Nonlinear Waves in Zonostrophic Turbulence

The Charney-Hasegawa-Mima equation applies to a broad variety of hydrodynamic systems ranging from the large-scale planetary circulations to small-scale processes in magnetically confined plasma. This equation harbors flow regimes that have not yet been fully understood. One of those is the recently discovered regime of zonostrophic turbulence emerging in the case of small-scale forced, barotropic two-dimensional turbulence on the surface of a rotating sphere or in its $\beta$-plane approximation. The commingling of strong nonlinearity, strong anisotropy and Rossby waves underlying this regime is highlighted by the emergence of stable systems of alternating zonal jets and a new class of nonlinear waves, or zonons. This Letter elucidates the physics of the zonons and their relation to the large-scale coherent structures.

Figure 1

The kinetic energy spectra for a friction-dominated regime ($n_{\mathrm{R}}=9.2$, $n_{\beta }=14.3$). The thick and thin solid lines show $E(n)$ and $\mathcal{E}(n,m)$, respectively. The dashed lines correspond to the spectra (3, 4).

Figure 2

The velocity correlator, $U(\omega ,n,m)\times {10}^7$, for the friction-dominated regime; $n_{\mathrm{R}}=9.2$, $n_{\beta }=12.3$. The filled triangles correspond to the RHWs dispersion relation (2).

Figure 3

The kinetic energy spectra for a zonostrophic regime ($n_{\mathrm{R}}=5.5$, $n_{\beta }=16.2$). The thick gray and black solid lines show $E_Z(n)$ and $E_R(n)$, respectively. The thin lines show $\mathcal{E}(n,m)$. The dashed lines correspond to the spectra (6a, 6b, 4).

Figure 4

The velocity correlator, $U(\omega ,n,m)$, for the regime of zonostrophic turbulence; $n_{\mathrm{R}}=5.5$, $n_{\beta }=16.2$. The filled triangles correspond to the RHWs dispersion relation (2) while the filled circles mark the zonons.

Figure 5

The correlator $U(\omega ,n,1)$ (solid lines) for a zonostrophic regime ($n_{\mathrm{R}}=5.5$, $n_{\beta }=16.2$). The filled triangles at the top of the panel correspond to the RHWs dispersion relation (2) with values of $n$ shown near the triangles. To ease tracing the lines at different $n$, they are shown at different gray scales.

Figure 6

Frequencies of RHWs and zonons as functions of $m$ for different $n$; $n_{\mathrm{R}}=5.5$, $n_{\beta }=16.2$. The empty circles and dashed lines correspond to the RHW dispersion relation (2); the filled circles and the solid black and gray lines show the dispersion relations for the zonons excited by RHWs with $n=4$ and 5, respectively.

Figure 7

Total energy of zonal modes as a function of time.

Figure 8

The Hovmöller diagrams of the stream function at three different latitudes $\theta$ in the zonostrophic regime. The white lines show the slope used to calculate the angular velocities.

Figure 9

Comparison of zonal phase speeds computed from the Hovmöller diagrams and the dispersion relation (2) for most energetic RHWs for a number of different simulations.