Four-dimensional integral equations for the MHD diffraction waves in plasma
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irk-123456789-785402015-03-19T03:02:19Z Four-dimensional integral equations for the MHD diffraction waves in plasma Alexandrova, A.A. Khizhnyak, N.A. Вeams and waves in plasma 2000 Article Four-dimensional integral equations for the MHD diffraction waves in plasma / A.A. Alexandrova, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2000. — № 6. — С. 109-113. — Бібліогр.: 5 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78540 533.51 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Вeams and waves in plasma Вeams and waves in plasma Alexandrova, A.A. Khizhnyak, N.A. Four-dimensional integral equations for the MHD diffraction waves in plasma Вопросы атомной науки и техники |
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Four-dimensional integral equations for the MHD diffraction waves in plasma |
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Four-dimensional integral equations for the MHD diffraction waves in plasma |
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Four-dimensional integral equations for the MHD diffraction waves in plasma |
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Four-dimensional integral equations for the MHD diffraction waves in plasma |
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four-dimensional integral equations for the mhd diffraction waves in plasma |
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Four-dimensional integral equations for the MHD diffraction waves in plasma / A.A. Alexandrova, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2000. — № 6. — С. 109-113. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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UDC 533.51
Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 109-113 109
FOUR-DIMENSIONAL INTEGRAL EQUATIONS FOR THE MHD
DIFFRACTION WAVES IN PLASMA
A.A. Alexandrova
Kharkov Military University. National Center
N.A. Khizhnyak
National Scientific Center KPTI,Ukraine Kharkov 61108, Akademicheskaja, 1
E-mail: khizh@khizh.kharkov.ua
Introduction
Let consider low frequency oscillations those can be
excited and propagate in plasma. Therefore we shall
limit ourselves sufficiently slow development of
macroscopic processes. Such a supposition is necessary
for the possible application in hydrodynamic description
that together with electromagnetic one for the field is
expressed by equations of magnetic hydrodynamics [1]
relatively to medium velocity u r( , )t , intensity of a
magnetic field b r( , )t and a density of media ρ( , )r t .
In linear magnetic hydrodynamics a wave packet
contains seven types of characteristics: two rapid
magnetoacoustic waves, two slow magnetoacoustic
waves, two Alfven waves and entropy wave. The state
vector Ψ describing the considered wave packet at
each spatial point M x y z( , , ) and at any time moment
t is determined for initial and boundary conditions of
the above waves.
Under definite physical conditions, for example, at
the meeting of the solar wind with the Earth magnetic
field, or at sudden inclusion of an electromagnetic field,
or during collisions of two gas masses and so on arise
strong discontinuities under which not only derivatives
of the MHD-values are discontinuous along spatial-
temporary coordinates, but these values themselves are
also discontinuous ones. Jumps of MHD-values on a
surface of discontinuities are determined according to
integral laws of conservation or integral balance
equations. As to differential equations of
magnetohydrodynamics the solutions of those are
inaccessible at differential of discontinuous values on
the surface of discontinuity they can be represented in
the integral form [2] completely equivalent to
differential equations (induction and Navier-Stores
equations) and also initial and boundary conditions
above mentioned.
The questions of evolution in
magnetohydrodynamics
Here it should be noted two factors connecting to the
problem of evolution in magnetic hydrodynamics occur.
The first factor may already refereed to the classical one
and it has been considered sufficiently well in literature
[3] and we named it conditionally evolution of
discontinuities in space. As it seemed setting of
boundary conditions for discontinuities is not sufficient
to determine discontinuity moving of the MHD-medium
by the only one method. One needs to take into account
an increase of entropy and also wave stability in
reference to splitting it into several discontinuous or
automodel waves. Such waves in magnetic
hydrodynamics are called evolutionary ones. For them
infinitely small disturbances of MHD values evolves
with time remaining small. The nonevolutionary wave is
instantly splitted (in case of an ideal medium).
The problem of evolution of initial disturbance has
here a unique solution if a number of expanding waves
(a number of unknown disturbances) is equal to a
number of independent boundary conditions. In this
case the initial discontinuity is evolutionary one.
Otherwise the problem has either innumerable quantity
of solutions or the solution of this problem is
inaccessible generally, i.e. discontinuity is non-
evolutionary and splitted.
The evolution conditions of shock wave easily to
find, analyzing the linear boundary conditions, written
down in laboratory system of coordinates. In brief these
evolutionary conditions can be formulated as follows.
