Canonical variables in MHD flows with shocks and other breaks
The rigorous method of introduction of hamiltonian variables in MHD flows with discontinuities is presented. Appropriate boundary conditions are obtained and proved to give conventional continuity conditions at the boundary.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Canonical variables in MHD flows with shocks and other breaks / A.V. Kats, V.N. Korabel // Вопросы атомной науки и техники. — 2000. — № 6. — С. 88-90. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860106149422956544 |
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| author | Kats, A.V. Korabel, V.N. |
| author_facet | Kats, A.V. Korabel, V.N. |
| citation_txt | Canonical variables in MHD flows with shocks and other breaks / A.V. Kats, V.N. Korabel // Вопросы атомной науки и техники. — 2000. — № 6. — С. 88-90. — Бібліогр.: 11 назв. — англ. |
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| description | The rigorous method of introduction of hamiltonian variables in MHD flows with discontinuities is presented. Appropriate boundary conditions are obtained and proved to give conventional continuity conditions at the boundary.
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| first_indexed | 2025-12-07T17:31:08Z |
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UDC 533.9
88 Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 88-90
CANONICAL VARIABLES IN MHD FLOWS WITH SHOCKS AND
OTHER BREAKS
A.V. KATS*, V.N. KORABEL**
*Kharkiv Military University, P.O. Box 8847,
61002 Kharkiv, Ukraine, e-mail: avkats@akfirst.kharkiv.com
** Kharkiv National University, Svobody sq. 4, 61005 Kharkiv, Ukraine,
e-mail: korabel@isc.kharkov.com
The rigorous method of introduction of hamiltonian variables in MHD flows with discontinuities is presented. Ap-
propriate boundary conditions are obtained and proved to give conventional continuity conditions at the boundary.
Introduction
The most general form of physical laws is those ex-
pressed by variational principle. A variational principle
for fluid motion description in terms of Lagrangian
variables has been developed, see, for instance, [1], [2]
and literature cited therein. Formulation of variational
principle in Eulerian description requires introduction of
subsidiary fields, known as Lagrange markers. Appar-
ently the most widely used procedure is the use of addi-
tional fields introduced by means of constraints, cf. [3]
that leads to the fluid velocity excluding from the set of
independent variables. The velocity representation aris-
ing is analogous to the Weber’s one [2]. In spite of sub-
sidiary variables appearing, the use of canonical vari-
ables leads to essential simplifications allowing to ex-
ploit highly-developed standard methods, cf. [4] – [8].
Hamiltonian variables for fluid with free boundary were
introduced in [9]. Lately this results were generalized
for any type of discontinuities, including shocks [10],
[11]. Here we develop the variational principle and pre-
sent corresponding canonical variables for MHD flows.
The general variational principle presented describes
any possible type of MHD discontinuity surfaces.
Variational principle
Let us assume that discontinuity surface separating
two regions of continuous motion is defined by equation
( ) 0tR =,r . (1)
Suppose the surface has no edges, thus locally Eq. (1)
may be written in explicit form:
( ) ( ),,, tztR ⊥ζ−= rr ( )y x,=⊥r . (2)
We start with the Hamiltonian
Σ+= HHH V , (3)
VV ddtH H∫ ∫= r , ,H Hdiv
8
H
2
v
ñ
22
Φ−
π
+
ε+=V (4)
∑⊥Σ ∫ ∫= HrddtH , { } { } ,H SkGu rotRR −∇−∇Ψ=∑ (5)
,SHvG mfñ ++Γ′≡ .g ημ+ϕ+γ=Γ (6)
Here ñ is the fluid density and ε is internal energy;
braces indicate the jump of the corresponding quantity
at the boundary 0R = : { } 21 XXX −≡ ; uvv −=′ de-
notes the fluid velocity relative to the boundary one,
( )t,⊥≡ ruu . The latter may be obtained from Eq. (1) by
differentiating it with respect to time:
0RtRdtdR =∇+∂∂= u , .dtd 0R=≡ ru VH repre-
sents the volume Hamiltonian and ΣH the surface one.
