Lateral Boundary Conditions

The treatment of the lateral boundary conditions is critical for a LAM (limited Area Model), especially (obviously) as the forecast period becomes longer. In order to compute the forcing at the lateral boundaries, the large scale prognostic variables are prescribed using atmospheric model data from an external source (usually an operational global scale model for example) at set time increments (for forecast model data, by default this is every 6 hours although 3 hour data is generally available, but obviously at a greater data processing cost). Within the model, the values of $u$, $v$, $T$, $q$ and $\pi _s$ from the large scale model are linearly interpolated in time to each LAM model (i.e. ASP) large time step (where $T$ and $q$ tendencies are converted to $\theta_\rho$ and $r$ tendencies for final prognostic updates, respectively).

Application of the lateral boundary conditions is treated using two methods. First, the values of $u$, $v$, $T$, $q$ and $\pi _s$ from the large scale model are used to compute the advection (and horizontal diffusion if activated) terms along the lateral boundaries. For high order advection or diffusion schemes, this implies adding additional rows and columns (for example, the fifth order advection scheme requires three additional columns along the lateral east-west boundaries): here for simplicity, the large scale values right at the boundaries are used to fill in the values outside of the domain. Note that just applying the values of the large scale variables at the boundary for NWP applications does not generally produce very good results (the entry and exit of weather systems for example), although some rather complex treatments can be found in the literature which can give reasonable results.

In contrast, a very efficient and conceptually simple method used by most RCMs and NWP-LAMs consists in a relaxation to the large scale flow which gradually increases as one approaches the boundaries: this essentially amounts to a simple data assimilation scheme. There is a transition “sponge” or “relaxation” zone, which generally consists in 5 to 9 grid points from the boundary, which is used to “relax” or “couple” the prognostic variables within the limited area model domain with the large scale flow. So the relaxation scheme first proposed by Davies (1976) is included in a prognostic equation as as

$${\frac{d\varphi}{dt}} = F_\varphi - K_r\left(\varphi-\varphi_{L}\right) + \kappa_r \,\nabla^2\left(\varphi-\varphi_{L}\right)$$ (106)

The second term on the RHS of Eq. 107 corresponds to a “Newtonian relaxation” of the LAM variables, $\varphi$, to the large scale flow variables, $\varphi_{L}$, while the third term corresponds to a second order diffusive relaxation.