June 15, 2011
Melchior van Wessem &Stephan de Roode
Mixed-Layer Model Solutions ofEquilibrium States of
Stratocumulus-Topped Boundary Layers
Equilibrium states of stratocumulus with a mixed layer model
Stevens 2002
Both temperature and humidity jumps are important for the evolution of a stratocumulus cloud deck
Exploring phase space with a stratocumulus mixed-layer model to investigate stratocumulus cloud-
climate feedback
• Tool: Mixed Layer model
• Set up: Boundary conditions
• Changing parameters: Free atmosphere potential temperature θ and humidity q
• Climate change: SST + 2K
Perturbing the “phase space” with SST+2K.
What happens with LWP ~ (zi-zb)2 ?
Mixed Layer Model
radiative cooling
surface flux surface flux
entrainment fluxentrainment flux
Radiative cooling
zi
zb
zi
zb
q
t∂∂θ
θmoisture temperature
zi∂qT,ML
∂t= Cd U qsat,sst − qT,ML( )+ w eΔqT
zi∂θL,ML
∂t= Cd U θ0,sst − θL,ML( )+ w eΔθL − Qrad
∂zi
∂t= w e + w ; w = −Dz
Radiative cooling
zi
zb
Equilibrium Solutions (Stevens 2006):
qT,ML = qsat,sst +w e(qfa − qsat,sst )
Cd U + w e
θL,ML = θ0,sst +w e(θfa − θ0,sst ) − Qrad
Cd U + w e
w e = Dzi
( )2
21
biql zzLWP −Γ=Liquid water path:
−≅
SST
ml
dv
vb q
qLSSTRz ln
2
γ
Entrainment we from Nicholls & Turton (1986)
Free atmosphere conditions
hei
ght
z
qT θL
qT,MLθL,ML
qT0
θ0
Γq =∂q∂z
FA
Γθ =∂θ∂z
FA
qref θref
qT,FA = qref + δq + Γqz θL,FA = θref + δθ + Γθz
Change free atmosphere conditions. Note: Free atmosphere is assumed to be constant with time
hei
ght
z
qT θL
qT,MLθL,ML
qT0
θ0qref θref
δq δθ
qT,FA = qref + δq + Γqz θL,FA = θref + δθ + Γθz
Γq =∂q∂z
FA
Γθ =∂θ∂z
FA
Change free atmosphere conditions. Free atmosphere is assumed to be constant with time
hei
ght
z
qT θL
qT,MLθL,ML
qT0
θ0qref θref
δq δθ
qT,FA = qref + δq + Γqz θL,FA = θref + δθ + Γθz
Γq =∂q∂z
FA
Γθ =∂θ∂z
FA
Free atmosphere conditions
Steady-state solutions for a range of
different free atmosphere conditions (change δq and δθ)
Boundary conditions
Θfa-ΘSST
qfa-qSST
fog
Decoupling
warmer free atmosphere
moister free atmosphere
Equilibrium solutions as a function of free atmosphere conditions:Inversion height (zi)
Θfa-ΘSST
qfa-qSST
fog
Decoupling
warmer free atmosphere
moister free atmosphere
Equilibrium solutions as a function of free atmosphere conditions:Cloud base height (zb)
Equilibrium solutions as a function of free atmosphere conditions:The Liquid Water Path (LWP)
Θfa-ΘSST
qfa-qSST
fog
Decoupling
warmer free atmosphere
moister free atmosphere
Equilibrium solutions as a function of SST and horizontal wind velocity:The Liquid Water Path (LWP)
Decoupling
SST + 2K, free atmosphere conditions the same
hei
ght
z
qT θL
qT,MLθL,ML
qT0 (SST+2)
Γq =∂q∂z
FA
Γθ =∂θ∂z
FA
qref
θref
qT,FA = qref + δq + Γqz θL,FA = θref + δθ + Γθz
θ0+2 K
SST = 2K, free atmosphere state not changedChanges in LWP
Θfa-ΘSST
qfa-qSST
warmer free atmosphere
moister free atmosphere
SST = 2K, free atmosphere state not changedChanges in LWP
Θfa-ΘSST
qfa-qSST
warmer free atmosphere
moister free atmosphere
1Thicker
SST = 2K, free atmosphere state not changedChanges in LWP
Θfa-ΘSST
qfa-qSST
warmer free atmosphere
moister free atmosphere
1Thicker
2
Thinner
SST = 2K, free atmosphere state not changedChanges in LWP
∼ Θfa-ΘSST
qfa-qSST
warmer free atmosphere
moister free atmosphere
1Thicker
2
Thinner3
Cumulus
θ L,M
L[K
]
[g/k
g]
qT,ML
qsat
Climate change (constant free atmosphere)
Mixed layer θL and qT ~ linearly dependent on SST
Saturation specific humidity increases exponentially
case + : cloud thickeningcase - : cloud thinning
Climate change (constant free atmosphere)
Inversion and cloud base increase for increasing SST
For red line zi increases more than zb
→ cloud thickens
For blue line zi increases slighty less than zb
→ cloud thins
case + : cloud thickeningcase - : cloud thinning
SST + 2K, free atmosphere conditions the same
hei
ght
z
qT θL
qT,MLθL,ML
qT0 (SST+2)
Γq =∂q∂z
FA
Γθ =∂θ∂z
FA
qref
θref+2
qT,FA = qref + δq + Γqz θL,FA = θref + δθ + Γθz
θ0+2 K
ΔLWP/ΔSST
Small increase of LWP
(small negative feedback)
Transition from Scu to Cu (strong positive feedback)
Free atmosphere increased with ΔSST
Conclusions
• Mixed Layer model shows (small) negative feedback if free atmosphere is also
becoming warmer (ΔθFA = ΔSST)
• Results depend on many factors: entrainment formulation, strength of radiative
cooling, lapse rate in free atmosphere, horizontal advection, horizontal wind
velocity
• Phase space gives very much useful information about cloud regime
• Similar excercise can be performed with SCM (see RACMO/EC-EARTH results by
Sara Dal Gesso)