On Arnold's Variational Principles in Fluid Mechanics: This Paper Is Dedicated To V.I.Arnold
On Arnold's Variational Principles in Fluid Mechanics: This Paper Is Dedicated To V.I.Arnold
On Arnold's Variational Principles in Fluid Mechanics: This Paper Is Dedicated To V.I.Arnold
V.A.Vladimirov
K.I.Ilin
Department of Mathematics
Hong Kong University of Science and Technology
Clear Water Bay, Kowloon
Hong Kong
1 Introduction
In the 1960s in the series of pioneering papers V.I.Arnold obtained a number
of fundamental results in the mathematical theory of the dynamics of an ideal
incompressible fluid, especially in the area of hydrodynamic stability (see Arnold
[1965a,b, 1966a,b]).
First, he has developed a new, very effective method in the hydrodynamic
stability theory and proved the theorem on the nonlinear stability of steady two-
dimensional flows that generalizes the well-known linear stability criterion of Rayleigh
(Arnold [1965a,1966a]). Since that time this method, now known as the Arnold
method (or Energy-Casimir method) has been successfully applied to a wide range
of the problems in fluid mechanics, astrophysics, plasma physics etc. (for review
see Holm et al [1985], Marsden and Ratiu [1994], Marchioro and Pulvirenti [1994],
Arnold and Khesin [1998]). Remarkably, the Arnold method have found applica-
tions not only in theoretical sciences but also in such an applied field as geophysical
fluid dynamics (see e.g. McIntyre and Shepherd [1987], Shepherd [1990], Cho et al
[1993], Mu et al [1996]).
Second, V.I.Arnold [1966b] has discovered a close connection between the sta-
bility properties of an ideal incompressible fluid and the geometry of infinite dimen-
sional Lie groups. In Arnold’s theory, the configuration space of ideal incompress-
ible hydrodynamics is identified with the Lie group G = V Dif f (D) of volume-
preserving diffeomorphisms of the domain D, and fluid flows represent geodesics on
G with respect to the metric given by the kinetic energy. One of the consequences
of this theory is that any steady flow of an ideal fluid corresponds to a critical
point of the energy functional restricted to the orbit of coajoint representation of
G. In physical terms, this means that, on the set of all isovortical velocity fields, a
1991 Mathematics Subject Classification. Primary 76C05, 76E99; Secondary 76M30, 76W05.
The second author was supported by Hong Kong Research Grants HKUST701/96P and
HKUST6169/97P..
c
°0000 American Mathematical Society
1
2 V.A.Vladimirov and K.I.Ilin
1. a dynamical system ‘rigid body + fluid’, which may be either a body placed
in an inviscid rotational flow or a body with a cavity containing a fluid;
2. flows of an ideal incompressible fluid with contact discontinuities and, in
particular, flows with discontinuities of vorticity;
3. magnetohydrodynamic flows of an ideal, incompressible, perfectly conduct-
ing fluid.
We shall closely follow the original work of Arnold [1965b] (see also the paper
by Sedenko and Yudovich [1978], who extended Arnold’s principle for free-boundary
flows of an ideal incompressible fluid, and the paper by Grinfeld [1984] who consid-
ered compressible barotropic flows). Our analysis will be based on simple physical
arguments rather than on the general but highly abstract geometric theory. We
shall not discuss the underlying Hamiltonian structures of the considered mechani-
cal systems – they are all known and may be found in the literature (see e.g. Arnold
[1966b], Holm et al [1985], Khesin and Chekanov [1989], Marsden and Ratiu [1994],
Arnold and Khesin [1998]).
We believe that the construction of the variational principles, based on phys-
ically understandable ideas (similar to those of Arnold [1965b]) rather than on
abstract machinery of the differential geometry, is interesting and important from
several viewpoints:
1. the proposed theory may shed some light on the physical meaning of the
related abstract theories as, for instance, in the case of ideal magnetohy-
drodynamics where the variational principle formulated in Section 4 of the
present paper (see also Friedlander and Vishik [1994], Vladimirov and Ilin
[1997b], Vladimirov et al [1998]) clarifies the physical meaning of coajoint
orbits in the related semi-direct product theory of Marsden, Ratiu and We-
instein [1984] (see also Khesin and Chekanov [1989]);
2. in a very simple way it may, in some situations (such as ideal magnetohy-
drodynamics, see Friedlander and Vishik [1994], Vladimirov et al [1998]),
result in stability criteria for general three-dimensional steady states;
3. it is applicable to the systems whose configuration space cannot be identified
with any Lie group (the examples are: free-boundary flows of an ideal fluid,
On Arnold’s variational principles in fluid mechanics 3
see Yudovich and Sedenko [1978] and Section 3 of the present paper, and
the system ‘rigid body + fluid’, see Section 2);
4. it may be modified so as to take account of the effects of dissipation (for
example, for the system ‘body + fluid’ one may include dissipation in finite
dimensional degrees of freedom corresponding to the rigid body).
