a. Objectives

We summarize first the requirements of the new scheme to be developed. We are in the course of developing a global climate model in the nonhydrostatic equations to operate with a ultramassive parallel computer with distributed memories. The proposed new scheme is for the purpose of this target. First, we require conservations of variables; in particular, conservations of mass and energy are important for the climate modeling. To satisfy the conservations, we discretized the equations in the flux forms with the finite volume method. Second, the use of a massive parallel computer gives a restriction on the choice of the time integration scheme. We are only considering a horizontally two-dimensional decomposition of the domain for the parallel computation, that is, the whole horizontal computational domain consists of tiles of a rectangular area, each of which is assigned to one of computer nodes. In order to efficiently run on massive parallel computers, data communications between nodes should be smaller. Particularly, we should avoid those methods that generate global communications among many nodes, or large transfer of memory between nodes. The spectrum method and the finite element method are counted as such methods. Use of the horizontally implicit method for sound and gravity waves or the anelastic equations should be reserved since they require elliptic equations to be solved. Solving an elliptic equation on massive parallel computers usually generates large communications between nodes, although efficient numerical methods such as a multigrid method are nowadays available.

Choice of the time integration scheme greatly depends on the treatment of sound waves. We choose the horizontally explicit and vertically implicit scheme for sound waves based on the reason described above. For a horizontally explicit scheme, one also can use the time splitting scheme (Klemp and Wilhelmson 1978; Skamarock and Klemp 1992) to accelerate the efficiency; only the terms related to sound waves are integrated by a small time step and the other slower terms such as advection terms are integrated by a large time step. The scheme shown below can incorporate the time splitting scheme, although the results with the time splitting will be omitted throughout the paper, since the conservation properties are the main concern of the present study.

We start from the Euler equations and make the least approximations to them. The Euler equations are written in the conservative forms of density, momentum, and total energy. The equation of total energy is converted to that of the internal energy, which has transformation terms with the kinetic and potential energies. We choose density ρ and internal energy e as prognostic variables. This choice is contrasted with that in the usual nonhydrostatic models where pressure p (or the Exner function π) and potential temperature θ (or temperature T) are used as prognostic variables. Here, the Exner function is defined as π = (p/p₀₀)^R_d/C_p = T/θ, where p₀₀ = 1000 hPa, R_d is the gas constant for dry air, and C_p is the specific heat at constant pressure. We will discuss these choices of the prognostic variables in terms of the “pressure equation” in section 2c. Furthermore, we must say that the conservation of internal energy does not mean the conservation of total energy. Even if the flux form discretization of the prognostic equation of internal energy is used, the volume integral of total energy is not conserved unless each of the transformation terms with other energies are properly discretized. We will discuss an alternative approach in section 3. In section 2e, we will modify the equations in the terrain-following coordinates for cases where topography exists.

b. The basic equations

We start from the Euler equations. The equations of density, momentum, and total energy are written as

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where υ_i(i = 1, 2, 3) is each a component of velocity vector v = (u, υ, w), e^tot is the total energy defined by

e^totev²(4)

where e is the internal energy and Φ is the gravitational potential. Here σ_ij is the stress tensor and q is the heat flux. We use the summation convention for the repeated indices of i or j with x₁ = x, x₂ = y, and x₃ = z. The Φ is a function of only z and ∂Φ/∂z = g is the acceleration due to gravity.

Subtracting the equations of kinetic energy and potential energy from the equations of total energy (3), we have the equation of internal energy (see section 3):

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where h = e + p/ρ is enthalpy and

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is the diabatic term. In the case of the moist atmosphere, the latent heat release Q_m is added to the right-hand side.

