Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids

Abstract

The growth rates of scalar gradients are studied numerically and analytically in a family of two-dimensional (2D) incompressible fluid equations related to the surface quasigeostrophic (SQG) equation. The active scalar is related to the stream function ψ by θ=(-△){α/2}ψ (0 ≤ α ≤ 2). A notable difference is observed in a comparison of the instantaneous growth rates in L{p} and in L{∞} norms, depending on the stage of the time evolution. The crux is the phase-shift effect of singular integral operators, which displaces the peak location of the scalar gradient from that of the strain rate. On this basis, a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow-up. Moreover, it is found in the long-time evolution that a solution of the SQG equation (whose regularity is not known) is less singular than that of the 2D Euler equations (known to be regular) on the time interval covered by this computation. This consistently expands an earlier observation by Majda and Tabak [Physica D 98, 515 (1996).] in some detail. A 1D model problem is discussed to illustrate the present method, and extensions to the 3D case are also are briefly discussed.