pier siebesma [email protected] knmi, the netherlands motivation fundamentals, models, equations dry...
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Pier [email protected]
KNMI, The Netherlands
MotivationFundamentals, Models, Equations
Dry Convective Boundary layerShallow Moist Convection (Friday)
Parameterizations of moist and dry convection (Saturday)
Atmospheric dry and shallow moist convection
1. Motivation
o0
Equator
N30
Northoregion
windTrade
Subsidence
~0.5 cm/s
10 m/s
vE vE
inversion
Cloud base
~500m
Tropopause 10km
•Stratocumulus
•Interaction with radiation
•Shallow Convective Clouds
•No precipitation
•Vertical turbulent transport
•No net latent heat production
•Fuel Supply Hadley Circulation
•Deep Convective Clouds
•Precipitation
•Vertical turbulent transport
• Net latent heat production
•Engine Hadley Circulation
The GCSS intercomparison project on cloud representation in GCM’s in the Eastern Pacific
ECMWF IFS overestimates Tradewind cumulus cloudiness:
Siebesma et al. (2005, QJRMS)
scuShallow cuDeep cu
Water vapour is the building material for clouds
name SymbolUnits
Definition
Near surfacevalues
Atmospheric column
spec. humidity
qv [g/kg] amount of water vapour in 1kg dry air
10 g/kg 20 kg/m2
Saturation spec. hum.
qs [g/kg] Max. amount of water vapour in 1kg dry air
15 g/kg
Liquid water
ql [g/kg] amount of liquid water in 1kg dry air
1 g/kg 200 g/m2
qsat
qt
Tip of the iceberg
water vapor clouds albedo lapse rate total
(Bony and Dufresne, GRL, 2005)
High-sensitivity GCMs
Low-sensitivity GCMs
Sensitivity of the Tropical Cloud Radiative Forcingto Global Warming in 15 AR4 OAGCMs
CRFSW
SST
Fundamentals , Models and Equations
Some fundamental notions on Turbulence (1)
2
2
3j
i
ii
j
ij
i
x
u
x
pg
x
uu
t
u
Conservation of momentum: Navier Stokes equations:
storage
term
advection
term
gravity
term
pressure gradient
term
viscosity
term
In order to discuss the non-linearity consider a simpler 1d version: The Burgers Equation:
2
2
x
u
x
uu
t
u
And treat both physical processes seperately:
Some fundamental notions on Turbulence (2)
1) The diffusion equation:
2
2
x
u
t
u
dissipation gradient weakening stabilizing
0
x
uu
t
u
2) The advection equation:
General solution:
)( utxfu
Advection term gradient sharpening instability
Some fundamental notions on Turbulence (3)
2
2
x
u
x
uu
t
u
•Competition between both processes determines
the solution.
•Compare both terms by making the equation dimensionless:
L
Utt
Lxx
Uuu
~
/~/~
2
2
2
2
~
~1~
~
~
~~
~~
x
u
Rex
u
ULx
uu
t
u
UL
Re Reynolds Number measures the ratio between the 2 terms.
1Re dissipation dominates flow is stable laminar
1Re Non-linear advection term dominates flow is unstable turbulent
Turbulence made by convection in the atmospheric boundary layer
L=1000m
U=10m/s
m2s-1
95
1010
10000 UL
Re
Macrostructure dominated by non-linear advection!!
Heat flux
Potential temperature profile
Large eddy simulation of the convective boundary layer
Poor man’s artist impression of the convective boundary layer
Some fundamental notions on Turbulence (3):
Energy Cascade
Some fundamental notions on Turbulence (4):
• Energy injection through buoyancy at the macroscale
Re dominated by non-linear processes
• Hence, Large eddies break up in smaller eddies
• that have less kinetic energy:
• and a lower “local” Reynold number
• until they are so small that :
• and viscosity takes over and the eddies dissipate.
UL
Relocal
1localRe
J.L Richardson (1881-
1953) Some fundamental notions on Turbulence (4):
Richardson 1926
free after Jonathan Swift (1733):
(Sloppy) Kolmogorov
(1941) Some fundamental notions on Turbulence (4):
Kinetic Energy (per unit mass) : E
Dissipation rate :
t
E
eddy size:
eddy velocity:
eddy turnover time: )(v
)( v
Kolmogorov Assumption:
Kinetic Energy transfer is constant and equal to the dissipation rate
)(v
cstvvE
)(
)(
)(
)(
)( 323/1~)( v
Some fundamental notions on Turbulence (5):Consequences of Kolmogorov
(1941)
3/~)( ppv Structure functions:
3/53/22 ~~))(( kdlelvF ik Fourier transform of kinetic energy Famous 5/3-law!!
