JPRS ID: 9322 JAPAN REPORT UPPER, LOWER HOUSE ELECTIONS

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JPRS L/9639
, 2 April 1981
= Trar~slc~tion ~
STUDY OF HYDRODYf~IAMIC
INSTABILITY BY N~UMERICAL METI-IODS ?
~ Ed. by ~
A.A. Samarskiy
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~ -
~ JPRS L/9639
,
2 April 1981
= STUDY OF HYDRODYNAMIC
_ INSTABI~ITY BY NUMERI"~AL METHODS `
= Moscow IZUCHEIJIYE GIDRODINAMICHESKOY NEUSTOYCHIVOSTI CHISLENNYMI
= METO~AMI in Russi.an 1980 signed to press 26 Dec 79 pp 1-227 ~
- [Translation of the collection of scienta.fic articles "Study of Hydro- ~
dynar.iic Instability nf Numerical Methods", edited by A.A. Samarskiy,
' Institut prikladnoy matematiki AN SSSR, 400 copies, 227 pages]
I ..r~`~"- CONTENTS
Editor's roreword 1
Heat Inerti a and Dissipative Structures 3
~
1. Introduction 3
- 2. Metastable Localization of Heat 3
3. Development of Thermal Structures 6 ~
4. Multidimension al Effects in the Heat Localization Phenomenon 8
5. Localization of Tieat in a DlasmaT~Jith n-Type Thermal Conductivity 9
6. Lo.r.alization of Thermonuclear Combustion in a Plasma With
n-Type Electrical ~onductivity 11 -
- Bibliography 12
~ Study of the Stability of the Comnression Pro cess of Thin Glass Shells 2G '
I Introduction 22
~ gl. General Statement of the Problem 22
~
~ �2. Nature of the Occurrence of Instability 24
�3. Test Calculations. Choice of the Finite Difference 25 `
�4. Analysis of the Instability in a'Corona' 31 -
�S. Free Flight Stage 37
�6. Instability of the Inside Boundary of the Shell 40
Bib liography 46
i�fathematical Models of the Formation of Tornadoes as a Result oi the
Development of Cas Dynamic Instabilities 49 -
- ~ - [I - USSR - L FOUO] _
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Introduction 49
Chapter I. Axisymmetric Instability 50 ~
; � 1. Statement of the Problem 50
, �2. Conservation Laws and Energy of. Instabilities 51
- ~3. Steady-State Axisymmetric Configuration 52
� 4. Linear Theory 53
~ �5. Variation Principle 55
�6. Results of the Nuznerical Calculation 5~
_y Chaptex II. Helical Instab ility of a Cylindrical Gas Jet 74 ~
�1. Statement of the Problem %4
. ~2, ~inear Theory 74
~ � 3. Energy Limitations 76
g~, Results of Solving the Nonlinear Problem 76
~ Conclusion
~ Bibliogranhy 82 `
Hydrudynamic Description of the Sel.f-Focusing of Light Beams in a
Cubic P:edium ~3 ~
Introduction 83 �
~ 1. Variation Statement of the Problem. Integrals of r?otion.. _
Hydrodynamic Analogy 85
, �2, Coordinates Connected With the Rays (Optical Analog of the =
Lagrange Mass Coordinates) 88 -
�3. Numerical Simulation of Self-Focusing. Conservativenes~~.
Method of rloving Finite-Difference Nets 92
~4. Asyrnptotic Behavior of thE Solution of the Problem of Self~ ~
- Focusing in the Vicinity of the Focal..Point 97
~ ~5. Results of the Numerical Integra ti~n 103 -
g6. Ray Equation and Its S~mF lification 105
�7~ General Solution of the Simplified Equation. Aberrations 111
During Self-Focusing of Gaussian Beams. Results of Numerical
Integr.ation ~
~ 8o Formula for the Focal Length 114
Bibliography 116
~ Variation Systems of r4agnetohydrodyna_mics in an Arbitrary Coordinate I29 _
System
- r
j Introduction 129 _
~l. Different~al Equltions 130 ~
�2. Discrete I~?odel 133
~3. Diff.erential-Difference Eouations of P~agnetohydrodynamics 136
_ �4. Some Properties of the bifferer~tial-Difference Equations o.f 141
_ Magnei.ohydrodynamics ,
Bibliography 146 ~
-b- -
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� Completely Neu~ral Dif.ference Scher~e for the Navier-Stokes Equatior.s 148
�1. Statement of t::c lnitial Problem 149
~2. F'inite-Difference Nets and Functions 150
�3. Balance Equation. Approximation of Flows 152
~4. Differen ce P rob lem in the Variabl~s Q 157
�5. Equivalent Difference Scheme in ~~ariables w 159
~6. Families of Neutral Systems. Entirely Neutral System 161
, ~7. Supplement 164
Bibliography 167
- Numerical Simulatj.~n of Thermal arid Concentrati.an Convection in 168
Chemical Reactors
Introduction 168
; I. Sti.atement of the Problem 168
= II. Solution Procedure 17~
_ III. Results of the Numerical Experiments 173
Bibliography 175
- c -
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~ [Text] Editor's Foreword =
Numerical simulation b ased on finite-difference methods oriented toward computer
applications is acquiring gre ater and gr?ater significance in the investiga~ion '
of physical phenomena. The nonlinearity of the processes and the correspon~ing
~ equations makes the computer experiment a powerftil, and in a number of cases,
_ the only possib le means of efficient solution of complex apglied and theoret-
ical problems~
- This publication is one of the thematic coilections on the urgent problems uf `
= applied mathematics published by the Institut~: of Applied Mathematics imeni
M. V. Keldysh.
.
Stability problems play an important role in the investigation of a number of
prob lems of modern physics and in a number of cases they play the defining role.
Solution of the prob lems connected with studying the development of instabi.li- !
ties of various types imposes rigid requirements on the numerical methods
- and algorithms. Therefore a number of articles in tlie collection are devoted
- t~ a discussion of the computer aspects connected with solving stabili.ty prob lems.
These are the paper by B. D. Moiseyenko, L. V. F'ryazir~ov, in whi_ch a prospective
_ algorithm for numerical simulation of the mo~ion of an incompressible medium is
discussed, and the paper by V. M. Goloviznin, T. K. Korshiy, A. A. Samarski_y, ~
V. F. Tishkin and A. P. Favorskiy which contains a generaltzarion of the variation
� approach to constructing completely conservative magnetohydrodyr~amic systems
to the case of three spatial measurements. ~
' The remaining articles contain examples of a numeric~-=.l solution and theoretical
~ ~tudy of instabili*_ies in a medium. `
In a paper by a group of authors, a brief survey is given of previously obtained
results pert aining to the numerical simulation of the Ra~leigh-Taylor instability
in experimental glass shells investigated at the Physics Institute of the USSR
Acade~ry of Sciences.
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I
FOR Ur~r~1~~aL U5~ UNLY
' A study is made of the results of the numerical simulation of the occurrQnce of
~ tarnadoe~ as a result of gas dynami.c instability performed by N. M. Zuqeva,
V. Paleychiic and L. S. Sol~v~yev.
a
It is known that th~ hydrodynamic approdch turns eut to be highly effective in
~ many problems with respect ro physical mean{ng. Accordingly, the collection con-
taias a suYVey by L. M.Degtyarev and V. V. Krylov of the algorithms and results
of numerical simulation cf the self-focusing of light in nonl~.near media obtained
using the hydrodynamic anal.ogy for the Schroedinger type eq~ation.
Recently the interest in s~udying the general laws of the development of instabil-
- ity in z continuous medium in the non].inear stage has intensified noticeably.
This has served as the basis for inclusion of new interesting results obtained
~by S. P. Kurdyumov, N. V. Zmitrenko, A. P. Mikh~ylov, et al., in the collection
pertaining to the formation and intera~tion of nonlinear structurea in the peaking
mode. The conclusions contained here have a v~ry broad range nf theoretical and
~ practical applicat~~ons.
A. A. Samarskiy
~
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HEAT INERTIA AND DISSIPATIVE STRUCTURES
[G. G. Yelenin, N. V. Zmitrenko, S. P. Kurdyumov, A. P. Mikhaylov, A. A. Samatskiy,
PP 5-27 ]
- Introduction
In this paper a study is made of the ph~nomena of heat inertia accompanied by
the localization of heat and thermar:uclear combustion in a d~nse plasma during
the development of processes in it ir. the peaking mode.
Analytical and multidimensional numerical solutions to the problems are presented
' in partial derivatives, explatning a number of the peculiarities of development
of superheated and other types of instabilities in such a plasma.
It is demonstrated that heat inertia can lead to the metastable existence of in-
~ stabilities having paradoxical for~4 ~i the region of localization ("thermal
_ j crystals") . The conditions of the exc{ tation of the combustion of a medium in
~ the peaking mode are formulated, leading to local.ization of the combustion 3.n
i individual sections iz~ the form of structures of different types. It is demon-
~ strated that the resonar~ce conditions of their excitation are determined by the
i eigenfunctions of the nonlinear self-similar problem. Estimates are pres~~nted of
the region of occurrence of these phenomena during the processes of heat~ng the
piasma by shaped laser radiation and during initiation of combustion in laser
targets. The relati~n of the described phenomena to the fundamental laws of the
- occurrence and complication of organization in nonlinear media is indicated.
- 2. Metastable Localization of Heat
The process of the propagation of heat in a stationary medium with nonlinear
thermal conductivity in the simplest one-dimensional case is described by the
equation:
~ T ~a ( K C r) a T
a
t- a~G a~~~ cl)
where T(r, t) is the temperature, t is the time, O,r0 is the coefficient of the~al conductivity.
- Let at the boundary of the unheated mediwn
t, ~ = 0 t2)
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- r'~x urrl~~.~ u~~ UNLY
the temperature increase by the law
T(o,t) =T (~f-t) h 0. (3~
On variation of time in the interval tp~r~ =LT*. The size of the region of locali-
zation of combustion ~rR is given by (17) by ~he for~nula: ~0.5/pT~m85 cm
(on variation of the initial amplitude i~.t~he range of 1-3 kev). The combustion
is localized durinQ the time ~tA~10-6/pT~m sec. When tr.e temperatures T~5 kev
are reached in the combustion pr.ocess, the size and t e localizatio ime of the
- combustion region are deterud.ned by the S-mode: ~r ~5~~0.2/p cm, ~t~s~~8�10-8/pT~mS
sec. With further increase in the temperature, its prof~le inside the localiza-
~ tion region begins to be rearranged to convex, and for T>10 kev an increase in the
combustion region begins.
