Cw. Hughes, THE USE OF TOPOGRAPHIC WAVE MODES TO SOLVE FOR THE BAROTROPIC MODE OFA RIGID-LID OCEAN MODEL, Journal of atmospheric and oceanic technology, 13(3), 1996, pp. 751-761
Topographic wave modes are defined for the barotropic mode of a rigid-
lid ocean, and the question is asked whether these might form an effic
ient basis for a description of the barotropic mode of a general ocean
flow. The modes are shown to be incomplete, most particularly in thei
r representation of barotropic potential vorticity over certain areas,
so extra functions must be added to ''patch up'' the wave modes descr
iption. With the aid of two simple, flat bottom beta-plane models, it
is shown that the form of the extra function required depends on the p
osition of the boundaries relative to contours of planetary barotropic
potential vorticity, f/H. Where all the boundaries are along contours
of f/H, the extra function is simply a function of f/H and time. Wher
e a finite stretch of boundary runs parallel to f/H contours, a furthe
r additional function can (at least sometimes) produce a complete set,
but when the boundary runs parallel to SIH contours for no finite dis
tance, there is no simple way to augment the wave modes to produce a c
omplete set. It is shown that :his incompleteness is only in the repre
sentation of barotropic potential vorticity at the boundary and causes
no finite error in the streamfunction, but it seems likely that the p
resence of this incompleteness spoils the efficiency of the sum of wav
e modes as a description of a general flow. It appears that topographi
c wave modes are the natural modes only for systems in which the bound
ary (at least partially) follows contours of planetary barotropic pote
ntial vorticity, f/H. The above deficiencies vanish when enough modes
are considered to resolve frictional boundary layers if the no-slip bo
undary condition is applied. When boundary layers are too thin to reso
lve, however, use of the modes to represent the difference between rot
ating and nonrotating responses suggests the possibility of a novel wa
y of modeling the approach to the inviscid limit. In both these cases,
however, current technology limits the practicality of the method to
cases where the spatial structure of the wave modes can be calculated
analytically.