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Numerical quadrature for calculating the mass transfer coefficient of a lognormal mode¶

As explained in the section on mass transfer rates, the mass transfer coefficient for gas species \(L\) and aerosol mode \(i\) is proportional to \(f(K_{n,L}, \alpha_{L}) D_{p}\) integrated over the size distribution of mode \(i\). MAM applies the Gauss–Hermite quadrature to a manipulated form of the integrant, aiming at achieving a good trade-off betwen accuracy and computational cost. The algorithm is explained below.

The integrand¶

Let us use \(J\left(D_p\right)\) to denote the integrand and insert the expression of \(f(K_{n,L}, \alpha_{L})\). This gives (see section on mass transfer rates)

\[ \begin{eqnarray} J\left(D_p\right) &=& f(K_{n,L}, \alpha_{L})D_p = \dfrac{0.75 \alpha_{L} \left[1+\dfrac{2\lambda_L}{D_p}\right] D_p} {0.75 \alpha_{L} + \dfrac{2\lambda_L}{D_p} \left[1+\dfrac{2\lambda_L}{D_p}+0.283 \alpha_{L}\right]} \,. \end{eqnarray} \]

We can see

\[ \begin{eqnarray} \lim_{D_p\rightarrow 0} J\left(D_p\right) &=& \dfrac{0.75\alpha_L}{2\lambda_L}D_p^2 \\[2mm] \lim_{D_p\rightarrow \infty} J\left(D_p\right) &=& D_p \end{eqnarray} \]

This gives the motivation to rewrite the integrand as

\[ J(D_p) = \left[\dfrac{J(D_p)}{D_p^\beta}\right] D_p^\beta \]

with \(\beta \in [1,2]\) being a constant for a specific lognormal mode at a specific location and time. It is likely that the part in square brackets will vary relatively slowly with respect to \(D_p\). Since \(D_p^\beta\) can be integrated with high accurracy using the Gauss-Hermite quadrature, this rewritten form has the potential of helping to obtain an overall high accuracy.

Specifying \(\beta\)¶

For brevity, we define

\[ G(D_p) = \dfrac{J(D_p)}{D_p^\beta}\,. \]

Since

\[ \beta = \dfrac{d\ln J(D_p)}{d\ln D_p} - \dfrac{d\ln G(D_p)}{d\ln D_p}\,, \]

ignoring the last term gives

\[ \begin{eqnarray} \beta &\approx& \dfrac{d\ln\left[ f(K_{n,L}, \alpha_{L}) D_p \right]}{d\ln D_p}\\ &=& 1 + \frac{d\left[ \ln f(K_{n,L}, \alpha_{L}) \right]}{d\ln D_p} \\ &=& 1 + \frac{d\ln f(K_{n,L},\alpha_{L})}{d\ln K_{n,L}} \frac{d\ln K_{n,L}}{d\ln D_p} \\ &=& 1 - K_{n,L} \left[ \frac{1}{1+K_{n,L}} - \frac{1+2K_{n,L}+0.283\alpha_{L}} {0.75\alpha_{L}+K_{n,L} \left( 1+K_{n,L}+0.283\alpha_{L} \right)} \right] \,. \end{eqnarray} \]

The numerical quadrature described below uses \(\beta\) estimated at \(D_{p} = D_{gn,w,i}\).

Numerical quadrature¶

Using the shorthand

\[ \begin{align} x &= \ln D_{p} \\ x_{gn,i}&= \ln D_{gn,w,i} \\ s_{i}&= \ln \sigma_{g,i} \end{align} \]

we can write

\[ \begin{eqnarray} J(D_p)\, n_{i}(\ln D_p) &=& G(D_p)D_{p}^\beta\, n_{i}(x)\\ &=& \frac{N_{t,i} G(D_p)}{\sqrt{2\pi}s_{i}} D_{p}^\beta\exp \left [ -\frac{(x-x_{gn,i})^2}{2s_{i}^2} \right ] \\ &=& \frac{N_{t,i} G(D_p)}{\sqrt{2\pi}s_{i}} \exp \left [\beta x-\frac{(x-x_{gn,i})^2}{2s_{i}^2} \right ] \end{eqnarray} \]

