Determination of Ocean Absolute Dynamic Topography Using the Minimum Energy State

Geostrophic balance represents the minimum energy state in linear Boussinesq primitive equations with conservation of potential vorticity. On the base of that, a new absolute dynamical topography equation (η-equation),

  -[∂_x(gH/f2)∂_xη + ∂_y(gH/f2)∂_yη ] +η = (g/f²)(∂C/∂x - ∂B/∂y)

is derived for the absolute dynamic topography (η) with H the water depth,  f the Coriolis parameter, g the gravitational acceleration, and coefficients (B, C) determined by temperature (T) and salinity (S). The η-equation provides the constraint to identify the geostrophy for the practically produced η fields. Along with satellite observations, in-situ ocean measurements of temperature, salinity, and velocity have also rapidly advanced such that the global ocean is now continuously monitored by near 4,000 free-drifting profiling floats (called Argo) from the surface to 2000 m depth with all data being relayed and made publicly available within hours after collection (http://www.argo.ucsd.edu/). This provides a huge database of (T, S) and in turns the (B, C) coefficients for the η-equation.

The η-equation is solved numerically in this study for the North Atlantic Ocean (100^oW-6^oW, 7^oN-72^oN) on 1^oX1^o grids with the coefficients (B, C) calculated from the three-dimensional (T, S) data of the NOAA National Centers for Environmental Information (NCEI) World Ocean Atlas 2013 version 2 (http://www.nodc.noaa.gov/OC5/woa13/woa13data.html) and H from the NOAA ETOPO5 (https://www.ngdc.noaa.gov/mgg/fliers/93mgg01.html) using simple Dirichlet (for rigid boundary sections) and Neumann (for open boundary sections) boundary conditions. The numerical solution η agrees well with the difference between the time-averaged sea surface height and the geoid from the NASA/JPL (http://gracetellus.jpl.nasa.gov/data/dot/,  also considered as the mean dynamic topography) in most part of the North Atlantic Ocean.  It offers an alternative way to estimate the absolute dynamic topography.

Further application of the η equation method on the high-precision altimetry measurements of SSH such as the Surface Water and Ocean Topography (SWOT) is also presented.