As was shown in [le], the scales reached ~tA and ~rA are such that for T3.7 kev. we set g(T) -
2 3
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~ rux urri~iew uon u1VLY
P~Pe+P~ ~ Pe=Pe(S,Te)~ P~ = Pt ~3~T:) (1.5)
&Q = &n~S~'l'e) , E1 � Fi ~ S~`l'~ ) (1.6) -
- ~
- ~Ig1~0~~' ~K(8,T4)~r (1.~>
_ ~ .
1A/e = 2e ~ 3,T e.) ~a.d Te (1. 8)
- 4 = Qo ~2 '~e
.~3,~ c~.9~
e
. Here p is the density of the substance,
- ~v is t::e hydrodynamic velocity,
F~, Pi, P are the electron, ion and total pressures,
� Te, Ti a~e the electron and ion temperatures, ,
ee, ei are the electron and ion specific internal energies.
The system (1.1)-(1.9) was solved in the approximation of axial symn?etry.
- As follaws from the experimental papers j6], [3] for the investigated energy flux
density the number of epithermal electrons is small, and it has no influence on
_ the campression. 'I'herefore the Yieating by the fast electrons was not taken into
- accotmt, and the th~rmal conducrivity was considered to be classical.
Thus, the investigated approximation quite completely gives a qualitatively and '
= quantitatively correct description of the processes.
�2. Nature of the Occurrence of Instability
1. There are two stages of the proc~ss where the motion is hydrodynamically -
unstable. The first stage is acceleration of the heavy unevaporated part of the
shell by a hot, low-density ablation layer. The second stage comes when the
p ressure in the compressed nucleus increases tc a degree such that it begins to
b rake the densEr she11. These s~ages are separated in time by the region of
stable flow with approximately ~onstant velocity [see [4]).
2. In a number of simplest cases, the est:Cmate for the rate of development of
disturbances can b e obtained analytically. Thus, for example, the analytical
solutions are obtained in the case where the disturbance wave length is large by
comparison with the characteristic dimen~ions of the investigated subject [20],
~ [21], [22]. There are also a number of papers devoted to the study of the
behavior of an it~compressib le liquid in a constant gravitational force field
L231, I241, LZSI, I26], L271, C28], (29), [38].
3. In contrast to ~he classical situation, on compression of the shells, the
development of the instability takes place against an essentially nonstationary =
background which is f~rmed as a result of interaction of the nonlinear thermal
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and hydrodynamic pr~cesses. The investigation of the instability in the first
stage near the ablation boundary is also complicated by the fact that the
evaporated substance flows through the instability zone with high velocity, and
the zone itself moves through the mass deep into the shell. "
The noted facte complicate the application of analytical methods and make numeri-
cal simulation~in practice the only method permitting successful solution of the _
- stated problem.
~3. Test Calculations. Choice of the Fin{.te=Difference
- 1. When investigating the dynamics of the development of the disturbances by
numerical methods, the question arises of to what degree the correctly used pro-
cedure transmits the quantitative and qualitative nature of the grawth of the dis- _
turb ances. In order to explain this question, a numober of test calculations were
made including comparison with the known analytical solutions and calculations on _
series of finfte-differences becoming denser. The calculations demonstrated that
the numerical sqlution quantibatively and qualitatively correctly reproduces the
dynamics of the development of the disturbances.
2. As one of the model problems, a stu~y was made of the prob lem of the stability
of a thin layer of incompressib le liquid under the effect of gravity [20]. The
gravitational acceleration is directed opposite to the y-axis (see Figure 3). _
~ n � ' . Q~N
, . . . .
. � � � -T u'l.
, Figure 3
On the lower boundary of the layer of liquid, the candition of nonpenetration
was given. The upper boundary was assumed to be free.
At the initial point in time the height of the free boundary of the liquid was
disturbed by the law =
h. � ~.a C 1+ a~Kx) (3.i)
here h~ is the height of the undisturbed layer,
a is the amplitude of the disturbance,
k=2~r/a is the wave number,
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r'ux urr~l~l~ u~r. UNLY
a is the disturbance wave length.
The speed of the li.~uid at the initial point in time is asstuned to be
~,t,= -a ~I~e ~~~?+.(ICx) (3.2)
~ = cx,K~ c~ ko GaS(Kx)
(3. 3)
For r:umerical solution of this problem, entirely satisfactory agreement was
obtained between the numPrical and the analytical solutions in the different
- stages of development of the instability.
I
t
~ ~
- i
~ I
~ I
r
~ '
o ~ e
A
~t
~
d
6
d
~
d
~
� t
0. O.~t~ ./.M 1.1St 1. .~n .1M ~M . IT
Figure 4 -
In the initial period of movement, the behavior of the liquid is described well _
by the Iinear approximation, according to which the disturbances must increase _
as exp(k ~t). In Figure 4 comparative graphs are presented for the growth of
the disturbance aiuplitude calculated by the linear theory and by the data from
the numerical calculation.
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- ! -
~
- a
~o
so
M6
= I6
30
26
20
' !6
!0 t
5 0
t
. ny a92 1U H4 4b li f (a ! n~ W w ts .a a~f op a!! a~o oa ~A
Figure 5
~1)
~ BPGMA-0.60775 �
ry
ry .
~
, I
~
r
( 3) ~ ' I,~~
� rj 1 ~ r~+!
1~,. ;.;6 -
/
~
8 -
~ 6
OCbX ~2~
Figure 6
Key: -
' 1. time; 2. x-axis; 3. y-axis
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r~~x urri~iEw u~~ UNLY
In the later stage, the linear approximation loses its correci:ness, bt~t for the
nonlinear problem the asymptotic regime exists [25], I26], j27], j38], for which
the solution "peaks" directed upward move with acceleration close to the gravi-
tational acceleration. Figure 5 shaws the arrival of the numerical solution at
this regime. In conclusion, Figure 6 gives the shape of the liquid--boundary at
the time close to the time of reaching the asy~ptotic motion regime.
3. Satisfactory agreement was also obtained on numerical simulation of the prob-
lem of Rayleigh-Taylor instability of a fine strand of thread (see [21J). The
thread was simulated by a layer of incompressible liquid, the thickness of which
was appreciably less than the wave length and the amplitude of the initial dis-
turbance (see Figure 7).
-
~J
~ P~0
~c~, ~~~r-? .
- P - ~9~
~
. ~
x
Figure 7
The gravitational acceleration was directed dawn along the y-axis. At the upper
and lawer boimds of the layer the pressure was given: P=0 at the upper bound
and P=pg at the lower bound, p is the linear density of the undisturbed layer.
The thread coordinates were disturbed by the formulas
= aC = aC o(. Su1 ( K~x. ) ( 3. 4)
- l~ = CY, f.~;, ( ) ( 3. 5)
x~ is the coordinate of the undisturbed thread. The velocities were assumed equal
to zero at the initial point in time.
If the thickness of the layer is sufficiently small, then the equations of motion
have analytical solution of the type [see [21]) :
.h= = :xo u, Si., ~K :~o) (3.6)
a. C~~ ( K x~) ( 3. 7)
Here they acquire the shape of a cycloid. This solution will be valid until
t~~: kacos ht~=1 when the cycloid forms a self-intersection.
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a
-
~
N -
O
N
N
.w �
O
- ~i
~
O
O ~
r~ Q 4 6 8 ~ t0
h
o �
0
m
~
0
~ N
ui
N 1 ~
O
r
_ , .
_ X .
- Figure 8 -
The results of the ntmmerical calculation satisfactorily reproduce the solutions
" of (3.6)-(3.7) to the time t* when the layer thickness increases, and cumulative
jets are formed (see Figure 8). The divergence of the numerical and the analyti-
cal solutions was 0.5%. Let us note that the appearance of cumulative ~ets was
predicted in [21],
~ 4. For numerical s3.mulation, the problem of the choice of the number of finite-
difference nodes is important. The use of dense- finite-difference nets
unj ustifiably increases the solution time of the problem, and when using ~a ~small
number of. nodes, significant deviations from t~e correct value can occux. In
order to determine the optimal number of nodes, several calculations were made
of the compression of a glass shell described in ~l, where disturbances were
introduced into the initial shape by the following laws:
R=RQ(l+a ~ sin n6) (g,g~ _
U
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rUx ~rr t~it~, U~~: UNLY
n is the harmonic number,
~ is the shell thickness,
a is the amplitude of the disturbance.
The disturbance amplitude a was taken equal to 0.01, and the number of the
harmonic, n=10.
f~max Rmi?~
R mcuc~ ~ n
- o.2Q '
0.;!0 ~
5 TO 15 20 ~
_ Figure 9
The calculations demonstrated that with an increase in the numb er of finite-
difference nodes, the disturbance grawth increases, but for a numbei of
nodes per disturbance wave length k;10 to 15, saturation of the growth rate
takes place (see Figure 9). Let us note that approximately the same criterion
was obtained in reference [31]. In the calculations described below, the number
of finite-difference nodes was selected beginning with this criterion.
5. The study of high harmonic numbers (n~20) is possible when performing the
calculations in the sector with angular dimensions less than ~r/2 under the
condition that the angular overflaw of plasma during compression is much less
th an the dimensions of the sector. On the la~teral boundaries of the sector, the
equality of the normal velocity component t~ 0 was given. In order to check
~ whether this influ~nces the nature of the growth of disturbances, the harmonic
was calculated with n=20 in the ~ector with its aperture angle ~r/2 and aperture
angle ~r~10 (see Figures 10, 11). A comparison shaws that the qualitative and
_ quantitative nature of the development of the instability did not change.
