1. Atmospheric dry and shallow moist convection
Pier Siebesma
KNMI, The Netherlands
Het komende uur zal ik proberen antwoord te geven op een 4-tal vragen:
Waarom zijn wolken belangrijk om mee te nemen in het klimaatsysteem
Wat zijn de problemen om wolken goed te representeren in klimaatmodellen
Hoe goed zijn klimaatmodellen momenteel in staat om wolken correct te representeren.
Wat zijn de toekomstige ontwikkelingen.
Motivation
Fundamentals, Models, Equations
Dry Convective Boundary layer
Shallow Moist Convection (Friday)
Parameterizations of moist and dry convection (Saturday)

2. 1. Motivation

3. Deep Convective Clouds Precipitation Vertical turbulent transport
Tropopause 10km
Subsidence
~0.5 cm/s
inversion
10 m/s
Cloud base
~500m
Deep Convective Clouds
Precipitation
Vertical turbulent transport
Net latent heat production
Engine Hadley Circulation
Shallow Convective Clouds
No precipitation
Vertical turbulent transport
No net latent heat production
Fuel Supply Hadley Circulation
Stratocumulus
Interaction with radiation

4. ECMWF IFS overestimates Tradewind cumulus cloudiness:
The GCSS intercomparison project on cloud representation in GCM’s in the Eastern Pacific
ECMWF IFS overestimates Tradewind cumulus cloudiness:
Deep cu
scu
Shallow cu
Siebesma et al. (2005, QJRMS)

5. building material for clouds
Water vapour is the
building material for clouds
name
Symbol
Units
Definition
Near surface
values
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
Tip of the iceberg
qsat qt

6. water vapor clouds albedo lapse rate total

7. Sensitivity of the Tropical Cloud Radiative Forcing
to Global Warming in 15 AR4 OAGCMs
High-sensitivity GCMs
Low-sensitivity GCMs
(Bony and Dufresne, GRL, 2005)

8. Fundamentals , Models and Equations

9. Some fundamental notions on Turbulence (1)
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:
And treat both physical processes seperately:

10. Some fundamental notions on Turbulence (2)
1) The diffusion equation:
dissipation  gradient weakening  stabilizing
2) The advection equation:
General solution:
Advection term  gradient sharpening  instability

11. Some fundamental notions on Turbulence (3)
Competition between both processes determines the solution.
Compare both terms by making the equation dimensionless:
Reynolds Number
measures the ratio between the 2 terms.
dissipation dominates flow is stable  laminar
Non-linear advection term dominates  flow is unstable turbulent

12. Turbulence made by convection in the atmospheric boundary layer
Some fundamental notions on Turbulence (3):
Turbulence made by convection in the atmospheric boundary layer
Large eddy simulation of the convective boundary layer
u = 10-5m2s-1
U=10m/s
L=1000m
Macrostructure dominated by non-linear advection!!
Poor man’s artist impression of the convective boundary layer
Potential temperature
profile
Heat flux

13. Energy Cascade Some fundamental notions on Turbulence (4):
Energy injection through buoyancy at the macroscale
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.

14. J.L Richardson (1881-1953) Some fundamental notions on Turbulence (4):
free after Jonathan Swift (1733):

15. (Sloppy) Kolmogorov (1941)
Some fundamental notions on Turbulence (4):
(Sloppy) Kolmogorov (1941)
Kinetic Energy (per unit mass) : E
Dissipation rate : e
eddy size:
eddy velocity:
eddy turnover time:
Kolmogorov Assumption:
Kinetic Energy transfer is constant and equal to the dissipation rate 

16. Consequences of Kolmogorov (1941)
Some fundamental notions on Turbulence (5):
Consequences of Kolmogorov (1941)
Structure functions:
Fourier transform of kinetic energy
Famous 5/3-law!!
Largest eddies are the most energetic
Kolmogorov scale : the scale at which dissipation begins to dominate:

