2. Basic Equations

2.1 A Family of Anelastic Approximations

As mentioned above in section  1.2, the size of time step that produces a stable numerical solution for explicit finite-difference approximation to the Navier-Stokes equations is restricted by the rapid propagation of sound waves. This time step limitation may be overcome by replacing the full Navier-Stokes equations by an approximation that filters out the sound waves. In search of such sound-proof systems, a wide family of anelastic approximation has been proposed, and all these systems assume that thermodynamic state variables exhibit only small departures from a suitably defined static reference state. The original form of the anelastic approximation was proposed by Batchelor (1953) and later refined by Ogura and Phillips (1962) as an improvement over the Boussinesq approximation more suitable for studying convection within deep atmospheric layers. This original anelastic system conserved modified forms of both total energy and Ertel potential vorticity (Nance and Durran, 1994; Bannon, 1995). Unfortunately, this original anelastic approximation only holds for an isentropic reference state, and is not really applicable to realistic meteorological investigation. Subsequently, this restriction was removed by Wihelmson and Ogura (1972) which replaced the constant reference-state potential temperature with a vertically varying reference-state potential temperature. This modified anelastic equation set improves the representation of deep convection but sacrifices potential vorticity and energy conservations (Bannon, 1995). A major improvement was provided by the Lipps and Hemler (1982) equation set which allows an accurate representation of deep troposheric phenomena, and simultaneously conserves modified forms of Ertel potential vorticity and total energy for finite amplitude disturbances (Bannon, 1995). Finally, Durran (1989) has introduced the pseudo-imcompressible approximation to obtain a better description of upper-level isothermal layers, while retaining a good conservation of appropriate forms of Ertel potential vorticty and total energy for finite amplitude displacements (Bannon, 1995).

The current formulation of Meso-NH model allows for the use of three of these analastic systems: the Lipps and Hemler (1982) system (LH hereafter), the traditional Wihelmson and Ogura (1972), a.k.a. the "Modified Anelastic Equations" (MAE hereafter ), and a simplified implementation of the pseudo-incompressible system of Durran (1989, DUR hereafter). The present chapter tries to describe these three implementations in an orderly way.

2.2 Preliminary Definitions

We will use the following notations:

the rotation velocity of the earth $\vec \Omega$,

the temperature $T$,

the pressure $P$,

the reference value of the pressure $P_{00}=100000 Pa$,

the total density of the moist air $\rho$,

the density of the dry fraction of the air $\rho_d$,

the specific heat at constant pressure of dry air $C_{pd}$,

the specific heat at constant pressure of water vapor $C_{pv}$,

the specific heat of liquid water $C_l$,

the specific heat of ice water $C_i$,

the gas constant for dry air $R_d$,

the gas constant for water vapor $R_v$,

the latent heat of vaporization $L_v$,

the latent heat of sublimation $L_s$,

the latent heat of melting $L_m$.

Various substances are measured by their mixing ratio $r_\bullet$, which is the mass of the substance within a given volume, divided by the mass of dry air within the same volume. In the current anelastic formulation, we assimilate the dry air mass and the reference state dry air mass (see below), thus the volumic mass may be recovered as $\rho_{d\,ref} r_{\bullet}$.

The treatment of the different phases of water is flexible. Up to seven forms of water may be considered if the user wishes so, the mixing ratios being $r_v$ for vapor, $r_c$ for cloud liquid water, $r_r$ for liquid rain, $r_i$ for cloud ice, $r_s$ for snow, $r_g$ for graupels, and $r_h$ for hail. The same mnemonics are used in the code, inasmuch as possible. If a user is not interested by such computations, he may decide to use only a subset of these variables. The computations concerning the remaining water variables will not be performed, resulting in lesser cost. The resulting equations may be obtained by setting the corresponding variables to zero in the following of this chapter.

