| | 90 | Collision and coalescence are computed using a statistical approach that allows the collision of all droplets that are currently located |
| | 91 | in the same LES grid volume. For this purpose, two quantities are predicted: the weighting factor, i.e., the number of droplets |
| | 92 | represented by a super-droplet, and the bulk mass of all droplets represented by a super droplet, ''m'',,n,,'' = A'',,n,,'' (4/3) π ρ'',,l,, ''r'',,n,,^3^. For the collision of a super-droplet with a super-droplet smaller in radius, we assume that the larger droplets merges with a certain amount of smaller droplets. Thereby, the weighting factor of the larger super-droplet is kept constant, while bulk mass and consequently radius increase (see Fig. 9a). On the other hand, the weighting factor and bulk mass of the smaller super-droplet decrease according to the amount of droplets lost to the larger super-droplet, keeping the smaller super-droplet's radius constant. As described in [#riechelmann2015 Riechelmann et al. (2015)], we allow the droplets represented by a single super-droplet to collide among each other. These internal collisions only decrease the weighting factor of the super-droplet but not the bulk mass. Consequently, internal collisions increase the super-droplet's radius (see Fig. 9b). The collision kernel ''K'', which describes the collision probability of two droplets, can either be a purely gravitational one ([#hall1980 Hall, 1980]) or including turbulence effects ([#ayala2008 Ayala et al., 2008]). |
| | 93 | |
| | 94 | We arrange the droplets by radius such that ''r'',,n,,'' ≤ r'',,n+1,,. The weighting factor after one collision/coalescence time step then reads as |
| | 95 | {{{ |
| | 96 | #!Latex |
| | 97 | \begin{align*} |
| | 98 | A_n^{\ast}=A_n - K(r_n,\,r_n)\,\frac{1}{2}\,\frac{A_n (A_n - |
| | 99 | 1)}{\Delta V}\,\Delta t - \sum \limits_{m=n+1}^{N_\mathrm{p}} |
| | 100 | K(r_m,\,r_n) \frac{A_n\,A_m}{\Delta V}\,\Delta t. |
| | 101 | \end{align*} |
| | 102 | }}} |
| | 103 | The asterisk denotes a quantity after one collision/coalescence time step. On the right-hand side, we consider the initial weighting factor |
| | 104 | (first term), the loss of droplets due to internal collisions (second term), and the loss of droplets due to collision with all larger droplets (third term). Note that collision with smaller droplets does not change the weighting factor of the larger droplet. |
| | 105 | |
| | 106 | Since the mass of all droplets represented by a~single super-droplet is not a~useful quantity, we predict the volume averaged radius of all |
| | 107 | droplets represented by a super-droplet directly: |
| | 108 | {{{ |
| | 109 | #!Latex |
| | 110 | \begin{align*} |
| | 111 | r_n^{\ast}&~= |
| | 112 | \left(\frac{m_n^{\ast}}{\frac{4}{3}\pi\rho_{\mathrm{l},0}A_n^{\ast}}\right)^{\frac{1}{3}}\\ |
| | 113 | &~=\left[\left(r_n^3 + \sum \limits_{m=1}^{n-1} K(r_{n},\,r_{m}) \frac{A_m}{\Delta V}\,r_m^3\,\Delta t |
| | 114 | -\sum \limits_{m=n+1}^{N_\mathrm{p}} K(r_{m},\,r_{n}) |
| | 115 | \frac{A_m}{\Delta V}\,r_n^3\,\Delta t \right)\right. \nonumber\\ |
| | 116 | &~~\left. \cdot \left(1 - K(r_n,\,r_n)\,\frac{1}{2}\,\frac{A_n - 1}{\Delta V}\,\Delta t |
| | 117 | - \sum \limits_{m=n+1}^{N_\mathrm{p}} K(r_m,\,r_n) \frac{A_m}{\Delta V}\,\Delta t \right)^{-1} \right]^{\frac{1}{3}}. |
| | 118 | \end{align*} |
| | 119 | }}} |
| | 120 | On the right-hand side, the nominator (first pair of round brackets) contains the initial mass (first term), the gain of mass due to |
| | 121 | collisions with all smaller droplets (second term), and the loss of mass due to collisions with all larger droplets (third term). The |
| | 122 | denominator (second pair of round brackets) is identical to the Eq. for ''A'',,n,,^∗^ above divided by ''A'',,n,,. |
| | 123 | |
| 106 | | * [=#grabowski2011] ''' Grabowski WM, Andrejczuk M, Wang L-P.''' 2011. Droplet growth in a bin warm-rain scheme with Twomey CCN activation. Atmos. Res. 99: 290–301. |
| | 140 | * [=#grabowski2011] '''Grabowski WM, Andrejczuk M, Wang L-P.''' 2011. Droplet growth in a bin warm-rain scheme with Twomey CCN activation. Atmos. Res. 99: 290–301. |
| | 141 | |
| | 142 | * [=#riechelmann2015] '''Riechelmann T, Wacker U, Beheng KD, Etling D, Raasch S.''' 2015. Influence of turbulence on the drip growth in warm clouds, part II: Sensitivity studies with a spectral bin microphysics and a lagrangian cloud model. Meteorol. submitted. |
| | 143 | |
| | 144 | * [=#hall1980] '''Hall WD.''' 1980. A detailed microphysical model within a two-dimensional dynamic framework: model description and preliminary results. J. Atmos. Sci. 37: 2486–2507. |
| | 145 | |
| | 146 | * [=#ayala2008] '''Ayala O, Rosa B, Wang L-P.''' 2008. Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New J. Phys. 10: 075016. [http://dx.doi.org/10.1088/1367-2630/10/7/075016 doi] |
