v.a. vladimirov and h.k. moffatt- on general transformations and variational principles for the...

8/3/2019 V.A. Vladimirov and H.K. Moffatt- On general transformations and variational principles for the magnetohydrodyna

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J . Fluid Mech. (1995), vol. 283, p p . 125-139Copyright 0 1995 Cam bridge University Press

125 @On general transformations and variational

principles for the magnetohydrodynamics of idealfluids. Part 1. Fundamental principles

b By V. A. VLA DIM IROV '? A N D H. K. MOFFATT'Department of Mathematics, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver Street, Cambridge CB3 9EW, UK(Received 28 December 1993)

A new frozen-in field w (generalizing vorticity) is constructed for ideal mag-netohydrodynam ic flow. In conjunction w ith the frozen-in magnetic field h, this is usedto ob tain a generalized Weber transfo rmation of the M H D equ ations, expressing thevelocity as a bilinear form in generalized Weber variables. This expression is alsoobtained from Hamilton's principle of least action, and the canonically conjugateHamiltonian variables for MHD flow are identified. Two alternative energy-typevariational principles for three-dimensional steady MHD flow are established. Bothinvolve a functional R which is the sum of the total energy and another conservedfunctional, the volume integral of a function @ of Lagrangian coordinates. It is shownthat the first variation S IR vanishes if @ is suitably chosen (as minus a generalizedBernoulli integral). Expressions for the second variation S2R are presented.1. Introduction

A variational principle that finds its origins in the work of Kelvin (1910) has beenformulated and developed, for three-dimensional steady flows of an ideal fluid, byFjortoft (1950) and Arnold (1965a), and has been used in a consideration of thestability of two-dimensional flows to finite-amplitude disturbances by Arnold (1965b).The underlying Ham iltonian structure of the Euler equations plays an im portant rolein such an approach, as do any invariants associated with the equations (H olm et al.1985; McIntyre & Shepherd 1987; Vladimirov 1987). The A rnold appr oach toproblems of stability is summarized by Saffman (1992, 0 14.2). Similar variationalprinciples have been developed for the treatment of the stability of magnetostaticequilibria of perfectly conducting fluids (Bernstein et al. 1958) and for the basiccharacterization of equilibrium states (Woltjer 1958; Tay lor 1974; see also the recentmonograph of Biskamp 1993, which gives a detailed treatment of these topics).There are subtle differences between the E uler flow problem an d the m agnetostaticproblem associated with the fact that in the former case it is the curl of the basicvelocity field (i.e. the vorticity) which is 'frozen' under unsteady Euler evolution,whereas in the latter case it is the basic field itself (the magnetic field) which is frozenund er arb itrary unsteady deform ations (the fluid being assumed perfectly conducting).These differences have been discussed by M offatt (1985, 1986) both in relation to theproblem of characterizing, or determining, steady states, and in relation to the stabilityof these states.t Also at : Lavrentyev Institute for Hydrodynamics, 630090 Novosibirsk, Russia.

www.moffatt.tc


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126 V.A . Vladimirov and H . K . MoffattThe problems become more acute when the stability of general solutions of theMHD equations (with non-zero velocity and magnetic field) is considered. In thissituation, the magnetic field is still frozen, but the vorticity field is not, the Lorentzforce being in general rotational. In order to construct a theory of stability of suchstates, it is important to identify the field that replaces vorticity as the frozen-in

dynamical field, for only then can the counterpart of isovortical perturbations beidentified. 4Earlier theories (Frieman & Rotenberg 1960; Moffatt 1989; Friedlander & Vishik1990) have considered virtual displacem ents under which the magnetic field is frozen,but have not addressed the problem of identifying the second frozen field which(together with the magnetic field) determines the structure and evolution ofperturbations from a given steady state. The existence of such a field is suggested bythe cross-helicity invariant (the integrated scalar product of velocity and magneticfield) (Woltjer 1958) which is topological in character (Moffatt 1969) and whichtherefore presum ably involves the m utua l linkage of two frozen fields; one of these ismagnetic field, but what is the other when the rotational Lorentz force is operative?In $ 2 of this paper, we identify this second field in the following way. First, we definea solenoidal vector field m ( x , ) with the property that the rate of change of the flux ofm across any material (Lagrangian) surface element 6s is equal to the flux of currentj across the same elem ent. We then define the vector field