Whenever necessary, we use the mixing ratio of total water substance

$$r_w=r_v+r_c+r_r+r_i+r_s+r_g+r_h$$ (2.1)

The specific heat at constant pressure of the moist air takes the general form

$$C_{ph}=C_{pd}+r_v C_{pv} + (r_c+r_r) C_l + (r_i+r_s+r_g+r_h) C_i .$$ (2.2)

An important quantity is the Exner function

$$\Pi = ( P/P_{00} )^{R_d / C_{pd}} .$$ (2.3)

We use the "dry" potential temperature

$$\theta = {T \over \Pi}.$$ (2.4)

The virtual temperature is defined as

$$T_v= T . (1+r_v R_v/R_d)/(1+r_w),$$ (2.5)

and the virtual potential temperature as

$$\theta_v= \theta . (1+r_v R_v/R_d)/(1+r_w).$$ (2.6)

$\vec{U}$ is the air velocity, and whenever necessary, its cartesian components are called $u$, $v$, and $w$.

2.3 Reference State

All anelastic approximation systems rely on the hypothesis that the atmosphere will not depart very far from a "reference state", defined as an atmosphere at rest, in hydrostatic equilibrium, with horizontally uniform profiles of temperature $T_{ref}(z)$ and water vapor $r_{v\,ref}(z)$. No condensed water is considered in the reference state. For our application, the reference profiles are often chosen as the initial horizontal averages of actual fields over the expected domain of simulation. Any profile, however, may be used, but the inaccuracy of the computation increases if the reference state is far from the actual mean state. In the following the dependency in $z$ only will be assumed for all quantities subscripted with $()_{ref}$.

The hydrostatic relation and the equation of state are used to derive the profiles of the virtual temperature and the Exner function of the reference state $T_{v\,ref}$, $\Pi_{ref}$:

$$T_{v\,ref}=T_{ref}(1+r_{v\,ref} R_v/R_d)/(1+r_{v\,ref} )$$ (2.7)

$$\frac{\displaystyle d(Log\Pi_{ref})}{\displaystyle dz} = -\frac{\displaystyle g}{\displaystyle C_{pd}T_{v\,ref}}$$ (2.8)

with the upper boundary condition $\Pi_{ref} = \Pi_{ref}^{top}$ at the model top $z=H$.

$\theta_{v\,ref}$ is then deduced from $\Pi_{ref}$ and $T_{v\,ref}$ as

$$\theta_{v\,ref} = \frac{\displaystyle T_{v\,ref}}{\displaystyle\Pi_{ref}}$$ (2.9)

Finally, $\rho_{ref}$ is deduced from the equation of state

$$\rho_{ref} = \frac{\displaystyle\Pi_{ref}^{C_{vd}/R_d}P_{00} }{\displaystyle R_d \theta_{v\,ref}}$$ (2.10)

and the density of the dry air fraction $\rho_{d\,ref}$ is deduced by

$$\rho_{d\,ref}={ \rho_{ref} \over (1+r_{v\,ref}) }$$ (2.11)

Note that both the dry air and the water vapor must be considered to build the profile of the Exner function of the reference state, since each gas is subject to the partial pressure of the other one.

2.4 Equation of State

A primary difference between the three anelastic systems used in Meso-NH is the form adopted for the equation of state. Both the modified anelastic (MAE) and Lipps-Hemler (LH) systems follow the usual framework of the Boussinesq approximation by linearizing the equation of state. On the contrary, the Durran system (DUR) keeps the full equation of state, without any linearization.

2.4.0.1 Modified-Anelastic and Lipps-Hemler Systems

For the MAE and LH cases, the equation of state $p=\rho R_d T_v$ is replaced by the linearized form:

$$\rho ' = \rho_{ref} \left( \frac{\displaystyle C_{vd}}{\disp... ...displaystyle\theta_v '}{\displaystyle\theta_{v\,ref}} \right),$$ (2.12)

where $\rho '= \rho - \rho_{ref}, \Pi '=\Pi - \Pi_{ref}$, and $\theta_v '= \theta_v - \theta_{v\,ref}$.

2.4.0.2 Durran System

In the DUR case, contrary to most anelastic systems, the equation of state

$$p=\rho R_d T_v$$ (2.13)

is not linearized, and thus is used without any approximation.

