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 ωK′, 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.
The shallow water equations, which describe the horizontal dynamics of a shallow layer of fluid with constant density, are given by
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 + Δh, where H is the layer height at rest, i.e. a constant, and Δh 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 Δh = Hη∗, where η∗ is non-dimensional. The relation between η in TM and η∗ in this paper is therefore given by
where , which is constant.
To solve the shallow water equations numerically, using the CASL algorithm, we re-write them such that they do not describe the evolution of (u,v,h), but instead of (q,δ,γ). These new variables are defined by
where q/H is the potential vorticity, δ the divergence, and ζ the vorticity defined by
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 = ∇ · Du/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
Calculating ∇· (1) gives
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
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
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
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 Dα in TM (5.43) and in re-written form by TM (7.1) and (7.2), which is
with the free parameter α = 2.5, as mentioned in the introduction, and
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.
Both models, the shallow water equations and the equatorial balanced model, are numerically solved in a zonally periodic channel on an equatorial β-plane using the CASL algorithm. The zonal periodicity allows to express the dynamics in terms of Fourier transforms, where de-aliasing is accomplished by a Broutman spectral filter (Broutman et al., 1997). The lateral boundaries are taken to be sufficiently far away from the equator so that we can take advantage of the property of the equatorial balanced model mentioned below (18) and thus set δ′|Σ = 0 and X′|Σ = 0, where Σ indicates the lateral boundaries. For consistency we adopt these boundary conditions also for the shallow water equations. At the initial time, (q,δ,γ) in the domain for the shallow water equations and only q in the domain for the equatorial balanced model as well as u at the lateral boundaries for both models are specified. The algorithm has two parts. One part is the inversion that determines (u,v,η∗) at a fixed time from given variables that are those specified at the initial time. In the case of the shallow water equations, an inversion becomes only necessary because the CASL algorithm advects q. If the CASL algorithm was not chosen, (u,v,η∗) could simply be directly evolved using (1) and (2). In the case of the equatorial balanced model, the inversion is the specific inversion of potential vorticity, or q, subject to its balance relations. The inversion is different for different models and therefore we develop the required algorithms within the CASL algorithm for both models. The other part is the advection that advects contours of q and in the case of the shallow water equations also evolves the two other independent variables δ and γ on a regular grid for one time step to a new fixed time. The algorithm for advecting contours of q is at the core of the CASL algorithm and is the same for all potential vorticity, or q, conserving models. We therefore adopt it for both models from Dritschel and Ambaum (1997).
The inversion determines (u,v,η∗) at a fixed time from the above-mentioned given variables, which are (q,δ,γ) and only q in the domain for the shallow water equations and the equatorial balanced model, respectively, as well as u at the lateral boundaries for both models. This is achieved using several iterative steps.
The first step determines and then corrects ζ. To determine ζ, we use (4). This requires η∗, which can generally only be estimated. An estimate of η∗ at the initial time can be the η∗ that is used in constructing the initial state and at subsequent time steps it can be the η∗ at the respective previous time step. A new and improved approximation of η∗ compared to the initial estimate is determined by the third step below, allowing this and the following two steps to be repeated until η∗ converges. The ζ obtained using (4) must be corrected such that the corrected ζ, denoted by ζc, is consistent with the zonal mean of the given zonal velocities at the southern and northern lateral boundaries, denoted by ūS and ūN, respectively. Using (7), this consistency means
where the double bar denotes the domain average and Ly the width of the channel. To determine ζc from ζ, we use (4) to obtain
where q is a function of ζ given by (4) and Δqc is taken to be a constant reflecting the change of q associated with the correction of ζ. Multiplying (20) by (1 + η∗) and taking the domain average, using the fact that by definition, gives
Substituting (21) into (20) and using (4), leads to ζc being a function of ζ, given by
where is given by (19). Although only the corrected ζc is used throughout the algorithm, q is not corrected correspondingly by replacing it with q + Δqc. This is justified because no other variable besides ζ is determined from q and because contour plots of q and q + Δqc are identical except for the difference in contour values of Δqc.
The second step determines u. Taking the zonal average of ζc and δ, using respectively (7) and (5), gives
Since the velocities at the lateral boundaries are given, (23) and (24) can be integrated to give in the domain.
