Relaxation formulation used in ASP

P. Marbaix and van Ypersele (2003) noted that one possible advantage of using a diffusive form relaxation with the Newtonian form is to reduce some noise within the sponge zone (notably in terms of sea level pressure), but they showed that the diffusive coefficient must be nearly an order of magnitude lower than the Newtonian form, otherwise the combined method performs worse than the Newtonian method alone (in particular, in terms of wave reflection). Note that models generally use either the Newtonian nudging (or Raleigh damping) or diffusion (or artificial viscosity) relaxation terms, or a combination of both. As an example, Skamarock et al. (2005) used ${K_r}^\ast=0.20$ and ${\kappa_r}^\ast=0.04$ in WRF (which is consistent with the idea that one term is significantly lower than the other for the best results) with a linear function for $f_R$ (i.e. a form akin to Eq. 110 with $a_R=1$). Tests using ASP have shown that the inclusion of a comparatively weak diffusion relaxation term has a nearly negligible impact, and therefore it's inclusion does not seem to be justified considering both theory and the computational expense: so currently in ASP it is assumed that

$$K_r \gg \kappa_r$$ (116)

For a flux form models such as ASP, the relaxation term is included in the prognostic equations as

$${\frac{\left( \partial\mu_d\,\varphi \right) }{\partial t}} + \na... ...{F}}_{u,v,w} \,+\,\mu_d \, {\dot{S}}_\varphi \,+\,\mu_d \,{\cal{D}}_{h,\varphi}$$ (117)

where ${\cal{F}}_{u,v,w}$ is used to represent sources/sinks from dynamics (pressure gradient terms and curvature terms), and

$${\cal{D}}_{h,\varphi} = K_r\,\mu_d \left(\varphi_L - \varphi\right)$$ (118)

so that the diffusion lateral boundary relaxation term is currently not included in the lateral relaxation term.