3. Coordinate Systems

In general, it will not be possible to develop the equations of the previous chapter in a simple cartesian coordinate system, because of the earth sphericity, and of the underlying topography. In meteorology, a natural coordinate system is defined by the longitude $\lambda$, the latitude $\varphi$, and the distance from the earth center $r$ (or the altitude above sea surface $z=r-a$, where $a$ is the earth radius). The vector basis associated with this natural system will be called $(\vec{i}_0, \vec{j}_0,\vec{k})$. $\vec{i}_0$ points towards the east, $\vec{j}_0$ towards the north, and $\vec{k}$ is vertical. In Meso-NH, we prefer to work with a conformal projection allowing for rotation with respect to this natural basis. This allows more flexibility in the direction of the coordinate lines to study particular processes. This also simplifies the initialization of model runs with products of operational weather prediction systems, like the Arpege system of Meteo-France, or the ECMWF system.

3.1 Conformal Projections and Terrain Following Coordinates

We use a system of curvilinear coordinates $\widehat{x},\widehat{y}, \widehat{z}$, defined in the following manner.

3.1.1 Gal-Chen and Sommerville vertical coordinate

$\widehat{z}$ is the Gal-Chen and Sommerville (1975) vertical coordinate:

$$\widehat{z} =H \frac{\displaystyle z - z_{s}}{\displaystyle H - z_{s} },$$ (3.1)

where $H$ is the height of the model top, and $z_{s}$ the height of the local topography. Since the transformation is purely vertical, $\vec{k}$ is also parallel to the $\widehat{z}$ lines in the physical space (see Fig. 3.1).

Figure: The Gal-Chen and Sommerville vertical coordinate and the cartesian ( $\vec{i},\vec{j}$), covariant ($\vec{e}_i$) and contravariant ($\vec{e}^i$) bases

On the other hand, $\widehat{x}$ and $\widehat{y}$ are the distances counted from an arbitrary origin in two arbitrary orthogonal directions on a conformal surface of projection (Fig. 3.2). The traces on the sphere of these coordinate lines define two orthogonal directions in each point. We will call hereafter $\vec{i}$ and $\vec{j}$ the horizontal, unit length vectors parallel to those directions. We note $\gamma$ the angle between from $\vec{i}$ to $\vec{i}_0$. In general, this angle will vary with $\widehat{x}$ and $\widehat{y}$ because of the earth sphericity.

Figure 3.2: Principles of Projections and notations on the sphere

Clearly, $\vec{i},\vec{j}$ and $\vec{k}$ form a local, cartesian basis which is particularly interesting to develop the wind velocity vector. In the following, we will call $u , v , w$ the components of ${\vec U}$ on this basis.

$$\vec{U}=u\vec{i}+v\vec{j}+w\vec{k}$$ (3.2)

It is the most natural decomposition of the wind in a local basis.

Three types of conformal projections are supported in Meso-NH: Lambert, Polar Stereographic, and Mercator. We recall here the formulae which allow to compute the coordinates $\widehat{x},\widehat{y}$ from the latitude $\varphi$ and the longitude $\lambda$ of a given point, and inversely.

3.1.2 Polar stereographic and Lambert Projections:

The projection is defined by the conicity parameter $K$ ($0<K<1$), the earth radius $a$, a reference latitude $\phi_0$, a reference longitude $\lambda_0$, an arbitrary angle of rotation $\beta$, and the coordinates of the pole in the projection $\widehat{x}_0, \widehat{y}_0$ (Fig. 3.3). The useful formulae are

$$\gamma = K (\lambda - \lambda _{0}) - \beta$$

$$R = \frac{\displaystyle a}{\displaystyle K} (\cos\varphi _{0})^{1-K} ... ...left(\frac{\displaystyle\cos\varphi }{\displaystyle 1 + \sin\varphi}\right)^{K}$$

$$\widehat{x} = \widehat{x}_0 + R sin\gamma$$ (3.3)

$$\widehat{y} = \widehat{y}_0 - R cos \gamma$$

The polar stereographic projection corresponds to $K=1$. The more general case of the Lambert conical projection is obtained for $0<K<1$.