17. Energy Spectra in the atmosphere
data: van Hove 1957
cyclones
microscale
turbulence
diurnal cycle
spectral gap
100 hours
1 hour
0.01 hour

18. energy spectra at z=150m below stratocumulus
U Spectrum
V Spectrum
W Spectrum
500m
Duynkerke 1998

19. Governing Equations for incompressible flows in the atmosphere
Continuity Equation
(incompressible)
with
NS Equations
gravity
term
coriolis
term
Heat equation
Moisture equation
Condensed water eq.
Gas law

20. The Zoo of Atmospheric Models
1 km
10 km
100 km
1000 km
10000 km 
turbulence
Cumulus
clouds
Cumulonimbus
clouds
Mesoscale
Convective systems
Extratropical
Cyclones
Planetary
waves
Subgrid
Large Eddy Simulation (LES) Model
Cloud System Resolving Model (CSRM)
Numerical Weather Prediction (NWP) Model
Global Climate Model

21. Global Climate and NWP models (Dx>10km)
Subgrid
To be parameterized

22. Large Eddy Simulation (LES) Model (Dx<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
Inertial Range
Resolution LES 5 3
ln(Energy)
DissipationRange
ln(wave number)

23. Dynamics of thermodynamical variables in LES
Resolved
turbulence
subgrid
turbulence
Subgrid turbulence:
Remark:
Richardson law!!

24. Cloud Scheme in LES
Simple: All or Nothing: { {

25. Turbulent Kinetic Energy (TKE) Equation
Definition:
Assume:
No mean wind
No horizontal flux terms
Reynolds Averaged budget TKE-equation:
Shear production
Buoyancy production
Transport
Dissipation S B T D
Laminar flow
Richardson Number:
Shear driven turbulence
Buoyancy driven turbulence

26. Mixed layer turbulent kinetic energy budget
dry PBL
Stull 1988
normalized

27. Conditions for Atmospheric Convection
Reynolds Number
Condition for fully developed turbulence
Richardson Number
Condition for buoyancy drive turbulence
Atmospheric Convection = Turbulence driven by Buoyancy

28. Objectives Tools Methods Application
To “understand” the various aspects of atmospheric convection
To find closures (for the turbulent fluxes and variances)
Tools
Observations, Large Eddy Simulation (LES) models
Methods
Dimension analysis, Similarity theory, common sense
Application
Climate and Numerical Weather Prediction (NWP) Models

29. (Simplified) Working Strategy Versions of Climate Models
See
Large Eddy Simulation (LES) Models
Cloud Resolving Models (CRM)
Single Column Model
Versions of Climate Models
3d-Climate Models
NWP’s
Global observational
Data sets
Observations from
Field Campaigns
Development
Testing
Evaluation

30. Dry Convective Boundary Layer
Phenomenology
Properties
Models and Parameterization for Convective Transport

31. 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)
tropopause we z Q0 q

32. Can we see the convective PBL?
July 2001
Downtown LA
PBL top
10km
(Courtesy Martin Kohler)

33. Typical Profiles of the convective BL
Entrainment Zone
Mixed layer
Surface layer
Stull 1988

34. LES View of the Dynamics: potential temperature
Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

35. LES View of the Dynamics: vertical velocity
Courtesy: Chiel van Heerwaarden, Wageningen University, Netherlands

36. Horizontal Crosscut Inversion layer Mixed Layer Surface layer
Irregular polygonal structures!
Moeng 1998

37. Dry Convective Boundary Layer
2. Properties
Surface Layer
Mixed Layer
Inversion Layer

38. Monin-Obukhov Similarity
Construct dimensionless gradient terms:
and evaluate this as a function of the stability parameter
unstable
stable
Fleagle and Businger 1980