. 3p
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I
~ ~i ~ . ~ j
. ~ ~ 1 I c.--t
t, l- ...._f
,
~ . ~ t�-~~-r-
�7.�. ~
`~'~Y~_
" ' i�.'"I.:i
. f~::.~; . ;
;..w~ - ~
L.-~ ' ~i ~ ~
.r ; I . t
I~ �`7
,:,1
, t~ . :
. , j ' ~ f,
11 ~
~ ~
.

~ is the disturbance amplitude,
_ K is the wave number.
36
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. _
~
i';~%i.%~,
, 'i ~ . ; ~
'i,,/~/:'
:
~ - .
,
' / .
_ .~i~i;-,
. , , . .
: ~ %f ~ , . . ~ : .
�:'y'/..: :~/,I:' '1 ~ . . .
~/I,�~iI ~~/1'~~ . . . c
' ~'y~i ~~.~'I~~.' . .
f ~~/i ~ .~i .
i' ' ' . . ~
;i:i'i i i' ii'''~.
' ~!~'i.^ , j'' 'i' .
. :
_ ~ ~'1� ':Y' _
. . . .
~ '%i" :v .�i:�
~l~:~.
J
.
' ~ . . ' ' l.` ~ l 1~ ~ . , _
Yll -
- Figure 16 -
For this mode the growth rate of the disturbances is saturated, and it is
- approximately identical for all numbers of harmonics. _
7. On the whole the instability in the corona is the standard Rayleigh-Taylor
instability. The amplitude of the disturbance increases by more than 100 times.
- The effect of the evaporation and dissipation leads to the fact that the
_ increments of the grcrwth of the disturbances turn out to be somewhat less than
- the Taylor modes. This is especially felt in the long-wave disturbances which
cannot be propagated even to the shell.
The short-wave disturbances penetrate to the cold part of the shell, but the
depth of penetration decreases rapidly with an in.^.rease in the harmonic number.
_ The harmonics with n~15 to 20 can have the strangest influence.
�5. Free Flight Stage
l. In the free flight st~ge (before the beginning of braking of the s~ell),
almost periodic fluctuations of the inside boundary of the shell occur, the
phase ~f which depends on the time, and the amplitude increases insignificantly -
(see [9), [33]). The indicated results of the numerical calculation are in
37
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rUr Vrrl~lcw u~L vlVLl
good agreement with the results of the analytical solution of the linearized
prob lem of instability of the spherical boundary of a gas bubble in an incompressi-
ble liquid [33]. These fluctuations occur as a result of adiabatic contraction
of the surface of the bubble on compression
_ R 1/4
d? = d?a ( Ro/~~~- ~o)GVS(It9) (5.1)
From (5.1) it follows that at the time determined f rom the equation
.i, 3 ~~Q R_.
(2 K+1) 2-~ 2~ IZ el2 ;~(t) (5.2)
the inside boundary will b e spherical. A comparison of the result of (1)-(2) with
th e data from the numerical calculati~n (Figure 18) reveals qualitative agreement.
.0~ _
, ~~0�~4 ~~r.)
: : ~ .
. � _
i. , '
- ~i. �
_ 'i~
/
~ � ; r,,t' ' � ~ .
_ i ; ;
J, '~i;;
_ ' . , .
- I .
; ' ~i~ _
.
r� t'L';.;.. -
. .:`;;'y~ .
-
~ � . ~
, . , . ~ ~ . .~5 ~ , . . ~ ' ' '
~~V
, Figure 17
38
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35.2o R n.~ .
~1) { = 1.ZS H[EK ~2~ -
. ,
35.18 y ~
e pa g
(3)
35. i6
30. 25 R~K.~t f =1.,3SHcEK
(1) ~2j
30�23
_ - 9 p4~, c3~
30.24
= Zti.S R,~K,~c t= g. 46 NcEK -
~1) � ~2)
n
zy s4 ~ ~ e -
P4~ (3)
24.52 ~ . -
Figure 18
Kev:
1. R, mi crons
- 2. nanoseconds
3. radians
39
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~Qa `R~,;~ . ~~i'
- R~wx ~ R.:n ~ f
/
i0~i ~i ~ / /
a ~
1
. ~
/
tio I
J
,
io 2 ~
i
i
~
.
' ~
~
_ R=10 I
r 1` ~
- , i0 3 R, 5 ~ ~
~~i
4
--ti
H~~I: '
O. O, S 1.0 1.5 2�0
(1) -
Figure 19
- Key:
1. nanoseconds
g6. Instability of the Inside Boundazty of the Shell
I. Instability of the inside boundary of the shell occurs when the gas included
inside begins to brake the shell. The relative amplitude of the disturbance as
a function of time for different harmonic numbers is presented in Figure 19. A
comparfson with the value calculated by the formula (see [4])
n a(u i/z
dn - Q no� eXp { S4~v"'R d t~t (6.1>
~
shows that the lo~w harmonics n~10 increase approximately with the same rate.
For n>12 the growth rate becomes less than the Taylor modes, and the harmonics with
n=20, 40 develop completely in the linear~~ode. The effect of the "nonlinear
~ saturation" of the disturbance grawth rate is especially clearly obvious if the -
relative amplitude of the disturbance is represented as a~function of the
harmonic n~ber, taking time as the parameter (Figure 20).
. 40
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~ .
R.~QY Rn~in . .
~mqx+~�'!in ~.q68 HCEK ~
~1~
{ ~ 1.95 N~~K
(1)
a.2 -
f - f. 9i25 Hcek
Q 10 , 2 0 3~7 yo SO h' -
Figure 20
Key: -
~ 1. nanoseconds
,
, 2. From Figures 19, 20 it is obvious that for harmonic 10 the deviation from
- linearity occurs onl} at the last paints in time, and for harmonics 20 and 40,
the nonlinear mode comes in the earlier stages of motion. Let us note that the
exponential growth of the disturbances in the linear mode takes p lace while
L1� K � 1 c6.2> -
~ is the amplitude of the disturb ance,
- K is the wave number.
The results of the calculations show that for the initial ampli t ude of the dis-
_ turbance a~0.01 this condition is violated already for n=10 by the time of maxi-
mum compression. Consequently, the saturation.of the incremen t with growth of
the wave number is due to the nonlinear effect.
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rUec urr 1~1~ uar, UNLY
U - - ~ ~ -
~ ~ J~ ~ ~
- ~ v' .
- -
_s_ �-=~-:f`'
- ~ ~ . ~
. 1
~ ~ ' ~ ` \ ~\v~ ~
~1 ~ ~ � .y ~ .
\
~
\ ~
\ , .
~ ~ ~1~ \ ~~.ti; .
~ . ' r~^~\~ \ .
r_� - . . \
~ ~ C 1 ~ ~
. ' ~ ~ ~tt\' ,,`V
~ ~\~r\\~~\~`~ ` .
) / t\`,' ~\\,~7,~~' ~ -
/ / ~ :
~ T' y.~\~\ \ ~ \ 9~
~ ! \ t~ i
~ ~ ~ ~ i,
~ ~ ~ ~
/ ~ / ~ / ~ ~ , \ ~
' v ' ~ f~/% / //i' f .X,.~ ~ 1 `r
; : -,,-=k._.=__=~ ~ 1 ~ ~ -
'i /i�; r~~!~; s ~ .
_ ~ ~ ~;'~i~~. i i/ ~~�~~-,J 1:i I~~y~., ~ 1 ~ I
~~///,/,'~~r �-l'i / i il,i ~ ~ .
' ~ _ T f; . : ~ ~ ,j ~
j;y~ f.�./~ ,~'r,i ? f ~ t 1 r i ~ .
~.t�., ~ ~ ~ - � , I
~_f:; . i 1 ,,.~_r~
r r I~ f l. r~
� _
.2 : :ti . :G ~ .
_ ~ F~'
_ Figure 21
Thus, for the investigated shells with as~ect ratio of 20, the maximum dis-
turbance growth rate is achieved for 15-20. The presence of short-wave components
in the spectrinn of the initial disturbances is not dangerous for the investigated
- shells .
3. In order to study the effect of the disturbances of the intensity of the
energy flux, a calculation was made of a number of versions where disturbances
were introduced into the energy flux by the formula
CI� (~�o ~t~11. ~-Gr?5128~ -
U
The disturbance amplitudes a and the harmonic numbers n were varied within
various versions. -
These calculations demonstrated that the disturbances of the ener..gy flux lead to
smaller distortions of the inside boundary than the shape asymmetry. An obvious
symmetrizing factor here is the heat conducting equalization in the "corona." ~
Thus, a comparison of the ma~dmum disturbance amplitude on the inside boundary
for 1% amplitude of the initial disturbance and the same wave length indicates
42
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, Y
that the shaped disturbances lead to amplitudes that are twice as high as the
nonunifor~ty of the fluxes (Figures 21, 22).
. \
- " ~
. : .
_ , ' ~ ~ ~
, , ; : : ~
; ,
, ' { ; f ,�l i . , .
; ~ , j ; .
; ~ � i -1,' ~ ; . � .
j ~ i~ J ! I~ . _ � (.i : .
i ; ~ ; i " ' . . ' , j _ r 1' ~ , �I;, .
i 1 ~ f ~ 1'''lil' . . .
~I~ r~! ~ : . ~ ' ~~~l~ .i~ ' ' . '
! - i ' `I .
~ . �
. _ . _ ~
~ ....r ' . _
~ .
Figure~22
4. Let us discuss what changes in the state of the shell and the gas the
instability effect lead~ to. The relative thickness of the shell in the axi-
symmetric case increases by approximately 8 times by the time of maximum com-
pression. However, at this time the disturb~nce amplitude (with initial dis-
turbance amplitude of 1~ of the shell thickness) becomes comparable to the t:tick-
ness. However, this fact still does not mean rupture of the shell. Indeed,
from the state of the shell at the time cZose to the time of maximum compression
it is obvioiis that the Lagrange lines ~orresponding to the inside boundary of
the shell are more strongiy distorted than the outside lines (see Figure 10).
Thus, a significant magnitude of the disturbance on the inside boundary of the
shell indicates that part of the shell material has reached the inner cavity.
