Modelling of Cirrus Clouds. (MOD 10) (MOD 11). Overview. MOD11: Numerical modelling of important microphysical processes in cirrus clouds MOD12: Stochastic cloud modelling. Numerical modelling of important microphysical processes in cirrus clouds. MOD 11. Overview.
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MOD11: Numerical modelling of important microphysical processes in cirrus clouds
MOD12: Stochastic cloud modelling
1. Cirrus clouds may heat or cool the Earth-Atmosphere System depending on
generation mechanism (incl. synoptic situation, geogr. location)
2. Complex ice crystal shapes (inter alia T- and Si- dependent) render calculation of radiative transfer a tough problem
3. Various modes of ice crystal formation
from Vali, 2004
cloudy airProblems special to cirrus modelling - Supersaturation
4. Cirrus clouds have only a loose relation to ice saturation, viz.
there is plenty of ice supersaturated,
yet clear air in the UT
(sometimes marked by persistent contrails)
Cirrus clouds are embedded in
supersaturated air masses
RHi pdfs within cirrus have long tails into the
Spichtinger et al., 2004
Freezing/nucleation thresholds are high above saturation
extremal states in the RHi field
Extremal states react much more sensitive to changes of background conditions than do averages.
Hence it is difficult to estimate, how the probability will change in a changing climate that in the RHi field the nucleation thresholds will be surpassed.
mean Si increases from 10 to 11%
but probability to surpass 40%
decreases by about 1/3
wrt to the earlier pdf
simple microphysics: bulk microphysics
Processes marked in red are currently included in the Spichtinger/Gierens bulk cirrus physics of EuLag.
Equations used in the two-moment bulk cirrus scheme by Spichtinger and Gierens. Note also the two forms of ice!
Nucleation rate J given as polynomial of awawi.
In equilibrium the water activity equals the relative humidity wrt liquid water. Non-equilibrium occurs in strong updraughts.
Integration over droplet size distribution:
Actual droplet volume derived by inversion of Köhler equation.
For a log-normally distributed dry aerosol mass Gauss-Hermite integration works fine (Gierens and Ström, JAS, 1998).Homogeneous nucleation of aqueous solution droplets
For a given ambient relative humidity the equilibrium size of a solution droplet is given by the Köhler equation. Simplest form:
S = A/r B/r3
A/r is the Kelvin term,
B/r3 is the Raoult term
The two-moment sedimentation scheme nicely obyes the principle that large crystals fall faster than smaller ones. This is not so in the one-moment scheme.
Effect on vertical distribution of
ice water mass and number
Effect on SW and LW
extinction per model
layer. Larger optical
thickness in the
Ni = 1L-1, w = 4.5 cm/s, RHihet = 130 %
Ni = 3L-1, w = 4.5 cm/s, RHihet = 130 %
Ni = 7L-1, w = 4.5 cm/s, RHihet = 130 %
Ni = 10L-1, w = 4.5 cm/s, RHihet = 130 %
Ni = 30L-1, w = 4.5 cm/s, RHihet = 130 %
Ni = 50L-1, w = 4.5 cm/s, RHihet = 130 %
growth - sedimentation - cooling
Moistening of the sub saturated layer
Collection of aerosols in this layer
Due to cooling cloud formation by heterogeneous nucleation
Secondary cloud formation
Ni = 5L-1, w = 4.5 cm/s, RHihet = 110 %
Ni = 5L-1, w = 4.5 cm/s, RHihet = 140 %
Main problem here: Parameterisation of cloud fraction (i.e. fractional cloud cover).
Problem for large scale models, not for cloud resolving models.
In a CRM a grid box is either cloudy or cloud free (binary or 0-1 scheme). Some old GCMs also use this binary assumption of total or zero cloud cover.
The 0-1 schemes neglect sub-grid variability. This leads to errors in all computations, where quantities depend nonlinearly on liquid or ice water path or concentration.
Statistical cloud schemes would allow to consistently treat sub-grid variability in cloud microphysical processes and in radiation.
A cloud fraction looks somewhat better, although problems of cloud overlap assumptions arise, in particular for radiative transfer.
Most (all?) models do not assume a variable cloud fraction in the vertical within one grid layer.
In some GCM schemes cloud cover is parameterised as a function of relative humidity, e.g. the so-called Sundqvist scheme of ECHAM.
Simple statistical scheme:
Clouds already form at Uc>100%, i.e. at sub-saturated conditions.
Fluctuations of RH in the grid box Supersaturation somewhere clouds form in a fraction of the box.
Some schemes use also vertical wind speed to parameterise C.
Consider a phase space (T,RH). In a certain part of the phase space clouds can form, in the remaining part not.
Water clouds: RH>100%
Ice clouds: RHi > RHicrit (T)
The model predicts at every time step and for each grid box a mean state <(T,RH)>.
If we know the probability density functionof fluctuations of the phase point around the grid-box mean value, we can compute, how probable it is that a fluctuation reaches into the supercritical regime.
I call this probability the Overlap Integral .
can be interpreted as the actual cloud coverage C.
For numerical reasons it might be better to compute d / dt and from that dC/dt.
Red line: critical supersaturation for homogeneous nucleation
Green dots: fluctuations of temperature and relative humidity around the grid mean state
= (number of dots above the red line) / (total number of dots)
(for two pressure levels).
MOZAIC data (one year)
Gierens et al., Ann. Geophys., 1997
The measured fluctuations (on a T42 grid scale, i.e. 250×250 km2) follow closely a Cauchy distribution (Lorentz line shape):
() = ( / ) / (2 + 2)
Cauchy distribution: no moments! (not even a mean value). Widely extended tails.
convolution of two Cauchy distributions yield another Cauchy distribution
1() 2() = 1+2()
sum of two random variables
evaluate convolution integral
Insert the two Cauchy distributions ().
Result is a product of the original Cauchy distribution for T
with a “rotated” Cauchy distribution for RH + AT
joint pdf of (T,RH) fluctuations constructed from MOZAIC data
The calculation of the
overlap integral effectively
smears out the boundaries
in the phase space.
0practical considerations – random numbers with certain pdf
random number x is:
x = F-1 (R)
where R is a random
number in [0,1) produced
by a generator.
shift the red line in ±T-direction by ±dT, count the number of points between the black lines, divide by total number of points and divide by 2 dT.
shift the red line in ±RH-direction by ±dRH, count the number of points between the black lines, divide by total number of points and divide by 2 dRH.