Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations

Akio Arakawa Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Celal S. Konor Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an f plane without the quasigeostrophic approximation and on a midlatitude β plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

Abstract

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an f plane without the quasigeostrophic approximation and on a midlatitude β plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

1. Introduction

The nonhydrostatic anelastic system of equations (called the anelastic system in this paper) is widely used in theoretical and numerical studies of small-scale nonacoustic motions, such as turbulence and convection, while most large-scale models use the compressible quasi-hydrostatic system (the “primitive equations,” called the quasi-hydrostatic system in this paper) as the dynamics core. Both of these systems filter vertically propagating sound waves, but they do so in quite different ways.

In the anelastic system (e.g., Ogura and Phillips 1962; Dutton and Fichtl 1969; Wilhemson and Ogura 1972; Lipps and Hemler 1982; Bannon 1996), the deviations of thermodynamic variables from a horizontally uniform reference state are assumed to be small, and the local time derivative of density is neglected in the continuity equation to filter the acoustic waves. Thus, the original continuity equation
i1520-0493-137-2-710-e11
is replaced by
i1520-0493-137-2-710-e12
where ρ is the density, is the three-dimensional del operator, V is the three-dimensional velocity, and the subscript zero denotes a reference state that varies only vertically. To maintain the internal consistency of the system from the point of view of scale analysis and/or energetics, either the momentum or thermodynamic equation is usually modified. For example, Ogura and Phillips (1962) chose an isentropic atmosphere as the reference state, while Lipps and Hemler (1982) assumed that the reference-state potential temperature is a slowly varying function of the vertical coordinate. The pressure gradient force in the momentum equation is then approximated to maintain the consistency. Bannon (1996), on the other hand, maintains the consistency by modifying the thermodynamic equation introducing the concept of the “dynamic entropy.” By “the anelastic system,” we mean the Lipps–Hemler system in the rest of this paper unless otherwise noted.
We note, however, that the use of the anelastic continuity equation in (1.2) is more than needed to filter acoustic waves. Using the equation of state and the definition of potential temperature θ, we find
i1520-0493-137-2-710-e13
where p is the pressure, γcp/cυ, and cp and cυ are the specific heat at constant pressure and volume, respectively. The first and second terms on the right-hand side of (1.3) represent the local effects of isentropic compressibility and isobaric entropy change, respectively. The anelastic continuity equation neglects both of these effects. Durran (1989) showed that the inclusion of a linearized effect of the second term yields
i1520-0493-137-2-710-e14
where the overbar denotes the horizontal mean state, which may have an arbitrary vertical structure, π is the Exner function given by (p/p00)κ, p00 is a constant reference pressure, κR/cp = 1 − 1/γ, R is the gas constant, and Q is the heating rate per unit mass. Acoustic waves are still filtered because the first term on the right-hand side of (1.3) is neglected. Durran (1989) called (1.4) “the pseudo-incompressible equation.” Nance and Durran (1994) showed that (1.4) becomes increasingly accurate as the flow becomes more nonhydrostatic. Durran (2008) further showed that the pseudo-incompressible equation is accurate if the Mach number is smaller than the Rossby number or 1, whichever is smaller. Durran and Arakawa (2007) showed that when (1.4) is used, energy is conserved with no modifications of the momentum and thermodynamic equations except for linearization. Durran (2008) presented further discussions of the pseudo-incompressible system and its generalizations.

The merit of using the anelastic or pseudo-incompressible approximation for small-scale motions is well recognized. Although our experience is rather limited, these approximations seem to hold well for most large-scale atmospheric motions as well. Nance and Durran (1994) pointed out that the errors incurred by using both the anelastic and pseudo-incompressible systems could be significantly less than the errors generated by the numerical methods. By analyzing the results of an anelastic model applied to the global domain, Smolarkiewicz et al. (2001) further pointed out, “the differences due to the higher-order truncation errors of legitimate modes of executing contemporary global models overwhelm the differences due to analytic formulation of the governing equations.” Through this analysis, they conclude that nonhydrostatic anelastic models derived from small-scale codes adequately capture a broad range of planetary flows. Smolarkiewicz and Dörnbrack (2008) presented integrations of the anelastic and pseudo-incompressible systems applied to baroclinic development in the midlatitudes.

There are, however, conflicting views. Based on normal-mode analyses of fully compressible, pseudo-incompressible, anelastic and quasi-hydrostatic systems of equations applied to an f plane, Davies et al. (2003) concluded, “whilst of key importance for small-scale and process modeling, the anelastic equations are not recommended for either operational numerical prediction or climate simulation at any scale.” They also pointed out that the pseudo-incompressible system appears to be viable for numerical weather prediction, but only at short horizontal scales.

A potentially more serious problem appears with the β effect when an anelastic model is applied to a hemispheric or global domain. With the anelastic continuity equation in (1.2), horizontal motions must inevitably be horizontally nondivergent. The situation is the same with the pseudo-incompressible equation in (1.4) without heating because the mean state is horizontally uniform. Then, as far as the barotropic mode with a fixed upper boundary is concerned, we are essentially back to the problem recognized during the early years of NWP. Wolff (1958) showed that forecast errors with a hemispheric nondivergent barotropic model are dominated by spuriously fast westward retrogression of ultra-long waves. This is anticipated from the retrogression speed of the nondivergent Rossby wave given by β/k2 (Rossby et al. 1939), which unlimitedly increases as k decreases. Here β is the meridional gradient of the Coriolis parameter f and k is the zonal wavenumber. Rossby et al. (1939) pointed out, however, that conservation of the absolute potential vorticity, ( f + ζ)/h, instead of conservation of the absolute vorticity, f + ζ, gives slower retrogression speeds. Here ζ is the vertical component of vorticity and h is the height of the interface between the lower dynamically active homogeneous layer and the upper dynamically inactive homogeneous layer. Based on this and the work by Bolin (1956), who used the height of tropopause for h, Cressman (1958) succeeded to reduce the errors in actual forecasts by introducing a correction term in the vorticity equation to represent “barotropic divergence.” Wiin-Nielsen (1959) pointed out that the problem exists also for the barotropic mode in tropospheric baroclinic models. He noted that Cressman’s choice of h is rather ambiguous and interpreted the required divergence term as a result of vertically varying static stability. It is difficult to see, however, how the vertical variation of static stability influences the barotropic (or external) Rossby wave. Wedi and Smolarkiewicz (2004, 2006), on the other hand, introduced divergence of the vertically integrated motion into their anelastic model by making the model top variable in space and time. In the present paper we point out that even purely horizontal motion can be divergent with compressibility so that its potential vorticity is given by ( f + ζ)/ρ, where the denominator represents the effect of compressibility on the change of f + ζ.

