Transit-Time Distributions: General Discussion and Applications - PowerPoint PPT Presentation

Transit-Time Distributions: General Discussion and Applications Timothy Hall, NASA GISS

Defined “age spectrum” in last lecture. Type of Green function. How is age spectrum related to more familiar Green functions? ∂c ∂ t + L ( c ) = 0 with inhomogeneous c ( W , t ) = f ( t ) ∂ G + L ( G ) = 0 G ( r with W , t | W , t ') = d ( t - t ') ∂ t t dt ' c ( W , t ') G Ú c ( r , t ) = ( r , t | W , t ') 0 Compare to more familar ∂c ∂ t + L ( c ) = S ( r , t ) with homogeneous BCs ∂ G + L ( G ) = d ( r - r ') d ( t - t ') ∂ t t Ú Ú c ( r , t ) = dt ' dr ' S ( r ', t ') G ( r , t | r ', t ') 0

Can we express G in terms of G? Yes. Look at simple example: (Morse and Feshbach, 1953) 1-D semi-infinite diffusion, point source x= e , homogeneous BC concentration Q/ k S ( x , t ) = Q e d ( x - e ) distance x e t • dx ' Q Ú Ú c ( x , t ) = dt ' d ( x - e ) G ( x , x ', t - t ') e 0 0 t t Q c ( e ) k d 1 Ú Ú dt ' e G ( x , e , t - t ') = dt ' dx ' G ( x , 2 e , t - t ') = 0 0

t More generally Ú Ú d 2 r ' k ˆ c ( r , t ) = dt ' c ( W , t ') n •— r' G ( r , t | r ', t ') 0 W So, quantity that acts as boundary propagator, or “age spectrum” is G ( r , t | W , t ') = Ú d 2 r ' k ˆ n •— r' G ( r , t | r ', t ') W So what? Hard to interpret because gradient acts on source variable. † ( r ', t '| r , t ) RECIPROCITY: with t’<t G ( r , t | r ', t ') = G G ( r , t | W , t ') = Ú d 2 n •— r' G † ( r ', t '| r , t ) r ' k ˆ W G G = flux into W in adjoint flow

Reciprocity cartoon Reciprocity: symmetry between source and response (Morse and Feshbach, 1953) forward flow r (t>t’) r’,t’ For simple adv-diff., get adjoint t by sign-change on velocities. t’ r, t adjoint flow r’ (t>t’)

Holzer and Hall Fig forward flow From Green function reciprocity, get similar relationship between explicit-source Green functions and boundary propagators (“age spectra”) adjoint flow

Now, physical interpretation G ( r , t | W , t ') = Ú d 2 n •— r' G † ( r ', t '| r , t ) r ' k ˆ W flux into W from unit source at r, t = in adjoint flow ∂ ∂ t ' { fraction remaining in domain at t’ = “following” unit source at r,t ∂ ∂ t ' { probability that particle is still in = domain at time t’ = first W -arrival time pdf in adjoint flow = last W -arrival time pdf in forward flow

Summary of equivalences first W -arrival time pdf last W -arrival time pdf = at r in adjoint flow at r in forward flow = = flux into W from unit source response at r to d (t) Dirichlet = at r in adjoint flow BC on W in orward flow

Applications: • Strategies for ocean storage of CO 2 . • Diagnostics of “gross” flux; e.g., STE. • Estimating ocean uptake of CO 2 .

Strategies for ocean storage of CO 2 Goal: inject industrial CO 2 into deep ocean to keep it from atmosphere for long time. Question: from where in ocean will injected CO 2 take longest time to return to surface? Answer: Use GCM to simulate tracer releases from lots of possible injection points. Run each tracer 1000 years. Problem: Computationally expensive. Solution: Simulate single full-surface G in adjoint model. Response at each point equals first arrival time pdf for tracer release from r in forward model.

Primeau Fig Primeau, JPO, 2004

STE Flux diagnostics for stratosphere-troposphere exchange. • Midlatitude STE complex, episodic. • Net flux constrained globally, but chemical constituents affected by local back-and-forth motion. • Studies try to compute “gross” (one-way) fluxes. • Sum up local met-data fluxes keeping up and down separate.

Wernli and Bourqui, JGR, 2002 trop-to-strat strat-to-trop 24 hr 48 hr 96 hr Gross fluxes computed with particle trajectories for different particle resident-time thresholds (i.e., don’t count particle tropopause crossing if less than threshold).

Flux diagnostics Flux diagnostics and stratosphere-troposphere exchange

Advection-diffusion in R 1 r’ domain R partitioned by R 2 surface W . Unit source at R 1 W r’, t’. BC=0 on W . fraction from r’, t’ still in Ú 3 rG ( r , t ' + t | r ', t ') d R 2 after time t, given r’,t’ R 2 1 fraction from R 2 at t’ that is Ú 3 r ' Ú 3 rG ( r , t ' + t | r ', t ') d d still in R 2 time t later M 2 R 2 R 2 flux into W of fluid Ê ˆ 1 F ( t ' +t | t ') = - ∂ Ú 3 r ' Ú 3 rG ( r , t ' +t | r ', t ') d d Á ˜ fraction that resided Á ˜ M 2 ∂t at least t in R 2 Ë ¯ R 2 R 2