The map scale factor, defined as the ratio of distances on the projection surface to distances on the sphere, is given by

$$m=\left( \frac{\displaystyle\cos \varphi _{0}}{\displaystyle... ...+ \sin\varphi _{0}}{\displaystyle 1 + \sin\varphi} \right)^{K}$$ (3.4)

Figure 3.3: Polar Sterographic and Lambert Projections: $M_0$ is the point at the intersection of the reference latitude and longitude. In $M_0$, the angle from $\vec{i}$ to $\vec{i}_0$ is $-\beta$. In any other point $M$, this angle has a different value. Note that the North pole may be situated inside the model domain in the case of the polar stereographic projection, but not in the case of the Lambert projection

3.1.3 Mercator Projection:

The projection is defined by a reference latitude $\phi_0$, a reference longitude $\lambda_0$, an arbitrary angle of rotation $\beta$, and the coordinates of the origin in the projection $\widehat{x}_0, \widehat{y}_0$ (Fig. 3.4). The useful formulae are

$$\gamma = - \beta$$

$$\widehat{x}' = \widehat{x}_0 + a \cos \varphi _{0}(\lambda - \lambda _{0})$$

$$\widehat{y}' = \widehat{y}_0 -a\cos\varphi _{0} \ln\vert\tan(\frac{\displaystyle\pi}{\displaystyle 4}-\frac{\displaystyle\varphi}{\displaystyle 2})\vert$$ (3.5)

$$\widehat{x} = \widehat{x}'\cos\gamma - \widehat{y}'\sin\gamma$$

$$\widehat{y} = \widehat{x}'\sin\gamma + \widehat{y}'\cos\gamma$$

For this projection, if $\beta$ is chosen equal to zero, the local cartesian basis coincides everywhere with the natural basis.

The map scale factor is given by

$$m = \frac{\displaystyle\cos\varphi _{0} }{\displaystyle\cos\varphi }$$ (3.6)

Figure: Mercator Projection: the angle between $\vec{i}$ and $\vec{i}_0$ is independent of the position in the domain

Note that allowing for $K=0$ in the expression of the map scale factor of the Lambert projection will give exactly the same result. As far as the local metrics is concerned, the only useful informations are therefore $m$, $K$, and $\gamma$. These quantities will be used in the following equations, without further necessity to distinguish between the three types of projection.

3.2 Calculus in Non Orthogonal Coordinates

In the physical space, the $\widehat{x}$ and $\widehat{z}$ coordinate lines are not orthogonal, because of the underlying topography $z_s$. Therefore, the coordinate system is not orthogonal and it will be necessary to introduce, beside the cartesian basis, the covariant and contravariant bases to develop the tensor operators of Chapter 2. In Meso-NH, we follow the ideas of Viviand (1974), and Vinokur (1974), and use alternatively the cartesian and covariant bases, in order to simplify the formulation of these operators. We now recall the main classical formulae to work with non orthogonal coordinate systems.

3.2.1 Metric Coefficients

Let us call $x,y,z$ the local distances on the sphere in the directions $\vec{i},\vec{j},\vec{k}$.

We will use the classical notations $\widehat{d}_{\star \star}$ for the metric coefficients. Their values in the case of our transformation are given by:

$$\widehat{d}_{xx} = \frac{\displaystyle\partial x}{\displaystyle\partial \widehat{x} } = { r \over a m }$$

$$\widehat{d}_{yy} = \frac{\displaystyle\partial y}{\displaystyle\partial \widehat{y} } = { r \over a m }$$

$$\widehat{d}_{zz} = \frac{\displaystyle\partial z}{\displaystyle\partial \widehat{z} } = 1 - {z_s \over H}$$

$$\widehat{d}_{zx} = \frac{\displaystyle\partial z}{\displaystyle\partial \widehat{x} } = {\partial z_s \over \partial \widehat{x} }(1-{\widehat{z}\over H})$$

$$\widehat{d}_{zy} = \frac{\displaystyle\partial z}{\displaystyle\partial \widehat{y} } = {\partial z_s \over \partial \widehat{y} }(1-{\widehat{z}\over H})$$ (3.7)

$$\widehat{d}_{xz} = \frac{\displaystyle\partial x}{\displaystyle\partial \widehat{z} } = 0$$

$$\widehat{d}_{yz} = \frac{\displaystyle\partial y}{\displaystyle\partial \widehat{z} } = 0$$

$$\widehat{d}_{xy} = \frac{\displaystyle\partial x}{\displaystyle\partial \widehat{y} } = 0$$

$$\widehat{d}_{yx} = \frac{\displaystyle\partial y}{\displaystyle\partial \widehat{x} } = 0$$