w = o + V x ( h x m ) , (1.1)where w and h are the vorticity and m agnetic fields, and we show that w is a frozen-in field.Recognition of the roles of the fields m and w allows us in $ 3 to obtain thegeneralization of Webers trans form ation of the Euler equations to the situation whenh =k 0 (equation (3.18) below) and, equivalently, the generalization of Cauchysformula in terms of w instead of w (equation (3.2)). It a lso enables us to demonstrate($4) hat the MHD equations can be obtained from a Hamiltonian variationalprinciple, and that the equations have Hamiltonian structure in which ( h , g ) arecanon ically conjugate variables, where g is a vector potential for the field m.In 95 we consider properties of steady solutions of the M H D equa tions augmentedby the equation for the auxiliary field m, nd in $6 we obta in a general energyvariational principle for three-dimensional steady MHD flows, which leads to theconstruction of the appropriate second variation of the energy functional R equations(6.13)-(6.15)). Fina lly, in $7 , we develop an alternative form of energy varia tionalprinciple that may be used when the unperturbed velocity field has Clebschrepresentation (7.1 l), the streamlines then being closed curves. Again, the secondvariation of energy is calculated by a clear procedure, and presented in a form(equation (7.39)) tha t will be useful in subsequent applications.

t

2. The frozen fields of ideal magnetohydrodynamicsSuppose tha t an incom pressible, inviscid, perfectly conducting fluid of unit density

is contained in a domain 9 ith fixed boundary 89. et u ( x , ) be the velocity field inthe fluid and let h ( x , ) be the m agnetic field (in AlfvCn velocity units). ThenV - u = V . h = O in 9, (2.1)

and we adop t the natural bounda ry conditionsn - u = n . h = O on i39.


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Magnetohydrodynamics of idea l ju ids . Par t 1 127The fields U and h evolve according to the equations

Du u , + u . V u = - V p + j x h,L h - h , - V x ( u x h ) = O ,

where p(x, ) is the pressure field, andj = V x h

is the current density. The operator D is the material, or Lagrangian, derivative, theequation Df= 0 implying that f s conserved o n particle trajectories. The o pera tor Lis a form of Lie derivative, the equation Lh = 0 meaning that the h-field is frozen inthe fluid, the flux of h through any closed material circuit being conserved.Equ ations (2.3) and (2.4) have three q uadra tic integral invariants, namely the energy

the magnetic helicity rZM= h - curl- h dr, (2.7)Jand the cross-helicity

Zc= I g u . h d r ( 2 . 8 )(Woltjer 1958). Conservation of lue, s associated with the invariance of the topologyof the h-field (M offatt 1969); likewise, conse rvation of Zc s associated with the factthat, althoug h the flux of vorticity throug h a n arbitrary closed material circuit is notconserved (the Lorentz force j x h being in general rotational), the flux of vorticitythrough a closed h-line (which is a particular material circuit) is conserved.It will be convenient to adopt the Poisson bracket notation (Arnold 1965)

{ A ,B}E V x ( A x B) = (B. V) A - A V) B (2.9)V - A = V * B = 0. (2.10)

Obviously, {A,Bl = -{B,A); (2.11){A,{B,c>>+{ B,{C,A} }+{C,{A ,B}} 0 (2 .12)

for any two vector fields A ,B satisfying the solenoidal conditions

moreover, the Jacobi identitymay be easily verified. With this notation, (2.4) becomes

h, = { U , h } . (2.13)Also, for a n arbitrary solenoidal field A ,

aatL{h,A } = - h ,A )-{U, { h ,AI}

= { h , A , } + { { u , h } , A } - { u , { h , A } } sing (2.13)= { h ,A ,}+ { h , A , }} using (2.1 l) , (2.12)= {h ,LA}. (2.14)