Let us note that the average density and temperature of the inside gas with
?'espect to the nonspherical voltmie differ slightly f rom the corresponding values
in the spherical case (Figures 23, 24). Hawever, it is not necessary to attach
great significance to this fact, for the penetration of the shell material into
- the nucleus obviously leads to mixing of the nonperipheral layers of the gas
with the shell material and to a. decrease in the partial density of the internal
gas.
43
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� rUic vrrl~trw u~~ l1lVLlC
0.5 T x3g ~1~ .
~ - - - - o,
~ ~
- 0.4 -
i~ ~
D.3 ~
~
i
~
0.2 , i ~
~
i
~
~
O.i -
1. 5 16 1.7 !.8 ~ 1.9 2.0
f tIC!'::
(2)
Figure 23 _
Key:
1. T, kev
2. nanoseconds
- In or3er to estimate the role of the instabilities it is useful to compar.e the
- energy of turbulent motion with the kinetic energy of the plasma in the given _
calculations. The energy dissipated in the turbulent motion per unit time can be
estimated (see [.34]) as
d ET ti s~T (6.3)
dt ~
here
vT=YkT is the characteristic turbulent velocity (see [35j),
~T is the characteristic scale of the turbulence (in the given case, the maximum
- amplitude of the disturbances),
Y is the buildup increment of the disturbances.
If the kinetic energy of the plasma
EKwN ~ S ~ ~ (6. 4)
Key: 1. kin
44
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v is the hydrodynamic velocity,
p is the gas density,
then -
FT 3er~o
EK~N (
Ij ?l p . ( 6 . S ) '
Key: 1. kin
t~ is the instability development time.
s (1) .
~ p ~3
_ 6 ~ ~1. _
S . /
. ,
y ~ _
.a .
_ ~
Q
_ i
. �~.s t.6 ~z !s ~ t.s ~.a
' t HCFY,
Figure 24 ~2~
Key:
l. g/cm3
2. nanoseconds
Defining the values in (6.5) from the calculation, it is possible to obtain that
in the given case this ratio reaches 10%.
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r'Ux ur r t~ tEU, u 5~ UNLY
BIBLIOGRAPHY
1. Af anas'gev, Yu. V. ; Basw, N. G. ; Volosevich, P. P.; Gamaliy, ~ Ye. G. ;
Krokhin, 0. N. ;~Kurdyumov, S. ~-P. ; Levanov, Ye. I. ; Rozanov, V. B. ;
Samarskiy, A. A. ; Tikhonov, A. N. DOKLADY NA ~I3 KONFERENTSII PO FIZIKE
PLAZMY I UPRAVLYAYEMOMU TERMOYADERNOMU SINTEZU ~[Reports at the
5th Conference on Plasma Physics and Controlled Nuclear Fusion], November _
1974, Tokyo.
2. Afanas'yev, Yu. Basov, N. G.; Volosev~ch, P. P.; Gamaliy, Ye. G.; -
Krokhin, 0. N. ; Kurdyumov, S.~ ~P.~; Levanov, Ye. I. ; Rozanov, V. B. ; _
Samarskiy, A. A.; Tikhonov, A. N. PIS'MA V ZHETF [Letters to the Journal
of Experimental and Theoretical Physics], No 21, 1975, p 150.
3. Afanas'yev, Yu. V. ; Volosevich, P. P. ; Gamaliy, Ye. G. ; Krokhin, 0. N. ;
Kurdyumov, S. P.; Levanov, Ye. I.; Rozanov, V. B. PIS'MA V ZHETF, No 23,
19 76 , p 4 70 . `
_ 4. Afanas'yev, Yu. V.; Basov, N. G.; Gamaliy, Ye. G.; Krokhin, 0. N.; j
Rozanov, V. B.; PIS'MA V ZHETF, No 23, 1976, p 617.
5. Afanas'yev, Yu. V.; Gamaliy, Ye. G.; Krokhin, 0. N.; Rozanov, V. B.;
KRATKIYE SOOBSHCHENIYA PO FIZIKE~ FIAN [Brief Reports on Physics of the
- Physics InstituCe of the USSR Academy of Sciences], 1975.
6. Basov, N. G.; Kologrivov, A. A.; Krokhin, 0. N.; Rupasov, A. A.;
Sklikov, G. Sh3kanov, A. S. PIS'MA V ZHETF, No 23, 1976, p 474.
7. Volosevich, N. P.; Gamaliy, Ye. G.; Gulin, A. V.; Rozanov, V. B.;
Samarskiy, A. A.; Tyurina, N. N.; Favorskiy, A. P. PIS'MA V ZHETF, No 24,
1976, p 283.
8. Bunatyan, A. A.; Neuvazhayev, V. Ye.; Strontseva, L. G.; Frolov, V. L. ~
PREPRINT IPM 9N SSSR [Preprint of the Institiute of Applied ~tihemar.ics of
the USSR Academy of Sciences], No 71, 1975.
9. Afanas' yev, Yu. V. ; Basov, N. G. ; Gamaliy, V. G. ; Gasilov, V. A. ;
Krokhin, 0. N. ; Lebo, I. G. ; Rozanov, V. B. ; Samarskiy, A. A. ;
Tishkin, V. F.; Favorskiy, A. P. PREPRINT FIAN SSSR [Preprint of the
Physics Institute of the USSR Academy o.f Sciences], No 167, 1977.
, 10. Gamaliy, Ye. G. ; Rozanov, V. B. ; Samarskiys A. A. , et al. PREPRr~1T IPM
_ AN SSSR, No 117, 1978.
11. Basov, N. G.; Krokhin, 0. N. ZHETF, No 46, 1964, p 171.
12. Basov, N. G.; Krokhin, 0.- N, uESTNIK AN SSSR jVestnik of the USSR Academy -
- of Sciences], No 6, 1970, p 55. -
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13. Volosevich, P. P.; Kurdyumov, S. P.; Levanov, Ye. I. PMTF (Applied Mechanical-
~ Thermal Physics) No 5, 1972, p 41. ~
14. Volosevich, P. P.; Kurdyumov, S. P.; Levanov, Ye. I. PREPRINT IPM AN S55It,
No 40, 1970.
15. Nuckolls, J. ; Wood, L. ; Thiessen, A. ; Zimmerman, G. "Laser Campression of
- Ma~ter to Superhigh Densittes," PROCEEDINGS IEEE QUANTUM ELECTRONICS
CONFERENCE, Montreal, 1972.
16. Zimmerman, G.; Wood, L.; Thiessen, A.; Nuckolls, J. "LASNIX, A General
Purpose Laser-Fusion-Simulation Code," PROCEEDINGS IEEE QUANTUM ELECTRONICS
CONFERENCE, Montreal, 1972.
17. Thiessen, A.; Nuckolls, J.; Zimmerman, G.; Wood,~L. "Computer Calculation
of Laser Implosion of DT to Sup er-High-Densities," PROCEEDINGS IEEE QUANTUM
- ELECTRONICS CONFERENCE, Montreal, 1972. _
18. Wood, L.; Nuckolls, J.; Thiess en, A.; Zimm~rman, G. "The Super-High
- Density Approach to Laser-Fusion CTR," PROCEEDINGS IEEE QU~+NTUM ELECTRONICS
CONFEREHCE, Montreal9 1972.
19. Volosevich, P. P.; Degtyarpv, I~. M.; Levanov, Ye. I.; Kurdyumov, S. P.;
Popov, Yu. P.; Samarskiy, A. A.; Favorskiy, A. P. FIZIKA PZAZMY
[Plasma Phy~ics], Vol 2, No 6, 1976, p 883. .
20. Book, D. L.; Ott, E.; Sutoh, A. L. PHYS. FLUIDS, No 19, 1974, p 676. _
- 21. Ott, E. PHYS. REV. LETT., No 29, 1972, p 1429.
22. Bashilov, Yu. V.; Pokrovskiy, S. V. PIS'MA V ZHETF, Vol 23, No 8, p 462.
_ 23. Taylor, G.I. PROC. R. SOC., Landon, A20 1, 1950, p 192. ~
24. Davies, R. M.; Taylor, G. I. P ROC. R. SOC., London, A200, 1950, p 375.
25. Frieman, E. A. ASTROPHYS. J. , No 8, 1954, p 120. -
26. Layzer, D. ASTROPHYS. J., No 1, 1955, p 122. -
~ 27. Birkhoff, G.; Carter, D. J. MATH. MECH., No 6, 1957, p 769.
28. Chang, G. T. PHYS. FLtTIDS, No 2, 1959, p 656.
' 29. C~handrecekhar, S. HYDRODYNANII C AND HYDROMAGNETIC STABILITY, Clarendon
- Press, Oxford, 1961.
30. Gasilov, V. A,; Goloviznin, Q. M.;~~Tishkin, V. F.; Favorskiy, A. P. _
PREPRINT IPM AN SSSR, No 119, 1977.
- 31. Mead, W. C.; Lindl, J. D. University of California. Preprint No UCRL-77057,
19 75 . _
47
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. r'UK Ur~r~ll:l[~L ua~ UNLY
32. Volosevich, P. P.; Gamaliy, Ye. G.; Gasilov, V. A.; Tishkin, V. F.
pREPRINT IPM AN SSSR, No 24, 1978.
33. Gamaliy, Ye. G. KRATKIYE SOOBSHCHENIYA PO FIZIKE, FIAN, No 5, 1976, p 23.
34. Landau, L. D.; Lifshits, Ye. M. MEKHANIKA SPLOSHNYKH SRED [Mechanics of
Continuous Media], Costekhizdat, Moscow, 1954.
35. Belen'kiy, S. Z.; Fradkin, Ye. S. TRUDY FIAN [Works of the Physics Institute
of the USSR Academy of Sciences], No 29, 1965.
36. Bokov, N. N.; Bunatyan, A. A.; I~ykov, V. A.; Neuvazhayev, V. V.;
Stroptseva, A. P.; Frolav, V. D. PIS'MA V ZHETF, Vol 26, No 9, 1977,
p 6 30 .
37. Goloviznin, V. M.; Tishkin, V. F.; Favorskiy, A. P. PREPRINT IPM AN SSSR,
No 16, 1977.
