ner_tags: O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O
tokens: and C + 1, with If we assume m C ≪ m j , then the coagulation between any grain of bin j and a grain of bin i with a mass lower than m C , − − m j produces a grain in bin C . The fraction υ of grains transferred to bin C is therefore given by However, if m C , − − m j m i , − , then the product of coagulation is too massive for bin C and υ = 0. Conversely, if m C , − − m j > m i , + then υ = 1. The integrals of Eq. ( B.21 ) need to be calculated analytically due to the computational cost of performing a numerical integration for each pair of bins at each time step. In particular, if the distribution is steep, one end of the bin may not be captured accurately, yielding large errors. For coagulation kernels that are in the form , the analytical calculation is tedious but straightforward. Fortunately, this is the case for the kernel used in this paper (see Eqs. ( 9 ) and ( 14 )), The analytical expression of υ is a function of m β , which can lead to inaccuracies due to numerical truncation errors where the distribution is steep (| β | is large). Fortunately, for high values of | β |, some terms can be simplified. This is not the case for specific lower values (if we have to integrate m −1 into a log function instead of a power-law), however, so that the simplification cannot be generalized to all β . We also have to ensure that the simplification can be applied to both the top and bottom of the fraction. We detail here the procedure used in Ishinisan to minimize numerical inaccuracy. We define the following functions: and r ( m 1, − , m 1, + , m 2, − , m 2, + , β 1 , β 2 ), so that We therefore have for the turbulent kernel where κ k = m k , − if s k −5, and κ k = 1 else, for k = 1, 2, to match the criteria of function q ( B.23 ). When the value of υ is known, the variations rates of mass density in the four involved bins are After summing all the variation rates, we chose the time-step Δ t so that the relative variation of mass in each bin does not exceed a number ε prec , typically smaller than one, to ensure numerical accuracy, The densities are then updated and the new slopes, average size, and average mass of bins are calculated according to the calculations detailed in Appendix B.1 . B.3. Tests We tested the coagulation algorithm using two simple kernels, namely the constant and additive kernels, Menon Pego (2004) showed that these two kernels possess self-similar solutions, The mass density in each bin i is given by where erf and erfc are the error function and the complementary error function. In each case, the two formulae are mathematically equivalent but are written in this form to avoid numerical truncation errors. The fraction υ has a similar form as the turbulent kernel presented in the previous section (Eq. ( B.24 )). The function q is the same, and function r for the constant kernel reads and for the additive kernel We used Ishinisan to reproduce these self-similar solutions. We set up 30 bins between m min = 4 × 10 −6 and m max