1. GLOBAL MODELS OF ATMOSPHERIC COMPOSITION Daniel J. Jacob Harvard University

2. HOW TO MODEL ATMOSPHERIC COMPOSITION? x HOW TO MODEL ATMOSPHERIC COMPOSITION? Solve continuity equation for chemical mixing ratios C i (x, t) FiresLand biosphere Human activity Lightning Ocean Volcanoes Transport Eulerian form: Lagrangian form: U = wind vector P i = local source of chemical i L i = local sink Chemistry Aerosol microphysics

3. EULERIAN MODELS PARTITION ATMOSPHERIC DOMAIN INTO GRIDBOXES Solve continuity equation for individual gridboxes Detailed chemical/aerosol models can presently afford -10 6 gridboxes In global models, this implies a horizontal resolution of ~ 1 o (~100 km) in horizontal and ~ 1 km in vertical This discretizes the continuity equation in space Chemical Transport Models (CTMs) use external meteorological data as input General Circulation Models (GCMs) compute their own meteorological fields

4. OPERATOR SPLITTING IN EULERIAN MODELS … and integrate each process separately over discrete time steps: Split the continuity equation into contributions from transport and local terms: These operators can be split further: split transport into 1-D advective and turbulent transport for x, y, z (usually necessary) split local into chemistry, emissions, deposition (usually not necessary) Reduces dimensionality of problem

5. QUESTIONS 1.The Eulerian form of the continuity equation is a first-order PDE in 4 dimensions. What are suitable boundary conditions for each of these dimensions? 1.Textbooks will often tell you that operator splitting (transport vs. local in the continuity equation) requires time steps that are much smaller than the time scales for change in the system, but it actually also works fine for species that are very short-lived relative to the time step. Error is largest for species that have lifetimes of magnitude comparable to the splitting time step. Explain.

6. SPLITTING THE TRANSPORT OPERATOR U Wind velocity U has turbulent fluctuations over time step  t : Time-averaged component (resolved) Fluctuating component (stochastic) Further split transport in x, y, and z to reduce dimensionality. In x direction: Split transport into advection (mean wind) and turbulent components: advection turbulence (1 st -order closure) advection operator turbulent operator

7. SOLVING THE EULERIAN ADVECTION EQUATION Equation is conservative: need to avoid diffusion or dispersion of features. Also need mass conservation, stability, positivity… All schemes involve finite difference approximation of derivatives : order of approximation → accuracy of solution Classic schemes: leapfrog, Lax-Wendroff, Crank-Nicholson, upwind, moments… Stability requires Courant number u  t/  x < 1 … limits size of time step Addressing other requirements (e.g., positivity) introduces non-linearity in advection scheme

8. VERTICAL TURBULENT TRANSPORT (BUOYANCY) Convective cloud (0.1-100 km) Model grid scale Model vertical levels updraft entrainment downdraft detrainment Wet convection is subgrid scale in global models and must be treated as a vertical mass exchange separate from transport by grid-scale winds. Need info on convective mass fluxes from the model meteorological driver. generally dominates over mean vertical advection K-diffusion OK for dry convection in boundary layer (small eddies) Deeper (wet) convection requires non-local convective parameterization

9. LOCAL (CHEMISTRY) OPERATOR: solves ODE system for n interacting species System is typically “stiff” (lifetimes range over many orders of magnitude) → implicit solution method is necessary. Simplest method: backward Euler. Transform into system of n algebraic equations with n unknowns Solve e.g., by Newton’s method. Backward Euler is stable, mass-conserving, flexible (can use other constraints such as steady-state, chemical family closure, etc… in lieu of  C  t )  But it is expensive. Most 3-D models use higher-order implicit schemes such as the Gear method. For each species

10. SPECIFIC ISSUES FOR AEROSOL CONCENTRATIONS A given aerosol particle is characterized by its size, shape, phases, and chemical composition – large number of variables! Measures of aerosol concentrations must be given in some integral form, by summing over all particles present in a given air volume that have a certain property If evolution of the size distribution is not resolved, continuity equation for aerosol species can be applied in same way as for gases Simulating the evolution of the aerosol size distribution requires inclusion of nucleation/growth/coagulation terms in P i and L i, and size characterization either through size bins or moments. Typical aerosol size distributions by volume nucleation condensation coagulation

11. LAGRANGIAN APPROACH: TRACK TRANSPORT OF POINTS IN MODEL DOMAIN (NO GRID) UtUt U’  t Transport large number of points with trajectories from input meteorological data base (U) + random turbulent component (U’) over time steps  t Points have mass but no volume Determine local concentrations as the number of points within a given volume Nonlinear chemistry requires Eulerian mapping at every time step (semi-Lagrangian) PROS over Eulerian models: no Courant number restrictions no numerical diffusion/dispersion easily track air parcel histories invertible with respect to time CONS: need very large # points for statistics inhomogeneous representation of domain convection is poorly represented nonlinear chemistry is problematic position t o position t o +  t

12. LAGRANGIAN RECEPTOR-ORIENTED MODELING Run Lagrangian model backward from receptor location, with points released at receptor location only Efficient cost-effective quantification of source influence distribution on receptor (“footprint”) Enables inversion of source influences by the adjoint method (backward model is the adjoint of the Lagrangian forward model)

13. EMBEDDING LAGRANGIAN PLUMES IN EULERIAN MODELS Release puffs from point sources and transport them along trajectories, allowing them to gradually dilute by turbulent mixing (“Gaussian plume”) until they reach the Eulerian grid size at which point they mix into the gridbox Advantages: resolve subgrid ‘hot spots’ and associated nonlinear processes (chemistry, aerosol growth) within plume Difference with Lagrangian approach is that (1) puff has volume as well as mass, (2) turbulence is deterministic (Gaussian spread) rather than stochastic S. California fire plumes, Oct. 25 2004

