Direct Statistical Simulation of Out-of-Equilibrium Jets

We present direct statistical simulation of jet formation on a $\beta$ plane, solving for the statistics of a fluid flow via an expansion in cumulants. Here we compare an expansion truncated at second order (CE2) to statistics accumulated by direct numerical simulations. We show that, for jets near equilibrium, CE2 is capable of reproducing the jet structure (although some differences remain in the second cumulant). However, as the degree of departure from equilibrium is increased (as measured by the zonostrophy parameter), the jets meander more and CE2 becomes less accurate. We discuss a possible remedy by inclusion of higher cumulants.

Figure 1

Snapshot density plot of vorticity together with zonal mean vorticity profile of jets found by DNS. The parameters are $\kappa ={10}^{-3}$, $\nu ={10}^{-4}$, $\beta =16$. For these parameters, $R_{\beta }=2.12$.

Figure 2

Hovmöller diagrams of zonal mean relative vorticity versus time from DNS simulations. (a) Parameters as for Fig. 1 with $R_{\beta }=2.12$. (b) Parameters as for (a) but $\beta =8$ and $R_{\beta }=1.98$; here $L_R$ is increased by a factor $3/2$ from that in (a). (c) Parameters as for (a) but $\kappa ={10}^{-2}$ and $R_{\beta }=1.24$; here $L_R$ is decreased by a factor $3/5$ from that in (a). A running time average commences at the midpoint of each diagram.

Figure 3

Hovmöller diagrams of relative vorticity from CE2. (a) Parameters as for Fig. 2a. (b) Parameters as for Fig. 2c.

Figure 4

Comparison of mean zonal velocity from DNS (dashed lines) and CE2 (solid lines) for parameters as in Figs. 2a, 2b, 2c for which $R_{\beta }=2.12$, 1.98, and 1.24.

Figure 5

Second cumulant $c_{\zeta \zeta }$ as calculated from DNS (left) with $R_{\beta }=2.12$ and the corresponding CE2 solution (right). Cross correlation with respect to a test point located at ($\pi$, $4.7$).