38. Garabedian. PROC. R. SOC. LONDON, A241, 1957, p 423.
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UDC 532.5; 519.6 ~
MATHEMATICAL MODELS OF THE FORMATION OF TORNADOES AS A RESULT OF THE DEVELOPMENT
OF GAS DYNAMIC INSTABILITIES
[N. M. Zuyeva, V. V. Paleychik, L. S. Solov'yev, pp 65-105]
A study is made of the development in time of axisy~etric convective and helica.l
- instabilities of an ideal gas. By numerical integration of the equations of
hydrodynamics it was demons trated that the development of th e instability can
serve as the mechanism of generation of high angular velocitias of the gas. A
study is made of the effect of the vari ation of the parameters of the initial
steady-state on the specifics of the process dynamics.
Introduction
If it is assumed that the rotating formations~ of the atmosphere such as tornadoes
. arise as a result of the devc:lopment of gas dynamic instabilities, then, in par-
ticular, convective instability~caused by the growth of entropy in the vertical
direction, Rayleigh ~nstability connected with a decrease in the rotational
moment with respect to radius and helical instability naused by a decrease in
verti~al velocity along the radius can be possible.
Each of these three problems can be formulated as the problem of develop~ni: of
an instability in the one- dimensional equilib rium configuration having axial
symmetry. If we limit ourselves to th~2 investigation of axisyimnetric motions
_ in the first tw o cases and helical motion in the third case, the prcblem of the
development of instability reduces to a two-dimensional problem for all three
cases .
In the investigated mathematical mc~dels it is assumed that during the development
of the instability it is possible to neglect all oi the dissipative processes and
the thermal conductivity and consider the gas to be idaal. In the inirial
equilibrium state the investigated volume of the gas is ass umed to be included
within an impermeable cylindrical cavity. In this case the prob lem reduces to the
solution of the equations of Euler motion with boimdary conditions of vanishing
of the normal velocity component. The time problem of the development of an
instability is solved on a computer with assignment of the two-dimensional initial _
- disturbance. As a result of the development of the instability, the initial _
equilibrium configuration b ecomes a"quasi-steady" configuration characterized b~~
- concentration of the rotat.ional moment and the presence of ineridional motion.
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r~ux urr~lt;tEU, u5r: UNLY
Chapter I. Axisymmetric Instability
~1. Statement of the Problem
tlnder the asstmiption of axial symmetzy, the equations of motion of an ideal gas
in the gravitational field -g~Z is described in the form [1] -
d q ~-I 'T r
y ~
~G ~ / 7r v,- ~ ~ ~r~~.:.-~' ~ ZJ~ v.;r= . ~ -
' ~1~
2
~ r, ~~N T' p'j,~z r
r
t11~b'l.~ _ -D'~.,~ ~3 , dt t j~~U~ _VN L_C__ f, , -
where p is the density, p is pressure, v is velocity, N=pp-Y, I=rv~,
~ Z ~ -,:~p f ~J~ n'f`1 ) ~i!!l ;C7?I�= ~ _,~~11� Zt7 t % 7i''. -
~J L~
y i.s the adiabatic exponent. Obviously, instead of N and I it is possible to use
arbitrary functions for them which satisfy the same equations.
~Jhen investigating the development of a- convective instauility, we give the
initial vertical equilibrium state, which depends on one constant parameter
YO in which the temperature T=1II p/p decreases linearly with respect to z:
. k
I ' ~
--p ~-,T,-= ~o ~ ~
r
o
~ 1 ~
� ~ ( 2 )
/ D ~O ~O
This equilibrium is unstable if y0>y. If we take the density p~ as one and the
speed ~f sound c~= Yp~
p~, then the initial density and pressure distributions
will be ~,o -
- fJ=f~ ~ ~ _ '1__r. c 3~
~
AS the boundary conditions let us take that vn vanishes at the boundaries
= r-a and z=c; b, from which it fullows that N=const for z=0, b, I=const for r=a.
For investigati~n of the development of the RaylQigh instability it is possible
to give the rotational velocity in the form -
r/"~ yn z(~ t~ ~~~.~,r~~~'_l c4) ~
In this case v~=0 for r-a=1, and for specially large n it is characterized by
one parameter y. If we neglect the gravitational force ~ and assu.me the entropy _
to be constant, that is, p=pYO /y, then the equilibrium density distribution
corresponding to (4) will be
F
~i: P/�~,"'
. ~ ~ l e't ~ _ ` ~ ' I Z. - z ; ~ I~. 7 ` ' ,C( ` I
J ~ 5 ~
_ - A1 r. rr( r .3 ' (~''K/ 2hf `d , . _ , -
~ . a~
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The initial veloc}ty disturbance satisfying the boundary conditions vn=0 and
_ the equation div v=0 will be given in the form
~
~
. ~ / ~ a~ z j~
_ - a i ~ ~
_ ~ G-~, ~ x " ~ ' 4t, ~ (6)
- T;ze parameter a characterizes the velocity disturbance amplitudee Z'he current -
lines `Y=const are shown in Figure Sa.
In the problem of convective iastability the initial rotation is considered as
a disturbance. If we set afl=1, then the problem will contain two dimensionless
parameters b=bp/a~ and g. Here if we take the dimensionless values for the -
speed of sound cp=340 m/sec and the gravitational acceleration gp=10 m/sec, the
dimensionless unit of time will correspond to the interval 6t~=34 g sec, and
the cylinder radius a~=11.6 g km, where g is the dimensionless parameter of the
p rob lem.
~ �2. Conservation Laws and Energy of Instabilities
Under the adopted boundary condition the system of equations of motion (1) con-
_ tains the laws of conservation of mass, energy, entropy and angular momentum
a ( 7)
~ p ~ - L'onst' ~ ~t _ C.nnst~` _
The angular brackets indicate averaging with respect to the volume V=~ra2b.
The development of the instabilities is characterized by the ~rowth of tre
kinetic energy of the instability
Z
r~~ = ~ ~ ~ _ ~ ~ n /~zZ ) ~ '
~ /
- where, as the results of the numerical calculation show, the curve,WI (t) has
a maximum, that is, the growth of W~ is limited. A cantribution to the energy
of the instability Wl can be made by the thermal energy WT=V< gravita- ~
Y
- tional energy Wg V an~ rotational energy WE=V.
Here we sha11 present some restrictions on the maximum possible consumptions
of potential energy Wm WT+Wg+W~ following from the conservation laws (7).
Fro~} the conservation of mass and entropy we have the Gonservation law
_ =const, by using which it is possible to obtain the limit on the t~ermal _
energy flaw rate [2]
-d 11J~-~= -~-~_`J' 1~`;a~'-0 (see [1]).
2. In the case of the rotational configuration with 8/8z=0, the solution of
(17) has the form
- ~22)
. -:1~Z~~oS K+~~ ~ ~ r(2~ Jin
- ~ ~y
where
7 ~?~~�~~~[/,~1~~~,~~,. ~23~
w ~f K2G ~~J"'"" ~ ,
~ ~ C~ 4
and f(r) satisfies the condition
_ ,`~~c 1 ~ ,J� _ ' v ~ K~Cr t / c Y
' . ~~t/_~O k t~l~`0 ~24~
. ~ . ( ,rtc8 ~ ,~~~'~w ~
- with the boundary conditions f(0)=f(a)=0, the parameter k='rrm/b, m=1,2,3...
From (24) we have the stability condition j4]
:,;~%`ll;)~
~Z U'~~Cay~' (25)
which, considering the equilib rium equation it is possib le, analogously to the
preceding case, to wriQe i terms of the "f rozen functions" I=rv~ and K=N-1~Y
=pp-1/Y in the form (I K) >0. I~ the case of constant entropy K=0, and also
for an incompressib le liquid Y-~, the stability condition becomes the Rayleigh
number (I2)'>0.
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~
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3. In conclusion, let us construct the class,of stable rotating configurations
localized in space. Let us represent the condition of stability (25) in the
form
- ~ r 3(i7 ` ~1 o
(25a)
and let us set
, a_. ~y_ ~ ~ -
~ ' t26>
where e>0 is the stability index equal to e=(1-1/y)p~v~2/2, where the character-
istic scale of the pressure variation p(r) is taken as the unit length. For
p~=1, p~=~y, according to (26) and the equation of equilibrium p'=prv2, we obtain
the distributions v~(r) and p(r) in the form ~
_ - ~ 1y~ ^ v 2~~
!J' ~i~'Z~l .
f ~ a � ~7t~ ~ '
L~t v~~~4~C~'~ - ~r Y; ~1 J (27) -
~ ,
112 ? ; 7 `~>'�r` ~ ~
~~~r.~~~~~ Az~~1-~ r' - z.~ ~ 1.~ j~ a~.r:,
4
where p(r) is an arbitrary function. In the obtained class of one-dimensional
configurations which depends on one dimensionless parameter v, with an increase
in v, the stability increases, the depth of the hole increases p(r), and its -
radial dimension decreases. If the characteristic radius of the configuration
; rp is dimensionless and R is dimensional, then the dimensional angular velocity
of rotation in the center v~=vc~r~/R, c~=YpO/PO'
In Figure 2 we have the stable equilib rium distrib utions of v(r), p(r)
described by �ormulas (27) for p=3/2, Y=3/2, 1 for values of ~he parameters
v2=1, 2. The configurations more concentrated and with greater angular velocity -
of rotation are more stable.