Relatively the Alfven disturbances exist two domains of
evolution u V u Vz
a
z
a
1 1 2 2> >, (over Alfven) and
u V u Vz
a
z
a
1 1 2 2< <, (up to Alfven). Here index 1
means domain ahead of a shock wave, and index 2
behind of it. V a
1 2, − the phase Alfven wave velocities
accordingly in medium 1,2.Two evolution domains of
shock waves the relation to magnetoacoustic and
entropy disturbances: the fast shock wave
V u V u Vz z1 1 2 2 2
+ − +< < <, and the slow shock
wave u V V u Vz z2 2 1 2 1< < <− − +, . Here V1 2,
± − the
phase velocities of the fast and slow magnetoacoustic
waves. As, V V Va− +≤ ≤ , then fast shock waves are
over alfven, and slow shock waves - up to Alfven .It is
necessary to note, that Alfven, tangential and contact
types of breaks are always evolutionary.
The second factor leading to the question of wave
evolution in magnetic hydrodynamics is connected with
taking into account initial conditions. It should be noted
that interaction of MHD waves with moving boundary
of two media is sufficiently dependent on moving
boundary formation (in other terminology - on the
surface of MHD-values discontinuity). It can be the
interface of motionless and moving media. In this case
motion itself is due to external sources. The velocity of
110
boundary movement can be relative arbitrarily to phase
velocities of waves in immovable medium. However, as
the boundary and medium move along one site from
non-uniformity the difficulties to determine the amount
of divergent waves do not arise, can be used the above
described principle of evolution of MHD waves.
The discontinuity surface itself can be the boundary
formed as a result of propagation of rumpling wave
changing medium properties but not actuating the latter
for movement. In this case the movement of the medium
in both sides is absent, the phase velocity of waves do
not depend on the velocity of the boundary movement
and the characteristics of a transmitted wave are
determined only by parameters of the medium.
Considered above two variants of forming boundary
movement reduce to varied non-stationary boundary-
value problems resulting in wave scattering patterns of
sufficiently different nature. There is a very important
non-stationary problem with taking into account the
evolution which may be according to above mentioned
description is called as the temporary MHD evolution.
In this connection the initial moment of the non-
stationarity is very important, that is the statement of the
initial conditions, which make sense to consider at final
temporary moment. Usually at research of interaction of
a packet of MHD waves with moving nonuniformities
assume, that the process if inclusion of a movement
occurs adiabatic on infinity. It for some processes can
be strong idealism. This idealism can set in loss
qualitatively of the new phenomena, connected directly
to occurrence of the of this motion. Any real
phenomenon begins in a final moment of time, the
initial conditions can be considered in a zero moment of
time.
Integral formulation of boundary-value
problem
In this case a differential formulation of the solution
of the boundary-value problem can leads to the
difficulties, connected, on the one hand, with the
indefinite discrepancy of the secondary waves to the
number of boundary conditions, that has coincided
fortunately for considering the spatial evolution. On the
other hand, it is due to purely mathematical difficulties
of mixed boundary-value problems solution.
Therefore it is meaningful to apply an integral
formulation of the problem, automatically including the
boundary and initial condition. This idea is confirmed
by N.A. Khizhnyak and Nerukh А.G. [4] successful
application of the non-stationary integral equations by
consideration similar of boundary-value problems in
electrodynamics. There chain of the integral Volterra
equations describe interaction of electromagnetic waves
with non-stationarity medium.
For analysis of non-stationarity boundary-value
problems in linear magnetic hydrodynamics let us use
the integral equations of non-stationarity magnetic
hydrodynamics in terms of the constant magnetic field
perturbation b r( , )t , medium perturbations velocity
u r( , )t and perturbations density ρ( , )r t [5]:
u r u r K u r K b r K r K u br t( , ) ( , ) ! ( , ) ! ( , ) ! ( , ) ! ( , ), ( ), , ,t t t t tr t
u
r t
b
s= + + + +0 1ρ ρ
where u r0 ( , )t is an incident field and
! , ! , !, , ,K K Kr t
u
r t
b
r t
ρ are the differential-integral operators
of distribution of the velocity, magnetic field and
density. Each of these operators can be represented as a
sum of suboperators, conditionally describing different
properties of MHD media, namely
! ! ! ! ;,K r t
u
S
u
A
u u= + +Γ Γ Γ ν
! !
,K r t U
ρ ρ= Γ ; ! ! ! !