Some further arithmetic maniputation gives

\[ \begin{eqnarray} && J(D_p)\, n_{i}(\ln D_p) \\ &=&\left[\frac{N_{t,i}}{\sqrt{2\pi}s_{i}} \exp \left ( \beta x_{gn,i} + \frac{\beta^2}{2}s_{i}^2 \right ) \right] G(D_p) \exp \left \{ -\frac{ \left [ x - \left (x_{gn,i}+\beta s_{i}^2 \right ) \right ]^2} {2s_{i}^2} \right \} \,, \end{eqnarray} \]

hence the mass transfer coefficient can be written as

\[ \begin{align} \label{kmt2} C_{L,i} &= 2 \pi \mathbb{D}_{g,L} \int_{-\infty}^{\infty} \frac{J(x)}{D_{p}^\beta} D_{p}^\beta n_{i}(x) dx \nonumber \\ &= \left[\frac{\sqrt{2 \pi} \mathbb{D}_{g,L} N_{t,i}} {s_{i}} \exp \left( \beta x_{gn,i} + \frac{\beta^2}{2} s_{i}^2 \right) \right] \int_{-\infty}^{\infty} G(D_p) \exp \left\{ -\frac{ \left[ x - \left( x_{gn,i} + \beta s_{i}^2 \right) \right]^2 }{2s_{i}^2} \right\} dx \,. \end{align} \]

To apply the Gauss-Hermite quadrature, we denote

\[ y = \frac{x - \left( x_{gn,i}+\beta s_{i}^2 \right)} {\sqrt2 s_{i}} \]

and write

\[ \begin{align} \label{kmt3} C_{L,i} &= \frac{\sqrt{2 \pi} \mathbb{D}_{g,L} N_{t,i}}{s_{i}} \exp \left( \beta x_{gn,i} + \frac{\beta^2}{2} s_{i}^2 \right ) \int_{-\infty}^{\infty} \exp^{-y^2} \frac{J(\sqrt2 s_{i} y+x_{gn,i}+\beta s_{i}^2)} {\exp \left [ \beta \left ( \sqrt2 s_{i} y+x_{gn,i}+ \beta s_{i}^2 \right ) \right ]} d(\sqrt2 s_{i} y+x_{gn,i}+\beta s_{i}^2) \nonumber \\ &= 2\sqrt{\pi} \mathbb{D}_{g,L} N_{t,i} \exp \left( \beta x_{gn,i} + \frac{\beta^2}{2} s_{i}^2 \right) \int_{-\infty}^{\infty} \exp^{-y^2} \frac{J(\sqrt2 s_{i} y+x_{gn,i}+\beta s_{i}^2)} {\exp \left[ \beta \left( \sqrt2 s_{i} y+x_{gn,i}+\beta s_{i}^2 \right) \right]} dy \,. \end{align} \]

MAM4 uses two quadrature points. The approximated mass transfer coefficient for species \(L\) and mode \(i\) is

\[ \begin{equation} C_{L,i} \approx 2\sqrt{\pi} \mathbb{D}_{g,L} N_{t,i} \exp \left ( \beta x_{gn,i} + \frac{\beta^2}{2} s_{i}^2 \right ) \sum_{k=1}^{2} \mathbb{W}_{k} \frac{J(\sqrt2 s_{i} \mathbb{R}_{k}+x_{gn,i}+ \beta s_{i}^2)}{\exp \left[ \beta \left( \sqrt2 s_{i} \mathbb{R}_{k}+x_{gn,i}+\beta s_{i}^2 \right) \right]} \,. \end{equation} \]

where \(\mathbb{R}_{k}\) is the \(k\)-th root of the Hermite polynomial (\(k = 1,2\)) and \(\mathbb{W}_{k}\) is its associated weight.