO,VYO,VZO

4 /MAGNET/ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO

5 /FFT/WORK,SYMFLAG

* LOCAL VARIABLES
INTEGER J,L,M
REAL VPSTART,W,R1,E1,D,SIGN,A,U,MM,W0,UO
REAL XFACE,ZFACE,SN,CSN
CHARACTER*8 BCFLAG

******************************************************************

* BCFLAG DETERMINES WHICH BC ARE USED
******************************************************************
BCFLAG='PISTONZ'
IF(BCFLAG.EQ.'XPISLAB')THEN
DO 5 J:XLO,XHI
SN=DZDL(J,1)/DZDL(J,2)
CSN=DZDL(J,NL)/DZDL(J,LM)
DO 6 M:1,8
R(J,1,M)=R(J,2,M}*SN
R(J,NL,M)=R(J,LM,M)*CSN

6 CONTINUE
5 CONTINUE
GOTO 999
ENDIF

* BORING BC IN Z
DO 20 J:XLO,XHI
:R(J,1,l)=R(J,2,1)*DZDL(J,1)/DZDL(J,2)
R(J,NL,1)=R(J,LM, 1) *DZDL (J,NL) /DZDL (J,LM)
R(J11:2)=R(J:2:2)*R(J,1:1)/R{J:2:1)
R(JINL12)=R(JIIMI2) *R(JINLl1) /R(JII-Ml1)
R(Jlll4)=-R(JI214) *R(JI1I1) /R(JI2I 1)
R(J,N'L,4)='R(J,IM,4)*R(J,NL,1)/R(J,LM,1)
R(J, 1, 5)=R(J, 2, 5) *R(J,1,1) /R(J,2,1)
R(JINLI 5)=R(Jl LMI 5) *R(JINLI1) /R(JIIMI1)
20 CONTINUE

IF(T.LT.6.2831853/OM)THEN
FOR MAGNETOACOUSTIC WAVES
DO 40 L=1,NL
R(1,L,1)=RO*(1.+W/A)*DZDL(1,L)
R(l, L, 4) =-ALFZ*ALFX/ (A*A-ALFZ*ALFZ) *W*R (1, L, 1)
R(1,L,3)=0.

0000 $0

160

 

R(1,L,2)=W*R(1,L,1)
R(1,L, 5)=EO* (1.0+GMl*W/A) *R(1,L, 1)
R(1,L,8)=BZO*(1.O+A*W/(ATAFALFZ*ALFZ))*DZDL(1,L)
R(1,L,7)=0.
R(1,L,6)=BXO*DZDL(1,L)
4O CONTINUE
ELSE
XLO=0
XHI=NR
ENDIF
IF(BCFLAG.EQ.'PISTONZ')THEN
BC FOR Z PISTON
IF(SYMFLAG.GE.4)THEN
DO 14 L=1,NL
DO 12 M=1,8
R(0,L,M)=R(1,L,M)*DZDL(0,L)/DZDL(1,L)
2*R(l,L,M)-R(2,L,M)
R(NR,L,M)=R(NRr1,L,M)*DZDL(NR,L)/DZDL(NRr1,L)
2*R(NRF1,L,M)-R(NR-2,L,M)

12 CONTINUE
R(0,L,2)=0.5*(R(0,L,2)-R(NR,L,2))
R(NR,L,2)=-R(0,L,2)

14 CONTINUE

ENDIF
DO 10 J=XLO,XHI
R(J, 1, l)=R(J,2, 1) *DZDL (J, 1) /DZDL(J,2)
R(J,l,2)=R(J,2,2)*DZDL(J,1)/DZDL(J,2)
C R(J,1,2)=-0.05*R(J,1,1)
R(J, 1, 4)=VP*R(J, 1, 1)
C R(J,1,4)=2*R(J,1,l)*VP-R(J,2,4)*DZDL(J,1)/DZDL(J,2)
R(J,1,5)=R(J,2,5)*DZDL(J,1)/DZDL(J,2)
R(J, 1, 6)=R(J,2, 6) *DZDL(J, 1) /DZDL(J,2)
R(J,1,8)=R(J,2,8)*DZDL(J,1)/DZDL(J,2)
R(J,NL,4)=0.0
10 CONTINUE
GOTO 999
ENDIF

:1- 0000000000

0

0

 

IF(BCFLAG.EQ.'ZPISLAB')THEN
* BC FOR Z PISTON IN SLAB
DO 100 J=XLO,XHI
R(J,1,1)=R(J,2,1)*DZDL(J,1)/DZDL(J,2)
R(J,1,2)=R(J,2,2)*DZDL(J,1)/DZDL(J,2)
R(J,1,4)=0.0
C R(J,1,4)=2*R(J,1,1)*VP-R(J,2,4)*DZDL(J,1)/DZDL(J,2)
R(J,1,5)=R(J,2,5)*DZDL(J,1)/DZDL(J,2)
R(J,1,6)=R(J,2,6)*DZDL(J,1)/DZDL(J,2)
R(J,1,8)=R(J,2,8)*DZDL(J,1)/DZDL(J,2)
R(J,NL,4)=0.0
100 CONTINUE
DO 110 J=XLO,XHI
C SPLIT+1,NX-SPLIT-1
R(J,1,4)=VP*R(J,1,1)
110 CONTINUE
GOTO 999
ENDIF
FOR ALFVEN WAVE
DO 20 J=XLO,XHI
R(J, 1,1)=R(J, 2, 1) *DZDL(J, 1) /DZDL(J,2)
R(J,1,2)=R(J,1,1)*W
R(J,1,4)=0.
R(J,1,5)=R(J,2,5)*DZDL(J,l)/DZDL(J,2)

161

00000 11-

 

C R(J,1,6)=(BXO-BZO*W/ALF)*DZDL(J,1)

C R(J,NL,4)=0.