,K r t
u
S
u
A
u u= + +Γ Γ Γ ν and ! ! ! .,K r t
b
A
b
U
b= +Γ Γ
Here !ΓS
u is responsible for pressure of conducting
media and operates as follows
( )! ! graddiv uΓ ΠS
u
s SV V Gu = −1
2
2
2 ; the operators
! , !Γ ΓA
u
A
b are responsible for a magnetic tension of MHD
media and look like
! ! , rotrot , , ! ! , rot ;Γ Π Γ ΠA
u
A
u
A
b A A bG
B
B
G
V
B
V
B t
u V s s s b s s2
2= −
= − −
1 1 1
2
1
2
1
2
1
1
2
2
2
∂
∂
the operator [ ]! ! , rot uΓ ∆Πν νu
m
AG
V
B
u s= 1
2
1
1 account
presence of magnetic viscosity of media; the operators
! , !Γ ΓU
b
U
ρ are connected with macromovement of the
discontinuity:
[ ] ( )! ! , , , ! ! grad ,grad .Γ Π Γ ΠU
b b
U
SG G
V
b
B
U U= −
= −1
0
2
2
1
04π ρ
ρ
ρ
ρ ρrotrot
Here, MHD-potentials of velocity, magnetic field
and density are defined by analogy to the Hertz
potentials in electrodynamics
( )
Π
Π
Π
u
b
V t
t
t
t
dt I t t
t
t
t
d
( , )
( , )
( , )
' ' , '
( ' , ' )
( ' , ' )
( ' , ' )
'
( ')
r
r
r
r r
u r
b r
r
r
ρ ρ
= − −
−∞
∞
∫ ∫ (2)
And, at last, the operator ! ( , )K u bS is defined by presence of surface currents.
111
The kernel of the integral equation (1) is the
Green’s function ( )! ' , 'G t tr r− − for a free space,
determined by parameters { }B 1 , , ,V VA S1 1 1ρ . Such a
representation is typical’s for the integral equations of
scattering problems. The Green‘s function of MHD
linear media is written down in a basis, connected with
an undisturbed magnetic field as follows:
e e e e s B 11 2 3 2 1 1, , , /= = B .The complete its
description has been given in [2].
Equation (1) is fully equivalent to the corresponding
differential equations of magnetohydrodynamics and the
initial and boundary conditions at the surface of
inhomogeneous, in laboratory system of coordinates.
Formally, the equation (1) can be considered as a
linearised equation of magnetic hydrodynamics with by
nonlocal boundary conditions, written in laboratory
coordinate system. At the boundary-value problem
solution in differential formulation the local boundary
conditions can either be satisfied for waves of the same
style, or their satisfaction needs usage of several modes.
For integral formulation this difficult question is solved
automatically, which is caused by physics of the
phenomena.
The integral equations (1) contain whole information
on scattered waves for the following problem
formulation. Assume, that some inhomogeneous, which
is characterized by parameters: B 2 , V VA S2 2, ,
ρ ν2 , m , occupies a volume V t( ) with time-dependent
in a general case boundary. Assume, that considered
inhomogeneous is placed and move uniformly with
velocity U 0 in unbounded MHD medium, described by
parameters B1 , , , ( )V VA S m1 1 1 0ρ ν = before its
excitation by the incident field u r b r0 0( , ), ( , )t t
correspondingly.
Let carry out an analysis of ratio (1) in frame of a
non-stationary problem of diffraction on the
inhomogeneous of the volume V t( ) using notations
and terminology assumed by A.G. Nerukh [4]. So,
general ideology is as follows.
Assume, that the object of diffraction before a
moment t = 0 is described by parameters
B1 , ,V VA S1 1 and ρ1 . At temporary moment t = 0 the
state of an object is changed for
B 2 2 2 2, , , ,V VA S mρ ν and U 0 . Change of the state
results in that, (1) disintegrates in the chain of
evolutionarily connected expressions.
There are three temporary intervals for the internal
field. At t < 0 the four-dimensional interval of
integration of equation (1) for temporary- spatial
coordinates is unlimited as the integration is made for
the crossing of the region of a transmitted light cone
with the top at a point ( )t ,r and into region given by
the characteristic function
( )χ t
V t
V t,
, ( )
, ( )
r
r
r=
∈
∉
1
0 .
After a zero temporary moment the part of the
region of integration limited by hyperplane appears
t ' = 0 . How the object is in a new state. And two
regions of integration appear. One of them is completely
in four-dimensional spatial region of events
corresponding to the object of diffraction. There is no
effect of boundaries of a diffraction object and one is
taking into account only change of medium properties in
pure appearance. If the equation (1) is used for all the
four-dimensional space intervals then it will describe
the field in unlimited medium with similar properties as
the medium within the object.