�5. Variation P rinciple
_ For investigation of the stability of steady-state rotating axisymmetric
configurations it is convenient to use the expression for the variation of the -
potential energy 8W=w2lpv21dv. Multiplying equation (17) by pv~ and integrating -
over the volume V, we obtain
. 1�~ ~'~J~1 ~:c-, { �-'-1'.... :'~V;) ~:"~~~f;{~'t '~z~;. , . ,
J 1' j0 r %'3 i ~ Y..t~ ~j; 1 ~28~
i ~
- Minimization with respect to c~;� G~ gives
r~'e'7l~ ,l~ - i.'~= ~T~;)J. ~~;r~,
U 1 j
f Sr~. r~~� ~ ' ,
iJ "�1l' .r G � ( 30 > -
_ '~l~ z. t z~ . ~s ~
In the limiting cases of one-dimensional equilib rium a/ar=o and a/aZ=o the
dependence of the eigenfunctions on r and on z is known (see (19) and (22)) and
equation (29) permits expression of v in terms of vz or vice versa
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r'ux urrl~iEw uar; UNLY
�
~ ~ . ~,v
~ ~ ( >
,~lf~(~?) _ . ,E L~ , L~:~ /1, 31 =
U'
Substituting (31) in (30), we obtain the variation expressions for the minimum ;
s quare frequency w2 for both limiting cases '
~ ~'r'J i,. ,~r,?~/
i.. 1 f:; ~ ~32)
l i~'". r ~ _1.~,c', ~ 3 �i
. �~.t;~ ----.._.11._.__. . _.a~.
Therefore we shall select the function
i ;_...lf~/.J-Z.Zi~~~"~~, (35)
~ . . ~ , , , ,
which corresponds to the ~asic mode with ~~~pect to r and find k from the condi-
tion of localization of w. 13eglecting p in the second expression of (32)
we ob t ain
� ~l!, rz ,
az
f ~ ~ / y'
- ~ . � ~
~ ~~rT
~ ~ .r..._. f ~
_ V~~'' ~ ~
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t,~~ ~ rI
1
~
~1
3 ~Y
~
_ 1
0 0,~ 2
1
- Figure 1
~ ~
~ - y=2, ~,4
, 7{, p �
>>~n ~ ~
_ ~ i'
i'
. ! ~ ~
f ~ !
/ .
~ ,
i 1~.. ~ ' ,
~ ,,~~f
.,I~ --___~'~,~{-%3
' ` ~ .~~~..~.~..-r~ . ,1s ~ ' v = ~ �
_ ~ � ~ ~ r e
~ �
~ 1/ . ~ " "'^r~...~ ~
,~~',.r~' ~T.r~
1 ~f~.~.~~.~ . ~y"-/y�4~
~ - - �__--~---7 ~O
~ ; . 3 5 fr ~
_ Fi.gure 2
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FoR or�r�1c;laL u5r: or1LY
The detelmination of the integrals for f(r) defining in (35) leads to the following:
- ~ ~
' ti 1 i{. ~ .
, ~ i~ � ) '
L-- ~.r
t . ~ l.I ~ '
[three words illegib le] Q gives Q=4(1+~) [letters illegib le].
Thus, setting [formula illegible] and negiecting the variation of p, we obtain
~-Gl~ ~ _ - V`~L/~ .
~~~z (36)
1 t~Q~z
By complete analogy with the preceding case of (33), the increments of the basic
mode not having nodes along the main direction r increase with an increase in
the number of nodes m along the second direct-~on z.
3. In the general case of two-dimensional equilibrium, assuming smallness of the
derivatives with respect to z and with respect ta r and selecting the dependence
of v on z in the form of (22) and also setting p~pY~, we obtain
_~Z_- ~2 A~ r~-=-~~'~`A/~
,t~cz~1 o~f~ z`'
-u~ _ - %Z e ~ 37~
= f
~ ~f ~~E ~ ~ z
where k=mrr /b .
a) For the case of uniform rotation I=vOr2, selecting the trial function
f (r)=rJ1(xlnr/a) , we obtain
a,z,' ~2 ~ i ~
z ~zma,~.(~"_~~~y~'� ~38)
_.GL~ = z s T ~
- ~j x>;~ ~ %J~' //I . a. ~
Thus, the uniform rotation is the stabi?izing facto~ and leads to a sharper
increase in the increment with an increase in xZn/m.
b) In the case of destabilizing rotation (34), the use of the test function (35)
with R,=10 leads to the expression
. f 2 Z 2
- -~.v~ L~~:~.~_. ~~_~~r~a/~~1
~~p 2 (39)
( r~o.i.:T~j'
Consequently, if the convective increment of the development of the stability
predominates (1-Y/YO~gZ~c2>vp2/2, then -w2 decreases with the mode number m, and
otherwise, -w2 increases with m.
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�6. Results of the Numerical Calculation
- In the simplest statement the problem of the development of a cortvective instabil-
ity contains three parameters: the mismatch characteristic y~-Y, the size rgtio
b/a and the gravitational accele~ation g.
In order to study the dynamics of the development of a convective instability,
determine the role of the above-indicated parameters and also to discover the
- effect of the initial, tmiform rotation on its development, the system of equa-
tions of ideal gas dynamics in the cylindrical coordinates (1)
- - ~~r' ~ ~r~~~ , ~i ; G~~ -
~C/1~'- n -~~T 7i ~',r G,
~ c~
~~~Lr _ _ f dL 12 C4~~ ~ _ ~ 9 .
`~~z 4 ~ ` ~~4 ~ V~ c;~~ ~:~.3 ~ vr.`~ v ~ ` ~ ~ ~~z' ~
~
were integrated numerically in the region:
~ ~ i~ ' ~ ~ ~
_ with the following botmdary conditions:
Fo r - G, 'r = ~?R j~ ~J. G}~'~ ~ ~~2: I l:~1 r.= G
~
For :'7~ 1~= G'~;�~Sl_' ,
Fbr z. ~ Zfa = C7 /L'=0, equation (43) becomes
,,�Z
f' i m 'r~2/~y~c ~~j1 ~l~s~tc~~~~~_ .~Y~
/ ~ GtJ Z ~ 71'~ A/ } (~s~a)
~
Hence, we have the required stability condition of I5]
Z ~~�~z22~~~~:z_ Jl,f/ 7 0
?l~ ~ ~y y?1.' ~ ~ ( 44 )
in which the term ..N' describes the stability condition with respect to the
convective transport to the centrifugal force field, and it disappears for an
incompressible liquid (Y-~7. The remaining terms a.re caused by instability as a
result of j=rot -'v-~0. The stability condition (44) can also be represented in
the torm
(Iz,/Y-~)~>D
' (44a)
where it~ is expressed in terms of the frozen fimctions N and I.
For the steady-state configuration p=const.
_ 1I~ =1- z2 ?J'~=d z/t z z~, P=/~o ~ 6 ~1-~1- zz~ 3 c>
J 45
we obtain the following expression for the stabilit_y criterion (44)
~
-.~~0~2 ~ z - ~-~2 ~ZJ'~~ O, ~46~
Here the first term describes the destabilization as a result of the decreasing
longitudinal velocity vZ(r), the second term, stabilization caused by an increase
- in r~~ (r), and the third, convective instability. If the inequality (46) is not
satis~ied in the entire interval 0
Separating the variables in (4.16), we obtain
a"a' _ _J~ (4. is)
- i_
~ _ u'~ ~x/' ( 4 .19 )
_ Dividing both sides of equation (4.19) by u2, using (4.17) and integrating with
respect to m, we have
_ + u a u'' Rz d
u--~ ~Ry+ 2J ~
u ~m a�~ - 1 a
where v2 is the integration constant. Finally, taking R as the independent
variable, we obtain the equation for the function u(R)
R Q R aR +(~iR~-2.~L+ ~a:~u = 0 (4.20)
This equation defines the amplitude structure of the solution
A~~r~x~-4ul a~ (4.21)
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Then, from (4.12) , (4.14) , we have
Ci~
Q r (4.22)
and, consequently, comparing the (2.4), (2.5) it is possible to define the
phase in the form
~
S(r x) = a i+ S~.~ dx + coni~ (4. 23)
0
Thus, the class of partial solutions represented in the coordinates (m, z) by
_ separation of variables is determined from (4.21), (4.23) where u(R) and a(z)
satisfy the equations (4.20) and (4.18). By analogy w~th hydrodynamics these
- solutions will be called self-similar [13]; :in essence, these solutions describe `
the spherical waves with variable radius of curvature j2], [21].
For self-focusing in a cubic medi~ when z=z~ let the focal point be formed on _
the axis. We shall assume that generally speaking, not the entire beam is
focused on the point, bu nly some part of it, ccmtaining the power mk. In the
coordinates (m,z) the pro s of focusing part of the beam is written in the form
rc~,2) 2~Y� (4.24) -
for all m from the interval 0
The f actor f is the compression coefficient (f1)
of the transverse coordinate of the given ray. Substitutin~ f(m,z) f.rom (6.8)
in (6.2)-(6.5) , we obtain the following problem. The equations (6.2)-(6. 4)
reduce (after exclusion of p from (6.2)) to one nonlinear, fourth-order equation
ad = F . a-,n' ~ a~~� ?n~ (6. 9) .
with the boundary conditions
S~m,a)=1
(6.10)
~ (~~Q) _ ~o(~x (6.11)
1", (rr?)
In this form the problem is much simpler than the initial (6.2)-(6.5). However, ~
let us turn attention co the fact that the boundary condition for f:(6.10) does
not depend on the specific form of the beam for z=0, and it has an especially
simple form. Let us consider the variation of the function f for z>0. Its
derivatives with respect to z obviously characterize the focusing or def ocusing
rate of the given part of the beam. At the same time its transverse derivatives -
(with respect to m) characterize the transverse nonimiformity of the self-f~cusing
process, that is, the dependence of f on m describes the aberrations themselves.
The special case of the aberration-free self-focusing corresponds to independence .
of f with respect to m
~(fx,xla ~~x) (6.12)
In tl~is special case the ray tra~ectories r(m,~) are represented by the f unction
witti sep arated varia~ les (are "similar"), and the solution of the initial prob lem -
is self-similar (g5). Near the bound ary of the nonlinear medium on the b asis of
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rux Urrll,lt~L u~~ ~~vLY
- the condition (6.10) with monotonic phase distribution uniform ~vith respect to
- radius, it is possible approximately to consider the self-focusin~ aberrationless.
The approximation by the condition (6.12) of the process on the whole leads to
aberration-free approximation [23].
