Program on Mathematical and Statistical Methods for Climate and the Earth System Opening Workshop, A Cloud of Numbers: Representing Physical Processes in the Earth System with Mathematics - Andrew Gettelman, Aug 24, 2017
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The document discusses the complexities of climate and weather modeling, emphasizing the limitations and uncertainties of these models, especially regarding parameterizations and sub-grid processes like cloud dynamics. It explains the significance of using statistical and stochastic methods to enhance model accuracy and represent small-scale variability effectively. The text also outlines the development and evaluation process of climate models, particularly the Community Atmosphere Model (CAM) series, while underscoring the role of uncertainty in climate modeling.
Program on Mathematical and Statistical Methods for Climate and the Earth System Opening Workshop, A Cloud of Numbers: Representing Physical Processes in the Earth System with Mathematics - Andrew Gettelman, Aug 24, 2017
- 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)