Parameterization of Cumulus Convection Dmitrii V. Mironov German Weather Service, Research and Development, FE14, Offenbach am Main, Germany
[email protected]
COSMO-CLM Training Course, Langen, Germany, 18 - 22 February 2013
Outline • Cumulus convection and the need for parameterizations • Convection parameterization schemes • Mass-flux schemes • The COSMO-model convection parameterization scheme(s) • Critical Issues
Phenomenology Deep Cumulus (ITCZ)
A great variety of convective clouds far and wide
Shallow Cumulus (Trade winds)
Stratus, stratocumulus (Sub-tropics)
P B L
Phenomenology (cont’d) Stratus
Broken stratocumulus
Cumulus
Berlin-St. Petersburg, 28 August 2007.
Phenomenology (cont’d)
Stratocumulus
Cumulus
Berlin-St. Petersburg, 28 August 2007.
Phenomenology (cont’d)
Supercell near Alvo, Nebraska, USA, 13 June 2004. (http://www.extremeinstability.com)
Phenomenology (cont’d)
Supercell off Burliegh, Australia, 31 December 2008. (http://www.sydneystormchasers.com)
The Need for a Parameterization Convection is a sub-grid scale phenomenon. It cannot be explicitly computed (resolved) by an atmospheric model. Hence, it should be parameterized.
∆x
∆y
Recall ... what a convection parameterization should do (it is not a mystery, it is just a model)
Transport equation for a generic quantity X
( )
SGS flux divergence
∂ui′ X ′ ∂X ∂ ui X + = ... − + Sx ∂t ∂xi ∂xi
Source terms
Splitting of the SGS flux divergence and of the source term
( )
∂X ∂ ui X ∂ui′ X ′ ∂ui′ X ′ + = ... − − + Sx + Sx conv other ∂t ∂xi ∂xi conv ∂xi other
What a convection parameterization should do (cont’d) Temperature and specific-humidity equations
( )
∂T ∂ uiT ∂ui′T ′ ∂ui′T ′ ∂Ri + =− − − ∂t ∂xi ∂xi conv ∂xi turb ∂xi
+ L(c − e ) conv + L(c − e ) grid − scale rad
( )
∂q ∂ ui q ∂ui′q′ ∂ui′q′ + =− − + (e − c ) conv + (e − c ) grid − scale ∂t ∂xi ∂xi conv ∂xi turb Here, L is the specific heat of vaporization, e is the rate of evaporation, and c is the rate of condensation. Apart from mixing (redistribution of heat and moisture), convection produces precipitation
Convection Parameterization Schemes • Moisture convergence schemes (e.g. Kuo 1965, 1974) • Convective adjustment schemes (e.g. Betts 1986, Betts and Miller 1986) • Mass-flux schemes (e.g. Arakawa and Schubert 1974; Bougeault 1985; Tiedtke 1989; Gregory and Rowntree 1990; Kain and Fritsch, 1990, 1993, Kain 2004; Emanuel 2001; Bechtold et al. 2001, 2004)
Mass-Flux Schemes. Basic Features A triple top-hat decomposition
X = au X u + ad X d + ae X e , au + ad + ae = 1, “u”, “d” and “e” refer to the updraught, downdraught and the environment, respectively, and a is the fractional area coverage. In terms of the probabilities (δ is the Dirac delta function)
X = Pu X u + Pd X d + Pe X e , P( X ′) = Puδ ( X ′ − X u ) + Pd δ ( X ′ − X d ) + Peδ ( X ′ − X e ) . Vertical flux of a fluctuating quantity X
ρ w′X ′ = ρ au ( wu − w )( X u − X ) + ρ ad ( wd − w )( X d − X ) + ρ ae ( we − w )( X e − X ) = M u ( X u − X ) + M d ( X − X d ) + M e ( X e − X ), M u = ρau ( wu − w ) is the updraught mass flux (similarly for the downdraught and for the environment).
Mass-Flux Schemes. Basic Features (cont’d) A top-hat representation of a fluctuating quantity Updraught
Only coherent top-hat part of the signal is accounted for
Environment
After M. Köhler (2005)
Mass-Flux Schemes. Basic Features (cont’d) Assumption 1: a mean over the environment is equal to to a horizontal mean (over a grid box),
Xe = X ,
au