We have developed a reduced gravity (RG) data assimilation scheme for mapping low-mode coherent internal tides [Egbert and Erofeeva (2014)], and applied this to a multi-mission dataset to produce preliminary global first-mode M2 and K1 solutions. Our scheme is based on the Boussinesq linear equations for flow over arbitrary topography with a free surface and horizontally uniform stratification. As in [ Tailleux and McWillims (2001)] and [ Griffithsand Grimshaw (2007) ] vertical dependence of the flow variables are described using flat-bottom modes (which depend on the local depth H(x, y)), yielding a coupled system of (2-dimensional) PDEs for the modal coefficients for surface elevation and horizontal velocity. Equations for each mode are coupled through interaction coefficients, which can be given in terms of the vertical mode eigenvalues following the approach of [ Griffithsand Grimshaw (2007) ]. Modes are decoupled wherever bathymetric gradients are zero, and for a flat bottom the system reduces to the usual single mode RG shallow water equations.

For our RG scheme we drop the vertical-mode coupling terms to obtain independent equations for the propagation of each mode with spatially variable reduced water depth, which we determine from local bathymetry and stratification. These simplified equations are identical to the linear SWE used in OTIS, allowing us to thus use the assimilation system to map internal tides by simply modifying depth, and fitting along-track harmonic constants as a sum over a small number of modes. With some extensions to OTIS, coupling terms for the first few modes can be included in the dynamics.

We have applied the OTIS-RG assimilation scheme to construct global maps of first mode temporally coherent internal tide elevations. All available exact repeat mission data were assimilated (TP/Jason, ERS/Envisat), with the AVISO weekly gridded SSH product used to reduce mesoscale variations before harmonic analysis. Solutions are computed in overlapping patches (~ 20 x 30 degrees), and then merged (via weighted average on overlaps) into a global solution. We note that adjacent solutions almost always match quite well even without this explicit tapering. Elevations for the intitial RG assimilation solutions are shown for M2 and K1, first mode.

!!! NOTE: these internal tide solutions are prliminary, and are not available for download !!! !!!