Most large-scale models of the atmosphere use the quasi-hydrostatic system of equations (the primitive equations) instead of the anelastic system. In the quasi-hydrostatic system, the vertical component of the momentum equation is replaced by the hydrostatic equation. This filters vertically propagating sound waves, but it is done in a totally different way from the anelastic system. The quasi-hydrostatic system uses no approximation in the continuity equation and, therefore, compressibility is fully included as far as quasi-hydrostatic motions are concerned. To our knowledge, catastrophic errors for ultra-long waves such as those observed by Wolff (1958) with the nondivergent barotropic model have not been reported with the primitive equation models. We can think of various reasons for this. For example, the purely barotropic mode may not be a significant component in such models when they are applied to realistic situations. It is also possible that the improved treatment of planetary-scale topography commonly used in those models might have hidden the problem. In our point of view, however, the effect of compressibility on those waves included in the primitive equation models is at least one of the possible causes for the success of those models in predicting ultra-long waves.

It is well known that the quasi-hydrostatic approximation breaks down for motions with horizontal scales of the order of 10 km or less. There have been attempts to overcome this deficiency by including approximate nonhydrostatic effects without introducing vertically propagating sound waves. One of the earliest attempts along this line is the approach proposed by Miller (1974) (see also Miller and Pearce 1974; Miller and White 1984; White 1989), which uses the pressure as the vertical coordinate and the approximation Dw/DtD(−ω/ρsg)/Dt in the vertical component of the momentum equation. Here D/Dt is the material time derivative, w is the vertical velocity, ωDp/Dt, ρs is the density of the reference state, and g is the gravitational acceleration. They used the standard form of the quasi-hydrostatic continuity equation with the p coordinate without introducing the anelastic approximation. This is also an approximation since p in their system is not necessarily hydrostatic. Miller (1974) showed that, when viewed with the z coordinate, this approximation is equivalent to the use of the hydrostatic equation for the time derivative of density in the continuity equation in (1.1).

In the approach proposed by Laprise (1992), on the other hand, the hydrostatic pressure is used as the vertical coordinate. Despite the use of the hydrostatic pressure for the vertical coordinate, no approximation is used in the momentum and continuity equations and, therefore, the system is nonhydrostatic and fully compressible. In his “alternative approach,” w is calculated using wDz/Dt. Thus, the vertical component of the momentum equation is not used as a prognostic equation for w. Instead, it is used as a diagnostic equation that determines the vertical gradient of the total pressure from known Dw/Dt. Bubnová et al. (1995) emphasized the merit of Laprise’s approach saying “…all the big investments that have been put into developing complex environments for primitive equation models can be used with profit to do nonhydrostatic research experiments and, in some future, operational forecasts.” Janjic et al. (2001) and Janjic (2003) extended Laprise’s alternative approach to the case of a sigma coordinate based on the hydrostatic pressure. They also emphasized the advantage of Laprise’s approach because the nonhydrostatic dynamics is introduced as an add-on module without interfering with the favorable features of the hydrostatic formulation.

The main thrust of this paper is to develop a system of dynamics equations that maintains close ties with both the primitive equation models for large scales and the anelastic (and Boussinesq) models for small scales, for each of which we have generations of valuable experience, while filtering vertically propagating sound waves of all scales. An obvious alternative to this approach is to use a fully compressible model with the split-explicit approach (Klemp and Wilhemson 1978; Skamarock and Klemp 1992, 1994; Klemp et al. 2007) or a semi-implicit scheme (e.g., Tanguay et al. 1990; Cullen et al. 1997; Côté et al. 1998). For a concise review of these methods, see Steppeler et al. (2003). In the approach presented in this paper, on the other hand, the vertically propagating sound waves are eliminated at their origin so that our effort in improving computational aspects can be more focused on motions of our interest.

The essence of the unified system presented in this paper is in the use of the continuity equation:
i1520-0493-137-2-710-e15
where ρqs is the quasi-hydrostatic density. This equation is a straightforward generalization of the anelastic continuity equation in (1.2) and the pseudo-incompressible equation in (1.4) and, when applied to quasi-hydrostatic motions, it includes both terms in the right-hand side of (1.3). The use of (1.5) obviously requires
i1520-0493-137-2-710-e16
where δρρρqs is the nonhydrostatic density. This assumption should be better justifiable than the corresponding assumption commonly used in the anelastic systems because δρ is the deviation of ρ from the local quasi-hydrostatic value rather than the value of a prescribed reference state that varies only vertically. This point is especially important when the system is applied to a large horizontal domain such as the entire globe. The assumption in (1.6) alone, however, does not automatically justify the use of (1.5) because the time derivative of δρ cannot be neglected for vertically propagating sound waves because of their high frequencies. The unified system filters these waves through omitting the ∂δρ/∂t term in (1.5). Since the ∂ρqs/∂t term is retained, this equation may still appear to be prognostic, but actually it is not, because ρqs is predicted not by this equation, but by the thermodynamic (and surface pressure tendency) equations.

