A cloud-following parametrization for a mountainous path
G.Bonafè
2017-10-25
Parametrization of wet deposition
The parameterization of wet deposition follows EMEP (2003) approach, including in-cloud and below-cloud scavenging of gas and particles, and is actually implemented in the AERO0 configuration of FARM model.
The in-cloud scavenging of a component1 valid only for soluble components? of concentration C is computed as: \[\Delta C_{wet} = -C \frac{W_{in}\cdot P}{\Delta z\cdot \rho_{w}}\]
where P is the precipitation rate (\(kg ~m^{-2} ~s ^{-1}\)), Δz is the scavenging depth (assumed to be 1000\(~m\)), \(\rho_{w}\) is the water density (1000\(~kg ~m^{-3}\)) and \(W_{in}=10^6\) is the in-cloud scavenging ratio.
In case of particles, below-cloud scavenging is computed by means of Scott (1979) relationship: \[\Delta C_{wet} = -C \frac{A\cdot P}{V_{dr}} \bar{E}\]
where \(A = 5.2 ~m^3 ~kg^{-1} ~s^{-1}\) is an empirical coefficient (a Marshall-Palmer size distribution is assumed for rain drops), \(V_{dr}\) is the raindrop fall speed (assumed to be \(5 ~m ~s^{-1}\)) and \(\bar{E}\) is the size-dependent collision efficiency of aerosols by the raindrops (0.1 for fine PM, 0.4 for coarse PM).
According to these equations, let’s define two R functions:
Given a precipitation rate2 About the precipitation rate: to convert mm/h to \(kg ~m^{-2} ~s ^{-1}\), we just need to divide by 3600. of 5\(~mm/h\) and an initial concentration of 100\(~ng/m^3\):
after one second, the removal in the cloud is
deltaC_in(C = 100, P = 5/3600)
## [1] -0.1388889
below the cloud, if 70% of the PM is fine
deltaC_below(C = 70, P = 5/3600, E=0.1)
## [1] -0.01011111
below the cloud, since the remaining 30% of the PM is coarse
deltaC_below(C = 30, P = 5/3600, E=0.4)
## [1] -0.01733333
Evolution over a time period
We can iterate the process, to parametrize a cloud with precipitation and wet deposition.
Let’s define the height of the cloud base \(h_C\), along its path \[h_C=\bar{h}_C + h'_C + h''_C\] where \(\bar{h}_C = 1200~m\) is the unperturbed height and the other are lifting terms. The first lifting term is a function of the smoothed ground elevation below the cloud
\[h'_C = h_{GS}-h_{C0}+d\cdot log \left( 1+e^{\frac{h_C0-h_{GS}-d/2}{d}} \right)\] where \(d=500~m\) and \(h_{GS}=smoothing(h_G)\), with \(h_G\) elevation of the ground.
The second lifting term avoids the cloud going underground
\[h''_C = max(0, h_G - \bar{h}_C - h'_C)\]
library(caTools)
h_G <- path$elevation
h_GS <- runmean(h_G, 21, align="center")
d <- 500
D_C <- 500 # cloud depth
h_C0 <- rep(1200, nrow(path)) # height of unperturbed cloud
h_C1 <- h_GS-h_C0+d*log(1+exp((h_C0-h_GS-d/2)/d)) # first lifting term
h_C2 <- pmax(0, h_G-h_C0-h_C1) # second lifting term
h_C <- h_C0 + h_C1 + h_C2 # cloud base
h_Ctop <- h_C + D_C # cloud top
plot(path$location$lat, h_G, type="l",
ylim=c(0,max(h_Ctop)))
lines(path$location$lat, h_C, lty=2)
lines(path$location$lat, h_Ctop, lty=2)
Precipitation from the cloud
Let say the precipitation rate is proportional to the part of cloud exceeding a given height above sea level, with an attenuation factor \(\alpha\) for the leeward precipitation3 equation 17 in Jiang, Q., & Smith, R. B. (2003). Cloud timescales and orographic precipitation. Journal of the atmospheric sciences, 60(13), 1543-1559.
\[\alpha = \frac{PE_{lee}}{PE_{ww}} = \frac{1}{\tau_a/\tau_f+\tau_a/\tau_{sub}}\] where \(\tau_a\), \(\tau_f\) and \(\tau_{sub}\) are timescales for advection, hydrometeor fallout and snow sublimation:
\[\tau_f \approx \frac{H_C+D_C/2 }{v_{prec}}\] where \(H_C\) is the cloud-base height above ground and \(D_C\) the cloud depth; \[\tau_a \approx \frac{a_m}{ v_{wind}}\] where \(a\) is half the horizontal dimension of the mountain and \(v_{wind}\) the mean wind speed; \[\tau_{sub} \approx \frac{H_{sub}\cdot a_m}{v_{wind} \cdot h_m}\] where \(H_{sub}=1~km\) is a “scale height” for sublimation and \(h_m\) the height of the mountain.
Now we can calculate the wet deposition rates \(D_{in}\), \(D_{below}^{fine}\) and \(D_{below}^{coarse}\) along the path. We assume the following initial concentrations (in \(ng/m^3\))