For discussion of further generalizations and applications of the general geo-
metric theory of V.I.Arnold in continuum mechanics we refer to the recent book by
Arnold and Khesin [1998] (see also Holm et al [1985], Simo et al [1991a,b], Marsden
and Ratiu [1994]).
We conclude this introduction with a statement of Arnold’s original variational
principle.
1 For existence of another invariant, the helicity, one must assume that the vorticity ! ≡ curlu
To find the general form of infinitesimal variations the field u satisfying this con-
dition we introduce family of vector fields ũ(x̃, ²) such that the value ² = 0 corre-
sponds to the steady solution (1.7), i.e. ũ(x̃, ²)|²=0 = U(x). For any ², ũ(x̃, ²) is
divergence-free and parallel to D. Assuming that ² is small we define the first and
the second variations of the field u as
¯ ¯
¯ ¯
δu ≡ ũ² ¯ , δ 2 u ≡ 21 ũ²² ¯ .
²=0 ²=0
we obtain
I n ³ ´ ³
1 2
² δu − ξ × Ω + 2² δ2u − χ × Ω
γ
´o
− ξ × δω − ξ × (δω − [ξ, Ω]) · dl + o(²2 ) = 0 ,
where α(x) and β(x) are scalar functions, which, in the case of singly-connected
domain D, are uniquely determined by the conditions
∇ · δu = ∇ · δ 2 u = 0 in D , δu · n = δ 2 u · n = 0 on ∂D .
Variational principle. Now we shall show that the first variation of energy (1.4)
with respect to variations of the velocity field u of the form (1.12), (1.13) vanishes.
We have
¯ Z Z
d ¯¯
δE ≡ = U · δudτ = U · (ξ × Ω − ∇α)dτ
d² ¯²=0 D D
Z Z
= ξ · (Ω × U)dτ = − ξ · ∇(P + 12 U2 )dτ = 0 .
D D
Then, it follows from (1.8) that Ω = Ω(Ψ) where Ω = −∇2 Ψ, and (1.14) reduces to
Z ³ ´
2 dΩ
δ E=2 1
(δu)2 − (ξ · ∇Ψ)2 dτ . (1.15)
D dΨ
Evidently, the second variation (1.15) is positive definite provided that dΩ/dΨ ≤ 0.
Equations (1.1), (1.2), (2.1)-(2.3) with boundary conditions (2.5) give us the com-
plete set of equations governing the motion of the system ‘body + fluid’.
The conserved total energy of the system is given by
E = Ef + Eb = const , Eb ≡ T + Π ,
Z
Ef ≡ 1
2 u2 dτ , dτ ≡ dx1 dx2 dx3 . (2.6)
Df
Basic state. Steady solutions of the problem (1.1), (1.2), (2.1)-(2.3), (2.5) given
by
vα = 0 , qα = Qα , r = R = 0 , u = U(x) , p = P (x) ,
w = W = 0 , σ = Σ = 0 , Pij = P0ij = δij (2.7)
satisfy the equations
Ω×U = −∇H , H ≡ P + 12 U2 , ∇ · U = 0 in Df 0 ; (2.8)
Z µ ¶
∂Π ∂R £¡ ¢ ¤ ∂Σ
− + n· + x−R ×n · P dS = 0 ; (2.9)
∂Qα ∂Db0 ∂Qα ∂Vα
and boundary conditions
U · n = 0 on ∂D and on ∂Db0 . (2.10)
8 V.A.Vladimirov and K.I.Ilin
2.2 Variational principle. We shall show that the total energy of the dy-
namical system ‘body + fluid’ has a stationary value at the steady solution (2.7) on
the set of all possible fluid flows that are isovortical to the basic flow. The isovortic-
ity condition is the same as in Arnold’s principle: we admit only such variations of
the velocity field u that preserve the velocity circulation over any material contour.
It is however more convenient to reformulate Arnold’s isovorticity condition in a
form first proposed in Vladimirov [1987b].
