Waves and turbulence on a beta-plane

Abstract

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

The structure of zonal jets in geostrophic turbulence

“…Regarding the spacing of the jets, it has long been understood that it is closely linked to a fundamental length scale of the flow, the Rhines scale, given by L Rh = √ U/β (Rhines 1975;Williams 1978), where U is a typical velocity scale of the flow and β is the local background latitudinal gradient of potential vorticity (comprising, in the simplest context, the relative vorticity and the vorticity due to the planetary rotation). The Rhines scale L Rh may be considered as the scale at which zonal motions become important, or, equivalently, the scale at which a background of quasi-two-dimensional turbulent motion begins to project significantly onto the lower frequencies of freely propagating Rossby waves (see e.g.…”

On the energetics of a two-layer baroclinic flow

“…The mixing acts to homogenise PV and sharpen jets, processes which are only enhanced by non-zonal variations (Dritschel & McIntyre 2008;Scott & Dritschel 2012). The mixing continues until most of the eddy energy (principally at higher wavenumbers where thermal damping is weak) is exhausted and converted into jets (qualitatively this is the argument behind the Rhines scale L Rh = U/β, where U characterises the eddy velocities and β is the background PV gradient, see Rhines (1975)). The low wavenumber part of the eddy energy gives rise to meanders, which unlike the high wavenumber part are affected by the thermal damping.…”

Zonal jets in equilibrating baroclinic instability on the polar beta‐plane: Experiments with altimetry

“…L Rh is the well-known Rhines scale and was defined as a dividing length scale between an isotropic turbulence (eddies) and linear Rossby waves [Rhines, 1975]. This transition scale has been viewed as an estimate of the meridional scale of the jets in different flows [e.g., Vallis and Maltrud, 1993;Huang et al, 2001;Danilov and Gurarie, 2000].…”