3.2.2 Covariant Basis

The basis of covariant vectors is defined as

$$\vec{e}_{i} = \frac{\displaystyle\partial \vec{r}}{\displaystyle\partial \widehat{x}^{i}}$$ (3.8)

(with $\widehat{x}^{i}=\widehat{x}$, $\widehat{y}$ or $\widehat{z}$). Those vectors are tangent to the coordinate lines (see Fig. 3.1). They can be projected to the cartesian basis, resulting in

$$\vec{e}_{1} = \frac{\displaystyle\partial \vec{r}}{\displaystyle\partial \widehat{x}} = \widehat{d}_{xx} \vec{i} + \widehat{d}_{zx} \vec{k}$$

$$\vec{e}_{2} = \frac{\displaystyle\partial \vec{r}}{\displaystyle\partial \widehat{y}} = \widehat{d}_{yy} \vec{j} + \widehat{d}_{zy} \vec{k}$$

$$\vec{e}_{3} = \frac{\displaystyle\partial \vec{r}}{\displaystyle\partial \widehat{z}} = \widehat{d}_{zz}\vec{k}$$ (3.9)

3.2.3 Jacobian

The Jacobian of the transformation from $\widehat{x},\widehat{y}, \widehat{z}$ to $x,y,z$ is given by

$$\widehat{J} = \vec{e}_{1} \cdot ( \vec{e}_{2} \wedge \vec{e}_{3})$$ (3.10)

or

$$\widehat{J} = \widehat{d}_{xx} \widehat{d}_{yy} \widehat{d}_{zz} = \left( {r\over am} \right)^2(1 - {z_s \over H})$$ (3.11)

It is the ratio of the volumes in the transformed and physical spaces.

3.2.4 Contravariant Basis

The basis of contravariant vectors is defined as

$$\vec{e}\, ^{i} = \vec{\nabla}\widehat{x}^{i}$$ (3.12)

These vectors are orthogonal to the surfaces $\widehat{x}^{i}=$Const (see Fig. 3.1).

Since $\vec{e}\, ^{i} \cdot \vec{e} _{j} = \delta_{ij}$, we also have:

$$\vec{e}\, ^{1} = \frac{\displaystyle 1}{\displaystyle\widehat{J}}(\vec{e} _{2} \wedge \vec{e} _{3}) = \frac{\displaystyle 1}{\displaystyle\widehat{d}_{xx}} \vec{i}$$

$$\vec{e}\, ^{2} = \frac{\displaystyle 1}{\displaystyle\widehat{J}}(\vec{e} _{3} \wedge \vec{e} _{1}) = \frac{\displaystyle 1}{\displaystyle\widehat{d}_{yy}} \vec{j}$$ (3.13)

$$\vec{e}\, ^{3} = \frac{\displaystyle 1}{\displaystyle\widehat{J}}(\vec{e} _{1} \we... ..._{zx} \widehat{d}_{yy}\vec{i} - \widehat{d}_{zy} \widehat{d}_{xx}\vec{j}\right)$$

These formulae allow to find the expression of the contravariant components of any vector, as a function of its cartesian components

$$\vec{A} = A^{c1} \vec{e} _{1}+ A^{c2} \vec{e} _{2} + A^{c3} \vec{e} _{3}$$

$$\vec{A} = a^{1} \vec{i}+ a^{2} \vec{j} + a^{3} \vec{k}$$

$$A^{c1} = \frac{\displaystyle a^1 }{\displaystyle\widehat{d}_{xx}}$$ (3.14)

$$A^{c2} = \frac{\displaystyle a^2 }{\displaystyle\widehat{d}_{yy}}$$

$$A^{c3} = \frac{\displaystyle a^3 }{\displaystyle\widehat{d}_{zz}}- \frac{\... ...splaystyle a^2 \widehat{d}_{zy}}{\displaystyle\widehat{d}_{yy}\widehat{d}_{zz}}$$