Consider a family of transformations
x̃ = x̃(x, ²) , q̃α = q̃α (²) . (2.11)
depending on a parameter ² ≥ 0 where the functions x̃(x, ²) and q̃α (²) are twice
differentiable with respect to ² and the value ² = 0 corresponds to the steady
solution (2.7):
x̃(x, 0) = x , q̃α (0) = Qα . (2.12)
The transformations defined by eqns. (2.11), (2.12) are similar to those introduced
in Section 1 and can be interpreted as a ‘virtual motion’ of the system ‘body +
fluid’ where ² plays the role of a ‘virtual time’, x̃(x, ²) is the position vector at the
moment of ‘time’ ² of a fluid particle whose position at the initial instant ² = 0
was x (in other words, x (x ∈ Df 0 ) serves as a label to identify the fluid particle,
while x̃(x, ²) represents its trajectory) and where the functions q̃α (²) determine
the position and the orientation of the rigid body at the moment of ‘time’ ². In
such a ‘motion’, the domain Df 0 = D̃f (0) evolves to a new one D̃f (²) which is
completely determined by the position and the orientation of the rigid body, i.e.
by the generalized coordinates q̃α (²).
Functions x̃(x, ²), q̃α (²) are specified through yet another set of functions
ξ(x̃, ²), hα (²) by the equations (cf (1.9))
dx̃/d² = ξ(x̃, ²) , dq̃α /d² = hα (²) , (2.13)
where hα (²) are arbitrary differentiable functions, while ξ(x̃, ²) is an arbitrary
divergence-free vector field differentiable with respect to ² and satisfying the con-
ditions
£ ¡ ¢¤
ξ · n = 0 on ∂ D̃ , ξ · n = r̃² + ϕ̃² × x̃ − r̃ · n on ∂ D̃b (²) . (2.14)
In (2.14),
∂r̃ ∂ P̃jl
r̃² ≡ hα , ϕ̃i² ≡ − 21 eijk P̃kl hα . (2.15)
∂ q̃α ∂ q̃α
In terms of ‘virtual motions’ the functions ξ(r̃, ²) and hα (²) entering equations
(2.13) have a natural interpretation as the ‘virtual velocities’ of the fluid and the
rigid body. The conditions (2.14) mean that in the ‘virtual motion’ there is no fluid
flow through the rigid boundaries.
The actual velocity field of the fluid and the actual generalized velocities of
the rigid body in the ‘virtual motion’ are described by twice differentiable (with
On Arnold’s variational principles in fluid mechanics 9
respect to ²) functions ũ(x̃, ²) and ṽα (²) such that the value ² = 0 corresponds to
the steady state (2.7):
¯ ¯
¯ ¯
ũ(x̃, ²) ¯ = U(x) , ṽα (²) ¯ = 0 . (2.16)
²=0 ²=0
where, as before, w̃, σ̃ are considered as functions of ṽα (²) and q̃α (²). The evolution
with the ‘time’ ² of the generalized velocities ṽα (²) is prescribed by the equation
dṽα /d² = gα (²) (2.18)
with some differentiable function gα (²). Note that the functions gα (²) and hα (²)
which determine the evolution in the ‘virtual motion’ of the generalized velocities
and coordinates are both arbitrary, so that ṽα (²) and q̃α (²) vary independently.
The evolution of the field ũ(x̃, ²) is defined through the evolution of vorticity
˜ × ũ by the equation
ω̃(x̃, ²) ≡ ∇
ω̃ ² = [ξ, ω̃] . (2.19)
Equation (2.19) means that the vorticity field ω̃ is considered as a passive vector
advected by the ‘virtual flow’ rather than as a field related with the ‘virtual velocity’
ξ by curl-operator; in other words, the evolution of ω̃ is the same as that of a
material line element δl or as the evolution of a frozen-in magnetic field in ideal
MHD. Yet another meaning of the equation (2.19) is that the circulation of the
velocity field ũ(x̃, ²) round any closed material curve is conserved in the ‘virtual
motion’, this, in turn, implies that equation (2.19) is equivalent to Arnold’s original
isovorticity condition (see Section 1).
On integrating equation (2.19) we obtain (cf (1.12))
˜
ũ² = ξ × ω̃ − ∇α (2.20)
with a certain function α(x̃, ²) which can be found from the conditions on ũ² that
follows from (2.17).
Remark. Though equation (2.20) also could be used as a primary condition for
defining the evolution of the field ũ(x, ²), from a view-point of physical interpreta-
tion equation (2.19) seems preferable.
Assuming that ² is small we define the first and the second variations of the
velocity field of the fluid u and the generalized velocities and coordinates of the
rigid body vα , qα as follows
δx ≡ ξ|²=0 , δu ≡ ũ² |²=0 , δ 2 u ≡ 12 ũ²² |²=0 , δvα ≡ vα² |²=0 etc. (2.21)
In (2.21), δx is the Lagrangian displacement of the fluid element whose position at
time t in the undisturbed flow was x. The first and the second variations of the
energy (2.6) considered as a functional of ũ(x̃, ²), ṽα (²), q̃α (²) are, by definition,
¯ ¯
¯ ¯
δE ≡ dE/d² ¯ , δ 2 E ≡ 12 d2 E/d²2 ¯ .