Lagrange multiplier Φ guarantees nondivergent charac-
ter of the magnetic field H . The fluid velocity v is
expressed via other fields by the representation
[ ] ,0rotsññ =+∇σ+∇+ϕ∇+ SHìëv (7)
where ϕ is scalar potential, μ is the vector field of
Lagrange markers and s is entropy density.1
The set of volume canonical coordinates Q and con-
jugated momenta P is as follows:
( )S Q,=Q , );s, ,(Q ìϕ= ( )H P,=P , ). , ,ñ(P σ= ë (8)
The surface Hamiltonian ΣH does not affect equa-
tions of motion but is essential for the boundary condi-
tions. The surface canonical coordinate is ζ with con-
jugated momentum Ψ , other surface variables are La-
grange multipliers for constraints that guarantee correct
boundary conditions. They also may be treated as ca-
nonical variables but with identically zero momenta.
Terms in Γ , Eq. (6), provide correspondingly the mass,
velocity potential and Lagrange markers fluxes continu-
ity.
Thus the total action containing both volume and
surface parts is as follows
[ ]+−∑=+= ∫ ∫∑ VV ddtAAA HQP !r [ ]Σ−ζΨ∫ ∫ ⊥ H!rddt (9)
Variation of the action with respect to the volume val-
ues of canonical variables Q , P , and Φ leads to the
set of Hamiltonian equations together with H nondi-
vergency condition:
,
2
vw
ñ
H
t
2
−ϕ∇−=
δ
δ=
∂
ϕ∂ v );div(ñH
t
ñ v−=
δϕ
δ−=
∂
∂ (10)
( ) ,ìvì ∇−=
δ
δ=
∂
∂
λ
H
t
);div(H
t m
m
m vλ−=
δµ
δ−=
∂
λ∂ (11)
,sH
t
s ∇−=
δσ
δ=
∂
∂ v );div(Tñ
s
H
t
σ−−=
δ
δ−=
∂
σ∂ v (12)
[ ],,rot
4
H
t
vSH
H
S −Φ∇+
π
=
δ
δ=
∂
∂ [ ];rotH
t
vH
S
H
=
δ
δ−=
∂
∂ (13)
0Hdiv =
Φδ
δ−=H , (14)
where T is temperature. From Eqs. (10) – (14) follow
the conventional MHD equations.
It should be noted that there exist some restrictions
on the boundary values of generalized potentials
1 The term with entropy is necessary for consideration of
shocks, cf. [10, 11], while the introduction of the term μλ∇
along with [ ]SHrot (in contrast with reduced representation,
cf., for instance, [4, 5]) guarantees correct passage to the limit
of conventional hydrodynamics.
89
λμ,,,σϕ and S that should be taken into account if the
boundary exists. These restrictions follow from the
jumps of the corresponding volume equations and de-
fine time evolution of the jumps in natural assumption
that corresponding surface sources are absent:
{ } { } { },2vwvt 2
nn −=ϕ∂′+∂ϕ∂ (15)
{ } { } ,0vt nn =∂′+∂∂ ìì (16)
{ } ( ){ } ,0ñvtñ nn =∂′+∂∂ ëë (17)
{ } ( ){ } { } ,Tñvtñ nn −=∂′+∂∂ σσ (18)
{ } [ ] ( ){ } .rot,4t SuSvHS ∇++Φ∇+π=∂∂ (19)
If some of these variables do not undergo jump (as, for
instance, it is for ϕ , ì , ñë in the shock case) then the
corresponding equation presents necessary time inde-
pendent condition on the boundary values of the volume
variables and their spatial derivatives.