Note further th at the curl of (2.3) give the equa tion for vorticity o = curl U in the form0,= {U,4 U ,4. (2.15)


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128 V.A . Vladimirov and H . K. Moffat tLet us now define an auxiliary dimensionless vector field m(x, ) by the equations

Lm m,-{u,m} =J , V - m= 0. (2.16)Equivalently, if SS( t ) is any area element moving with the fluid, then

d- (m.SS) =j.SS.dt (2.17)The physical interpretation of m is thus that it is the time-integrated current densityacross such a (Lagrangian) area element. Note that m is not uniquely determinedunless m ( x ,0) is specified; however, the p roperties (2.16) are sufficient for our presentpurpose.Let us now define a modified velocity field

U = u + h x m, (2.18)and a corresponding modified vorticity fieldThen Lw = Lo+L{h,m)w = V x u = o+ {h ,m} .

= L o + { h , Lm} using (2.14)= o t - { u , o } + { h , j } using (2.16)

(2.19)

= 0 using (2.15). (2.20)Hence w is now a frozen field; since w reduces to o when h = 0, the field w evidentlyprovides the appropriate frozen-field generalization of vorticity when h + 0. We shalltherefore describe w as the generalized vorticity fie ld.We can now give a new a nd m ore transp arent interpretation of the cross-helicitySc.Fo r, using (2.18), (2.8) beomes r (2.21)Since both w(= curl U) and h are now frozen fields, their mu tual topology is conserved;for exam ple, the flux of w throu gh any closed h-line is conserved. By the argum ents ofMoffatt (1969, 1981),Scprovides a m easure of the mutua l linkage of the fields h and

0 of (2.20). This corresponds to U = Va for som e scalarW.Note the trivial solution wfield a, or, from (2.18), (2.22)We may describe this as a generalized poten tial f l o w .Finally, note that the essential property Lw = 0 is unaffected if we add to m any fieldm, satisfying L{h,m,} = { h ,Lm,} = 0. Thus, if m+m+m, here m, is any solution ofLm, =fwith {h,f)= 0, then w + w +w, where w 1 { h ,m,}, butand the modified field w + w1 s still therefore frozen.

u = Va+m x h .

.L(w+w,) = L w+ L w, = 0

3. Generalized Weber transformationa = (a,, a,, as).The label of the particle passing through x at time t is thenSuppose now that the fluid particles are labelled by Lagrangian coordinates

a = a ( x , ) , (3.1)


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Magnetohydrodynamics of ideal ju id s. Part 1 129( 3 4aaand DU= - + ( u . V ) Ut = 0.

The particle paths are given by the inverse functionx = x ( a , t ) ,and the initial position of the particle labelled a is

x (u , 0) = X ( a ) , say.A natural choice of label would be a = X; owever it proves convenient to retain theextra freedom represented by the ' rearrangement function' X ( a ) .The m apping X ( a )+ ( a , t ) is induced by the flow

U = DX= ( ~ x / ~ t ) , , c o n s t , , (3.5)and is a volume-preserving diffeomorphism with the property

We shall suppose that the mapping a + X ( a ) is also a volume-preserving dif-feomorphism, so that

also.We now seek to transform the equation of motion (1.3) to integrable form, bygeneralization of the a rgum ent of W eber (1858) (see, for example, Serrin 1958). First,multiply (1.3) by ax,/aa,, giving

Now

Also we have the identity

( D u , ) ~X = D~ ZU ~ ) - - ( $ ~ ) .aai aa,

Hence, using (2.18), (3.8) may be written

where a = S i U 2 - P ) la-const. d t*0Integrating along a particle path a = const., we obtain