C 20 CONTINUE* SURFACE WAVES IN THE X DIRECTION

IF(BCFLAG.EQ.'SURFACEX')THEN
IF (2 . 0*T*OM.GT. 6 .2831853) THEN
SN=0.0
CSN=0.0
XLO=1
XHI=NC
ELSE
SN=SIN(OM*T)
CSN=COS(OM*T)
XLO=0
XHI=NR
ENDIF
J=l

C ZFACE=0.5*(Z(J,SPLIT)+Z(J,SPLIT+1))
ZFACE=Z(J,SPLIT)

* THE FIELD-FREE SIDE OF THE INTERFACE
W0=ME*CSURF*VP/((1.0-CSURF**2)*OM)*SN
U0=VP*CSN
DO 200 L=1,SPLIT-1

W=W0*EXP(ME*(Z(J,L)-ZFACE))
=U0*EXP(ME*(Z(J,L)-ZFACE))

R(J,L,1)=RMO*(1.0+CSURF*W)*DZDL(J,L)
R(J,L,4)=R(J,L,1)*U
R(J,L,2)=R(J,L,1)*W
R(J,L,5)=(RMO*EO+RMO*CSURF*W/GM1)*DZDL(J,L)
R(J,L,6)=0.0

C (-MM*IW/OM)*BXO*DZDL(J,L)
R(J,L, 8)=0.0

C (BZO+BXO*W/CSURF)*DZDL(J,L)

200 CONTINUE
* ON THE INTERFACE
R(J,SPLIT,2)=0.0
R(J,SPLIT,4)=U0*R(J,SPLIT,1)
R(J,SPLIT,8)=(0.5*BXO*U0/CSURF)*DZDL(J,SPLIT)
* MAGNETIC SIDE OF INTERFACE
C W0=-CSURF*MO*VP/((1-CSURF**2)*OM)*SN
W0=-MO*CSURF*VP/((1.0-CSURF**2)*OM)*SN
DO 210 L=SPLIT+1,NL
=W0*EXP(-MO*(Z(J,L)-ZFACE))
U=U0*EXP(-MO*(Z(J,L)-ZFACE))
R(J,L,1)=RO*(1.0+CSURF*W)*DZDL(J,L)
R(J,L,4)=R(J,L,1)*U
R(J,L,2)=R(J,L,1)*W
R(J,L,5)=(R0*E0+R0*CSURF*W/GM1)*DZDL(J,L)
R(J,L,6)=BXO*(l.0-(1-CSURF**2)*W/CSURF)*DZDL(J,L)
R(J,L,8)=(-BXO*U/CSURF)*DZDL(J,L)
210 CONTINUE
GOTO 999
ENDIF
* SURFACE WAVES IN THE +Z DIRECTION
IF(BCFLAG.EQ.'SURFACEZ')THEN

IF(T*OM.GT.6.2831853)THEN
SN=0.0
CSN=1.0

ELSE
SN=SIN(OM*T)
CSN=COS(OM*T)

ENDIF

L=1

162

 

XFACE=LX*SPLIT+0.5*LX

* THE FIELD-FREE SIDE OF THE INTERFACE

W0=ME*CSURF*VP/((1.0-CSURF**2)*OM)*SN
U0=VP*CSN
DO 300 J:0,SPLIT

W=W0*EXP(ME*(LX*J‘XFACE))
=UO*EXP(ME*(LX*J‘XFACE))

R(J,L,1)=RMO*(1.0-CSURF*W)*DZDL(J,L)

R(J,L,2)=R(J,L,1)*U

R(J,L,4)=R(J,L,1)*W

R(J,L,5)=(RMO*E0-RMO*CSURF*W/GM1)*DZDL(J,L)

R(J,L,6)=0.0

C (-MM*IW/OM)*BXO*DZDL(J,L)

R(J,L,8)=0.0

C (BZO+BXO*W/CSURF)*DZDL(J,L)

300

CONTINUE

* MAGNETIC SIDE OF INTERFACE

310

W0=-MO*CSURF*VP/((1.0-CSURF**2)*OM)*SN
DO 310 J=SPLIT+1,NR

W=W0*EXP(-MO*(LX*J‘XFACE))
U=U0*EXP(-MO*(LX*J-XFACE))
R(J,L,1)=R0*(1.0-CSURF*W)*DZDL(J,L)
R(J,L,2)=R(J,L,1)*U

R(J,L,4)=R(J,L,1)*W
R(J,L,5)=(R0*E0-R0*CSURF*W/GM1)*DZDL(J,L)
R(J,L,8)=BZO*(1.0-(1-CSURF**2)*W/CSURF)*DZDL(J,L)
R(J,L,6)=(-BZO*U/CSURF)*DZDL(J,L)

CONTINUE

GOTO 999
ENDIF
IF(BCFLAG.EQ.'JPISTON')THEN

* BC FOR Z PISTON

410

D0 410 L=1,NL

R(0,L,1)=R(1,L,1)*DZDL(O,L)/DZDL(1,L)

R(0,L, 4)=R(1,L, 4) *DZDL(O,L) /DZDL(1,L)
R(1,L,2)=-0.05*R(1,L,1)

R(0,L,2)=VP*R(0,L,1)

R(0,L,5)=R(1,L,5)*DZDL(0,L)/DZDL(1,L)