Fulfilling volume equations we obtain the following
residual (surface) variation of the action
,A)A(|AA boundsurf ∑δ+δ≡δ=δ
=δ bound)A( { } +δζ−
⊥∫ ∫ HQP !rddt (20)
( ) ( ) ( )( ){ }RRQPRddt ∇δ−∇δΦ+δ∇′+
⊥∫ ∫ uSHHvr , .
Here the second integral arises from integrations by
parts, it contains boundary values of the volume mo-
menta P and coordinates Q . The first integral corre-
sponds to the variation of the discontinuity surface.
The variational principle formulation should be sup-
plemented by the set of quantities that are varied inde-
pendently at the surface. The suitable choice is not
unique, we accept it2 consisting of i) all surface vari-
ables; ii) both sides boundary values of variations of all
volume momenta, entropy, and of spatial derivatives of
all volume coordinates; iii) boundary values of varia-
tions of ϕ , ì , S from one side of the surface. Thus we
suppose absence of jumps when crossing the surface for
the latter quantities, i.e.,
{ } { } { } 0=δ=δ=δϕ Sμ .
In this assumptions the functional derivatives of the
action with respect to the boundary values of all hydro-
dynamic momenta P , spatial derivatives of hydrody-
namic coordinates Q∇ and Srot gives us equations:
0,RPQ =∇Γ⇒∇δ (21)
0,RQP =∇∇Γ⇒δ
[ ] 0Rärot =∇Γ⇒ ,HS
that are fulfilled simultaneously from each side of the
surface if 0=Γ . The case 0≠Γ is trivial one, because
all hydrodynamic momenta P vanish on the boundary
as it follows from (21), consequently 0ñ = and the
fluid is absent.
2 This choice allows to consider all possible types of disconti-
nuities in the framework of one variational principle. If one
restricts to the special types of discontinuities, then less sub-
sidiary constraints required and more simple principles can be
formulated
Variation with respect to the rest variables leads to
the set of boundary conditions:3
{} ,0j =⇒δγ ( ),Rñj ∇′≡ v (22)
( )( ){ } ,0Rñä =∇′+⇒ vçëì (23)
( ){ } ,0Rñ =∇′⇒δ vìη (24)
( ){ } ,0Rñg =∇′ϕ⇒δ v (25)
( ) ,0Rs =∇′σ⇒δ v (26)
{ } ,0Hf n =⇒δ (27)
{ } ,0Sm n =⇒δ (28)
{ } ,0rot =⇒ Skδ (29)
,0f =Φ+⇒Hδ (30)
{ } ,0RmHv nn =∇+−′⇒δ vHS (31)
{ } ,0=∇Γρ−∇Ψ−⇒δ RRu (32)
{ } .)(HQP 0div =Ψ∇−Ψ−+−⇒δζ ⊥⊥ uG !! (33)
Two last equations for ζ and Ψ are of the Hamiltonian
form:
,RH ∇=Ψδδ=ζ u!
{ } ( ).HQP Ψ∇−+−=δζδ−=Ψ ⊥⊥ uGdivH !!
Note here that Eq. (32) with (21) taken into account
leads to vanishing of the conjugated to the boundary
displacement ζ momentum: 0=Ψ , for all types of dis-
continuities and thus the last equation takes a form
{ } 0div =+− GHQP ! .
From the presented boundary conditions follow con-
ventional MHD conditions, namely fluid momentum
flux continuity:
{ } ,08H8vñp 2
n
22
n =π−π+′+ τH (34)
{ } ,04Hvñ nn =π+′′ ττ Hv (35)
and the energy flux continuity:
( ) [ ]{ } .)( 04HHvw2vvñ n
2
n
2
n =π′−′++′′ Hv (36)
Here p denotes fluid pressure. Although Eqs. (22) – (33)
involve large number of subsidiary variables, neverthe-
less it may be checked they do not contain extraneous to
the conventional MHD cases (for short we will not dis-
cuss this point in details), but describe all possible types
of discontinuities intrinsic to MHD: slide, contact,
shock, and rotational discontinuity. Leaving out the pro-
cedure of splitting Eqs. (21), (22) – (33) let us discuss
below the special cases.
