On the adjoint of Laplace’s tidal equations

The concept of the adjoint is regularly used in studies of both the resonances of physical systems and their response to external forcing. This report reviews the underlying theory involved in the adjoints of both differential equations and matrices and shows how the theory may be used to derive a physically meaningful adjoint to Laplace’s Tidal Equations. Numerical models of the tides usually use a finite difference form of the tidal equations. The report investigates the adjoint properties of the finite difference equations. It shows that the are not necessarily symmetrical, i.e. the finite difference form of the adjoint tidal equations may not be the same as the adjoint of the normal finite difference equations. It also shows that, with a suitable choice of the way the boundary conditions and Coriolis terms are represented, the finite difference forms can be made symmetric.