(3.9)

(3.10)

(3.11)(3.12)

(3.13)


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130where the c onstant of integration is given from the initial condition by

V .A . Vladimirov and H . K . M o f a t t

(3.14)Here b , defined in this way as a function of a = a ( x , ) , also satisfies the condition ofLagrangian invariance, D b = 0. (3.15)In equation (3.13), the terms are best regarded as functions of a and t . We may howeverimmediately convert to Eulerian form by multiplying by aa,/ax,, giving

U = V a+ b, V a , , (3.16)in which now a, and a are regarded a s functions of x and t . Here and subsequentlywe use a mixed vector/suffix notation in which, for example,

a- ,ax ,b , V a ) = b -ak, (3.17)the repeated suffix k being summed. F ro m (2.18) and (3.16) it follows immediately that

u = V a + m x h + b , V a , . (3.18)Together with the equations

D a = D b = 0 ,L h = 0 , L m = j

D a = $ u 2 - p , (3.19)this provides the required genera lization of Webers (1 868) transformation of thegoverning equations (2.3)-(2.5). Conversely, by app lying the opera tor D to (3.18) andusing (3.19), the equatio n of m otion (2.3) is recovered. Equatio ns (3.18) and (3.19) arethus equivalent to (2 .3t(2 .5).The field w is now given by

W = v X U = v b k X Va, , (3.20)a formula that, in conjunction with D a = D b = 0, again exhibits the frozen-incharacter of w.This result is precisely equivalent to the Cauchy representation

ax. aa,w g ( x , ) = w,(a, 0) -2- *aa, ax,The Eulerian form (3.20) is however more useful in what follows.In the special situation when a = X , (3.18) and (3.20) reduce t o

I= V a + r n x h + & V X , ,w = vv,x vx,.If moreover we a do pt initial conditions

m ( x ,0) = 0, u ( x ,0) = uo(x ) ,then U is given bythe initial conditions fo r a and X being

U = V a + m x h+u, ,VX, ,

a ( x ,0) = 0, X ( x ,0) = x .If uo (x ) 0, then (3.24) reduces to the generalized potential flow (2.22).

(3.21)

(3.22)

(3.23)(3.24)(3.25)


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Magnetohydrodynamics of ideal fluid s. Part 1 1314. Hamiltonian variational principle and canonically conjugate variablesvariational principle. Letbe the Lagrangian density, and consider the variational problem

We now show that equations (3.18), (3.19) may be obtained from a HamiltonianY ( u ,) = ;(u2- 2 ) (4.11

6IgOT Y ( u , ) d7 dt = 0,subject to the constraints

(4.3)I- U= 0 (incompressibility),Lh = 0 (frozen-in condition fo r h ) ,D a = 0 (identity of particles).Note that Lh = 0 implies tha t D ( V .h ) = 0, so that V - h= 0 should be regarded as aninitial condition, an d n ot as an additional constraint. Introd ucing L agrange multipliersa ( x , ) , g ( x , ) , b ( x , ) , the variational principle (4.2), (4.3) may be written6 sf(9aV . - g Lh - . D a )d7 d t = 0, (4.4)

where now U, and a are to be varied subject only to boundary conditionsn . S u = n . S h = O on U, (4.5)

6u = 6h = 6a = 0 a t t = 0, T. (4.6)and initial and final conditions

If we carry out the variation of ( 4 4 , and integrate by parts a s necessary using (4.5) and(4.6), we may then equa te to zero the coefficients of the independent variations &U, 6h,Sa, giving&U: U = V a - h x ( V x g ) + b , V a , , (4.7)6h: h = g t - u x (V X g) , (4.8)& a : D b = 0. (4.9), m = V x g . (4.10)Note that (4.7) is identical with (3.18) provided

Moreover the curl of (4.8) then gives immediately j = Lrn as in (3.19), and theremaining equations of (3.19) are contained in (4.3) and (4.9). The initial conditionV - h = 0 a t t = 0 is satisfied providedaat- ( V . g ) = V . ( u x m ) a t t = 0.

