## 1. Turbulence and Diffusion in Flows With a *β*-Effect

[2] Large scale planetary and terrestrial circulation are studied under the paradigm of the geostrophic turbulence introduced by *Charney* [1971]to model a turbulent, rotating, stably stratified fluid in near geostrophic balance. This model, based on the three-dimensional inviscid conservation equation for the potential vorticity, presents several analogies with purely 2D turbulence, e.g., the conservation of two quadratic invariants. Nevertheless, important differences exist [*Read*, 2001]. Here, among these differences, we will focus on turbulence modification by the meridional variation of the Coriolis parameter, i.e., the so-called*β*-effect, defined approximately as*f* = *f*_{0} + *βy*, where *f*_{0} is twice the rotation angular velocity Ω and *y* is the latitude coordinate. Rhines considered this problem in the context of unforced barotropic flows [*Rhines*, 1975] demonstrating that the *β*-effect induces the anisotropization of the upward energy transfer preferentially directing it towards zonal modes. The emergence of the flow anisotropy was explained in terms of a competition between nonlinear and*β* terms and anisotropy of the dispersion relation. The associated characteristic scale separating the regions of wave vector space where either *β* or nonlinear effects respectively dominate is known as the Rhines wavenumber, *n*_{R}, and can be expressed in terms of the root mean square of the velocity and *β.*Similar ideas can be applied to geostrophic turbulence where the baroclinic instability takes the role of the small-scale forcing and the non-linear interaction between eddies and the mean flow plays an important role in flow anisotropization; nevertheless, for small Burger numbers, the inverse cascade develops in the barotropic mode due to the barotropization mechanism [*Salmon*, 1980; *Rhines*, 1994; *Read*, 2001; *Galperin et al.*, 2010].

[3] The concept of the cascade arrest was introduced by *Rhines* [1975] for unforced flows and recently revisited by *Sukoriansky et al.* [2007]in the context of small-scale forced, large-scale damped, barotropic vorticity equation on the surface of a rotating sphere. They show how, in the presence of a*β*-effect, 2D turbulence features a complex interplay with waves at large scales and develops an anisotropic inverse energy cascade. As a consequence, several steady-state regimes characterized by different degree of anisotropy and energy spectra may be attained. These flow regimes can be classified in terms of relationship between the wavenumbers associated with the forcing,*n*_{f}, the small-scale dissipation,*n*_{d}, the *β*-effect,*n*_{β}, and the large-scale drag,*n*_{fr}. A prominent role in characterization of flow regimes plays the non-dimensional parameter*R*_{β} = *n*_{β}/*n*_{R}, termed the zonostrophy index. *Sukoriansky et al.* [2007] and *Galperin et al.* [2010] studied two regimes: friction dominated and zonostrophic. In the former, *R*_{β} < 1.5 while in the latter, *R*_{β} > 2. The intermediate range refers to a transitional regime [*Sukoriansky et al.*, 2009]. In the friction dominated regime, the flow presents features similar to classical 2D turbulence, i.e., a Kolmogorov *n*^{−5/3} scaling for the total spectrum and a nearly isotropic inverse cascade for *n* > *n*_{β}. The zonostrophic regime is characterized by a strong spectral anisotropy, e.g., the zonal spectrum follows a *n*^{−5} scaling while the residual spectrum follows the classical *n*^{−5/3} scaling. The most distinguished feature of zonostrophic turbulence is the formation of a nearly stable system of eastward/westward zonal jets. Jets related to the zonostrophic regime were studied in laboratory experiments [*Whitehead*, 1975; *Armi*, 1989; *Sommeria et al.*, 1991; *Aubert et al.*, 2002; *Rasmussen et al.*, 2006; *Read et al.*, 2007; *Slavin and Afanasyev*, 2012], as well as in numerical simulations [*Galperin et al.*, 2010]. The outputs of Oceanic General Circulation Models [*Nakano and Hasumi*, 2005; *Berloff et al.*, 2009] and the analysis of satellite altimetry observations [*Maximenko et al.*, 2005; *Sokolov and Rintoul*, 2007] also yield zonally elongated structures resembling zonal jets [*Galperin et al.*, 2004; *Nadiga*, 2006].

[4] Turbulent meridional diffusion is strongly influenced by the presence of a *β*-effect. An analogy with the modification imposed on the diapycnal diffusion by the presence of a strong stable stratification has been recently proposed by*Sukoriansky et al.* [2009]. In particular, in the friction dominated regime, for *R*_{β} ≤ 1, the meridional (or, equivalently, radial) diffusivity, *D*_{rad}, follows the Richardson law behavior *D*_{rad} ≃ 2*ε*^{1/3}*n*_{E}^{−4/3}, where *ε* is the rate of the inverse energy cascade and *n*_{E} is the wavenumber corresponding to the most energetic scale. In the range 1 < *R*_{β} < 1.5, the threshold of the transitional regime, the meridional diffusion coefficient follows the law *D*_{rad} ≃ 0.3*ε*^{1/3}*n*_{β}^{−4/3}. This law holds for values of *R*_{β} belonging to either the transitional or the zonostrophic regime, indicating that scales of order of magnitude larger than 1/*n*_{β} do not contribute to meridional transport [*Galperin et al.*, 2010]. *Sukoriansky et al.* [2009]found some confirmation for these predictions analyzing field measurements of diffusion in deep large-scale oceanic circulation.