3.2.5 Spatial derivatives of the Cartesian basis

We will also use the expression of the spatial derivatives of the vectors of the cartesian basis, due to the earth sphericity:

$$\frac{\displaystyle\partial \vec{i}}{\displaystyle\partial \widehat{x}} = \cos\gamma \; \frac{\displaystyle\widehat{d}_{xx}}{\displaystyle ... ...K) \;\vec{j} - \frac{\displaystyle\widehat{d}_{xx}}{\displaystyle r} \; \vec{k}$$

$$\frac{\displaystyle\partial \vec{i}}{\displaystyle\partial \widehat{y}} = \sin\gamma \; \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r \cos\varphi} \; (\sin\varphi - K) \;\vec{j}$$

$$\frac{\displaystyle\partial \vec{j}}{\displaystyle\partial \widehat{x}} = -\cos\gamma \; \frac{\displaystyle\widehat{d}_{xx}}{\displaystyle r \cos\varphi} \; (\sin\varphi - K) \;\vec{i}$$ (3.15)

$$\frac{\displaystyle\partial \vec{j}}{\displaystyle\partial \widehat{y}} = - \sin\gamma \; \frac{\displaystyle\widehat{d}_{yy}}{\displaystyl... ...K) \;\vec{i} - \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r} \; \vec{k}$$

$$\frac{\displaystyle\partial \vec{k}}{\displaystyle\partial \widehat{x}} = \frac{\displaystyle\widehat{d}_{xx}}{\displaystyle r} \;\vec{i}$$

$$\frac{\displaystyle\partial \vec{k}}{\displaystyle\partial \widehat{y}} = \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r} \;\vec{j}$$

3.2.6 Gradient

For any scalar quantity $\alpha$,

$$\vec{\nabla} \alpha = \frac{\displaystyle\partial \alpha }{\... ...ial \alpha }{\displaystyle\partial \widehat{z}}\vec{e} \, ^{3}$$ (3.16)

This vector may be projected on $(\vec{i},\vec{j},\vec{k})$ resulting in the cartesian components of the gradient

$${\partial \alpha \over \partial x } = \vec{i} \cdot \vec{\nabla} \alpha = \frac{\displaystyle 1}{\displaystyle\widehat{d}_{xx}} \frac{\disp... ...}_{zz}} \frac{\displaystyle\partial \alpha }{\displaystyle\partial \widehat{z}}$$

$${\partial \alpha \over \partial y } = \vec{j} \cdot \vec{\nabla} \alpha = \frac{\displaystyle 1}{\displaystyle\widehat{d}_{yy}} \frac{\disp... ...}_{zz}} \frac{\displaystyle\partial \alpha }{\displaystyle\partial \widehat{z}}$$ (3.17)

$${\partial \alpha \over \partial z } = \vec{k} \cdot \vec{\nabla} \alpha = \frac{\displaystyle 1}{\displaystyle\widehat{d}_{zz}} \frac{\displaystyle\partial \alpha }{\displaystyle\partial \widehat{z}}$$

3.2.7 Divergence

For any vector quantity $\vec{A}$, the contravariant components

$$\vec{A} =A^{c \, 1}\vec{e} _{1} + A^{c \, 2}\vec{e} _{2} +A... ...,^{2})\vec{e} _{2} + (\vec{A} \cdot\vec{e}\, ^{3})\vec{e} _{3}$$ (3.18)

allow to compute easily the divergence

$$\vec{\nabla}\cdot \vec{A} = \frac{\displaystyle 1}{\displays... ...splaystyle\partial \widehat{z}}(\widehat{J} A^{c\, 3}) \right)$$ (3.19)

3.2.8 Divergence of a Tensor Product

In order to avoid generating a large number of Christoffel symbols (quantities involved in the derivatives of the basis vectors), we use the contravariant components of $\vec{B}$ and the cartesian components of $\vec{A}$

$$\vec{B} = B^{c \, 1}\vec{e} _{1} + B^{c \, 2}\vec{e} _{2} +B^{c \, 3}\vec{e} _{3}$$

$$\vec{A} = {\cal A}^{1} \vec{i} + {\cal A}^{2}\vec{j}+{\cal A}^{3} \vec{k}$$

The result projected on the cartesian basis reads

$$\vec{i} \cdot \vec{\nabla}\cdot (\vec{B} \otimes \vec{A}) = \frac{\displaystyle 1}{\displaystyle\widehat{J}}\left(\frac{\disp... ...}{\displaystyle\partial \widehat{z}}(\widehat{J} B^{c \, 3}{\cal A}^{1})\right)$$