²=0 ²=0
for any function F (x̃, ²) (see e.g. Batchelor [1967]). With help of this formula we
obtain
Z n ³ Z
d ¯¯ ´ o
1 2
¡ ¢
¯ Ef = ξ · Ω × U + U · ∇α dτ + 2 U ξ · n dS .
d² ²=0 Df 0 ∂ D̃b0
By using (2.8), Green’s theorem and the boundary conditions (2.14), this can be
transformed to
Z
d ¯¯ £ ¡ ¢¤
¯ E f = − P δr + δϕ × x − r · ndS . (2.24)
d² ²=0 ∂ D̃b (0)
The comparison of (2.25) with (2.9) then shows that δE = 0. Thus, we have proved
the following.
Proposition 2.1 The energy of the system ‘body + fluid’ has a stationary
value at any steady solution of the form (2.7) provided that we take account only of
‘isovortical’ fluid flows.
This result is a natural generalization of Arnold’s variational principle to the
dynamical system ‘body + fluid’.
2.3 The second variation. The second variation of the energy (2.6) evalu-
ated at the stationary point is given by the expression Vladimirov and Ilin [1997a]
δ2 E = δ 2 EA + δ 2 Ec + δ 2 Eb ,
Z n
¡ ¢2 ¡ ¢o
δ 2 EA ≡ 1
2 δu + U · δx × δω dτ ,
Df 0
Z n o¡ Z
2 1
¢ 1
¡ ¢³ ´
δ Ec ≡ 2 2U · δu − δy · ∇P δy · n dS + 2 δy · n δx · ∇H dS
∂Db0 ∂Db0
Z n £ o
¤
− 12 P n · δr × δϕ + Aαβ δqα δqβ + Bαβ δqα δqβ dS ,
∂Db0
∂2Π
δ 2 Eb ≡ 1
2 M δwi δwi + 12 Iik δσi δσk + 1
2 δqα δqβ , (2.26)
∂Qα ∂Qβ
On Arnold’s variational principles in fluid mechanics 11
Now qα = (r, φ), vα = (ṙ, φ̇) where we use the notation φ = (φ1 , φ2 , φ3 ) ≡ (ψ, θ, φ).
The expression for the second variation given by eqns. (2.26) remains almost un-
changed except that now δϕ = δφ, δw = δ ṙ, δσ = δ φ̇ = (δ ψ̇, δ θ̇, δ φ̇), Aαβ = 0 and
Bαβ δqα δqβ = B̃ik δφi δφk where matrix [B̃ik ] is given by
£ ¤ 0 −ez · (x × n) ey · (x × n)
B̃ik ≡ −ez · (x × n) 0 −ex · (x × n)
ey · (x × n) −ex · (x × n) 0
Moreover, with help of the equilibrium condition (2.9) it can be shown that
Z
− 21 P B̃ik δφi δφk dS = Πψ δθδφ − Πθ δψδφ + Πφ δψδθ
∂Db0
variation simplifies to
δ2 E = δ 2 E + δ 2 Ec + δ 2 Eb ,
Z An
¡ ¢2 ¡ ¢o
2δ 2 EA = δu + U · δx × δω dτ ,
Df 0
Z n o¡ Z
¢ ¡ ¢³ ´
2δ 2 Ec = 2U · δu − δr · ∇P δr · n dS + δr · n δx · ∇H dS ,
∂Db0 ∂Db0
2
∂ Π
2δ 2 Eb = M δ ṙi δ ṙi + δri δrk . (2.29)
∂Ri ∂Rk
If, in addition, the basic flow is such that Ω · n = 0 on ∂Db0 , then it can be shown
from eqn. (2.8) that H = const on ∂Db0 , and δ 2 Ec in (2.29) reduces to the equation
Z n ¡ ¢o¡ ¢
2δ 2 Ec = 2U · δu + δr · ∇ 12 U2 δr · n dS .
∂Db0
Rigid body with fluid-filled cavities. All the results described above were ob-
tained for a rigid body placed in an arbitrary rotational inviscid flow. However it
is easy to see that these results are equally valid for a rigid body with a cavity
containing an ideal fluid. The only difference between these two problems lies in
interpreting the boundary ∂Db , namely, for a body with a fluid-filled cavity we con-
sider the surface ∂Db as an internal (for the body) boundary which represents the
boundary of the cavity, i.e. ∂Db is an outer boundary of the fluid domain Df which
is completely filled with a fluid. With this interpretation the basic state given
by equations (2.7)-(2.9) represents an equilibrium of a rigid body with a cavity
containing a fluid which in turn is in a steady motion with velocity field U(x).
