The Jablonowski and Williamson (2006) baroclinic wave test case has proven to be highly beneficial for the validation of atmospheric dynamical cores (Lauritzen et al., 2010; Williamson et al., 2009). However, it is restricted to shallow-atmosphere models in spherical geometry, because of the intrinsic assumption of an exact hydrostatically balanced basic state, which then violates the quasi-hydrostatic balance of the deep-atmosphere equations. (White et al., 2005, gives a discussion of quasi-hydrostatic and hydrostatic balance for the deep- and shallow-atmosphere equations).

With the advent of more powerful computers, numerical models based on the deep-atmosphere equations have become operationally feasible (e.g. Davies et al., 2005). It is therefore desirable to develop test cases, similar in spirit to those introduced in Jablonowski and Williamson (2006), but applicable not only for shallow-atmosphere models, but also for deep-atmosphere ones, and not only in spherical geometry, but also in more general geometries, such as β–γ-plane (Dellar, 2011, D11; Staniforth, 2012, S12) and spheroidal (White et al., 2008). An essential preparatory step towards this is the development of sufficiently complex, exact, axisymmetric solutions of the deep- and shallow-atmosphere equations in curvilinear and plane geometries. This is the goal of the present work.

The article is organised as follows: exact solutions in general curvilinear geometry are derived in section 2; an illustrative example of an exact solution in spherical geometry is presented in section 3 for both deep and shallow atmospheres; and conclusions are drawn in section 4.

2. Derivation

2.1. Governing equations

White and Wood (2012), hereafter WW12, presented a particular form of the three components of the Euler momentum equation for a general axisymmetric orthogonal geopotential coordinate system (their equations (A.10)–(A.12)). These equations, in the absence of any forcing, are:

(1)

(2)

(3)

Here ξ_{3} is the coordinate whose corresponding unit vector is aligned with the vertical, defined by the direction of the gradient of the assumed geopotential; the unit vector corresponding to the coordinate ξ_{2} lies in the plane containing both the vertical and the Earth's rotation vector; and ξ_{1} is the remaining orthogonal coordinate forming a right-handed system. In spherical polar coordinates, ξ_{1}, ξ_{2} and ξ_{3} correspond, in standard notation, to λ, ϕ and r, and are the coordinates in the zonal, meridional and upward vertical directions, respectively. The velocity components and metric factors corresponding to the coordinates ξ_{1}, ξ_{2} and ξ_{3} are respectively: u_{1}, u_{2}, u_{3}; and h_{1}, h_{2}, h_{3}. Due to the assumed axisymmetry of the coordinate system, the metric factors are functions only of ξ_{2} and ξ_{3}. The density is denoted by ρ and pressure by p. The Earth's rotation rate is given by Ω and g(ξ_{3}) is the magnitude of the acceleration due to gravity.

Equations (1)–(3) (equivalently equations (A.10)–(A.12) of WW12) are only valid for curvilinear geometry in a general axisymmetric coordinate system. As a result they are not appropriate when h_{1} is a constant (as would be the case for a Cartesian coordinate system). In fact, in that case, the Coriolis terms of (1)–(3) (those terms proportional to Ω) all vanish. Therefore an alternative, more general form of these equations is sought that includes the case of planar geometry.

A notable aspect of equations (1)–(3) is that the Coriolis terms are written in terms of the metric factors. This is achieved by making use of the relations

(4)

(5)

where ϕ is the geographic latitude (the angle between the local vertical and the equatorial plane). From these equations (and using the definition of curl in a general orthogonal system, e.g. (3) of WW12) the Earth's rotation vector, Ω, can be written as

(6)

where

(7)

is a vector potential for the unit vector parallel to the Earth's axis of rotation divided by h_{1}. The use of a vector potential to represent (and suitably approximate) the Coriolis terms is the approach followed by D11, as summarised in S12.

These observations therefore prompt generalising (7) to define R as

(8)

where is not necessarily constrained to equal h_{1}. Consistent with this, (4) and (5) can be generalised by defining

(9)

(10)

Note that with this definition ϕ is no longer necessarily geographic latitude, nor indeed does it necessarily correspond to a physical angle. (This is true also of the forms (4) and (5). Consider the case of a shallow atmosphere for which h_{1} has no dependence on ξ_{3}: from (5), cosϕ then vanishes yet –cf. (4) –the magnitude of sinϕ is not constrained to be unity.)