Canonically conjugate variablesConsider now the first variation of energy

(4.1 1)

(4.12)


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132 V.A . Vladimirov and H . K . Moffattin which U is given by (4.7), bu t now the independently varied fields are h,g , b,a , so that

&U = V S o l - ~ h X ( V X g ) - h X ( V X 6 g ) + 6 b , V a k + b k V S U k , (4.13)where Sa is determined so that V - S u= 0. Using the conditions (2.1) and (2.2), (4.12)is easily transformed to

SE = { S h , ( h + u x(V xg) ) , -Sg , [V x ( u x h) ] , -Sa i (u . V )b , + 6b , (u -V )a , } d~ .(4.14)s,Hence, from (4.3), (4.8) and (4.9), we have immediately

a g , p = hi+[U x (v g ) ] ,=%,/at = [V x (U x h) ] ,= -SE/&,,

I,/at = -(U. 0) , = - E/Sb,,a b , p = -(U. V ) , = SE/Ga,,and so ( h ,g ) and (a,b ) are canonically conjugate variables.(4.15)(4.16)

5. Steady MHD flowsConsider now steady solutions of (2.3), (2.4) and (2.16)

U = U ( x ) , h = H ( x ) , m = M(x), p = P ( x ) , (5.1)and the associated fields

0 = a ( x ) = v x U , w = W ( x )= s z+ (H , M} . (5.2)The Lagrangian variables a,b are in general unsteady:a = A ( x , ) , b = B ( x , ) , (5.3)

(5.4)dDoA = DoB= 0, Do = -+ U . V .atwhereFr om (2.20), (2.13) an d (2.16), we have

{U,W }= { U ,H } = { U , M } + V x H = 0, ( 5 . 5 )and hence U XW = V Z , U x H = V J , U x M + H = V K , (5.6)where Z, J and K may be described a s generalized Bernoulli functions. Note that

vz= uxa+ux [V x ( H x M ) ]= V ( P + i U 2 )-H X V x ( U x M ) ]+ U x [V x ( H xM ) ]

from (2.3) and (5.4). Hence it may be shown thatZ = P + i U 2 + M * ( U xH ) . (5.7)

Fr om (5.5), U . V Z = U . V J = 0, (5.8)and so the streamlines lie bo th on surfaces Z = const. and J = const., and are thereforethe curves of intersection of these surfaces. Hence, unless VZ = 0 and/or V J = 0 insome subd om ain of the flow, the streamlines a re closed curves. By virtue of (5.8) bo thZ and J may be expressed as functions of the Lagrangian coordinates

Z = Z(A,B ) , J = J ( A ,B ) . (5.9)


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Magnetohydrodynamics of ideal Puid s. Part 1 133(There is considerable freedom here since, as note d earlier, B is expressible, via (3 .14),as a function of A . )No w, from (3.20), W = V B , x V A k , so that the first of (5.6) becomes

U X V B , x V A , ) = V I ( A ,B ) ,or equivalently aI aA , -( U . V ) B , + - - 0.aA ,1axiClearly, this is satisfied provided (5.10)

(5.1I )and we may use the freedom in the specification of the fields A and B to ensure thatthese equations are satisfied.