Ê ˆ 1 F ( t ' + t | t ') = - ∂ Ú 3 r ' Ú 3 rG ( r , t ' + t | r ', t ') d d Á ˜ Á ˜ M 2 ∂t Ë ¯ R 2 R 2 1 Ú 3 r ' Ú 2 r k ˆ d d n •— r G ( r , t ' + t | r ', t ') = M 2 R 2 W lim t Æ 0 G ( r , t ' + t | r ', t ) = d ( r - r ') i.e., the source r’ adjacent to W dominates contribution at small t. But, G=0 for r on S by the BC. So, at small t, G is discontinuous in r. lim t Æ 0 F ( t ' + t | t ') = • ˆ So, n •— r G = • and “Gross” flux dominated by smallest scales of motion.

Estimating ocean uptake of anthropogenic carbon Difficult to measure directly: order 1% background, natural in-situ biochemical source and sinks. Good approximation: anthropogenic C perturbation propagates as inert tracer responding to surface BC (biochemistry remains preindustrial, nutrient limited). Approach: use another inert tracer with no natural background as proxy for anthropogenic C transport.

t Tracer Ú Ú d 2 r c ( r , t ) = d ¢ t S c ( r S , ¢ t ) G ( r , t | r S , ¢ t ) machinery S -• S s r G s (r, x ) Isopycnal, c(t) uniform on outcrop, V steady-state ( x = t-t’) x t c s ( t - x ) G s ( r , x ) Ú c ( r , t ) = d x G V (r,x) t 0 c s ( t - x ) G V ( x ) d x Ú c V ( t ) = t x Ú c ( r , t ) = c s ( t - t ) = 0 c s ( t - x ) d ( x - t ) d x 0

Application to Indian Ocean CFC-12 on sample isopycnal ( s 0 = 26.7) isopycnals 26.7 WOCE CFC-12 observations gridded on isopycnal surfaces. CFC-12 contours define northern boundaries of series of volumes, V(CFC-12), for G V ( x ) analysis.

t c S ( t - t ') G V ( t ') dt ' Ú c ( t ) = 0 Invert parametrically. Choose two-parameter form for G V ( t) . Peclet number P, mean “age” t. CFC constrains ∆C to range. Upper bound (weak-mixing), Lower bound (strong-mixing).

Ê ˆ Ê ˆ Ê ˆ 1 ) + 1 Pt P e - Pt /4 t - e - P ( t - t ) 2 /4 t t ( G V ( t ) = 2 t erf ˜ + erf 4 t t ( t - t ) Á ˜ Á Á ˜ Á ˜ 4 t p P t t Ë ¯ Ë ¯ Ë ¯ • Related to “inverse Gaussian” distribution. • Two parameters: (1) t: mean transit time (2) P: measure of diffusive mixing Functional form can mimic G simulated in GCMs

CO 2 at equilibrium with dissolved C in For each water-density class: surface waters CO 2 flux by air-sea difference. ( ) F ( t ) = k CO 2 ( t ) - pCO 2 ( t ) Anthropogenic component: ( ) d F ( t ) = k d CO 2 ( t ) - d pCO 2 ( t ) Also, flux drives rate change of mass in domain. Ê ˆ t d F ( t ) = d ) G V ( ¢ Ú ( dt V f eq d pCO 2 ( t - ¢ t ) t ) d ¢ t Á ˜ Ë ¯ 0 Note: assumed linear C dissolved = f eq (pCO 2 ), but more precise treatment makes only small difference. G V (t) is V-averaged boundary propagator: last-contact time pdf for entire constant-density volume

Indian Ocean Mass ∆DIC Sabine et al (1999) (Gruber C*) McNeil et al (2003) (CFC age) Indian Ocean Net air-sea flux

BC spatial variation More general case of boundary propagator: spatial variation of boundary condition and non-steady flow. t d 2 r ' c ( r ', t ') G ( Ú Ú c ( r , t ) = dt ' r , t | r ', t ') -• W Still have probabilistic interpretation: G (r,t|r’,t’) d 2 r’ d t’ = probability that parcel at (r,t) made last W contact time t’ to t’+ d t’ and made the contact on d 2 r’. G (r,t|r’,t’) = joint PDF in source time and space. Note: much more difficult to compute, now. Need d (t) BC for each t’ and r’ to resolve on source .

Troposphere PDF Example: surface origins and transit-times for tropospheric air observation point

Ocean illustration: simulate G (r,r’,t-t’) in North Atlantic MYCOM. (Haine and Hall, 2002) Tile the domain. For i th tile, tracer has BC = d (t) and zero elsewhere

Summary • Green function (boundary propagator) has interpretation as transit-time pdf (first-passage time, age spectrum). • Alternative transport discription to velocities and diffusivities. • Generally, G doesn’t have simple relationship to u and k. But, offers more direct translation one tracer to another. • Three examples: (1) Artificial carbon sequestration strategies in ocean. (2) Stratosphere-troposphere exchange diagnostics. (3) Estimating anthropogenic CO 2 uptake by ocean. • Other uses: (1) Propagation of T and S anomalies in DWBC diagnosed with tracers. (2) Evolution of total chlorine in stratosphere. (3) Model intercomparison studies.