Shallow Moist Convection
•Basic Moist Thermodynamics
•Remarkable Features of Moist Convection
•Shallow Cumulus
•(Stratocumulus)
Courtesy: Dave Stevens
Basic Moist Thermodynamics
Grid Averaged Budget Equations



L
c  e  Qrad
  v    w
  w  
t
z
z
cp
qv
q

  v  qv  w v   wqv   c  e 
t
z
z
ql
q

  v  ql  w l   wql   c  e   Pr
t
z
z
Large scale
Large scale
advection
subsidence
Vertical
turbulent
transport
Net
Condensation
Rate
Schematically:
for   ql , qv , :
 
   

 
 
t  t  large scale  t


subgrid
Objectives
•
Understand Moist Convection….
•
Design Models…..
•
 

But ultimately design parameterizations of:  t


 subgrid
Moist Conserved Variables
qv
:Specific Humidity (g/kg)
Condensation occurs if qv exceeds the saturation value qs(T,p)
Usually through rising motion
ql
:Liquid Water (g/kg)
qt = qv + ql
:Total water specific humidity
(Conserved for phase changes!!)
Used Temperature Variables
  T  (g cp ) z
 l  T  ( g c p ) z  ( L c p )ql
•Potential Temperature
•Conserved for dry adiabatic changes
•Liquid Water Potential Temperature
•Conserved for moist adiabatic changes
Energy equivalent:
hl  c pT  gz  Lql
 v   (1  0.61qv  ql )
•Liquid Water Static Energy
•Virtual Potential Temperature
•Directly proportional to the density
•Measure for buoyancy
Grid averaged equations for moist conserved variables:
 l
 l

  v   l  w
  w l  Qrad
t
z
z
qt
qt

  v  qt  w
  wqt  Pr
t
z
z
Parametrization issue reduced to a
convective mixing problem!
A Unique Feature of Moist Convection
Moist Adiabatic Lapse Rate
A saturated ascending parcel will conserve hl :
hl
 0 with hl  c pT  gz  Lqt  qs ( p, T ) 
z
Leads to a moist adiabatic lapse rate :
 T 
m  

.

z

 sat
parcel
Remarks:

Lq s 
1 

g  Rd T 

q 
cp 
1  L s 
T 

m  4.4K / km
g
 9.8 K / km
•Temperature decrease less than for dry parcels m  d  
cp


•Difference between m and
d becomes progressively smaller for lower temperatures
•Example: T=290K,p=1000mb 
d
d
m
z
z
T(K)
Q (K)
m
Absolute Instability
•Lift a (un)saturated parcel from a sounding at z0 by dz
•Check on buoyancy with respect to a mean profile:
Bg
Tv, p  T v
Tv
Example 1:
d
m
Unstable for saturated and unsaturated parcels
Absolute Unstable
z
sounding
zo
Tv(K)
Absolute Stability
•Lift a (un)saturated parcel from a sounding at z0 by dz
•Check on buoyancy with respect to the sounding:
Bg
Tv, p  T v
Tv
Example 2:
sounding
d
m
Stable for saturated and unsaturated parcels
Absolute stable
zo
Tv(K)
Conditional Instability
•Lift a (un)saturated parcel from a sounding at z0 by dz
Bg
•Check on buoyancy with respect to the sounding:
Tv, p  T v
Tv
Example 2:
sounding
d
m
Stable for unsaturated parcels
Unstable for saturated parcels
Conditionally Unstable!!!
z
zo
d 
Tv(K)
T v
 m
z
The Miraculous Consequences of conditional Instability
Mean profile
or: the “Cinderalla Effect” (Bjorn Stevens)
“Level of zero kinetic energy”
Inversion
Level of neutral buoyancy (LNB)
height
Level of free convection (LFC)
Lifting condensation level (LCL)
conditionally
unstable
layer
well mixed layer
θ v (K)
Non-local integrated stability funcions:
CAPE, CIN
Define a work function:
zlnb
LNB
W ( z 0 , z lnb ) 