The required generalisation of (1)–(3) is therefore:

(11)

(12)

(13)

When = h_{1}, these equations are identical to (1)–(3) ((A.10)–(A.12) of WW12). But, by instead setting h_{1} = h_{2} = h_{3} = 1, (ξ_{1},ξ_{2},ξ_{3}) = (x,y,z), and choosing, for example, = −y^{2}/(2a), (11)–(13) deliver the equatorial β-plane equations (the shallow-water version of which is given in Gill (1982), for example).

However, of particular interest to the present study are the pseudo-Cartesian β–γ-plane equations of D11, as these are effectively the ‘deepish’-atmosphere generalisation of the β-plane equations (S12). The β–γ-plane equations are obtained by starting from a non-Cartesian curvilinear geometry, and consistently retaining certain aspects of the curvature as it impacts on the Coriolis terms, whilst approximating all the metric factors by unity, i.e. : hence why D11 refers to them as ‘pseudo-Cartesian’ equations.

Specifically, (11)–(13) reduce to D11's equations when h_{1} = h_{2} = h_{3} = 1 and

(14)

Here (ξ_{1},ξ_{2},ξ_{3}) = (x,y,z) and is related to D11's vector potential R_{x} by = aR_{x}/2 (where a is the mean radius of the Earth). Evaluating (9)–(10) for this equation set, gives

(15)

(16)

where ϕ_{0} is the reference latitude of the β–γ plane. Note that the following analysis, and resulting exact solutions, apply to any approximation of (regardless of whether it has a sound physical basis): as noted above, = −y^{2}/(2a) delivers the equatorial β-plane equations, and other examples are given in section 6 of D11, and in section 2.4 of S12.

It is important to note that (11)–(13) are only generally valid (in the sense of delivering consistent equation sets) for two cases:

1.that presented in WW12, for which h_{1}, h_{2} and h_{3} are non-trivial, and = h_{1};

2.the one corresponding to the equations presented in D11 and S12 for which, after approximation, h_{1} = h_{2} = h_{3} = 1 and

(17)

where (17) is S12's redefinition of (14) through the introduction of binary switches δ_{y}, δ_{z}, δ_{yz}, and .

2.2. Axisymmetric steady flow

Axially symmetric, steady solutions are sought that satisfy: u_{2} = u_{3} = 0; all prognostic variables are functions only of ξ_{2} and ξ_{3}; and D/Dt = 0. Upon use of the perfect gas law, p = RTρ (where T is temperature and R is the specific gas constant), (12)–(13) reduce (after multiplying the u_{2} equation by a factor of −h_{2}, and the u_{3} equation by −h_{3} ) to:

(18)

(19)

where q ≡ lnp. Now, following White and Staniforth (2008) and Staniforth and White (2011) (hereafter WS08 and SW11, respectively), define

(20)

where the switch δ_{M} determines whether the metric term is retained, i.e. δ_{M} = 1, (as required when = h_{1}), or it is absent, i.e. δ_{M} = 0, (as in the pseudo-Cartesian case). The dimensions of U are those of acceleration. Noting the constraint that when δ_{M} = 1, , (18)–(19) can then be written in the form

(21)

(22)

When δ_{M} = 1, (21) expresses geostrophic and cyclostrophic balance in the horizontal, whereas (22) expresses quasi-hydrostatic balance in the vertical.

2.3. Exact solutions

Equations (21)–(22) are a pair of nonlinear equations that constrain the dependent variables U, T and q. However, as noted in S12 in the context of β–γ-plane geometry, they can also be viewed as being a pair of linear equations for the three dependent variables U/T, 1/T and q, where U/T can be considered to be a proxy variable for u. Therefore specifying any one of these three dependent variables in an arbitrary manner, the remaining two may be obtained as a direct consequence. However, because q appears everywhere as a differentiated quantity, it is natural to make it the specified dependent variable from which all other dependent variables are derived; this avoids having to integrate to obtain q, which would otherwise be required.

