On the use of the Proudman--Heaps tidal theorem

Abstract

The Proudman-Heaps tidal theorem provides, in principle, one way of computing tides across the open boundary of a bay, given measurements round the shore; however, attempts to use the theorem numerically have so far failed. We describe in this paper the results of a theoretical and numerical investigation into the difficulties which are encountered. We show first that:(1) The mathematical model being solved is ill-posed; hence, M Z~ method, and in particular a direct use of the tidal theorem, must expect numerical difficulties.( 2 ) The direct use of the tidal theorem yields the solution of a problem close to that posed; thus, discrepancies in the solution reflect directly the ill-posedness of the problem.The ill-posedness can be countered by imposing suitable constraints, which may be either mathematical (regularity conditions on the solution) or physical (spot tidal measurements across the open mouth of the bay). We investigate the extent to which using such constraints can improve the predictive power of the method; the results obtained are not encouraging.

The M2 Internal Tide off Oregon: Inferences from Data Assimilation

“…In reality, uncertainty exists first of all about the magnitude of the errors, both for the model and the data. The relative amplitude of the assumed errors in the in- (Foreman et al 1980;Kivman 1997). In our case, spurious small scale baroclinic vertical modes appear in the solution for low w o .…”