The paper is organized as follows. Section 2 presents the definitions of the quasi-hydrostatic pressure and density and the equations for their predictions, including the condition on the time change of the quasi-hydrostatic pressure at the model top, and section 3 presents the dynamics of the unified system including the problem of determining nonhydrostatic pressure. Section 3 also presents an equation that governs the time change of total energy. Section 4 gives a computational procedure of the unified system that can be followed when the height coordinate is used. Section 5 presents the unified system when the quasi-hydrostatic pressure is used as the vertical coordinate. For the purpose of comparing the unified system with other systems, section 6 discusses small-amplitude perturbations on a resting horizontally uniform atmosphere in view of the dispersion relation and vertical structure of the normal modes on an f plane. The analysis is then extended to the midlatitude β plane with the quasigeostrophic approximation. Section 7 presents a summary and further discussions. The form of energy conserved in this system is presented in appendix A. A version of the unified system based on the vector vorticity equation instead of the momentum equation is presented in appendix B.

2. Quasi-hydrostatic pressure

In this section we define quasi-hydrostatic values of pressure and density and then discuss how those values are predicted in the unified system. Throughout this paper, the virtual temperature effect is neglected for simplicity. Using the equation of state p = ρRT and the definition of potential temperature θT/π, we may write the momentum equation as
i1520-0493-137-2-710-e21
Here Ω is the earth’s angular velocity and k is the vertical unit vector. Replacing the vertical component of (2.1) by the hydrostatic equation, we define the vertical derivative of πqs for a given vertical structure of θ by
i1520-0493-137-2-710-e22
Integrating (2.2) with respect to z, we obtain
i1520-0493-137-2-710-e23
where the subscript S denotes the earth’s surface. Replacing π in p = p00π1/κ by πqs, we define pqs by
i1520-0493-137-2-710-e24
and ρqs by
i1520-0493-137-2-710-e25
These quasi-hydrostatic values do not necessarily represent a reference state because the vertical distribution of θ in (2.3) is arbitrary and, therefore, pqs and ρqs do not necessarily have characteristic vertical structures. From the definitions of the quasi-hydrostatic state given by (2.2), (2.4), and (2.5), we find
i1520-0493-137-2-710-e26
This equation and the assumption in (1.6) implies that we are also assuming
i1520-0493-137-2-710-e27
though it is not formally used in the equations given in the text.
Equation (2.3) shows that (πqs)S and θ must be predicted to determine the time evolution of the quasi-hydrostatic state. To predict θ, we use the following thermodynamic equation:
i1520-0493-137-2-710-e28
To predict (πqs)S, we apply the time derivative of (2.3) to z = zT to obtain
i1520-0493-137-2-710-e29
where the subscript T denotes the model top. Here zT (as well as zS) is assumed to be constant in time. The first term on the right-hand side, however, remains to be determined because ∂(πqs)T/∂t = 0 is not a correct condition at the model top even when zT → ∞. To see this, let us consider a small perturbation denoted by a prime on a horizontally uniform basic state denoted by an overbar. The perturbation part of (2.2) is given by
i1520-0493-137-2-710-e210
If θ′ = 0 (i.e., barotropic) at all heights, (2.10) shows that πqs is constant throughout the entire vertical column. The assumption of (πqs)T = 0 thus means πqs = 0 at all heights. Barotropic modes are then eliminated. Since πqsκpqs(qs/pqs) = (κ/p00κ)(pqs/pqs1 −κ), constant πqs means that pqs decreases in height as pqs1 −κ( = pqs1/γ) does. This is consistent with the fact that free quasi-hydrostatic oscillations in an isothermal atmosphere, such as the Lamb wave (modified by rotation) and the barotropic Rossby wave (modified by compressibility), have the equivalent depth γH, where H is the scale height (e.g., Siebert 1961).
We see that (2.9) can be closed if we consider the mass budget of the entire vertical column. For this purpose, we rewrite the continuity equation in (1.5) as
i1520-0493-137-2-710-e211
Hereinafter the subscript H denotes the horizontal component and w is the vertical velocity. Integrating the time derivative of (2.5) with respect to z from zS to zT, using (2.11) and wS = VH · zS, and assuming zT = const. in space as well as in time, we obtain the surface-pressure tendency equation given by
i1520-0493-137-2-710-e212
Though not fully justifiable, we have neglected (ρqsw)T in deriving (2.12) as is done in many models.
Both (2.12) and (2.9) are obtained through vertically integrating the time derivative of the hydrostatic equation. Their physical meanings are different, however, because (2.12) relates the integral to mass budget while (2.9) relates it to thermodynamics. Naturally they must be consistent in view of (2.4). The time derivative of (2.4) applied to the earth’s surface and the model top are given by
i1520-0493-137-2-710-e213
and
i1520-0493-137-2-710-e214
respectively. Substituting (2.13) and (2.14) into (2.12) and eliminating ∂(πqs)S/∂t using (2.9), we obtain
i1520-0493-137-2-710-e215
Since the two terms in the brackets do not necessarily cancel, ∂(πqs)T/∂t is generally finite even when (pqs)T = 0. Since pqs/πqs = p00κpqs1 −κ → 0 as pqs → 0, however, (2.14) shows that ∂(pqs)T/∂t = 0 holds when (pqs)T = 0, as expected, but not when (pqs)T ≠ 0. With (2.15), (2.9) is closed and may be rewritten as
i1520-0493-137-2-710-e216

3. Dynamics of the unified system and determination of the nonhydrostatic pressure

In this section we discuss the dynamics equations of the unified system, in which the continuity equation takes the form of (1.5). Since the unified system is a generalization of the anelastic system, the procedure is parallel to that of the anelastic system except that the continuity equation is exact for quasi-hydrostatic motion. When the momentum equation is used as the basic dynamical equation (instead of the vector vorticity equation as discussed in appendix B), the predicted three-dimensional velocity must satisfy the continuity equation. Thus, in parallel to the anelastic system, an elliptic equation must be solved for the nonhydrostatic pressure.