2.5 Anelastic Constraint

For all three anelastic system used in Meso-NH, the continuity equation is may be written as the approximated form:

$$\vec{ \nabla} \cdot (\rho_{d\,eff} \vec{U}) = 0,$$ (2.14)

where $\rho_{d\,eff}$ is a convenient compact notation for:

$$\rho_{d\,eff} = \left\{ \begin{array}{r@{\quad}cc} \rho_{d\,... ... (1+r_{v\,ref}). & \quad &\mbox{(Durran)} \end{array}\right.$$

$$\$$ (2.17)

Stricly speaking, the Durran (1989) form of the anelastic constraint (equation 2.17) only holds in the adiabatic limit, and the r.h.s of (2.17) should be written $\frac{\displaystyle\cal H}{\displaystyle C_{pd}\Pi_{ref}}$ in the general case where $\cal H$ is the heating rate per unit volume. However, Durran (1989, p. 1454) argues that this generalized form reduces to (2.17) in the usual case where the anelastic assumption itself holds.

Also note that the dimensional definition of $\rho_{d\,eff}$ is slightly inconsistent. It corresponds to the dry air density of the reference state when the Lipps-Hemler deep convection anelastic (LH) or the Modified Anelastic Equation (MAE) approximations are used; but it includes an extra potential temperature factor when the pseudo-incompressible anelastic approximation (DUR) of Durran (1989) is used.

Also note that equation (2.14) represents a strong geometrical constraint on the wind field, called the anelastic constraint. This constraint is enforced by solving an elliptic equation for some pressure deviation function, that results from the combination of the continuity and momentum equations (see below).

2.6 Conservation of Momentum

The system is written in a referential frame linked to the earth, with a rotation velocity $\vec{\Omega}$, and includes a yet un-specified momentum source term $\vec{\cal F}$. In Meso-NH, unlike most atmospheric models, the total density of the air will vary in function of the precipitation and evaporation of water. Therefore a consistent flux form of the equations can only be obtained by combining the Lagrangian forms and the equation of continuity for dry air. Furthermore, we want to allow for an easy transition to the compressible system, should this one be adopted in the future. Therefore, we use an equation of conservation for the momentum of the dry air fraction of the fluid:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\rho_{... ...\,eff} \vec{\Omega} \wedge\vec{U}= \rho_{d\,eff} \vec{\cal F},$$ (2.18)

where $\rho_{d\,eff} \vec{{\cal F}_{\Pi}}$ is the pressure gradient force, which takes different forms in the three systems:

$$\rho_{d\,eff} \vec{{\cal F}_{\Pi}} = \left\{ \begin{array}{l... ...ec{ \nabla} \Pi '. & \quad &\mbox{(Durran)} \end{array}\right.$$

$$\$$ (2.21)

A distinctive property of the Lipp and Hemler (1982) system is that the reference state virtual potential temperature is pulled inside the pressure gradient term (2.19), whereas this factor appears outside of the gradient for both the MAE and Durran (1989) systems (2.20 and 2.21). For the MAE system also note that the virtual potential temperature of the reference state is used, whereas the Durran (1989) system directly retains the un-approximated virtual potential temperature in (2.21).

It is also worth noticing that the form adopted for the Durran (1989) system is identical to the momentum equation in the complete compressible system. Although the hydrostatic reference state have been removed, the pressure-gradient term has not been linearized. The Durran (1989) system is therefore expected to provide a particularly accurate description of the (compressible) momentum balance.

Apart from the differences in expressing the pressure gradient term, the other terms retain their usual form, respectively representing the time evolution, the advection, the buoyancy force, the Coriolis force, and the diabatic effects. Also note that the flux form of the equation is consistently kept throughout the model, insuring an integral conservation of momentum to a very good accuracy. For convenience, the model further uses $\theta _v$ rather than $\theta_v '$, therefore, in the following equations, we will use $\theta_v-\theta_{v\,ref}$ to denote the fluctuation of buoyancy.

