### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

The equatorial counterpart of the quasi-geostrophic model that has recently been derived and is referred to as the equatorial balanced model is compared to the shallow water equations. This comparison allows for the assessment of the accuracy with which the equatorial balanced model approximates the low-frequency, or balanced, component of the dynamics described by the shallow water equations. In order to be able to consider fully nonlinear dynamics, numerical implementations of both models are developed for a zonally periodic channel on the equatorial *β*-plane using the Contour-Advective Semi-Lagragian (CASL) algorithm. The CASL algorithm is chosen because it has beneficial qualities compared to other algorithms by taking full advantage of the property that both models materially conserve potential vorticity. The particular dynamics chosen for the assessment is that of an initially unstable westward flow that freely evolves. The assessment also shows that the accuracy of the equatorial balanced model is comparable to that of the quasi-geostrophic model outside the equatorial region and to that of one of the most accurate balanced models with its balance relations linearized in the equatorial region. Copyright © 2011 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

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.

### 2. Equations

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

The shallow water equations, which describe the horizontal dynamics of a shallow layer of fluid with constant density, are given by

- (1)

- (2)

The independent variables are (**u***,h*) = (*u,v,h*), where *u* is the velocity component in the zonal direction, denoted by *x*, *v* the velocity component in the meridional direction, denoted by *y*, and *h* the height. Time is denoted by *t*, the Coriolis parameter by *f*, the acceleration due to gravity by *g*, the vertical unit vector by , and the gradient operator (*∂/∂x,∂/∂y*) by ∇. We consider an equatorial *β*-plane, i.e. *f* is approximated by *f* = *βy*, where *β* is the meridional derivative of *f* at the equator. The height *h* can be expressed as *h* = *H* + Δ*h*, where *H* is the layer height at rest, i.e. a constant, and Δ*h* the height perturbation. In TM, the height perturbation is expressed as so that the matrix, in terms of which the linear form of the shallow water equations can be expressed, becomes Hermitian. This is essential in TM, but in this paper, it is more suitable to express the height perturbation as Δ*h* = *Hη*_{∗}, where *η*_{∗} is non-dimensional. The relation between *η* in TM and *η*_{∗} in this paper is therefore given by

- (3)

where , which is constant.

To avoid confusion, *q* = (*ζ* + *βy*)/(*c* + *η*) in TM and thus it is equal to 1*/c* times *q* in this paper. By using *q* in (4) rather than the potential vorticity *q/H* itself, both the shallow water equations and the equatorial balanced model are dependent on *H* only through *c*, which simplifies their numerical implementations. A shorter form of the definition of *γ* is given by *γ* = ∇ · *D***u***/Dt*, which gives (6), using (1). The variable *γ* is therefore referred to as the acceleration divergence, which in the special case of an *f*-plane outside the equatorial region is identical to the ageostrophic vorticity. It is often used in the study of balanced dynamics (e.g. Lynch, 1989; Mohebalhojeh and Dritschel, 2001). To obtain the shallow water equations in terms of the new variables (*q,δ,γ*), we eliminate *δ* between ∇× (1) and (2) (Salmon, 1998, p. 61) and multiply the result by *H*, which gives

- (8)

Calculating ∇· (1) gives

- (9)

and taking the time-derivative of *γ* in (6), using (1) and (2) to eliminate the resulting time-derivatives on the right-hand side, gives

- (10)

The equatorial balanced model, derived in TM from the shallow water equations is given by an evolution equation and two balance relations. The variables representing the dynamics described by the equatorial balanced model are labeled by a subscript *R* in TM. Since in this paper there is no need to indicate this explicitly, the subscript is dropped. The evolution equation is given by TM (5.35), which multiplied by *c* gives (8). The two balance relations take different forms for the zonal mean of the flow and the perturbation from this zonal mean. We therefore introduce the decompositions

- (11)

- (12)

The bar and prime on **u** and *η*_{∗} as well as on any other variable in this paper indicates the zonal mean and the perturbation from this zonal mean, respectively. The two balance relations for the zonal mean are given by TM (5.10) and (5.11), which are

- (13)

- (14)

The two balance relations for the perturbation from the zonal mean are given by TM (5.17) and (5.18) with *D* in TM (5.19) replaced by *D*_{α} in TM (5.43) and in re-written form by TM (7.1) and (7.2), which is

- (15)

- (16)

where

- (17)

with the free parameter *α* = 2.5, as mentioned in the introduction, and

- (18)

The balance relations (15) and (16) have the benefit of being given in terms of *δ*′ and *X*′, which have the property *δ*′ 0 and *X*′ 0 as the distance to the equator increases. This property is shown in TM section 6 and used below.