Thus exact solutions of (21)–(22) can be found by specifying q as any differentiable function of ξ_{2} and ξ_{3}, i.e. q = q(ξ_{2},ξ_{3}). Having done so, (21) determines U/(RT) which can then be used in (22) to give 1/T, and thence U. The result is

(23)

(24)

Finally u_{1} is obtained by solving (20) as a quadratic for given U, with the appropriate choice of root, as discussed below. This assumes that the common denominator in (23) and (24) (i.e. the Jacobian of and q) does not vanish. (It has also been assumed in this derivation that does not vanish, as would happen if this method were applied to the f-plane equations. In this case it follows that ∂q/∂ξ_{2} must also vanish and the only constraint on U and T is that they are linearly related via (24) to ensure that hydrostatic balance is satisfied.)

One can simply stop here. However, simply specifying q in this way does not give any insight into how any particular solution for one configuration (spherical deep-atmosphere, say) should be modified to obtain the appropriate corresponding solution for an alternative configuration (spheroidal deep, or spherical shallow, etc.). For example, WS08 derive both deep- and shallow-atmosphere exact solutions for appropriate specific forms of (23) and (24). However, naïvely setting all occurrences of r in the deep solution (i.e. Eq. (33) of WS08 for the temperature field) to the constant value a does not produce the corresponding shallow solution (WS08's (71)). Doing so of course removes all vertical variation in the solution; in fact it is only those factors of r that are associated with cosϕ (and also the specification of g; WW12 gives a discussion of appropriate forms for g in the various cases) that need to be replaced by a to obtain the shallow solution.

Motivated therefore by the solutions of WS08, and also those of SW11, and by the form of (21)–(22), consider an alternative choice of variables for the specification of q, with replacing ξ_{2} so that . Then (21) reduces to

Then, given U, u_{1} is finally obtained by solving (20) as a quadratic in u_{1}. The requirement for non-singular solutions to (21) and (22), i.e. that the Jacobian of and q does not vanish, reduces for (27) and (28) to requiring that does not vanish, which is far simpler to achieve.

When the metric term is retained, i.e. δ_{M} = 1 and , (20) is a quadratic for u_{1} and (as argued in WS08 and SW11) the appropriate root is the positive one, giving

(29)

The key aspect of these exact solutions is that once a specific functional form for has been chosen, that functional form can be straightforwardly applied to all geometries for which (11)–(13) apply. For example, deep and shallow spherical/spheroidal geometries, cylindrical geometry, and all flavours of the (pseudo-) Cartesian geometries, are all encompassed by this framework. To specify the solutions in these various geometries, all that is needed (in addition to the physical parameters Ω and R) are: the coordinates ξ_{2} and ξ_{3} (ξ_{1} does not explicitly play a rôle in these solutions); ; and the product g(ξ_{3})h_{3}. The appendix gives the definition of these quantities for a variety of geometries.

These solutions pay no specific heed to any boundary conditions. It is implicit in the nature of the (axisymmetric) solutions sought that the domain is periodic in the ξ_{1} direction. For spherical, spheroidal and cylindrical domains, the metric factors ensure the appropriate behaviour of the solutions (for example, in spherical polar coordinates, h_{1} –and hence also u_{1} from (29) –vanishes at the two poles of the coordinate system). However, if further restrictions need to be imposed on u_{1}, then specific restrictions on the form of would need to be made on a case-by-case basis. For example, if it is required that u_{1} = 0 at the boundaries of a channel then, from (20), U must vanish there. Then, from (28), this means that (assuming h_{3} is not zero at the boundaries) q must be chosen so that vanishes there.

2.4. Generalised thermal wind equation

Herein, and also in S12, the starting point for deriving solutions to (18)–(19) is to specify the solution for q ≡ lnp, with T and U (and thence u_{1}) then being derived from it. However, if instead the solution for T is chosen as the starting point (as in WS08 and SW11), then the equation for U in terms of T is obtained by eliminating q between (21) and (22). This results in what WS08 referred to as a compatibility constraint. They discussed its relation to the thermal wind equation for axisymmetric flows, and also termed the constraint equation a ‘generalised thermal wind equation for zonal flows’.