Using D/Dt = ∂/∂t + V · and (2.2), we rewrite the momentum equation in (2.1) as
i1520-0493-137-2-710-e31
where δπππqs. Combining (3.1) with the continuity equation in (1.5), we obtain
i1520-0493-137-2-710-e32
where · (ρqsVV) is the divergence of the dyadic tensor ρqsVV. When υi is the ith component of V in the Cartesian coordinates (x1, x2, x3), the ith component of · (ρqsVV) can be written as
i1520-0493-137-2-710-e33
Taking the divergence of (3.2) and using the continuity equation (1.5) again, we obtain
i1520-0493-137-2-710-e34
This is an elliptic equation to determine δπ. In the anelastic models, a similar elliptic equation is solved, but usually for the deviation of pressure from a horizontally uniform hydrostatic state. In the unified system, on the other hand, (3.4) governs the deviation of pressure from the local quasi-hydrostatic pressure. Another important difference from the anelastic system is the existence of the last term in (3.4), which originates from the time derivative term in the continuity equation in (1.5). It thus represents a correction to the anelastic system. An expression for this term in the time-discrete case is presented in section 4.

Equation (3.4) requires boundary conditions. For vertical boundary conditions, it is a common practice in the anelastic models to use the vertical derivative of pressure obtained from the vertical component of the momentum equation applied to the upper and lower boundaries. We can do the same in the unified system, but in this way the spatially constant part of δπ cannot be determined. While this constant part does not matter for dynamics (Ogura and Charney 1962), it does matter for cloud microphysics as Schlesinger (1975) pointed out. Bannon et al. (2006) showed that this ambiguity could be removed by requiring total mass conservation. On the other hand, P. Smolarkiewicz (2008, personal communication) suggests using this freedom to conserve energy. Unless we enforce such kind of constraint on the constant part, the time sequence of δπ diagnostically determined at individual time steps may not be physical.

Recall that only the spatially varying part of δπ matters for dynamics and, therefore, only that part needs to be constrained for filtering vertically propagating sound waves. Then, if we are concerned with the most general filtered system, the spatially constant part of δπ should be predicted as is done (or effectively done) in a fully compressible nonhydrostatic model. Here we show that, by predicting the spatially constant part of δπ, the system can conserve a properly defined energy. Appendix A derives the following equation from the equations of the unified system:
i1520-0493-137-2-710-e35
In (3.5), Eqs is the quasi-hydrostatic energy per unit mass given by
i1520-0493-137-2-710-e36
where
i1520-0493-137-2-710-e37
and δT is defined by
i1520-0493-137-2-710-e38
The cpδT term in (3.5) represents the enthalpy change per unit mass due to the change of δp through an adiabatic process. Equation (3.5) shows that
i1520-0493-137-2-710-e39
if
i1520-0493-137-2-710-e310
where the double overbar denotes the volume mean over the entire domain. Then the mass-weighted mean of the energy Eqs + cpδT is conserved. Let (δπ)* represent the solution of (3.4) with = 0. We see that δπ given by
i1520-0493-137-2-710-e311
satisfies both (3.4) and (3.10) if is determined by
i1520-0493-137-2-710-e312
Equation (3.12) means that is prognostically determined. For an expression of (3.12) for a time-discrete case, see the next section.
Once δπ is determined, we can predict the horizontal velocity VH using the horizontal component of (3.1) or (3.2). We could also predict w using the vertical component of (3.1) or (3.2). However, since the vertical component has already been used in deriving (3.4), w can be more simply determined from (2.11) with known ∂ρqs/∂t. In this way, it is guaranteed that the continuity equation is exactly satisfied in a time-discrete case. We can show that this procedure is closely related to the determination of w using the Richardson equation (Richardson 1922, p. 118) for the quasi-hydrostatic system. To show this, we first rewrite (2.11) with (2.6) as
i1520-0493-137-2-710-e313
Further manipulating (3.13) using (2.5), (2.11), (ρqsw)T = 0, and (2.5), we finally obtain
i1520-0493-137-2-710-e314
If the first term in the brackets is neglected, (3.14) essentially becomes the Richardson equation, which is still complicated. The complication is partly due to the use of the pressure tendency equation at all levels, as pointed out by Ooyama (1990), while only its application to the earth’s surface given by (2.12) is needed. A more fundamental problem is that the Richardson equation is not physically illuminating. Recall that the anelastic approximation neglects the term on the left-hand side of (2.11). It is then appropriate to regard that term as a generally small correction to the anelastic approximation. Equation (3.14) splits this correction term into the sum of larger terms that exactly compensate each other when the motion is anelastic.

