SLIDE 1
Transit-Time Distributions: General Discussion and Applications
Timothy Hall, NASA GISS
SLIDE 2 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 ∂t + L(G) = d(r - r')d(t - t') ∂c ∂t + L(c) = S(r,t) with homogeneous BCs ∂G ∂t + L(G) = 0 with G(r
W,t |W,t') = d(t - t')
c(r,t) = dt' dr'
Ú
S(r',t')G
t
Ú
(r,t | r',t') c(r,t) = dt'c(W,t')G
t
Ú
(r,t |W,t')
Compare to more familar
SLIDE 3 Can we express G in terms of G? Yes. Look at simple example: 1-D semi-infinite diffusion, point source x=e, homogeneous BC
e
distance x concentration Q/k
S(x,t) = Q e d(x -e) c(x,t) = dt' dx' Q e
t
Ú
d(x -e)G(x,x',t - t') = dt'
t
Ú
Q e G(x,e,t - t') = dt'
t
Ú
c(e)k d dx' G(x,
1 2e,t - t')
(Morse and Feshbach, 1953)
SLIDE 4 More generally
c(r,t) = dt'c(W,t') d2r' k
W
Ú
t
Ú
ˆ n •—r'G(r,t | r',t')
So, quantity that acts as boundary propagator, or “age spectrum” is
G(r,t |W,t') = d2
W
Ú
r' k ˆ n •—r'G(r,t | r',t')
So what? Hard to interpret because gradient acts on source variable.
G(r,t | r',t') = G
†(r',t'| r,t)
RECIPROCITY:
G(r,t |W,t') = d2
W
Ú
r' k ˆ n •—r'G†(r',t'| r,t)
G G = flux into W in adjoint flow
with t’<t
SLIDE 5 Reciprocity cartoon
r r’,t’ t r, t r’ t’ Reciprocity: symmetry between source and response
forward flow (t>t’) adjoint flow (t>t’)
For simple adv-diff., get adjoint by sign-change on velocities.
(Morse and Feshbach, 1953)
SLIDE 6 Holzer and Hall Fig
From Green function reciprocity, get similar relationship between explicit-source Green functions and boundary propagators (“age spectra”) forward flow adjoint flow
SLIDE 7 G(r,t |W,t') = d2
W
Ú
r' k ˆ n •—r'G†(r',t'| r,t)
Now, physical interpretation flux into W from unit source at r, t in adjoint flow
∂ ∂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
∂ ∂t'{fraction remaining in domain at t’
“following” unit source at r,t
= = = = =
SLIDE 8
first W-arrival time pdf at r in adjoint flow last W-arrival time pdf at r in forward flow flux into W from unit source at r in adjoint flow response at r to d(t) Dirichlet BC on W in orward flow
=
Summary of equivalences
= = =
SLIDE 9 Applications:
- Strategies for ocean storage of CO2.
- Diagnostics of “gross” flux; e.g., STE.
- Estimating ocean uptake of CO2.
SLIDE 10
Goal: inject industrial CO2 into deep ocean to keep it from atmosphere for long time. Question: from where in ocean will injected CO2 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. Strategies for ocean storage of CO2
SLIDE 11 Primeau Fig
Primeau, JPO, 2004
SLIDE 12 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.
SLIDE 13
Wernli and Bourqui, JGR, 2002
strat-to-trop trop-to-strat
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).
SLIDE 14 Flux diagnostics
Flux diagnostics and stratosphere-troposphere exchange
SLIDE 15 r’ R2 R1 W R1
d
3rG(r,t'+t | r',t') R2
Ú
fraction from r’, t’ still in R2 after time t, given r’,t’
1 M2 d
3r' R2
Ú
d
3rG(r,t'+t | r',t') R2
Ú
fraction from R2 at t’ that is still in R2 time t later Advection-diffusion in domain R partitioned by surface W. Unit source at r’, t’. BC=0 on W.
