1. Introduction

The Met Office’s Unified Model (MetUM; Martin et al. 2006), used for climate and weather prediction, is one of the few general circulation models (GCMs) based on the general, nonhydrostatic primitive equations. Another is the nonhydrostatic icosahedral atmospheric model (NICAM) described by Tomita and Satoh (2004). Before suitable semi-implicit numerical methods were developed, use of the general equations in GCMs was not practicable and the hydrostatic primitive equations (HPEs) were the preferred option. The HPEs, given in log-pressure coordinates by Andrews et al. (1987, hereafter AHL), for example, can be derived from the general equations by making the approximations described in White et al. (2005) and given later in this paper. They are still used today by most GCMs.

The general nonhydrostatic primitive equations, unlike the HPEs, are formulated in a Euclidean space, which is advantageous since this better represents the actual atmosphere. Further, the greater accuracy of the general primitive equations allows the modeling of subtle physical processes that are not captured when using the HPEs. An example (White and Bromley 1995) is the zonal velocity arising from conservation of absolute axial angular momentum of a fluid parcel moving in the vertical. We therefore expect that as computing power continues to increase, more GCMs will switch to using the general equations rather than the HPEs.

Conceptually, the meridional circulation from equator to pole can be split into two parts: a balanced part due to differential heating and a wave-driven part.¹ The primitive equations can be rearranged to form the so-called transformed Eulerian mean (TEM) equations in which the wave-driven (hereafter residual) circulation is made explicit. The wave driving (at least, that by resolved waves) is studied using the Eliassen–Palm (EP) diagnostics. The TEM equations and EP diagnostics are given by AHL for the HPEs in log-pressure coordinates, following the pressure-coordinate versions derived by Andrews and McIntyre (1978a, hereafter AM). The formalism given in AHL is that commonly used by the stratospheric community.

An important aspect of the TEM formalism is that it includes eddy forcing by the divergence of the EP flux, which in turn appears in a generalized EP relation:

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where ∇ · F is the EP flux divergence, A is the wave activity density, D denotes dissipation due to nonconservative (i.e., frictional and diabatic) effects, and α is a measure of wave amplitude (AHL, p. 131). This relation, when used with the TEM equations, leads to the nonacceleration theorem: ∇ · F = 0 in a steady, conservative flow, and under such conditions

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is a solution of the TEM equations under appropriate boundary conditions (AHL, p. 132). Here, u is the zonal mean zonal wind (see the appendix), θ is the zonal mean potential temperature, and (υ*, w*) is the residual mean circulation. The residual mean circulation can thus, as desired, be considered to be the wave-driven part of the mean circulation. Equations for A and D in the HPE case are given in AHL, and the general, nonhydrostatic, version of the nonacceleration theorem is given in AM [see their Eq. (6.23a)].

AM also derive the TEM equations and EP diagnostics in the general, nonhydrostatic case (appropriate for the MetUM) and in a coordinate-independent form. The present paper shows that a naive application of AHL’s HPE diagnostics, in log-pressure coordinates, to a GCM based on the general, nonhydrostatic equations, in geometric coordinates, gives results that are significantly different from those obtained with the exact diagnostics of AM. Since there is a significant difference, the residual mean circulation calculated from the AHL equations will not satisfy the nonacceleration theorem for steady, conservative waves and thus could give misleading results in such GCMs. Further, we show physically why this difference occurs. The aim of this paper, then, is to provide advice to modeling groups as to which formulation of the TEM equations can be meaningfully applied to their model output. The paper, therefore, includes the equations of AM presented in a way that is directly applicable to general, nonhydrostatic GCM output.

One common use of the EP diagnostics in the modeling community is to compute the so-called downward control streamfunction (Haynes et al. 1991). In this paper, we therefore extend the downward control formulation to the general, nonhydrostatic case and consider differences between this general streamfunction and its more familiar approximate counterpart.