Deriving exact solutions of the form sought by specifying q has the advantage that, provided it is appropriately differentiable, the compatibility constraint, or the generalised thermal wind equation, is automatically satisfied. However, it is straightforward to derive a generalisation of this constraint. This is achieved by eliminating q between (21) and (22) to give

(30)

i.e.

(31)

This relation is true for all steady axisymmetric flows governed by (11)–(13). For the specific exact solutions given by (25) and (26), for which , (31) takes the even simpler form

(32)

This is seen to have a very similar form to that of the generalised thermal wind equation (24) of WS08 for spherical geometry. (Indeed, one might be tempted to term (32) the generalised, generalised thermal wind equation!)

3. Illustrative example

As an illustrative example, consider

(33)

This choice is inspired by the functional form used in S12 for exact solutions in z coordinates on a β–γ plane, which was itself inspired by that of Ullrich and Jablonowski (2012) for their channel-flow test problems in pressure-based coordinates on a β plane.

3.1. Deep spherical atmosphere

Taking (33) as the generic solution, a specific deep-atmosphere solution is constructed in spherical geometry. In this case

(34)

(35)

(36)

In principle, g(r) in (36) is an arbitrary function. However, for physical reasons related to the spherical geopotential approximation (White et al., 2005; WW12), its appropriate functional form is g(r) = g_{0} (a/r)^{2}, where g_{0} is the value of g at r = a, i.e. at the surface. Then (33) becomes

(37)

where q_{0} is an arbitrary constant value of the natural logarithm of pressure, T_{0} is a representative constant value of temperature, Γ is an assumed lapse rate, H_{0} ≡ RT_{0}/g_{0} is the scale height of the atmosphere, b is a half-width parameter, k is a positive integer, and A, B and C are arbitrary constants, subject only to physical realisability. The remaining fields are direct consequences of (34)–(37), and are obtained as follows.

The arbitrary parameter A can be expressed in terms of a more meaningful physical quantity by considering the special case when B = C = 0. Evaluating (40) at the surface with B = C = 0 gives T(r = a) = T_{0}/(AΓ) there. It is therefore natural to set

(33)

and T_{0} is then to be interpreted as being a representative surface value of temperature. The physical interpretation of Γ now becomes apparent by setting A = 1/Γ and B = C = 0 in (40), differentiating the resulting equation with respect to r, and evaluating it at r = a, to give (dT/dr)|_{r=a} = −Γ. Thus Γ can be interpreted as being a representative value of the lapse rate of temperature at the Earth's surface.

Evaluating (40) at the surface, and then at the Equator and at the poles, i.e. at r = a, and at ϕ = 0,±π/2, allows the two arbitrary parameters B and C to be expressed in terms of the two more-meaningful quantities and . Thus

(44)

Although T_{0} can be set to any representative value, for simplicity set it to

(45)

Using this value in (44) then gives B and C in terms of and as

(46)

This illustrative example happens to also be a special case of the SW11 solution where the temperature is assumed to have the functional form

(47)

Setting

(48)

(49)

and setting SW11's arbitrary integration function F(rcosϕ) to zero, then leads to the solution constructed above.

To visualise the solution constructed above, set the parameter values to

(50)

The corresponding temperature (T), zonal wind (u_{1}), potential temperature (θ) and Brunt–Väisälä frequency (N) fields are displayed, as functions of z ≡ r − a, in Figure 1, and some vertical profiles of the temperature and zonal wind fields in Figure 2.

3.2. Shallow spherical atmosphere

Again taking (33) as the generic solution, but using

(51)

(52)

(53)

instead of (34)–(36), an analogous shallow-atmosphere solution can be constructed following the above procedure. Because r/a varies so little over the combined depth of the troposphere and stratosphere, the corresponding fields and profiles for a shallow atmosphere are graphically almost indistinguishable from those in Figures 1–2, and are therefore not shown. This is consistent with WS08 in which it was demonstrated analytically that two of their particular, deep-atmosphere, exact solutions reduced to the equivalent ones for a shallow atmosphere to leading order in z/a.