'ihe above-substituted problem of aberrations, hawever, requires a more detailed
pi cture and consideration of the variation of the function across the beam. The
nonuniformities of f, sniall for z-0, are accumulated during the self-focusing
process, naticeably distorting the shape of both the amplitude profile and the
phase front. -
The problem of calculating the beam trajectories can be simplified significantly
if we consider the nature of the process. Let us estimate the transverse
derivatives, entering into the righthand side of equation (6.9). We have -
a~ ~~f~~.sr a`{, ~n~~~+~r (6.13~ -
a
M. ~ ' ~ ~ ' a m' M~- _
where ~ f = ~ ~~rn,2) - ~ ~m~~)~ � ~ f ; - o
~~M~f I
- In (6.13) estimates are presented for the derivatives with respect to the cross -
section of the entire beam as a whole. Hawever, the b eam experiences the greatest
distortion for formation of the focal region and the derivative of f with respect
to m are larger here than in (6.13). At the same time, on the periphery of the
beam the transverse variations are smoother; correspondingly, the derivatives of
f with respec.t to m are smaller here than fram the estimates of (Fi.13) .
Then we proceed as follaws. In the region of peripheral rays we estimate the
derivatives 8f/am, 84f/8m4 by relations analogous to (6.13), assigning them, ~
however, a defined order of smallness ~
a (e )�a,x a~ ~a f )M4x (6 .14)
_ a ~~t M aM~ ~ M~
Then let us substitute these derivatives in the equations for f and let us drop ~
the terms of higher order of smallness. Thus, we obtain the truncated equation
describing the trajectories of the peripheral rays.
From equations (G.8) and (5.2) ~ae have
_ S = ~ _ Qo
~2aM z ~ i,r~ a3L ~
� am
where
Z
_ ~gorL a~ _
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~
In the last equality we estimate the term ~per~'~~..~. . For beams of monotonic
~
smooth profile (in particular, for the gaussian beams) the functions
(1!2) p0 (m) r~ (m) has a unique maximum at a distance on the order of the radiua
of the beam a from the axis, and it reaches the values of
a ! 4. !
~~4.r ~ ~ E,a =2M
L � 1~
(ED is the intensity on the a:sis). On the other hand, setting (~f2)m~~f2, we _
have
a ~Z
a r?t N t"9
Thus ,
~
- ~=f~'+~~o:''a~=~+~M~yM~'_~z~ -
and p $r~ -
('v.15)
- Now let us consider the derivative ap/am
a4 _ ~ a~ _ ~ a~
am - ~r aM g am
Let us estimate 8g/am. We have
/ ~ z a~~i
.a-a~ =a
~~~+a
il2.s�r� 2~�r� an?L
Inasmuch as
- a4. _ _ E'' -
a~
- M
- we obtain
i+ �,~i4~o'' =2+ir''a4a ~ 1 2 EQ
~ 2�~-2-~r,-~`
The initial quasicoptical approximation is vaZid if r~ exceeds the radius a by
no more than a few times (r~�ka2) . Consequently, .
2- 2 C;'' M2 = 2- 2 ctz E; M=~
Thus ,
am = a~,y+ Zp�r~ a~ = ~
~
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- C~mparing the terms in the expression for ap/am, we have (inasmuch as g=f2)
~aso__~ Ea. a~~ = E~' J~ ~Z
~r ~ -~i t~' \ g~i am ~4a ~ M ~
and for the derivative 8p/8m we obtain
aQ ~ ! ~a4o ~6.16) -
' a r?~
As a result, the first term in the righthand side of the beam equation (6.3)
corresponding to nonlinear ref raction assumes the form
i ap~~ a~
2 yr a~ - 2~' a m

- Analogously, it is possible to obtain the approximation for the diffraction
"force" =
_ , ~4" ~?~s M"1~ +~~"aM~2]
For this purpose it is necessary to estimate the derivatives 82p/8m2 and -
83p/8m3 entering into it similarly to haw this was done ~or the first derivative ~
8p/ 8m. As a result, we obtain -
a 1 e a~ a~Q,
1aS1 ~ a -
a rK ~ a rn f -
and, cons~quently,
(6.17)
ra~,' = p_' aw-�
~ ~ ~a am
where ~ 2
- '~e =1~(?'?~~o) _ +~[4.+~p~ a~n. ~~"am. ~
The appro~mate equation for f assumes the form -
d'~ i aw~ (6. ~s)
~=~4�~
The truncated beam equation will be obtained, returning from f to r(m,z)=r~(m)f
- (m,z) in (6.18).
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= �7. General Solution of the Simplified Equation. Aberrations During Self- -
Focusing of Gaussian Beams. Results of Numerical Integration
The simplification of the equation (6.3) of the ray trajectory performed in the
precedi.ng section consisted in replacing the differential dependence on m
entering into the equation by the parametric function (6.18). Returning from f
to the ray tra~ectories and denoting them as before by r(m,z), let us write the
truncated equation of the ray trajectories
daz ~ (~.l)
r
where
~ G(~) _ - :~,Q~ ~
( 7. 2) ,
The first approximatio~ of (6.5) was written for the intensity p(m,z). We obtain
the next approximation, solving (7.1) and defining from equation (6.2)
4�E�~1-M~, M=~EoaZ>Mk ~
' ~rQ~m)=- 2~Qh~~.- (7.8) -
M~
1!', (M ) = p
The solution of (7.7) is written in the form
r(m~z)=r;(rn,~)~Q (rn_M~)Za.+11 (7.9)
J
where M' =M-1 (see (6. 7) , (7. 2) ) .
2
For 0M' and dr/dz
y where
rk=ro~M~l, 1J'~=~o~M~), SK=p,~Mk), Ur4=ur,(~~~
and z is determined from (8.2). As an example let us again consider the self-
focus~ng of the gaussian beam by a plane phZSe frant. In this case (8.2) has the
form (see 7.9).
Q~~~MK+MK)-M)z? ~ ~ = O
- and for the focal length we have
A~
2 = ;
? 2JMA-i.25
(8. 3)
_ In Table 4a the values of z~ obtained by formula (8.3) and from the numerical
calculations for gaussian beams of different power are nresented. In Table~4b
an analogous comparison is made for the focal length of the beam (7.13). The
nwnerical values and the values obtained from (8.2) are compared. The formula
for z~ is the following here
_ zt=a'' z.
i{yt (8.4)
2 ~ y i }y,,
where
y-~a~2-}~~ M"
(8.5)
Setting for M/Mk=2-10
y-2 M
= from (8.4) we have
~ .V a''(l+y~)~ !+y
_ (s.6>
? - 4 i-~y~
The results of the comparison indicate that iii the range of powers of practical
interest M/Mk=2 to 10, formulas (8.3) and (8.4) or (8.6) give values which are
equal to the numerical values with very good accuracy. The calculations for the
axial beatns of power significantly exceeding critical, do not appear to be of.
great practical interest inasmuch as the disturbances of such a beam lead to
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breakdown of it into "filaments" each of which contains a~ower on the order of
critical [28-30] . Comparing formulas (8. 3) and (8.4) , it is possible to draw the _
following conclusion: the focal length is determined as a function of the power
- to a great extent by the f orm of the amplitude profile.
- The f ormula for the focal length (the self-focusing length) is of unconditional _
interest for applied�problems inasmuch as it i~cl~des the values experix?entally
known or sub~ect to detennination (the power P, the distance to the focal region
z~, the nonlinearity coefficient e2, the beam radius a).
A study was made above of the beams with monotonic smooth distributions of
intensity and phase. The nonmonotanic profiles correspond to annular structures
- which can also be generated by diffraction on the edge of the "clipped" smooth
amplitude distribution. In these cases the behavior of the beam differs, -
generally speaking, from the above-investigated case of "monotonic" self-focusing
[31J. Using the solution to problem (6.1)-(6.5) here, we find that the defined
beam trajectories (7.7) for some z' begin to intersect, wnich corresponds to
going beyond the framework of the approximation adopted in g6. For z
= Jointly with the continuity equation (1.2) condition (1.10) defines the law of
variation of e _
,P d t- - P d
t ~ ;11
~ ~ (1.11) -
5. According to the principle of least effect according to Hamilton, the motion
of the medium takes place in such a way that the functional of the effect =
_ r`~~ ~ f assumes a stationary value [5 that is,
J
e,
t`(' ' 1S~ ~'V`+ ~~~e~~`~ S~'K- _
s~' = f ~1 ~.~�te ~Q ~X~ -
b'E]~` ~ N`H~ - ~~~F~~~fl`- (1.12)
- sc~~")
- gs- 4T
- ~ F~`f~` ~ ~X" ~ dR~ d - t' _
'~c ,~'rc
Usir~g the additional conditions (1.2), (1.7), (1.10) from (1.12) it is possible to ~
exclude the versions 6e , dvi, BHi, a(J~) , expressing them in terms of the _
variation dxl.
Equating the factors to zero for independent variations, we arrive at the eqvation =
of motion of magnetohydrodynamics: _
r.
' 13z .
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rcl U
~ ~ K ?f ~ ~ _ _ ~p ~l
- '~u l c~ i 2 ,r d ~ l 1'` ~ ~ _
'ii }~`.rj~c~ Nic ~f~e ,7~( (1.13) ~
1 -i- - _ !I_1_~ J Y
� 8~c~ ,-~X~ -
i
~ where p*=p+ ~iH .
8n
The equations (1.2), (1.7) or (1.8), (1.11) and (1.2) agree with the kinematic _
relations dxl/dt=vi and the equation of state p=p(p, e) completely define the
- behavior of the dissipativeless MHI3-medium for the corresponding initial and
b oundary conditions.
For distrib ution of the initial magnetic field it is necessary to s atisfy
obse rvation of the condition of solenoidality -
~i ~ )
d i er ~I - H;`c = y~---g-- ~
x; 1-I ~ 1.14
which can be expressed in terms of the fluxes ~i as follows
~c _ 0 ~1.15) -
~
g2. Liscrete Model -
1. We ~hall assume that S~q is a unit cube in the space of the Lagrangian
variables ql. In S~Q let us introduce the rectang,ular difference net with the
steps ~q1=h.. We shall index the finite-difference values by r_he Greek letters. -
- Let us place a triplet of natural numbers in correspondence to each node:
= ~a,,~. ~ ) E W h = 1 ~�~;p~~l ) % = 0, i, N~ f3 = 0, i~ , h,
4~...~ 0,~ .
