Nonlinear Waves in Zonostrophic Turbulence Semion Sukoriansky * and Nadejda Dikovskaya Department of Mechanical Engineering/Perlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, Beer-Sheva, Israel Boris Galperin College of Marine Science, University of South Florida, St. Petersburg, Florida, USA (Received 14 May 2008; published 24 October 2008) The Charney-Hasegawa-Mima equation applies to a broad variety of hydrodynamic systems ranging from the large-scale planetary circulations to small-scale processes in magnetically confined plasma. This equation harbors flow regimes that have not yet been fully understood. One of those is the recently discovered regime of zonostrophic turbulence emerging in the case of small-scale forced, barotropic two- dimensional turbulence on the surface of a rotating sphere or in its -plane approximation. The commingling of strong nonlinearity, strong anisotropy and Rossby waves underlying this regime is high- lighted by the emergence of stable systems of alternating zonal jets and a new class of nonlinear waves, or zonons. This Letter elucidates the physics of the zonons and their relation to the large-scale coherent structures. DOI: 10.1103/PhysRevLett.101.178501 PACS numbers: 92.60.e, 92.10.Hm, 92.10.Lq, 92.10.Ty The Charney-Hasegawa-Mima equation (CHME) [1,2] is a simple model of planetary and plasma turbulence. Charney [1] showed that for the former, CHME describes the regime of geostrophic turbulence pertinent to the larg- est planetary scales. Those scales are much larger than the scales at which the flow undergoes two-dimensionalization [3,4]. Even in its simplified, barotropic version (infinite Rossby deformation radius), the commingling of strong nonlinearity, strong anisotropy and Rossby waves gives rise to complicated dynamics. In flows with a small-scale forcing, the inherent anisotropic inverse energy cascade may lead to the development of the regime of zonostrophic turbulence, a subset of geostrophic turbulence [5,6]. This regime is distinguished by an anisotropic spectrum and stable systems of alternating zonal (east-west) jets. In this Letter, we show that another important attribute of zonostrophic turbulence is a new class of nonlinear waves coined zonons. Zonons may form coherent structures ob- servable in physical space. The barotropic vorticity equation on the surface of a rotating sphere (BVES) is an example of the CHME with the infinite Rossby radius. The small scale forced and linearly damped version of this equation is an efficient simple model of the barotropic mode of planetary circu- lations [5]. Numerical experiments with such a system are at the focus of this Letter. The small-scale forced BVES is given by @ @t þ Jð c ; þ fÞ¼ r 2p    þ ; (1) where  is the vorticity; c is the stream function, r 2 c ¼  ; f ¼ 2 sin is the Coriolis parameter (the planetary vorticity);  is the angular velocity of the sphere’s rota- tion;  is the latitude,  is the longitude;  is the hyper- viscosity coefficient; p is the power of the hyperviscous operator (p ¼ 4 in this study), and  is the linear friction coefficient. The scale at which the flow frequency is equal to 2 is reciprocal to the wave number of the large-scale friction, n fr . The small-scale forcing, , acting on the scales around n 1  , pumps energy into the system at a constant rate. Part of this energy becomes available for the inverse cascade at a rate . The Jacobian, Jð c ; þ fÞ, where JðA; BÞ¼ðR 2 cosÞ 1 ðA  B   A  B  Þ and R is the radius of the sphere, represents the nonlinear term. Equation (1) was solved numerically using the decomposition of the stream function in spherical harmonics Y m n ðsin; Þ, m being the zonal index. Conventionally, n and m are non- dimensional. Setting R ¼ 1 eliminates the difference be- tween the indices and wave numbers. The setup of the simulations was the same as in [6]. Equation (1) admits a class of linear Rossby-Haurwitz waves (RHWs) [7] with the dispersion relation ! R ðn; mÞ¼2 m nðn þ 1Þ : (2) The consistency with the -plane approximation is pre- served by setting  ¼ =R [8]. The energy spectrum for flows in spherical geometry is EðnÞ¼ P n m¼n Eðn; mÞ¼ nðnþ1Þ 4R 2 P n m¼n hj c m n j 2 i, where the modal spectrum, Eðn; mÞ, is the spectral energy density per mode (n, m), and the angular brackets indicate an ensemble or time average [9,10]. The spectrum EðnÞ can be represented as a sum of the zonal and nonzonal, or residual components, EðnÞ¼ E Z ðnÞþ E R ðnÞ, where the zonal spectrum is E Z ðnÞ¼ Eðn; 0Þ. If the characteristic time scales of turbulence and RHWs are  t ¼½n 3 EðnÞ 1=2 and  R ¼½! R ðn; mÞ 1 , respec- tively, then turbulent processes prevail on relatively small PRL 101, 178501 (2008) PHYSICAL REVIEW LETTERS week ending 24 OCTOBER 2008 0031-9007= 08=101(17)=178501(4) 178501-1 Ó 2008 The American Physical Society