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Quasilinear theory of the 2 D Euler equation

Motivated by the numerical investigations of [1] Laval et al. (1999), we develop a quasilinear theory of the 2D Euler equation and derive an integro-differential equation for the evolution of the coarse-grained vorticity ω(r, t). This equation respects all the invariance properties of the Euler equation and conserves angular momentum in a circular domain and linear impulse in a channel (as well as in an infinite domain). The explicit energy is not rigorously conserved as it is partly transfered into fine-grained fluctuations but the total energy is conserved. We prove a H-theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence. PACS numbers: 47, 47.10.+g, 47.27.Jv, 47.32.Cc A fundamental difficulty in fluid turbulence is the developement of motion at very small scales, down to the dissipation scale, while numerical computations are limited by the scale of the numerical mesh. One cannot simply ignore the contribution of the small scales, otherwise energy would accumulate at the grid mesh and produce a bottle neck instability. In general, a turbulent viscosity (or hyperviscosity) is introduced in order to provide a source of dissipation and regularize the equations. However, this additional term is relatively ad hoc and breaks the conservation laws of the Euler equation. A small-scale parametrization of 2D turbulence can be constructed on the basis of thermodynamical considerations. Using a Maximum Entropy Production Principle (MEPP), [2] Robert & Sommeria (1992) obtained an “optimal” relaxation equation for the coarse-grained vorticity which respects all the conservation laws of the inviscid dynamics. This approach was extended by [3] Chavanis & Sommeria (1997) who derived a set of equations respecting in addition the invariance properties of the Euler equation. However, this thermodynamical approach is based on the assumption that the system evolves towards a maximum entropy state or a collection of maximum entropy “bubbles” [4]. In this letter, we obtain a small-scale parametrization of 2D turbulence starting directly from the Euler equation. We prove a H-theorem and recover the results of the statistical theory as an approximation of our model. For a two-dimensional incompressible and inviscid flow, the Euler equation can be written: