The Parameter Dependence of Eddy Heat Flux in a Homogeneous Quasigeostrophic Two-Layer Model on a β Plane with Quadratic Friction

Abstract

This study investigates the parameter dependence of eddy heat flux in a homogeneous quasigeostrophic two-layer model on a β-plane with imposed environmental vertical wind shear and quadratic frictional drag. We examine the extent to which the results can be explained by a recently proposed diffusivity theory for passive tracers in two-dimensional turbulence. To account for the differences between two-layer and two-dimensional models, we modify the two-dimensional theory according to our two-layer f -plane analyses reported in an earlier study. Specifically, we replace the classic Kolmogorovian spectral slope, −5/3, assumed to predict eddy kinetic energy spectrum in the former with a larger slope, −7/3, suggested by a heuristic argument and fit to the model results in the latter. It is found that the modified theory provides a reasonable estimate within the regime where both β˜=βkd−2U−1 and the strength of the frictional drag, c˜D=cDkd−1, are much smaller than unity (here, cD is the nondimensional drag coefficient divided by the depth of the layer, kd is the wavenumber of deformation radius, and U is the imposed background vertical wind shear). For values of β˜ and c˜D that are closer to one, the theory works only if the full spectrum shape of the eddy kinetic energy is given. Despite the qualitative, fitting nature of this approach and its failure to explain the full parameter range, we believe its documentation here remains useful as a reference for the future attempt in pursuing a better theory.

A Quantitative Scaling Theory for Meridional Heat Transport in Planetary Atmospheres and Oceans

“…A second key observation is that, on the logarithmic plot of Figure 4, the curves corresponding to various values of the drag seem to converge to values of order one when itself becomes . In other words, as increases, the diffusivity drops from the large values of the vortex gas scaling regime to the values characteristic of the weakly nonlinear regime that arises in the vicinity of the = 1 frictionless threshold for baroclinic instability (see also Chang & Held, 2021; Thompson & Young, 2007).…”

“…. For B ≳ 1 we expect the zonostrophic arrest mechanism to be fully operational and the drag coefficient to become irrelevant (this idea is at the core of the theory of HL and Chang & Held, 2021). The scaling between X 1 and X 2 indeed appears to follow a drag-independent power-law…”

“…We focus on the strongly turbulent regime that arises for values of E   significantly lower than the threshold value E   = 1 at which the undamped system is no longer unstable. As E   approaches one, the diffusivity E D  drops to values of order one: this is the weakly nonlinear regime, where the precise parametric dependence of E D  becomes more complex as documented by Thompson and Young (2007) and Chang and Held (2021). The present suite of numerical simulations thus focuses on E   ∈ [0; 0.5], which corresponds to the strongly turbulent, large-diffusivity, and large-mixing-length regime for which we derived the vortex gas scaling theory in GF.…”

“…In other words, as GALLET AND FERRARI 10.1029/2020AV000362 9 of 21 E   increases, the diffusivity drops from the large values of the vortex gas scaling regime to the (1) E  values characteristic of the weakly nonlinear regime that arises in the vicinity of the E   = 1 frictionless threshold for baroclinic instability (see also Chang & Held, 2021;Thompson & Young, 2007).…”

“…Beyond this transition point, that is, for B ≳ 1, the zonostrophic regime connects X 1 to the weakly nonlinear regime characterized by the point X 2 of approximate coordinates ( ∼ 1, ∼ 1). For B ≳ 1 we expect the zonostrophic arrest mechanism to be fully operational and the drag coefficient to become irrelevant (this idea is at the core of the theory of HL and Chang & Held, 2021). The scaling between X 1 and X 2 indeed appears to follow a drag‐independent power‐law , the exponent of which we now predict.…”