Shocks
Let us start with the shock discontinuity, where 0j ≠
and { } 0ñ ≠ . First of all, if the mass flux is nonzero, it
should be continuous due to (22). Along with equations
(24), (25), (27) – (29) that are introduced by surface
constraints
3 Note that it is convenient to vary the surface term
{ }R∇G included in ∑H in the form of volume integral,
namely { } ,divdRd GrGr ∫∫ =∇
⊥
assuming the surface
functions m,f,,g, çγ be prolonged to the volume along nor-
mal RR ∇∇≡n , i.e.,
.0mfg nnnnn =∂=∂=∂=∂=γ∂ ç
90
{ } { } { } { } { } 0rotSH nn ====ϕ= Sì , (37)
we get the continuity of the tangential component of the
electric field, Eq. (31),
{ } { } .Hñj n ττ ′= vH (38)
Eq. (23) defines the variable ç value and, in turn, leads
to { } .0ñ =ë For 0j ≠ Eq. (26) leads to the boundary
value of entropy momentum vanishing, .0=σ ζ The
normal component of Eq. (31) yields .0m = Eq. (33)
with identity
{ } { })div(8Hp 2
HΦ+π+=− τHQP ! ,
which follows from the volume equations, together with
Eqs. (22) – (33) and (19) leads to the continuity of the
normal component of the fluid momenta's flux
{ } 08vñp 22
n =π+′+ τH .
Let us prove that the tangential component of the
momenta's flux (35) is also continuous. The first of the
volume equations (13) and Eq. (38) can be presented in
the following form:
( ){ } { } SH ττ =+ rotñ1HñaHj
2
nn ,
{ } { } ( ){ } SHH τττ −=−π rotñ1aHjña4 n
2
with .rota nS= Excluding ñ1 jump from these equa-
tions we obtain that jumps of vectors τH and ñτH
are parallel:
{ } { }ñj4H 22
n ττ =π HH . (39)
It may be checked that substituting of tangential veloc-
ity jump { } { } [ ]{ }ñrot τττ −==′ SHvv , to Eq. (35) and
subsequent excluding of { }ñ1 leads to the same equa-
tion (39). Thus the tangential component of the mo-
menta's flux is also continuous.
Analogously, it can be checked that the energy flux
is also continuous (we suppress proof for short).
Rotational discontinuities
Let us consider another type of discontinuities, ,0j≠
,0H n ≠ but without density jump, { } 0ñ = , conse-
quently, 0v n =′ , { } 0v n =
Note that Eq. (38) yields equation
{ } ( ){ }ττ ′′= vH nn vH (40)
that means that velocity and magnetic filed tangentional
components jumps are codirectional. Using (40) and
(35) we get ñ4Hj n π= . Boundary conditions for
the variables reiterate the previous case with { } 0ñ1 =
taken into account, so, evidently, the proof of the hy-
drodynamic fluxes continuity is analogous.
Contact discontinuities
If the mass flux vanishes from one side of the dis-
continuity surface 0j= , than, as it is clear from it's
continuity, it equals zero from the another one. This
case corresponds to contact discontinuity if 0H n ≠ .
Boundary conditions become much more simpler in
these cases, as Eqs. (22) – (26) are satisfied identically.
From Eqs. (31), (35), and (33) follow the continuity of
the tangential components of velocity, magnetic field,
and pressure: { } 0=′τv , { } 0=τH , { } .0p = These equa-
tions are known to describe contact discontinuity. Note
that surface tension effects are easily described by in-
cluding the corresponding term to the surface Hamilto-
nian density:
( )
−ζ∇+α=+→ ⊥⊥ααΣΣ ∫ 11d 2rH,HHH ,
where α denotes surface tension coefficient.
