Physical processes in the earth system are modeled with mathematical representations called parameterizations. This talk will describe some of the conceptual approaches and mathematics used do describe physical parameterizations focusing on cloud parameterizations. This includes tracing physical laws to discrete representations in coarse scale models. Clouds illustrate several of the complexities and techniques common to many physical parameterizations. This includes the problem of different scales, sub-grid scale variability. Discussions of mathematical methods for dealing with the sub-grid scale will be discussed. In-exactness or indeterminate problems for both weather and climate will be discussed, including the problems of indeterminate parameterizations, and inexact initial conditions. Different mathematical methods, including the use of stochastic methods, will be described and discussed, with examples from contemporary earth system models.

1. A Cloud of Numbers: Representing
Physical Processes in the Earth
System with Mathematics
Andrew Gettelman, NCAR

3. All models are wrong. But some are useful.
-George E. P. Box, 1976 (Statistician)
What is a model?
A model is an imperfect representation

4. Ceci n’est pas un nuage

5. How does a model see clouds?

6. Cloud Resolving Model (CRM)
Convective Cloud

7. Outline
• What is a Model?
• What is climate? What is weather?
• Physical Laws
• What is a parameterization? (Especially Clouds)
• Problem of Scales: Sub-grid variability
• From Empirical to Stochastic parameterizations
• Putting it all together
• Summary/Conclusions/State of the Art

8. What is Climate?
“Climate is What you Expect, Weather is what you get”
• Climate = distribution (probability) of possible weather
• Weather = Chaos
• Chaos theory ‘developed’ by a meteorologist (Lorenz, 1961)
• Simple model to explain why the weather is not predictable

9. Chaos Theory: Lorenz Attractor
Climate
Weather
Weather
Pattern (climate) is stable
Nearby trajectories can be different (weather)

10. Physics of Climate
Laplace
Newton
Carnot
ClausiusKelvin Maxwell Arrhenius

11. Fundamentals of Climate
(and climate modeling)
• Conservation of Mass and Energy
• Laws of Motion on a rotating sphere
• Fluid Dynamics
• Mathematical integration & Statistics
• Radiative transfer
• Absorption and Scattering of Solar Energy
• Thermal Transfer, Conduction & Convection
• Thermodynamics: humidity, salinity
• Chemistry

12. Hydrostatic Primitive Equations
‘Hydrostatic’ = ‘larger’ scale (> 10km)
(parameterizations)

13. Parameterization is a Series of Processes
Example: Cloud Microphysics
• 6 class, 2 moment scheme
• Seifert and Behang 2001
• Processes
• Maybe a matrix better?
• Break down by processes
S = Σ(Si)
Q = Σ(Qi)
Seifert, Personal Communication

14. Community Atmosphere Model (CAM5)
Dynamics
Boundary Layer
Macrophysics
Microphysics Shallow Convection
Deep Convection
Radiation
Aerosols
Clouds (Al),
Condensate (qv, qc)
Mass,
Number Conc
A, qc, qi, qv
rei, rel
Surface Fluxes
Precipitation
Detrained qc,qi
Clouds & Condensate:
T, Adeep, Ash
A = cloud fraction, q=H2O, re=effective radius (size), T=temperature
(i)ce, (l)iquid, (v)apor
Finite Volume Cartesian
3-Mode
Liu, Ghan et al
2 Moment
Morrison & Gettelman
Ice supersaturation
Diag 2-moment Precip
Crystal/Drop
Activation
Park et al: Equil PDF Zhang & McFarlane
Park &
Bretherton
Bretherton
& Park
CAM5.1-5.3: IPCC AR5 version (Neale et al 2010)
RRTMG

