Mapping M2 Internal Tides Using a Data-Assimilative Reduced Gravity Mode

We have developed data assimilation methods for mapping low-mode phase-locked internal tides from altimetry data, using a reduced gravity (RG) approach. Dynamical equations are derived following the approach of Griffiths and Grimshaw (2007), with the vertical dependence of pressure and velocities in the linear Boussinesq 3D equations expanded in basis functions derived from local 1D modes for a stratified ocean. This results in a system of coupled 2D PDEs for the coefficients of the modal expansion. Excluding coupling terms between modes (which arise in the presence of variable bottomt opography) the resulting equations for each mode are analogous to the usual shallow water equations for the barotropic tide. With modest changes to the OSU tidal inversion software (OTIS) an assimilation scheme for this linear model is readily implemented. The coupling terms can be used to derive the forcing (by the barotropic tide), and also can be used to quantify the component of model error associated with unmodeled topographic scattering. Because the inversion yields currents as well as elevations, mode energy fluxes can obtained with minimal further calculation. Relatively high spatial resolution (at least 1/30 degree) is required for the RG dynamical model, so the inversion must be done in modest sized overlapping patches, which can then be merged to obtain global maps of phase-locked low-mode internal tides. To obtain reliable results some care with preliminary data processing has proved necessary, including correction for lower frequency SSH variations in areas of strong mesoscale activity, and filtering to reduce Long wavelength error, especially in ERS/Envisat data. Because the assimilation essentially derives dynamically consistent spatio-temporal basis functions for data fitting, the RG scheme may also provide a means to better quantify at least slowly varying incoherent internal tides.