2. The TEM equations and EP diagnostics

a. The general, nonhydrostatic equations

We derive EP diagnostics for the fully compressible, nonhydrostatic equations without making any “shallow atmosphere” assumption. We follow the procedure outlined in section 6 of AM but work in standard meteorological spherical coordinates λ, ϕ, and r, with corresponding velocity components u, υ, and w, rather than in the cylindrical coordinates used by AM.

We start with the zonal momentum equation in the form of White et al. (2005)’s Eq. (2.17), multiplied through by the density ρ:

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where Ω is the earth’s rotation rate, p is pressure, G_λ represents any external zonal force, and

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Using mass conservation,

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where u is the velocity vector; thus, the term on the lhs of Eq. (3) can be written

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where subscripts preceded by a comma denote derivatives. This expression can be substituted in Eq. (3) to give the “flux form” of the zonal momentum equation; after collecting the terms in uυ and uw in the latter, and taking the zonal average at fixed ϕ, r, and t (denoted by an overbar), we obtain

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Following AM, we put ρu = ρ u + ρ′u′, where u′ = u − u so the primes denote eddy terms (deviations from the zonal mean at fixed ϕ, r, and t). We separate the zonal means of triple products into “mean” and “eddy” parts as follows:

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Equation (5) can therefore be written

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where the eddy terms have been put on the rhs together with the forcing term. It is straightforward to show that the zonal mean of the mass conservation equation [Eq. (4)] can be written as follows:

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Equation (7) can be multiplied by u and then used to rewrite the lhs of Eq. (6) as

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where

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is the zonal-mean axial absolute angular momentum per unit mass, so that

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Following AM, we proceed from the above “Eulerian mean” versions of the zonal-mean zonal momentum equation to the TEM version by introducing a “difference streamfunction” ψ defined in terms of the component of the mean eddy potential temperature flux along the mean isentropes:

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where e_λ is a unit vector in the λ direction, and θ is the potential temperature. Expanding the scalar triple product, we obtain

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As noted by AM in their section 6c, there is flexibility in the choice of ψ; this particular definition is independent of the (ϕ, r) coordinates. An alternative definition in terms of the northward potential temperature flux, that is, of the form

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has been used extensively, especially for studies based on the hydrostatic primitive equations.² Figure 1 shows that for the month of January in the MetUM, ψ and ψ₁ are virtually identical. The difference between them is six orders of magnitude smaller than their absolute value (not shown). Difference streamfunctions are shown for January, which is the month that we simulate in this study (see section 3).

Following AM, we define the residual meridional velocity u* = (0, υ*, w*) by

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and the EP flux F = (0, F^(ϕ), F^(r)) by

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Then, after some manipulation, we obtain the TEM zonal momentum equation

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where

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Note that the divergence of the EP flux ∇ · F appears as part of the eddy forcing in the TEM equation [Eq. (15)]. As shown by AM, the nonacceleration theorem, mentioned in section 1 of the present paper, follows from this TEM equation.

b. Approximations to the general equations

1) Hydrostatic primitive equation version, in geometric coordinates

Starting from the zonal momentum equation [Eq. (3)], two approximations are made. One is the “shallow atmosphere approximation,” which we define as in White et al. (2005):

• use geometric height z = r − a as the vertical coordinate, where a is the mean radius of the earth,

• omit all metric terms not involving tanϕ,

• omit Coriolis terms that vary as cosϕ, and

• replace r with a in all terms that remain.

Equation (3) thus takes the form of Eq. (3.8) of White et al. (2005):

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where

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The second approximation is the “hydrostatic approximation” in which the vertical component of the momentum equation is replaced by a precise balance between gravity and the vertical component of the pressure gradient force (full details are given in White et al. 2005).

The corresponding versions of the EP diagnostics for the HPEs can then be derived in the same manner as in section 2a. We just quote the results here.

The zonal-mean axial absolute angular momentum per unit mass is now defined by

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so that

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The residual meridional velocity components are defined by

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and the EP flux vector components are

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where ψ is defined by

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The EP flux divergence is given by

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The TEM zonal momentum equation [Eq. (15)] is replaced by

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Below, this formulation is referred to as HPE(z).