15. Climate Model Schematic
Dynamical Core
Plumbing that Connects Tendencies
Parameterizations
(Tendency Generators)

16. Putting Parameterizations Together
Dynamical core
Connections
Deep Convection
Microphysics
Condensation
/Fraction

17. How do we develop a model?
• Parameterization development
• Evaluation against theory and observations
• Constrain each process & parameterization to be
physically realistic
• Conservation of mass and energy
• Other physical laws
• Connect each process together (plumbing)
• Coupled: connect each component model
• Global constraints
• Make the results match important global or emergent
properties of the earth system
• “Training” (optimization, tuning)

18. Why doesn’t it work?
• Process rates are uncertain at a given scale
“all models are wrong…” (uncertainty v. observations)
• Problems with the dynamical core
• Problems connecting processes
• Each parameterization is it’s own ‘animal’ & performs
differently with others (tuning = training, limits)
Each parameterization needs to contribute to a self
consistent whole

19. Parameterization
Component (ocean)
Earth System
Component (atmos)
Component (land)
Process
Stages of Optimization (Training)

20. Stochastic Nature of Parameterization
• Climate is a distribution: how is it sampled?
• Distribution = PDF in space rather than time
• Variability occurs at the small scale: at all scales.
• How to represent it?

21. Scales of Atmospheric Processes
Resolved Scales
Global Models
Future Global Models
Cloud/Mesoscale Models
Large Eddy Simulation Models
DNS Turbulence Models

22. What is ‘Sub-Grid’ scale
What is resolved at different scales?
Usually ’resolved’ is 4∆x (resolve a wave)
• Global Scale (15-400km)
• Resolve ‘synoptic’ systems and the general circulation
• Regional/Mesoscale (0.5-20km)
• Start to resolve stratiform clouds, mesoscale circulations
• ‘LES’ scale (10m – 200m)
• Clouds, updrafts (convection)
• Turbulence (1-50m)
• ‘Full’ representation of convection

23. Scales and parameterization
• If processes have a large separation from the grid scale:
this is usually okay
• Statistical (empirical) treatments often work: can represent
small scale uniquely with state of large scale
• When the scales get close together: this often creates
problems
• Example: representation of moist convection, or cloud
dynamics in general
• Convective equilibrium is a large scale process
• Stochastic (sampling) methods can represent the small scale
• Another way to phrase the issue: proper representation
of sub-grid variability

24. Making Models ‘Stochastic’
Stochastic: randomly determined; having a random probability distribution or
pattern that may be analyzed statistically but may not be predicted precisely.
Methods:
A. Build in sub-grid statistics
• Cloud Fraction, Size distributions, Statistically based
turbulence schemes, co-variance
B. Sample Sub-grid statistics
• Cloud overlap, sub-columns
C. Perturb Initial Conditions
• Ensemble forecasting
D. Perturb parameterization tendencies, perturb
parameterization ‘unknowns’

25. Sub-Grid Humidity and Clouds
Liquid clouds form when RH = 100% (q>esat)
But if there is variation in RH in space, some
clouds will form before mean RH = 100%
Horizontal fraction of Grid Box
RH
100%
Mean RH
0.5 1.0
Clear
(RH < 100%)
Cloudy
(RH = 100%)
0.0
Humidity in a grid box
with sub-grid variation

26. Sub-Grid Humidity and Clouds
Liquid clouds form when RH = 100% (q>esat)
But if there is variation in RH in space, some
clouds will form before mean RH = 100%
Fraction of Grid Box
RH
100%
Mean RH
0.5 1.0
Clear
(RH < 100%)
Cloudy
(RH = 100%)
0.0
Assumed Cumulative
Distribution function of
Humidity in a grid box
with sub-grid variation

27. Sub-Grid Cloud Assumptions
Pincus et al 2006, Fig 1: Schematic of stochastically generated clouds.
• Fractional Cloudiness is useful
• Can even break it down into
subcolumns
• Microphysics and Radiation are
non-linear
• Σ f(x) ≠ f(x)
• Usually uses stochastic elements
to distribute cloud, water, etc