Finally, enforcing the anelastic constraint (2.14) leads to the improperly called "pressure problem" to determine $\Pi '$ (DUR and MAE) or $C_{pd} \theta_{v} \Pi '$ (LH), respectively. In order to simplify the expression of this problem, we introduce the dynamical source of momentum:

$$\vec{S} = - \vec{ \nabla} \cdot (\rho_{d\,eff} \vec{U}\otime... ...eff} \vec{\Omega} \wedge\vec{U} + \rho_{d\,eff} \vec{\cal F} ,$$ (2.22)

whereby the momentum conservation equation may be rewritten:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\rho_{... ...ff} C_{pd} \theta_{v} \vec{ \nabla} \Pi ' \end{array}\right\}.$$

$$\begin{array}{c@{\quad}c} \quad &\mbox{(Lipps-Hemler)} \\ \... ...ox{(Mod. Anelastic Eq.)} \\ \quad &\mbox{(Durran)} \end{array}$$

$$\$$ (2.25)

2.7 Hydrostatic Equation

When an hydrostatic approximation of the vertical velocity equation is required, for instance within initialization routines or to compute diagnostic quantities, the hydrostatic form to be used also differs for the three equations systems.

With the pseudo-incompressible system of Durran (1989), as the vertical momentum equation is written without any approximation, the hydrostatic relation is exact and reads:

$$\frac{\displaystyle\partial \Pi '}{\displaystyle\partial z} ... ...ystyle\theta_v-\theta_{v\,ref}}{\displaystyle\theta_{v\,ref}}.$$ (2.26)

With both the MAE and Lipp-Hemler systems, the usual form of the hydrostatic relation $dp=-\rho g dz$ should not be used to establish diagnostic quantities related to this model. For the Lipps-Hemler (1982) sytem, by consistency with (2.18-2.19), one should use instead:

$$\frac{\displaystyle\partial (C_{pd} \theta_{v\,ref} \Pi ')}{... ...ystyle\theta_v-\theta_{v\,ref}}{\displaystyle\theta_{v\,ref}}.$$ (2.27)

Whereas the MAE system leads to a formula similar to (2.26), except for the use of the reference state virtual potential temperature instead of the virtual potential temperature:

$$\frac{\displaystyle\partial \Pi '}{\displaystyle\partial z} ... ...ystyle\theta_v-\theta_{v\,ref}}{\displaystyle\theta_{v\,ref}}.$$ (2.28)

2.8 Thermodynamic Equation

For the time being, the prognostic energy variable is the dry potential temperature $\theta$. In order to account for the effects of moisture, its equation takes the following original form

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\rho_{d\,ef... ...ph}}-1 \right] {\theta \over \Pi_{ref}} w {\partial \Pi_{ref} \over \partial z}$$

$$+ {\rho_{d\,eff}\over \Pi_{ref} C_{ph}} \left[ L_m {D(r_i+r_s+r_g+r_h)\over Dt} - L_v {Dr_v\over Dt} + {\cal H} \right].$$ (2.29)

The terms on the right hand side represent respectively the moist correction in absence of any phase change (derived from the conservation of total energy, with some approximations), the effects of phase changes, and the other diabatic effects (radiation and diffusion). The use of the reference state Exner function instead of the total one in the above equation allows for an effective decoupling of the pressure problem from the thermodynamical problem, leading to a great simplification of the resolution, while retaining an excellent accuracy.

2.9 Conservation of Moisture

For any of the water substances $r_{\star}$, the conservation equation is written:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\rho_{... ..._{d\,eff} r_{\star} \vec{U}) = \rho_{d\,eff} {\cal Q}_{\star},$$ (2.30)

where ${\cal Q}_{\star}$ stands for the effects of phase changes, sedimentation and diffusion. Note the flux form, and the use of $\rho_{d\,eff}$ (not $\rho_{ref}$), insuring the existence of integral conservation properties.

2.10 Conservation of Passive Scalars

Similarly, the model can carry an arbitrary number of passive scalars, following the equation:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\rho_{... ..._{d\,eff} s_{\star} \vec{U}) = \rho_{d\,eff} {\cal S}_{\star},$$ (2.31)

where ${\cal S}_{\star}$ stands for the effects of diabatic and chemical processes.

