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Re: Positive definite advection schemes

Dear Jonathan,

A presentation which I gave at AGU (Fall Meeting 1996) touched on this
subject, but my sensitivty tests were only for simulations of bomb
C-14 in the LODyC OGCM (see abstract below).  I was most interested in
comparing later years, when bomb C-14 behavior is most like that of
anthropogenic CO2. Similar sensitivity tests for anthropogenic CO2
itself are ongoing.  Future work may include the same for natural CO2.

Something not detailed in the abstract are regional-scale differences.
Comparisons were made relative to the reference scheme (MPDATA, a
positive definite scheme from Smolarkawicz), which is the most
realistic of the three schemes tested. The other two schemes are
UPSTREAM and CTCS--what you call Cox centered-differencing.  Relative
to MPDATA, the air-sea flux of bomb C-14 in 1990 is everywhere larger
with the upstream scheme and everywhere smaller with the CTCS scheme.
The largest differences are found in the low latitudes.  With the
upstream scheme, the air-sea flux is nearly doubled in the tropics. At
30S and 30N, the flux is large and directed out of the ocean with the
upstream scheme; with the other two schemes, the flux is directed in
the opposite direction and is small. Differences between the CTCS and
the MPDATA scheme reach about 20% in the low latitudes; however, the
CTCS-MPDATA differences reach nearly 100% at 30S and 30N.  

These are not small differences, so investing more in your advection
advection scheme would probably not be a waste of time.  I hear that
Robbie Toggweiler and co-workers find some large differences when
using the FCT vs. the CTCS scheme in their regional ecological
modeling of the equatorial Pacific.  Especially if you plan on doing
point source studies, such as proposed for CO2 disposal, you really
need a positive definite scheme.  As for the QUICK scheme, which you
mentioned,I've yet to do any testing with that one.  However, that
scheme does look promising, better than CTCS, and nearly as fast
(much faster than FCT).  Technically speaking, though, the QUICK
scheme is not positive definite.

- -- Jim


Tracer Advection Schemes and Uptake of Bomb C-14 in a 3-D Global Ocean Model

Presented at the AGU Fall 1996 meeting (San Francisco) 
by J. C. Orr, O. Marti, G. Madec, P. Monfray
Advection dominates transport of carbon and other tracers throughout
much of the ocean.  Advection can be implemented in numerous fashions
in 3-D ocean models.  Differences in advection schemes may help
explain disparities between model CO2 uptake estimates; however, no
studies are available that quantify if such differences are important.
As a first investigation, we separately use three advection schemes
(upstream, centered difference, and flux-corrected transport) in
otherwise identical simulations focused on ocean uptake of bomb C-14 .
These schemes are the same as those used by most 3-D ocean modelers
within the carbon-cycle community.  Relative to the reference scheme
(flux-corrected transport), the numerically diffusive upstream scheme
results in higher global mean penetration depths (7\% more during
GEOSECS; 16\% more in 1995) and smoothing of sharp gradients.  The
centered difference approach (numerically dispersive) produces
negative levels of bomb C-14 as well as biases where gradients are
most intense.  The flux-corrected transport scheme is little diffusive
and never lets the tracer go negative, but only at the expense of
being three times as costly (i.e., for our tracer-transport runs in
the LODyC global ocean model, which use two iterations to correct for
numerical diffusion).  A greater investment in the advection scheme
may be necessary when modeling tracers with sharp gradients (e.g.,
bomb C-14 , anthropogenic CO2, and CFC's), especially when wanting to
compare model results to data.