As in the formulations of the usual nonhydrostatic models, we define perturbation pressure from a basic reference state that satisfies the hydrostatic balance,

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The pressure and the density of the basic state p_s and ρ_s are determined if an appropriate temperature profile T_s is given. Using the perturbations defined by p′ = p − p_s and ρ′ = ρ − ρ_s, the continuity equation (1) and the equation of motion (2) are rewritten as

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Thus far, no approximation is introduced by the subtraction of the basic state. However, the transformation terms of energy become different forms as

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The first term (v · ∇p′ + ρ′wg) on the right is the transformation with the kinetic energy, whereas the second term −ρwg is the transformation with the potential energy.

c. The pressure equation

In the fully compressible nonhydrostatic models, the perturbation of pressure p′ or the Exner function π′ is usually used as a prognostic variable instead of that of density ρ′. Normally, p′ or π′, T or θ, and three components of velocities are chosen for the five prognostic variables of the models. To choose p′ as a prognostic variable, one need to use the pressure equation instead of the equation of density. From the thermodynamic equation, tendency of density is related to that of pressure as

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where c_s is the speed of sound, which is given, for the ideal gas, by c²_s = γR_dT, where γ = C_p/C_υ and C_υ is the specific heat at constant volume. From Eqs. (13) and (8), one obtains

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This is one of the forms of the pressure equation.

The advantage of the use of the pressure equation is that the propagation of sound waves is directly incorporated. This is typically shown by the linearized equation in an isentropic gas with no gravity. Substituting the divergence of linearized Eqs. (9)–(11) into the time derivative of Eq. (14) with g = 0 and θ′ = 0, one obtains the equation for the sound waves:

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In the nonhydrostatic model, in practice, one needs to implicitly treat the terms related to the sound waves. If the pressure equation is used, p′ can be implicitly calculated by solving a Helmholtz equation.

The following problems arise, however, if the pressure equation is used as a prognostic equation. First, the conservation of mass is not exactly satisfied. In fact, in the case that c_s is not constant, a volume integral of p′ does not conserve when Eq. (14) is discretized in the flux form. Even when c_s is constant, it is not the total mass but the domain integral of pressure that is conserved.

The second problem, which has not been relatively noticed, is that the pressure equation is not compatible with the conservation of energy. For instance, if one integrates internal energy ρe in the flux form, it is equivalent to use pressure as a prognostic variable, since

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where the equation of state for the ideal gas is used. Since most of the past nonhydrostatic models use T or θ as a prognostic variable, the above problem has been avoided. For our purpose, however, the conservation of energy is required so that the use of the pressure equation should be abandoned.

In order to resolve the above two problems, we use alternative prognostic variables, ρ′ and ρe, which are directly suggested from the original Euler equations. Although the pressure equation is not used, an equation for the implicit calculation of sound waves can be formulated if one notes the relation (16). That is, substituting Eq. (16) into the equation of internal energy (12) and dividing by C_υ/R_d, we obtain

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where

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is used for the ideal gas. Formally, Eq. (17) is analogous to the pressure equation (14). The conservation of internal energy is satisfied if Eq. (17) is discretized in the flux form even when c_s is not a constant. We have chosen a pair of the prognostic variables (ρ, ρe), which correspond to a pair of (ρ, p). In this sense, our dynamical framework is different from that with the customary used variables (p, T) or (π, θ).

In the case of no gravity, the contribution from the right-hand side of Eq. (17) vanishes in the linearized equation around the basic reference state. The convergence of the advective flux c²_sρv on the left-hand side, which is originally the advection of enthalpy, plays a role in the compressibility. In order to implicitly calculate propagation of sound waves, we need to treat this convergence term implicitly.

d. Time discretization

The governing equations of the nonhydrostatic model in this study are based on the flux form equations with prognostic variables being density ρ′, three components of momentum ρu, ρυ, ρw, and internal energy ρe. We define symbols as follows:

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For the time integration, we use the forward scheme for U and V, and the backward scheme for R, W, and E. That is, the horizontally explicit and vertically implicit (HE–VI) scheme is used. Furthermore, we can use the time splitting scheme by partitioning terms of fast and slow motions (Klemp and Wilhelmson 1978; Skamarock and Klemp 1992).