R(O, L, 6)=R(1,L, 6) *DZDL(O,L) /DZDL(1,L)
R(0,L,8)=R(1,L,8)*DZDL(O,L)/DZDL(1,L)
R(NR,L,2)=0.0

CONTINUE

GOTO 999
ENDIF

PERIODIC (IN Z) BC

100

D0 100 M=1,8

DO 100 J=XLO,XHI
R(J, 1,M)=R(J, LM3,M) *DZDL (J, 1) /DZDL(J, LM3)
R(J,2,M)=R(J,LM2,M)*DZDL(J,2)/DZDL(J,LM2)
R(J,LM,M)=R(J,3,M)*DZDL(J,LM)/DZDL(J,3)
R(J,NL,M)=R(J,4,M)*DZDL(J,NL)/DZDL(J,4)
CONTINUE

999 CONTINUE
THE END OF THE CASE STRUCTURE

RETURN

**********************‘k**‘k*********************‘k'k'k'k*************************

* THE ROUTINES TO SMOOTH THE X 1 CELL VARIATIONS

163

 

****************************************************************************

SUBROUTINE SMOOTH(R)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1 /GRID/DZDX,DZDL,DZDT,DX
* LOCAL VARIABLES
REAL A(O:31,1:50)
INTEGER M, J, L
* BEGIN SMOOTH
DO 10 M;1,8
DO 20 J=XLO,XHI
DO 20 L=1,NL
A(J,L)=R(J,L,M)/DZDL(J,L)
20 CONTINUE
CALL SMOOTH1(A)
DO 30 J=XLO,XHI
DO 30 L=1,NL
R(J,L,M)=A(J,L)*DZDL(J,L)
30 CONTINUE
10 CONTINUE
RETURN
END

**********************************************************************

SUBROUTINE SMOOTH1(V)
IMPLICIT NONE
*GLOBAL VARIABLES
REAL WORK(0:31,1:50,8),TRIGS(451)
INTEGER IFAX(13),NPTS,INC,JUMP,NVEC
INTEGER SYMFLAG
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
COMMON /FFT/WORK,SYMFLAG
$ /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
* PASSED VARIABLES
COMPLEX V(0:NC,NL)
* LOCAL VARIABLES
INTEGER.L,M,ISIGN,NVEC3

C CALL FFT99(V,WORK,TRIGS,IFAX,INC,JUMP,NPTS,NVEC3,ISIGN)
DO 50 L=1,NL
50 V(NC,L)=0.0
C CALL FFT99(V,WORK,TRIGS,IFAX,INC,JUMP,NPTS,NVEC3,ISIGN)
RETURN
END

164

 

 

 

set dzmin=0

 

for all cells

 

 

 

 

 

 

dzdt:fv*vz
dzm1n=m1n(dzmin,z ”-1- 2L)
if dzmin<0
then else
print "points crossed” a

exceptional stop

 

use ddl to calculate dzdl
use ddx to calculate dzdx

 

 

*****************************************************************

* THE

GRID ROUTINE CONTROLS THE MOTION OF THE GRID.

* DZDL,DZDX, & DZDT ARE CALCULATED HERE

*****************************************************************

SUBROUTINE GRID(R,Z,T,DT)
IMPLICIT NONE

* PASSED VARIABLES

REAL R(0:31,1:50,8),Z(O:31,1:50),T,DT

* GLOBAL VARIABLES

INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF

REAL TMAX,TPRINT,PFLAG,TVIEW

REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)

COMPLEX DX(0:31)

OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
2 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW
3 /GRID/DZDX,DZDL,DZDT,DX

* LOCAL VARIABLES

INTEGER J,L,M
REAL DZl,DZMIN,A(0:31,1:50)

* BEGIN GRID

20
10

DZMIN=LZ

DO 10 J=XLO,XHI
DZDT(J,1)=R(J,1,4)/R(J,1,1)
DZDT(J,2)=FV*R(J,2,4)/R(J,2,1)
DZDT(J,LM)=FV*R(J,LM,4)/R(J,LM,1)
DZDT(J,NL)=1.*R(J,NL,4)/R(J,NL,1)

DO 20 L=3,LM2
DZDT(J,L)=FV*R(J,L,4)/R(J,L,1)
DZl=Z(J,L+1)-Z(J,L)
DZMIN=AMIN1(DZMIN,D21)
CONTINUE

CONTINUE

IF(DZMIN.LE.0.)THEN

165

 

PRINT *,' POINTS CROSSED AT T=',T
CALL OUTPUT(R,T,Z)
STOP
ENDIF

CALL DDL(Z,A)

DO 30 J=XLO,XHI

DO 30 L=ZLO,ZHI

DZDL(J,L)=A(J,L)

30CONTINUE

CALL DDX(1,Z,A)

DO 40 J2XLO,XHI

DO 40 L=ZLO,ZHI

DZDX(J,L)=A(J,L)/DZDL(J,L)
40CONTINUE

IF(DT.LT.lE-7)THEN
PRINT *,‘DT TOO SMALL AT DT=',DT
CALL OUTPUT(R,T,Z)
STOP
ENDIF
RETURN
END
****-k1k******************************1k*******************************inki-*int-
SUBROUTINE OUTX1D(R,Z,TIME)
COMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
$ /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI
$ /GRID/DZDX,DZDL,DZDT,DX
s /PRINTS/TMAX,TPRINT,PFLAG,TVIEW
$ /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
REAL G,QH,GMl,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
INTEGER I,J,V,VMAX,VAXIS,L,M,NX,NL,NR,NC,LM,LM2,LM3,LM4,
VD,VV,VT,XLO,XHI,ZLO,ZHI,SPLIT
REAL R(0:31,1:50,8),MIND,MAXD,DY,OVER8PI,DZDL(0:31,1:50),
DZDX(0:31,1:50),D(0:31),VX(O:31),T(0:31),MAXVZ,MINVZ,
MAXT,MINT,Z(O:31,1:50),DD,DVZ,DTT,DZDT(0:31,1:50),
TMAX,TPRINT,PFLAG,TVIEW,VY,VZ,BX,BY,BZ
COMPLEX DX(0:31)

mmm {D

PRINT *,' '
PRINT *,'@FTF'
PRINT *,NR,‘ NUMBER OF POINTS FOR PLOTTER'
WRITE(6,1000)TIME
1000 FORMAT(' TIME= ',F11.4,F10.5)

