Governing Equations

TOSCA solves the incompressible Navier–Stokes equations for a flow with Coriolis forces augmented by the Boussinesq approximation for buoyancy. The buoyancy term is evaluated through the modified density \(\rho_k\), derived from a transport equation for the potential temperature. In Cartesian coordinates the equations read

(1)\[\frac{\partial u_i}{\partial x_i} = 0\]

(2)\[\begin{split}\frac{\partial u_i}{\partial t} + \frac{\partial}{\partial x_j}\left(u_j u_i\right) &= -\frac{1}{\rho_0}\frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left[\nu_\text{eff}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\right] \\ & - \frac{1}{\rho_0}\frac{\partial p_{\infty}}{\partial x_i} + \frac{\rho_k}{\rho_0}g_i - 2\epsilon_{ijk}\Omega_j u_k + f_i + s^v_i + s^h_i\end{split}\]

(3)\[\frac{\partial\theta}{\partial t} + \frac{\partial}{\partial x_j}\left(u_j\theta\right) = \frac{\partial}{\partial x_j}\left(\kappa_\text{eff}\frac{\partial \theta}{\partial x_j}\right)\]

where \(u_i\) is the Cartesian velocity, \(p/\rho_0\) is the kinematic pressure, and \(\theta\) is the potential temperature defined as \(\theta = T(p_0/p)^{R/c_p}\) (with \(T\) the absolute temperature, \(R\) the gas specific constant, \(c_p\) the specific heat at constant pressure, and \(p_0\) a reference pressure). The vector \(g_i\) is the gravitational acceleration, and \(\Omega_j\) is the planetary rotation rate vector, defined as \(\omega\cos\phi\,\hat{y}+\omega\sin\phi\,\hat{z}\) at latitude \(\phi\) in a local frame with \(\hat{z}\) pointing against gravity, \(\hat{x}\) tangent to Earth’s parallels, and \(\hat{y}\) completing the right-handed triad. Source terms \(f_i\), \(s^v_i\), and \(s^h_i\) are body forces due to wind turbines, and vertical and horizontal damping regions, respectively.

The modified density \(\rho_k\) is

(4)\[\frac{\rho_k}{\rho_0} = 1 - \left(\frac{\theta-\theta_0}{\theta_0}\right)\]

where \(\theta_0\) is a reference potential temperature chosen as the ground temperature. The effective viscosity \(\nu_\text{eff} = \nu + \nu_t\) and effective thermal diffusivity \(\kappa_\text{eff} = \kappa + \kappa_t\) include both molecular and sub-grid scale (SGS) contributions; the SGS quantities \(\nu_t\) and \(\kappa_t\) are described in Sub-grid Scale Models. The Prandtl number \(Pr = \nu/\kappa\) is set to 0.7 by default. The term \(-\rho_0^{-1}\partial p_\infty/\partial x_i\) is a prescribed horizontal pressure gradient that drives the flow and represents the velocity controller described in Flow Controllers and Mesoscale Coupling.

The discretized form of the governing equations solved by TOSCA — including the curvilinear coordinate transformation, contravariant flux variables, staggered grid arrangement, and fractional-step projection — is described in Numerical Method.