MODEL
The density anomalies below the 18 degree water were modeled as adiabatic displacements of isopycnals, with the overall flow in quasi-geostrophic balance. The vertical structure of the streamfunction was assumed to be a superposition of the first three planetary wave vertical structure functions (“modes”), calculated using the buoyancy profile from the basic state and assuming a flat bottom with a depth of 5500m. The annual average Levitus climatology [Levitus, 1982] was averaged over the region to construct a single buoyancy frequency profile, which was used to calculate the modes. The stratification was set to zero above the 18 degree water to assume no predictability near the surface due to the QG modes. This is because of the changing stratification in the region over the course of the year.
The temperature and salinity perturbations consistent with these dynamic modes were subtracted from the observed temperature and salinity values to obtain residual stations. The residual temperature and salinity signals were used to make a covariance matrix for the both temperature and salinity at all depths. This matrix was factored to give empirical orthogonal functions (EOFs) of the temperature and salinity perturbations together. The salinity perturbations were multiplied by a factor of 8 to give them roughly equal weight with temperature in the EOFs. The first two eigenvalues of this weighted covariance matrix accounted for over 80% of the combined weighted variance, and the temperature and salinity structures of the associated eigenvectors were used to supplement the dynamical modes.
In addition, a‘spiciness’mode, meant to account for density-compensated temperature and salinity variability at thermocline depth (700 m) was included, since the temperature and travel time measurements include significant contributions from this mode.
The horizontal structure of all modes is expressed as a superposition of sines and cosines in the 1200 by 1200 km domain, chosen to be compatible with the psuedo-spectral model used to predict the dynamical evolution of the quasi-geostrophic streamfunction. Only the quasi-geostrophic part of the ocean representation can be forecast properly; persistence (time-independence) is assumed for the other modes. Both the 3 quasi-geostrophic modes and the 3 empirically derived modes were given low-passed horizontal structure using up to 8 Fourier harmonics in either direction in the 1200 km square domain, for a minimum scale of about 75 km.
Detailed exposition of the quasi-geostrophic approximation can be found in [Pedlosky, 1979] and [Flierl, 1981]. The ocean dynamics were approximated by a limited-area, pseudo-spectral, QG model originally developed by Geoff Vallis (Vallis, 1983, Vallis and Maltrud, 1993, Maltrud and Vallis, 1993) set for these examples to use three flat-bottom planetary modes as vertical basis functions.
The model uses three modes (the barotropic and the first two baroclinic modes), with horizontal structure of each mode specified by a Fourier expansion with period 1200 km in each direction. The wavenumber grid is given the circular truncation recommended by Patterson and Orszag [1971] to eliminate aliasing in the Jacobian. The model was run with a flat bottom and without forcing, or bottom friction, although some runs used hyperviscosity.
The model contains higher wavenumbers than are solved for using the data; these are assumed to be zero at the start of the evolution, and are allowed to evolve normally during the model integration. This was used to cut down on the size of the inverse problem, and because the high wavenumbers were more nonlinear (and less predictable) than the lower wavenumbers.