Substituting the symbols defined above, the flux form equations (8), (9), (10), (11), and (12) become

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We have placed the fast terms related to sound waves and gravity waves on the left-hand side of the equations. The choice of the implicit terms is based on the linearized equations around the reference state at rest; the terms responsible for the propagation of sound waves in the linearized equations are treated as implicit terms. Note that the buoyancy term is also treated implicitly in order to take account of the buoyancy effect on sound waves. The stability analysis of the implicit scheme of is given in the appendix. As noted when Eq. (17) is derived, the advection of enthalpy in the equation of internal energy is related to the compressibility of sound waves. This means that the advection terms in Eq. (27) should be treated implicitly.

Based on the forward–backward scheme, therefore, the time discretized forms of Eqs. (23)–(27) become

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The superscripts n and n + 1 represent the quantities of the nth and (n + 1)th step in the time integration and we express

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In Eq. (32), we have introduced u^†, υ^†, and w^†, whose discretization should be determined so as to keep the consistency of the transformation between the internal energy and the kinetic energy. We will argue about the transformations of energy and the possible forms of u^†, υ^†, and w^† in section 3. Here, we proceed using a practical approach.

Equations (28)–(32) are integrated via two steps. First, the (n + 1)th step values of U, V are calculated from Eqs. (29) and (30), respectively. The equation of the internal energy (32) is transformed to the equation of P using Eq. (16). A set of Eqs. (28), (31), and (32), therefore, consists of the implicit equations for the three variables, Rⁿ⁺¹, Wⁿ⁺¹, and Pⁿ⁺¹. It is troublesome to solve for the (n + 1)th step variables, however, if u^†, υ^†, and w^† contains either Rⁿ⁺¹, Wⁿ⁺¹, or Pⁿ⁺¹. To avoid the nonlinearity of the (n + 1)th step variables, we choose

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Under the condition that the change in density is small, the transformation with the kinetic energy of the horizontal winds is consistent with the equation of momentum as will be discussed in section 3. We also replace the term in Eq. (32) as

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Although the left-hand side contains nonlinear terms of (n + 1)th, step variables to keep the conservation (section 3), this replacement makes it linear and thus the implicit calculation can be easily performed. This term is an effect of stratification on sound/gravity waves. The use of the right-hand side means that the nth step stratification is used as a basic state and does not affect linear propagation of sound/gravity waves. On the other hand, the transformation of energy becomes incomplete with this approximation. We will make a correction when the integration of E is performed [see Eq. (47) below]. Using Eq. (16), the equation of internal energy (32) is transformed to

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where

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In the case of the hydrostatic balance, g̃ agrees with the acceleration due to gravity g.

We are in a position of describing how the equations are integrated. First, from Eqs. (29) and (30),

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The remaining variables Rⁿ⁺¹, Wⁿ⁺¹, Pⁿ⁺¹ are solved by the implicit calculation of the three equations (28), (31), and (36). They are formally solved for the (n + 1)th step:

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where

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Substituting Rⁿ⁺¹ and Pⁿ⁺¹ of Eqs. (42) and (44) into Eq. (43), we obtain a Helmholtz equation for Wⁿ⁺¹:

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If we use the first-order discretization in the vertical gradient, this equation is discretized as a form of a tridiagonal matrix multiplied by the vector Wⁿ⁺¹. This type of equation is easily solved.

The density Rⁿ⁺¹ is integrated with a conservative flux form by substituting Wⁿ⁺¹ into Eq. (42). Similarly, Pⁿ⁺¹ can be integrated by using Eq. (44). The replacement of Eq. (35), however, introduced inconsistency in the transformation of energy. Then, we recover Eq. (32) instead of (44) to integrate E, that is,

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where A = (Aⁿ + Aⁿ⁺¹)/2. It is clear that Eⁿ⁺¹ obtained by (47) and Pⁿ⁺¹ obtained by (44) do not satisfy the relation (16). Precisely, one needs to iterate for Eⁿ⁺¹ and Pⁿ⁺¹; number of iterations should be determined by the relative importance between the accuracy and the computation time. For our purpose, however, no iteration is needed as long as the integration is stable, since it is thought that the accuracy of sound waves is unimportant.