Slide discontinuities
This case corresponds to 0j= , .0H n = Then from
Eq. (33) we obtain the continuity of normal component
of the fluid momenta's flux
{ } .08p 2 =π+ τH
It is rather simple to check up and momenta's flux tan-
gential component continuity (35). Thus, we may assert
that the set of boundary conditions describes slides.
References
1. V.L. Berdichevskii, Variational Principles in the Me-
chanics of Continuous Medium, Moskow: “Nauka”,
1983 (in Russian), 447 p.
2. Lamb, Hydrodynamics, Cambridge: “Univ. Press”,
1932, pp. 1–20.
3. C.C. Lin, Liquid helium, Proc. Int. School of physics,
Course XXI, N.Y.: “Acad. Press”, 1963. H.
4. V.E. Zakharov, E.A. Kuznetsov, Hamiltonian for-
malism for nonlinear waves // Uspechi Fizicheskich
Nauk (167), 1997, № 11, 1137 p. (in Russian).
5. V.P. Goncharov, V.I. Pavlov, The problems of hydro-
dynamic in Hamiltonian description, Moskow: “Izd.
MGU”, (1993) (in Russian), 196 p.
6. V.E. Zakharov, V.S. L'vov, G. Falkovich, Kolmo-
gorov Spectra of Turbulence. Wave Turbulence, N.Y.:
“Springer–Verlag”, 1992.
7. H.D.I. Abarbanel, R. Brown, Y.M. Yang, Hamilto-
nian formulation of inviscid flows with free boundaries
// Phys. Fluids (31), 1983, № 10, pp. 2802 – 2809.
8. V.A. Vladimirov, H.K. Moffatt, On General Trans-
formations and variational Principles in Magnetohydro-
dynamics. Part I. Fundamental Principles. // J. Fl. Mech.
(283), 1993, pp. 125–138.
9. V.E. Zakharov, N.N. Filonenko, Weak turbulence of
capillary waves // Zh. Prikl. Mekh. i Fiz., N5, 1967
10. A.V. Kats, V.M. Kontorovich, Hamiltonian descrip-
tion of the motion of discontinuity surfaces // Low
Temp. Phys., (23), 1997, № 1, pp. 89–95.
11. A.V. Kats, Variational principle and canonical vari-
ables in hydrodynamics with discontinuities // Physica
D, 2000 (in press)
A.V. KATS*, V.N. KORABEL**
|
| id | nasplib_isofts_kiev_ua-123456789-78549 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:31:08Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kats, A.V. Korabel, V.N. 2015-03-18T18:48:51Z 2015-03-18T18:48:51Z 2000 Canonical variables in MHD flows with shocks and other breaks / A.V. Kats, V.N. Korabel // Вопросы атомной науки и техники. — 2000. — № 6. — С. 88-90. — Бібліогр.: 11 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78549 533.9 The rigorous method of introduction of hamiltonian variables in MHD flows with discontinuities is presented. Appropriate boundary conditions are obtained and proved to give conventional continuity conditions at the boundary. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma dynamics and plasma-wall interaction Canonical variables in MHD flows with shocks and other breaks Article published earlier |
| spellingShingle | Canonical variables in MHD flows with shocks and other breaks Kats, A.V. Korabel, V.N. Plasma dynamics and plasma-wall interaction |
| title | Canonical variables in MHD flows with shocks and other breaks |
| title_full | Canonical variables in MHD flows with shocks and other breaks |
| title_fullStr | Canonical variables in MHD flows with shocks and other breaks |
| title_full_unstemmed | Canonical variables in MHD flows with shocks and other breaks |
| title_short | Canonical variables in MHD flows with shocks and other breaks |
| title_sort | canonical variables in mhd flows with shocks and other breaks |
| topic | Plasma dynamics and plasma-wall interaction |
| topic_facet | Plasma dynamics and plasma-wall interaction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78549 |
| work_keys_str_mv | AT katsav canonicalvariablesinmhdflowswithshocksandotherbreaks AT korabelvn canonicalvariablesinmhdflowswithshocksandotherbreaks |