2.11 Conservation of Total Mass

The total mass inside the model domain is the sum of the mass of dry air and the mass of water (other substances are neglected for the time being)

$${\cal M}={\cal M}_d + {\cal M}_w$$ (2.32)

The anelastic constraint (see above) does not supply the variation of the total mass of dry air ${\cal M}_d (t)$ inside the model domain, since it uses the reference density profile instead of the actual density. It is therefore necessary to use an additional equation. This one depends on the general conditions of the experiment:

• If periodic or wall boundary conditions are assumed, the total mass of dry air may not vary during a simulation. It is therefore specified once and for all at the beginning ( ${\cal M}_d(t)={\cal M}_d(t_0)$).
• If the model is driven by larger-scale meteorological information (forecasts or analyses), the time variation of the total mass of dry air is assumed to be entirely governed by the larger-scale fields. This is consistent with procedures currently used in other limited area models. In that case, the value ${\cal M}_d (t)$ of the total mass of dry air within the simulation volume is regularly updated by linear interpolation in time between the values deduced from the larger-scale fields.

On the other hand, the total mass of water ${\cal M}_w(t)$ can be computed at each time during the simulation, by a simple volume integration

$${\cal M}_w = \int_V \rho_{d\,ref} r_w dV$$ (2.33)

where $V$ is the total volume of the model domain. Note that this results in the total mass varying with precipitation and evaporation of water, as it should, contrary to what was assumed in most models in the past. Also, when the model is forced by larger-scale fields, the evolution of the total mass of water in the model may be different from that in the forcing fields, since the representation of physical processes is different.

2.12 Pressure Equation

For all three anelastic system implemented in Meso-NH, combining the anelastic constraint 2.14) and the momentum equations (2.23-2.25), some appropriate "pressure function" can be retrieved by solving an elliptic problem:

$$\vec{\nabla} \cdot \left\{ \begin{array}{c} \rho_{d\,eff} \v... ...nabla} \Pi ' \end{array}\right\} = \vec{\nabla} \cdot \vec{S},$$

$$\begin{array}{c@{\quad}c} \quad &\mbox{(Lipps-Hemler)} \\ \... ...ox{(Mod. Anelastic Eq.)} \\ \quad &\mbox{(Durran)} \end{array}$$

$$\$$ (2.36)

the details of the procedure, however, are system-depended.

2.12.0.1 Lipps-Hemler System

With the Lipps-Hemler (1982) system, it is convenient to define the pressure function in (2.34), as:

$$\Phi= C_{pd} \theta_{v\,ref} \Pi ',$$ (2.37)

and to solve directly (2.34) for $\Phi$, which can be interpreted as a geopotential perturbation.

As the boundary condition for the elliptic equation (2.34) are Neuman ones (see below and Chapter 7) the solution $\Phi$, hence the pressure, is determined only to an arbitrary constant. This is sufficient for the dynamical problem; however, for thermodynamic computations, it is often necessary to use the absolute value of the pressure. Let us assume that $\Phi_{fg}$ is a particular solution of the above elliptic problem. We want to determine the additional constant $\Phi_0$ that will insure the correct absolute value of the pressure. This is achieved by using the total mass as defined above, which may be developed following:

$${\cal M} = \int _{V} \rho \; dV = \int _{V} \rho_{ref} \; dV... ... _{V} \rho ' \; dV = {\cal M}_{ref} + \int _{V} \rho ' \; dV,$$ (2.38)

where ${\cal M}_{ref}$ is the mass of the reference state (including both dry air and water), computed once and for all at the beginning of each experiment.