Remark. Evidently, the theory developed in the previous sections can be easily
modified to cover the situation when there are n rigid bodies in a fluid or the
situation when a cavity in the rigid body contains fluid and other rigid bodies.
For a general three-dimensional basic state (2.7) the second variation given
by (2.26) (and by (2.29) for a spherical body) is indefinite in sign. Nevertheless,
for some particular situations (such as a body in an irrotational flow, a force-free
rotation of a body with fluid-filled cavity and some two-dimensional problems), it
is possible to find sufficient conditions for sign-definiteness of δ 2 E and, hence, to
prove the linear stability of corresponding steady states (see Vladimirov and Ilin
[1994], Vladimirov and Ilin [1997a]).
closed curves γ (see (1.10)) that do not intersect the contact surface S, or, in other
words, that entirely lie either in D+ or in D− . Then, from (1.12), (1.13), we have
δu± = ξ ± × Ω± − ∇α± or δω ± = [ξ ± , Ω± ] , (3.13)
2 ± ± ± ± ± ±
δ u = ξ × δω + χ × Ω − ∇β . (3.14)
Scalar functions α(x) and β(x) are determined by the conditions that ∇ · δu± =
∇ · δ 2 u± = 0 in D0± , δu± · n = δ 2 u± · n = 0 on S ± and and by the£ boundary
¤
conditions on S0 that may be obtained by differentiating the condition ũ · n = 0
on S̃ with respect to ² at ² = 0.
Variational principle. Let us show that the first variation of the energy
Z ³ ´ Z ³ ´
E= ρ+ 21 |u+ |2 + Φ dτ + ρ− 12 |u− |2 + Φ dτ (3.15)
D+ D−
with respect to variations of the form (3.13), (3.14) vanishes in the steady state
(3.5).
We have
XZ Z
£ ¤
δE = ρ± U± · δu± dτ + (ξ · n) 12 ρU2 + ρΦ dS .
D± S0
P
Here denotes the sum of the corresponding integrals over the domains D± .
Substitution of (3.13) in this equation results in
XZ ³ ´ Z
£ ¤
± ± ± ±
δE = ρ U · ξ × Ω − ∇α dτ + (ξ · n) 12 ρU2 + ρΦ dS
D± S0
XZ Z
£ ¤
=− ρ± ξ ± · ∇H ± dτ + (ξ · n) 12 ρU2 + ρΦ dS
D ± S0
Z ³£ ¤ £ Z
1 2
¤´ £ ¤
=− (ξ · n) H + 2 ρU + ρΦ dS = − (ξ · n) P = 0 .
S0 S0
In general, this second variation is indefinite in sign. There are however certain
particular situations (including particular classes of variations) for which it is def-
inite in sign. We shall not discuss all of them here. Instead, we shall concentrate
our efforts on one important subclass of flows with contact discontinuities - on flows
with discontinuous vorticity.
general situation considered above is that, in addition to (3.3), we impose one more
restriction: tangent to S(t) components of velocity are also continuous, i.e.
£ ¤
u · σ α = 0 (α = 1, 2) on S(t) , (3.17)
where σ α (α = 1, 2) are independent unit vectors tangent to S(t).
Steady flows. Consider now a steady solution (3.5) of the problem (3.1)-(3.3),
(3.17) that satisfy (3.6)-(3.8) and, in addition, the following conditions
£ ¤
U · σ α = 0 (α = 1, 2) , on S0 . (3.18)
Boundary conditions (3.8) and (3.18) impose a certain restriction on possible dis-
continuities of vorticity. Note first that, in view of (3.7), (3.8) and (3.18), [H0 ] = 0
on S0 , and therefore [σ α · ∇H0 ] = 0 on S0 . On taking scalar product of equation
(3.6) with σ α and using (3.8), we obtain
¡ ¢ ¡ ¢
U · σ β ρΩ± · σ β × σ α = −σ α · ∇H0± .
whence, with help of the formula
σα × σβ
eαβ n =
|σ 1 × σ 2 |
(where eαβ is a unit alternating tensor), we find that
¡ ¢ £ ¤ £ ¤
eβα U · σ β ρ Ω · n /|σ 1 × σ 2 | = − σ α · ∇H0 = 0 . (3.19)
Therefore, in the steady flow (3.5) the vorticity field can have only tangent discon-
tinuity on S0 :
£ ¤ £ ¤
Ω · n = 0, Ω · σ α 6= 0 (α = 1, 2) on S0 . (3.20)
Similarly, it can be shown that another consequence of (3.7), (3.8) and (3.18) is
£ ¤
n · ∇P = 0 on S0 . (3.21)
One more formula useful formula
£ ¤ £ ¤
n · ∇H0 = −ρ|σ 1 × σ 2 |eαβ Ω · σ α (U · σ β ) (3.22)
is obtained by taking scalar product of equation (3.6) with n.
