| | 19 | The growth of a particle by diffusion of water vapor, i.e., condensation and evaporation, is described by |
| | 20 | {{{ |
| | 21 | #!Latex |
| | 22 | \begin{align*} |
| | 23 | r \frac{\mathrm{d}r}{\mathrm{d}t} = \frac{f_\mathrm{v}}{F_\mathrm{D} + F_\mathrm{k}} |
| | 24 | \left(S - S_{\text{eq}} \right), |
| | 25 | \end{align*} |
| | 26 | }}} |
| | 27 | with the coefficients |
| | 28 | {{{ |
| | 29 | #!Latex |
| | 30 | \begin{align*} |
| | 31 | F_\mathrm{D} =\frac{R_\mathrm{v}T}{K_\mathrm{v}p_\text{v, |
| | 32 | s}(T)}\,\rho_{\mathrm{l},0}~\text{and~}\,F_\mathrm{k} = |
| | 33 | \left(\frac{L_\mathrm{V}}{R_\mathrm{v} T}-1\right) |
| | 34 | \frac{L_\mathrm{V}}{\lambda_\mathrm{h},T}\,\rho_{\mathrm{l},0}, |
| | 35 | \end{align*} |
| | 36 | }}} |
| | 37 | depending primarily on the diffusion of water vapor in air and heat conductivity of air, respectively. ''f'',,v,, is the ventilation factor, which accounts for the increased diffusion of water vapor, particularly the accelerated evaporation of large drops precipitating from a cloud ([#pruppacher1997 e.g., Pruppacher and Klett, 1997, Chap. 13.2.3]): |
| | 38 | {{{ |
| | 39 | #!Latex |
| | 40 | \begin{align*} |
| | 41 | & f_\mathrm{v} = \begin{cases} |
| | 42 | 1 + 0.09 \cdot {Re}_\mathrm{p} & \text{for~} \quad {Re}_\mathrm{p} < 2.5,\\ |
| | 43 | 0.78 + 0.28 \cdot\,{Re}_\mathrm{p}^{0.5} & \text{otherwise}. |
| | 44 | \end{cases} |
| | 45 | \end{align*} |
| | 46 | }}} |
| | 47 | Here, ''Re'',,p,, is particle Reynolds number. The relative water supersaturation ''S'' is computed from the LES values of ''θ'' and ''q'',,v,,, tri-linearly interpolated to the particle's position. The equilibrium saturation term ''S'',,eq,, considers the impact of surface tension as well as the physical and chemical properties of the solute aerosol on the equilibrium saturation of the droplet. In order to take into account these effects, the optional activation model for fully soluble aerosols must be switched on: |
| | 48 | {{{ |
| | 49 | #!Latex |
| | 50 | \begin{align*} |
| | 51 | S_{\text{eq}} = |
| | 52 | \begin{cases} |
| | 53 | 0 &\text{without activation},\\ |
| | 54 | A_{\text{eq}} r^{-1} - B_{\text{eq}} r^{-3} &\text{with activation}, |
| | 55 | \end{cases} |
| | 56 | \end{align*} |
| | 57 | }}} |
| | 58 | with coefficients for surface tension |
| | 59 | {{{ |
| | 60 | #!Latex |
| | 61 | \begin{align*} |
| | 62 | & A_{\text{eq}}=\frac{2 |
| | 63 | \vartheta}{\rho_{\mathrm{l},0}\,R_\mathrm{v}\,T}, |
| | 64 | \end{align*} |
| | 65 | }}} |
| | 66 | and physical and chemical properties |
| | 67 | {{{ |
| | 68 | #!Latex |
| | 69 | \begin{align*} |
| | 70 | B_{\text{eq}}=\frac{F_{\text{vH}}\,m_\mathrm{s}\,M_\mathrm{l}}{\frac{4}{3}\,\pi\,\rho_{\mathrm{l},0}\,M_\mathrm{s}}. |
| | 71 | \end{align*} |
| | 72 | }}} |
| | 73 | Here, ''ϑ'' is the temperature-dependent surface tension, and ''M'',,l,,'' = 18.01528'' g mol^-1^ the molecular mass of water. Depending on the simulation setup (e.g., continental or maritime conditions), the physical and chemical properties of the aerosol, its mass ''m'',,s,,, molecular mass ''M'',,s,,, and the van't Hoff factor ''F'',,vH,,, indicating the degree of the solute aerosol's dissociation, are prescribed. As discussed by [#hoffmann2015a Hoffmann et al. (2015)], the aerosol mass (or equivalently aerosol radius) can be specified by an additional particle feature allowing the initialization of aerosol mass distributions, i.e., varying aerosol masses among the simulated particle ensemble. |
| | 74 | |
| | 75 | In summary, diffusional growth is the major coupling between the LES and LCM model. The change of water vapor during one time step is |
| | 76 | considered in the prognostic equations for potential temperature (see Eq. three in Sect. [wiki:/doc/tec/gov governing equations]) and specific humidity (see Eq. four in Sect. [wiki:/doc/tec/gov governing equations]) by |
| | 77 | {{{ |
| | 78 | #!Latex |
| | 79 | \begin{align*} |
| | 80 | \Psi_{q_\mathrm{v}}=\frac{1}{\Delta |
| | 81 | t}\,\frac{\frac{4}{3}\,\pi \rho_{\mathrm{l},0}}{\rho_0\Delta |
| | 82 | V}\,\sum\limits_{n=1}^{N_\mathrm{p}} A_n (r_n^{\ast\,3}-r_n^3). |
| | 83 | \end{align*} |
| | 84 | }}} |
| | 85 | Here, ''r'',,n,, and ''r'',,n,,^∗^ are the radius of the ''n''th droplet before and after diffusional growth, respectively. Since the |
| | 86 | diffusional growth (see first Eq. in Sect. [/doc/tec/lcm#Diffusionalgrowth diffusional growth]) is a stiff differential equation, we use a 4th-order Rosenbrock-method ([#press1996 Press et al., 1996;] [#grabowski2011 Grabowski et al., 2011]), adapting its internal time step for both a computationally efficient and numerically accurate solution. |
| | 87 | |
