Spectra of local and nonlocal two-dimensional turbulence

Abstract

Abstract--We propose a family of two-dimensional incompressible fluid models indexed by a parameter o~e [0,~], and discuss the spectral scaling properties for homogeneous, isotropic turbulence in these models. The family includes two physically realizable members. It is shown that the enstrophy cascade is spectrally local for o~ < 2, but becomes dominated by nonlocal interactions for cr > 2. Numerical simulations indicate that the spectral slopes are systematically steeper than those predicted by the local scaling argument.Incompressible fluid turbulence in two dimensions is of interest because of its applications in geophysical problems [1] and because of its role as a computationally tractable testbed for theories, Spectrally local similarity theories of the inertial range spectrum based on Kolmogorov's phenomenology [2,3] often provide a starting point for such studies. The enstrophy cascading inertial range in 2D turbulence has a pathology in this regard; assuming the local similarity spectrum, the aggregate straining effects of eddies larger than a given scale L just fail to be dominated by the effects of eddies with scales similar to L. Hence, the enstrophy range of conventional 2D turbulence is precisely on the threshold of locality [3,4]. In this note we examine a family of modified 2D fluid equations, which span the range between spectrally local and strongly nonlocal behavior.The equation for advection of a conserved scalar q in a velocity field with streamfunction ~p is: 5,q + J(% q) = 0 (la)where J(A, B) = S~AOyB -axB~yA. Advection of even a passive scalar decoupled from ~p embraces a rich variety of phenomena [5,6], but the 2D hydrodynamic equations are distinguished by coupling between q and ~p, which renders the equation nonlinear. The form of the coupling determines the degree of locality. We consider the family of couplings defined in Fourier transform space by 1111

Tracer microstructure in the large-eddy dominated regime

“…The squared-gradient PDFs for the turbulence models of Pierrehumbert et al [23] show behavior that is quite similar to those we have seen for the passive tracer case. Some representative curves are shown in Fig.…”

“…The substantial independence of o~ indicates that although o~ = 1 is formally more local spectrally than the other cases, and although the tracer filaments curdle rather than remaining lamellar (see ref. [23]), the curdling process is evidently intermittent enough that the gradient field over most of the domain is still controlled by large-scale processes.…”

“…A counterpoint to these results is provided by analysis of the pseudo-vorticity PDFs for the family of turbulence models simulated in Pierrehumbert et al [23]. These are also equilibrium runs, but rather than the tracer (in this case pseudo-vorticity) values being reset to a fixed value at each time-step, only the large-scale projection is re-set, and even then the phase is randomized.…”

Instability of surface‐temperature filaments in strain and shear

“…Here we investigate the striking 'curdling' at small scales that is apparent in many SQG turbulence simulations (Pierrehumbert, et al, 1994;Held, et al, 1995;Juckes, 1995). This curdling is an explanation for the instability of filamentary stratospheric intrusions as well as surfacetemperature features.…”

A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow

“…(212) 998-3088; fax: (212) 995-4121; e-mail: tabak@cims.nyu.edu. men in [2], a study continued by Pierrehumbert et al [12]. Constantin, et a1.…”