The second variation of the energy. Variational principle of previous subsection
still holds for steady flows with vorticity discontinuities. But now we do not need
to consider discontinuous fields ξ(x̃, ²) and ũ(x̃, ²), so that we assume that they are
continuous
£ ¤ £ ¤ £ ¤ £ ¤
ξ · n = ξ · σ α = ũ · n = ũ · σ α = 0 on S̃ ,
and, hence,
£ ¤¯ £ ¤¯ £ ¤ £ ¤
ξ · n ¯²=0 = ξ · σ α ¯²=0 = δu · n = δu · σ α = 0 on S0 . (3.23)
In view of (3.18), (3.20)-(3.23), the second variation (3.16) simplifies to
XZ ³ ¡ ¢´
δ 2 E = 12 ρ (δu)2 + δω ± · U × ξ dτ
±
ZD
£ ¤
− 21 ρ(ξ · n)2 |σ 1 × σ 2 |eαβ Ω · σ α (U · σ β )dS . (3.24)
S0
If there is no discontinuity of vorticity then, evidently, (3.24) reduces to Arnold’s
second variation (1.14). The second variation (3.24) is, in general, indefinite in sign
because of the volume integrals in (3.26). As in Arnold’s case, if both the basic
flow and the perturbation have a symmetry then there are situations when δ 2 E is
definite in sign.
16 V.A.Vladimirov and K.I.Ilin
Two-dimensional problem. Let both the basic steady flow and the variations
be two-dimensional, i.e. the fields U, ξ, have only two non-zero components and
depend only on two coordinates on the plane of motion, then
U = (U1 (x, y), U2 (x, y), 0) , Ω ± = Ω±
0 ez , F0 = F0 (x, y) ,
σ 1 = ez , σ 2 = n × ez n = ∇F0 /|∇F0 | at F0 = 0 . (3.25)
Let Ψ be stream function for U such that U1 = Ψx , U2 = −Ψy . Then, the vorticity
Ω± 2 ± ±
0 = −∇ Ψ and stream function Ψ are functionally dependent Ω0 = Ω (Ψ) and
the second variation (3.24) takes the form
XZ ³ dΩ± ´
δ 2 E = 21 ρ (δu)2 − 0
(ξ · ∇Ψ)2 dτ
D± dΨ
Z
£ ¤
1
− 2 Ω0 ρ(ξ · n)2 |U|dS . (3.26)
S0
2
It is clear that δ E is positive definite provided that
£ ¤
dΩ± 0 /dΨ < 0 in D ,
±
Ω0 < 0 on S0 . (3.27)
Thus, we can formulate the following.
Proposition 3.2 A two dimensional steady flow with discontinuity of vorticity
along a contact line S0 is linearly stable to two-dimensional isovortical perturbations
provided that the conditions (3.27) are satisfied.
In a particular case of a flow with piecewise constant vorticity (Ω+ 0 = const in
D , Ω−
+
0 = const in D −
), these sufficient conditions for stability
¤ reduce to only one
condition on the sign of the vorticity jump across S0 : [Ω0 < 0.
More examples of stable two-dimensional flows with discontinuous vorticity,
can be found in Vladimirov [1988].
4 Ideal magnetohydrodynamics
Here we discuss a variational principle for a steady three-dimensional magne-
tohydrodynamic flow of an ideal incompressible fluid which is a generalization of
Arnold’ principle for a steady three-dimensional inviscid flow. We formulate a cer-
tain ‘generalized isovorticity condition’ and then show that on the set of all possible
velocity fields and magnetic fields satisfying this condition the energy has a critical
point in a steady solution of the governing equations. The second variation of the
energy is calculated. The ‘modified vorticity field’ introduced by Vladimirov and
Moffatt [1995] and its connection with present analysis is also discussed.
