| | 293 | The sedimentation of rain water is implemented following [#stevens2008 Stevens and Seifert (2008)]. The sedimentation velocities are based on an empirical relation for the terminal fall velocity after [#rogers1993 Rogers et al. (1993)]. They are given by |
| | 294 | {{{ |
| | 295 | #!Latex |
| | 296 | \begin{align*} |
| | 297 | & w_{N_\mathrm{r}} = \left(9.65\,\text{m\,s}^{-1} - |
| | 298 | 9.8\,\text{m\,s}^{-1} \left(1+ |
| | 299 | 600\,\text{m}/\lambda_\mathrm{r}\right)^{-(\mu_\mathrm{r} + 1)} |
| | 300 | \right), |
| | 301 | \end{align*} |
| | 302 | }}} |
| | 303 | and |
| | 304 | {{{ |
| | 305 | #!Latex |
| | 306 | \begin{align*} |
| | 307 | & w_{q_\mathrm{r}} = \left(9.65\,\text{m\,s}^{-1} - |
| | 308 | 9.8\,\text{m\,s}^{-1} \left(1+ |
| | 309 | 600\,\text{m}/\lambda_\mathrm{r}\right)^{-(\mu_\mathrm{r} + 4)} |
| | 310 | \right). |
| | 311 | \end{align*} |
| | 312 | }}} |
| | 313 | The resulting sedimentation fluxes ''F'',,Nr,, and ''F'',,qr,, are calculated using a semi-Lagrangian |
| | 314 | scheme and a slope limiter (see [#stevens2008 Stevens and Seifert, 2008], their Appendix A). The resulting tendencies read as |
| | 315 | {{{ |
| | 316 | #!Latex |
| | 317 | \begin{align*} |
| | 318 | & |
| | 319 | \left.\frac{\partial q_\mathrm{r}}{\partial t} \right|_\text{sed, r}= -\frac{\partial F_{q_\mathrm{r}}}{\partial z},~ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{sed, r}= -\frac{\partial F_{N_\mathrm{r}}}{\partial z},\;\text{and}~ \left.\frac{\partial q}{\partial t} \right|_\text{sed, r}= |
| | 320 | \left.\frac{\partial q_\mathrm{r}}{\partial t} \right|_\text{sed, r}. |
| | 321 | \end{align*} |
| | 322 | }}} |
| | 323 | |