L=NL/2
WRITE(*,ZOOO)'J','X','DENSITY',’VX','VZ','TEMP','BX',
$ 'BZ','VY','BY'
WRITE(*,3100) ' 0 ',0.,1.,0.,0.,1.,0.,0.,0.,0.
3100 FORMAT(1X,A4,1X,9(F7.4,1X))
2000 FORMAT(1X,A4,1X,9(A7,1X))
DO 20 J=XLO,XHI
D(J)=R(J,L,1)/DZDL(J,L)
VX(J)=R(JILI2)/R(JILI1)
VY=R(J,L,3)/R(J,L,1)
VZ=R(J,L,4)/R(J,L,1)
T(J)=R(J,L,5)/R(J,L,1)/O.9DO
BX=R(J,L,6)/DZDL(J,L)
BY=R(J,L,7)/DZDL(J,L)
BZ=R(J,L,8)/DZDL(J,L)
WRITE(*,3000)J,J*LX,D(J),VX(J),VZ,T(J),BX,BZ,VY,BY
3000 FORMAT(lX,I4,1X,F8.4,1X,9(F7.4,1X))

166

 

20 CONTINUE
RETURN
END

SUBROUTINE OUTZlD(R,Z,TIME)

COMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

$ /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI

$ /GRID/DZDX,DZDL,DZDT,DX

$ /PRINTS/TMAX,TPRINT,PFLAG,TVIEW

$ /ZEROES/BXO,BYO,BZO,E0,R0,VXO,VYO,VZO

REAL BXO,BYO,BZO,EO,RO,VXO,VYO,VZO

CHARACTER*1,A(-l:132)

INTEGER I, J,V, VMAX, VAXIS,L,M, NX, NL,NR,NC, LM,LM2, LM3, LM4,

VD,VV,VT

REAL R(0:31,1:50,8),MIND,MAXD,DY,OVER8P1,DZDL(0:31,1:50),
DZDX(0:31,1:50),D(1:50),VZ(1:50),T(1:50),MAXVZ,MINVZ,
MAXT,MINT,Z(0:31,1:50),DD,DVZ,DTT,DZDT(0:31,1:50),VY,BY
TMAX,TPRINT,PFLAG,TVIEW

COMPLEX DX(0:31)

(1)-(DU) {D

PRINT *,' '
PRINT *,'@FTF'
PRINT *,NL,’ NUMBER OF POINTS FOR PLOTTER'
WRITE(6,1000)TIME
1000 FORMAT(' TIME= ',F11.4,F10.5)

J=NC
WRITE(*,ZOOO)'L','Z','DENSITY','VX','VZ','TEMP',
$ IBXI,IBZ"!VYIIIBYI
WRITE(*,3100) '-0',0.,1.,0.,0.,l.,BX0,0.0
3100 FORMAT(1X,A4,2X,9(F7.4,1X),1A25)
2000 FORMAT(1X,A4,2X,9(A7,1X))
DO 20 L=1,NL
D(L)=R(J,L,1)/DZDL(J,L)
VX=R(J,L,2)/R(J,L,1)
VY=R(J,L,3)/R(J,L,1)
VZ(L)=R(J,L,4)/R(J,L,l)
T(L)=R(J,L,5)/R(J,L,1)/0.9D0
BX=R(J,L,6)/DZDL(J,L)
BY=R(J,L,7)/DZDL(J,L)
BZ=R(J,L,8)/DZDL(J,L)
WRITE(*,3000)L,Z(J,L),D(L),VX,VZ(L),T(L),BX,BZ,VY,BY

3000 FORMAT(1X,I4,2X,F8.4,1X,9(F7.4,1X))
20 CONTINUE

RETURN

END

LOGICAL FUNCTION CHECKDT(R,R1,R2,T,DT)

IMPLICIT NONE
* PASSED VARIABLES

REAL R(0:31,1:50,8),Rl(0:31,1:50,8),R2(O:31,1:50,8),T,DT
* GLOBAL VARIABLES

INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

REAL FlD15,F16D15,0VER8PI,PI,FOURPI

REAL TMAX,TPRINT,PFLAG,TVIEW

OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI

2 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW

* LOCAL VARIABLES
167

 