To determine u′ in the domain, we apply the general algorithm developed for an arbitrary domain by Chen and Kuo (1992a, 1992b) to our zonally periodic channel domain. By expressing u′ in the form of the Helmholtz decomposition, given by
where ψ is called the streamfunction and χ the velocity potential, we have
This set of equations is separated into two sets of equations by introducing
where the subscript h labels the so-called harmonic parts and the subscript I the so-called inner parts. The corresponding velocities, denoted by u′h and u′I, are defined using (25). The separation into two parts allows the harmonic parts to be determined by one set of equations, given by
subject to the Neumann boundary conditions
where the subscript y denotes ∂/∂y, and the inner parts by another set of equations, given by
subject to the Neumann boundary conditions
The inner parts ψI and χI are determined by expressing all variables in (34)-(37) in terms of their Fourier transforms in the x-direction. Thus, (34) with (36) becomes
and (35) with (37) takes on an equivalent form, where k is the wavenumber and the tildes denote the Fourier transforms. The first and second derivatives are then descretized by centered finite differences accurate to second order. The left-hand side of (38) therefore becomes, using (39), a tridiagonal matrix. This matrix is inverted using the Thomas algorithm, giving the discretized values of ψI. This method is also adopted further below to solve (6), (15), and (16). The determined ψI and χI, allow us to determine also u′I, using (25).
The harmonic parts ψh and χh are determined iteratively because the right-hand sides of (32) and (33) are unknown. To determine these right-hand sides, we use (25) to express u′h|Σ as
The unknown right-hand sides of (32) and (33) are the first term on the right-hand side of (40) with the opposite sign and the second term on the right-hand side of (41), respectively. To determine them, the remaining terms in (40) and (41) must therefore be known or determined. The left-hand sides, expressed as u′h|Σ = u′|Σ − u′I|Σ, are known because u′|Σ is given and u′I|Σ is determined above. The second term on the right-hand side of (40) is unknown and therefore set to zero, which thus gives us an approximation of the unknown right-hand side of (32). This allows us to solve (30) with (32). As above for ψI, we express ψh in terms of its Fourier transform, denoted by ψh. This leads to equations equivalent to (38) and (39), where the right-hand sides are given by only k2ψh and the Fourier transform of ψy(Σ), respectively. The solution of the equivalent to (38) is given in explicit form by ψh = aexp(ky) + bexp(−ky), where the constant coefficients a and b are determined by the equivalent to (39). We thus obtain an approximate ψh, denoted by . The result is substituted for ψh in (41), which thus gives us an approximation of the unknown right-hand side of (33). This allows us to solve (31) with (33), using the method that is used to solve (30) with (32) above, resulting in an approximate χh, denoted by . The result is substituted for χh in (40), allowing us to calculate an improved approximate ψh, denoted by , which in turn is substituted for ψh in (41), allowing us to calculate an improved approximate χh, denoted by . This is repeated until respectively ψh and χh converge, allowing us to determine u′h, using (25). The velocity is then determined by .
The third step determines a new and improved approximation of η∗. Since we have for both models X′|Σ = 0 and v′|Σ = 0, (18) implies η∗′|Σ = 0. Using this as well as and δ′|Σ = 0 in evaluating the mass continuity equation (2) at the lateral boundaries, gives . This together with η′∗|Σ = 0 means that η∗|Σ must be constant both along each lateral boundary and in time to satisfy the mass continuity equation (2) and thus ensure that mass is conserved locally. However, mass must also be conserved globally, i.e. , which is achieved by adding a constant to η∗ as a correction at each time step. This changes η∗|Σ in time and thus exact local mass conservation is not ensured. In the case of the shallow water equations, this represents only a small, inevitable numerical error and η∗ is determined by solving (6) with Dirichlet boundary conditions given by setting η∗|Σ to the same value as for the equatorial balanced model at the initial time determined below. However, in the case of the equatorial balanced model and many other balanced models, including the quasi-geostrophic model, the fact that exact local mass conservation is not ensured is not a small numerical error, but their property. A considerable effort must be undertaken to construct balanced models based on potential vorticity conservation that also do conserve mass locally (Mohebalhojeh and McIntyre, 2007a,b). To determine η∗ of the equatorial balanced model, and η′∗ are determined separately. Because of the periodicity in the zonal direction and (14), . The zonal average of (6) with the initial estimate as a Dirichlet boundary condition is therefore solved, giving . This is corrected to ensure global mass conservation. The corrected includes a corrected , which is used as a new and improved Dirichlet boundary condition in solving the zonal average of (6) for again. This is repeated until converges. The resulting at the initial time provides , which is used in the numerical implementation of the shallow water equations as a Dirichlet boundary condition for η∗|Σ in solving for η∗ at every time step, as described above. To determine η∗′, (18) is solved, which requires X′ to be estimated initially. We choose X′ = 0 and a new and improved approximation compared to this initial estimate is determined further below.
The above three steps are repeated with the new and improved approximation of η∗, obtained in the last step, until η∗ converges. We consider η∗ to have converged if the difference in its values obtained from two successive iterations has become smaller than 10−6. This three-step iteration results in (u,v,η∗).