In the above formula, we derived the Helmholtz equation for Wⁿ⁺¹. The other choice of the equation for Rⁿ⁺¹ or Pⁿ⁺¹ may be considered. One of the advantages of solving for W is that the top and the bottom boundary conditions are easily incorporated; in the case when the kinematic condition is applied at the boundaries, the boundary values of Wⁿ⁺¹ are directly given. In addition, round-off errors will be smaller if Rⁿ⁺¹ and Eⁿ⁺¹ are directly integrated using the flux form equations. In particular, by using the energy equation (47), we can make a correction for the errors introduced when Eq. (36) is derived. Furthermore, if the specific heat depends on temperature or humidity, Eq. (36) is only an approximation. Equation (47) satisfies the conservation since the advection term is written in the flux form.

e. Terrain-following coordinate

The conservation form of the equations is extended to the terrain-following coordinate when the surface has a topography. We summarize here the flux form equations in the terrain following coordinate. We assume that a new vertical coordinate ξ is steady and monotonic in the vertical direction. A simplest example of ξ is

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where z_T is the top of the model domain and z_s is the height of the surface. This coordinate is used for the simulation of topographic waves in section 4d.

The metrics are defined as

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Mass element within a vertical length δz is δm = ρδz. Denoting the corresponding length in the ξ coordinate by δξ, we have

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Hence, the prognostic variable of the equation of ϕ is transformed to G^1/2ρϕ in the flux form equation. We also have the transformation of the derivatives:

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Thus, the divergence of a flux F = (F_x, F_y, F_z) is expressed as

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where

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The vertical velocity in the ξ coordinate is

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where

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Therefore, multiplying G^1/2 to the equation,

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we have the flux form equation in the (x, y, ξ) coordinates as

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By transforming Eqs. (23)–(27) using this formula, we obtain a set of the equations in the terrain following coordinate.

a. Transformation of energy

In section 2d, we described the time discretization of the model equations. We have pointed out that each of the transformation terms of energy should be consistently discretized to satisfy the conservation of energy, and have presented one possible approach. In this section, we consider how the conservation of energy is satisfied in the numerical scheme and argue about alternative forms of the equation of energy.

The total energy is defined as the sum of kinetic energy, potential energy, and internal energy as Eq. (4). The change in each energy is written as

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The sum of Eqs. (61), (62), and (63) gives the conservation of total energy Eq. (3). Let us denote the transformation term from the potential energy to the internal energy by C_PI, and that from the kinetic energy to the internal energy by C_KI:

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Note that the contribution of the stress tensor is not considered here. To satisfy the conservation of total energy, we need to consistently discretize these transformation terms.

The form of C_PI depends on the discretization of the equation of density. We have chosen Eq. (28) for the equation of density. In this case, the corresponding change in the potential energy is written as

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Thus, the transformation from the potential energy to the internal energy should be expressed as

C_PIWⁿ⁺¹g.(67)

As for the transformation term C_KI, the exact conservation is not satisfied since the discretization is not complete as shown below. When each of the velocity components are chosen as the prognostic variables, the change in the kinetic energy is derived from the discretized equations of momentum. Actually, the equation of kinetic energy (61) is given by the inner product of the equations of momentum (2) and the velocity vector using the equation of density (1). By using a formula,

Aδ_tBBδ_tAδ_tAB(68)

we have the discretized relation for the change in kinetic energy:

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where the kinetic energy is defined by

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The second term on the right-hand side of Eq. (69) is a discretization error. From Eqs. (28), (29), (30), and (31), the discretized equation of kinetic energy is given by

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The corresponding discretized form of the transformation term is

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which is introduced in Eq. (47).

It is clear from Eq. (71) that the conservation of kinetic energy is not exactly satisfied, since ρ is time dependent and the last term on the right-hand side of Eq. (71) remains as a discretization error. The other difficulty comes from the advection of the momentum. The advection of kinetic energy in the third line on the right-hand side of Eq. (71) is not be rewritten as a flux form, since the velocities of the nth and (n + 1)th step values are contaminated. In general, when the equation of kinetic energy is constructed from the equations of momentum, the advection of kinetic energy is not expressed in the flux form; this will be another source of the change in total energy. As for the incompressible fluid, for instance, Morinishi et al. (1998) have proposed a tactful discretization of the advection of momentum so that the advection of kinetic energy is written in the flux form. It is widely believed, however, that the contribution of the advection of kinetic energy is small in the budget of total energy. In the numerical scheme presented in section 2d, therefore, the total energy is approximately conserved in the case that the contribution of density change and the advection of kinetic energy are negligible.

