Generalized Quasilinear Approximation: Application to Zonal Jets

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.

Figure 1

Triad interactions between modes. Long wavelength fields $\overline{\mathbf{q}}$ with zonal wave number $|m|\le \mathrm{\Lambda }$ and short wavelength fields $q^{\prime }$ with $|m|>\mathrm{\Lambda }$ are shown. Diagrams (a)–(c) in the top row are retained by the GQL approximation and triad interactions (d)–(f) in the bottom row are omitted.

Figure 2

Time- and zonal-averaged zonal velocity as a function of latitude. Time averaging over 5000 days (50 000 days for QL) commences after a spin-up of 500 days. NL is the fully nonlinear simulation. In the QL limit ($\mathrm{\Lambda }=0$), there is no mechanism to transfer angular momentum from low latitudes, where it is forced, to high latitudes. GQL corrects this defect.

Figure 3

Second cumulant (two-point correlation function of the vorticity). One point is centered along the prime meridian and at latitude 0°. The nonlocal nature of the correlations (or “teleconnections”) is evident.

Figure 4

Color-coded Hovmöller diagram (space-time plots with time $t$ along the horizontal axis and $0\le y\le 2\pi$ along the vertical axis) for the mean vorticity $\overline{\zeta }(y,t)$ for (a) NL DNS ($\mathrm{\Lambda }=1024$), (b) $\mathrm{\Lambda }=3$, (c) $\mathrm{\Lambda }=1$, (d) $\mathrm{\Lambda }=0$ (equivalent to QL).

Figure 5

Time-averaged vorticity power spectra on a ${\log }_{10}$ scale for fully NL DNS ($\mathrm{\Lambda }=1024$), GQL with $\mathrm{\Lambda }=3$ and $\mathrm{\Lambda }=1$, and QL ($\mathrm{\Lambda }=0$). In all cases, power scatters in the meridional y direction via the nonlinear interaction. Like NL, GQL also redistributes power in the zonal direction. Small power seen in the QL spectrum outside of the forced zonal wave numbers is due to the initial condition and vanishes rapidly with time.