28. Size Distributions as Statistics
Microphysical Schemes
• ‘Explicit’ or Bin Microphysics
• Bulk Microphysics
• Bulk Moment based microphysics
Represent the number of particles in each size ‘bin’
One species(number) for each mass bin
Computationally expensive, but ‘direct’
Represent the total mass and number
Computationally efficient
Approximate processes
Represent the size distribution with a function
Have a distribution for different ‘Classes’
(Liquid, Ice, Mixed Phase)
Hybrid: functional form makes complexity possible

29. Using Sub-Grid Statistics
Auto-conversion (Ac) & Accretion (Kc)
Khairoutdinov & Kogan 2000: regressions from LES experiments with explicit bin model
• Auto-conversion an inverse function of drop number
• Accretion is a mass only function
Balance of these processes (sinks) controls mass and size of cloud drops
Ac =
Kc=

30. Autoconversion and Accretion & Sub Grid
• If cloud water has sub-grid variability, process rate will not
be constant.
• Autoconversion/accretion: depends on co-variance of cloud
& rain water
• Assuming a distribution (log-normal) a power law M=axb
can be integrated over to get a grid box mean M
and vx is the normalized variance vx = x2/σ2
E = Enhancement factor
E.g.: Morrison and Gettelman 2008, Lebsock et al 2013

31. Observing co-variance of cloud & rain
Lebsock et al 2013
• Observe Cloud/Rain from satellites (CloudSat)
• Calculate variance, mean & normalized variance (v) or
homogeneity
• Observational estimate of Ac & Au enhancement factors

32. Enhancement Factors
• More enhancement in drier
regions, and regions with more
variance
• Good example of observing
higher order effects and sub-
grid scale variability from Space
• Also an example of how to use
observations to constrain
microphysical process rates.
Lebsock et al 2013

33. Sample Sub Grid
• McICA: Monte Carlo Approach to
radiation and cloud overlap
• Pincus et al 2003,2006: sample
individual bands
• Noise often ‘small’
• Sometimes extreme in SW (200Wm-2)
• How much ‘noise’ is okay for climate?
For Weather?
Pincus et al 2006, Fig3

34. Dynamics-Based PDFs for Cloud Parameterization: Motivation
• Moisture-based PDFs (widely used to represent cloud cover in GCMs)
are not linked to dynamics of cloud formation and dissipation
RH
• Key cloud processes (drop activation,
entrainment, and precip. Formation) are closely
linked to vertical motions
• Need joint distribution of thermodynamics and
dynamics (vertical motion)
• Assume a form of the distribution (Double
Gaussian) and predict the moments

35. CLUBB: Cloud Layers Unified By Binormals
Advance prognostic moment equations
Select PDF from functional
form to match
moments
Use PDF to close higher-order
moments, buoyancy terms
Diagnose cloud fraction,
liquid water from PDF.
Pass to cloud
microphysicsAdapted from Golaz et al. 2002a,b (JAS)
Higher Order Closure

36. Sub-columns
Sub-column Generator
Example: SILHS (Larson et al)
(Sub-grid Importance Latin Hypercube Sampling)
• Sample PDF cloud for cloud microphysics.
• Draw subcolumns from a PDF
• Run the microphysics on each sub-column
• Average back
• Can also ‘weight’ unequally
• Goal: run with all physics (including radiation)
• Simpler forms exist.

37. Adding Variance: Clouds
Liquid clouds form when RH = 100% (q>esat)
But if there is variation in RH in space, some
clouds will form before mean RH = 100%
Fraction of Grid Box
RH
100%
Mean RH
0.5 1.00.0
RH + Variance
Variance changes the mean cloud fraction
Other examples: ‘sub-grid’ velocities, wave perturbations

