The notion of the cascade arrest in a ␤-plane turbulence in the context of continuously forced flows is revised in this paper using both theoretical analysis and numerical simulations. It is demonstrated that the upscale energy propagation cannot be stopped by a ␤ effect and can only be absorbed by friction. A fundamental dimensional parameter in flows with a ␤ effect, the Rhines scale, L R , has traditionally been associated with the cascade arrest or with the scale that separates turbulence and Rossby wave-dominated spectral ranges. It is shown that rather than being a measure of the inverse cascade arrest, L R is a characteristic of different processes in different flow regimes. In unsteady flows, L R can be identified with the moving energy front propagating toward the decreasing wavenumbers. When large-scale energy sink is present, ␤-plane turbulence may attain several steady-state regimes. Two of these regimes are highlighted: friction-dominated and zonostrophic. In the former, L R does not have any particular significance, while in the latter, the Rhines scale nearly coincides with the characteristic length associated with the large-scale friction. Spectral analysis in the frequency domain demonstrates that Rossby waves coexist with turbulence on scales smaller than L R thus indicating that the Rhines scale cannot be viewed as a crossover between turbulence and Rossby wave ranges.

The gradient Richardson number, Ri, defined asis a measure of the relative strength of the density gradient in stably stratified shear flows. Here, N 2 = −g(∂ρ/∂z )/ρ 0 is the square of the buoyancy, or Brunt-Väisälä frequency, g is the gravity acceleration, ρ is the fluid density, ρ 0 is the reference density, S is the vertical shear of the horizontal velocity, and z is the vertical coordinate. The critical Richardson number, Ri c , appears in the classical papers by Miles (1961) Gage (1971).The increase of the Reynolds number causes nonstationarity of the basic field, bifurcations and transition to turbulence, and the analysis of the effect of stable stratification in this case needs to take into account the increasing complexity. Laval et al. (2003) considered the effect of the Reynolds number, Re = VL h /ν, on forced, stably-stratified shear layers (here, V and L h are the horizontal velocity and length scales, respectively, and ν is the kinematic viscosity). In order to be able to deal with shearless, freely decaying flows, Laval et al. (2003) also used a Froude number defined as Fr = V /NL v , where L v is a vertical length scale. Using the Kolmogorov scaling for the rate of viscous dissipation , = V 3 /L v , one can re-define the Froude number as Fr = /NK , where K = V 2 /2 is the turbulence kinetic energy. An increasing strength of stable stratification corresponds to decreasing Fr. With increasing Re, a simulated flow underwent a series of successive transitions to states with increasing anisotropy dominated by horizontal pancake-like vortices. These vortices, however, could become unstable with increasing Re. The study concluded that "for any Froude number, no matter how small, there are Reynolds numbers large enough so that a sequence of transitions to nonpancake motions will always occur and, conversely, for any Reynolds number, no matter how large, there are Froude numbers small enough so that these transitions are suppressed."From these results, one can surmise that the adaptation of the critical Richardson number to turbulent flows is not straightforward. Since turbulence is an unstable, stochastic phenomenon, it lacks

[1] Recent eddy-permitting simulations of the North Pacific Ocean have revealed robust patterns of multiple zonal jets that visually resemble the zonal jets on giant planets. We argue that this resemblance is more than just visual because the energy spectrum of the oceanic jets obeys a power law that fits spectra of zonal flows on the outer planets. Remarkably, even the non-dimensional proportionality coefficient, C Z , determined by data under that spectral law, appears to be constant for all cases and approximately equal to 0.5. These results indicate that the multiple jet sets in the ocean and in the atmospheres of giant planets are governed by the same dynamics characterized by an anisotropic inverse energy cascade, i.e., the flow of energy from isotropic small-scale eddies to anisotropic large-scales structures, as well as the unique anisotropic spectrum. Implications of these results for climate research and future designs of observational missions are discussed.

A new spectral closure model of stably stratified turbulence is used to develop a K À model suitable for applications to the atmospheric boundary layer. This K À model utilizes vertical viscosity and diffusivity obtained from the spectral theory. In the equation, the Coriolis parameter-dependent formulation of the coefficient C 1 suggested by Detering and Etling is generalized to include the dependence on the Brunt-Va¨isa¨la¨frequency, N. The new K À model is tested in simulations of the ABL over sea ice and compared with observations from BASE as simulated in large-eddy simulations by Kosovic and Curry, and observations from SHEBA.