F(t'+t | t') = - ∂ ∂t 1 M2 d
3r' R2
Ú
d
3rG(r,t'+t | r',t') R2
Ú
Ê Ë Á Á ˆ ¯ ˜ ˜
flux into W of fluid fraction that resided at least t in R2
SLIDE 16 F(t'+t | t') = - ∂ ∂t 1 M2 d
3r' R2
Ú
d
3rG(r,t'+t | r',t') R2
Ú
Ê Ë Á Á ˆ ¯ ˜ ˜ = 1 M2 d
3r' R2
Ú
d
2rk ˆ
n •—rG(r,t'+t | r',t')
W
Ú
limt Æ0G(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.
ˆ n •—rG = •
and So,
limt Æ0 F(t'+t | t') = •
“Gross” flux dominated by smallest scales of motion.
SLIDE 17
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.
SLIDE 18 c(r,t) = d ¢ t d2r
S c(r S, ¢
t ) G(r,t | r
S, ¢
t )
S
Ú
Ú
x Gs(r,x)
c(r,t) = cs (t -t) = cs (t -x) d(x -t) dx
t
Ú
s S
x GV(r,x)
cV (t) = cs (t -x)GV (x) dx
t
Ú
V Tracer machinery r
c(r,t) = cs (t -x)Gs (r,x)
t
Ú
dx
Isopycnal, c(t) uniform on outcrop, steady-state (x = t-t’)
SLIDE 19 CFC-12 on sample isopycnal (s0 = 26.7) isopycnals
Application to Indian Ocean
WOCE CFC-12 observations gridded on isopycnal surfaces. CFC-12 contours define northern boundaries of series of volumes, V(CFC-12), for GV(x) analysis.
26.7
SLIDE 20 Invert parametrically. Choose two-parameter form for GV(t). Peclet number P, mean “age” t. CFC constrains ∆C to range. Upper bound (weak-mixing), Lower bound (strong-mixing).
c(t) = cS(t - t')GV (t') dt'
t
Ú
SLIDE 21 GV (t) = 1 pPtt e-Pt /4t - e-P(t-t)2 /4tt
( ) + 1
2t erf Pt 4t Ê Ë Á ˆ ¯ ˜ + erf P 4tt (t - t) Ê Ë Á ˆ ¯ ˜ Ê Ë Á Á ˆ ¯ ˜ ˜
- 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
SLIDE 22 dF(t) = k dCO2(t) -dpCO2(t)
( )
dF(t) = d dt V feq dpCO2(t - ¢ t )
( )GV ( ¢
t ) d ¢ t
t
Ú
Ê Ë Á ˆ ¯ ˜
F(t) = k CO2(t) - pCO2(t)
( )
CO2 flux by air-sea difference. Anthropogenic component: Also, flux drives rate change of mass in domain. For each water-density class:
Note: assumed linear Cdissolved = feq(pCO2), but more precise treatment makes only small difference.
GV(t) is V-averaged boundary propagator: last-contact time pdf for entire constant-density volume
CO2 at equilibrium with dissolved C in surface waters
SLIDE 23 Sabine et al (1999) (Gruber C*) McNeil et al (2003) (CFC age)
Indian Ocean Mass ∆DIC Indian Ocean Net air-sea flux
SLIDE 24 BC spatial variation
c(r,t) = dt' d2r'c(r',t')G(
W
Ú
Ú
r,t | r',t')
More general case of boundary propagator: spatial variation of boundary condition and non-steady flow. Still have probabilistic interpretation: G(r,t|r’,t’)d2r’dt’ = probability that parcel at (r,t) made last W contact time t’ to t’+dt’ and made the contact on d2r’. 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.
SLIDE 25 Troposphere PDF
Example: surface origins and transit-times for tropospheric air
SLIDE 26
Ocean illustration: simulate G(r,r’,t-t’) in North Atlantic MYCOM.
(Haine and Hall, 2002)
Tile the domain. For ith tile, tracer has BC = d(t) and zero elsewhere
SLIDE 27
SLIDE 28 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.
(1) Artificial carbon sequestration strategies in ocean. (2) Stratosphere-troposphere exchange diagnostics. (3) Estimating anthropogenic CO2 uptake by ocean.
(1) Propagation of T and S anomalies in DWBC diagnosed with tracers. (2) Evolution of total chlorine in stratosphere. (3) Model intercomparison studies.