REAL EPD, EPE, EPV, DLIM, ELIM, VLIM, CRIT, DTMAX, DTVIEW, CURRENT
INTEGER J,L,M,MSFLAG
DATA DLIM,ELIM,VLIM,DTMAX/.00001,.00001,.00001,2.0/
EPD=0.
EPE=0.
EPV=0.
MSFLAG=O
DTVIEw=-DT+TVIEW*0.5
IF (DTVIEW.LE.0.0)THEN
DTVIEw=DTMAX
ENDIF
DO 10 J=1,NR-1
DO 10 L=2,LM2
EPD=AMAX1(EPD,ABS((R1(J,L,1)-R2(J,L,1))/R2(J,L,l)))
EPE=AMAX1(EPE,ABS((R1(J,L, 5)-R2 (J,L, 5) ) /R2 (J,L, 5) ))
EPv=AMAX1(EPv,ABS((R1(J,L,4)-R2(J,L,4))/R2(J,L,1)))
10 CONTINUE
CRIT=AMAX1(EPD/DLIM,EPE/ELIM,EPV/VLIM)
IF(CRIT.GE.1.)THEN
CHECKDT=.TRUE.
DT=.9*DT*(CRIT**(-0.25))
PRINT *,‘REPEATING STEP AT T=',T,' DT=',DT
ELSEIF(CRIT.GT.0.)THEN
CHECKDT=.FALSE.
CURRENT=T+2.*DT
DT=AMIN1(.8*DT*(CRIT**(-0.2)),DTMAx,DTVIEW)
IF(MSFLAG.NE.O)PRINT *,‘SETTING DT=',DT,' AT T=',CURRENT
ELSE
CHECKDT=.FALSE.
ENDIF
RETURN
END

'k***‘k****************************‘k‘k*********************************

* THIS SUBROUITNE CALCULATES THE WAVES SPEEDS

*‘k‘k'k*‘k‘k********‘A'*******‘k***'k*‘k‘k************************************'k

SUBROUTINE SOMMER(R,NEWR,DT,Z,NEWZ)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8),NEWR(0:31,1:50,8),NEWZ(0:31,1:50),DT
REAL Z(0:31,1:50)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)
COMPLEX DX(0:31)
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
REAL C(0:31,1:50,8)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1/GRID/DZDX,DZDL,DZDT,DX
2 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
3/SOMMER/C
* LOCAL VARIABLES
INTEGER J,L,M
REAL A(0:31,1:50),DADL(0:31,1:50),Q
* BEGIN SOMMER
DO 10 J:XLO,XHI
NEWR(J,NL,1)=R(J,LM,1)*DZDL(J,NL)/DZDL(J,LM)
10CONTINUE
DO 20 M=6,8
DO 20 J=XLO,XHI
0 C(J,LM,M)=AMIN1(LZ/DT,AMAX1(1.0,-(NEWR(J,LM,M)/DZDL(J,LM)-
1R(J:LM,M)/DZDL(J,LM))*(Z(J,NL)'Z(J,LM2))/

168

 

2(R(J,NL,M)/DZDL(J,NL)-R(J,LM2,M)/DZDL(J,LM2)+lE-20)))
20CONTINUE
DO 30 M=2,5
DO 30 J=XLO,XHI
C(J,LM,M)=AMIN1(LZ/DT,AMAX1(1.0,-(NEWR(J,LM,M)/NEWR(J,LM,1)-
1R(JILMPM)/R(JILM11))/DT*(Z(JINL)-Z(JILM2))/
2(R(J,NL,M)/R(J,NL,1)-R(J,LM2,M)/R(J,LM2,1)+1E-20)))
30 CONTINUE
DO 40 M=6,8
DO 40 J=XLO,XHI
Q=C(J,LM:M)*DT/(Z{J:NL)'Z(J:LM))
NEWR(J,NL,M)=R(J,LM,M)*DZDL(J,NL)/DZDL(J,LM)*Q+(1.-Q)*R(J,NL,M)
40CONTINUE
DO 50 M=2,5
DO 60 J=XLO,XHI
Q=C(J:LMJM)*DT/(Z(J:NL)‘Z{J:LM))
NEWR(J}NLIM)=R(JILM4M)*R(J1NLpl)/R(JpLM11)*Q+(1--Q)*R(JINLrM)
60CONTINUE ‘
50 CONTINUE
RETURN
END
*********************************************************************

*ALL OF THE INITIALIZATION ROUTINES FOLLOW

**************************************************************

* THIS SUBROUTINE INITIALIZES THE PARAMETERS AND VARIABLES
********************************************************************
SUBROUTINE INIT(R,Z,DT)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8),Z(0:31,1:50),DT
* GLOBAL VARIABLES
REAL TMAX,TPRINT,PFLAG,TVIEW
REAL BXO,BYO,BZO,E0,R0,VXO,VYO,VZO
REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
REAL F1D15,F16D15,0VER8PI,PI,FOURPI
REAL ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /ZEROES/BXO,BYO,BZO,E0,R0,VXO,VYO,VZO

2 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
3 /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI

4 /MAGNET/ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO

* LOCAL VARIABLES
REAL T,RATIO,W,E1,R1,PMO,PM,S,QO,RADIUS
INTEGER J,L,MIDJ,MIDL,SPLIT2
CHARACTER*B INITFLAG
INITFLAG='Y TUBE'
WRITE(*,1001)INITFLAG
1001 FORMAT(‘l GMHDI INITIAL CONDITIONS= ',A8)

 

* SET PARAMETERS
CALL INITNUM
CALL INITZER
CALL INITPAR
CALL SETCON
IF((BXO+BYO+BZO.NE.0.0))CALL INITMAG

 

* SET UP INITIAL ATMOSPHERE

 

DO 10 J=0,NR

169

10

R(J,L,1)=R0
R(J,L,2)=VXO
R(J,L,3)=VYO
R(J,L,4)=VZO

R(J,L,5)=EO
R(J,L,6)=BXO
R(J,L,7)=BYO
R(J,L,8)=BZO
CONTINUE

* SPECIAL INIT CONDITIONS

IF (INITFLAG.EQ.'Z TUBE')THEN

* Shock tube initial conditions

20

30

RATIO=0.02

PRINT *,‘SHOCK TUBE WITH INITIAL PRESSURE RATIO OF ',RATIO

SPLIT=NL*.5

J=NC

DO 30 J=XLO,XHI

DO 20 L=1,SPLIT
R(J,L,1)=R0*RATIO
R(J,L,5)=E0*RATIO
CONTINUE

R(J,SPLIT,1)=R0*.5*(RATIO+1)