z0
CAPE
Bdz g
z lnb

z0
Tv,p  Tv
Tv
dz
Positive part:
CAPE = Convective Available Potential Energy.
CAPE  W ( zlfc , zlnb )
CIN
z1
Negative part:
z0
CIN  W ( z1 , zlcl )
wmax  2CAPE
CIN allows the accumulation of CAPE
CIN
= Convection Inhibition
CAPE and CIN: An Analogue with Chemistry
1) Large Scale
Forcing:
• Horizontal Advection
Activation (triggering)
• Vertical Advection
(subs)
LS-forcing
Surf Flux
• Radiation
CIN
2) Large Scale
Forcing:
CAPE
slowly builds up CAPE
Free
3) CAPE
Energy
•Consumed by moist
convection
RAD
LS-forcing
Mixed Layer
LFC
Parcel Height
LNB
• Transformed in Kinetic
Energy
•Heating due to latent
heat release (as
measured by the
precipitation)
•Fast Process!!
Free after Brian Mapes
Quasi-Equilibrium
dCAPE
 JM b  FLS  0
dt
LS-Forcing that slowly builds up slowly
The convective process that stabilizes
environment
Quasi-equilibrium: near-balance is maintained even when F is
varying with time, i.e. cloud ensemble follows the Forcing.
Forfilled if : tadj << tF
wu
au
Used convection closure (explicit or implicit)
JMb ~ CAPE/ tadj
tadj : hours to a day.
Mb=au wu r :Amount of convective vertical motion at cloud base (in an ensemble sense)
Quasi-Equilibrium: An Earthly Analogue
Free after Dave Randall:
•Think of CAPE as the length of the grass
•Forcing as an irrigation system
•Convective clouds as sheep
•Quasi-equilibrium: Sheep eat grass no
matter how quickly it grows, so the grass is
allways short.
•Precipitation………..
Typical Tradewind Cumulus
Strong horizontal variability !
Horizontal Variability and Correlation
Mean profile
height
θ v (K)
•Schematic picture of cumulus moist convection:
Cumulus convection:
1. more intermittant
2. more organized
than
Dry Convection.
Mass flux concept: tomorrow more!!
e
 M (c   )
c
w   (1  a) w   a w   awc (c   )
wc
a
a
a
Shallow Cumulus Convection
Photo courtesy Bjorn Stevens
Observational Characteristics : Trade wind shallow Cu
non well-mixed cloud layer
Surface heat-flux: ~10W/m^2
Surface Latent heat flux : 150~200W/m^2
Mixing between Clouds and Environment
(SCMS Florida 1995)
ql
ql ,ad
 0.3 ~ 0.4
adiabat
Due to entraiment!
Data provided by: S. Rodts, Delft University, thesis available
from:http://www.phys.uu.nl/~www.imau/ShalCumDyn/Rodts.html
adiab
at
Liquid water potential temperature
Total water (ql+qv)
Entrainment Influences:
1. Vertical transport
2. Cloud top height
The simplest Cloud Mixing
Model
4.1 lateral mixing bulk model
for    l , qt  :
qt,e θl ,e
qt,c θ l ,c
wc
hc
dc
 Fmixing
dt
c
(c  e )
wc

z
t
c
1
1
  (c  e ) where  

z
wct hc
Fractional entrainment rate
Diagnose
c
 
(c  e ) for    l , qt 
z
through conditional sampling:
Typical Tradewind Cumulus Case (BOMEX)
Data from LES: Pseudo Observations
Trade wind cumulus: BOMEX
LES
  1 ~ 3 103 m-1
Observations
Cumulus over Florida: SCMS
Siebesma JAS 2003
Horizontal or vertical mixing?
Lateral
mixing
Cloud-top
mixing
Adopted in cloud parameterizations:
Observations
(e.g. Jensen 1985)
However: cloud top mixing needs
substantial adiabatic cores within the clouds.
(SCMS Florida 1995)
No substantial adiabatic
cores (>100m) found
during SCMS except near
cloud base. (Gerber)
adiabat
Does not completely
justify the entraining
plume model but………
It does disqualify a
substantial number of
other cloud mixing models.
Backtracing particles in LES: where does the air in
the cloud come from?
Cloudtop
Cloudtop entrainment
Entrance level
Cloudbase
Inflow from subcloud
Measurement level
Cloudtop
May 16, 2007
Courtesy Thijs Heus
Height vs. Source level
May 16, 2007
Virtually all cloudy air comes from below the
observational level!!
5.Dynamics, Fluxes and other
stuff that can’t be measured
accurately
•No observations of turbulent fluxes.
•Use Large Eddy Simulation (LES)
based on observations
BOMEX ship array (1969)
 
   
    
t  t large  t
scale
observed


0
subgrid
observed
To be modeled by LES
•10 different LES models
•Initial profiles
•Large scale forcings prescribed
•6 hours of simulation
Is LES capable of
reproducing the steady state?
•Large Scale Forcings
•Mean profiles after 6 hours
•Use the last 4 simulation hours for analysis of …….
To do analyses of the dynamics using the
LES results
How is the steady state achieved?
forcing
forcing
rad
turb
turb
c-e
c-e
How is the steady state achieved?
•Cloud cover
•Turbulent Fluxes of wql and w v
Subcloud layer looks similar than dry PBL!!
•Turbulent Fluxes of the conserved
variables qt and l
w l  w  
L
wql
cp
Cloud layer looks like a enormous entrainment layer!!
Vertical Velocity in the cloud and the total vertical velocity variance
Dry PBL velocity variance profile
Conditional Sampling of:
•Total water qt
•Liquid water potential temperature l
Virtual potential temperature:
v
•Cloud Liquid water
Shallow Cumulus Growth, (an idealized view)
Bjorn Stevens accepted for JAS
Extension from the dry PBL growth, but now…..
Non-precipating cumulus
Adding Moisture
dry
cloudy
Temporal Evolution
Temporal Evolution
cloud top height
t
dry pbl growth
 t 1/ 2
cloud base height
 t0
Energetics
evaporation
condensation
w l  w  
L
wql
 cp
Stabilisation of Cloud Base (1)
Mass flux
h
 we  w  M
t
Growth through dry top-entrainment
Negative in the presense of subsidence
Mass leaking out of PBL through clouds
M  ac wc
Stabilisation of Cloud Base height (2)
h
 we  w  M
t
Bjorn Stevens
(accepted JAS)
cloud top height
Pbl-height
Many Parameterization of Shallow Cu still give poor results
Role of Precipitation
Mesoscale-Organisation
Momentum Transport