The anisotropic characteristics of small-scale forced 2D turbulence on the surface of a rotating sphere are investigated. In the absence of rotation, the Kolmogorov k−5/3 spectrum is recovered with the Kolmogorov constant CK≈6, close to previous estimates in plane geometry. Under strong rotation, in long-term simulations without a large-scale drag, a −5 slope emerges in the vicinity of the zonal axis (kx→0), while a −5/3 slope prevails in other sectors far away from the zonal axis in the wave number plane. This picture is consistent with the new flow regime recently simulated by Chekhlov et al. [Physica D 98, 321–334 (1995)] and Smith and Waleffe [Phys. Fluids 11, 1608–1622 (1999)] on the beta plane. The concentration of energy in the zonal components and breaking of isotropy are caused by the strongly anisotropic spectral energy transfer and the stabilization of zonal mean flow by the meridional gradient of the planetary vorticity. The sharp tilt-up of the spectrum along the zonal axis was qualitatively understood through the scale-dependent stability property of the zonal flow. Under planetary rotation, the capacity for the zonal jets to hold energy and remain stable sharply increases with an increase of the meridional scale of the jets. In our simulations that were virtually inviscid at the large scales, the energy spectrum along the zonal axis tilts up all the way to the largest possible scale, indicating an apparent up-scale energy “cascade” along the zonal axis. This apparent up-scale cascade corresponds to a process of continuous mergers of zonal jets that does not cease until reaching the largest scale. This picture is consistent with the inviscid scenario for jet merging discussed by Manfroi and Young [J. Atmos. Sci. 56, 784–800 (1999)]. It contrasts the viscous scenario (for flows under the influence of a constant bottom drag) simulated in several previous studies, in which a distinct and finite jet scale emerges asymptotically.

A new spectral model for turbulent flows with stable stratification is presented. The model is based on a quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations and employs a recursive procedure of small-scale mode elimination that results in a coupled system of differential equations for effective, horizontal and vertical, viscosities and diffusivities. With increasing stratification, the vertical viscosity and diffusivity are suppressed while their horizontal counterparts are enhanced thus explicitly accounting for the anisotropy introduced by stable stratification. The new model is used to derive various spectral characteristics of stably stratified turbulent flows. It accounts for energy accumulation in the horizontal components at the expense of the energy reduction in the vertical component. The scale elimination algorithm explicitly accounts for the combined effect of turbulence and internal waves. A modified dispersion relation for internal waves, a relationship for internal wave frequency shift, and a threshold criterion for internal wave generation in the presence of turbulent scrambling are derived. It is shown how the model can be utilized to derive parameters needed for Reynolds-averaged Navier-Stokes modeling. Implemented in the K-ϵ format, the new model produces results that agree very well with the observational data on stably stratified atmospheric boundary layers and that are difficult to obtain using traditional two-equation turbulence modeling.

The Kolmogorov-Batchelor-Kraichnan (KBK) theory of two-dimensional turbulence is generalized for turbulence on the surface of a rotating sphere. The energy spectrum develops considerable anisotropy; a steep -5 slope emerges in the zonal direction, while in all others the classical KBK scaling prevails. This flow regime in robust steady state is reproduced in simulations with linear drag. The conditions favorable for this regime may be common for giant planets' atmospheric circulations; the same steep spectra are found in their observed zonal velocity profiles and utilized to explain their basic characteristics.

The effect of small scale forcing on large scale structures in β-plane twodimensional (2D) turbulence is studied using long-term direct numerical simulations (DNS). We find that nonlinear effects remain strong at all times and for all scales and establish an inverse energy cascade that extends to the largest scales available in the system. The large scale flow develops strong spectral anisotropy: k −5/3 Kolmogorov scaling holds for almost all φ, φ = arctan(k y /k x ), except in the small vicinity of k x = 0, where Rhines's k −5 scaling prevails. Due to the k −5 scaling, the spectral evolution of β-plane turbulence becomes extremely slow which, perhaps, explains why this scaling law has never before been observed in DNS. Simulations with different values of β indicate that the β-effect diminishes at small scales where the flow is nearly isotropic. Thus, for simulations of β-plane turbulence forced at small scales sufficiently removed from the scales where β-effect is strong, large eddy simulation (LES) can be used. A subgrid scale (SGS) parameterization for such LES must account for the small scale forcing that is not explicitly resolved and correctly accommodate two inviscid conservation laws, viz. energy and enstrophy. This requirement gives rise to a new anisotropic stabilized negative viscosity (SNV) SGS representation which is discussed in the context of LES of isotropic 2D turbulence.