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Time for model equilibration

Dear colleagues,

As per our discussion of model equilibration time scales at the US OCMIP
meeting at NCAR last week, the simplest approach is to look at a one-box
reservoir equilibrating with an overlying atmosphere:

(1)     d a / d t = k (a0 - a(t)) ,

which has the solution

(2)    a(t) =  a0 + a1 E^(-k t) .

If we then add radioactive decay,  we then have

(3)    d b / d t = k (b0 - b (t)) - kdecay b(t) ,

the solution is

(4)    b(t) = (k / (k + kdecay)) b0 + b1 E ^ (- ( k + kdecay) t)

If we think that the e-folding time for the equilibration of a stable
tracer is on the order of 1 kyr, and the decay of 14C is about 5.7 kyr,
then the e-folding time for the combined system to approach steady-state is
about 85% (i.e., 1/(1/(1 kyr) + 1/(5.7 kyr)) of that of the non-decaying

If Jorge's results converged only 92 % after 6000 yr, this suggests an
e-folding time-scale of about 2400 yrs for the central North Pacific. With
radioactive decay considered, the e-folding time to approach a 14C solution
should be about 1700 yr, which means that the 14C would be 97 % converged
after 6000 yr. (Does this mean there could still be a 30 o/oo error?)



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