2) Hydrostatic primitive equation version, in log-pressure coordinates

Starting from the HPE(z) approximation, a transformation from geometric height coordinates to log-pressure coordinates is required to obtain the familiar TEM equations and EP diagnostics of section 3.5 of AHL. To avoid confusion with geometric height z, we denote log-pressure height by the symbol Z, defined by

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where H = RT_rg⁻¹ is a suitable pressure scale height for the stratosphere (H = 6800 m is used in this paper), p_r is a reference pressure (1000 hPa here), T_r is a reference temperature (232.4 K corresponds to the value of H used here), R = 287 J K⁻¹ kg⁻¹ is the specific gas constant for air, and g is acceleration due to gravity. It is convenient to introduce a quantity ρ₀(Z) defined by

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where ρ_r = p_r(RT_r)⁻¹ (≈1.5 kg m⁻³ here), chosen to ensure that the mass element ρdxdydz in geometric coordinates equals ρ₀dxdydZ in log-pressure coordinates, under hydrostatic balance. Note that ρ₀ is proportional to p; although its contribution in log-pressure coordinates is analogous to the density ρ in geometric coordinates, ρ₀ is not in general an approximation to ρ. Indeed, it follows from the ideal-gas law p = RTρ that ρρ₀⁻¹ = T_rT ⁻¹; thus, ρ₀ differs significantly from ρ wherever the temperature T differs significantly from T_r (e.g., at the ground).

Then AHL’s section 3.5, with z replaced by Z, gives the following definitions. The residual mean meridional circulation is defined by

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Note that the overbar here denotes a zonal average at fixed ϕ, Z, and t; since Z surfaces are generally different from z or r surfaces, this average is not the same as that defined above, in sections 2a and 2b(1). In the same way, the deviations (…)′ differ from those defined above, which will introduce differences of O(α²) (see the appendix).

The residual circulation corresponds to the definition of the difference streamfunction in terms of the northward potential temperature flux:

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(As noted in section 2a, other definitions of the residual circulation are also possible, with different definitions of ψ. Figure 1 showed, for the middle atmosphere, that the equivalent of ψ₁ used here is virtually identical to the form of ψ used in the general case.)

The TEM zonal-mean momentum equation is

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and the components of the EP flux are given by

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and

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Below, this formulation is referred to as HPE[ln(p)].

3. Numerical results

b. Model results—EP diagnostics

The EP diagnostics for the fully general TEM equations, denoted “general,” are shown for the middle atmosphere in Figs. 2a,c,d,e,f; w* and υ* are calculated from Eq. (12), F^(ϕ) and F^(r) from Eqs. (13) and (14), and ∇ · F from Eq. (16). We define SDIVF = ∇ · F/ρr cosϕ as the scaled EP flux divergence (or force per unit mass) as it appears on the rhs of the momentum equation [Eq. (15)]. Here, z is the vertical coordinate. The EP diagnostics corresponding to the hydrostatic primitive equations, as given in AHL, are shown in Figs. 3a,c,d,e,f and denoted HPE[ln(p)]; υ*, w*, F^(ϕ), F^(Z), and ∇ · F are as defined by Eqs. (22), (23), (25), (26), and (27), respectively. In this HPE[ln(p)] case we define SDIVF = ∇ · F/ρ₀a cosϕ as on the rhs of the momentum equation [Eq. (24)]. Here, Z(p) is the vertical coordinate:

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The diagnostics, shown in the stratosphere and lower mesosphere, look qualitatively similar between the two cases; however, the residual circulation is larger in the general case.