b. Integration of the sum of internal energy and kinetic energy

As a totally different approach to the exact conservation of total energy, one could choose the sum of internal energy and kinetic energy as a prognostic variable. We have implicitly solved for the variables R, W, and P from Eqs. (42), (43), and (44) for the vertical propagation of sound waves. In general, the internal energy E obtained by the following method does not satisfy the relation (36) with P. Nevertheless, as in the same thought of using Eq. (47) for the prediction of E, the following method may be used as an iterative process, if the difference of E obtained by the two methods is small.

From Eqs. (61) and (63), the change in the sum of internal energy and kinetic energy is written as

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where Eq. (6) is used. The discretized form may be given as

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In the advection term, the (n + 1)th step velocity Vⁿ⁺¹ is used; it is required from the stability of the propagation of sound waves. The transformation term with the potential energy is expressed by using Eq. (67). Since Kⁿ⁺¹ is already given from the (n + 1)th step values of U, V, W, and R, Eⁿ⁺¹ is given by subtracting Kⁿ⁺¹ from (E + K)ⁿ⁺¹, which is integrated by the above equation.

c. Integration of entropy

In the last part of this section, we comment on the conservation of entropy in comparison to the conservation of energy. In many nonhydrostatic models, the potential temperature is used as a prognostic variable. For instance, Ooyama (1990) argues that the choice of entropy as a prognostic variable is preferable. Klemp et al. (2000) follows the argument of Ooyama in their flux form equations. The formulation of Ooyama (1990), however, is based on the adiabatic process, so that his form is insufficient for the cases that the energy input is important as in the climate modeling. The change in potential temperature is written as

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The diabatic heating Q is given by Eq. (6). For the climate modeling, the convergence of the heat flux −∇·q has an important contribution. If one uses the equation of the internal energy (5), the domain integral of the source term ∫ Q dV leaves only the boundary values of the heat flux, for example, the sensible heat flux from the surface and the radiative fluxes at the top and the bottom boundaries of the atmosphere. On the other hand, if one uses the equation of potential temperature based on a discretized form of Eq. (75), the change in the domain integral of total energy is generally different from energy input from the boundaries, since temperature is diagnostically calculated form potential temperature. This will cause an error of the energy budget. As for the discretization of general circulation models, for instance, the conservation of energy is formulated based on the change in enthalpy as described by Arakawa and Lamb (1977) and Arakawa and Suarez (1983). They use an additional requirement that the domain integral of potential temperature remains constant under the adiabatic condition to eliminate an arbitrary choice of the vertical discretization. We also follow the Arakawa's concept that the first principle is the conservation of energy, that is, internal energy or enthalpy, but not the conservation of entropy.

The similar approach is adopted by Taylor (1984) who followed Arakawa and Lamb (1977) and devised the vertical discretization scheme using the height z as the vertical coordinate. Taylor uses the equation of enthalpy as a prognostic equation, and guarantees the conservations of potential temperature and entropy by appropriate vertical interpolations of thermodynamic variables. If we use such a vertical discretization as Taylor's, potential temperature and entropy will be also conserved even when the internal energy is chosen as a prognostic variable.

a. Model

We show some results for simple simulations using the numerical scheme described in sections 2 and 3. We have proposed three methods of the energy integration. The first method is based on Eq. (44), from which Eⁿ⁺¹ is given by multiplication of C_υ/R_d to Pⁿ⁺¹: this is referred to as the “noncorrection” method. The second is based on Eq. (47) where the correction is made to the transformation terms of energies: this method is referred to as the “correction” method. In the third method, the conservation of the total energy is guaranteed based on Eq. (74): this will be called the “conservative” method. The results of using these methods are compared in some of the following experiments.