4. Computational procedure with the height coordinate

This section discusses a procedure that can be followed in practical applications of the unified system with the z coordinate. The prognostic variables of the unified system are (πqs)S, θ, and (ρqsVH). The major diagnostic variables are πqs, pqs, ρqs, δπ, and (ρqsw) determined by (2.3), (2.4), (2.6), (3.4), with (3.12) and (2.11), respectively. Let the integer n denote a time level. Suppose that we know all variables except for δπ at time level n (and at past time levels if necessary) and we have a time-difference scheme for advancing (πqs)S(n), θ(n), and (ρqsVH)(n) to (πqs)S(n+1), θ(n+1), and (ρqsVH)(n+1) based on (2.16), (2.8), and the horizontal component of (3.2), respectively. We write these schemes symbolically as
i1520-0493-137-2-710-e41
i1520-0493-137-2-710-e42
and
i1520-0493-137-2-710-e43
where Δt is the time step and the term that depends on (δπ)n is explicitly written. For convenience, the value at the time level n is used for (ρqscpθ). To derive a time-discrete version of (3.4), we also need to specify a time difference scheme for the vertical component of (3.2), which may be symbolically written as
i1520-0493-137-2-710-e44
From (πqs)S(n+1) and θ(n+1) predicted by (4.1) and (4.2), respectively, ρqs(n+1) can be determined by (2.3), (2.4), and (2.6). To obtain a discrete version of (2.11), let us formally use a backward time-difference scheme to obtain
i1520-0493-137-2-710-e45
This time discretization has only the first-order accuracy. This is probably acceptable because the last term is supposed to represent a relatively small correction to the anelastic continuity equation. Moreover, ρqs is likely to change in time rather slowly compare to purely nonhydrostatic variables for which Δt is chosen. Applying H · and ∂/∂z to (4.3) and (4.4), respectively, taking the sum, and using (4.5), we obtain the following:
i1520-0493-137-2-710-e46
This is a time-discrete version of (3.4). Note that the right-hand side including ρqs(n+1) is known.
When we desire to fully determine (δπ)(n) including , which is the volume mean of (δπ)(n), we use the discrete version of (3.12) given by
i1520-0493-137-2-710-e47
After solving (4.6) for (δπ)(n), (ρqsVH)(n+1) can be determined by (4.3). Then (ρqsw)(n+1) can be found by a downward integration of (4.5) assuming (ρqsw)T(n+1) = 0.

5. The unified system based on the quasi-hydrostatic pressure coordinate

One of the main points of the unified system is that it reduces to a quasi-hydrostatic model when the nonhydrostatic pressure is neglected. In this way, the system maintains a close tie with the existing primitive equation models. But practically all existing primitive equation models use the pressure coordinate or its variants, and thus there is an advantage of using such a coordinate in the unified system to have the same vertical structure as the conventional large-scale models. The merit of Laprise’s approach of using the quasi-hydrostatic pressure as the vertical coordinate in nonhydrostatic models (see section 1) can be even greater for the unified system because the system explicitly deals with the quasi-hydrostatic values of thermodynamic state variables. In this section, we present the unified system based on the quasi-hydrostatic pressure coordinate.

Using the definition of pqs given by (2.4), we may rewrite the hydrostatic equation [(2.2)] as
i1520-0493-137-2-710-e51
Integrating (5.1) vertically with respect to pqs, we obtain
i1520-0493-137-2-710-e52
This corresponds to (2.3). The time derivative of (5.2) gives
i1520-0493-137-2-710-e53
Thus, (pqs)S and θ must be predicted, as in the z-coordinate case, but this time to determine the time evolution of the height field. The thermodynamic equation in (2.8) to predict θ is now written as
i1520-0493-137-2-710-e54
where ω is defined by
i1520-0493-137-2-710-e55
Using the hydrostatic equation in (2.5), the continuity equation in (2.11) can be rewritten as
i1520-0493-137-2-710-e56
It should be noted that, unlike in the usual quasi-hydrostatic p-coordinate system, (5.6) is a consequence of the definitions of pqs and ω, and not of the quasi-hydrostatic approximation. Let us assume that the model top is a material surface with a constant pqs. Then we have ωT = 0. The vertical integral of (5.6) then gives
i1520-0493-137-2-710-e57
Since the earth’s surface is a material surface, ωS can also be written as
i1520-0493-137-2-710-e58
Equating (5.7) and (5.8), we obtain
i1520-0493-137-2-710-e59
Equation (5.3) is now closed. This procedure is simpler than that with the z coordinate mainly because we now have ( ∂pqs/ ∂t)T = 0.
As in the z-coordinate case, the spatially varying part of the nonhydrostatic pressure δp can be determined by requiring that the velocity field predicted by the momentum equation satisfy the continuity equation, which now takes the form of (5.7). We begin with the momentum equation written in the following form:
i1520-0493-137-2-710-e510
Here the assumption in (1.6) and the definition of ρqs given by (2.5) have been used. Transforming the vertical coordinate in (5.10) from z to pqs and taking the horizontal and vertical components, we obtain
i1520-0493-137-2-710-e511
and
i1520-0493-137-2-710-e512
where the three-dimensional vector J is defined by
i1520-0493-137-2-710-e513
From (5.11), we obtain
i1520-0493-137-2-710-e514
From w = (∂/∂t + VH · )pqsz + ωz/∂pqs = (∂/∂t + VH · )pqszω/ρqsg, on the other hand,
i1520-0493-137-2-710-e515
where
i1520-0493-137-2-710-e516
Substituting (5.15) into the continuity equation in (5.6) results in
i1520-0493-137-2-710-e517
Since wC includes the term ∂z/∂t, (5.17) could be viewed as a prognostic equation for z. The point of the unified system is, however, z is predicted through (5.3) and, therefore, the continuity equation in (5.17) is used as a diagnostic equation. In this way, vertically propagating sound waves are filtered.
Taking the time derivative of (5.17) and then using (5.14) and (5.12), we finally obtain
i1520-0493-137-2-710-e518
Time discretization of (5.18) can follow (4.6) using the already predicted values of ρqs(n+1) and z(n+1). Determination of the spatially constant part of δp can follow the procedure for determining the spatially constant part of δπ described in sections 3 and 4.

If desired, the vertical coordinate used in the equations presented here can further be transformed to a sigma coordinate as Janjic et al. (2001) and Janjic (2003) did in their nonhydrostatic model.