In the case of the equatorial balanced model, δ′ and X′ are only approximated in contrast to the δ and γ of the shallow water equations, which are given. To obtain new and improved approximations of δ′ and X′ we use (u,v,η∗) obtained by the above three-step iteration to solve (15) and (16) with δ′|Σ = 0 and X′|Σ = 0 as Dirichlet boundary conditions. The above three-step iteration must then be repeated with these new and improved approximations of δ′ and X′ until they converge. We consider δ′ and X′ to have converged if the difference in the value of X′ obtained from two successive iterations has become smaller than 10−4. Once δ′ and X′ have converged, the above three-step iteration is repeated one more time to obtain the corresponding (u,v,η∗).
The advection mainly advects contours of q, and in the case of the shallow water equations it also evolves δ and γ on a regular grid, for one time step. The part of the CASL algorithm that advects contours of q is identical for all potential vorticity conserving models, which include both the shallow water equations and the equatorial balanced model. We therefore adopt this part of the CASL algorithm from Dritschel and Ambaum (1997). A summary of it is given in section 2.c. in Dritschel et al. (1999) and therefore it is only briefly explained in the following.
The initial state is typically specified on a grid and therefore q on the grid must be converted into a set of contours of q, which are represented by nodes on the contours. We achieve this in a slightly different way than described in section 2.c. in Dritschel et al. (1999) by following Mohebalhojeh and Dritschel (2009). We specify contours of q at levels qj = [j − 1/2(nc + 1)]Δq, where j = 1,…,nc with nc being an even number and Δq the contour interval, and consider q in between the contours to be uniform.
The first step is to obtain (u,v,η∗) on a grid of size nx × ny. This is achieved by interpolating q given on the nodes of the contours onto a fine grid of size mnx × mny, where m is typically chosen to be 4. Subsequent iterative averaging results in a q on the grid of size nx × ny, which is smoother than that on the fine grid. The inversion, described in section 3.1, uses q, and in the case of the shallow water equations also δ and γ, to determine (u,v,η∗) on the grid.
The second step is to advect the nodes for one time step of size Δt. Because u is given on a grid, bilinear interpolation is used to determine u at the locations where it is required. The advection is governed by dx/dt = u(x,t), where x is the location of a node, and numerically implemented using the midpoint method (Temperton and Staniforth, 1987; Bates et al., 1995), given by
together with a linear time extrapolation, given by
The locations xa, xd, and xmp = (xa + xd)/2 denote the arrival, departure, and midpoint locations, respectively, and tn the time at time step n. Because xmp depends on the unknown xa, it is first approximated by xmp = xd + 1/2Δtu(xd,tn), then this approximation is used as xa in (42) and (43) with Δt replaced by 1/2Δt to determine a more accurate approximation of xmp, which is then used in (42) and (43) to determine xa. A Robert-Asselin time filter is used to damp spurious high-frequency modes (Robert, 1966; Asselin, 1972).
Besides the advection of contours of q in the case of the shallow water equations, δ and γ are evolved on the grid using (9) and (10), respectively, where an explicit damping of δ, given by ν∇2δ, is added to (10) with the constant ν chosen such that the damping time is 0.01 day for the shortest resolvable wave. This damping removes small-scale gravity waves generated in particular by the effect of sharp gradients of q on δ (Mohebalhojeh and Dritschel, 2000, 2004). These waves can be due to numerical errors or be part of a physical process that adjusts the dynamics to its balanced state (Mohebalhojeh and McIntyre, 2007b). The damping allows for a better presentation of the results by only minimally affecting them.
The third step ensures that the contours of q are adequately represented by nodes and also removes small scale features in q. The advection of contours of q in the previous step has the effect of stretching and compressing the contours. As a result some sections of the contours are overrepresented and other underrepresented by nodes and therefore the nodes are redistributed and if necessary new ones are added. The advection of contours can also result in very small scale features in q such as fine filaments. They need to be removed as the increasing number of nodes that would be needed to accurately represent them would become unmanageable. This is accomplished by so-called contour surgery (Dritschel, 1989), which reconstructs adjacent sections of contours if they come too close to each other and deletes contours if the area they encompass has become sufficiently small. Contour surgery can be illustrated by imaging a single circular contour. Let us assume the advection has the effect of indenting the circular contour on two opposing sides until it has the shape of a dumbbell. When the two sections of the contour that represent the narrow part of the dumbbell come too close, contour surgery cuts through this narrow part, thus cutting the single contour in two locations. The resulting two contour fragments are then re-connected to form two separate contours. If one or both of these two new contours encompass an area that is sufficiently small, the respective contour is removed. Removing very small scale features is a numerical necessity and mimics molecular diffusion in the continuous limit. In most other numerical implementations this is achieved by the unphysical hyperdiffusion, which has the disadvantageous effect of diffusing gradients of q. Contour surgery, however, preserves these gradients. Another advantage of contour surgery is that the surgery scale can be chosen to be much smaller than the grid size, typically 10 times smaller, thus retaining features in q that are smaller than the grid size.