As the linearized equation of state (2.12) replaces the usual ideal-gas law for density fluctuations in the Lipps-Hemler system, we have then

$${\cal M}(t) = {\cal M}_{ref} + \int _{V} \rho_{ref} \; \left... ...a_v -\theta_{v\,ref}}{\displaystyle\theta_{v\,ref}} \right) dV$$ (2.39)

which is solved as

$$\Phi_{0} = \frac{\displaystyle{\cal M}(t) - 2 \times {\cal M... ...}\frac{\displaystyle 1}{\displaystyle C_{pd}\Pi_{ref}} \; dV }$$ (2.40)

The resulting field of $\Phi_{fg} + \Phi_0$ is used to retrieve the absolute pressure

$$P= P_{00} \left(\Pi_{ref} + \frac{\displaystyle\Phi_{fg} + \Phi_0}{\displaystyle C_{pd} \theta_{v\,ref} }\right)^{C_{pd}/R_d}$$ (2.41)

2.12.0.2 Modified Anelastic System

With the MAE system, solving the elliptic equation (2.35) directly provides the Exner function deviation $\Pi '$ from the reference state, instead of the above defined $\Phi$ function.

However, the rest of the solution procedure follows the principles outlined for the Lipps-Hemler system. As the boundary condition for the elliptic equation (2.34) are Neuman ones, the Exner function $\Pi '$ is determined to an arbitrary constant only, and the absolute value of the pressure is obtained as above. Assuming that $\Pi '_{fg}$ is a particular solution of the elliptic problem, the additional constant $\Pi_0$ that will insure the correct absolute value of the pressure is derived from the total mass budget, written using (2.38) without changes.

As the linearized equation of state (2.12) replaces the usual ideal-gas law for density fluctuations in the MAE system, we have then

$${\cal M}(t) = {\cal M}_{ref} + \int _{V} \rho_{ref} \; \left... ...a_v -\theta_{v\,ref}}{\displaystyle\theta_{v\,ref}} \right) dV$$ (2.42)

which is solved as

$$\Pi_{0} = \frac{\displaystyle{\cal M}(t) - 2 \times {\cal M}... ...f}} \; \frac{\displaystyle C_{vd}}{\displaystyle R_d} \; dV }$$ (2.43)

The resulting field of $\Pi '_{fg} + \Pi_0$ is used to retrieve the absolute pressure

$$P= P_{00} \left(\Pi_{ref} + \Pi '_{fg} + \Pi_0 \right)^{C_{pd}/R_d}$$ (2.44)

2.12.0.3 Durran System

Finally, with the Durran (1989) system, solving the elliptic equation (2.36) directly provides the Exner function deviation $\Pi '$ from the reference state. As it was the case for other systems in previous sections, the Neuman boundary conditions only determine $\Pi '$ to an arbitrary constant, and this constant has to be obtained from a total mass budget to retrieve the absolute value of the pressure field. However, as the Durran (1989) system use the equation of state without linearization, a modified procedure is followed.

Let us assume that ${\Pi '}_{fg}$ is a particular solution of the above elliptic problem. We want to determine the additional constant $\Pi_0$ that will insure the correct absolute value of the pressure. This is achieved by using the total mass as defined above:

$${\cal M} = \int _{V} \rho \; dV.$$ (2.45)

Combining the ideal-gas equation of state and the Exner function definition:

$$\rho = \frac{\displaystyle\Pi^{C_{vd}/R_d}P_{00} }{\displaystyle R_d \theta_{v}}$$ (2.46)

the total mass of the model can be written as:

$${\cal M}(t) = \int _{V} \frac{\displaystyle P_{00} }{\displ... ...eft( \Pi_{ref} + {\Pi '}_{fg} + \Pi_0 \right)^{C_{vd}/R_d} dV.$$ (2.47)

As $\Pi_{ref} + {\Pi '}_{fg} \gg \Pi_0$, a linearization of the $\Pi_0$ term allows to solve the above equation as

$$\Pi_{0} = \left( \frac{\displaystyle {\cal M}_{d}(t) + {\ca... ...d}}{\displaystyle R_d^2 \theta_{v}} dV } \right)^{R_d/C_{vd}}$$ (2.48)

Eventually, a second iteration is performed using the same equation to get an accurate value of $\Pi_0$. The resulting constant is used to retrieve the absolute pressure:

$$P= P_{00} \left( \Pi_{ref} + {\Pi '}_{fg} + \Pi_0 \right)^{C_{pd}/R_d}$$ (2.49)