R(J,SPLIT,5)=E0*.5*(RATIO+1)

CONTINUE

GOTO 999
ENDIF

(INITFLAG.EQ.'X FACE')THEN

* x magnetic interface

140

150

160

split=nl/2
PMO=(BXO*BXO+BYO*BYO+BZO*BZO)*OVERBPI
RMO=R0*(1+PMO/(GM1*EO))
DO 150 J=XLO,XHI
DO 140 L=1,SPLIT

R(J,L,6)=0.0

R(J,L,7)=0.0

CONTINUE
R(J,SPLIT,6)=0.5*(R(J,SPLIT-l,6)+R(J,SPLIT+1,6))
R(J,SPLIT,7)=O.5*(R(J,SPLIT-1,7)+R(J,SPLIT+1,7))
R(J,SPLIT,8)=0.5*(R(J,SPLIT-l,8)+R(J,SPLIT+1,8))

CONTINUE

DO 160 J=XLO,XHI
DO 160 L=1,SPLIT
PM;PMO-(R(J,L,6)**2+R(J,L,8)**2)*OVER8PI
R(J,L,1)=R0+PM/(GM1*E0)
R(J,L,5)=R(J,L,1)*E0
CONTINUE
GOTO 999
ENDIF

IF (INITFLAG.EQ.'Z FACE')THEN

* x magnetic interface

240

split=NC

PM=(BXO*BXO+BYO*BYO+BZO*BZO)*OVER8PI

RMO=RO*(1+PM/(GM1*E0))

DO 250 J=0,SPLIT

DO 240 L=1,NL
R(J,L,6)=0.0
R(J,L,7)=0.0
R(J,L,8)=0.0
R(J,L,5)=E0+R0*PM/GM1
R(J,L,1)=RMO
R(J,L,5)=E0*R(J,L,1)

CONTINUE

170

250 CONTINUE
GOTO 999
ENDIF
IF (INITFLAG.EQ.'X SLAB')THEN
* x magnetic interface
split=NL
PMO=(BXO*BXO+BYO*BYO+BZO*BZO)*OVER8PI
RMO=RO*(1+PMO/(GM1*EO))
DO 350 J=XLO,XHI
DO 340 L=1,SPLIT
R(J,L,6)=0.0
R(J,L,7)=0.0
340 (CONTINUE
SPLIT2=NL-SPLIT+1
DO 345 L=SPLIT2,NL
R(J,L,6)=0.0
R(J,L,7)=0.0
345 CONTINUE
R(J,SPLIT,6)=0.5*(R(J,SPLIT-l,6)+R(J,SPLIT+1,6))
R(J,SPLITZ,6)=0.5*(R(J,SPLITZ-l,6)+R(J,SPLIT2+1,6))
350 CONTINUE
DO 360 J:XLO,XHI
DO 360 L=1,NL
PM:PMO-(R(J,L,6)**2+R(J,L,8)**2)*OVER8PI
R(J,L,1)=R0+PM/(GM1*E0)
R(J,L,5)=R(J,L,1)*E0
360 CONTINUE
* TO PRODUCE SHOCK IN X DIRECTION
DO 370 L=1,NL
DO 370 J=O,NC
R(J,L,2)=-VP*R(J,L,1)
R(NR-J,L,2)=VP*R(NR-J,L,1)
370 CONTINUE
GOTO 999
ENDIF

IF (INITFLAG.EQ.'Z SLAB')THEN
* x magnetic interface
split=NC—4
PMO=(BXO*BXO+BYO*BYO+BZO*BZO)*OVER8PI
RMO=R0*(1+PMO/(GM1*E0))
DO 450 L=1,NL
DO 440 J=O,SPLIT
R(J,L,8)=0.0
440 CONTINUE
SPLIT2=NX-SPLIT
DO 445 J=SPLIT2,NR
R(J,L,8)=0.0
445 CONTINUE
R(SPLIT2,L,8)=0.5*(R(SPLIT2-1,L,8)+R(SPLIT2+1,L,8))
R(SPLIT,L,8)=O.5*(R(SPLIT-1,L,8)+R(SPLIT+1,L,8))
450 CONTINUE
DO 460 J=XLO,XHI
DO 460 L=1,NL
PM=PMO-(R(J,L,6)**2+R(J,L,8)**2)*OVER8PI
R(J,L,1)=R0+PM/(GM1*E0)
R(J,L,5)=R(J,L,1)*E0
460 CONTINUE
GOTO 999
ENDIF
IF(INITFLAG.EQ.'TEST')THEN
* TO PRODUCE Z SHOCK

171

490

495

D0 495 J=XLO,XHI

DO 490 L=1,15
R(J,L,4)=VP*R(J,L,1)

DO 495 L=15,NL
R(J,L,4)=-VP*R(J,L,1)

GOTO 999

ENDIF

IF(INITFLAG.EQ.'Y TUBE')THEN

* A CYLINDRICAL FLUX TUBE EXTENDING IN THE Y DIRECTION

1000

S=0.50

RADIUS=4.0

MIDJ=NC

MIDL=NL*.4
PMO=(BXO*BXO+BYO*BYO+BZO*BZO)*OVER8PI
WRITE(*,1000)MIDJ,MIDL,RADIUS,S

FORMAT(' CENTER: ',I3,',',I3,' RADIUS= ',F6.3,

$ ' THICKNESS=', F7.4)

