Atmospheric Chemistry and Transport - Continuity Equation and Basic Structures

Atmospheric Physics & Chemistry COMPONENTS OF ATMOSPHERIC CHEMISTRY MODELS DYNAMICAL EQUATIONS MASS CONTINUITY EQUATION Winds , eddy diffusion coefficients MOMENTUM EQUATIONS Temperature THERMODYNAMIC EQUATION CHEMICAL Rates for Heating CONTINUITY EQUATIONS .([i]V) ~ ph Rates for Chemistry UV r Fo Flu xe s n tio ia oc iss od es ot rat { = Pi - Li - EQUATIONS For unpredicted green house gases use scenarios or extrapolations For source gases use predictions, extrapolations or scenarios ∂t ∆ RADIATION { ∂ [i] Concentrations (O3, etc.) Interactions Between Air Pollution and Climate Stratosphere UV O2 NO2 O3 O2 UV OH NO HNO3 O(1D) OH N2O Lightning H 2O CO2 Hydrosphere H2SO4 HO2 O2 CO OH CH4 CO2 OH CFCs BC SO2 Biosphere & Human Activity Greenhouse Gases Primary & Secondary Pollutants Absorbing Aerosols (BC) Reactive Free Radical/Atom Less Reactive Radicals Reflective Aerosols 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 Combining Chemistry and Transport: The Continuity Equation Physical picture: (Eulerian) i Oz dxdy i Outputs Oy dzdx i Ox dydz Ixi dydz Iyi dzdx Inputs Izi dxdy Internal Net Production (Pi - Li)dxdydz Notation: Inputs = Iix dydz + Iiy dzdx + Iiz dxdy Outputs = Oix dydz + Oiy dzdx + Oiz dxdy ⎡ ∂Iiy ⎤ ⎡ ⎤ ⎡ ∂Ii ∂Ii ⎤ = ⎢ Iix + x dx ⎥ dydz + ⎢ Iiy + dy ⎥ dzdx + ⎢ Iiz + z dz ⎥ dxdy ∂x ⎦ ∂y ⎦⎥ ∂z ⎦ ⎣ ⎣ ⎣⎢ i i i I x = [i ] u, I y = [i ] v, I z = [i ] w (input fluxes) dx dy dz , v= , w= (wind velocities) dt dt dt Pi , Li = rates of chemical production, loss u= [i] = concentration of i Local rate of change of [i] given by: ∂ [i ] Inputs - Outputs + Internal net production ∂t dxdydz ∂ ∂ ∂ = Pi − Li − ([i ] u ) − ([i ] v ) − ([i ] w ) ∂x ∂y ∂z G = Pi − Li − ∇ ⋅ [i ] V = ( ) (1) Continuity equation for i For total molecular concentration [ M ] , PM − L M  0 so: G ∂ [M] = −∇ ⋅ [ M ] V ∂t ( ) (2) Continuity equation for M Defining mixing ratio = [i ] [ M ] = Xi : ∂ [i ] ∂ [M] [M] − [i ] ∂X i ∂ ∂t = ([i ] [ M ]) = ∂t 2 ∂t ∂t [M] G G Pi − Li − ∇ ⋅ X i [ M ] V [ M ] + ∇ ⋅ [ M ] V X i [ M ] (Using equations (1) and (2)) = 2 [M] G G G Pi − Li − X i∇ ⋅ [ M ] V − [ M ] V ⋅∇X i [ M ] + ∇ ⋅ [ M ] V X i [ M ] == 2 [M] G Pi − Li − [ M ] V ⋅∇X i [ M ] = 2 [M] P − Li G (3) = i − V ⋅∇X i [M] ( ( ( )) ( ( ) ( ) ) ( ) ) Continuity Equation for i (mixing ratio form) Theorem: If there is no gradient in the mixing ratio of i ( ∇Xi = 0 ) then there can be no local changes in i due to transport. Rate of change of Xi traveling with the air given by ( Lagrangian view ) : (a) dXi d = ⎡⎣ X i ( x, y, z, t ) ⎤⎦ dt dt ∂X i dx ∂X i dy ∂X i dz ∂X i = + + + ∂t ∂x dt ∂y dt ∂z dt ∂Xi ∂X ∂X ∂X u+ i v+ i w+ i = ∂x ∂y ∂z ∂t G G P − Li = V ⋅∇X i + i − V ⋅∇X i [M] P − Li = i [M] (chain rule) (using equation (3)) (4) Theorem: If there is no “net chemical production” ( Pi − Li = 0 ), then the mixing ratio of i is conserved moving with the air. (b) (s*, t*) s* dX i dt ds dt ds 0 X it* = X i0 + ∫ s* = X i0 + ∫ 0 (s, t) Pis − Lis ds [M] us (0, 0) (using equation (4)) (5) Theorem: The change in mixing ratio in an air mass from its initial value is a line integral of the “net chemical production” over the trajectory of the air mass. A steady state exists when the local rate of change is zero: ∂ [i ] =0 ∂t ∂ [M] =0 ∂t ∂X i =0 ∂t G i.e. Pi − Li = ∇ ⋅ [i ] V ⎫ ⎪⎪ ⎬ ( 6) G ⎪ i.e. ∇ ⋅ [ M ] V = 0 ⎪ ⎭ (6) Pi − Li G i.e. = V ⋅∇X i (7) [M] ( ( ) ) One-Dimensional (Horizontal) Model Source Region Downwind Region [Constant wind speed u , Pi = 0, Li = [Constant Xi] 0 z=0 Equation (7) with v = w = 0 gives: X Pi − Li = 0 − [ M ] i τi dX i dx d ln X i 1 i.e. =− dx uτ i = [M] u ⎛ x ⎞ i.e. X i ( x ) = X i ( 0 ) exp ⎜ − ⎟ ⎝ uτi ⎠ (8) [i] [M]X i ] = τi τi x [chemical (e-folding) distance, h = uτi ] [advection time = x u ] i.e. Xi(x) Xi(0) (a) Inert Case (x<> xu ( (b) Reactive Case (x>h, τi < xu ( X