The model is developed in the three-dimensional Cartesian coordinate, but only the results for the two-dimensional calculations are shown here. The definition points of the variables are based on the Arakawa C grid; R, E, and P are defined at the same points, and U, V, and W are defined at the staggered points shifted half point in the x, y, and z directions, respectively. The Lorenz grid is used in the vertical direction. Rigid walls are placed at the bottom and the top of boundaries and free slip and insulating conditions are used at the boundaries. Periodicity is assumed at the lateral boundaries. The time integration of sound waves is based on the forward–backward scheme; the horizontally explicit and vertically implicit scheme as described in section 2d. The model is constructed following the time splitting method (Klemp and Wilhelmson 1978; Skamarock and Klemp 1992), but we simply set the small time step equal to the large time step. The large time step is integrated with the leapfrog scheme with the time filter to suppress the computational mode. We denote the time step of the leapfrog scheme as 2Δt, where 2Δt is equal to the time step of the forward–backward scheme of sound waves. We use a simple second-order centered-in-space scheme for the advections.

b. One-dimensional vertical propagation of sound waves

First, we calculate vertically propagating sound waves to see the stability of the scheme. A perturbation of pressure is given in a layer at the initial state under a horizontally uniform condition. The initial state is uniform temperature 250 K at rest in the hydrostatic balance except for the perturbation of pressure p′ = 100 hPa between the heights 2.5 and 5 km. The top of the atmosphere is z_T = 15 km, and the layer difference is uniform with Δz = 500 m. No numerical smoothing such as numerical diffusion, Rayleigh damping, or the time filter is introduced in this case. Note that, however, the fully implicit scheme has a damping characteristic (see the appendix).

First, we use the correction method for the energy integration. Figures 1 and 2 show the vertical distributions of pressure and vertical velocity, respectively, at t = 10, 20, and 30 s. We compare the profiles with the time step Δt = 0.1, 1, and 10 s. It can be confirmed that the case with Δt = 10 s, which is larger than the Courant–Friedrichs–Lewy (CFL) criterion for the sound wave, is stably integrated. Since the fully implicit scheme is used for the vertical propagation of sound waves, the amplitude of pressure perturbation rapidly decreases as Δt becomes larger (see the appendix). The characteristics of these figures are almost the same for the three methods of the energy integration. The difference between the methods is shown if the energy budget is examined.

The conservation properties are investigated for the case Δt = 1 s. The total mass should be constant as long as no coding error is introduced. In the present calculation, the change in the mean density is within 10⁻¹⁵ kg m⁻³ until t = 1000 s, which comes from the numerical round-off errors. Figure 3 shows time sequences of energies until t = 100 s: internal energy ρe, potential energy ρΦ, and kinetic energy ρw²/2. Here, we define the total energy e^tot and the available potential energy e^avail:

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We use the name “available energy” since we will consider only the time change in the volume integral of e^avail; the change in e^avail must be canceled out with the change in kinetic energy in the analytic form. In these figures, the differences of the volume-averaged values from those initial values are shown. Figure 3a shows that the change in ρe is almost balanced by the change in ρΦ. The change in the potential energy is due to the change in the height of the center of gravity. The small difference between ρe and ρΦ is reflected in the available potential energy ρe^avail, which is transformed with the kinetic energy ρw²/2. In this set up, the total energy is analytically conserved since there is no energy input. The conservation of total energy in the numerical scheme is not guaranteed, however, unless the transformation between kinetic energy and available potential energy is taken care of. The changes in the total energy for the three methods of the energy integration are compared in Figs. 3b,c, and 3d. In the case of the noncorrection method, in which no special care is taken for the transformation, the change in the kinetic energy is very different from the change in the available potential energy (Fig. 3b). In the case of the correction method and the conservative method, on the other hand, the change in the kinetic energy is almost canceled out by the change in the available potential energy and the total energy is almost conserved (Figs. 3c and 3d), although the change in the total energy of the correction method is slightly different from zero. This result implies that the noncorrection method is inappropriate in terms of the conservation of the total energy. Since the nonlinear advection terms are not important in this experiment, the results of the correction method and the conservative method are almost similar.