DO 570 L=1,NL
DO 560 J=XLO,XHI

*HYPERBOLIC TANGENT IN r

560
570

580

999

QO=SQRT(((JTMIDJ)*LX)**2+((L-MIDL)*LZ)**2)
R(J,L,7)=0.5*BYO*(1.0+TANH((RADIUS-QO)/S))
PRINT *,J,L,QO,R(J,L,7)
CONTINUE
CONTINUE
DO 580 J=XLO,XHI
DO 580 L=1,NL
PM:PMO-(R(J,L,6)**2+R(J,L,7)**2+R(J,L,8)**2)*OVER8PI
R(J,L,1)=R0+PM/(GM1*E0)
R(J,L, 5)=R(J,L,1) *EO
CONTINUE
GOTO 999
ENDIF
CONTINUE
CALL SETGRID(R,Z,DT)
CALL INITFFT(R)
RETURN
END

***fr************************‘k****************************

SUBROUTINE INITNUM
IMPLICIT NONE

* GLOBAL VARIABLES

INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
INTEGER FXSYM(8)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
1 /SYMS/FXSYM

NX=32
NC=(NX-2)/2
NR=NX-1
NL=50

PRINT *,‘NX= ',NX,’ = ',NL
LM=NL-1
LM2=NL-2
LM3=NL-3
LM4=NL-4
XLO=0
XHI=NR
ZLO=1
ZHI=NL
FXSYM(1)=-l

172

 

FXSYM(2)=+1

FXSYM(3)=-1
FXSYM(4)=-1
FXSYM(S) =-1
FXSYM(6)=-l
FXSYM(7)=+1
FXSYM(8)=+1
RETURN

END

*************************************************************************

SUBROUTINE INITZER
IMPLICIT NONE
* GLOBAL VARIABLES
REAL BXO,BYO,BZO,EO,R0,VXO,VYO,VZO
COMMON /ZEROES/BXO,BYO,BZO,E0,R0,VXO,VYO,VZO

R0=1.0
VXO=0.0
VYO=0.0
VZO=0.0
E0=0.9
BXO=0.0
BYO=3.0
BZO=0.0
RETURN
END

*************************************************************************

SUBROUTINE INITPAR
IMPLICIT NONE
* GLOBAL VARIABLES

REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
REAL on,BY0,Bz0,E0,Ro,vxo,VYo,vzo '

REAL TMAX,TPRINT,PFLAG,TVIEW

OCOMMON /PARAM/G,QH,GMl,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
1 /ZEROES/BXO,BYO,BZO,E0,RO,VXO,VYO,VZO

2 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW

GM1=2.0/3.0
G=0.0
QH=G*RO/(GM1*E0)
IF(G.EQ.0.)THEN
PRINT *,' No Gravity'
ELSE
PRINT *,'G= ',G,'SCALE HEIGHT= ',1./QH
ENDIF
C READ (*,*)VP
VP=0.1
OM=0.0
FV:1.0
WRITE(*,900)VP,OM,FV
900 FORMAT(1X,'VP= ',F10.S,' OM; ',F10.5,' FV: ',F10.5)
ZETA2=9*(.05+VP)
ZETAl=ZETA2/3
IF(ZETA1+ZETA2.EQ.0.)THEN
PRINT *, 'No Viscosity'
ELSE
WRITE(*,1000)ZETA1,ZETA2/3.
1000 FORMAT(1X,'Zeta1= ',F6.4,' Zeta2= ',F6.4)
ENDIF
ETAO=0.0

173

IF(ETAO.EQ.0.)THEN

PRINT *,'Infinite Conductivity'
ELSE

PRINT *,'Eta=',ETAO

ENDIF

ERF=0.

LX=1.5

LY=1.0

LZ=1.0

PRINT *,'LX= ',LX,' LZ= ',LZ

TMAX=50.0

tprint=10.0

PFLAG=25.0
* REMEMBER TO CHANGE FLAG3D

PRINT *, ' = ' ,TMAX, ' TPRINT= ' ,TPRINT
C TVIEWfiTPRINT
C TVIEW=0.

TVIEW:30.

SETTING TVIEw=0.0 CAUSES THE INITIAL CONDITONS TO BE PRINTED
WHILE TVIEw=TPRINT CAUSES IT TO SKIP THE FIRST PRINT

* *

RETURN
END

*************************************************************************

SUBROUTINE SETCON
IMPLICIT NONE
* GLOBAL VARIABLES
REAL F1D15,F16DlS,OVER8PI,PI,FOURPI
COMMON /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI
PI=4.*ATAN(1.0)
FOURPI=4.*PI
OVER8PI=.125/PI
F16D15=16.D0/15.D0
F1D15=1.D0/15.D0
RETURN
END

*************************************************************************

SUBROUTINE INITMAG
IMPLICIT NONE
* GLOBAL VARIABLES
REAL ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO
REAL F1D15,F16D15,0VER8PI,PI,FOURPI
REAL BXO,BYO,BZO,E0,R0,VXO,VYO,VZO
REAL G,QH,GMl,VP,OM,LZ,RMO,FV,ZETAZ,ZETA1,ETAO,LX,LY,ERF
OCOMMON /MAGNET/ALF,ALFX,ALFY,ALFZ,BETA,CFM,CSM,CT,CSURF,ME,MO

1 /CONST/F1D15,F16D15,0VER8PI,PI,FOURPI

2 /ZEROES/BXO,BYO,BZO,E0,RO,VXO,VYO,VZO

3 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
BETA=1E20

IF(BZO.NE.0..OR.BXO.NE.0..OR.BYO.NE.0.)
$ BETA=.6/((Bzo*Bzo+BYo*BYo+on*Bx0)*OVERBPI)
ALFZ=BZO/SQRT(FOURPI)

