UDC 533.95

HYDROMECHANICS

V. E. ZAKHAROV, E. A. KUZNETSOV

VARIATIONAL PRINCIPLE AND CANONICAL VARIABLES IN MAGNETOHYDRODYNAMICS

(Presented by Academician R. Z. Sagdeev, 22 XII 1969)

B. I. Davydov (¹) formulated a variational principle for the hydrodynamics of an ideal fluid and showed that the canonical variables are the pairs ((\lambda,\mu)) and ((\rho,\Phi)), where

[
\mathbf{v}=\frac{\lambda}{\rho}\nabla\mu+\nabla\Phi .
]

In the present work the analogous problem is solved for the magnetohydrodynamics of an ideal fluid (MHD).

Let us consider the MHD equations:

[
\rho\left(\frac{\partial}{\partial t}\mathbf{v}+(\mathbf{v}\nabla)\mathbf{v}\right)
=-\nabla p+\frac{1}{4\pi}[\operatorname{rot}\mathbf{H},\mathbf{H}];
\tag{1}
]

[
\partial\rho/\partial t+\operatorname{div}\rho\mathbf{v}=0;
\tag{2}
]

[
\partial\mathbf{H}/\partial t=\operatorname{rot}[\mathbf{v},\mathbf{H}];
\tag{3}
]

[
\omega=\frac{\partial}{\partial\rho}(\rho\varepsilon)=\int dp/\rho .
]

Here (\omega) is the enthalpy, and (\varepsilon) is the internal energy.

We make the change of variables:

[
\mathbf{v}=\frac{1}{\rho}[\mathbf{H},\operatorname{rot}\mathbf{S}]+\nabla\Phi .
\tag{4}
]

For arbitrary (\rho,\mathbf{v},\mathbf{H}), from formula (4) one can reconstruct (\mathbf{S}) and (\Phi), moreover nonuniquely, up to the addition to (\Phi) of a scalar (\Phi_0) and to (\mathbf{S}) of a vector (\mathbf{S}_0) satisfying the equation

[
\frac{1}{\rho}[\mathbf{H},\operatorname{rot}\mathbf{S}_0]+\nabla\Phi_0=0.
]

After substituting formula (4) into equation (1), we obtain

[
\nabla\left{\frac{\partial\Phi}{\partial t}+(\mathbf{v}\nabla)\Phi-\frac{\mathbf{v}^2}{2}+\omega(\rho)\right}
+
\left[\frac{\mathbf{H}}{\rho},\operatorname{rot}\left{
\frac{\partial\mathbf{S}}{\partial t}-[\mathbf{v},\operatorname{rot}\mathbf{S}]
+\frac{\mathbf{H}}{4\pi}\right}\right]=0.
\tag{5}
]

Consider the system of equations

[
\frac{\partial\Phi}{\partial t}+(\mathbf{v}\nabla)\Phi-\frac{\mathbf{v}^2}{2}+\omega(\rho)=0;
\tag{6}
]

[
\frac{\partial\mathbf{S}}{\partial t}+\frac{\mathbf{H}}{4\pi}-[\mathbf{v},\operatorname{rot}\mathbf{S}]+\Delta\Psi=0
\tag{7}
]

together with equations (2), (3). In equation (7), (\Psi) is an arbitrary function of the coordinates and time.

By virtue of formula (5), every solution of the system (2), (3), (6), (7) gives rise to some solution of the system of MHD equations (1)—(3). Under the assumption of uniqueness of the Cauchy solution for the system (1)—(3) and the system (2),

(3), (6), (7) the converse is also true: to any solution of the system (1)—(3) one can assign a certain class of solutions of the system (2), (3), (6), (7). For this it is sufficient, from the set of values of (\mathbf v, \mathbf H, \rho) at some instant (t_0), to construct all possible sets of the quantities (S, \Phi) satisfying formula (4), and to take them as the initial conditions for (2), (3), (6), (7).

Thus, the system of equations (2), (3), (6), (7) is equivalent to the system of MHD equations.

Let us choose (\Psi) in such a way that the relation

[
\frac{\partial}{\partial t}\operatorname{div}\mathbf S=0.
\tag{8}
]

is satisfied. Obviously, this gives

[
\Psi=\frac{1}{\Delta}\operatorname{div}[\mathbf v,\operatorname{rot}\mathbf S]+\Psi_0,
]

where (\Psi_0) is an arbitrary solution of Laplace’s equation (\Delta\Psi_0=0). In the particular case when, as (|\mathbf r|\to\infty), (\mathbf v\to0,\ \rho\to\rho_0,\ \mathbf H\to\mathbf H_0), it is convenient to choose the quantity (\Psi_0) so that (\partial\mathbf S/\partial t\to0) as (|\mathbf r|\to\infty). Obviously, then

[
\Psi_0=-\frac{1}{4\pi}(\mathbf H_0,\mathbf r).
]

Consider the functional

[
E=\int\left{\rho\frac{\mathbf v^2}{2}+\rho\varepsilon+\frac{\mathbf H^2}{8\pi}-\Psi\operatorname{div}\mathbf H\right}d\mathbf r .
\tag{9}
]

By direct variation of this functional we find that the equations

[
\frac{\partial\rho}{\partial t}=\frac{\delta E}{\delta\Phi};\qquad
\frac{\partial\Phi}{\partial t}=-\frac{\delta E}{\delta\rho};\qquad
\frac{\partial\mathbf H}{\partial t}=\frac{\delta E}{\delta\mathbf S};\qquad
\frac{\partial\mathbf S}{\partial t}=-\frac{\partial E}{\partial\mathbf H}
\tag{10}
]

coincide with equations (2), (6), (3), (8). Thus, the variables ((\rho,\Phi)) and ((\mathbf H,\mathbf S)) are pairs of canonically conjugate quantities, and the functional (E) is the Hamiltonian.

Introduce the Lagrangian function

[
\mathcal L=\int\left{ \left(\mathbf S,\frac{\partial\mathbf H}{\partial t}\right)+\Phi\frac{\partial\rho}{\partial t}\right}d\mathbf r-E
]

and construct the action functional

[
S=\int \mathcal L\,dt,
]

whose minimization gives a variational principle for the MHD equations. Note that for real motion (\operatorname{div}\mathbf H=0), and the functional (E) coincides with the total energy of the fluid.

In the case of an incompressible fluid the quantity (\Phi) is eliminated with the aid of the continuity equation

[
\Delta\Phi=-\operatorname{div}\frac{1}{\rho_0}[\mathbf H,\operatorname{rot}\mathbf S].
]

The canonical variables are ((\mathbf H,\mathbf S)), and the Hamiltonian has the form

[
E=\int\left{\frac{\rho_0\mathbf v^2}{2}+\frac{\mathbf H^2}{8\pi}-\Psi\operatorname{div}\mathbf H\right}d\mathbf r .
]

Received
27 X 1969

REFERENCES

1. B. I. Davydov, DAN, 69, No. 2, 165 (1949).