In the geometric framework, the balance relations only describe an approximate Rossby manifold because in their derivation in TM section 5.1 some approximations are made.

### 4. Configuration

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

We consider an initially unstable westward flow on the equatorial *β*-plane. All variables are non-dimensionalized, using one day as the time scale, the radius of the earth as the length scale, and consequently the radius of the earth divided by one day as the velocity scale. The variable *η*_{∗} is already non-dimensional, as described above (3). The constant *c*, defined below (3), is evaluated by choosing *H* = 1000 m and also non-dimensionalized. The channel is chosen to have a non-dimensional length of *L*_{x} = 2*π*, ranging from −*π* to *π*, and a width of *L*_{y} = 2*πr*, ranging from −*πr* to *πr*, where *r* is the ratio of the channel width and channel length and taken to be 1/4. This implies that at rest the non-dimensional value of *q* ranges from −*π*^{2} at the southern boundary to *π*^{2} at the northern boundary. The parameters introduced in the previous section are given the values *n*_{x} = *n*_{y} = 256*,n*_{c} = 100, and Δ*q* = *π/*16.

The meridional profile of the unstable westward flow at the initial time is taken to be (2) in Galewsky *et al.* (2004) adopted for the equatorial *β*-plane. In the range *y*_{S}*< y < y*_{N}, where *y*_{S} = −*L*_{y}/4 and *y*_{N} = *L*_{y}/4 with the subscripts S and N indicating south and north of the equator, it is given by

- (44)

and outside this range by *u*_{ini} = 0. The parameter *c*_{norm} is a constant and chosen such that at the mid-point *y* = (*y*_{S} + *y*_{N})/2, where *u*_{ini} has its maximum absolute value, we have *u*_{ini} = *u*_{mid} and choose *u*_{mid} = −40 m/s. We require this initial state to be a solution of the equatorial balanced model and therefore it must satisfy the balance relations (13) and (14). To obtain *η*_{∗ini} that corresponds to *u*_{ini}, we integrate (14) with *ū* = *u*_{ini} and set the integration of constant such that the resulting has a zero domain average. The Rossby number tends to infinity at the equator and the maximum of the Froude number is 0.4.

In order to trigger an instability of the initially unstable westward flow, a small disturbance is added. We take it to be that given by (4) in Galewsky *et al.* (2004), which is

- (45)

where *η*_{∗a} = 10^{−3}*H*, *α*_{p} = 1/3, and *β*_{p} = 1/20 and the constant *η*_{∗c} is chosen such that *η*_{∗} has a zero domain average. Inserting **u** = (*u*_{ini},0) and *η*_{∗} = *η*_{∗ini} + *η*_{∗p} into (4) and non-dimensionalizing the result gives the non-dimensional initial *q* shown in Figure 2.

### 5. Assessment

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

The equatorial balanced model approximates by construction the low-frequency, or balanced, component of the dynamics described by the shallow water equations. The accuracy of the equatorial balanced model is therefore assessed by determining how closely the dynamics that it describes resembles that described by the shallow water equations. To implement this assessment, we apply the two tests used in McIntyre and Norton (2000). One is a diagnostic test that singles out the inversion to assess its accuracy independently from the advection. Both inversion and advection comprise the algorithm used to solve the equatorial balanced model, described in section 3. This test thus assesses the accuracy with which the balance relations of the equatorial balanced model describe the balanced component of the dynamics at a fixed time. The other is a prognostic test that assesses the cumulative accuracy of the repeated use of the inversion and advection that evolves the dynamics in time.

To carry out these tests, we first determine the initial state from the initial *q*, specified in the previous section, subject to the equatorial balanced model's balance relations. It is obtained by the inversion described in section 3.1, which gives among other variables **u**, *η*_{∗}, *δ*, and *X*. To obtain also *γ*, we recall that because of (14) and the periodicity in the zonal direction and therefore *γ*′ can be determined by taking the *x*-derivative of *γ* in (6), which gives

- (46)

Obtaining *γ* in this way turns out to be more accurate than determining it from (6) directly. The complete initial state thus obtained satisfies the balance relations of the equatorial balanced model. In the geometric framework, this means that the initial state is represented by a point on the approximate Rossby manifold mentioned at the end of section 2.