2.13 Boundary Conditions

Top Boundary

The model is assumed to be limited by a rigid horizontal lid, exerting a free slip condition on the atmosphere. The height $H$ of this lid may be chosen by the user. In order to obtain realistic results, it is recommended to place $H$ somewhere in the stratosphere. At $z=H$, the conditions imposed on the model are:

$$\vec{U}\cdot\vec{n} = 0$$ (2.50)

$$\vec{\nabla} \left\{ \begin{array}{c} \Phi \\ \Pi ' \\ \Pi ' \end{array}\right\} \cdot \vec{n} = \vec{S}\cdot \vec{n}$$

$$\begin{array}{c@{\quad}c} \quad &\mbox{(Lipps-Hemler)} \\ \... ...ox{(Mod. Anelastic Eq.)} \\ \quad &\mbox{(Durran)} \end{array}$$

where $\vec{n}$ is a vertical unit vector.

In order to prevent the reflexion of gravity waves on this lid, an absorbing layer may be activated, where the model prognostic variables are relaxed towards the large-scale values.

Bottom Boundary

In the adiabatic formulation of the model, the lower boundary condition is defined as an insulated rigid lid, with free slip. This may be again formulated as

$$\vec{U}\cdot\vec{n} = 0$$ (2.54)

$$\vec{\nabla} \left\{ \begin{array}{c} \Phi \\ \Pi ' \\ \Pi ' \end{array}\right\} \cdot \vec{n} = \vec{S}\cdot \vec{n}$$

$$\begin{array}{c@{\quad}c} \quad &\mbox{(Lipps-Hemler)} \\ \... ...ox{(Mod. Anelastic Eq.)} \\ \quad &\mbox{(Durran)} \end{array}$$

but $\vec{n}$, a unit vector normal to the earth surface, may not be vertical.

In the physical package of the model, the turbulent fluxes of heat, moisture, and momentum are computed in the usual way.

Lateral Boundary

This is a more complicated problem. We allow for several types of conditions:

• Periodic conditions
• Rigid wall
• Free propagation of waves out of the simulation domain
• Evolution of the prognostic variables imposed by external information from larger scale
• In some cases, a mixture of the previous conditions can be used

This will be discussed in detail in Chapter 5.

2.14 Energy and Potential Vorticity Conservation

A thorough discussion of the conservation properties for all three anelastic systems implemented in Meso-NH is given in Bannon (1995). Only a summary of the main conclusions is therefore given below, readers being referred to the original paper for derivations and details. Following Bannon (1995), we restrict the scope of this discussion to a dry atmosphere, and consider a finite amplitude flow with $\cal H$ representing the heating rate per unit volume, and $\vec{\cal F}$ the frictional forces. When a conservation property is existing, it will be given in a flux form, assuring that the volume integral of the relevant quantity is conserved for adiabatic inviscid flows in a closed domain for which the normal component of the velocity field vanishes on the boundaries.

2.14.0.1 Modified Anelastic System

When the potential temperature of the reference state varies with height, neither a closed form of the energy, nor a consistent statement of potential vorticity conservation are possible. The case of the constant reference state potential temperature corresponds to the original anelastic equations (Ogura and Phillips, 1962), which conserve both an energy form and an Ertel potential vorticity form (see Bannon 1995, and Durran, 1989, p 1459).

2.14.0.2 Lipps-Hemler System

This system conserves both a form of energy, and an Ertel-like potential vorticity. Bannon (1995, p 2305) writes the total energy equation in flux form as:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(E_{LH}... ...ec{U}] = {\cal H} + \rho_{d\,ref} \vec{U} \cdot \vec{\cal F},$$ (2.58)

where $P '$ is the pressure deviation from reference state, and $E_{LH}$ is the appropriate form of total energy in the LH system:

$$E_{LH} = \rho_{d\,ref} \left\{ { \frac{1}{2}\, \vec{U} \cdot \vec{U}} + g z + C_{pd} \theta \right\}$$ (2.59)

Note the use of the reference state density. Bannon (1995) further notes that the presence of $C_{pd}$ instead of $C_{vd}$ in the internal energy term reveals that in the LH system heating occur at constant pressure, as is required by this system's approximations.

