## 1. Introduction

The quasi-geostrophic model has widely been used to study geophysical fluid dynamics, from its linear to strongly nonlinear features at a broad range of scales (e.g. Salmon, 1998). It is a balanced model and as all balanced models it approximates the low-frequency, or balanced, component of the dynamics described by the primitive equations. Since a shallow layer of fluid with constant density is considered in this paper, the primitive equations reduce in this case to the shallow water equations. The quasi-geostrophic model is, however, invalid when the Rossby number is large, which is particularly the case in the equatorial region. This motivated the derivation of an equivalent model that is valid in the equatorial region by Theiss and Mohebalhojeh (2009) (thereafter referred to as TM). It is referred to as the equatorial balanced model and can be considered the equatorial counterpart of the quasi-geostrophic model because the same derivation that leads to the quasi-geostrophic model on an *f*-plane leads to the equatorial balanced model on the equatorial *β*-plane. This derivation, however, is not the standard one based on an asymptotic expansion in terms of the Rossby number (e.g. Salmon, 1998, section 2.10), but one presented by Leith (1980) based on the geometric framework of nonlinear normal mode initialization.

The objective of nonlinear normal mode initialization is to determine those solutions of the primitive equations that represent balanced dynamics, i.e. low-frequency dynamics whose excitation of high-frequency dynamics is minimized. This is achieved within a geometric framework, where each unique state of the dynamics is represented by a unique point in the so-called phase space. A complete set of mutually orthogonal vectors that span the phase space are the eigenvectors that are associated with the normal modes of the primitive equations. Because the normal modes can be distinguished between low-frequency Rossby modes and high-frequency gravity modes, the associated eigenvectors can be considered spanning the so-called Rossby and gravity subspaces, or manifolds, respectively. The states that constitute the balanced dynamics are represented by points within a subspace of the phase space, known as the slow manifold. The notion of the slow manifold as a subspace is practical, but not exact as it is rather a thin stochastic layer (Ford *et al.*, 2000). The slow manifold can be approximated by an iterative method (Baer, 1977; Baer and Tribbia, 1977; Machenhauer, 1977). Leith (1980) considers the approximation of the slow manifold that results from the first iterative step for the case of an *f*-plane. This approximate slow manifold represents states that constitute an approximate balanced dynamics. Leith shows that the dynamics constrained to the approximate slow manifold and then projected onto the Rossby manifold is the dynamics described by the quasi-geostrophic model. TM repeat Leith's derivation, but for the case of an equatorial *β*-plane, and derive in this sense the equatorial counterpart of the quasi-geostrophic model, which they call the equatorial balanced model. Since they consider a shallow layer of fluid with constant density, the primitive equations reduce in this case to the shallow water equations.

The geometric framework also illustrates the general fact that because the dynamics described by the primitive equations are not constrained to a subspace in phase space the variables that completely describe it are all independent while the dynamics described by any balanced model is constrained to the slow manifold, or in most cases to an approximation of it, resulting in only one variable being independent and all others being related. These relations are given by the balance relations of the balanced model and therefore describe the corresponding slow manifold. In the case of the quasi-geostrophic model, the relations are the geostrophic balance relations and they describe the Rossby manifold.

