| | 5 | PALM offers an embedded bulk cloud microphysics representation that takes into account the liquid water specific humidity and warm (i.e., no ice) cloud-microphysical processes. Therefore, PALM solves the prognostic equations for the total water content |
| | 6 | {{{ |
| | 7 | #!Latex |
| | 8 | \begin{align*} |
| | 9 | & q = q_\mathrm{v} + q_\mathrm{l}, |
| | 10 | \end{align*} |
| | 11 | }}} |
| | 12 | instead of ''q'',,v,,, and for a linear approximation of the liquid water potential temperature ([#emanuel1994 e.g., Emanuel, 1994]) |
| | 13 | {{{ |
| | 14 | #!Latex |
| | 15 | \begin{align*} |
| | 16 | \theta_\mathrm{l} = \theta - \frac{L_\mathrm{V}}{c_p \Pi} |
| | 17 | q_\mathrm{l}\,, |
| | 18 | \end{align*} |
| | 19 | }}} |
| | 20 | instead of ''θ'' as described in Sect. [wiki:doc/tec/gov governing equations]. Since ''q'' and ''θ'',,l,, are conserved quantities for wet adiabatic processes, condensation/evaporation is not considered for these variables. |
| | 21 | |
| | 22 | Liquid phase microphysics are parametrized following the two-moment scheme of [#seifert2001 Seifert and Beheng (2001],[#seifert2006 2006)], which is based on the separation of the droplet spectrum into droplets with radii < 40 μm (cloud droplets) and droplets with radii ≥ 40 μm (rain droplets). The model predicts the first two moments of these partial droplet spectra, namely cloud and rain |
| | 23 | droplet number concentration (''N'',,c,, and ''N'',,r,,, respectively) as well as cloud and rain water specific humidity |
| | 24 | (''q'',,c,, and ''q'',,r,,, respectively). Consequently, ''q'',,l,, is the sum of both ''q'',,c,, and ''q'',,r,,. The moments' corresponding microphysical tendencies are derived by assuming the partial droplet spectra to follow a gamma distribution that can be described by the predicted quantities and empirical relationships for the distribution's slope and shape parameters. For a detailed derivation of these terms, see [#seifert2001 Seifert and Beheng (2001],[#seifert2006 2006)]. |
| | 25 | |
| | 26 | We employ the computational efficient implementation of this scheme as used in the UCLA-LES ([#savic2008 Savic-Jovcic and Stevens, 2008]) and DALES ([#heus2010 Heus et al., 2010]) models. We thus solve only two additional prognostic equations for ''N'',,r,, and ''q'',,r,,: |
| | 27 | {{{ |
| | 28 | #!Latex |
| | 29 | \begin{align*} |
| | 30 | \frac{\partial N_\mathrm{r}}{\partial t} = - u_j \frac{\partial N_\mathrm{r}}{\partial x_j} - \frac{\partial}{\partial x_j}\left(\overline{u_j^{\prime\prime}N_\mathrm{r}^{\prime\prime}}\right) + \Psi_{N_\mathrm{r}},\\ |
| | 31 | \frac{\partial q_\mathrm{r}}{\partial t} = - u_j |
| | 32 | \frac{\partial q_\mathrm{r}}{\partial x_j} - \frac{\partial}{\partial |
| | 33 | x_j}\left(\overline{u_j^{\prime\prime}q_\mathrm{r}^{\prime\prime}}\right) |
| | 34 | + \Psi_{q_\mathrm{r}}, |
| | 35 | \end{align*} |
| | 36 | }}} |
| | 37 | with the sink/source terms ''Ψ'',,Nr,, and ''Ψ'',,qr,,, and the SGS fluxes |
| | 38 | {{{ |
| | 39 | #!Latex |
| | 40 | \begin{align*} |
| | 41 | & \overline{u_j^{\prime\prime}N_\mathrm{r}^{\prime\prime}} = -K_\mathrm{h} \:\frac{\partial q_\mathrm{r}} {\partial x_{i}}\,\\ |
| | 42 | & \overline{u_j^{\prime\prime}q_\mathrm{r}^{\prime\prime}} = |
| | 43 | -K_\mathrm{h} \:\frac{\partial N_\mathrm{r}} {\partial |
| | 44 | x_{i}}\, |
| | 45 | \end{align*} |
| | 46 | }}} |
| | 47 | with ''N'',,c,, and ''q'',,c,, being a fixed parameter and a diagnostic quantity, respectively. |
| | 48 | |
| | 49 | In the next subsections we will describe the diagnostic determination of ''q'',,c,,. From Sect. [wiki:doc/tec/microphysics#autoconversion autoconversion] on, the microphysical processes considered in the sink/source terms of ''θ'',,l,,, ''q'', ''N'',,r,, and ''q'',,r,,, |
| | 50 | {{{ |
| | 51 | #!Latex |
| | 52 | \begin{align*} |
| | 53 | & \Psi_{\theta_\mathrm{l}} = - \frac{L_\mathrm{v}}{c_p \Pi} \varphi_q,\\ |
| | 54 | & \Psi_{q} = \left.\frac{\partial q}{\partial t} \right|_\text{sed, c} + \left.\frac{\partial q}{\partial t} \right|_\text{sed, r},\\ |
| | 55 | & \Psi_{N_\mathrm{r}} = \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_{\text{auto}}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{slf/brk}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_{\text{evap}}+ \left.\frac{\partial N_\mathrm{r}}{\partial t} \right|_\text{sed, r},\\ |
| | 56 | & \Psi_{q_\mathrm{r}} = \left.\frac{\partial |
| | 57 | q_\mathrm{r}}{\partial t} \right|_{\text{auto}} + |
| | 58 | \left.\frac{\partial q_\mathrm{r}}{\partial t} |
| | 59 | \right|_{\text{accr}}+ \left.\frac{\partial q_\mathrm{r}}{\partial |
| | 60 | t} \right|_{\text{evap}}+ \left.\frac{\partial |
| | 61 | q_\mathrm{r}}{\partial t} \right|_\text{sed, r}, |
| | 62 | \end{align*} |
| | 63 | }}} |
| | 64 | are used in the formulations of [#seifert2006 Seifert and Beheng (2006)] unless explicitly specified. Section [wiki:doc/tec/microphysics#turbulenceclosure turbulence closure] gives an overview of the necessary changes for the turbulence closure |
| | 65 | (cf. Sect.~\ref{sec:closure}) using ''q'' and ''θ'',,l,, instead of ''q'',,v,, and $θ$, respectively. |
| | 66 | |
| | 67 | == Diffusional growth of cloud water == |
| | 68 | |
| | 69 | == Autoconversion == |
| | 70 | |
| | 71 | == Accretion == |
| | 72 | |
| | 73 | == Self-collection and breakup == |
| | 74 | |
| | 75 | == Evaporation of rainwater == |
| | 76 | |
| | 77 | == Sedimentation of rainwater |
| | 78 | |
| | 79 | == Turbulence closure == |
| | 80 | |
| | 81 | == Recent applications == |
