Closure of two-dimensional turbulence: The role of pressure gradients

Abstract

Inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations. Numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recently proposed self-consistent Gaussian model. An analogous low order closure model for the longitudinal pressure gradient is proposed and its validity is numerically examined. In this case numerical evidence for the presence of higher order terms in the closure is found. The fundamental role of conditional statistics between longitudinal and transverse components is highlighted. PACS number(s) : 47.27.Ak, The existence of two simultaneous inertial ranges in twodimensional turbulence, as a consequence of coupled energy and enstrophy conservation, is one of the most important phenomena in statistical fluid mechanics [1]. At variance with 3D-turbulence, the energy injected into the system at scale ℓ f flows toward the large scales, while the enstrophy cascades down on the small scales. Because of the inverse energy cascade, the Navier-Stokes equations,which rule the evolution of an incompressible (∂ i u i = 0) velocity field, cannot reach a steady-state unless an energy sink at large scales is added. Alternatively one can consider an ensemble of solutions of (1) with a fixed energy value below the condensation level [2], i.e. with an integral scale L(t) (growing in time as t 3/2 ) still much smaller than the system size. Because of the scaling of the characteristic times, the small scales (inertial range) in the system ℓ f ≪ r ≪ L can be considered in a stationary state. One of the most challenging problems is to understand the statistics of velocity fluctuations ∆u(r) = u(x + r) − u(x) [3]. In homogeneous and isotropic turbulence it amounts to study the joint probability density function (PDF) P (U, V, r) of longitudinal U and transverse V velocity differences where ∆u = Ux + Vŷ andx = r r . Recently experimental [4] and numerical [5] investigations in two dimensions have shown that the probability distribution of the pure longitudinal P (U, r) and transversal P (V, r) velocity differences at inertial scales display a close-toGaussian statistics with undetectable intermittency corrections to structure function exponents. Although the establishment of normal scaling in all inverse cascades seem to be generic [6] nevertheless the Gaussianity of the statistics in inverse cascade of the forced two dimensional turbulence is remained to be understood. From (1), a set of equations for generic mixed structure functions, i.e. S n,m (r) ≡ U n V m = A n,m r ξn,m have been obtained [7,8]. In [8] those equations are elaborated from the joint PDF equation. Unfortunately, the PDF equation is not closed, resembling the well known closure problem in turbulence. In the inverse energy cascade regime, dissipative contributions can be neglected so the remaining unclosed terms are the longitudinal and transversal pressure gradients increments. Recently Yakhot [8,9] suggested a selfcon...

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