Kolmogorov scale : the scale at which dissipation begins to dominate:
m
vlocal
34
1
3
354
133
1010
10)(1Re
Largest eddies are the most energetic
1 hour100 hours 0.01 hour
microscaleturbulence
spectral gap
diurnal cycle
cyclones
data: van Hove 1957
Energy Spectra in the atmosphere
energy spectra at z=150m below stratocumulus
Duynkerke 1998
U Spectrum
V Spectrum
W Spectrum500m
Governing Equations for incompressible flows in the atmosphere
2
2
j
i
iu
j
ij
i
x
u
x
pF
x
uu
t
ui
2
2
jjj
xF
xu
t
2
2
jqq
jj
x
qF
x
qu
t
q
0
i
i
x
u
vdTRp
Continuity Equation
(incompressible)
NS Equations
Heat equation
Moisture equation
Gas law
jcijiu ufFi 33 with
gravity
term
coriolis
term
Condensed water eq.lq
j
lj
l Fx
qu
t
q
10 m 100 m 1 km 10 km 100 km 1000 km 10000 km
turbulence Cumulus
clouds
Cumulonimbus
clouds
Mesoscale
Convective systems
Extratropical
Cyclones
Planetary
waves
Large Eddy Simulation (LES) Model
Cloud System Resolving Model (CSRM)
Numerical Weather Prediction (NWP) Model
Global Climate Model
The Zoo of Atmospheric Models
Global Climate and NWP models (x>10km)
Qz
w
t
'
v
z
wuvvfu
t
ugc
v
z
wvuufv
t
vgc
v
vdTRp
gdz
pd
t
w
0 0
z
w
y
v
x
u
Subgrid
To be parameterized
Large Eddy Simulation (LES) Model (x<100m)
High Resolution non-hydrostatic Model: 10~50m Large eddies explicitly resolved by NS-equations inertial range partially resolved Therefore: subgrid eddies can be realistically parametrised
by using Kolmogorov theory Used for parameterization development of turbulence,
convection, clouds
ln(Energy)
ln(wave number)-11 1mm
dl-11
0 1km ~ l
Inertial Range
53
DissipationRange
Resolution LES
Dynamics of thermodynamical variables in LES
ltqforwzz
w
t
,...... terms"horizontal"
ResolvedResolved
turbulenceturbulence
subgrid subgrid
turbulenceturbulence
3 with
: with
dxdydzleclK
zKw
Subgrid turbulence:
Remark: 3/4)( llvcleclK
Richardson law!!
Simple: All or Nothing:
0),(
,1
0,0
,0
sltsltl
sltl
qqifqqq
a
qqifq
a
c
c
{
{
Cloud Scheme in LES
• Definition:
Turbulent Kinetic Energy (TKE) Equation
2225.0 wvue
pwz
ewz
wg
z
vwv
z
uwu
t
ev
o
1'''''
Reynolds Averaged budget TKE-equation:
Assume:
•No mean wind
•No horizontal flux terms
Shear production Buoyancy production Transport Dissipation
S B T D
Richardson Number: S-
BfRi
1-
1
1
f
f
f
Ri
Ri
Ri Laminar flow
Shear driven turbulence
Buoyancy driven turbulence
Mixed layer turbulent kinetic energy budget
normalizedStull 1988
dry PBL
Conditions for Atmospheric Convection
1
ULReReynolds Number Condition for fully developed turbulence
Richardson Number 1
S
BRi Condition for buoyancy drive turbulence
Atmospheric Convection = Turbulence driven by Buoyancy
Objectives
Tools
• To “understand” the various aspects of atmospheric convection
• To find closures (for the turbulent fluxes and variances)
• Observations, Large Eddy Simulation (LES) models
Methods
• Dimension analysis, Similarity theory, common sense
Application
• Climate and Numerical Weather Prediction (NWP) Models
(Simplified) Working Strategy
Large Eddy Simulation (LES) Models
Cloud Resolving Models (CRM)
Single Column Model
Versions of Climate Models
3d-Climate Models
NWP’s
Observations from
Field Campaigns
Global observational
Data sets
Development Testing Evaluation
See http://www.gewex.org/gcss.html
Dry Convective Boundary Layer
1. Phenomenology
2. Properties
3. Models and Parameterization for Convective Transport
The Place of the Atmospheric Turbulent Boundary Layer
Depth of a well mixed layer: 0~5km
Determined by:
•Turbulent mixing in the BL
•Large Scale Flow (convergence, divergence)
we
Q0
z
tropopause
Can we see the convective PBL?
