Statistics of the general circulation from cumulant expansions.

Climate varies on decadal and longer time scales, yet climate simulations advance at intervals measured in minutes or even seconds. In 1967, however, Edward Lorenz observed: “More than any other theoretical procedure, numerical integration is also subject to the criticism that it yields little insight into the problem. The computed numbers are not only processed like data but they look like data, and a study of them may be no more enlightening than a study of real meteorological observations. An alternative procedure which does not suffer this disadvantage consists of deriving a new system of equations whose unknowns are the statistics themselves.”[1] We show here that such direct statistical simulation (DSS) is able to reproduce statistics obtained by the traditional route of time averaging direct numerical simulations (DNS). DSS of damped and driven systems may be formulated by a Reynolds decomposition of the dynamical variables qi into the sum of a mean value and a fluctuation (or eddy): qi = 〈qi〉 + q′ i with 〈q′ i〉 = 0. The first two equal-time cumulants are then ci ≡ 〈qi〉 and cij ≡ 〈q′ iq j〉 = 〈qiqj〉 − 〈qi〉〈qj〉. The equation of motion for the cumulants form an infinite hierarchy that must be truncated. The simplest closure, CE2, is to set the third and higher cumulants equal to zero, amounting to the neglect of eddy-eddy interactions. Here we extend the work of Ref. [2] to a two-layer quasigeostrophic model originally formulated on the β-plane[3] but generalized to the rotating sphere. DNS is carried out in real space on a spherical geodesic grid.[4] A typical instantaneous flow is shown in Fig. 1. Fig. 2 compares the two-point vorticity correlation function as calculated by DNS and CE2. Storm tracks are immediately apparent from the strong mid-latitude correlations and the jets fluctuate mainly at zonal wavenumber 5 and 6. CE2 captures long-range correlations (“teleconnection patterns”) and inter-hemisphere correlations are weak, as expected. A quantitative comparison of the zonal velocities, as calculated by DNS and CE2, is presented in Fig. 3 where the midlatitude westerlies and the tropical trade winds are evident. The comparison suggests that the large-scale flows are not so nonlinear as FIG. 2. The two-point second cumulant of the relative vorticity field with respect to a reference field point at latitude 45◦ and along the prime meridian. Top: DNS. Bottom: CE2.