39. MO theory allows to formulate the turbulent fluxes in a diffusivity form:
Diffusion Eq.

40. Dry Convective Boundary Layer
2. Properties
Surface Layer
Mixed Layer
Inversion Layer

41. Scaling Parameters for the convective mixed layer
Relevant parameters:
form
units
1) TKE production through buoyancy:
2) Depth of the boundary layer:
Construct a convective velocity scale:
Interpretation: velocity that results if all potential energy is converted into kinetic energy in an eddy of size z*

42. Typical Numbers

43. Dimensionless vertical velocity variance (in the free convective limit)
Garrat 1992)

44. Mixed layer turbulent kinetic energy budget (LES)
Shear production
Buoyancy production
Dissipation
Transport D B T S
Pino 2006

45. Dry Convective Boundary Layer
2. Properties
Surface Layer
Mixed Layer
Inversion Layer

46. 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

47. Entrainment Flux Free Convection:
Entrainment flux directly related to surface buoyancy flux
Observations suggest
(Tennekes 1972)

48. Dry Convective Boundary Layer
Phenomenology
Properties
Models and Parameterization for Turbulent Transport

49. Prototype: Dry Convection PBL Case
Initial Stable Temperature profile: qs=297 K ; g = K m-1
No Moisture ; No Mean wind.
Prescribed Surface Heat Flux : Qs = K ms-1
h (km) 1 5
x(km)

50. Mean Characteristics of LES (virtual truth)^
Non-dimensionalise : z  z/z*
w  w/w*
t  t/t*
Q  Q/Qs

51. Growth of the PBL
PBL height : Height where potential temperature has the largest gradient

52. Mixed Layer Model of PBL growth
Assume well-mixed profiles of Q.
Use simple top-entrainment assumption. q
Boundary layer height grows as:

53. Simplest Model of PBL growth: Encroachment
Assume well-mixed profiles of Q.
No top-entrainment assumed.
Boundary layer height grows as:
time

54. Internal Structure of PBL
Rescale profiles

55. Classic Parameterization of Turbulent Transport in de CBL
Eddy-diffusivity models, i.e.
Natural Extension of MO-theory
Diffusion tends to make profiles well mixed
Extension of mixing-length theory for shear-driven turbulence (Prandtl 1932)

56. 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:
z/zinv 1
0.1
K w* /zinv
(Operational in ECMWF model)

57. “flux against the gradient”
A critique on the K-profile method (or an any eddy diffusivity method) (1)
Diagnose the K that we would need from LES:
K>0
Forbidden area
“flux against the gradient”
K<0
K>0
Down-gradient diffusion cannot account for upward transport in the upper part of the PBL

58. 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.

59. “Standard “ remedy Add the socalled countergradient term:
Long History:
Ertel
Priestley
Deardorff ,1972
Holtslag and Moeng
Holtslag and Boville
B. Stevens
And many more…………….
zinv

60. Can we understand the characteristics of this system?
B. Stevens Monthly Weather Review (2000)
Non-dimensionalise:
(leave the ^ out of the notation from now on)
And let’s find quasi-steady state solutions for

61. Quasi-Steady Solutions (1)
That is to say to find steady state solutions of:
Which is to say solutions for which the shape of is not changing with time.
This implies a linear flux!
Use as boundary conditions:
(Remember we work in non-dimensionalised variables)

62. Quasi-Steady Solutions (2)
Solution for the gradient
Where K-profile is given by:
and g is constant
Solution of Q:
Top-entrainment
Surface fluxes
Non-local
processes
Surface fluxes
Top-entrainment B A

63. Quasi-Steady Solutions without countergradient (1)
No countergradient:
A=0 no top-entrainment
A=-0.2 typical top entrainment value
K-profile without countergradient
LES

64. Quasi-Steady Solutions without countergradient (2)
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!!!)
K-profile without countergradient
LES

65. Quasi-Steady Solutions with countergradient term
Height where
A = -0.2, k = 0.675,
gk =1.6, 3.2, 4.8

66. 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?