The Lipps and Hemler (1982) anelastic form of Ertel's conservation theorem is:

$$\frac{\displaystyle D}{\displaystyle Dt}\left({PV}_{LH}\righ... ...{\displaystyle\vec{\cal F}}{\displaystyle\rho_{d\,ref}}\right)$$ (2.60)

where $\vec{\omega_a} = 2 \vec{\Omega} + \vec{\nabla}\times\vec{U}$ is the absolute vorticity, $\dot{\Theta}$ the diabatic warming rate:

$$\dot{\Theta} = \frac{\displaystyle\theta\; \cal H}{\displaystyle\rho_{d\,ref} C_{pd} T}$$ (2.61)

and the appropriate form of the potential vorticity is (Bannon, 1995, p 2306):

$${PV}_{LH} = \frac{\displaystyle\vec{\omega_a} \cdot \vec{\nabla} \theta}{\displaystyle\rho_{d\,ref}}.$$ (2.62)

Note that the reference state density has to be used, not the true density.

2.14.0.3 Durran System

The pseudo-incompressible system of Durran (1989) also exhibits conservation properties for both a total energy and an Ertel-like potential vorticity.

Defining the pseudo-incompressible density:

$$\rho^{\star} = \rho_{d\,ref} \frac{\displaystyle\theta_{ref}}{\displaystyle\theta}$$ (2.63)

The flux form of the total energy equation is (Durran, 1989):

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(E_{D})... ...vec{U}] = {\cal H} + \rho^{\star} \vec{U} \cdot \vec{\cal F},$$ (2.64)

where $P^{\star} = P_{ref} + \rho_{d\,ref} C_{pd} \theta_{ref} \Pi '$, and $E_{D}$ is the appropriate form of total energy in the Durran (1989) system:

$$E_{D} = \rho^{\star} \left\{ { \frac{1}{2}\, \vec{U} \cdot \vec{U}} + g z \right\} + \rho_{d\,ref} C_{vd} T_{ref}$$ (2.65)

The pseudo-incompressible version of Ertel's conservation theorem is:

$$\frac{\displaystyle D}{\displaystyle Dt}\left({PV}_{D}\right... ...imes\frac{\displaystyle\vec{\cal F}}{\displaystyle\rho}\right)$$ (2.66)

where ${PV}_{D}$ is the appropriate form of the potential vorticity (Bannon, 1995, p 2308):

$${PV}_{D} = \frac{\displaystyle\vec{\omega_a} \cdot \vec{\nabla} \theta}{\displaystyle\rho^{\star}}.$$ (2.67)

Note that the pseudo-incompressible density has to be used, not the reference state one.

2.15 References

Bannon P. R., 1995: Potential vorticity conservation, hydrostatic adjustment, and the anelastic approximation, J. Atmos. Sci., 52, 2302-2312

Batchelor G. K., 1953: The condition for dynamical similarity of motions of frictionless perfect-gas atmosphere, Quart. J. Roy. Meteor. Soc., 79, 224-235.

Durran D. R., 1989: Improving the anelastic approximation, J. Atmos. Sci., 46, 1453-1461.

Lafore J.P., J. Stein, N. Asencio, P. Bougeault, V. Ducrocq, J. Duron, C. Fischer, P. Héreil, P. Mascart, V. Masson, J. P. Pinty, J. L. Redelsperger, E. Richard, J. Vilá-Guerau de Arellano, 1998: The Meso-NH atmospheric simulation system. Part I: adiabatic formulation and control simulations, Ann. Geophysicae, 16, 90-109.

Lipps F., and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci., 39, 2192-2210.

Lipps F., 1990: On the anelastic approximation for deep convection, J. Atmos. Sci, 47, 1794-1798.

Nance L. B., D. R. Durran, 1994: A comparison of the accuracy of three anelastic systems and the pseudo-incompressible system, J. Atmos. Sci., 51, 3549-3565.

Ogura Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173-179.

Wilhelmson, R., and Y. Ogura,1972: The pressure perturbation and the numerical modelling of a cloud, J. Atmos. Sci., 29, 1295-1307.