| | 72 | The diagnostic estimation of ''q'',,c,, is based on the assumption that water supersaturations are immediately removed by the diffusional growth of cloud droplets only. This can be justified since the bulk surface area of cloud droplets exceeds that of rain drops considerably ([#stevens2008 Stevens and Seifert, 2008]). Following this saturation adjustment approach, ''q'',,c,, is obtained by |
| | 73 | {{{ |
| | 74 | #!Latex |
| | 75 | \begin{align*} |
| | 76 | & q_\mathrm{c}=\max{\left(0, q - q_\mathrm{r} - q_\mathrm{s} |
| | 77 | \right)}, |
| | 78 | \end{align*} |
| | 79 | }}} |
| | 80 | where ''q'',,s,, is the saturation specific humidity. Because ''q'',,s,, is a function of ''T'' (not predicted), ''q'',,s,, is computed from the liquid water temperature ''T'',,l,, = ''Π θ,,l,, in a first step: |
| | 81 | {{{ |
| | 82 | #!Latex |
| | 83 | \begin{align*} |
| | 84 | q_\mathrm{s}(T_\mathrm{l}) = \frac{R_\mathrm{d}}{R_\mathrm{v}} |
| | 85 | \frac{p_\text{v, s}(T_\mathrm{l})}{p-\left(1-R_\mathrm{d}/R_\mathrm{v}\right)\,p_\text{v, s}(T_\mathrm{l})}, |
| | 86 | \end{align*} |
| | 87 | }}} |
| | 88 | using an empirical relationship for the saturation water vapor pressure ''p'',,v,s,, ([#bougeault1981 Bougeault, 1981]): |
| | 89 | {{{ |
| | 90 | #!Latex |
| | 91 | \begin{align*} |
| | 92 | & p_\text{v, s}(T_\mathrm{l}) = 610.78 \text{Pa} \cdot |
| | 93 | \exp{\left(17.269\,\frac{T_\mathrm{l}-273.16\,\text{K}}{T_\mathrm{l}-35.86\,\text{K}} |
| | 94 | \right)}. |
| | 95 | \end{align*} |
| | 96 | }}} |
| | 97 | ''q'',,s,,(''T'') is subsequently calculated from a 1st-order Taylor series expansion of ''q'',,s,, at ''T'',,l,, ([#sommeria1977 Sommeria and Deardorff, 1977]): |
| | 98 | {{{ |
| | 99 | #!Latex |
| | 100 | \begin{align*} |
| | 101 | & q_\mathrm{s}(T)=q_\mathrm{s}(T_\mathrm{l})\frac{1+\beta\,q}{1+ |
| | 102 | \beta\,q_\mathrm{s}(T_\mathrm{l})}, |
| | 103 | \end{align*} |
| | 104 | }}} |
| | 105 | with |
| | 106 | {{{ |
| | 107 | #!Latex |
| | 108 | \begin{align*} |
| | 109 | & \beta = \frac{L_\mathrm{v}^2}{R_\mathrm{v} c_p |
| | 110 | T_\mathrm{l}^2}. |
| | 111 | \end{align*} |
| | 112 | }}} |
| | 113 | |