$$- B^{c \, 1} {\cal A}^{3}\frac{\displaystyle\widehat{d}_{xx}}{\di... ...c{\displaystyle\widehat{d}_{xx}}{\displaystyle r \cos\varphi} (\sin\varphi - K)$$

$$- B^{c \, 2}{\cal A}^{2} \sin \gamma \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r \cos\varphi} (\sin\varphi - K)$$

$$\vec{j} \cdot \vec{\nabla}\cdot (\vec{B} \otimes \vec{A}) = \frac{\displaystyle 1}{\displaystyle\widehat{J}}\left(\frac{\disp... ...l}{\displaystyle\partial \widehat{z}}(\widehat{J} B^{c\, 3}{\cal A}^{2})\right)$$

$$+ B^{c \, 2}{\cal A}^{3}\frac{\displaystyle\widehat{d}_{yy}}{\dis... ...ac{\displaystyle\widehat{d}_{xx}}{\displaystyle r cos\varphi} (\sin\varphi - K)$$

$$+ B^{c \, 2}{\cal A}^{1} \sin \gamma \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r cos\varphi} (\sin\varphi - K)$$

$$\vec{k} \cdot \vec{\nabla}\cdot (\vec{B} \otimes \vec{A}) = \frac{\displaystyle 1}{\displaystyle\widehat{J}}\left(\frac{\disp... ...}{\displaystyle\partial \widehat{z}}(\widehat{J} B^{c \, 3}{\cal A}^{3})\right)$$

$$- B^{c \, 1}{\cal A}^{1} \frac{\displaystyle\widehat{d}_{xx}}{\di... ... - B^{c \, 2}{\cal A}^{2} \frac{\displaystyle\widehat{d}_{yy}}{\displaystyle r}$$ (3.20)

For each equation, the first line contains the dominant terms, and the second and third lines show the additional terms due to the spatial variation of the vectors of the local basis. These terms are the curvature terms. The influence of non-orthogonality is also present through the use of the contravariant components of $\vec{B}$.

3.2.9 Coriolis Force

Both vectors are expressed in basis $(\vec{i},\vec{j},\vec{k})$. We call $f=2\Omega \sin\varphi$ and $f_{\star}=2\Omega \cos\varphi$.

$$2\vec{\Omega}\wedge\vec{U} = (f_{\star} \cos\gamma w - f v ) \vec{i}$$

$$+ (f u + f_{\star} \sin \gamma w ) \vec{j}$$

$$- (f_{\star} \cos\gamma u + f \sin \gamma v ) \vec{k}$$ (3.21)

3.3 Model Equations in Non Orthogonal Coordinates

3.3.1 Prognostic Variables

Given the form of the divergence operator, the most convenient choice for the model prognostic variables is the product of the wind components, potential temperature, and mixing ratios by the dry density of the reference state, and by the Jacobian of the coordinate system.

To simplify the notations, we define $\widehat{\rho}=\rho_{d\,ref}\widehat{J}$. The prognostic variables of the model are therefore $\widehat{\rho}u, \widehat{\rho}v, \widehat{\rho}w, \widehat{\rho}\theta, \widehat{\rho}r_{*}$, and $\widehat{\rho} s_{*}$.

For the water and passive scalars, the prognostic variable represents therefore (to the extent that $\rho_{d\,ref}$ is a good approximation of $\rho_d$) the mass of substance within the grid volume, which is a very simple and safe quantity to carry in a model.

3.3.2 The contravariant components of the wind

The contravariant components of the velocity vector $\vec{U} = U^{c} \vec{e} _{1}+ V^{c} \vec{e} _{2} + W^{c} \vec{e} _{3}$ will be needed to express the advection operator. They may be computed as

$$U^{c} = \vec{U} \cdot \vec{e}\, ^{1}$$ (3.22)

$$V^{c} = \vec{U} \cdot \vec{e}\, ^{2}$$ (3.23)

$$W^{c} = \vec{U} \cdot \vec{e}\, ^{3}$$ (3.24)

Since $( u , v )$ are the horizontal wind components in the basis $( \vec{i}, \vec{j} )$ we have

$$U^{c} = \frac{\displaystyle u }{\displaystyle\widehat{d}_{xx}}$$ (3.25)

$$V^{c} = \frac{\displaystyle v }{\displaystyle\widehat{d}_{yy}}$$ (3.26)

$$W^{c} = \frac{\displaystyle w}{\displaystyle\widehat{d}_{zz}}- \frac{\dis... ...displaystyle v \widehat{d}_{zy}}{\displaystyle\widehat{d}_{yy}\widehat{d}_{zz}}$$ (3.27)

This simple computation is performed at the beginning of each model time step.