Figure 4a shows the difference in SDIVF between the general and HPE[ln(p)] cases. A greater negative SDIVF in the general case is consistent with a larger residual circulation in that case. This difference, although subject to interannual variability, is remarkably similar when calculated for any of the five months simulated here (not shown) and therefore can be considered generic across the five simulations in this paper. The important point to note here is that there is a significant difference in SDIVF seen throughout the winter stratosphere. In fact, there is a difference in all the EP diagnostics between the general and HPE[ln(p)] cases throughout the winter lower stratosphere that is of the same magnitude as their absolute values (not shown).

Figures 4b and 4c show that these differences are due almost entirely to the differences in SDIVF between the HPE(z) and HPE[ln(p)] cases. The HPE(z) and HPE[ln(p)] cases are compared (Fig. 4b) by interpolating the HPE(z) diagnostics to pressure levels using Eq. (28). The interpolation could also have been done using the zonal mean of the modeled pressure field, p(ϕ, z, t); however, p(ϕ, z, t) is found to be essentially time and latitude independent, apart from a weak poleward downslope of pressure surfaces in the winter hemisphere extratropical stratosphere.³ Using p(ϕ, z, t) is found to make almost no difference to using Z(p) (not shown), so Z(p) is used for simplicity.

The difference between the general and HPE(z) cases is negligible (Fig. 4c), and this is true for all the EP diagnostics (not shown). In other words, the shallow atmosphere approximation and the hydrostatic approximation make essentially no difference to the EP diagnostics. The major differences between the general and HPE[ln(p)] cases are thus due to the vertical coordinate used. Both the general and HPE[ln(p)] formulations are self-consistent and exact in their respective coordinate systems (geometric and log-pressure height). However, recall that taking the zonal mean of quantities will lead to O(α²) differences when transforming between coordinate systems (see the appendix). A difference between the general and HPE[ln(p)] cases is thus expected.

Figure 5 shows the difference in the EP diagnostics between the HPE(z) case (essentially identical to the general case, as noted in the previous paragraph), and the HPE[ln(p)] case. The much larger negative SDIVF in the HPE(z) case (Fig. 4b) is found to be due largely to a decreased ∂F^(Z)/∂Z (Fig. 5b) in the lower stratosphere. This causes a poleward and downward circulation in the extratropics in the difference fields (Figs. 5c,d), leading to a stronger residual circulation in the HPE(z) case.

The work of this section shows that the EP diagnostics appropriate to the general and HPE[ln(p)] formalisms of the primitive equations differ by the same order of magnitude as their absolute values throughout the winter stratosphere. For a given GCM, to compute the residual circulation that satisfies the nonacceleration theorem in that GCM, it is essential to compute EP diagnostics formulated within the same vertical coordinate system as that of the particular equations on which the GCM is based.

Note that this conclusion has implications for the validation of GCM output with reanalysis data, since reanalysis data is often only available on pressure levels. As discussed, the general or HPE(z) formulation of the TEM equations should be used for model level output in the case that a model is based on a geometric height coordinate, and this cannot be directly compared to the HPE[ln(p)] formulation, which should be used for pressure level-based reanalysis data. This is because of the differences that arise from the interpolation of TEM diagnostics from height coordinates to pressure coordinates and from the taking of zonal means in the two different coordinate systems. A solution is to interpolate the model dynamical fields from model to pressure levels at each time step during the model run and using the model pressure field p(λ, ϕ, r). This will give the most accurate dynamical output on pressure surfaces, which can then be meaningfully used in the HPE[ln(p)] formulation of the TEM equations and thus compared directly to the reanalysis data for use in model validation.

