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14C time constants, yet again


Rick is right that considering the gas equilibration can introduce a longer
time constant.

The effective residence time of water in the mixed-layer (relative to
gas-exchange-isotopic equilibration time-scales) can have a big effect on
the 14C equilibration time-constant for the deep ocean, with short
residence times (i.e., lack of equilibration with the atmosphere) leading
to long deep ocean equilibration time scales ...

In Mathematica, I analytically solved a simple two-box model with the
following equations, where 'a' is the 14C  in the surface box, and 'b' is
the 14C for the deep box. I ignored radioactive decay in the surface box to
make the equations more tractable.

d a / d t = kgas (a0 - a[t]) + km (b[t] - a[t])

d b / d t = kd (a[t] - b[t]) - kdecay b[t],


a0 is the atmospheric delta 14C
kgas is the inverse time constant for 14C equilibration (~1/(8 yr))
km is the inverse time constant for surface mixing with the deep ocean
(~1/(10 yr) ???)
kd is the inverse time constant for deep mixing with the surface ocean
(~1/(2400 yr) ???)
kdecay is the C14 decay inverse time constant (~1/(5700 yr))

Solving this system, the dominant inverse time constant for deep ocean
equilibration is:

kequil =  (sqrt((kd + kdecay + kgas + km)^2 - 4 (kd kgas+kdecay (kgas+km)))
- - (kd + kdecay + kgas + km))/2 ,

which for the above parameter values yields ~1/(2459 yr), slightly longer
than the deep-North-Pacific equilibration time for a non-decaying tracer
that equilibrates rapidly with the atmosphere!

With the other values above held constant,

if km = 1/(1yr),  1/kequil = 4511 yr

if km = 1/(100 yr), 1/kequil = 1782 yr

if kgas = 1/(0.8 yr), 1/kequil = 1782 yr

if kgas = 1/(80 yr), 1/kequil = 4524 yr

The upshot of this is that, as Rick Murnane pointed out, that gas exchange
and vertical structure can introduce longer time constants than my previous
1-box  result indicated, and a more sophisticated model is probably needed
to predict actual convergence time scales.

Has anybody run Maier-Reimer's model for a very long time and examined
convegence rates for 14C?

What does this mean for the simulated time required to reach a meaningful
deep ocean 14C result?



Ken Caldeira
Climate System Modeling Group
Lawrence Livermore National Laboratory
7000 East Ave., L-103
Livermore CA 94550 USA

tel: (925)  423-4191 (new area code!)
fax: (925)  422-6388
e-mail:  kenc@LLNL.gov