3.3.3 Momentum Equation

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} u ) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...tyle\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; u )$$

$$+ \widehat{\rho} u v \frac{\displaystyle \cos\gamma}{\displaystyle ... ...} \frac{\displaystyle \sin\gamma }{\displaystyle r\cos\varphi} (\sin\varphi -K)$$

$$- \widehat{\rho}\frac{\displaystyle u w}{\displaystyle r} \, - \, \... ...t{d}_{zz}} \frac{\displaystyle\partial \Phi}{\displaystyle\partial \widehat{z}}$$

$$- \widehat{\rho} f^{*} \cos\gamma w \, + \, \widehat{\rho} f v \,+ \, \widehat{\rho} \vec{F}_{v} \cdot \vec{i}$$ (3.28)

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} v ) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...tyle\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; v )$$

$$- \widehat{\rho} u ^{2} \frac{\displaystyle \cos\gamma}{\displaysty... ...u v \frac{\displaystyle\sin\gamma}{\displaystyle r\cos\varphi} (\sin\varphi -K)$$

$$- \widehat{\rho}\frac{\displaystyle v w}{\displaystyle r} \, - \, \... ...t{d}_{zz}} \frac{\displaystyle\partial \Phi}{\displaystyle\partial \widehat{z}}$$

$$- \widehat{\rho} f^{*} \sin\gamma w \, - \, \widehat{\rho} f u \,+ \, \widehat{\rho} \vec{F}_{v} \cdot\vec{j}$$ (3.29)

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} w) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...style\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; w)$$

$$+ \widehat{\rho} u ^{2} \frac{\displaystyle 1}{\displaystyle r}+ \w... ...idehat{\rho} g \frac{\displaystyle\theta_v ' }{\displaystyle\widehat{\theta}_v}$$

$$+ \widehat{\rho} f^{*}(\sin\gamma v + \cos\gamma u ) \,+ \, \widehat{\rho} \vec{F}_{v} \cdot \vec{k}$$ (3.30)

The right hand side of these equations has been rewritten in Chapter 2 as the sum of a dynamical source $\vec{S}_V$ and the pressure gradient. This is still valid, except that the source now contains the Jacobian. Note that those are still the cartesian components of $\vec{S}_V$, i.e. the true horizontal and vertical components of the acceleration.

3.3.4 Thermodynamic Equation

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} \theta) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; \theta)$$

$$+\widehat{\rho} \left[ {R_d+r_vR_v\over R_d}{C_{pd} \over C_{ph}}... ...f}} {w \over \widehat{d}_{zz} } {\partial \Pi_{ref} \over \partial \widehat{z}}$$

$$+ {\widehat{\rho}\over \Pi_{ref} C_{ph}} \left[ L_m {D(r_i+r_s+r_g+r_h)\over Dt} - L_v {Dr_v\over Dt} + {\cal H} \right]$$ (3.31)

3.3.5 Water and Other Scalars

The equations have exactly the same form as the thermodynamic equation, except for the source terms.

3.3.6 Continuity Equation

It has the simple form, using the contravariant velocity components

$$\frac{\displaystyle\partial }{\displaystyle\partial \widehat... ...\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c }) =0.$$ (3.32)

3.3.7 Pressure Equation

The divergence operators appearing in the pressure equation may be evaluated with the contravariant components of the vectors $\vec{\nabla}\Phi$ and $\vec{S}_V$. The result is an elliptic equation, involving a quasi-laplacian operator. This is discussed in Chapter 7.

3.4 Degenerated Forms

3.4.1 Thin shell approximation

The depth of the atmosphere ($\simeq 30$ km) is much smaller than the earth radius ($a \simeq 6000$ km). Therefore, it is customary in meteorology to assume that the atmosphere is a thin shell ($z \ll a$ and $r \simeq a$). However, if this hypothesis is retained without care, the principle of conservation of angular momentum may be violated (Phillips, 1966). It is necessary to make the additional hypothesis that the horizontal component of the earth rotation is negligible. In the Meso-NH model, we decided not to retain the thin shell approximation. However, for the purpose of comparison with models making this approximation, we introduce a flag $\delta_1$, taking the value 1 in the general case and 0 in the case of the thin shell approximation.