c. Horizontal propagation of gravity waves

Second, we calculate horizontal propagation of gravity waves by giving potential temperature perturbations initially. The purpose of this experiment is to examine whether the propagation of gravity waves is properly calculated in our scheme in which the buoyancy term is implicitly treated. The experimental condition is the same as that given by Skamarock and Klemp (1994). The top of atmosphere is z_T = 10 km, and the horizontal domain size is 300 km in the x direction. The grid interval is Δx = Δz = 1 km and the time step is Δt = 1 s. The coefficient of the time filter is 0.05. We do not introduce divergence damping or any other numerical smoothing. The correction method is used for the energy integration. The initial thermal state has a constant Brunt–Väisälä frequency with N² = 10⁻⁴ s⁻². The temperature at the bottom of the atmosphere is 300 K. The initial profile of potential temperature perturbation is given by

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where Δθ₀ = 0.01 K, x_m = 100 km, and x_r = 5 km. The uniform horizontal wind 20 m s⁻¹ is given. Note that this experiment is different from that given by Skamarock and Klemp (1994) where vertical variation of density is not considered, whereas we are using the ideal gas. The similar experiment with the ideal gas is shown by Bonaventura (2000).

Figure 4 shows the distributions of potential temperature at the initial state and t = 3000 s. Although a slight asymmetry emerges in the vertical direction, the horizontal propagation of the gravity waves almost agrees with the analytic solution shown by Skamarock and Klemp (1994).

Figure 5 shows time sequences of the volume-averaged values of energies. The differences from the initial values are shown. Similar to Fig. 3, the conservation of total energy is well satisfied. In particular, as shown by Fig. 5b, the transformation between the kinetic energy and the available energy is appropriately calculated. The change in mass is negligibly small though the mass is increasing due to the round-off error: the change in the mean density is below 10⁻¹³ kg m⁻³.

d. Topographic waves

Next, in order to examine the conservative property in the terrain following coordinate, we perform simulations of propagation of topographic waves induced by a surface topography in a uniform zonal wind field. The configuration of topography is given by

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Durran and Klemp (1983) give the maximum height h = 1 m to show an almost linear response. We also have calculated the same condition with h = 1 m and confirmed that our result is almost the same as Durran and Klemp (figures are not shown here). We also simulate the case with a finite amplitude h = 1000 m in order to examine how the energy is conserved. The horizontal length of the mountain is a = 10 km. The top of the model domain is z_T = 16 km, and the horizontal domain length is 180 km. The terrain-following coordinate ξ given by (48) is used. The grid intervals are Δx = 2 km and Δξ = 200 m. Initially, temperature is constant 250 K, and the zonal wind is uniform u = 20 m s⁻¹. In the layers above ξ = 8 km, we introduce damping terms that relax velocities and temperature to those of the basic initial fields. Similar to Durran and Klemp (1983), the relaxation rate of the damping has a cosine profile in the vertical. The damping time takes its shortest value 20Δt at the top of the model domain. The energy integration is based on the correction method. We will show results for a time step Δt = 1 s below. Note that we found that the time step can be extended as Δt = 3 s (the leapfrog time step is 6 s), which gives a sufficiently large Courant number for the vertical propagation of sound wave.

Figure 6 shows the structure of the perturbation of horizontal velocity u′ and the vertical velocity w at t = 3000 s. The amplitudes of the velocities are about 1000 times larger than those of the analytic solution shown by Durran and Klemp (1983, their Figs. 1 and 2). At this time t = 3000 s (ut/a = 6), the flow is not yet in the steady state. However, the further integration shows that turbulence develops due to the nonlinearity since this is the finite amplitude calculation; numerical diffusions are required to continue a stable calculation.