We consider the evolution of the dynamics from this initial state as described by the shallow water equations. At the initial non-dimensional time *t* = 0, *q* is shown in Figure 2, at *t* = 45 in Figure 3a, and at *t* = 100 in Figure 4a. In all three figures only two of the 100 contour levels are displayed, representing two values of *q* with the same magnitude but opposite signs. At *t* = 45, fine scales are already apparent. At *t* = 100, the fine scales are so complex that even by showing only two of the 100 contour levels, individual contour sections are indiscernible. A magnification of the area encompassed by the white box in Figure 4a is therefore given in Figure 4b. It clearly shows the very fine-scale nature of *q*. The corresponding **u**, *η*_{∗}, and *δ* are more broad-scale and shown in the top panels in Figures 5 and 6 for *t* = 45 and *t* = 100, respectively. This evolution of the dynamics as described by the shallow water equations serves as a benchmark in the assessment of the equatorial balanced model.

The diagnostic test, which assesses the accuracy of only the inversion, is carried out first. This requires at each time step to perform the inversion used to solve the equatorial balanced model, but using *q* as described by the shallow water equations, resulting in particular in **u**, *η*_{∗}, and *δ*. For *t* = 45 and *t* = 100, the resulting variables are presented in the middle panels of Figures 5 and 6, respectively. They differ from those described by the shallow water equations shown in the respective top panels of these figures. This is because in the case of the shallow water equations, **u**, *η*_{∗}, and *δ* are obtained from *q*, *δ*, and *γ*, which are all evolved independently, while in the case of the equatorial balanced model they are obtained by the inversion from *q* alone, subject to the balance relations of the equatorial balanced model. Comparison of the respective panels indicates that the difference is minimal, which implies that the inversion is accurate, i.e. that the balance relations of the equatorial balanced model accurately describe the low-frequency, or balanced, dynamics from *q* alone.

To quantify this accuracy, we introduce a measure of the difference given by

- (47)

where *s* is a variable as described by the shallow water equations and *s*_{b} is the corresponding variable that is either the result of the inversion or *q* as described by the equatorial balanced model, and ||*…*|| denotes the *L*_{2} norm of a variable. Another way to quantify this accuracy is by calculating the correlation coefficient, denoted by *r*, between *s* and *s*_{b}. Figures 7 and 8 show and *r*, respectively. They therefore show in particular and *r* of the comparison between the top and middle panels of Figures 5 and 6, which are (*u*) = 0.45 × 10^{−2}, *r*(*u*) = 1.00, (*v*) = 1.76 × 10^{−2}, *r*(*v*) = 1.00, (*η*_{∗}) = 0.33, *r*(*η*_{∗}) = 0.95, (*δ*) = 0.90, and *r*(*δ*) = 0.97 at *t* = 45 and (*u*) = 0.19, *r*(*u*) = 0.98, (*v*) = 0.05, *r*(*v*) = 1.00, (*η*_{∗}) = 0.26, *r*(*η*_{∗}) = 0.97, (*δ*) = 1.05, and *r*(*δ*) = 0.84 at *t* = 100. Both measures indicate an overall high accuracy of the inversion and also the same relative accuracy of the individual variables, where for late times the sequence of variables with the highest to lowest accuracy is given by *v*, *ζ*, *u*, *η*_{∗}, and *δ*.

Some of the inaccuracy is not due to the inherent inaccuracy of the equatorial balanced model, but a natural physical difference of its dynamics compared to that described by the shallow water equations. The inversion, which is a part of the equatorial balanced model, is constructed such that it determines only the low-frequency, or balanced, component of the dynamics described by the shallow water equations. The high-frequency, or unbalanced, component described by the shallow water equations therefore represents this physical difference. Nevertheless, this physical difference contributes to the inaccuracies as determined by the two measures above. For example, the shallow water equations describe the equatorial Kelvin wave, an unbalanced feature, while the equatorial balanced model by construction does not. As pointed out at the end of section 3.2, the CASL algorithm is not as diffusive as conventional algorithms and therefore damping of the generation of unbalanced dynamics is weaker. The use of the CASL algorithm thus makes the tests more stringent. The unbalanced component is most prominently represented by *δ* and therefore the low accuracy expressed by *δ* in Figures 7 and 8 is mainly due to this physical difference (e.g. McIntyre and Norton, 2000).

