| | 116 | In the following Sects. [wiki:doc/tec/microphysics#Autoconversion Autoconversion] - [wiki:doc/tec/microphysics#Self-collectionandbreakup Self-collection and breakup] we describe collision and coalescence processes by applying the stochastic collection equation ([#pruppacher1997 e.g., Pruppacher and Klett, 1997, Chap. 15.3]) in the framework of the described two-moment scheme. As two species (cloud and rain droplets, hereafter also denoted as c and r, respectively) are considered only, there are three possible interactions affecting the rain quantities: autoconversion, accretion, and selfcollection. Autoconversion summarizes all merging of cloud droplets resulting in rain drops |
| | 117 | (c + c → r). Accretion describes the growth of rain drops by the collection of cloud droplets (r + c → r). Selfcollection denotes the merging of rain drops (r + r → r). |
| | 118 | |
| | 119 | The local temporal change of ''q'',,r,, due to autoconversion is |
| | 120 | {{{ |
| | 121 | #!Latex |
| | 122 | \begin{align*} |
| | 123 | & \left.\frac{\partial q_\mathrm{r}}{\partial t} |
| | 124 | \right|_{\text{auto}}=\frac{K_{\text{auto}}}{20\,m_{\text{sep}}}\frac{(\mu_\mathrm{c} +2) |
| | 125 | (\mu_\mathrm{c} +4)}{(\mu_\mathrm{c} + 1)^2} q_\mathrm{c}^2 |
| | 126 | m_\mathrm{c}^2 \cdot \left[1+ |
| | 127 | \frac{\Phi_{\text{auto}}(\tau_\mathrm{c})}{(1-\tau_\mathrm{c})^2}\right] |
| | 128 | \rho_0. |
| | 129 | \end{align*} |
| | 130 | }}} |
| | 131 | Assuming that all new rain drops have a radius of 40 μm corresponding to the separation mass ''m'',,sep,, ''= 2.6 x 10^-10^'' kg, the local temporal change of ''N'',,r,, is |
| | 132 | {{{ |
| | 133 | #!Latex |
| | 134 | \begin{align*} |
| | 135 | & \left.\frac{\partial N_\mathrm{r}}{\partial t} |
| | 136 | \right|_{\text{auto}}= \rho \left.\frac{\partial |
| | 137 | q_\mathrm{r}}{\partial t} \right|_{\text{auto}} |
| | 138 | \frac{1}{m_{\text{sep}}}. |
| | 139 | \end{align*} |
| | 140 | }}} |
| | 141 | Here, ''K'',,auto,, ''= 9.44 x 10^9^'' m^3^ kg^-2^ s^-1^ is the autoconversion kernel, ''μ'',,c,,'' = 1'' is the shape parameter of the cloud droplet gamma distribution and |
| | 142 | ''m'',,c,, ''= ρ q'',,c,, ''/ N'',,c,, is the mean mass of cloud droplets. ''τ'',,c,, ''= 1 - q'',,c,,'' / (q'',,c,,'' + q'',,r,,) is a dimensionless timescale steering the autoconversion similarity function |
| | 143 | {{{ |
| | 144 | #!Latex |
| | 145 | \begin{align*} |
| | 146 | & |
| | 147 | \Phi_{\text{auto}}=600\,\cdot\,\tau_\mathrm{c}^{0.68}\,\left(1-\tau_\mathrm{c}^{0.68}\right)^3. |
| | 148 | \end{align*} |
| | 149 | }}} |
| | 150 | The increase of the autoconversion rate due to turbulence can be considered optionally by an increased autoconversion kernel depending on the local kinetic energy dissipation rate after [#seifert2010 Seifert et al. (2010)]. |