Figure 7 shows time sequences of energies. Since the damping terms are included above ξ = 8 km, the change in the total energy should be equal to the accumulation of the energy sinks due to the damping terms. In Fig. 7b, the accumulated energy change due to the damping terms is shown by the dashed–dotted curve E_R. It is found that E_R is almost explained by the damping term due to temperature perturbations (not shown). Figure 7b shows the difference between the change in the total energy and E_R by the dotted curve. It is confirmed that the change in the total energy is almost equal to E_R. Since this simulation is essentially linear, the energy conservation is well satisfied if the correction method is used. The mass is also almost conserved; the change in the mean density is about 10⁻¹⁴ kg m⁻³ for the 3000-s integration. This comes from the round-off errors.

e. Cold-bubble simulation

As a final example, we simulate the finite amplitude density current induced by a cold bubble given at the initial state. This experiment is suggested as a benchmark simulation of a nonlinear density current by Straka et al. (1993), and also used by Gallus and Rančić (1996). The basic thermal state is neutral and the initial perturbation of temperature, that is, a cold bubble, is given by

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where ΔT₀ = −15.0 K, x_r = 4.0 km, z_r = 2.0 km, and z_m = 3.0 km. The domain size is 51.2 km in the horizontal, and 6.4 km in the vertical. We have tested the simulations for the grid intervals with Δx = Δz = 50, 100, and 200 m. The corresponding time step is Δt = 0.05, 0.1, and 0.2 s, respectively. The second-order diffusion is applied to the velocity and the temperature fields with the diffusion coefficient 75 m² s⁻¹. Note that the energy change due to the diffusion terms are almost negligible in this calculation.

The distributions of the perturbation of potential temperature at t = 0, 300, 600, and 900 s are shown in Fig. 8 for the 50-m grid simulation. The gravity current is propagating along the lower boundary and three rotors associated with the Kelvin–Helmholtz shear instability are well captured. The distribution of potential temperature at t = 900 s is comparable to the corresponding resolution experiment shown by Straka et al. (1993). Figure 9 compares the distributions of the perturbation of potential temperature at t = 900 s in the cases with the coarser resolutions 100 and 200 m. Since we only use the second-order centered-in-space scheme for the advection, the internal structure of vortices are not well simulated particularly in the case with the 200-m grid. However, as the resolution becomes finer, the representation of the rotors becomes better; this results indicate the confidence of the present scheme. The maximum and minimum values of perturbation of potential temperature at t = 900 s is 0.007000 K, −12.1084 K for the 50-m grid, 0.2178 K, −16.9435 K for the 100-m grid, and 2.2038 K, −21.6496 K for the 200-m grid. These values tend to converge to those of the reference solution presented by Straka et al. (1993).

Time sequences of energies for the simulation with the 50-m grid are displayed in Fig. 10. The nonlinear advective terms play important roles in this simulation. Strictly, the conservation of total energy is not satisfied in the correction scheme, if the advection of the kinetic energy is important. Reflecting this fact, the total energy is decreased by 5.12 J m⁻³ for the 900-s calculation as shown by Fig. 10b.

Gallus and Rančić (1996) developed a nonhydrostatic model with an energy conserving scheme using the pressure in the hydrostatic balance as a vertical coordinate. Gallus and Rančić (1996) have considered only the transformation term due to the pressure gradient force between the internal energy and the kinetic energy. Their scheme does not guarantee the exact conservation of total energy, hence it has the same limitation with the correction method. They also have shown the result of the cold-bubble experiment by using a grid interval of 100 m. In their calculation, their total energy is defined as the sum of the internal energy and the kinetic energy [Eq. (A.5) of Gallus and Rančić]. They stated that the change in the total energy is about 0.05% for the 40 000 iterations in the 900-s integration. In our calculation, since the initial value of the internal energy is 175 868 J m⁻³, the corresponding change in the energy is 0.003% in the 18 000 steps integration. It may be said our result is superior to that of Gallus and Rančić in terms of the conservation of total energy.

As an alternative approach, we have performed the similar calculation of the cold-bubble experiment based on the conservative method. We found that the simulation is successfully performed without any instability. Figure 10c shows time sequences of energies, which are compared with Fig. 10b. The change in the total energy during 900 s is only −0.005 428 J m⁻³ for the 50-m grid, −0.010 84 J m⁻³ for the 100-m grid, and −0.025 28 J m⁻³ for the 200-m grid; the conservation is much improved. This result suggests that the use of the conservation method is very encouraging.