Finally note that since U - V J = H - V J = 0, the surfaces J = const. contain both U-lines and H-lines. In particular, the boundary c?9s one such surface, i.e.J = J,(const.) on 89. (5.12)

6. A general energy variational principle for 3D steady MHD flows(5.6). We adopt the generalized Weber representation (3.18), so thatConsider now a steady M H D flow of the form (5.1), (5.2), satisfying the cond itions

u(X)= Va,,+M X H + B , V A ,, (6.1)V . U = O in 9, . U = O on c?9, (6.2)

where a, s the field that ensures that

We now consider smooth instantaneous independent variations Sh, Sm, Sa, Sb of thefields H , M , A , B where Sh, Sm satisfyV . S m = V - S h = O in 9, . S h = O on c?9. (6.3)

S'U = Vs'ct +M x Sh+6m H + Sb, V A ,+B , V&,, (6.4)Let S1u be the corresponding first variation of U , given from (3.18) bywhere again Sla is chosen so that

V . S 1 u = O in 9, n.S 'u=O on 8 9 .Consider now the functional

R = E+ @ (U , b)d7 = [$(u2 h 2 )+@(U, b ) ]d7, (6.5)s, Lwhere @(a, ) is an arbitrary function of a , b satisfying

D@= D u .a @ s / a a + D b .a @ / a b= 0. (6.6)This condition guarantees tha t @(a, )d7, and so R, s invariant under the evolution(2.3), (2.4). In the above steady state, R s given byR = R, = [+(U 2+H 2 )+ @ ( A ,B )]d7,s


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134and its first variation is given by

V.A . Vladimirov and H . K. MofSatt

S'R = l[ .S1u+H . Sh+Sa - a @ / a A+Sb - a@/aB]7.Substituting for Slu from (6.4) and rearranging, this becomesS I R = [ { ( H + U x M ) . 6 h - ( U x H ) . S m

+(- U.VB+ a@/?lA).Sa+(U*VA +a@/aB)*Sb}dT. (6.9)If we now choose the arbitrary function @ ( A ,B ) to be

@ ( A ,B ) = -&4, B ) (6.10)then, by v irtue of (5.1 l) , the coefficients of the variations Sa,Sb vanish. Moreover,using (5.6), (6.9) then reduces tor r

Hence, since Sh-n = 0 and J = J , (const.) on 3 9 ,S I R = - J , l a 9 n . S m d S = - J , s 9 V . S m d i = 0. (6.11)

Hence the functionalR = [ i (u2+ h2)- (u,b)]s (6.12)is stationary under variations ab ou t a steady state U ( x ) ,H ( x ) ,provided that m, h, a ,b are regarded as the independent variables, U being given by (3.18).We can now construct the second variation of R und er the same con ditions ; this is

6 - {;(S1u2 +Sh2)+U .S2u- '1) d7, (6.13)(6.14)

2 R- s,where(8% being chosen so that V.S2u = 0, n - S 2 u= 0 on a 9 ) , and

6% = VS2a+Sm x Sh+Sb, V 6 a ,Sa, Sb, + Sb, Sb,.a, Sa, +2S21= (6.15)2 12 1aAi aA, aAi aB, C?B,aB ,

According to general principles (Arnold 1965a), the steady solution ( U ( x ) ,H ( x ) ) sstable if S2R is definite in sign for all variations (Sh, Sm, Sa, Sb) satisfying (6.3). It isdifficult however to use this condition, because of the dependence of the integrand in(6.13) on the time-dependent fieldsA(x, ) , B ( x , ) . We seek a n alternative simpler formof variational principle in the next section.7. Alternative form of energy variational principle for 3D steady MHDflows

Let a ( x , ) be a single generalized Lagrangian coo rdina te satisfyingD a = 0 , (7.1)


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Magnetohydrodynamics of ideal fluids . Part 1 135and let b ( x , ) now be defined by

b ( x , t ) = ( w . V ) a , (7.2)with w still defined by (2.19). In view of the conditions (2.20) and (7.1), b is alsoconstant on particle paths, i.e.

D b = 0.In considering steady solutions ( 5 . ), (5.2), we may now suppose tha t a and b are alsosteady, i.e.

