2.5 Gas phase chemistry

 

2.5.1 Numerical Solver

Metphomod includes a module for the simulation of homogeneous reactions of gaseous constituents. The chemical reaction equations can be defined in a text file, and are directly read by the included chemical interpreter. Chemical reaction schemes therefore can be changed rapidly by creating or editing this text file. The module uses the method of Gong and Cho (1993) for solving the chemistry system. The method distinguishes between slow and rapid species. The slow ones are integrated, using an explicit integration step, the fast ones are integrated by a fully implicit integration procedure. The solving procedure is as follows:

1.

Calculate the concentration of the slow species at time step t + 1 with an explicit time step, 

$$\bf$$S ^* = $$\bf$$S ^t + $$\bf$$f($$\bf$$S ^t,$$\bf$$F ^t) $$\cdot$$ $$\Delta$$t

where

$\bf$S t	vector of the concentrations of the slow species at time t ,
$\bf$F t	vector of the concentrations of the fast species at time t ,
$\bf$f ()	vector of production minus loss rates of all species.

2.

Calculate the concentrations of the fast species fully implicitly (involving an iterative procedure), using the concentrations obtained in step (1).  The equation

$$\bf$$F ^{t + 1} = $$\bf$$F ^t + $$\bf$$f($$\bf$$S ^*,$$\bf$$F ^{t + 1})

can be solved with an iterative Newton-Raphson procedure:

$$\bf$$F ^{t + 1}_{k + 1} = $$\bf$$F ^{t + 1}_k - $$\left( {\bf 1} - \frac{\partial{\bf f}({\bf S}^*,{\bf F}^{t+1}_k)}{\partial {\bf F}^{t+1}_k} \right)^$$-1 $$\cdot$$ $$\left ( {\bf F}^{t+1}_k - {\bf F}^t - {\bf f}({\bf S}^*,{\bf F}^{t+1}_k) \right)$$

where $\bf$F ^{t + 1}_{k + 1} is the k ^th estimation of the concentration vector of fast species at time t + 1 and 1 is the unity matrix.

3.

Recalculate the concentrations of the slow species using the concentrations of the fast ones obtained by step (2). 

$$\bf$$S ^{t + 1} = $$\bf$$S ^t + $$\bf$$f($$\bf$$S ^t,$$\bf$$F ^{t + 1}) $$\cdot$$ $$\Delta$$t

Gong and Cho (1993) stated in their paper that step 3 could be omitted, but we find that this step is essential to guarantee mass conservation. In addition, it must be ensured that the same production rates are used in step 3 as were used in the last iteration of step 2.

The following simple algorithm to distinguish between slow and rapid species was developed for Metphomod : All production terms can be approximated by

f_i(...) = P_i - C_i $$\cdot$$ L_i,

where P_i is the production of species i , C_i is its concentration, and L_i is its loss terms divided by the concentration. Numerical instabilities are mainly due to big L_i values which give rise to an exponential loss of species i . A species therefore is considered as fast if

 

$$\Delta \cdot$$ (36)

This condition will find the ``fast'' species in most cases, but in some situations step 3 of the Gong and Cho procedure produces negative concentrations even when applying (36). The integration procedure then has to be repeated and the species which got negative before has to be considered as ``fast''.

The Gong and Cho procedure has the disadvantage of requiring the derivatives of all production and loss terms of all species concentrations. Since Metphomod 's chemical interpreter automatically calculates these derivatives, they are no problem for the user.

2.5.2 Chemical mechanisms

Metphomod's built-in chemical interpreter makes it easy to implement new chemical mechanisms or change existing ones. The model therefore is not implicitly linked to one single mechanism. Up to now, there exist chemical input files for RACM (Stockwell et al.; 1997), RADM I (Stockwell; 1986), and KOREM (Moussiopoulos; 1989).

All simulations presented in this article were carried out with the RACM mechanism. RACM is the successor of the well known and widely accepted RADM 2 (Stockwell et al.; 1990) mechanism. It includes 73 species and 232 reactions. Its main advantages in comparison to RADM 2 are: (1) An updated treatment of alkane and alkene reaction products; (2) the reactions of biogenic compounds were highly improved; (3) better treatment of peroxy radicals; (4) new photolysis reactions were included.

Both RADM I and KOREM are relatively simple mechanisms which are less accurate than RACM, but they have the advantage of being less expensive in computing time and computer memory consumption.