4.1 Basic equations. Consider an incompressible, inviscid and perfectly con-
ducting fluid contained in a domain D with fixed boundary ∂D. Let u(x, t) be the
velocity field, h(x, t) the magnetic field (in Alfven velocity units), p(x, t) the pres-
sure (divided by density), and j = ∇ × h the current density. Then the governing
equations are
³ ´
Du ≡ ∂/∂t + u · ∇ u = −∇p + j × h , (4.1)
ht = [u, h] ≡ ∇ × (u × h) , (4.2)
∇ · u = ∇ · h = 0. (4.3)
Equation (4.2) implies that h is frozen in the fluid, its flux through any material
surface is conserved. We suppose that the boundary ∂D is perfectly conducting
On Arnold’s variational principles in fluid mechanics 17
and therefore the magnetic field h does not penetrate through ∂D. The boundary
conditions are then
We suppose further that at t = 0, the fields u and h are smooth and satisfy (4.3)
and (4.4), but are otherwise arbitrary.
The equations (4.1)-(4.3) with boundary conditions (4.4) have three quadratic
integral invariants: the energy
Z ³ ´
E=2 1
u2 + h2 dτ , (4.5)
D
(Woltjer 1958). By arguments of Moffatt [1969], the helicities HM and HC are both
topological in character.
Taking curl of equation (4.1) we obtain
where ω = ∇ × u is the vorticity field. Equation (4.8) implies that vortex lines
are not frozen in the fluid unless the Lorentz force j × h is irrotational. However,
the flux of vorticity through any material surface bounded by a closed magnetic
line (which, according to (4.2), is also a material line) is conserved. This fact has a
consequence that the circulation of the velocity round any closed h-line is conserved:
I
Γh = u · dl = const . (4.9)
γh (t)
In (4.9), γh (t) is a closed h-line. The invariants Γh will play the key role in the
subsequent analysis.
Steady MHD flows. We now consider a steady solution of (4.1)-(4.4)
where S is any surface bounded by the curve γ and g ² S is its image under
the transformation g ² ;
2. the circulation of the velocity u1 round the original closed h-line γh is equal
to the circulation of u2 round its image g ² γh under the transformation g ² :
I I
u1 · dl = u2 · dl . (4.14)
γh g ² γh
To find the general form of infinitesimal variations of the fields u and h that satisfy
the ‘generalized isovorticity condition’ (expressed by (4.13), (4.14)) we introduce
another family of transformations ũ(x̃, ²), h̃(x̃, ²) such that the value ² = 0 corre-
sponds to the steady solution (4.10):
¯ ¯
¯ ¯
ũ(x̃, ²) ¯ = U(x) , h̃(x̃, ²) ¯ = H(x) .
²=0 ²=0
For small ² the generalized isovorticity conditions (4.13), (4.14) reduce to (cf (1.11))
¯ Z ¯ Z
d ¯ d2 ¯
² ¯¯ h̃ · dS + 12 ²2 2 ¯¯ h̃ · dS + o(²2 ) = 0 , (4.15)
d² ²=0 g² S d² ²=0 g² S
¯ Z ¯ Z
d ¯¯ 1 2 d
2 ¯
¯
² ¯ ũ · dl + 2 ² ũ · dl + o(²2 ) = 0 . (4.16)
d² ²=0 g² γh d²2 ¯²=0 g² γh
From (4.15), using the formula (see e.g. Batchelor [1967])
Z Z ³ ´
d
h̃ · dS = h̃² + (ξ · ∇)h̃ − (h̃ · ∇)ξ · dS ,
d² g² S g² S
On Arnold’s variational principles in fluid mechanics 19
we obtain
Z n ³
¡ ¢ 1 2
² δh − [ξ, H] + 2² δh − [χ, H]
S
£ ¤´o
− [ξ, δh] − ξ, δh − [ξ, H] · dS + o(²2 ) = 0 .
Whence, using the fact that S is an arbitrary material surface, we deduce that
¯
¯
δh = [ξ, H] , δ 2 h = [ξ, δh] + [χ, H] , χ ≡ ξ ² ¯ . (4.17)
²=0
2
Note that the variations δh, δ h satisfy the conditions
∇ · δh = ∇ · δ 2 h = 0 in D , δh · n = δ 2 h · n = 0 on ∂D .
From (4.16), we obtain
I n ³ ´ ³
² δu − ξ × Ω + 21 ²2 δ 2 u − χ × Ω
γh
´o
− ξ × δω − ξ × (δω − [ξ, Ω]) · dl + o(²2 ) = 0 . (4.18)
2 To satisfy the condition (4.18) it is not necessary for to be a divergence-free field, so that
this property is our assumption. We shall use it below while calculating the first variation of the
energy functional.
20 V.A.Vladimirov and K.I.Ilin
Variational principle. Now we shall show that the first variation of the energy
(4.5) vanishes with respect to variations of the fields h, u of the form (4.17), (4.19).