(7 3)

a = A ( x ) , b = B ( x ) , (7.4)where ( U * V ) A= ( U . V ) B = 0, B = ( W . V ) A . (7.5)These equations imply that the streamlines of the flow Ulie both on surfaces A = const.and on surfaces B = const., and are therefore the curves of intersection of these twofamilies of surfaces. Each streamline is identified by the correspond ing pair ( A ,B ) . TheBernoulli functions I , J defined by (5.6) are con stant on these streamlines a nd are hencefunctions of A and B :Consider now the first of (5.6):

I = I ( A ,B ) , J = J ( A ,B ) . (7.6)

U XW = V I ( A , B ) = I A V A + I B V B , (7.7)where the suffices A , B correspond to partial differentiation. Tak ing the vector prod uctfirst with V A , then with V B , gives

B U = IBVA x V B , {( W * V ) ) U = -IA V A x V B , (7.8)and hence ( W . 0 )B = - I A B / I B . (7.9)Defining Y ( A ,B ) such thatequations (7.8) both become = lB /B ,U = V A x V Y ,

(7.10)(7.11)

and the pair ( A ( x ) ,Y(x)) constitute the generalized stream-function of the flow U.We now consider the problem (2.1)-(2.5) togethe r with (2.18)-(2.20) and (7.1F(7.3).(Note that we do not here use the representation (3.18) for u.) These equations stilladmit the energy integral E = const., and, for arbitrary @(a,b), we also have theintegral 1% ( a , )d7 = const., (7.12)by virtue of (7.1), (7.3). Hence we may now construct the invariant functional

E E {$(u2+h 2 )+@(a,b)} 7,s,which has steady-state value

E , = 1 $(U2 H 2 )+@ ( A ,B ) }d7.D(7.13)

(7.14)Let us now regard U, , a and m as the indep enden t variable fields which we may subjectto smooth variations Su, Sh, Sa, Sm satisfying as usual

V . S u = V . S h = V - S m= 0, (7.15)


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136 V. A . Vladimirov an d H . K. Mofaattand n . S u = n . S h = O on c79. (7.16)The fields w and b are now the dependent fields defined by (2.19) and (7.2), and withthe first a nd second variations

S 1 w = 6 u + V x ( H + 6 m + S h x M ) ,62w= V x (Sh x Sm ),6lb = ( W . V ) S a + ( S l w . V ) A ,6'b = (6'~. )Sa+ S 2 w .V )A .

(7.17 a )(7.17 b )(7.17 c)(7.17d)

We now show that, with an appropriate choice of @(a,b), the steady flow (5.1), (5.6),(7.4) is a stationary point of the functional R. To this end, we calculate the firstvariationSIR" = lg U.Su+ .h + @ A Sa+ QB S1b)d7. (7.18)The term involving 8% may be treated as follows:

Qi, 6% d7 = QB[( W - V ) a+ (V x S1w ) .VA ] 7s, s,QB ( W S a - V A x S l u ) . n d S - Sa(W . 0 )Qi,d7+

(7.19)where G = V x ( G B V A )= - @ , , V A x V B , (7.20)and S l u = S u + H x S m + S h x M . (7.21)Hence, we obtain

= JI, s,

+ {(U+ ) Su+ H - G x M ) Sh+ G x H ) Sm + Q iA- W - )G B ] u}d7. (7.22)s,Here, the coefficient of Su in the volume integral isU + G = B-'(IB - B Q B B ) A x V B = B-'FB V A x V B , (7.23)where F ( A , B ) E @ + I - B @ , =Moreover, the coefficient of Sa in the volume integral is then

@ A - B Q A B + L @ B BB = &--FB.*I B I B

Hence if we choose @ so that F 0, i.e.I

then both coefficients (7.23) and (7.25) vanish.

(7.24)

(7.25)

(7.26)


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Magnetohydrodynamics of i dea l ju id s . Part 1 137With G = - U , the term s involving Sh and Sm in (7.22) become

J { ( H + U x M ) . S h - ( U x H).Sm)dT, (7.27)

exactly as in (6.9), and these vanish for the same reason as in $6 , using (5.6) an d theconditions (7.15), (7.16) and J = J , (const.) on U.Hence, finally, (7.22) reduces to the surface integralSIR"= J QB( W6a-VA x S1v).ndS.