PBL top
Downtown LA
10km
July 2001
(Courtesy Martin Kohler)
Typical Profiles of the convective BL
Surface layer
Mixed layer
Entrainment Zone
Stull 1988
LES View of the Dynamics: potential temperature
Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands
LES View of the Dynamics: vertical velocity
Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands
Horizontal Crosscut
Surface layer
Mixed Layer
Inversion layer
Irregular polygonal structures!
Moeng 1998
Dry Convective Boundary Layer
2. Properties
•Surface Layer
•Mixed Layer
•Inversion Layer
Monin-Obukhov Similarity
Construct dimensionless gradient terms:
*u
z
z
um
L
z
(von Karman constant) is defined such that 1 for / 0m z L
SLh
z
z *,
and evaluate this as a function of the stability parameter
stableunstable
mh
u
qwq
u
w
wuu
srfSL
srfSL
srf
,
,
Fleagle and Businger 1980
MO theory allows to formulate the turbulent fluxes in a diffusivity form:
zK
zz
w
t h
'
**
)''(
u
w o
)/(*
Lzz
zh
z
Kz
uzkw hh
1
0
Diffusion Eq.
Dry Convective Boundary Layer
2. Properties
•Surface Layer
•Mixed Layer
•Inversion Layer
Scaling Parameters for the convective mixed layer
Relevant parameters:
1) TKE production through buoyancy: srfv
buoyancybyproduction
wg
t
e
0
2) Depth of the boundary layer: z
unitsform
3
2
s
m
m
Construct a convective velocity scale:
3/1
0*
zw
gw srf
Interpretation: velocity that results if all potential energy is converted into kinetic energy in an eddy of size z*
*w
z
Typical Numbers
K
msg
mz
msKwmWwc srfsrfp
300
81.9
1000
15.0'/150'
0
2
12
smzwg
w srf /7.13/1
0*
sec600*
w
z
2/13/1
9.01
z
z
z
z
ww
Dimensionless vertical velocity variance (in the free convective limit)
Garrat 1992)
Mixed layer turbulent kinetic energy budget (LES)
pwz
ewz
wg
z
vwv
z
uwu
t
ev
o
1'''''
Shear production Buoyancy production Transport Dissipation
S B T D
Pino 2006
Dry Convective Boundary Layer
2. Properties
•Surface Layer
•Mixed Layer
•Inversion Layer
Turbulent Entrainment
quiet non-turbulent air
turbulent air
One-way entrainment: less turbulent air is entrained into more turbulent air
Mixed layer erodes into the Free atmosphere and is growing as a result of the entrainment proces
Entrainment Flux
Free Convection: Entrainment flux directly related to surface buoyancy flux
Rsrfv
invv Aw
w
,
,
Observations suggest 3.0~15.0RA
vevveinvv www ,
MLsrfv ww ,, Ri
AA
w
w RR
e
(Tennekes 1972)
Dry Convective Boundary Layer
1. Phenomenology
2. Properties
3. Models and Parameterization for Turbulent Transport
Prototype: Dry Convection PBL Case
•Initial Stable Temperature profile: qs=297 K ; = 2 10-3 K m-1
•No Moisture ; No Mean wind.
•Prescribed Surface Heat Flux : Qs = 6 10-2 K ms-1
h (km)
x(km)
0
51
Mean Characteristics of LES (virtual truth)
^Non-dimensionalise : z z/z*
w w/w*
t t/t*
Q Q/Qs
^
^
^
1700,0
04.0,0
5.1,0
*
*
z
w
Growth of the PBL
PBL height : Height where potential temperature has the largest gradient
Mixed Layer Model of PBL growth
Assume well-mixed profiles of
Use simple top-entrainment assumption.
