Mass drift

The pressure equation also includes a simple mass conservation relaxation (this is included in this section because the form is mathematically similar to that for the lateral boundaries). This simply acts as a bias correction (compared to the large scale mass within the limited area domain: it is not used when the global grid is activated): the spatially-averaged mass within in domain doesn't drift very far from that in the large scale forcing (model) in LAM mode (this is not used in the global domain application). The mass continuity equation including sources or sinks is defined by Eq. 28 which includes a source/sink sink term defined as $D_{\mu_d}$: the mass drift term effects described here are included therein.

A mass drift term can be defined as

$\displaystyle {\cal{D}}_{h,\pi_d} = K_p\,\left({\overline{\mu}}_{d,L} -
{\overline{\mu}}_d \right)$ (120)

Generally $K_p= 4/86400$ s$^{-1}$ (i.e. a relaxation time of 6 hours). The over-bars in Eq. 121 represent spatial averages over the entire domain. Thus this relaxation has no impact on the pressure gradients, but the average value of the mass stays close to that of the large scale model. Note that this term is turned off (i.e. $K_p=0$) in the global grid application. Integrating the mass drift term in the vertical gives

$\displaystyle {\frac{\partial \pi_{d,s}}{\partial t}} =
- \int_0^1 \nabla \cdot...
...\eta
\,+\,
K_p\,\left({\overline{\pi}}_{d,s,L} - {\overline{\pi}}_{d,s} \right)$ (121)

Note that again since the flux form equations are being used, for consistency, all of the other prognostic equations should include the relaxation term thus the mass diffusion/filter term is expressed (defined in Eq. 123) as

$\displaystyle D_{\mu_d} = {\cal{D}}_{h,\pi_d} =
K_p\,\frac{\partial B_\eta}{\partial\eta}
\left({\overline{\pi}}_{d,s,L} - {\overline{\pi}}_{d,s} \right)$ (122)

Note again that this mass drift term can be turned off by setting $K_p=0$. If the mass drift term is off, then $\varphi \,D_{\mu_d} = 0$ (vanishes) in the prognostic equations.