2 9(7.28)

We now identify three distinct circumstances in which this surface integral vanishes.(i) Obviously SIR"= 0 if QB = 0 o n i39, nd from (7.24) (with F = 0) this holdsprovidedQ(A,B)= -Z(A,B) on a9. (7.29)

Now the condition n . ( V A x V B ) = O on a.9 (7.30)implies that, provided n x V A =+ 0 , the fields A and B restricted to i39 are functionallyrelated, i.e.

B = B,(A), say, on c7.9. (7.31)Moreover, it is evident from (7.26) that if Q o ( A , B ) s a solution of (7.26), then@ ,+ B g ( A ) is also a solution, for arb itrary g ( A ) . Provided B + 0 on 8 9 , his freedomallows us to satisfy (7.29) by choosing

g ( A ) = - I ( A ,B)+@,(A,B))/B, with B = B,(A). (7.32)Hence, providedwe can always choose @(A, ) so that SIR"= 0.(ii) The function A ( x ) may be chosen so that A = const. on 3 9 , i.e. n x V A = 0 a tall points of a.9). We may suppose further that the variation 6a(x) of a satisfies thebound ary condition Sa = 0 on 8 9 so that the boundary value a = A , is maintained inthe perturbed situation). Then obviously SIR"= 0 again.(iii) The streamlines of U ( x )on a9 are closed curves on which A = const. Let r ( A )be the circulation of v round A = const. (which is conserved since w = curl v is a frozenfield). F ro m (7.28)

n x V A + O , B + O for x E a 9 , (7.33)

-SIR = J Q,SaW.ndS- J QB(nx VA).S1udS.2 9 2s (7.34)

Now QB(A,B),with B = B,(A), is a function of A on i39,nd n x V A s parallel to Uon 2 3 . If (7.35)then the second integral may be treated by first integrating round curves A = const.,

n x V A =A A )U / \UI,and then integrating over A. The first operation involves

6 l v . d ~ S1r(A).f -const.Hence sufficient conditions for the vanishing of 6lR"are

Wen = 0 , In x VAI =AA), 61q.4)= 0

(7.36)

on a9. (7.37)


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138 V . A. Vladimirov and H . K . M o f a t tThese conditions are convenient when the flow is either two-dimensional, oraxisymmetric, or has some comparable symm etry.For S2R" we have

PI? = [Su,Su,+Sh, Sh,+ QAA(Sa)'+@BB(S1b)22 G A B a 6% +2Q B S'b] d7, (7.38)where S'w, S'w, S'b, S2b are presented in the form (7.17). By simple operations andusing (7.26) we obtains,22S'k = [a {1I A a + I B S1b)'-- Sa(8'w. V) + U(8mx Sh)}d7BIB B

{2@,[6aS1w+ Sm(Sh.0) ]+ W(Sa) ' } .n dS. (7.39)+s,, B

The flow is stable if S2k s definite in sign for arbitrary choice of Su, Sh, Sa, Sm withS'w, Slb given by (7 .17a, c).8. Conclusions

Starting with the identification of the frozen fields ( h , w ) of ideal magneto-hydrodynamics, we have given a systematic development of the transformationproperties of the MHD equations (thus generalizing the Weber transformation of theEuler equations) and of the underlying Hamiltonian structure of these equations. Wehave then considered steady-state properties, an d have constructed energy variationalprinciples characterizing the steady states. In a subsequent paper, we shall discuss theapplication of these principles to flows and fields with particular symmetries.

This work was carried out during the tenure by V. A. Vladimirov of a VisitingFellowship at DA M TP , Cambridge, under S ERC Research G rant GR/J39748 .R E F E R E N C E S

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v.a. vladimirov and h.k. moffatt- on general transformations and variational principles for the...

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