Map Factors

The dynamic equations can be expressed using a generalized map factor, $m$, which is defined as the ratio of the model grid distance to the true distance on the Earth's surface. Map factors are used conjunction with the horizontal derivative terms within the dynamics equations. In this model, they can be used for conformal Lambert or Mercator projections (usual limited area domain applications) or for a latitude-longitude (Plate Carrée) projection (global scale applications). In this latter case, the horizontal component is stretched so that the map factor must be considered separately for the two horizontal directions ($m_x$ and $m_y$). See for example, Haltiner and Williams (1980) and Richardson et al. (2007) for detailed discussions of map factors.

The the actual distance along a grid cell wall is defined as $\Delta x = d/m_x$ (the analogous expression applies for $\Delta y$), where $d$ is a constant grid cell distance (usually $\Delta x=d$ at the equator for the Plate Carrée projection, or at the center of a conformal projection). The mass fluxes across the grid faces can then be expressed using map factors as

$$U = {\frac{\mu_d \, u}{m_y}}$$ (284)

$$V = {\frac{\mu_d \, v}{m_x}}$$ (285)

Again note that for Cartesian coordinates (no projection), $m_x=m_y=m=1$, so that Eq.s 286-287 are equivalent to Eq.s 124-125.

The continuity equation (Eq. 28) neglecting sources can be expanded using a general map factor as

Next, the prognostic flux-form equation for scalars (Eq. 48), momentum (Eq.s 30-32), and geopotential (Eq. 35), again neglecting diabatic terms, can be expanded, respectively, using a general map factor as

$$\begin{align*}\begin{split} {\frac{\partial \varphi\mu_d }{\partial t}} \,\,= &\... ...}\mu_d \varphi\right) }{\partial\eta}} + \mu_d D_\varphi \end{split}\end{align*}$$ (287)

$$\begin{align*}\begin{split} {\frac{\partial u\mu_d }{\partial t}} \,\,= &\,\, - ... ...rtial\Pi}{\partial x}} + \mu_d \,f_{u,crv} + \mu_d \,D_u \end{split}\end{align*}$$ (288)

$$\begin{align*}\begin{split} {\frac{\partial v\mu_d }{\partial t}} \,\,= &\,\, - ... ...rtial\Pi}{\partial y}} + \mu_d \,f_{v,crv} + \mu_d \,D_v \end{split}\end{align*}$$ (289)

$$\begin{align*}\begin{split} {\frac{\partial w\mu_d }{\partial t}} \,\,= &\,\, - ... ... \mu_d \, g\, \epsilon + \mu_d \,f_{w,crv} + \mu_d \,D_w \end{split}\end{align*}$$ (290)

$$\begin{align*}\begin{split} {\frac{\partial\Phi}{\partial t}} =& - m_x \, u {\fr... ...- {\dot\eta} {\frac{\partial\Phi}{\partial\eta}} + g \,w \end{split}\end{align*}$$ (291)

where $\varphi$ represents a scalar (passive tracer, TKE, temperature, cloud water, etc...). Note that the horizontal diffusion terms, $D$, include map factors. They are expressed as

$$D_\phi = \kappa^o \, \nabla^o \phi = \kappa_x^o \, m_x^o \, {\fra... ...partial x^o}} \,+\, \kappa_y^o \, m_y^o \, {\frac{\partial \phi}{\partial y^o}}$$ (292)

where $\phi$ corresponds to the quantity being diffused and $o$ represents the order of the diffusion operator, and $\kappa$ represents the corresponding diffusivity.

In addition to horizontal and vertical diffusion, the $u$ and $v$ momentum equations also include divergence damping terms (Eq. 77, for more details see Section 2.2.1). The default damping is fourth order, and as an example, it is given for the $u$-momentum equation by

$$D_{u\,d}^4 = - {\frac {\kappa_{d4\,u}}{\mu}} {\frac{\partial^3 }{... ...eft[ {\frac{\partial }{\partial x}} \left(\nabla \cdot{\bf V}\mu\right) \right]$$ (293)

thus it amounts to taking the second order derivative of the gradient of the horizontal divergence. The divergence (defined at mass points) is expressed in finite difference form by Eq. 126, where map factors are introduced using the definitions in Eq.s 286-287 so that

$$\nabla\cdot{\left( {\bf V}\mu\right)}_{i,j} = m_{x\,i,j} \, m_{y\,i,j} \left(U_{i+1,j}-U_{i,j}+V_{i,j+1}-V_{i,j}\right)/d$$ (294)

where $d$ is the reference grid distance (m). The gradient of the divergence in the $x$-direction as

$${\frac{\partial \, \nabla\cdot{\left( {\bf V}\mu\right)}_{i,j}}{\... ...\right)}_{i,j} \,-\, \nabla\cdot{\left( {\bf V}\mu\right)}_{i-1,j} \right)/(2d)$$ (295)

where $\varphi$ represents the divergence gradient in the $x$-direction and is centered on the $u$ points of the C-grid. Next, the second order diffusion of the divergence gradient in the $x$-direction is computed as

$${\frac{\partial^2 \varphi}{\partial x^2}}\Big\vert_{i,j} = {\frac... ...ac{m_{x\,i-1,j}}{d}} \left( {\varphi}_{i,j} - {\varphi}_{i-1,j} \right) \right]$$ (296)

Finally, the fourth order damping coefficient (defined in Eq. 85) has a spatial dependence owing to the map factor and is expressed for the $u$ momentum equation as

$$\kappa_{d4\,u} = {\frac{k_{d}\, d^4 }{16 \, \Delta t \, m_x^4 }} \hskip1.in \left( 0 \leq k_d < 1 \right)$$ (297)

where $k_d$ must be less than 1 for numerical stability. The value is selected to optimize the balance between effectively filtering high frequency noise and avoiding too much damping of the divergence (see Section 2.2.1 for a detailed discussion). The same form is used in the $y$-direction for the damping in the $v$ momentum equation.

Finally, for the lat-lon grid, the map factors are defined as

$$m_y \,\,= \,\, \frac{d}{R_e \, \Delta\phi_r}$$ (298)

$$m_x \,\,= \,\, \frac{d}{R_e \, {\rm cos}\left(\phi_r\right) \Delta\lambda_r}$$ (299)

where $R_e$ represents the radius of the Earth, $\phi_r$ represents the latitude in radians, and $\Delta\phi_r$ and $\Delta\lambda_r$ are the constant latitude and longitude grid thicknesses, respectively, in radians. Thus, $m_x$ varies strongly as a function of latitude.