The set of all nodes defining the f?nite-difference cell (elementary p arallele-
piped) will be denoted by W1, considerin~ that the index of the cell is equal to
the nodal index (a, S,y~ Wl, in which min (a+s+~y) is reached, The set of all
cells containing the ~iven node (a~s,y) as the anex will be called W2 (a,~,Y).
' Let us introduce the set of cells wn and all the in~ernal nodes wn, and also
the space of the finite-difference functions Rn and I~ defined in c~ and
respectively. -
The values of xl, vl, vi and gik belong to the nodes of the difference net,
den~ting them, respectively {(xl)asY} and so on. Then the relation between the _
co variant and contravariant components v will be written in the usual way for
cach node:
1J~~ = ~.;R 2I~K for (a~~i ) ~ ~.J~ (2 .1)
- (for simplification of the notation the index (a,S,Y) is omitted in the formula). -
133
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The thermodynamic values and also Hi, Hi, J and Jm will be referred to the
centers of the Lagrangiun cell and marked by the cell index. Since
{(gik)aSY} C-R~,t~ the relation between (Hi)aBY and (Hl)asY will be establ,ished .
by the expression
I~f;.1 1
~ ~ ~a`~~ _ ~ ~"iK ~dJ,y~' ~ K/a(5N (2.2~
where f~~i :K ~r ~j E~~ is an approximation of gik ar the center of the
cell, for example, of the type -
- ~ \ - ~ ~ ~~IK~J `
~~K/a~~^ 8 Je ~ji~a~pg) (2.3)
2. Let us define the difference analogs for the partial derivatives 8f/8qi.
For the Lagrangian cell (a, S,y) let us iiitroduce the expressions :
J yh~ ~ 7 a,~ci,~y~i~ tf*~'i ~�iri j
� f 2 . 4 1
( )
r L ~ L 1 ~i+i
~2 7~ 4hQ J ot~d~,~rJ3l, ~'+~'i t ~ -
ai,~~~ ~1, pi
~g ~ y h ~ f a+dt, ~,~l, ~,rs (-1 } 11 i
' ~+~,,p~, ~s = o t i
. i
where {fasY} ER,~' Expression (2.4) approxiMates 8f/aqi at the center of the cell. !
For sufficientiy smooth f:
I
I i
ac ~f a~~ ~ ~f12~ (2.4)
;
here ~he bar over the index indicates that the approximation is made in the center !
of the cell, h2=hl+h2+h3, ;
i
Let us note that for ~f the formula that follows is valid:
XK1 (2.S) i~
- 't~~ f - 3 ~ ~K~J ~ 1 ;
~E ll/, ~~t,~a~) i
The difference analog of the derivative 8f/8ql defined at the node of the diff~r- '
~ ence net will be introduced as follows : ,
) _ ~ ~ x ^ '
f Jd,P~ - - 3 ~ ~`~'~(X") ~ ~ (2.6) I
v E ~/Y(d~i~)
It is easy to see that 8if (f C-Rh) is an anproximation of af/aql at the node ;
with second order. +
_ ~
134
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~ .
3. Using let us carry out the digitalizatioi~ of the variables Ji and Ji.
- In accordance with Ji and J let us substitute the difference expre$s~~-ns
_ S'- and S obtained byiformal replacement of the derivative 8/aqi by
;
s~ = 2 ~i"'"e;K~ x'~ ~h x~
_ s_ C erm?i~J~e t~1_~ X K t~n X P t~r X ~ (2. 8)
!
Here J~ I~~r + O( h4~ ; S: ~ I~~ + 0(ht~ and the dif ference analogs of the
identities (1.6) o.ccurring in the differential case can be satisfied
( ~i X?na O;Mh'
sf (2.9)
in addition, let us note the equality
_ ~ ~ r 0 -
~ C x
~e W;
- Using (2.9) and (2.4), it is possiblp to demonstrate that f GRh and ~ERh, then f

, s t
` ~ ~
- ~ '~(xpJu SK a ~ ~,~X ~ ~~h
~E (2. io>
~ ~J ~ s
K a = ~ ` 7 S K ~J.r _ X + t7~ht) _
J F Wt ~d~, r~ ~,~r ,~r d~r
_ Consequently, the ex~ressions~ f J~~ and ~ c~~'sJ
[Ul ~~X~)J LL/R ~~~~.(ft
; can be considered as difference analogs of the partial derivatives 8/8-t~` reduced,
- correspondingly, to the cell and the node of the finite-difference net,
4. Let us discuss the pr~blem of approximation of the expression ~ in the cell.
Inasmuch as gi~ pertains to the nodes, the value of ~ at the center of the cell
_ can be defined, for example, as follows -
~ -
~E w, c�,p~~ rE w! ~c~~~
K~,~~ 7 J~. ~XK~ I~ * o fh`) t
+ S ~ (X K)v
t s~/,U K1U J~~~'i
X
r t 7l K~~ ) r 0( ht ~
8 1 ~~X'~~~ ~ x la~
J~ W~ f jt)
3. Let us digitalize the freezing equation for the magnetic field. The
difference expression
~ H K SR _ (3.8) -
on the basis of (2.8), (2.12) approximates the freezing condition (1.5) at the
center of the cell with accuracy 0(h2). It is possible to show that ~1 is
the "difference" fluxes through the planes passing through ttie center of the
cell perpendicular to qi. Performing the convolution of ~l~xk cansideri_ng
(2.9) we convert equations (3.8) to the difference analog of ~quation (1.7)
137
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~HK= xK (3.s~)
Differentiation of (3.8) with respect to time leads to the difference equation of
induction corresponding to (1.9)
K
dHR K dv t~c a~ cs.9>
(3.9)~ d~ _ - f~
- here by dV/dt we mra:. nne of the expressions (3.5) or (3.6).
From (3.9) and from the i.nduction equation written for the covariant components Hk
by staridard transformatic~ns we have the equation for the eRergy of the magnetic
fi~ld of the cell in the form
d H"N~ _ N"N~ dY + NK`~_~r
v; YH~H" d< (3.10)

- dt ~ 8r Y~ Bsi at 4~r tr d t
As is easily seen, (3.10) approximates (1.10) with accuracy o(h2). For
d/dt, just as for the derivative dV/dt, different approximations are permissi-
ble. In the general case, obviously _
tK~ _ ~ ~,U~ (3.11)
�L~- ~ l~
~ { J e ur! ~~p~)
If. is calculated by formula (2.3), then (3.11) becomes _
ol > ~ ~R~,
dc //r? r~~~;
- ~
Since l r tI (f1' ~ the second-order approximation for eauations
_ ~ . � )
(3.20) is easily estabZished on the basis of the formulas for ~2. In equation
(3.19) it is necessary to estimate the order of the approximation of the
expression
~
~ yl; ~a r ~ : ~ ! f, i~
~y f./ v' / 7 r/ ~~(J f~~(,r i ~ V r~ y/
.v ~
~Fll?~ f�~tj~' JE[lSl~~3y) " -
Using (2.10) and the fact that n i~ t olht) -
~ "P~ ~ ~6W~~aD~r)
we ob t ain
~~Y,, ~ p~ r- ~~P I t o(h`1=
~ Pv ~x~ = g ~x~ s ~x% ~~r
oEWf ~ ~ P� J"
- _ jPy~~~ P'~ ~ ~ o(~?`) - - ~ ~ 'I ~ o cti')
- l ~x~ ~
x~
Thus, the approximation of the dynamic equations (3.19), (3.20) is demonstrated.
3. The equation for the specif.ic internal energy can be obtained analogously to
- how this is done in -referer.~e I6] :
140
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dE dY m d P -
. at - Pdt - ,p` a~r (3.2~>
Equation (3.21) has entropy form; for dV/dt it is possible to use any of the
expressions (3.5), (3.6).
For calculation of the currents with shock waves accompanied by an increase in
entropy, it is necessary to introduce artificial dissipative processes. This can
be done by the recommendations proposed in references [13], j14]. We shall not
discuss this problem, �or it is a sub~ect of special study.
The digitalization technique with respect to time does not differ in any way from
the one developed in j6 j 7].
�4. Some Properties of the Differential-Dif.ference Equations of Plagnetohydro-
dynamics
In this section a study is made of the properties of the difference system of
- MHD equations for cases where the dynamic equation (3.19) is used, and the
expression for dV/dt is given from (3.5). All of the results obtained are
extended without difficulty to the case with equations (3.20) and (3.6).
1. Let us c~rite� the comp lete s~stem of differential-difference equations of
magne tohy drodynami cs :
a~ ~rK~r~a~
- M~dt - 2 ~x! ) -
- ` p~ _ ~x$~~>~ ( YH KNP)r _'aK 4r ~K ( 4.1>
~ )
Jtuilt~c~r) ~X 85;
~V K (4.2)
pY dt = - pdt = -P ~'a(xKJ~~~ ~d
~eWjc~~r)
YNK = a~ xK (4.3>
dH ~ ; dV , ~Kax,~~
- V ~f - - H dt
dt ` H~ KY / 9A ~dt + H"~~`?-' YK- gH"N~ d (4.4)
. 4 ~ ~ 4! a-`` _
- ~P . ( 4 . 5 )
~
~tx P= P r P? f~ c4. 6> -
_ Let us note that the system of equations uses the mixed components of the velocity
vectors and the magnetic field intensity vectors Gihich complicates their notation.
However, it is easy to reduce these equations to the form of notation in which
, ~
141 =
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_ only the covariant or contravariant comoonents were used; for this purpose it is
necessary to resort to the transformation formulas (2.1) and (2.1').
2. Let us consider the problem of the conservation laws for the system (4.1)-
(4.6). As follows f rom the differential dynamic equation ('.13), the law of
variation of the pulse of isolated volume of the liquid S2' < S2 has the form:
,c < R ~
, ~t Q. ~r ds1 = f p. ( z'~ + 8~}' ~ dlZ%
. ~~4 ~
(4.7)
~
- f P~~---~ ~ d..Rg _~(P�3' ~~,--N ~ d S R
'~x~ - 8~
~~g
- Let SZ' in the difference case correspond to the set
p7ti ={ai