We have
¯ Z ³ ´
dE ¯¯
δE ≡ = U · δu + H · δh dτ
d² ¯ D
Z ³ ²=0 ´
= U · (ξ × Ω + η × H − ∇α) + H · (∇ × (ξ × H)) dτ
ZD ³ ´
= ξ · (Ω × H − J × H) − η · (U × H) dτ
ZD ³ ´
= −ξ · ∇K + η · ∇I dτ = 0 . (4.23)
D
Proposition 4.1 On the set of all possible fields h and u satisfying the gen-
eralized isovorticity conditions (4.13), (4.14) the energy functional (4.5) has a sta-
tionary value in the steady state (4.10).
4.3 The second variation. . Let us now calculate the second variation of
the energy at the stationary point. We have
¯ Z ³ ´
2 1 d2 E ¯¯ 1 2 1 2 2 2
δ E≡ = (δu) + (δh) + U · δ u + H · δ h dτ .
2 d²2 ¯²=0 D
2 2
After substitution of the equations (4.17), (4.22) and integration by parts, it may
be shown that all the terms containing χ and f vanish due to the equations (4.12)
and the boundary conditions on ∂D for the fields χ, U and H and the second
variation takes the form
Z ³ ´
2
δ E=2 1
(δu)2 + (δh)2 + δω · (U × ξ) + δh · (U × η + J × ξ) dτ . (4.24)
D
Suppose now that δu and δh are identified with infinitesimal perturbations to the
basic steady state (4.10) whose evolution is governed by the appropriate linearized
equations. Then the following statement holds.
4.4 Another form of variational principle for steady MHD flows. The
theory developed above heavily uses the fact that the circulation of velocity round
any closed h-line is conserved. It is known, however, that the situation when
magnetic lines are all closed is very particular, usually even in steady MHD flows
almost all magnetic lines are not closed. It is necessary therefore to modify our
theory so as to cover such situations.
The field η turns out to be closely related with a certain generalization of the
vorticity for MHD flows, namely with the ‘modified vorticity field’ w introduced by
Vladimirov and Moffatt [1995]. We therefore start with a new approach (different
from that of Vladimirov and Moffatt [1995]) to introducing the field w.
Modified vorticity field. The variational principle of section 4.2 was based on
the fact that the circulation Γh of velocity round any closed h-line is conserved. It
is easy to see that Γh is invariant with respect to transformations of the form
u → v = u + h × m + ∇c (4.25)
w ≡ ∇ × v = ω + [h, m] . (4.26)
wt = [u, w] . (4.27)
On substituting ω t from equation (4.8) and using the Jacobi identity we obtain
This means that up to an arbitrary field commuting with h the field m satisfies the
equation
mt = [u, m] + j , (4.29)
which is exactly the same as that of Vladimirov and Moffatt [1995]. Thus, in our
approach m appeared as a generator of transformations that leave the circulations
Γh unchanged, while the equation (4.29) is a consequence of the requirement that
the circulation of the ‘modified velocity’ v round any material contour is conserved.
22 V.A.Vladimirov and K.I.Ilin
The second variation. It can be shown by standard calculations that the second
variation of the energy evaluated in the steady state (4.10) is given by
Z ³
δ 2 E = 21 (δu)2 + (δh)2 + δω · (U × ξ)
∂D
¡ ¢´
+ δh · U × (δm − [ξ, M]) + J × ξ dτ . (4.35)
Comparing this formula with equation (4.24), we conclude that they coincide pro-
vided that
η = δm − [ξ, M] . (4.36)
The relation between the fields η given by (4.36) is the same as obtained in
Vladimirov and Ilin [1997b] from the analysis of corresponding linearized equations.
Note that if we identify variations δu, δh, δm with infinitesimal perturbations to
the basic state (4.10) that obey the corresponding linearized equations, then the
relation (4.36) gives us an evolution equation for the field η (see Vladimirov and
Ilin [1997b]).
5 Conclusion
We started with formulation of Arnold’s variational principle for steady three-
dimensional flows of an ideal incompressible fluid. Then we established the analo-
gous variational principles for steady states of a system ‘body + fluid’, for steady
flows of an ideal incompressible fluid with contact discontinuities and for steady
magnetohydrodynamic flows of ideal, perfectly conducting fluid.
We should note that all these variational principles can be generalized so as
to cover the situations when the basic state is unsteady provided that it is steady
relative to coordinate system which either moves along a fixed axis with constant
velocity or rotates around a fixed axis with constant angular velocity. For a system
‘body + fluid’ such principles have been established and exploited for obtaining
stability conditions in Vladimirov and Ilin [1997a].
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