The prognostic test is carried out next. It compares the dynamics as described by the shallow water equations, which is also used in the diagnostic test above, with that described by the equatorial balanced model, starting from the same initial state of which *q* is shown in Figure 2. This means that for this prognostic test, the inversion uses *q* as described by the equatorial balanced model instead of as described by the shallow water model, which is the case for the diagnostic test. At *t* = 0, the values of all variables of both models are respectively identical by construction, as described in section 4. Both models therefore advect the same *q*, using the same **u** for the first time step. At the new time, **u** is determined by the shallow water equations from *q*, *δ*, and *γ*, where *δ* and *γ* are evolved to this new time independently from *q* and from each other, and by the equatorial balanced model from the inversion of *q* alone. The diagnostic test shows that the two resulting **u**'s differ. The difference at this first time step is indiscernible as shown in Figures 7 and 8, but still causes *q* to be advected slightly differently by both models for the next time step. At the new time, the *q* of both models is thus different and this difference then adds to the difference caused by the fact that each model at the new time determines **u** differently. This is also the case for every following time step, resulting in an accumulation of differences.

The difference between the *q*'s is apparent at *t* = 45 as shown in Figure 3 and the accumulation of this difference over time is measured by and *r*, which are shown as (*q*_{EBM}) and *r*(*q*_{EBM}) in Figures 9 and 10, respectively. The accumulation of the differences is also evident in Figures 5 and 6 as respectively the bottom panels, showing the dynamics as described by the equatorial balanced model, differ greater from the top panels, showing the dynamics as described by the shallow water equations, than the middle panels, showing the result of the inversion used in the diagnostic test for which *q* is taken to be that corresponding to the **u**, *η*_{∗}, and *δ* shown in the respective top panels. The difference between the top and bottom panels is quantified again by calculating the correlation coefficient, which gives *r*(*u*) = 1.00, *r*(*v*) = 0.83, *r*(*η*_{∗}) = 0.94, and *r*(*δ*) = 0.79 for *t* = 45 and *r*(*u*) = 0.82, *r*(*v*) = 0.12, *r*(*η*_{∗}) = 0.78, and *r*(*δ*) = 0.19 for *t* = 100. Comparing them with their counterparts of the diagnostic test clearly shows the accumulation of the differences.

The differences, however, do not accumulate indefinitely as indicated by (*q*_{EBM}) and *r*(*q*_{EBM}) in Figures 9 and 10, respectively. The reason is that the inaccuracy of the inversion is sufficiently small so that the difference between the dynamics of both models as measured by and *r* does not change once the maximum difference in a statistical sense is reached, which is when the maximum degree of nonlinearity is reached. This occurs at about *t* = 50 as indicated by the eddy kinetic energy (not shown).

Another comparison of the equatorial balanced model that allows the assessment of its accuracy is with a highly accurate balanced model. Such a model is the so-called third-order plain-*δδ* potential vorticity-based balanced model, or short third-order *δδ* model, a summary of which is provided in the appendix. It is one of the most accurate balanced models as shown by e.g. McIntyre and Norton (2000), Mohebalhojeh and Dritschel (2001), and Mohebalhojeh and McIntyre (2007b). However, its balance relations are nonlinear whereas those of the equatorial balanced model are linear. Since balanced models with nonlinear balance relations are almost always more accurate than those with linear balance relations, a more instructive comparison of the equatorial balanced model would be with the third-order *δδ* model with its balance relations linearized.