38. Putting it all Together
Community Atmosphere Model (CAM5)
Dynamics
Boundary Layer
Macrophysics
Microphysics Shallow Convection
Deep Convection
Radiation
Aerosols
Clouds (Al),
Condensate (qv, qc)
Mass,
Number Conc
A, qc, qi, qv
rei, rel
Surface Fluxes
Precipitation
Detrained qc,qi
Clouds & Condensate:
T, Adeep, Ash
A = cloud fraction, q=H2O, re=effective radius (size), T=temperature
(i)ce, (l)iquid, (v)apor
Finite Volume Cartesian
3-Mode
Liu, Ghan et al
2 Moment
Morrison & Gettelman
Ice supersaturation
Diag 2-moment Precip
Crystal/Drop
Activation
Park et al: Equil PDF Zhang & McFarlane
Park &
Bretherton
Bretherton
& Park
CAM5.1-5.3: IPCC AR5
version (Neale et al 2010)
RRTMG

39. Community Atmosphere Model (CAM6)
Dynamics
Unified Turbulence
Radiation
Aerosols
Clouds (Al),
Condensate (qv, qc)
Mass,
Number Conc
A, qc, qi, qv
rei, rel
Surface Fluxes
Precipitation
Clouds & Condensate:
T, Adeep, Ash
A = cloud fraction, q=H2O, re=effective radius (size), T=temperature
(i)ce, (l)iquid, (v)apor
4-Mode
Liu, Ghan et al
2 Moment
Morrison & Gettelman
Ice supersaturation
Prognostic 2-moment Precip
Crystal/Drop
Activation
Zhang-
McFarlane
CMIP6 model
Deep Convection
CLUBB
Sub-StepMicrophysics
Finite Volume Cartesian

40. Community Atmosphere Model (CAM6+)
Dynamics
Unified Turbulence
Microphysics
Sub Columns
Radiation
Aerosols
Mass,
Number Conc
A, qc, qi, qv
rei, rel
Surface Fluxes
Clouds & Condensate:
T, Adeep, Ash
A = cloud fraction, q=H2O, re=effective radius (size), T=temperature
(i)ce, (l)iquid, (v)apor
4-Mode
Liu, Ghan et al
2 Moment
Morrison & Gettelman
Ice supersaturation
Prognostic 2-moment Precip
Crystal/Drop
Activation
Now in development: Sub-columns
CLUBB
Averaging
Sub-Step
Precipitation

41. Model Ensembles: Statistics
• Single Model Initial
Condition Ensembles
• Multiple Scenario
Ensembles
• Multi—Model
Ensembles (perturbing
models)
Sanderson et al 2015, Climatic Change
Initial Condition Ensemble: Climate

42. Model Perturbations
• Parameter perturbations to a model/emulators
• Sometimes used for climate model sensitivity tests
• Or single model ensembles
• Adding ‘noise’ to initial conditions
• Like cloud example, but for ensemble spread
• Adding ‘noise’ to models
• Stochastic backscatter: (tendency noise)
• Used for ensemble prediction
• ’Adding variance’ to get right up-scale behavior (cloud
fraction example)

43. Summary
• Models Are Uncertain
• Sub-grid scale processes (clouds) are critical
• Makes parameterizations uncertain
• Statistics and ’stochastic’ processes can help
• Several different methods…
100%
0.5 1.00.0

44. Statistical Methods are our friend
1. Build in sub-grid statistics
When scale separation exists (Microphysics, Turbulence)
2. Sample Sub-grid statistics
When near to the variability scale (cloud updrafts)
3. Adding variance
Perturb Initial conditions [(Forecasting)
Perturb sub-grid scales (velocity, adding ’noise’)
4. Perturb parameterization ‘unknowns’, emulators
Statistical tests

45. Current/Future Prospects
Stochastic Parameterization in climate models
Goals for Climate Modeling
• Going to refined mesh simulations
• Unified Weather to Climate
How to use stochastic methods?
• Sub-grid turbulence, predict moments
• PDFs collapse as variance goes down
• Sampling
• Use distributions to represent updrafts
• Use of ensembles and stochastic methods for
optimization (parameter sweeps, emulators)