14. THE INVERSE MODELING PROBLEM Optimize values of an ensemble of variables (state vector x ) using observations: THREE MAIN APPLICATIONS FOR ATMOSPHERIC COMPOSITION: 1.Retrieve atmospheric concentrations ( x ) from observed atmospheric radiances ( y ) using a radiative transfer model as forward model 2.Invert sources ( x ) from observed atmospheric concentrations ( y ) using a CTM as forward model 3.Construct a continuous field of concentrations ( x ) by assimilation of sparse observations ( y ) using a forecast model (initial-value CTM) as forward model a priori estimate x a +  a observation vector y forward model y = F(x) +  “MAP solution” “optimal estimate” “retrieval” Bayes’ theorem

15. BAYES’ THEOREM: FOUNDATION FOR INVERSE MODELS P(x) = probability distribution function (pdf) of x P(x,y) = pdf of (x,y) P(y|x) = pdf of y given x a priori pdf observation pdf normalizing factor (unimportant) a posteriori pdf Maximum a posteriori (MAP) solution for x given y is defined by solve for  P(x,y)dxdy  Bayes’ theorem

16. SIMPLE LINEAR INVERSE PROBLEM FOR A SCALAR use single measurement used to optimize a single source a priori bottom-up estimate x a  a Monitoring site measures concentration y Forward model gives y = kx “Observational error”   instrument fwd model y = kx   Max of P(x|y) is given by minimum of cost function Solution: where g is a gain factor Alternate expression of solution: where a = gk is an averaging kernel  solve for Assume random Gaussian errors, let x be the true value. Bayes’ theorem: Variance of solution:

17. GENERALIZATION: CONSTRAINING n SOURCES WITH m OBSERVATIONS Linear forward model: A cost function defined as is generally not adequate because it does not account for correlation between sources or between observations. Need vector-matrix formalism: Jacobian matrix

18. JACOBIAN MATRIX FOR FORWARD MODEL Consider a non-linear forward model y = F(x) Use of vector-matrix formalism requires linearization of forward model and linearize it about x a : is the Jacobian of F evaluated at x a with elements If forward model is non-linear, K must be recalculated iteratively for successive solutions K T is the adjoint of the forward model (to be discussed later) Construct Jacobian numerically column by column: perturb x a by  x i, run forward model to get corresponding  y

19. GAUSSIAN PDFs FOR VECTORS A priori pdf for x : Scalar x Vector where S a is the a priori error covariance matrix describing error statistics on ( x-x a ) In log space:

20. OBSERVATIONAL ERROR COVARIANCE MATRIX observation true value instrument error fwd model errorobservational error Observational error covariance matrix is the sum of the instrument and fwd model error covariance matrices: How well can the observing system constrain the true value of x ? Corresponding pdf, in log space:

21. MAXIMUM A POSTERIORI (MAP) SOLUTION Bayes’ theorem: MAP solution: miminize cost function J : Solve for Analytical solution: with gain matrix bottom-up constraint top-down constraint

22. PARALLEL BETWEEN VECTOR-MATRIX AND SCALAR SOLUTIONS: Scalar problemVector-matrix problem MAP solution: Gain factor: A posteriori error: Averaging kernel: Jacobian matrixsensitivity of observations to true state Gain matrix sensitivity of retrieval to observations Averaging kernel matrix sensitivity of retrieval to true state

23. A LITTLE MORE ON THE AVERAGING KERNEL MATRIX A describes the sensitivity of the retrieval to the true state and hence the smoothing of the solution: smoothing error retrieval error MAP retrieval gives A as part of the retrieval: Other retrieval methods (e.g., neural network, adjoint method) do not provide A # pieces of info in a retrieval = degrees of freedom for signal (DOFS) = trace(A)

24. APPLICATION TO SATELLITE RETRIEVALS Here y is the vector of wavelength-dependent radiances (radiance spectrum); x is the state vector of concentrations; forward model y = F(x) is the radiative transfer model Illustrative MOPITT averaging kernel matrix for CO retrieval MOPITT retrieves concentrations at 7 pressure levels; lines are the corresponding columns of the averaging kernel matrix trace(A) = 1.5 in this case; 1.5 pieces of information

25. Analytical solution to inverse problem requires (iterative) numerical construction of the Jacobian matrix K and matrix operations of dimension (mxn); this limits the size of n, i.e., the number of variables that you can optimize Address this limitation with Kalman filter (for time-dependent x ) or with adjoint method 

26. BASIC KALMAN FILTER to optimize time-dependent state vector a priori x a,0 ± S a,0 toto time observations state vector y0y0 Evolution model M for [t 0, t 1 ]: t1t1 y1y1 Apply evolution model for [t 1, t 2 ]… etc.

27. ADJOINT INVERSION (4-D VAR) ° ° ° ° aa 22 11 33 x2x2 x1x1 x3x3 xaxa Minimum of cost function J Solve numerically rather than analytically 1. Starting from a priori x a, calculate 2. Using an optimization algorithm (BFGS), get next guess x 1 3. Calculate, get next guess x 2 4. Iterate until convergence

28. NUMERICAL CALCULATION OF COST FUNCTION GRADIENT Adjoint model is applied to error-weighted difference between model and obs …but we want to avoid explicit construction of K and since ( AB) T = B T A T, Apply transpose of tangent linear model to the adjoint forcings; for time interval [t 0, t n ], start from observations at t n and work backward in time until t 0, picking up new observations (adjoint forcings) along the way. Construct tangent linear model describing evolution of concentration field over time interval [ t i-1, t i ] Sensitivity of y (i ) to x (0 ) at time t 0 can then be written of forward model adjoint “adjoint forcing”