2.0,
,
Rsrfv
invv Aw
w
Boundary layer height grows as:
)(tz
tAQtz
12)( 0
Q2.0
Simplest Model of PBL growth: Encroachment
Assume well-mixed profiles of
No top-entrainment assumed.
time
srfw )( 1t
1t
06.0
Boundary layer height grows as:
)(tz tQ
tz 02)(
Internal Structure of PBL
Rescale profiles
z
zz
ˆ min
Classic Parameterization of Turbulent Transport in de CBL
Eddy-diffusivity models, i.e.
zKw
zK
zw
zt
•Natural Extension of MO-theory
•Diffusion tends to make profiles well mixed
•Extension of mixing-length theory for shear-driven turbulence (Prandtl 1932)
K-profile: The simplest Practical Eddy Diffusivity Approach (1)
The eddy diffusivity K should forfill three constraints:
•K-profile should match surface layer similarity near zero
•K-profile should go to zero near the inversion
•Maximum value of K should be around: 1.0max zwK
z/zinv
0
1
0.1
K w* /zinv
2
0*
* 1ˆ
iih
i
hh z
z
z
z
w
uk
zw
KK
(Operational in ECMWF model)
A critique on the K-profile method (or an any eddy diffusivity method) (1)
0w
0w
0w
0
z
0
z
0
z
Diagnose the K that we would need from LES:
K>0
K<0
K>0
Forbidden area
“flux against the gradient”
Down-gradient diffusion cannot account for upward transport in the upper part of the PBL
Physical Reason!
•In the convective BL undiluted parcels can rise from the surface layer all the way to the inversion.
•Convection is an inherent non-local process.
•The local gradientof the profile in the upper half of the convective BL is irrelevant to this process.
•Theories based on the local gradient (K-diffusion) fail for the Convective BL.
“Standard “ remedy
NLwz
Kw
Add the socalled countergradient term:
zinv
Long History:
Ertel 1942
Priestley 1959
Deardorff 1966,1972
Holtslag and Moeng 1991
Holtslag and Boville 1993
B. Stevens 2003
And many more…………….
K
zK
Can we understand the characteristics of this system?
B. Stevens Monthly Weather Review (2000)
zK
ztAnd let’s find quasi-steady state solutions for
Non-dimensionalise: 1ˆ,ˆ,ˆ0
0 Q
z
zz
(leave the ^ out of the notation from now on)
0ˆ,ˆ
Q
zw
zw
KK
Quasi-Steady Solutions (1)
That is to say to find steady state solutions of:
0
z
0
z
0
z
This implies a linear flux!
Use as boundary conditions:
(Remember we work in non-dimensionalised variables)
Azzz
KQ
w
1
Which is to say solutions for which the shape of is not changing with time.
02
2
zK
zzt
Quasi-Steady Solutions (2)
Solution for the gradient
K
AzzK
z
1
Where K-profile is given by: 675.01)( zzzK and is constant
cstz
A
z
zzz
11ln
1)(
Solution of
Non-local
processesSurface fluxesTop-entrainment
AB
Surface fluxes
Top-entrainment
Quasi-Steady Solutions without countergradient (1)
cstz
A
z
zz
11ln
1)(
No countergradient:
A=0 no top-entrainment
A=-0.2 typical top entrainment value
LES K-profile without countergradient
Quasi-Steady Solutions without countergradient (2)
LES K-profile without countergradient
•The system tends to make quasi-steady solutions (in the absence of large scale forcings)
•So it allways produces linear fluxes
•It will find a quasi-steady profile that along with the K-profile provides such a linear flux
•So it is the dynamics that determines the profile (not the other way around!!!)
Quasi-Steady Solutions with countergradient term
cstz
A
z
zzz
11ln
1)(
A = -0.2, = 0.675,
=1.6, 3.2, 4.8
Height where
Countergradient: Conclusions
•Addition of a countergradient gives an improved shape of the internal structure
•But…
•How does it affect the interaction with the free atmosphere, i.e. what happens if we do not prescribe the top-entrainment anymore.?
•Can it be used in the presence of a cloud-topped boundary layer?
•Are there other ways of parameterizing the non-local flux?