# Time for model equilibration

• Subject: Time for model equilibration
• From: kenc@llnl.gov (Ken Caldeira)
• Date: Tue, 26 May 1998 17:59:37 -0700

```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
tracer.

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?)

Cheers,

Ken

<--------------------------->
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

```