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Binary nucleation¶

The binary nucleation parameterization of Vehkamaki et al. (2002, 2013) provides an empirical expression of the nucleation rate \(J^*\) as a function of \(T\), \(RH\) and \(c_{n,H_2SO_4}\). Here, \(J^*\) is the number of stable new clusters formed per unit time and unit volume of air (cm\(^{-3}\) s\(^{-1}\)). The parameterization is valid for atmospheric conditions within the following ranges:

\(T \in [230.15, 305.15]\);
\(RH \in [0.0001, 1.0]\);
\(c_{n,H_2SO_4} \in [10^4, 10^{11}]\).

\(T\) and \(RH\) values that are outside the ranges of validity are clipped to the corresponding bounds.

When \(c_{n,H_2SO_4} \le 10^4\) or \(q_{v,H_2SO_4,avg} \le 4 \cdot 10^{-16}\) nucleation is assumed to not occur (i.e., \(J^* = 0\) and no further calculation is done.

The original paper also states that the parameterization is expected to be valid only in the range of \(J^* \in [10^{-7}, 10^{10}]\). This conditions is not checked in MAM.

Under atmospheric conditions within the range of validity, this parameterization will compute the following quantities in order:

Mole fraction of sulfric acid in a critical cluster¶

The mole fraction \(x^*\) is a function of T and RH.

\[ \begin{align} x^* &= 0.740997 - 0.00266379 T - 0.00349998 \ln c_{H_2SO_4} + 0.0000504022 T \ln c_{H_2SO_4} \nonumber \\ &+ 0.00201048 \ln RH - 0.000183289 T \ln RH + 0.00157407 (\ln RH)^2 \nonumber \\ &- 0.0000179059 T (\ln RH)^2 + 0.000184403 (\ln RH)^3 \nonumber \\ &- 1.50345 \cdot 10^{-6} T (\ln RH)^3 \,. \end{align} \]

Formation rate of critical clusters¶

\[ \begin{align} J^* &= \exp \Big[ a(T,x^*) + b(T,x^*) \ln RH + c(T,x^*) (\ln RH)^2 + d(T,x^*) (\ln RH)^3 \nonumber \\ &+ e(T,x^*) \ln c_{H_2SO_4} + f(T,x^*) (\ln RH) \ln c_{H_2SO_4} + g(T,x^*) (\ln RH)^2 \ln c_{H_2SO_4} \nonumber \\ &+ h(T,x^*) (\ln c_{H_2SO_4})^2 + i(T,x^*) (\ln RH) (\ln c_{H_2SO_4})^2 + j(T,x^*) (\ln c_{H_2SO_4})^3 \Big] \,, \label{vk2002_nucrate} \end{align} \]

where

\[ \begin{align} a(T,x^*) &= 0.14309 + 2.21956 T - 0.0273911 T^2 \nonumber \\ &+ 0.0000722811 T^3 + \frac{5.91822}{x^*} \,, \\ b(T,x^*) &= 0.117489 + 0.462532 T - 0.0118059 T^2 \nonumber \\ &+ 0.0000404196 T^3 + \frac{15.7963}{x^*} \,, \\ c(T,x^*) &= -0.215554 - 0.0810269 T + 0.00143581T^2 \nonumber \\ &- 4.7758 \cdot 10^{-6} T^3 - \frac{2.91297}{x^*} \,, \\ d(T,x^*) &= -3.58856 + 0.049508 T - 0.00021382 T^2 \nonumber \\ &+ 3.10801 \cdot 10^{-7} T^3 - \frac{0.0293333}{x^*} \,, \\ e(T,x^*) &= 1.14598 - 0.600796 T + 0.00864245 T^2 \nonumber \\ &- 0.0000228947 T^3 - \frac{8.44985}{x^*} \,, \\ f(T,x^*) &= 2.15855 + 0.0808121 T - 0.000407382 T^2 \nonumber \\ &- 4.01957 \cdot 10^{-7} T^3 + \frac{0.721326}{x^*} \,, \\ g(T,x^*) &= 1.6241 - 0.0160106 T + 0.0000377124 T^2 \nonumber \\ &+ 3.21794 \cdot 10^{-8} T^3 - \frac{0.0113255}{x^*} \,, \\ h(T,x^*) &= 9.71682 - 0.115048 T + 0.000157098 T^2 \nonumber \\ &+ 4.00914 \cdot 10^{-7} T^3 + \frac{0.71186}{x^*} \,, \\ i(T,x^*) &= -1.05611 + 0.00903378 T - 0.0000198417 T^2 \nonumber \\ &+ 2.46048 \cdot 10^{-8} T^3 - \frac{0.0579087}{x^*} \,, \\ j(T,x^*) &= -0.148712 + 0.00283508 T - 9.24619 \cdot 10^{-6} T^2 \nonumber \\ &+ 5.00427 \cdot 10^{-9} T^3 - \frac{0.0127081}{x^*} \,. \end{align} \]