4. Effect on the wave-driven circulation

One of the most common ways in which the EP diagnostics are used is to compute the so-called downward control streamfunction (Haynes et al. 1991) and thus obtain a measure of the strength of the wave-driven meridional circulation. We now consider the extension of the downward control principle (Haynes et al. 1991) to the general case and the differences between the streamfunctions as computed in both general and HPE[ln(p)] cases.

a. The downward control equations

1) General, nonhydrostatic version

The approach of Haynes et al. (1991) is extended to the general, nonhydrostatic equations. From Eqs. (7) and (12) we obtain the TEM mass-conservation equation

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In a steady state, or after performing a time average, we can take ρ_,t = 0 and may therefore define a streamfunction Ψ(ϕ, r) for the residual circulation by

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These expressions can be used to substitute for υ* and w* in Eq. (15); again assuming a steady state, so that u_,t = 0, we obtain

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where F denotes the rhs of Eq. (15), and the first partial derivative in the middle expression is taken at constant M. It follows that

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where the integral is taken along a contour ϕ(r), say, of constant M. The residual velocity components can then be found by substituting for Ψ in Eq. (30), for example,

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2) Hydrostatic version in geometric coordinates

Here, the streamfunction Ψ for the residual circulation can be defined by

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Then proceeding as before, we obtain

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where the integral is taken along a contour ϕ(z), say, of constant M, and F denotes the rhs of Eq. (21); M is now given by Eq. (18). Substituting for Ψ in Eq. (33) as before,

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3) Hydrostatic version in log-pressure coordinates

In this case the relevant expressions were given by Haynes et al. (1991). We define the streamfunction Ψ for the residual circulation by

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Our Ψ equals Haynes et al.’s ψ multiplied by a to maintain the same dimensions as in the previous cases. Now,

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where the integral is again taken along a contour ϕ(Z), say, of constant M, F denotes the rhs of Eq. (24), and

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Equation (37) is equivalent to Haynes et al.’s Eq. (2.6).

5. Discussion and conclusions

In this paper it has been demonstrated that in a general circulation model (GCM) based on the general, nonhydrostatic primitive equations and based on a geometric height vertical coordinate, naive use of the approximated form of the transformed Eulerian mean (TEM) equations and Eliassen–Palm (EP) diagnostics given in AHL, denoted by the HPE[ln(p)] case, leads to misleading features in the EP diagnostics in the stratosphere that are of the same magnitude as the absolute values of those diagnostics. These features are largely due to the vertical coordinate system of the model output and TEM formalism used: geometric height in the general case and log-pressure height in the HPE[ln(p)] case. Differences between zonal mean quantities calculated in geometric and in log-pressure coordinates are on the order of wave amplitude squared. In the example shown in this paper, these differences lead to a slightly larger wave-driven circulation in the general case than is seen in the HPE[ln(p)] case. The point is made that to validate such models against reanalysis data, which is often available only on pressure surfaces, the model dynamical fields should be interpolated to pressure levels at each time step during the model run and using the modeled pressure field. The HPE[ln(p)] case could then be used with both model and reanalysis pressure level data. Reanalysis data is sometimes available only at coarse resolution, but little difference is found between the EP flux diagnostics as calculated from 1.25° latitude by 1.875° longitude resolution dynamical data and those calculated from 2.5° latitude by 3.75° longitude resolution dynamical data (not shown).

The downward control principle of Haynes et al. (1991) is found to extend easily to the general case. Differences between the general case and the HPE[ln(p)] case in the mass streamfunction, computed both from the residual vertical velocity w* and from downward control, are found to occur for the same reasons as those stated above for the EP diagnostics. Mass conservation as deduced from the streamfunction calculated from w* is found to hold better in the general case than the HPE[ln(p)] case in the Met Office GCM (the MetUM).

Modeling groups with GCMs based on the general primitive equations will, strictly speaking, only obtain EP diagnostics that obey mass conservation and the nonacceleration theorem if they use the general TEM equations of AM—Eqs. (15) and (16) here. However, the work of this paper demonstrates that using the so-called HPE(z) approximation [Eqs. (20) and (21)] gives virtually the same results as using the general TEM equations.

The implication for the modeling community is that to produce accurate EP diagnostics from geometric height-based model output, existing EP diagnostics written using the HPE equations of AHL [their Eqs. (3.5.2) and (3.5.3)] can be modified in a straightforward way to use the form of the HPEs appropriate for geometric height coordinates [Eqs. (20) and (21)]. Further generalization toward the AM equations gives little extra gain.