CABARET in the ocean gyres S.A. Karabasov a,d, * , P.S. Berloff b,c , V.M. Goloviznin d a University of Cambridge, Department of Engineering, Whittle Laboratory, Cambridge, UK b Imperial College London, Grantham Institute for Climate Change and Department of Mathematics, London, UK c Woods Hole Oceanographic Institution, Physical Oceanography Department, Woods Hole, USA d Moscow Institute of Nuclear Safety of Russian Academy of Science, Moscow, Russia article info Article history: Received 12 January 2009 Received in revised form 18 June 2009 Accepted 23 June 2009 Available online xxxx Keywords: Mesoscale ocean dynamics Eddy resolving simulations High-resolution schemes abstract A new high-resolution Eulerian numerical method is proposed for modelling quasigeostrophic ocean dynamics in eddying regimes. The method is based on a novel, second-order non-dissipative and low-dis- persive conservative advection scheme called CABARET. The properties of the new method are compared with those of several high-resolution Eulerian methods for linear advection and gas dynamics. Then, the CABARET method is applied to the classical model of the double-gyre ocean circulation and its perfor- mance is contrasted against that of the common vorticity-preserving Arakawa method. In turbulent regimes, the new method permits credible numerical simulations on much coarser computational grids. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction In many aspects mesoscale oceanic eddies, operating on the lengthscales of O(1–100) km are analogous to the cyclones and anticyclones that constitute the atmospheric weather phenome- non. The problem of resolving these eddies in a dynamically con- sistent way is very important for ocean modelling and, therefore, for global climate predictions. For achieving high Reynolds number (Re) simulations, which are required for accurate modelling of the ocean, the models have to account for all important scales of motion. Modern ocean models enter a new phase in which eddies will be, at least, permitted in the numerical simulation. For such mod- els advection scheme is a very important component. A crucial ele- ment of numerical advection scheme is its ability to propagate finite-amplitude and -phase disturbances on a discrete grid either without generating spurious short-wave oscillations, because of not preserving the correct dispersion relation i.e., dispersion error, or any considerable damping of the amplitude i.e., dissipation error (e.g., Kravchenko and Moin, 1997; Pope, 2000). Note, that the gen- eration of short-wave oscillations is particularly detrimental in case an inverse energy cascade takes place, and the small scales af- fect large scales (e.g., Tabeling, 2002; Vallis, 2006). In this paper, the effect of spurious small-scale dispersion and dissipation on important large-scale properties of the solution are captured by comparing numerical predictions obtained with two different advection methods implemented within the same quasigeostroph- ic ocean modelling code (Berloff et al., 2007), which solves for the classical double-gyre problem (Holland, 1978). The original model implements standard eddy viscosity for parameterising effects of the unresolved scales of turbulent diffusion and conservative sec- ond-order Arakawa scheme for advection (Arakawa, 1966). In this paper, we consider a few versions of the original code based on different advection methods. Comparisons with the converged fine-grid solutions are made to investigate effects of numerical advection schemes on the coarse-grid solutions. Solving ‘convection-dominated’ problems is a longstanding challenge for computational fluid dynamics (e.g., Rozhdestvensky and Yanenko, 1978; Roache, 1982; Hirsch, 2007). One of the diffi- culties is that the conventional second-order finite-difference schemes have large dispersion errors, which generate spurious rip- ples in the solutions. To counterbalance this effect, a numerical dis- sipation, such as the classical von Neumann and Richtmyer (1950) artificial viscosity for compression-type pressure waves or such as the eddy viscosity in ocean circulation models, is added to the gov- erning equations. However, a common negative side of this ap- proach is the associated spurious dissipation that smears the large eddies along with the spurious ripples. There are three general approaches for solving ‘convection- dominated’ problems: Eulerian, Lagrangian, and mixed Eulerian– Lagrangian. For the Eulerian methods, significant presence of both numerical dissipation and dispersion is common drawback. It is partially overcome in the Lagrangian and Eulerian–Lagrangian methods, which describe flow advection by following fluid particle coordinates, rather than by considering fixed coordinates on the 1463-5003/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ocemod.2009.06.009 * Corresponding author. Address: University of Cambridge, Department of Engineering, Whittle Laboratory, 1 JJ Thompson Avenue, Cambridge CB3 0DY, UK. Tel.: +44 1223 337599. E-mail address: sak36@eng.cam.ac.uk (S.A. Karabasov). Ocean Modelling xxx (2009) xxx–xxx Contents lists available at ScienceDirect Ocean Modelling journal homepage: www.elsevier.com/locate/ocemod ARTICLE IN PRESS Please cite this article in press as: Karabasov, S.A., et al. CABARET in the ocean gyres. Ocean Modell. (2009), doi:10.1016/j.ocemod.2009.06.009