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One-Dimensional (Vertical) Chemistry-Transport Model Take horizontal average of equation (5) and denote horizontal average with overbar and deviation from horizontal average with a prime: G d d Pi − Li = ∇ ⋅ [i ] V = ([i ] w ) = ( X i [ M ] w ) dz dz d = Xi '[ M ] w + Xi [ M ] w dz d = X i ' [ M ] ' w ' + [ M ]X i ' w ' + wX i ' [ M ] ' + X i '[ M ]w (net vertical flux of air dz [M] w = 0 ) ( ) ( ( ) ) ( ( d X i ' [ M ] ' w ' + [ M ]X i ' w ' dz d [ M ]Xi ' w ' dz ⎞ dX d ⎛ − ⎜ [ M ] i δz w ' ⎟ dz ⎝ dz ⎠ = =− ) ( Xi ' = 0 and w = 0 ) ([ M ] ' [ M ]) ) (eddy diffusion approximation*) d ⎛ dX i ⎞ Kz ⎟ ⎜ [M] dz ⎝ dz ⎠ (9) ( K z = eddy diffusion coefficient = δz w ' ) *eddy diffusion approximation: z z+ |δz| (air moving down from above) -|w'| Xi' w' = z +|w'| z- |δz| dXi (+|δz| )(-|w'| ) dz dXi (-|δz|)(+|w'| ) dz = - dXi |δz||w'| dz (air moving up from below) Figure by MIT OCW. Consider case when loss only: Pi − Li = − Li = − =− [i ] ( τ = chemical lifetime of i) τ ([M] ' X ' + [M]X ) i i τ ( X i ' = 0 and [ M ] ' = 0 ) − [ M ]Xi ([M] 'X ' [M]X ) τ i i For brevity drop subscripts i and overbars and assume K z = K is independent of altitude and temperature is constant: X [M] d ⎛ dX ⎞ = K ⎜ [M] ⎟ τ dz ⎝ dz ⎠ ⎛ d [ M ] dX d2X ⎞ = K⎜ + [M] 2 ⎟ dz ⎠ ⎝ dz dz d [M] ⎛ [ M ] dX [ M ] for d2X ⎞ =− (In hydrostatic equilibrium: = K⎜− + [M] 2 ⎟ dz H dz ⎠ ⎝ H dz constant temperature) Rearranging: d 2 X 1 dX X (10) − − =0 dz 2 H dz Kτ General solution is: ⎛ z ⎞ ⎛ z ⎞ X = A exp ⎜ ⎟ + Bexp ⎜ ⎟ ⎝ h+ ⎠ ⎝ h− ⎠ 1 1 1 ⎛ 1 1 ⎞2 = ±⎜ + ⎟ h ± 2H ⎝ 4H 2 Kτ ⎠ (Note that h + > 0 and h − < 0 ) Determine A and B from boundary conditions. Say X Æ 0 as z Æ ∞ , then A = 0 and X = X ( 0 ) at z = 0 is given so B = X ( 0 ) . Thus specific solution is: 1 ⎡ ⎛ ⎞⎤ 2 1 1 1 ⎛ ⎞ ⎢ ⎜ ⎟⎥ − + X = X ( 0 ) exp z ⎢ ⎜ 2H ⎝⎜ 4H 2 Kτ ⎠⎟ ⎟ ⎥ ⎠⎦ ⎣ ⎝ (11) Consider two cases: 4H 2 τ denoted REACTIVE SPECIES case, K i.e. (vertical transport time) (chemical lifetime) z ⎞ ⎛ Then X X ( 0 ) exp ⎜ − {rapid decreases in mixing ratio with z} ⎟ Kτ ⎠ ⎝ (a) (b) 4H 2 τ denoted INERT SPECIES case K ⎛ ⎛ 1 1 ⎞⎞ Then X X ( 0 ) exp ⎜ −z ⎜ − ⎟⎟ ⎝ ⎝ 2H 2H ⎠ ⎠ = X(0) {mixing ratio constant with z} (i.e. h − H ) z Reactive (stratosphere) Inert (troposphere) X(z) X(0) Example: surface source and stratospheric sink (such as N 2 O , CFCl3 , CF2Cl2 , etc.) Coupled Chemistry-Transport 3D Models 1. Basic Equations Want to solve the 3D Eulerian continuity equation as an initial value problem: G ∂ [i ] = Pi − L1 − ∇ ⋅ V [i ] (“concentration” form) ∂t ∂X i Pi − Li G = − V ⋅∇X i (“mixing ratio” or “mole fraction” form) ∂t [M] G subject to upper and lower boundary conditions. But do not know V as continuous function of space and time. Thus express the flux as the sum of “mean advective” and “eddy diffusive” parts: G G G V [i ] = V [i ] + V ' [i ] ' G V [ i ] − K∇ [ i ] ( ) where denotes time and/or space average, ( ) ' denotes deviation from , and K is a 3x3 matrix containing “eddy diffusion” (or “turbulent exchange”) coefficients. The G average winds V can be obtained in principle from general circulation models (gcm’s), observations, or gcm’s “corrected” through assimilation of observations (“forecast” or G “analyzed observed” winds). In this case V are Eulerian averages appropriate to the grid spacing and time step in the g.c.m. and K refers to unresolved “sub-grid-scale” winds. K may be determined by empirical (e.g. fitting observed [i ] ), semi-empirical, or theoretical approaches. The latter two approaches involve so-called “parameterizations.” 2. Prognostic and diagnostic continuity equations It is not usually necessary to consider transport of all chemical species. Consider the prognostic (time dependent) continuity equation in mixing ratio form: G ∂X Pi L = i + V ⋅∇X i + i ∂t [M] [M] ⎛1 G [i ] and τ = [i ] ) ∂⎞ (using [ M ] = = ⎜ + V ⋅∇ + ⎟ X i i Xi Li ∂t ⎠ ⎝ τi ⎛1 u v w 1 ⎞ (assuming Δu u, Δv v, Δw w ) ⎜ + + + + ⎟ Xi ⎝ τi Δx Δy Δz Δt ⎠ Xi u v w 1 1 if , , , Δx Δy Δz Δt τi τi L [chemical steady state; diagnostic equation] = i [M] where τi = chemical time scale = (8) [i ] Li Δx Δy Δz , , = transport (advection) time scales u v w Δt = integration time scale The diagnostic equation is much faster to solve. Chemical families: τi transport time (for loss by conversion of one family member to another) τi ≥ transport time (for loss of overall family) [O x ] = [ O] + [O3 ] = odd oxygen [ HO x ] = [ H ] + [OH ] + [ HO2 ] = odd hydrogen [ NO x ] = [ NO] + [ NO2 ] = odd nitrogen [Clx ] = [Cl] + [ ClO] = odd (reactive) chlorine Without chemical families and diagnostic equations, atmospheric chemical models are G invariably “stiff” systems. Specifically if X is a vector of chemical mixing ratios Xi and G ∂X G G = R X, t then the ratio of the largest and smallest eigenvalues λi of the Jacobian ∂t ( ) G ∂R matrix G is typically 1 (equivalently the ratios of the largest to smallest “lifetimes” ∂X −1 λi 1) 3. Spatial representations a. Finite difference schemes (truncated Taylor expansion at J grid-points) b. Spectral techniques (express variables using truncated series of N orthogonal harmonic functions and solve for N coefficients of expansion;) see c. Interpolation schemes (interpolates between grid points e.g. using a polynomial) d. Finite element schemes (minimizes error between actual and approximate solutions using a “basis function”, good for irregular geometries, c.f. (b) above which is good for regular geometries) 4. Explicit and Implicit time stepping t +Δt t Explicit: ( )x = f ⎡⎣...., ( )x* ,...⎤⎦ t +Δt t t +Δt Implicit: ( )x = f ⎡⎣..., ( )x* , ( )x* ,....⎤⎦ (Implicit methods more stable (but often less accurate) than explicit methods for longer time steps)