Local and nonlocal dispersive turbulence

Abstract

We consider the evolution of a family of two-dimensional ͑2D͒ dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, and as per convention, the locality of the streamfunction-scalar relation is denoted by ␣, with smaller ␣ implying increased locality ͑␣ = 1 gives traditional 2D dynamics͒. The dispersive nature arises via a linear term whose strength, after nondimensionalization, is characterized by a parameter ⑀. Setting 0 Ͻ ⑀ Յ 1, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward ͑inverse͒ transfer of enstrophy ͑energy͒ under large-scale ͑small-scale͒ random forcing along with the geometry of the scalar field. Straightforward arguments suggest that for small ␣ the scalar field should consist of progressively larger isotropic eddies, while for large ␣ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism, much like that of a small-scale passive field when advected by a large-scale smooth flow. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly nonlocal to local conditions ͑i.e., ␣ Յ 1͒. In all cases we see the establishment of well-defined spectral scaling regimes. For ⑀ ϳ 1, there is quantitative agreement between nondispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding nondispersive estimate. At present resolution, additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term: specifically, as ⑀ decreases, quite expectedly the inertial-range shortens but we also observe that the slope of the power law decreases. On the other hand, for the same range of ⑀ values, the forward regime scaling is observed to be fairly universal.

Surface Ocean Enstrophy, Kinetic Energy Fluxes, and Spectra From Satellite Altimetry

“…Noting the significance of surface buoyancy gradients (a fact missed in the aforementioned first baroclinic mode framework), it has been suggested that surface QG (SQG) dynamics (Blumen, 1978;Held et al, 1995;Lapeyre, 2017) is a more appropriate framework for the oceans' surface Lapeyre & Klein, 2006;Sasaki & Klein, 2012), and is reflected in the altimeter measurements (Lapeyre, 2009). Though the variance of buoyancy is transferred downscale (Pierrehumbert et al, 1994;Sukhatme & Pierrehumbert, 2002), even in the presence of an ambient buoyancy gradient (Sukhatme & Smith, 2009), surface KE actually flows upscale in SQG dynamics (Capet et al, 2008a;Smith et al, 2002), consistent with the flux calculations using altimetry data. Recognizing the importance of both these approaches, there are ongoing efforts to represent the variability of the ocean surface and interpret the altimetry data in terms of a combination of interior and surface QG modes (Lapeyre, 2009;Scott & Furnival, 2012;Smith & Vanneste, 2013).…”

Anomalous eddy viscosity for two-dimensional turbulence

“…We have not used any approximation for the transformation of (8) into forms (13) or (14). That is, the forms of (13) and (14) are independent of the approximations of the statistical theory of turbulence.…”

“…This system, which also includes three realizable members of 2D geophysical fluid systems, 5,6 has been actively investigated both theoretically and numerically over the past decade. [7][8][9][10][11][12][13][14][15][16][17][18][19]26 When α = 2, q is the familiar vorticity, and governing equation (1) reduces to the vorticity equation for a 2D incompressible barotropic fluid (2D NS system for D = ν∇ 2 q, where ν is the kinematic viscosity coefficient, and a 2D Euler system for D = 0). For α > 3, the velocity induced by a point vortex, which is a delta-functional distribution of q, increases with the distance from the point vortex, as shown by Iwayama and Watanabe.…”

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

“…The generalized system (3)-(4) has been considered in e.g. [6][7][8][9]. It reduces to the 2D Euler equations if α = 2 and to the 2D SQG equation if α = 1, and trivially to a steady state if α = 0.…”

“…First, it should be noted that the structure of the nonlinear terms in (8), the stretching of the scalar gradient is given by a product of the gradient and its singular integral transform we have that…”