In the geometric framework presented in the introduction, linear balance relations describe flat subspaces in phase space. The most accurate linear balance relations describe the Rossby manifold. This is because the Rossby manifold is spanned by the eigenvectors associated with the low-frequency normal mode of the primitive equations, which in this paper are the shallow water equations. The states it represents are therefore those of purely low-frequency dynamics, unlike all other flat subspaces described by linear balance relations, which represent states of a dynamics that is mainly low-frequency, but to a minimized extent also high-frequency. In the derivation of the quasi-geostrophic model by Leith (1980), the dynamics constrained on the approximate slow manifold is projected onto the Rossby manifold. The approximate slow manifold is described by balance relations, which for both the quasi-geostrophic model and the equatorial balanced model are given in general form by TM (2.2) with **x**^{s} replaced by on its right-hand side. If their nonlinear terms are neglected, i.e. the right-hand side of TM (2.2) set to zero, they give the linear balance relations that describe the Rossby manifold. In other words, the projection onto the Rossby manifold is the geometric equivalent of the linearization of the balance relations that describe the approximate slow manifold. The derivation of the equatorial balanced model is equivalent to that of Leith (1980), except that it involves approximations. The Rossby manifold and thus the balance relations that describe it are only approximate, as mentioned at the end of section 2, and in addition the projection and thus the evolution equation is also approximate. The equatorial balanced model therefore describes an approximation of a dynamics constrained on the Rossby manifold. This is not necessarily the case for other balanced models, but it is also the case for the third-order *δδ*-model. This is because according to Browning *et al.* (1980), Hinkelmann (1969) shows that as the *δδ* model's order tends to infinity its linearized balance relations describe a flat subspace that tends to the Rossby manifold. Both models therefore approximate the Rossby manifold. The third-order *δδ*-model is one of the most accurate balanced models. This suggests that with its balance relations linearized, it is most likely also one of the most accurate balanced models among all balanced models with linearized balance relations. The third-order *δδ*-model with its balance relations linearized is therefore an excellent model to compare with the equatorial balanced model.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Equations
- 3. Algorithm
- 4. Configuration
- 5. Assessment
- 6. Conclusions
- Acknowledgements
- Appendix: The third-order
*δδ* model - References

The equatorial balanced model is derived in TM as a balanced model that represents the equatorial counterpart of the quasi-geostrophic model because the quasi-geostrophic model itself is not valid in the equatorial region. It is considered the counterpart because its derivation is equivalent to that of the quasi-geostrophic model by Leith (1980). As any other balanced model, it is designed to describe the low-frequency, or balanced, component of the dynamics as described by the primitive equations, which reduce to the shallow water equations in the case of a shallow layer of fluid with constant density, as considered in this paper. Its accuracy is assessed in TM, but only of its description of linear dynamics, by determining its dispersion relation and comparing it to that of the shallow water equations. This shows that it approximates strikingly well the linear balanced dynamics described by the shallow water equations.

The equatorial balanced model's accuracy in approximating the fully nonlinear balanced dynamics described by the shallow water equations is best assessed numerically. The specific dynamics considered is that of an unstable westward flow in a zonally periodic channel on the equatorial *β*-plane. For the numerical implementation of both models, the shallow water equations and the equatorial balanced model, the CASL algorithm is adopted because it uses the fact that both models materially conserve potential vorticity by advecting contours of *q*, which is proportional to the potential vorticity, and because it has several advantages over more conventional models. The CASL algorithm consists of two parts. One part is the inversion, which at each time step determines the velocity. In the case of the shallow water equations, the velocity is obtained from *q* and two other variables that are being evolved independently from *q* and from each other. In the case of the equatorial balanced model, the velocity is obtained from *q* alone, subject to its balance relations. The other part is the advection, which advects *q*, using the velocity determined by the inversion, for one time step and in case of the shallow water equations also evolves independently the above-mentioned other two variables. The inversion is unique to each balanced model and domain and therefore a unique inversion is developed in this paper, while the required advection is generic and is therefore adopted from other model studies.

The accuracy is assessed by two tests, one diagnostic and the other prognostic. The diagnostic test considers the dynamics as described by the shallow water equations and at each time step uses its *q* to perform the inversion used to solve the equatorial balanced model. This inversion of *q* is subject to the balance relations of the equatorial balanced model, which results in various variables, including the velocity and height, whose values are different to the values as determined by the shallow water equations. This difference is therefore a measure of the accuracy of the inversion alone and thus of the balance relations. The test shows that the inversion is accurate as indicated, for example, by the fact that the correlation between the respective variables of both models is mostly larger than 0.95. The prognostic test assesses the cumulative accuracy as the differences accumulate because of the repetition of the inversion and advection. However, this accumulation ceases at about the time when the degree of nonlinearity reaches its maximum, as is seen for many other balanced models. This is because the differences introduced at each time step are sufficiently small so that the difference between the dynamics of both models does not further increase in a statistically sense.

Because the equatorial balanced model is the equatorial counterpart of the quasi-geostrophic model, the accuracy of both models is compared and found to be comparable. The equatorial balanced model is even comparable to one of the most accurate balanced models, the so-called third-order *δδ* model, with its balance relations linearized. This means that as a balanced model with linear balance relations the equatorial balanced model must be for the equatorial region one of the most accurate balanced models with linear or linearized balance relations.