6. Small-amplitude perturbations on a resting atmosphere

To compare the unified system with other commonly used systems, this section discusses small-amplitude perturbations on a resting, horizontally uniform atmosphere. For simplicity, the motion is assumed to be adiabatic and frictionless. The standard z coordinate is used for this analysis.

a. Linearized equations

Let an overbar and a prime denote the basic state and perturbation, respectively. Linearizing the equation of state and the definition of θ applied to the perturbation, we obtain
i1520-0493-137-2-710-e61
where cs is the speed of sound given by cs2γRT. Also, we obtain π′ ≈ κ(/p)p′ from the definition of π. Then, using the equation of state applied to the basic state written in the form p = ρRθ, we can show
i1520-0493-137-2-710-e62
Linearizing the horizontal and vertical components of the momentum equation [(2.1)] with the “traditional approximation” (Eckart 1960; Phillips 1966, 1968) and using (6.2), we obtain
i1520-0493-137-2-710-e63
and
i1520-0493-137-2-710-e64
Here δ = 1 and δ = 0 represent the nonhydrostatic and quasi-hydrostatic systems, respectively. The anelastic systems proposed by Ogura and Phillips (1962) and Lipps and Hemler (1982) drop the double-underlined term. Linearization of (2.8) without heating gives
i1520-0493-137-2-710-e65
On the other hand, linearization of (1.1) and the use of (6.1) and (6.5) give
i1520-0493-137-2-710-e66
The anelastic continuity equation neglects both the single- and double-underlined terms while the pseudo-incompressible equation (Durran 1989) neglects only the single-underlined term.

b. Normal-mode analysis on an f plane

In the rest of this section, we analyze the dispersion relation and vertical structure of the normal modes for various systems of equations using a Cartesian horizontal coordinate (x, y), first on an f plane without the quasigeostrophic approximation and then on a midlatitude β plane with the quasigeostrophic approximation. For simplicity, we assume that the motion is uniform in y as in Rossby et al. (1939). An isothermal resting atmosphere is used as the basic state.

Our analysis on an f plane is almost parallel to that performed by Davies et al. (2003) except that we use a different transformation of the dependent variables. From (6.3) with p′ = pqs + δp′, we can derive the divergence and vorticity equations as
i1520-0493-137-2-710-e67
and
i1520-0493-137-2-710-e68
where u and υ are the x and y components of velocity, respectively, and f is a constant Coriolis parameter. Using d ln θ/dz = κ/H in (6.4), where H is the scale height, the vertical component of the momentum equation may be written as
i1520-0493-137-2-710-e69
where b′ ≡ ′/θ. From the definition of pqs, we have
i1520-0493-137-2-710-e610
Then (6.9) gives
i1520-0493-137-2-710-e611
Equations (6.5) and (6.6), on the other hand, give
i1520-0493-137-2-710-e612
and
i1520-0493-137-2-710-e613
Equations (6.7)(6.8) and (6.10)(6.13) form a closed system for the dependent variables u′, υ′, w′, pqs/ρ, b′, and δp′/ρ. Recall the following definitions:
  • Fully compressible: all underlined terms are retained with ε = 1, δ = 1;

  • Unified: all underlined terms are retained with ε = 0, δ = 1;

  • Pseudo-incompressible: terms with single underline are omitted with δ = 1;

  • Anelastic (Lipps–Hemler): terms with single and double underlines are omitted with δ = 1;

  • Quasi-hydrostatic: all underlined terms are retained with δ = 0.

From (6.11), (6.10), and (6.13), it is obvious that the Lipps–Hemler anelastic model applied to an isothermal atmosphere requires κ (∼0.286) << 1 at least for deep motions. As Bannon (1995) pointed out, this condition is also required for the approximation θ′/θ0T ′/T0 used in their model. This requirement suggests that applications of the anelastic system to the stratosphere need some caution. The pseudo-incompressible model of Durran (1989) is free of this requirement.

We now consider normal modes governed by (6.7)(6.8) and (6.10)(6.13) that have the following form:
i1520-0493-137-2-710-e614
where k and m are the horizontal and vertical wavenumbers, respectively, and ν is the frequency, We consider barotropic and baroclinic modes separately.

1) Barotropic mode

By “barotropic mode,” we mean nonbuoyant motions (b′ = 0). Equation (6.12) shows that such motions are horizontal (w′ = 0). Then (6.11) and (6.10) show that both δp′/ρ and pqs/ρ vary exponentially in height. Thus, if we assume that these variables are zero at the model top, they are zero at all height. This is not acceptable at least for pqs/ρ as we discussed in section 2. Using (6.14) in (6.7), (6.8) and (6.13) with w′ = 0, we find the following dispersion relation:
i1520-0493-137-2-710-e615
A solution of (6.15) is ν = 0, representing the stationary barotropic geostrophic motion.
(i) The pseudo-incompressible and anelastic systems

These systems neglect the underlined term in (6.15). Consequently, ν = 0 is the only solution of (6.15).

(ii) The fully compressible, unified, and quasi-hydrostatic systems
The assumption of ε = 0 for the unified system and that of δ = 0 for the quasi-hydrostatic system do not influence the dispersion relation (6.15). Thus, the fully compressible, unified and quasi-hydrostatic systems have identical solutions given by
i1520-0493-137-2-710-e616
which gives the frequency of the Lamb wave modified by the Coriolis force.

2) Baroclinic modes

Using the transformation of the dependent variables given by (6.14), we obtain the dispersion relation for baroclinic modes. As in the case of barotropic modes, ν = 0 is a solution, representing the stationary baroclinic geostrophic motion. For other modes, the dispersion relation is given by
i1520-0493-137-2-710-e617
where
i1520-0493-137-2-710-e618
For solutions satisfying wS = wT = 0 at the upper and lower boundaries, the vertical wavenumber of the solutions is constrained to the form given by
i1520-0493-137-2-710-e619
where n is the integer vertical wavenumber and zT is the height of the upper boundary as previously defined. In (6.19), we assume that zS = 0 at the lower boundary.