4. Conclusion

A wide family of exact closed-form axisymmetric solutions to the deep- and shallow-atmosphere Euler equations has been derived. These solutions are valid in general curvilinear coordinates and subsume those obtained in WS08 and SW11 for spherical geometries, and S12 for β-plane and β–γ-plane geometries. SW11 noted that their exact solutions hold in the presence of any orography of the form r_{H} = r_{H} (ϕ), where the arbitrary function r_{H} (ϕ) is distance of the orographic surface from Earth's centre, and any imposed lid located at r = r_{T} (ϕ). The same is also true for the exact solutions derived herein, but with r_{H} = r_{H} (ξ_{2}) and r = r_{T} (ξ_{2}).

The enhanced generality of the present solutions provides more flexibility in the specification of challenging initial conditions for numerical model validation than has hitherto been possible. In particular, exactly balanced states for deep- and shallow-atmosphere global models can be defined that not only have similar functional forms, but are also quantitatively very close to one another. (This is because the difference between their functional forms involves the quantity r/a to some power, which hardly varies over the combined depth of the troposphere and stratosphere.)

This then opens up the way to defining a unified baroclinic wave test problem, of the more elaborate and physically realistic (albeit not analytical) type introduced in Jablonowski and Williamson (2006) for shallow-atmosphere models, that is applicable to both deep- and shallow-atmosphere models in spherical geometry. Furthermore, it also facilitates the development of analogous test problems in β-plane, β–γ-plane, cylindrical and spheroidal geometries.

Acknowledgements

The authors would like to thank Prof. Andy White for the derivation of the oblate spheroidal metrics of the Appendix, as well as Dr Markus Gross for help with the Maple software package. Additionally Prof. Andy White and an anonymous referee made insightful comments that improved our understanding and the manuscript.

Appendix. Metric factors

For a given coordinate system, the parameters that are needed to specify the equations and solutions of section 2 are: the coordinates (ξ_{1},ξ_{2},ξ_{3}); the metric factors (h_{1},h_{2},h_{3}); ; and g(ξ_{3}). The velocity components (u_{1},u_{2},u_{3}) are given by u_{i} = h_{i}Dξ_{i}/Dt for i = 1, 2, 3.

These parameters are now given for a selection of coordinate systems.

A1. Deep-atmosphere spherical

(A1)

(A2)

(A3)

A2. Shallow-atmosphere spherical

(A4)

(A5)

(A6)

6.3. Pseudo-Cartesian β–γ plane and Cartesian f–F-plane

(A7)

(A8)

(A9)

The standard Cartesian f–F-plane equations (Thuburn et al., 2002) are, for example, obtained by setting δ_{y} = δ_{z} ≡ 1 and .

A4. Cylindrical

(A10)

(A11)

(A12)

Here the direction associated with z is aligned with the axis of the cylinder (also the axis of rotation) and the choice of (ξ_{1},ξ_{2},ξ_{3}) is motivated by considering a cylinder whose axis is aligned with the Earth's rotation axis and which is tangential to the Earth at the Equator.

A5. Deep-atmosphere similar oblate spheroidal

(A13)

(White et al. (2008) and Staniforth et al. (2010) provide definitions of the meridional, φ, and radial, R, coordinates, where here R replaces their R to avoid confusion with the gas constant)

(A14)

(A15)

(A16)

(A17)

(A18)

Here g_{a} is the surface, equatorial value of g, i.e. the value of gravity at R = a, ϕ = 0, and

(A19)

where ϕ is the geographic latitude (ψ in the notation of Staniforth et al., 2010). This is the angle at which the normal to the oblate spheroid is inclined to the equatorial plane and is distinct from the latitudinal coordinate φ. In particular, it depends (weakly) on the vertical coordinate R and is given by the implicit relation (A19) with the sign of ϕ chosen to be the same as the sign of φ. The eccentricity is where L_{major} and L_{minor} are respectively the semi-major and semi-minor axes of the family of similar ellipses.

The expressions for the metric factors are obtained by rewriting those given in Appendix A of Staniforth et al. (2010) in terms of ϕ, instead of the eccentric angle θ, and using e instead of .