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14C time constants, yet again
- Subject: 14C time constants, yet again
- From: firstname.lastname@example.org (Ken Caldeira)
- Date: Thu, 28 May 1998 16:26:26 -0700
Rick is right that considering the gas equilibration can introduce a longer
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
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?
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