TM conduct a first assessment of the equatorial balanced model by analytically deriving its dispersion relation and comparing it to that of the shallow water equations. Their comparison of the two models is presented in Figure 1. The solid curves represent the dispersion relation of the shallow water equations, given by TM (5.7) or (5.8), and the dashed curves, three of which are below label 1 and overlaid by three solid curves, that of the equatorial balanced model, given by TM (5.44) with *α* = 2.5. The free parameter *α* is set to this specific value in TM and in this paper because it allows the equatorial balanced model to approximate well the dispersion relation of the shallow water equations. The dispersion relations of the Rossby waves of meridional modes 1,2, and 3 of both models are depicted below the label 1 and are hardly distinguishable. This remarkably accurate approximation of the Rossby wave dispersion relations even improves with increasing meridional mode number. This is not surprising. As all balanced models, the equatorial balanced model approximates the low-frequency, or balanced, component of the dynamics described by the primitive equations and because the frequency of the Rossby wave decreases with increasing meridional wavenumber, the accuracy of the approximation of the corresponding dispersion relation is therefore expected to increase. The dispersion relation of the mixed Rossby-gravity, or Yanai, wave of the shallow water equations is depicted by the solid curve consisting of parts a and d and the corresponding one of the equatorial balanced model by the top dashed curve. It shows that for large negative non-dimensional zonal wavenumber *k*′, at which the mixed Rossby-gravity wave of the shallow water equations has the characteristics of a Rossby wave, the approximation is good. As *k*′ increases, the mixed Rossby-gravity wave of the shallow water equations takes on increasingly more the characteristics of a gravity wave while the corresponding one of the equatorial balanced model retains the characteristics of a Rossby wave. This can simply be interpreted as a bad approximation. However, balanced models are expected to describe low-frequency dynamics and therefore it is rather remarkable that the dispersion relation of the equatorial balanced model that corresponds to the mixed Rossby-gravity wave of the shallow water equations retains the characteristics of a Rossby wave. Alternatively, this dispersion relation of the equatorial balanced model can also be interpreted not to approximate the solid curve consisting of parts a and d, but instead the one consisting of parts a and b. This would be in accordance with Matsuno (1966), who interprets the curve consisting of parts a and b as a Rossby wave dispersion relation and that consisting of parts c and d as a gravity wave dispersion relation rather than parts a and d as the mixed Rossby-gravity wave dispersion relation and parts b and c as the westward-propagating Kelvin, or anti-Kelvin, wave dispersion relation, which has become the conventional interpretation. The remaining dispersion relations of the shallow water equations are not approximated by the equatorial balanced model because they represent almost entirely high-frequency dynamics. They are the dispersion relations of the Kelvin wave, labeled by *ω*_{K}′, all gravity waves of which those of the first meridional mode are depicted and labeled , part c, and either part b and d depending on which of the above two interpretations is chosen.

TM's comparison of the dispersion relation of both models only assesses an aspect of the linearized form of the equatorial balanced model. To be able to assess the equatorial balanced model more comprehensively, we develop in this paper numerical implementations of both models for a zonally periodic channel on the equatorial *β*-plane. Both models have the property of materially conserving potential vorticity, i.e. potential vorticity is conserved on fluid particles. This allows us to adopt the Contour-Advective Semi-Lagrangian (CASL) algorithm (Dritschel, 1989; Dritschel and Ambaum, 1997) for the numerical implementations because it uses this property by advecting contours of potential vorticity, which has various benefits compared to more conventional numerical methods. The advection is a Lagrangian process, but the CASL algorithm is only semi-Lagrangian because the potential vorticity is interpolated onto a regular grid on which the corresponding velocity is determined by an Eulerian process. In the case of the shallow water equations, the velocity is determined from the potential vorticity and two other independent variables, specified in section 3, which are being evolved on the grid. In the case of the equatorial balanced model, the velocity is determined by the so-called inversion of potential vorticity, subject to its balance relations. For both models, the velocity on the regular grid is then interpolated onto the contours of potential vorticity, allowing the contours to be advected for a time step. This results in a new potential vorticity and the procedure of inversion and advection is repeated.

The CASL algorithm has the benefit of resolving features in the potential vorticity field that are much smaller than the grid interval. Further benefits are related to the necessity of every numerical model of nonlinear fluid dynamics to remove the smallest features in order to mimic molecular diffusion in the continuous limit as it can never be resolved because the grid size is always finite. The removal of small features is usually achieved by so-called hyperdiffusion. This, however, is not used by the CASL algorithm. It instead restructures and removes potential vorticity contours, a process referred to as contour surgery (Dritschel, 1989). The benefits represented by contour surgery are that the features being removed are much smaller than those removed by a typical hyperdiffusion using the same grid size and, unlike hyperdiffusion, it does not remove gradients in potential vorticity.

In order to assess the degree of accuracy with which the equatorial balanced model approximates the balanced component of the dynamics described by the shallow water equations, we consider an initially unstable westward flow that evolves freely and apply the tests of McIntyre and Norton (2000). One is a diagnostic test that assesses the degree of accuracy of only the inversion at a fixed time and thus assesses the accuracy of only the balance relations of the equatorial balance model. The other is a prognostic test that assesses the cumulative accuracy of the iteration of the inversion and advection, which evolves the dynamics, and thus assesses the cumulative degree of accuracy over time of the equatorial balanced model.

In the following section, we introduce the equations that comprise the shallow water equations and the equatorial balanced model. The algorithm used to solve both models numerically is described in section 3 and details of the configuration are given in section 4. The results are presented in section 5 and section 6 concludes the paper.