Metriplectic relaxation to equilibria

Abstract

Metriplectic dynamical systems consist of a special combination of a Hamiltonian and a (generalized) entropy-gradient flow, such that the Hamiltonian is conserved and entropy is dissipated/produced (depending on a sign convention). It is natural to expect that, in the long-time limit, the orbit of a metriplectic system should converge to an extremum of entropy restricted to a constant-Hamiltonian surface. In this paper, we discuss sufficient conditions for this to occur. Then, we construct a class of metriplectic systems inspired by the Landau operator for Coulomb collisions in plasmas, which is included as special case. For this class of brackets, checking the conditions for convergence reduces to checking two usually simpler conditions, and we discuss examples in detail. We apply these results to the construction of relaxation methods for the solution of equilibrium problems in fluid dynamics and plasma physics.

Figures (22)

• Figure 1: Example of solution of Eq. (\ref{['eq:parallel-diffusion']}) with Hamiltonian (\ref{['eq:islands-h']}). Upper panels: initial condition and final state, compared to the contours of $h$ (black circular curves). Middle panel: visualization of the functional relation between $h$ and $u$ obtained by plotting the points $(h_{ij},u_{ij})$, with $h_{ij}$ and $u_{ij}$ being the values of $h$ and $u$, at the node $(i,j)$ of the computational grid. Lower panel: relaxation time $\tau_h$, computed from Eq. (\ref{['eq:relaxation-time']}) on the contours of the two central (full) islands, as a function of $h$. (For clarity, in the color maps we display the solution $u$ only where $u \geq 10^{-4}$.)
• Figure 2: Metriplectic relaxation of a vortex toward an equilibrium of the reduced Euler equations, using (\ref{['eq:paired-bracket-L2']}) with (\ref{['eq:Euler-S-H']}) and $s(y) = y^2/2$. Top row: initial and final state of the system; the color scheme represents the vorticity $\omega = u - u_\Omega$; white lines represent the contours of the potential $\phi$. Bottom row: relative error of the Hamiltonian and the value of entropy during the evolution. The thick horizontal line indicates the constrained entropy minimum $\mathcal{S}_\eta = \mathcal{H}_0$, cf. Eq. (\ref{['eq:Seta-Euler_periodic']}).
• Figure 3: Visualization of the relation between the potential $\phi$ and the vorticity $\omega$, for the initial and final state of the calculation. The green crosses mark the linear relation for a minimum entropy state, Eq. (\ref{['eq:u-eta_Euler_periodic']}). The data marked "averages" represent the average of the initial condition on the contours of the corresponding potential $\phi$.
• Figure 4: The same as in Fig. \ref{['fig:2']} but for the metric bracket (\ref{['eq:projector-brackets']}).
• Figure 5: Left-hand-side panel: the relation between $\phi$ and $\omega = u - u_\Omega$ for the case of Fig. \ref{['fig:4']}. Right-hand-side panel: difference between the best fit of the exact solution (\ref{['eq:u-eta_Euler_periodic']}) and the relaxed state.

Theorems & Definitions (24)

• Theorem 1: Lyapunov stability
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof