On the Influence of a Simple Microphysics Parametrization on Radiation Fog Modelling: A Case Study During ParisFog

Abstract

A detailed numerical simulation of a radiation fog event with a single column model is presented, which takes into account recent developments in microphysical parametrizations. One-dimensional simulations are performed using the computational fluid dynamics model Code_Saturne and the results are compared to a very detailed in situ dataset collected during the ParisFog campaign, which took place near Paris, France, during the winter 2006–2007. Special attention is given to the detailed and complete diurnal simulations and to the role of microphysics in the fog life cycle. The comparison between the simulated and the observed visibility, in the single-column model case study, shows that the evolution of radiation fog is correctly simulated. Sensitivity simulations show that fog development and dissipation are sensitive to the droplet-size distribution through sedimentation/deposition processes but the aerosol number concentration in the coarse mode has a low impact on the time of fog formation.

Appendix

1.1 Model Equations

1.1.1 Dynamic Equations

The dynamic equations are written as

$$\begin{aligned} \frac{\partial U}{\partial t}=\frac{\partial }{\partial z}\left( {K\frac{\partial U}{\partial z}} \right) +C_n \left( {U-U_\mathrm{obs} } \right) , \end{aligned}$$

(17)

where $U$ is the horizontal wind component, $K$ is the wind exchange coefficient, $C_{n}$ is the nudging coefficient (inverse of relaxation time: $1/\tau _{n})$, $U_\mathrm{obs}$ is the driving wind field derived from observations.

1.1.2 Thermodynamic Equations

Prognostic equations for the liquid-water potential temperature, $\theta _\mathrm{l}$, and for the total specific cloud water content $q_\mathrm{w}$ are

$$\begin{aligned} \rho \frac{\partial \theta _\mathrm{l} }{\partial t}&= \frac{\partial }{\partial z}\left[ {\left( {\frac{\lambda _\mathrm{c} }{C_\mathrm{p} }+\frac{\mu _\mathrm{t} }{P_\mathrm{r}}} \right) \frac{\partial \theta _\mathrm{l} }{\partial z}} \right] -\frac{\theta }{TC_\mathrm{p} }\frac{\partial F_\mathrm{rad} }{\partial z}-\rho \frac{L_\mathrm{v} \theta }{TC_\mathrm{p} }\left( {\frac{\partial q_{_\mathrm{l} } }{\partial t}} \right) _\mathrm{SED} +\rho C_n (\theta _\mathrm{l} -\theta _{l_\mathrm{obs} } ),\nonumber \\ \end{aligned}$$

(18)

$$\begin{aligned} \rho \frac{\partial q_\mathrm{w} }{\partial t}&= \frac{\partial }{\partial z}\left[ {\left( {\frac{\lambda _\mathrm{c} }{C_\mathrm{p} }+\frac{\mu _\mathrm{t}}{P_\mathrm{r} }} \right) \frac{\partial q_\mathrm{w} }{\partial z}} \right] -\rho \left( {\frac{\partial q_{_\mathrm{l} } }{\partial t}} \right) _\mathrm{SED} +\rho C_n (q_\mathrm{w} -q_{w_\mathrm{obs} } ), \end{aligned}$$

(19)

where $\rho$ is the air density, $\lambda _\mathrm{c}$ is the thermal diffusivity, $\mu _\mathrm{t}$ is the turbulent viscosity, $P_\mathrm{r}$ is the turbulent Prandtl number, $F_\mathrm{rad}$ is the vertical divergence of net radiative fluxes, and $\theta _\mathrm{lobs},\; q_\mathrm{wobs}$ are derived from observations.

The equation for the cloud droplet number, $N_\mathrm{d}$, is written as

$$\begin{aligned} \rho \frac{\partial N_\mathrm{d} }{\partial t}=\left[ {\left( {\frac{\lambda _\mathrm{t}}{C_\mathrm{p} }+\frac{\mu _\mathrm{t}}{P_\mathrm{r} }} \right) \frac{\partial N_\mathrm{d} }{\partial z}} \right] +\rho \left( {\frac{\partial N_\mathrm{d} }{\partial t}} \right) _\mathrm{C/E} +\rho \left( {\frac{\partial N_\mathrm{d} }{\partial t}} \right) _\mathrm{NUC} +\rho \left( {\frac{\partial N_\mathrm{d} }{\partial t}} \right) _\mathrm{SED},\nonumber \\ \end{aligned}$$

(20)

where the subscript SED refers to the rate of change due to sedimentation; C/E to condensation/evaporation; NUC to cloud droplet nucleation respectively. The sink/source terms on the right-hand side are parametrized in terms of the prognostic variables themselves $(q_\mathrm{l}$ and $N_\mathrm{d})$.

1.1.3 Determination of the maximum supersaturation $s_\mathrm{max}$

$$\begin{aligned} s_{\max } =\sum _{i=1}^3 {\frac{1}{s_i^2 }} \left[ {f_i \left( {\frac{\varsigma }{\eta _i }} \right) ^{3/2}+g_i \left( {\frac{s_i^2 }{\eta _i +3\varsigma }} \right) ^{3/4}} \right] , \end{aligned}$$

(21a)

with

$$\begin{aligned} f_i =0.5\exp \left( {2.5\ln ^{2}\sigma _i } \right) ,\end{aligned}$$

(21b)

$$\begin{aligned} g_i =1+0.25\ln \sigma _i,\end{aligned}$$

(21c)

$$\begin{aligned} s_i =\frac{2}{\sqrt{B}}\left( {\frac{A}{3r_{\mathrm{a}_i } }} \right) ^{3/2}, \end{aligned}$$

(21d)

and

$$\begin{aligned} \varsigma =\frac{2A}{3}\left( {\frac{A_1 W+A_3 {\partial F_\mathrm{rad} }/{\partial z}}{A_4 }} \right) ^{1/2},\end{aligned}$$

(22a)

$$\begin{aligned} \eta _i =\frac{\left[ {{\left( {A_1 W+A_3 {\partial F_\mathrm{rad} }/{\partial z}} \right) }/{A_4 }} \right] ^{3/2}}{2\pi \rho _\mathrm{w} A_2 N_{\mathrm{a}_i } }, \end{aligned}$$

(22b)

where $\rho _\mathrm{w}$ is the water density, $A_{1},\; A_{2},\;A_{3}$ are the constants defined in Eq. 6, and $A_{4},\; A,\; B$ can be found in Abdul-Razzak et al. (1998).