Figure 1 shows frequencies of these modes (solid lines) as well as that of the Lamb wave (dashed line) as functions of the horizontal wavenumber for selected values of n for (Fig. 1a) the fully compressible, (Fig. 1b) anelastic, (Fig. 1c) pseudo-incompressible, (Fig. 1d) unified, and (Fig. 1e) quasi-hydrostatic systems. Only positive frequencies are shown. The fully compressible system (Fig. 1a) yields three distinct modes, one representing vertically propagating sound waves, one representing inertia-gravity waves and one representing the Lamb wave. Vertically propagating sound waves are filtered by all the systems (Figs. 1b–e). The unified, pseudo-incompressible, and anelastic systems do the filtering without significant distortions in the dispersion of the inertia–gravity mode while the quasi-hydrostatic system seriously distorts the dispersion of that mode with large horizontal wavenumbers.

It is evident in Fig. 1 that the fully compressible, unified, pseudo-incompressible, and anelastic systems produce virtually identical dispersion relation for the inertia–gravity mode. The n = 1 case of the pseudo-incompressible system is, however, worse than the anelastic system. This does not mean, however, that the solutions for u′, υ′, w′, b′, pqs, and δp′ of the anelastic system are better than those of the pseudo-incompressible system. To show this, we define the vertical phase angle φ by
i1520-0493-137-2-710-e620
In the definition of μ given by (6.18), all terms are kept in the fully compressible, unified, pseudo-incompressible, and quasi-hydrostatic systems, while the term with double underline is omitted in the anelastic system. Consequently, the vertical phase angle is different for the anelastic system from the others. The difference is maximum (approximately 30°) for n = 1 and decreases with increasing n. This is due to the failure of the anelastic system in correctly recognizing the effect of static stability.

c. Normal-mode analysis on a midlatitude β plane with the quasigeostrophic approximation

The normal-mode analysis presented in section 6b is extended to a midlatitude β plane. Since our focus here is on the Rossby wave, we use the quasigeostrophic (and quasi-hydrostatic) approximations for clarity of the results. In this analysis, (6.7), (6.8), and (6.13) are replaced by
i1520-0493-137-2-710-e621
i1520-0493-137-2-710-e622
and
i1520-0493-137-2-710-e623

1) Barotropic mode

(i) The pseudo-incompressible and anelastic systems
These systems neglect the underlined term in (6.13), which gives u′ = 0 for horizontal motion. The dispersion relation then becomes
i1520-0493-137-2-710-e624
This frequency gives the westward retrogression speed of the prototype Rossby wave, which becomes infinite as k→0.
(ii) The fully compressible, unified, and quasi-hydrostatic systems
All of these systems have the dispersion relation given by
i1520-0493-137-2-710-e625
In a sharp contrast to (6.24), (6.25) gives ν → 0 as k → 0.

2) Baroclinic modes

For these modes, the dispersion relation is given by
i1520-0493-137-2-710-e626
With the upper boundary at a height comparable to the scale height H, the second term in the brackets dominates over the first term so that, unlike the barotropic mode, the differences of the anelastic/pseudo-incompressible systems from the others are relatively minor.

Figure 2 shows frequencies of the barotropic (dashed lines) and baroclinic (solid lines) Rossby modes for the fully compressible, unified, and quasi-hydrostatic system (Fig. 2a), the pseudo-incompressible system (Fig. 2b), and the anelastic system (Fig. 2c). The overall performance of the unified, pseudo-incompressible, anelastic, and quasi-hydrostatic systems relative to the fully compressible system is summarized in Table 1. In the table, “not modified” and “modified” are relative to the fully compressible system. In summary, as far as the normal modes are concerned, the unified system maintains the characteristics of the fully compressible system almost exactly except that it filters vertically propagating sound waves.

7. Summary and conclusions

This paper presents a system of equations that can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation of the system includes the time derivative of quasi-hydrostatic density, which can be predicted using the thermodynamic equation and the tendency equation for the quasi-hydrostatic surface pressure. The system can therefore be viewed as a generalization of the anelastic system while it is fully compressible for quasi-hydrostatic motions. The system can also be viewed as a generalization of the quasi-hydrostatic (usually simply called “hydrostatic”) system since no approximation is introduced into the momentum equation. In this way, the system maintains close ties with both the primitive equation models for large scales and the anelastic (and Boussinesq) models for small scales. As in the anelastic system, the spatially varying part of the nonhydrostatic Exner function is determined through solving an elliptic equation. A computational procedure that can be followed in a time-discrete model is presented. The paper also presents the unified system with the quasi-hydrostatic pressure as the vertical coordinate. Appendix B shows that the unified system can also use the vector vorticity equation instead of the momentum equation.

Through normal-mode analysis, it is shown that the unified system reduces the westward retrogression speed of the barotropic Rossby wave through the inclusion of horizontal divergence due to compressibility. It also removes the large systematic error of the anelastic system in the vertical structure. While vertically propagating sound waves are filtered, the Lamb wave is included in the unified system as in the usual primitive equation models. Because of the close analogy between the Lamb wave and shallow-water gravity waves, we hope that the multipoint explicit differencing (MED) technique originally developed for shallow-water gravity waves by Konor and Arakawa (2007) will be effective in stabilizing the Lamb wave with high Courant numbers.

It is shown that a properly defined energy can be conserved in this system with no heating and friction. Whether the energy is conserved or not, however, depends on how we determine the spatially constant part of the Exner function, which does not influence the dynamics of the system. Conservation also depends on the definition of nonhydrostatic temperature, which appears only in the right-hand side of (2.8) representing the diabatic effect. Thus energy conservation in this system is a matter of interpretation as far as adiabatic cases are concerned.