| | 151 | |
| | 154 | The increase of ''q'',,r,, by accretion is given by: |
| | 155 | {{{ |
| | 156 | #!Latex |
| | 157 | \begin{align*} |
| | 158 | & \left.\frac{\partial q_\mathrm{r}}{\partial t} |
| | 159 | \right|_{\text{accr}}= |
| | 160 | K_{\text{accr}}\,q_\mathrm{c}\,q_\mathrm{r}\,\Phi_{\text{accr}}(\tau_\mathrm{c}) |
| | 161 | \left(\rho_0\,\rho \right)^{\frac{1}{2}}, |
| | 162 | \end{align*} |
| | 163 | }}} |
| | 164 | with the accretion kernel ''K'',,accr,,'' = 4.33'' m^3^ kg^-1^ s^-1^ and the similarity function |
| | 165 | {{{ |
| | 166 | #!Latex |
| | 167 | \begin{align*} |
| | 168 | & \Phi_{\text{accr}}=\left(\frac{\tau_\mathrm{c}}{\tau_\mathrm{c} + |
| | 169 | 5 \times 10^{-5}}\right)^4. |
| | 170 | \end{align*} |
| | 171 | }}} |
| | 172 | Turbulence effects on the accretion rate can be considered after using the kernel after [#seifert2010 Seifert et al. (2010)]. |
| | 173 | |
| | 175 | |
| | 176 | Selfcollection and breakup describe merging and splitting of rain drops, respectively, which affect the rain water drop number concentration only. Their combined impact is parametrized as |
| | 177 | {{{ |
| | 178 | #!Latex |
| | 179 | \begin{align*} |
| | 180 | & \left.\frac{\partial N_\mathrm{r}}{\partial t} |
| | 181 | \right|_\text{slf/brk}= |
| | 182 | -(\Phi_{\text{break}}(r)+1)\,\left.\frac{\partial |
| | 183 | N_\mathrm{r}}{\partial t} \right|_{\text{self}}, |
| | 184 | \end{align*} |
| | 185 | }}} |
| | 186 | with the breakup function |
| | 187 | {{{ |
| | 188 | #!Latex |
| | 189 | \begin{align*} |
| | 190 | & \Phi_{\text{break}} = |
| | 191 | \begin{cases} 0 & \text{for~} \widetilde{r_\mathrm{r}} < 0.15 \times 10^{-3}\,\mathrm{m},\\ |
| | 192 | K_{\text{break}} (\widetilde{r_\mathrm{r}}-r_{\text{eq}}) & |
| | 193 | \text{otherwise}, |
| | 194 | \end{cases} |
| | 195 | \end{align*} |
| | 196 | }}} |
| | 197 | depending on the volume averaged rain drop radius |
| | 198 | {{{ |
| | 199 | #!Latex |
| | 200 | \begin{align*} |
| | 201 | & |
| | 202 | \widetilde{r_\mathrm{r}}=\left(\frac{\rho\,q_\mathrm{r}}{\frac{4}{3}\,\pi\,\rho_{\mathrm{l},0}\,N_\mathrm{r}} |
| | 203 | \right)^{\frac{1}{3}}, |
| | 204 | \end{align*} |
| | 205 | }}} |
| | 206 | the equilibrium radius ''r'',,eq,, ''= 550 x 10^-6^'' m and the breakup kernel ''K'',,break,, ''= 2000'' m^-1^. The local temporal change of ''N'',,r,, due to selfcollection is |
| | 207 | {{{ |
| | 208 | #!Latex |
| | 209 | \begin{align*} |
| | 210 | & \left.\frac{\partial N_\mathrm{r}}{\partial t} |
| | 211 | \right|_{\text{self}}= K_{\text{self}}\,N_\mathrm{r}\,q_\mathrm{r} |
| | 212 | \left(\rho_0\,\rho \right)^{\frac{1}{2}}, |
| | 213 | \end{align*} |
| | 214 | }}} |
| | 215 | with the selfcollection kernel ''K'',,self,, ''= 7.12'' m^3^ kg^-1^ s^-1^. |
