Review of the Mathematics of Climate Analysis (JC)

GLOBAL CLIMATE CHANGE: ECONOMICS, SCIENCE AND POLICY Review of the Mathematics of Climate Analysis Lecture Contents 1. 2. 3. 4. 5. 6. 7. What is a model (of the climate)? What variables do we consider? How to equate these variables? How to solve the equations? Issues of numerical solutions & example Simplified Models & 1D example Addressing Uncertainty What is a model of the climate? 1. A simplified representation of the Earth's atmosphere, land, oceans, and ice. 2. How these systems change and interact with each other over time scales of interest. 3. Is based on physical laws and uses real observations to make predictions. 4. Ideally, it is computationally efficient, and provides useful, precise, and accurate results. What variables do we consider? Atmospheric Models Use: 1. 2. 3. 4. 5. 6. 7. 8. 9. Radiation Temperature Wind Speed (u,v,w) Pressure Density Liquid Water Ice Water Aerosols Radiatively Active Gases 2003 Class Notes How to equate these variables? Conservation Laws (usually ODEs and PDEs) **There is nothing to be scared of!!!** Continuity ∂ ρ = − d (uρ ) − d (vρ ) − d ( wρ ) ∂t Energy dx dy dz ∂T d (uT ) d (vT ) d ( wT ) 1 ⎛ D (1 / ρ = − − − + ⎜J− p ∂t dx dy dz cv ⎝ Dt Momentum )⎞ ⎟ ⎠ ∂u d (uu ) d (vu ) d ( wu ) 1 dP = − − − − + 2Ω sin(φ )v − 2Ω cos(φ ) w + F ∂t dx dy dz ρ dx ∂v d (uv ) d (vv ) d ( wv ) 1 dP = − − − − − 2Ω sin(φ )u + F ∂t dx dy dz ρ dy ∂w d (uw) d (vw) d ( ww) 1 dP = − − − − − g + 2Ω cos(φ )u + F ∂t dx dy dz ρ dz Chemical Continuity ∂χ d (uχ ) d (vχ ) d ( wχ ) = − − − + E+ P− L ∂t dx dy dz How to solve the equations? 1. There is no analytic solution. 2. Discretize the equations in both time and space, treating each point as a distinct “box”. − Once a time step is chosen, any faster process will not be representable. − Each value in a box is constant over the entire box. − Once a spacial resolution is chosen, any spatially smaller process will not be representable. In the Atmospheric column: Wind vectors, humidity, clouds, temperature, and chemical species Horizontal exchange between columns Geography and orography Ocean grid Atmospheric grid Vertical exchange between levels At the surface: Ground temperature, water and energy, momentum and CO2 fluxes Bathymetry Within the Ocean column: Current vectors, temperature and salinity Issues of numerical solutions 1. Speed of Information (eg: CFL condition) − − We do not want information travelling faster than 1 model box per model step in time umax=150m/s ; dx=2.8ox2.8o(T42) ; Thus dt<30 min − Note: 230,000 boxes & 1,750,000 times 2. Numerical Stability & Precision 3. Subgrid Scale Processes − Cloud and Aerosol Microphysics − Deep Convection, Fronts, Eddies − Fast Chemistry, Photolosys, and Biology − Ice Formation and Melt, Snowfall, Rainfall Simplified models 1. 2D Models (Zonal or Latitudinal Averaged) − MIT IGSM Atmosphere − Show the Equator to Pole Temperature Gradient 2. 1D Models − Vertical Shells − Useful for planetary Energy Balance 3. These models aim to provide some level of reality, while being computationally efficient (eg: attempting to analyze uncertainty).  1D example 1. 2 box model: transport and decay − 3 and 30 year decay constants − Threshold and No Threshold cases on transport between boxes 2. What do you think will happen?  1D example (3 Year Decay)  1D example (30 Year Decay) Addressing uncertainty 1. Results of models can be compared against each other using the same set of inputs. − Does not address things left out of model formulation or model bias. − Works well for comparing different parameterizations. 2. Uncertainty in the model inputs can be formulated in a probabilistic sense and the results compared using the same model. − More computationally demanding