In conclusion, the unified system seems to be a promising system as the dynamics core of global cloud-resolving models although its computational efficiency relative to that of fully compressible models is yet to be assessed.

Acknowledgments

We thank David Randall for his encouragement and support throughout this work. The first author benefited from collaborating with Dale Durran on a similar subject. We also thank Piotr Smolarkiewicz and an anonymous reviewer for a number of suggestions to improve the paper. This work was funded by the U.S. DOE under Cooperative Agreement DE-FC02-06ER64302 to Colorado State University and CSU Contracts G-3818-1 and G3045-1 to UCLA.

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APPENDIX A

Derivation of the Energy Equation

Rewriting the definition of θ using the equation of state, we obtain
i1520-0493-137-2-710-ea1
while from (2.6), which is a consequence of the definition of the quasi-hydrostatic state, we obtain
i1520-0493-137-2-710-ea2
From the definition of Tqs given by (3.7) and (A.2), we can also express lnθ as
i1520-0493-137-2-710-ea3
From (A.2) and (A.3) with the adiabatic thermodynamic equation D ln θ/Dt = 0, we may write
i1520-0493-137-2-710-ea4a
i1520-0493-137-2-710-ea4b
Subtracting (A.2) from (A.1) and using (1.6) and (2.7), on the other hand, we obtain
i1520-0493-137-2-710-ea5
We also have
i1520-0493-137-2-710-ea6
Linearizing the pressure gradient force −p/ρ using (1.6), the momentum equation without the friction force may be written as
i1520-0493-137-2-710-ea7
Multiplying (A.7) by ρqsV, using w = Dz/Dt, and substituting · V obtained from the continuity equation in (1.5) rewritten in the following form:
i1520-0493-137-2-710-ea8
we obtain
i1520-0493-137-2-710-ea9
Using p = pqs + δp, (A.4b) for −pqsD lnρqs/Dt, (A.4a) for −δpD lnρqs/Dt, and then (A.5), (A.9) may be rewritten as
i1520-0493-137-2-710-ea10
where Eqs is the quasi-hydrostatic energy defined by
i1520-0493-137-2-710-ea11
Using (3.8) and (A.6), the last term in (A.10) may be rewritten as
i1520-0493-137-2-710-ea12
From (2.4), the sum of the terms in the last pair of parentheses vanishes. Using this result and the Eulerian form of (A.10), we finally obtain
i1520-0493-137-2-710-ea13
This is (3.5) in the text.

APPENDIX B

Computational Procedure with the Vector Vorticity Equation

In the cloud-resolving model developed by Jung and Arakawa (2008), the horizontal component of the three-dimensional vorticity equation is used instead of the momentum equation. From the curl of (2.1), we can derive the vector vorticity equation as
i1520-0493-137-2-710-eb1
where ω is the three-dimensional vorticity, × V. The horizontal component of (B.1) is
i1520-0493-137-2-710-eb2
In the Jung–Arakawa model, the vertical component of ω is diagnosed from its horizontal component using the identity
i1520-0493-137-2-710-eb3
To derive equations that relate w to the horizontal components of velocity or vorticity, we first rewrite the continuity equation in (2.13) as
i1520-0493-137-2-710-eb4
where (D/Dt)H ≡ ∂/∂t + VH · H. Differentiating (B.4) with respect to z, adding H2w to both sides, and using H × ωHk∇H·(∂VH/∂zHw), we obtain
i1520-0493-137-2-710-eb5
When the last term is dropped, (B.5) becomes a diagnostic equation that relates w to ωH. This diagnostic equation is used in the anelastic model presented by Jung and Arakawa (2008), which replaces the elliptic equation for the Exner function. In the unified system based on the vector vorticity equation, (B.5) is used to update w from the predicted ωH and ρqs. Using the backward scheme to express the last term as in section 4, (B.5) may be discretized as
i1520-0493-137-2-710-eb6
Using already known w(n+1) and ρ(n+1) in the continuity equation in (4.5), we can diagnose · VH(n+1)at an arbitrary level (e.g., at the model top). Then, the horizontal divergence equation applied to the model top determines (δπ)T(n+1) except for a horizontally constant part through a Poisson equation. A downward integration of the vertical component of the momentum equation in (3.3) from this temporary value of (δπ)T(n+1) determines (δπ)(n+1) at all height except for a spatially constant part. We can then follow (3.11) and a time-discrete version of (3.12) to obtain the final value of (δπ)(n+1).

Fig. 1.
Fig. 1.

Frequencies of normal modes on an f plane as functions of horizontal wavenumber for (a) the fully compressible, (b) anelastic, (c) pseudo-incompressible, (d) unified, and (e) quasi-hydrostatic systems. See the text for more details.

Citation: Monthly Weather Review 137, 2; 10.1175/2008MWR2520.1

Fig. 2.
Fig. 2.

Frequencies of normal modes on a midlatitude β plane with the quasigeostrophic approximation as functions of horizontal wavenumber for (a) the fully compressible, unified, and quasi-hydrostatic, (b) pseudo-incompressible, and (c) anelastic systems. See the text for more details.

Citation: Monthly Weather Review 137, 2; 10.1175/2008MWR2520.1

Table 1.

A summary of the normal-mode analysis.

Table 1.
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  • Fig. 1.

    Frequencies of normal modes on an f plane as functions of horizontal wavenumber for (a) the fully compressible, (b) anelastic, (c) pseudo-incompressible, (d) unified, and (e) quasi-hydrostatic systems. See the text for more details.

  • Fig. 2.

    Frequencies of normal modes on a midlatitude β plane with the quasigeostrophic approximation as functions of horizontal wavenumber for (a) the fully compressible, unified, and quasi-hydrostatic, (b) pseudo-incompressible, and (c) anelastic systems. See the text for more details.

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