CHAPTER 3: SIMPLE MODELS The atmospheric evolution of a species X is given by the continuity equation deposition emission transport (flux divergence; U is wind vector) local change in concentration with time chemical production and loss (depends on concentrations of other species) This equation cannot be solved exactly e need to construct model (simplified representation of complex system) Improve model, characterize its error Design model; make assumptions needed to simplify equations and make them solvable Design observational system to test model Define problem of interest Evaluate model with observations Apply model: make hypotheses, predictions

Questions 1. The Badwater Ultramarathon held every July starts from the bottom of Death Valley (100 m below sea level) and finishes at the top of Mt. Whitney (4300 m above sea level).  This race is a challenge to the human organism! By what percentage does the oxygen number density decrease between the start and the finish of the race? 2. Consider a pollutant emitted in an urban airshed of 100 km dimension. The pollutant can be removed from the airshed by oxidation, precipitation scavenging, or export. The lifetime against oxidation is 1 day. It rains once a week. The wind is 20 km/h. Which is the dominant pathway for removal?

X ONE-BOX MODEL Atmospheric “box”; spatial distribution of X within box is not resolved Chemical production Chemical loss Inflow Fin Outflow Fout P X L D E Deposition Emission Lifetimes add in parallel: Loss rate constants add in series:

NO2 has atmospheric lifetime ~ 1 day: strong gradients away from combustion source regions Satellite observations of NO2 columns

CO has atmospheric lifetime ~ 2 months: mixing around latitude bands Satellite observations

CO2 has atmospheric lifetime ~ 100 years: global mixing Assimilated observations

Using a box model to quantify CO2 sinks Pg C yr-1 On average, only 60% of emitted CO2 remains in the atmosphere – but there is large interannual variability in this fraction

SPECIAL CASE: SPECIES WITH CONSTANT SOURCE, 1st ORDER SINK Steady state solution (dm/dt = 0) Initial condition m(0) Characteristic time t = 1/k for reaching steady state decay of initial condition If S, k are constant over t >> t, then dm/dt g 0 and mg S/k: quasi steady state

LATITUDINAL GRADIENT OF CO2 , 2000-2012 Illustrates long time scale for interhemispheric exchange; use 2-box model to constrain CO2 sources/sinks in each hemisphere http://www.esrl.noaa.gov/gmd/ccgg/globalview/

TWO-BOX MODEL defines spatial gradient between two domains Mass balance equations: (similar equation for dm2/dt) If mass exchange between boxes is first-order: e system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)

EULERIAN RESEARCH MODELS SOLVE MASS BALANCE EQUATION IN 3-D ASSEMBLAGE OF GRIDBOXES The mass balance equation is then the finite-difference approximation of the continuity equation. Solve continuity equation for individual gridboxes Models can presently afford ~ 106 gridboxes In global models, this implies a horizontal resolution of 100-500 km in horizontal and ~ 1 km in vertical

IN EULERIAN APPROACH, DESCRIBING THE EVOLUTION OF A POLLUTION PLUME REQUIRES A LARGE NUMBER OF GRIDBOXES Fire plumes over southern California, 25 Oct. 2003 A Lagrangian “puff” model offers a much simpler alternative

PUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WIND CX(x, t) In the moving puff, wind CX(xo, to) …no transport terms! (they’re implicit in the trajectory) Application to the chemical evolution of an isolated pollution plume: CX,b CX In pollution plume,

COLUMN MODEL FOR TRANSPORT ACROSS URBAN AIRSHED Temperature inversion (defines “mixing depth”) Emission E In column moving across city, CX L x

LAGRANGIAN RESEARCH MODELS FOLLOW LARGE NUMBERS OF INDIVIDUAL “PUFFS” C(x, to+Dt) Individual puff trajectories over time Dt ADVANTAGE OVER EULERIAN MODELS: Computational performance (focus puffs on region of interest) DISADVANTAGES: Can’t handle mixing between puffs a can’t handle nonlinear processes Spatial coverage by puffs may be inadequate C(x, to) Concentration field at time t defined by n puffs

LAGRANGIAN RECEPTOR-ORIENTED MODELING Run Lagrangian model backward from receptor location, with points released at receptor location only backward in time Efficient cost-effective quantification of source influence distribution on receptor (“footprint”)