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 Lx = 2π, ranging from −π to π, and a width of Ly = 2πr, 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 −π2 at the southern boundary to π2 at the northern boundary. The parameters introduced in the previous section are given the values nx = ny = 256,nc = 100, and Δq = 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 yS< y < yN, where yS = −Ly/4 and yN = Ly/4 with the subscripts S and N indicating south and north of the equator, it is given by
and outside this range by uini = 0. The parameter cnorm is a constant and chosen such that at the mid-point y = (yS + yN)/2, where uini has its maximum absolute value, we have uini = umid and choose umid = −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 uini, we integrate (14) with ū = uini 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
where η∗a = 10−3H, αp = 1/3, and βp = 1/20 and the constant η∗c is chosen such that η∗ has a zero domain average. Inserting u = (uini,0) and η∗ = η∗ini + η∗p into (4) and non-dimensionalizing the result gives the non-dimensional initial q shown in Figure 2.
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
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
where s is a variable as described by the shallow water equations and sb is the corresponding variable that is either the result of the inversion or q as described by the equatorial balanced model, and ||…|| denotes the L2 norm of a variable. Another way to quantify this accuracy is by calculating the correlation coefficient, denoted by r, between s and sb. 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 (qEBM) and r(qEBM) 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 (qEBM) and r(qEBM) 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).
The equatorial balanced model on the equatorial β-plane is derived by TM in the equivalent way Leith (1980) drives the quasi-geostrophic model on an f-plane. Because the equatorial balanced model can in this sense be considered the equatorial counterpart of the quasi-geostrophic model, it is of interest to compare their respective accuracies. Mohebalhojeh and McIntyre (2007b) consider the quasi-geostrophic model and determine of its q compared to that of the shallow water equations by describing an unstable zonal flow with the Froude and Rossby numbers having maximum values of about 0.5 and 1, respectively, on an f-plane in a doubly periodic domain, using not the CASL, but a standard semi-Lagrangian algorithm. The differences in the initial state, domain, and algorithm compared to those of the study presented in this paper do not substantially effect the accuracy of the models. This therefore allows us to compare the of the equatorial balanced model with that of the quasi-geostrophic model as determined by Mohebalhojeh and McIntyre (2007b). Both are shown in Figure 9, denoted by (qEBM) and (qQG), indicating clearly that the accuracy of both models is comparable.
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 xs 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.
The numerical implementation of the third-order δδ model uses the same configuration as that used for the equatorial balanced model described in section 4. The difference between the q's of the shallow water model and the third-order δδ model is, as in the case of the equatorial balanced model, measured by and r and shown as (qδδ) and r(qδδ) in Figures 9 and 10, respectively. Furthermore, what is shown in Figures 7 and 8 for the equatorial balanced model is shown in Figures 11 and 12 for the third-order δδ model. The comparison of the figures show that the accuracy of both the equatorial balanced model and the third-order δδ model as measured by and r are comparable.
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.
This paper is based upon work supported by the National Science Foundation under Grant Nos. AGS-0827210 and OCE-0851493. ARM would like to thank the UK Natural Environment Research Council for a research fellowship and the Universities of St. Andrews and Tehran for support during this research.
Appendix: The third-order δδ model
The third-order plain-δδ potential vorticity-based balanced model, or short third-order δδ model, materially conserves potential vorticity as the shallow water equations and the equatorial balanced model. Its evolution equation is therefore given by (8). It is first defined in McIntyre and Norton (2000), who refer to it as third-order direct inversion. They define its balance relations for an f-plane in their (3.6a)-(3.6i). Mohebalhojeh and Dritschel (2001) rename it third-order δ balance, or (δ(2),δ(3)) balance, and give its balance relations for an f-plane in their (24). Its name used in this paper is introduced in Mohebalhojeh and McIntyre (2007a, 2007b). They refer to the balance relations as third-order δδ truncation and specify them in their (4.1)-(4.3) with M = 2 in Mohebalhojeh and McIntyre (2007a) and in their (2.3)-(2.5) with M = 2 in Mohebalhojeh and McIntyre (2007b). For this paper the balance relations are formulated for an equatorial β-plane and linearized. They thus become
where δ(n) and u(n) with n = 1,2 are diagnostic estimates of ∂nδ/∂tn and ∂nu/∂tn, as defined in McIntyre and Norton (2000) and Mohebalhojeh and Dritschel (2001).