Properties of critical clusters¶

Total number of molecules in a critical cluster, \(n_{tot}\)

\[ \begin{align} n_{tot} &= \exp \Big[ A(T,x^*) + B(T,x^*) \ln RH + C(T,x^*) (\ln RH)^2 + D(T,x^*) (\ln RH)^3 \nonumber \\ &+ E(T,x^*) \ln c_{H_2SO_4} + F(T,x^*) \ln RH \ln c_{H_2SO_4} + G(T,x^*) (\ln RH)^2 \ln c_{H_2SO_4} \nonumber \\ &+ H(T,x^*) (\ln c_{H_2SO_4})^2 + I(T,x^*) \ln RH (\ln c_{H_2SO_4})^2 + J(T,x^*) (\ln c_{H_2SO_4})^3 \Big] \,, \label{vk2002_ntot} \end{align} \]

where

\[ \begin{align} A(T,x^*) &= -0.00295413 - 0.0976834 T + 0.00102485 T^2 \nonumber \\ &- 2.18646 \cdot 10^{-6} T^3 - \frac{0.101717}{x^*} \,, \\ B(T,x^*) &= -0.00205064 - 0.00758504 T + 0.000192654 T^2 \nonumber \\ &- 6.7043 \cdot 10^{-7} T^3 - \frac{0.255774}{x^*} \,, \\ C(T,x^*) &= 0.00322308 + 0.000852637 T - 0.0000154757 T^2 \nonumber \\ &+ 5.66661 \cdot 10^{-8} T^3 + \frac{0.0338444}{x^*} \,, \\ D(T,x^*) &= 0.0474323 - 0.000625104 T + 2.65066 \cdot 10^{-6} T^2 \nonumber \\ &- 3.67471 \cdot 10^{-9} T^3 - \frac{0.000267251}{x^*} \,, \\ E(T,x^*) &= -0.0125211 + 0.00580655 T - 0.000101674 T^2 \nonumber \\ &+ 2.88195 \cdot 10^{-7} T^3 + \frac{0.0942243}{x^*} \,, \\ F(T,x^*) &= -0.038546 - 0.000672316 T + 2.60288 \cdot 10^{-6} T^2 \nonumber \\ &+ 1.19416 \cdot 10^{-8} T^3 - \frac{0.00851515}{x^*} \,, \\ G(T,x^*) &= -0.0183749 + 0.000172072 T - 3.71766 \cdot 10^{-7} T^2 \nonumber \\ &- 5.14875 \cdot 10^{-10} T^3 + \frac{0.00026866}{x^*} \,, \\ H(T,x^*) &= -0.0619974 + 0.000906958 T - 9.11728 \cdot 10^{-7} T^2 \nonumber \\ &- 5.36796 \cdot 10^{-9} T^3 - \frac{0.00774234}{x^*} \,, \\ I(T,x^*) &= 0.0121827 - 0.00010665 T + 2.5346 \cdot 10^{-7} T^2 \nonumber \\ &- 3.63519 \cdot 10^{-10} T^3 + \frac{0.000610065}{x^*} \,, \\ J(T,x^*) &= 0.000320184 - 0.0000174762 T + 6.06504 \cdot 10^{-8} T^2 \nonumber \\ &- 1.42177 \cdot 10^{-11} T^3 + \frac{0.000135751}{x^*} \,. \end{align} \]

Total number of H\(_2\)SO\(_4\) molecules in a critical cluster is, per definition,

\[ \begin{equation} n_{H_2SO_4} = n_{tot} \,x^* \,. \end{equation} \]

Radius of critical cluster \(r^*\) (unit: nm)

\[ r^* = \exp \left[ -1.6524245 + 0.42316402 x^* + 0.3346648 \ln n_{tot} \right] \,. \]