Starting from the moment ( )t
u
d=
1
r , where
( )d r is a minimal distance from a point of r to the
boundary of volume V ( ),0 a transmitted light cone
will already cross the boundary of four-dimensional
domain ( )χ t ,r = 1 and a boundary of V t( ) will effect
on formation of the internal field.
As a result we obtain the following evolution of
process of the field interaction with a object of
diffraction.
A point of observation is within the region V t( ) . 1)
Before the zero temporary moment the incident field
u r b r0 0( , ), ( , )t t generates the internal fields
u r b r1 1( , ), ( , )t t . 2) After changing of state of an
object at zero temporary there is no effect of the object
boundary of an object in the region and incident field
does not obviously take part in creation of the internal
field, that is the field u r b r2 2( , ), ( , )t t is directly
generated by the field of u r b r1 1( , ), ( , )t t , formed
within the object before changing of the state and
conditioned by the field of u r b r0 0( , ), ( , )t t that is
the memory of medium. 3) To form the field
u r b r3 3( , ), ( , )t t together with the field
u r b r1 1( , ), ( , )t t the incident field
u r b r0 0( , ), ( , )t t crossing the boundary and field
u r b r2 2( , ), ( , )t t take part in the process above
mentioned.
A point of observation is beyond the region of
V t( ) . Then from the expression (1) we receive the
square formula. In this case two temporary intervals are
allocated. 1). At ( )t
u
l<
1
r , where ( )l r − distance
from a observation point at moment t=0 up to the
nearest point of domain V t( ) , the external field
defined by nondisturbance condition of diffraction
object by founding field u r b r1 1( , ), ( , )t t . 2). At
( )t
u
l≥
1
r , an external field, on which already
influence and the new condition, will be defined in view
of a field u r b r3 3( , ), ( , )t t .
In this connection the following algorithm to solve
the problem is considered. Firstly the solution of non-
disturbance problem that can be considered as steady-
112
state one with corresponding choice of point parameters
is sought. Secondary is sought the solution of the
disturbance problem, but without the effect of
boundaries of a diffraction object, that is only the
disturbance of the medium itself is present. At the third
stage the problem is solved with taking into account the
boundary-value effects.
If the non-stationarity dissipation problem of MHD
waves on the elementary plane boundary it is possible to
consider theoretically stage by stage investigated, in
sense mathematical realization there is the set of
difficulties. First of all transformation of waves an each
other on the boundary of interface a much complicates
problem. It is characteristic just of magnetic
hydrodynamics. The transformation is displayed already
by consideration of a three-dimensional stationary
diffraction problem even on the elementary
inhomogeneous is half-space. In this case the expression
(1) is not divided on the separate equations, describing
only Alfven and only magnetoacoustic waves. Here it is
possible to look after complete transformation of MHD
waves, as was made. That is wave as though
«interplace». More simple two-dimensional diffraction
problem allows to consider separately of a boundary-
value problem for Alfven and in common for fast and
slow magnetoacoustic waves. Here it is possible to look
after transformation of magnetoacoustic waves. And
that, at last, completely to exclude transformation of
waves, it is meaningful to consider a spatially single-
dimensional problem separately for an each type of
waves.
In short a problem of scattering of Alfven wave by
plasma half-space (MHD-inhomogeneity) after a
initiating its movement is considered.
The model in which one type of MHD waves is
coupling with another through a plane boundary
between two media is an important first approximation
for the description of the propagation of small
perturbations in strongly inhomogeneous MHD media.
Let the plane Alfven wave
( )u r
b r
u
b
0
0
0
0
0 0
( , )
( ,
exp
t
t ik z i tA A
=
− ω
falls onto a plane boundary (z=0) of two nondissipation
medium having parameters B i Ai Si iV V i, , , , , .ρ = 1 2
Let internal medium begins a uniform moment with
velocity U0 to perpendicularly its boundary in moment
t=0. Prior to the beginning a movement an
inhomogeneous occupied domain (z>0). Then the law of
boundary movement will be set by the formula
( )z t U t( ) = 0θ . Here ( )θ t is the Heaviside
function.
For such a setting a diffraction problem in case of
consideration only the Alfven wave is of scalar form.
