The development of linear instabilities on a geostrophic surface front in a two-layer primitive equation model on an ƒ-plane is studied analytically and numerically using a highly accurate differential shooting method. The basic state is composed of an upper layer in which the mean flow has a constant potential vorticity, and a quiescent lower layer that outcrops between a vertical wall and the surface front (defined as the line of intersection between the interface that separates the two layers and the ocean's surface). The characteristics of the linear instabilities found in the present work confirm earlier results regarding the strong dependence of the growth rate (σi) on the depth ratio r (defined as the ratio between the total ocean depth and the upper layer's depth at infinity) for r ≥ 2 and their weak dependence on the distance L between the surface front and the wall. These earlier results of the large r limit were obtained using a much coarser, algebraic, method and had a single maximum of the growth rate curve at some large wavenumber k. Our new results, in the narrow range of 1.005 ≤ r ≤ 1.05, demonstrate that the growth rate curve displays a second lobe with a local (secondary) maximum at a nondimensional wavenumber (with the length scale given by the internal radius of deformation) of 1.05. A new "fitting function" 0.183 r-0.87is found for the growth rate of the most unstable wave (σimax ) for r ranging between 1.001 and 20, and for L > 2 Rd (i.e.where the effect of the wall becomes negligible). Therefore, σimax converges to a finite value for |r — 1 | << 1 (infinitely thin lower layer). This result differs from quasi-geostrophic, analytic solutions that obtain for the no wall case since the QG approximation is not valid for very thin layers. In addition, an analytical solution is derived for the lower-layer solutions in the region between the wall and the surface front where the upper layer is not present. The weak dependence of the growth rate on L that emerges from the numerical solution of the eigenvalue problem is substantiated analytically by the way L appears in the boundary conditions at the surface front. Applications of these results for internal radii of deformation of 35– 45 km show reasonable agreement with observed meander characteristics of the Gulf Stream downstream of Cape Hatteras. Wavelengths and phase speeds of (180 –212 km, 39 –51 km/day) in the vicinity of Cape Hatteras were also found to match with the predicted dispersion relationships for the depth-ratio range of 1+ < r < 2.
Haza, Angelique C., Nathan Paldor, and Arthur J. Mariano. 2004. "Linear instabilities of a two-layer geostrophic surface front near a wall." Journal of Marine Research 62, (5). https://elischolar.library.yale.edu/journal_of_marine_research/57