Quantitatively mapping the Eady model onto a two-layer quasi-geostrophic model

Abstract

The two-layer quasigeostrophic model (2LQG) and the Eady model are two idealized systems illustrating the baroclinic instability of atmospheric jets and ocean currents. The two setups share many ingredients -- background vertically sheared zonal flow of density-stratified fluid in a rapidly rotating frame -- while differing in complexity and dimensionality. The Eady model has a continuous vertical direction, with baroclinic turbulence induced by boundary potential vorticity (PV) gradients at top and bottom. By contrast, the 2LQG sytem typically models baroclinic instability induced by interior PV gradients. This distinction challenges our ability to clearly identify a couple of 'modes' through which the Eady dynamics could be inferred from a simpler 2LQG system. In the present study, we show that this difficulty can be circumvented in the turbulent regime arising for weak bottom drag. Namely, guided by the common organization of both systems into a gas of coherent vortices, we identify a quantitative mapping between the Eady and the 2LQG models. The mapping allows for parameter-free predictions of the eddy diffusivity of the Eady model based on the knowledge of the 2LQG diffusivity. We illustrate these results using numerical simulations of the Eady and 2LQG models with linear or quadratic bottom drag.

Figures (6)

• Figure 1: Two idealized models for the study of horizontally homogeneous baroclinic turbulence on the $f$-plane. (a) The Eady model features a continuous vertical direction $z$. (b) By contrast, the two-layer quasi-geostrophic model consists of two stacked shallow fluid layers. In both models equilibrated turbulence results from a balance between potential energy release from the thermal-wind-balanced base state and dissipation by bottom drag.
• Figure 2: Energy cascades in baroclinic QG turbulence, as described in salmon1978two. According to this scenario, energy conversion from the baroclinic mode to the barotropic mode arises near the Rossby deformation radius $\lambda$.
• Figure 3: Premultiplied cross-spectra $S_{\tau \rightarrow \psi }(k)$ from the 2LQG model, rescaled using the power law $D_* \sim \mu_*^{-4/3}$ of the overall energy conversion rate. Color codes for the dimensionless drag $\mu_*$, with values visible in the inset. The gray-shaded interval corresponds to the range spanned by the injection wavenumber $k_\Omega$ obtained from the ratio of barotropic enstrophy injection rate to barotropic energy injection rate (see text). The inset shows the wavenumber $k_{0.9}$ as a function dimensionless drag, defined such that $90$% of energy injection into the barotropic flow arises from wavenumbers $k \leq k_{0.9}$.
• Figure 4: Snapshots from all three models. Left panels of (a) and (b) show the baroclinic streamfunction $\tau$ from the 2LQG and reduced models, to be compared with their Eady counterpart $\tilde{\tau}=\sqrt{12}(p_0-p_{-1})$, shown in the left panel of (c). The right-hand panels of (a) and (b) show the barotropic vorticity $\Delta \psi$ from the 2LQG and reduced models, to be compared with their Eady counterpart $b_0-b_{-1}$, shown in the right-hand panel of (c). The drag is $\mu_*=3 \times 10^{-4}$ in (a) and (b) and $\mu_\text{Eady}=6 \times 10^{-4}$ in (c). The Eady simulation maps onto an effective friction $\mu_\text{Eady}/\sqrt{3}\simeq 3.5 \times 10^{-4}$ for the equivalent 2LQG system, in an effective simulation domain that is larger than that of simulations (a) and (b). Hence the smaller apparent flow scales in (c).
• Figure 5: Effective diffusivity versus quadratic drag coefficient from numerical simulations of the Eady model (dark blue), of the expanded Eady model using inversion relation (\ref{['eq:inversionEadyexp']}) (light blue), of the 2LQG model (red) and of the reduced model (\ref{['eq:psireduced']}-\ref{['eq:taureduced']}) (black circles). Left: raw data for all models. Right: The mapping suggests plotting the data from the 2LQG and reduced models under the form $D_*/12$ vs. $\sqrt{3}\mu_*$, which collapses the data of all models in the low-drag regime. The inset shows the same data compensated by a $-4/3$ power-law. In each panel, the solid line represents the low-drag theoretical prediction (\ref{['eq:predictionDEadylinear']}) deduced from the mapping procedure.