ALFY=BYO/SQRT(FOURPI)

ALFX=BXO/SQRT(FOURPI)
ALF=SQRT(ALFX*ALFX+ALFY*ALFY+ALFZ*ALFZ)
CSM=SQRT(.5*(l.+ALF*ALF-SQRT((1+ALF*ALF)**2-4.*ALFZ*ALFZ)))
CFM=SQRT(.5*(1.+ALF*ALF+SQRT((1+ALF*ALF)**2-4.*ALFZ*ALFZ)))
PRINT *,' BETA: ',BETA

PRINT *, ' ALFVEN VELOCITY (A,AX,AY,AZ)= ',ALF,ALFX,ALFY,ALFZ

174

 

PRINT *,' SLO MODE SPEED= ',CSM
PRINT *, ' FAST MODE SPEED= ',CFM

CSURF=SQRT((SQRT((ALF**2+2)**2+(GM1+1)**2*ALF**4+4*GM1*ALF**2)
$ -(ALF**2+2))/(0.5*(GM1+1)**2*ALF**2+2*GM1))
ME=SQRT(OM**2*(1/CSURF**2-1.0))

CT=SQRT(ALF**2/(1+ALF**2))
MO=OM/CSURF*SQRT((1-CSURF**2)*(ALF**2-CSURF**2)

$ /(1+ALF**2)/(CT**2-CSURF**2))

WRITE(*,1000)CSURF,MO,ME

1000 FORMAT(1X,'Csurf= ',f10.5,' Mo= ',F10.5,' Me= ',F10.5)

RETURN
END

*************************************************************************

* THIS ROUTINE INITIALIZES THE GRID AND CONVERTS

* R TO MOVING GRID
*************************************************************************

SUBROUTINE SETGRID(R,Z,DT)
IMPLICIT NONE

* PASSED VARIABLES

REAL T,R(O:3l,1:50,8),Z(0:31,1:50),DT

* GLOBAL VARIABLES

REAL G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

REAL TMAX,TPRINT,PFLAG,TVIEW

REAL F1D15,F16D15,0VER8PI,PI,FOURPI

REAL DZDX(0:31,1:50),DZDL(0:31,1:50),DZDT(0:31,1:50)

COMPLEX DX(0:31)
OCOMMON /NUMBER/LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT

1 /PARAM/G,QH,GM1,VP,OM,LZ,RMO,FV,ZETA2,ZETA1,ETAO,LX,LY,ERF
2 /CONST/FlDlS,F16D15,0VER8PI,PI,FOURPI

3 /PRINTS/TMAX,TPRINT,PFLAG,TVIEW

4 /GRID/DZDX,DZDL,DZDT,DX

* LOCAL VARIABLES

COMPLEX II
INTEGER J,L,M
REAL A(0:31,1:50)

* CREATE DX FOR DERIVATIVES

10

30
20

50

60

II=(0.,1.)
IF(NX.GT.O)THEN
DO 10 J:0,NC
DX(J)=-2.*PI*II*J/(LX*NX)
CONTINUE
ENDIF
DT=0.01
DO 20 J=0,NR
DO 30 L=1,NL
Z(J,L)=LZ*(L-l)
CONTINUE

DO 50 J=O,NR

DO 50 L=1,NL
DZDT(J,L)=O.

CONTINUE

CALL DDL(Z,A)

60 J=0,NR

DO 60 L=1,NL
DZDL(J,L)=A(J,L)
CONTINUE

175

 

MULTIPLY BY DZDL TO CHANGE GAS VARIABLES AS DESIRED
FOR DYNAMICAL ZONING

* * fi 4

 

DO 80 M=1,8

DO 80 J=0,NR

DO 80 L=1,NL

80 R(J,L,M)=R(J,L,M) *DZDL(J,L)

T=0.0
C CALL BC(R,Z,T)

RETURN

END
*************************************************************

* THIS SUBROUTINE SETS UP THE FFT PARAMETERS
*************************************************************
SUBROUTINE INITFFT(R)
IMPLICIT NONE
* PASSED VARIABLES
REAL R(0:31,1:50,8)
* GLOBAL VARIABLES
INTEGER LM,LM2,LM3,LM4,NC,NL,NR,NX,XLO,XHI,ZLO,ZHI,SPLIT
REAL WORK(0:31,1:50,8),TRIGS(451)
INTEGER IFAX(13),NPTS,INC,JUMP,NVEC
INTEGER SYMFLAG

COMMON / FFT/ WORK, SYMFLAG
$ /NUMBER/LM, LMZ, LM3, LM4, NC, NL,NR, NX, XLO, XHI, ZLO, ZHI, SPLIT
SYMFLAG=1
PRINT *, 'SYMFLAG= ',SYMFLAG
IF ((NX.LT.4) .AND. (SYMFLAG.NE.3) )SYMFLAG=0
IF ((SYMFLAG.NE.O) .AND. (SYMFLAG.NE.3) )THEN
IF(NL.NE.(2*(NL/2)))THEN
PRINT *, 'NL must be even. '
STOP
ENDIF
ELSE
XHI=O
ENDIF
RETURN
END

176

 

Acknowledgements

I would like to express my gratitude to Dr. Robert Stein for his patient guidance. I
would also like to thank Dr. Sheridan Simon and Dr. Rex Adelberger for their support
and assistance— both professional and personal. I would also like to especially
thank Charlie White and the rest of the Guilford College computer services staff for
their patience and assistance.

This research was supported in part by NSF grant AST 83-16231.

The encouragement of Guilford College and their support, both economic and moral,
has made the completion of this work possible.

177

 

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180

 

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