Then integral relationship (1) to find the Alfven field
incident on the MHD inhomogeneous and reflected
from the latter is as follows:
( ) ( ) ( ) ( )u t u t
B
V V
z t
dt b t G t t dx x A A x
A
V t
( , ) , ' ' , ' ' , ' '
( ')
r r r r r r= + − − − −
−∞
∞
∫ ∫0
1
1
2
2
2
21 ∂
∂ ∂
( ) ( )− −
− −
−∞
∞
∫ ∫V
B
B t
dt u t G t t dA x
A
V t
1
2 2
1
2
21
∂
∂
' ' , ' ' , ' ',
( ')
r r r r
where G A is the Green function of Alfven component given by
( ) ( )G
V
x x x x t t
x x
VA
A A
= − − − −
−
1
2 1
1 1 2 2
3 3
1
δ δ θ' ' '
'
,
here ( )θ t − the Heaviside function, ( )δ x − the
Dirac function.
Three-dimension integral on spatial variable easily
to reduce to single-dimension integral. For internal
medium we have
( ) ( ) ( )
( )
( )
u t G t t d u z t t
z
V
t
z
V
dz
u z t t
z
V
t
z
V
dz
x
A
x
A AV t
x
A Az
r r r r
z t
z
' , ' ' , ' ' ' , ' '
'
'
' , ' '
'
' .
'( ')
− − = − − +
+
+ − −
∫∫
∫
∞
θ
θ
1 1
1 1
(4)
And for external medium we have
( ) ( ) ( )
( )
u t G t t d u z t t
z
V
t
z
V
dzx
A
V t
x
A Az t
r r r r' , ' ' , ' ' ' , ' '
'
' .
( ') '
− − = + − −
∫ ∫
∞
θ
1 1
(5)
On structure (4,5) practically coincides with the
appropriate integral equation in a dissipation problem of
electromagnetic waves by plasma half-space. Omitting
the intermediate mathematical calculatious one can note
the sufficiently new results of magnetic hydrodynamics
arising in the solution of the non-stationary boundary-
value problem for the Alfven waves in case of taking
into account the initial temporary moment of origin of
spatial boundary movement starting under the condition
of U VA0 1< .
For the analysis of the internal field after initial
113
movement ( )t > 0 it follows that the field in the region
of V t zA1 < does not have either frequency and wave
number and the amplitude existed before starting of
movement of the Alfven wave is changed and the
constant component due to jump of velocity of medium
appears. It is connected with that boundary-value effects
for the moving boundary do not influence on this field.
In the region of U t z V tA0 1≤ ≤ it is necessary to
consider separately cases of «run-away» U 0 0> and
"encouter" U 0 0< movement with for each case we
obtain the Volterra two-dimensional equation of the
second type with the Fredholm kernel for that was
developed sufficiently well the approach fort solution of
problem as uniformly convergent Neiman series.
The reflected field is easily restored by means of the
quadrature formulas (5) for the already known internal
field. Thus it should be noted the sufficient difference of
a spectrum of the scattering field in case of the finite
temporary moment from the spectrum of the scattered
field for the adiabatic inclusion at infinity. The
spectrum of this field consists of waves with different
frequencies propagating as from the medium boundary
as towards boundary itself then as in case of inclusion at
infinity there is only the wave reflected from the
boundary in transmitted field.
Conclusions
The superficial analysis of the boundary-value non-
stationary problem for Alfven wave has shown the
principal possibility of using the method of evolutionary
integral equations of non-stationary macroscopic
electrodynamical in a case of MHD description of
waves in plasma. With the importance of strict
mathematical solutions obtained for simple model
problems that is the diffraction of one separately taken
Alfven wave is that it can be the basis for construction
of the approximate solutions of more complex
boundary-value problems.
References
1. S.I. Syrovatski. Magnetic hydrodynamics
//Advances of physical sciences. 1957, v.LXII, N3,
p.247-303.
2. A.A. Alexandrova, N.A. Khizhnyak. Boundary-
value problems of magnetic hydrodynamics. Kharkov,
NPO Test-Radio, 1993. p.230.
3. R.V. Polovin, V.P. Demuzhki. Bases of magnetic
hydrodynamics. M. Energatomisdat, 1987. p.206.
4. A.G. Nerukh, N.A. Khizhnyak. Modern problems
of non-stationary state macroscopic electrodynamics.
Kharkov, NPO Test-Radio, 1991. p.278.
5. A.A. Alexandrova. Integral equations in problems
of the MHD wave diffraction in plasma media
//Advances of Modern Radioelectronics. 1999, N 3,
p.25-41.
I
Introduction
The questions of evolution in magnetohydrodynamics
Integral formulation of boundary-value problem
Conclusions
|