3.4.2 Cartesian Coordinate System

For many purposes, one may want to work in the much simpler cartesian frame on a plane tangent to the sphere, and neglect the curvature terms (e.g. study of very small scale processes, or idealized studies of meso-scale processes). This may be obtained easily by setting $f=2 \Omega \sin \varphi _{0}$ , $f^{*}=2 \Omega \cos \varphi _{0}$, $\widehat{d}_{xx}=1$, $\widehat{d}_{yy}=1$, $m=1$, $r=a=\infty$. Note that contrary to the previous case, we may keep here $f^{*} \neq 0$ without violating the conservation of angular momentum. In order to obtain this very simple form, we introduce another flag $\delta_2$ in the general system, taking the value 1 for the general case, and 0 for the cartesian case.

The combination of the thin shell approximation and the cartesian frame will supply the traditional f-plane approximation (no horizontal component of the earth rotation).

3.4.3 Flagged Equations

The $\delta_1$ and $\delta_2$ flags modify the momentum equation in the following way:

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} u ) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...tyle\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; u )$$

$$+ \delta _{2}\widehat{\rho} u v \frac{\displaystyle \cos\gamma}{\di... ...\frac{\displaystyle \sin\gamma }{\displaystyle r \cos \varphi} (\sin\varphi -K)$$

$$- \delta _{2}\delta _{1} \widehat{\rho}\frac{\displaystyle u w}{\di... ...t{d}_{zz}} \frac{\displaystyle\partial \Phi}{\displaystyle\partial \widehat{z}}$$

$$- \delta _{1} \widehat{\rho} f^{*} \cos\gamma w \, + \, \widehat{\rho} f v \, + \, \widehat{\rho} \vec{F}_{v} \cdot \vec{i}$$ (3.33)

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} v ) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...tyle\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; v )$$

$$- \delta _{2}\widehat{\rho} u ^{2} \frac{\displaystyle \cos\gamma}{... ...v \frac{\displaystyle\sin\gamma}{\displaystyle r \cos \varphi} (\sin\varphi -K)$$

$$- \delta _{2}\delta _{1} \widehat{\rho}\frac{\displaystyle v w}{\di... ...t{d}_{zz}} \frac{\displaystyle\partial \Phi}{\displaystyle\partial \widehat{z}}$$

$$- \delta _{1}\widehat{\rho} f^{*} \sin\gamma w \, - \, \widehat{\rho} f u \, + \, \widehat{\rho} \vec{F}_{v} \cdot \vec{j}$$ (3.34)

$$\frac{\displaystyle\partial}{\displaystyle\partial t}(\widehat{\rho} w) = \, - \, \frac{\displaystyle\partial }{\displaystyle\partial \wide... ...style\partial }{\displaystyle\partial \widehat{z}} (\widehat{\rho} W^{c } \; w)$$

$$+ \delta _{2}\delta _{1}\widehat{\rho}\frac{\displaystyle u ^{2}+ v... ...idehat{\rho} g \frac{\displaystyle\theta_v ' }{\displaystyle\widehat{\theta}_v}$$

$$+ \delta _{1}\widehat{\rho} f^{*}(\sin\gamma v + \cos\gamma u ) \, + \, \widehat{\rho} \vec{F}_{v} \cdot \vec{k}$$ (3.35)

with

$$\widehat{d}_{xx} = \frac{\displaystyle a+\delta_1 \delta_2 z}{\displaystyle am}$$

$$\widehat{d}_{yy} = \frac{\displaystyle a+\delta_1 \delta_2 z}{\displaystyle am}$$

$$r = a + \delta_1 z$$ (3.36)

$$\widehat{J} = \left( \frac{\displaystyle a+\delta_1 z}{\displaystyle am} \right)^2 (1 - {z_s \over H })$$

The other equations of the model do not involve the flags.

3.5 References

Gal-Chen, T., and R. C. J. Sommerville, 1975: On the Use of a Coordinate Transformation for the Solution of the Navier-Stokes Equations. J. Comput. Phys., 17, 209-228.

Vinokur, 1974:?

Viviand, H., 1974: Formes Conservatives des Equations de la Dynamique des Gaz. La Recherche Aérospatiale, 1974-1, 65-66.