Finite speed propagation in the relaxation of vortex patches

Abstract

The paper is dedicated to the memory of Louis-Hubert Rosier. Abstract.A degenerate parabolic equation has been proposed by Robert and Sommeria to describe the relaxation towards a statistical equilibrium state for a two-dimensional incompressible perfect fluid with a vortex patch as initial vorticity. In this paper, flows obtained by numerical integration of the Robert-Sommeria equation over a long-time interval are compared with those obtained for the Navier-Stokes equation at high Reynolds number. A finite speed propagation for the extremal values of the vorticity is numerically shown to hold for the Robert-Sommeria equation. A rigorous proof of this (fine) property is also provided.

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

“…We shall now introduce a reduced variational problem equivalent to (36) but expressed in terms of a generalized entropy S[ω] associated with the coarse-grained flow instead of a functional S χ [ρ] associated to the full vorticity distribution. Initially, we want to determine the vorticity distribution ρ * (r, σ) that maximizes S χ [ρ] with the robust constraints E[ω] = E, Γ[ω] = Γ and the normalization condition ρ dσ = 1.…”

“…Therefore, ρ * (r, σ) = ρ 1 [ω * (r), σ] maximizes S χ [ρ] at fixed E, Γ and normalization iff ω * (r) maximizes S[ω] at fixed E and Γ. Therefore, (36) and (52) are equivalent but (52) is simpler to study because it is expressed in terms of the vorticity field ω(r) instead of the full vorticity distribution ρ(r, σ). The equivalence between the stability conditions (38) and (58) is shown explicitly in Appendix B.…”

“…By construction, the relaxed vorticity field ω(r) solves the maximization problem (52). Then, the corresponding distribution (62)-(64) solves the maximization problem (36). Therefore, these relaxation equations tend to the statistical equilibrium state corresponding to the EHT approach.…”

“…The relaxed vorticity field ω(r) solves the maximization problem (52). Then, the corresponding distribution (62)-(64) solves the maximization problem (36). Remark: Since the maximization problem (52) provides a refined criterion of nonlinear dynamical stability for the 2D Euler equations [12], the relaxation equations of this section can also be used as numerical algorithms to construct nonlinearly dynamically stable steady states of the 2D Euler equation independently from the statistical mechanics interpretation [23].…”

“…III C, we note that these maximixation problems also provide sufficient conditions of MRS thermodynamical stability (this is the presentation adopted by Bouchet [22] and Chavanis [23]). Therefore, the study of the maximization problems (36) and (52), and the corresponding relaxation equations, is interesting in these two perspectives.…”