Turbulent diffusion in the geostrophic inverse cascade

Abstract

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

Spectral non-locality, absolute equilibria and Kraichnan–Leith–Batchelor phenomenology in two-dimensional turbulent energy cascades

“…This argument is for inviscid unforced decaying turbulence, and it was extended to general α by Smith et al (2002). Global conservation together with the assumption that the energy and enstrophy distributions broaden around the initial wavenumber allow one to prove that the energy and enstrophy centroid wavenumbers move to larger and smaller scales, respectively, for α > 0.…”

“…Shallow-water quasigeostrophic dynamics in the limit of strong rotation, i.e. the asymptotic model limit of the Charney-Hasegawa-Mima equation (Larichev & McWilliams 1991;Iwayama, Shepherd & Watanabe 2002) corresponds to α = −2 (Smith et al 2002). Surface quasigeostrophic (SQG) dynamics, a simplified model for edge waves on the tropopause or temperature advection near the Earth's surface (Blumen 1978;Held et al 1995), is given by α = 1, and 2D Navier-Stokes dynamics corresponds to α = 2, as noted above.…”

Scaling of Off-Equatorial Jets in Giant Planet Atmospheres

“…The effects of drag on geophysical turbulence have been studied extensively using two-dimensional and shallowwater models (Danilov and Gurarie 2002;Smith et al 2002;Galperin et al 2006;Scott and Dritschel 2013) and in the quasigeostrophic two-layer model (Thompson and Young 2007). In general, drag removes energy from the system and contributes to halting the inverse cascade (Vallis 2006).…”

The Relative Humidity in an Isentropic Advection–Condensation Model: Limited Poleward Influence and Properties of Subtropical Minima

“…We also use double the spatial resolution (256 2 ) and twice the Rayleigh drag coefficient [0.6 in units of inverse model time; the eddy time scale based on enstrophy defined as in O'Gorman and Schneider (2006) is 0.78]. The smoothing filter F is applied to both the vorticity and the moisture fields and is only active at k $ 50 (Smith et al 2002;O'Gorman and Schneider 2006). The kinetic energy spectrum peaks at the forcing wavenumbers and displays a power-law range at higher wavenumbers that is roughly consistent with the steepness of the spectrum found in observations of the troposphere at large scales (Boer and Shepherd 1983); however, the tropospheric spectrum results from a more complicated range of processes than just the nonlinear eddy-eddy interactions that are important in the turbulent flow considered here (Schneider and Walker 2006;O'Gorman and Schneider 2007).…”