Recent studies indicate that altimetric observations of the ocean's mesoscale
eddy field reflect the combined influence of surface buoyancy and interior
potential vorticity anomalies. The former have a surface-trapped structure,
while the latter have a more grave form. To assess the relative importance of
each contribution to the signal, it is useful to project the observed field
onto a set of modes that separates their influence in a natural way. However,
the surface-trapped dynamics are not well-represented by standard baroclinic
modes; moreover, they are dependent on horizontal scale.
Here we derive a modal decomposition that results from the simultaneous
diagonalization of the energy and a generalisation of potential enstrophy that
includes contributions from the surface buoyancy fields. This approach yields a
family of orthonomal bases that depend on two parameters: the standard
baroclinic modes are recovered in a limiting case, while other choices provide
modes that represent surface and interior dynamics in an efficient way.
For constant stratification, these modes consist of symmetric and
antisymmetric exponential modes that capture the surface dynamics, and a series
of oscillating modes that represent the interior dynamics. Motivated by the
ocean, where shears are concentrated near the upper surface, we also consider
the special case of a quiescent lower surface. In this case, the interior modes
are independent of wavenumber, and there is a single exponential surface mode
that replaces the barotropic mode. We demonstrate the use and effectiveness of
these modes by projecting the energy in a set of simulations of baroclinic
turbulence
