Stratified Inertial Flow in the Gulf Stream

Citation

Blandford, Robert Roy (1964) Stratified Inertial Flow in the Gulf Stream. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/JSQN-6Y34. https://resolver.caltech.edu/CaltechETD:etd-10142002-113121

Abstract

Earlier theoretical models of the Gulf Stream have treated the motion of a single fluid layer of constant density and vertically uniform flow velocity. As a step toward models with continuous stratification, the present work analyses inviscid, steady-state, purely inertial flow using two moving layers of different density and velocity.

The first type of Gulf Stream model analysed consists of two layers of different densities flowing over a denser layer at rest (baroclinic model). The second has two layers of different densities flowing over a rigid, horizontal bottom (mixed barotropic-baroclinic model).

In both models there exist, at any latitude, either 0, 2, or 4 theoretical solutions to the flow problem. Only one such solution, however, is realistic and satisfies the boundary condition of vanishing northward velocity at the southern latitude boundary of the flow region considered. This is called the correct solution, while the others are called incorrect solutions. As the parameters of the two-layer models converge to limiting values corresponding to one-layer models (for example, vanishing density difference between the upper and lower layers), the solutions may or may not converge to the one-layer solutions. If a correct solution converges uniformly, the limit is called a correct limit. If convergence is non-uniform at some value of the latitude coordinate, the limit is called an incorrect limit. If no solutions exist as the limit is approached, it is an impossible limit.

The most important limits discussed are as follows:
1. As the density contrast between the upper and lower moving layers becomes large in a baroclinic model for which the upper layer increases in thickness with latitude in the interior of the ocean to the east of the stream, the upper layer goes, via a correct limit, to the one-layer baroclinic model.
2. As the density contrast between the upper and lower layers becomes small in a baroclinic model, the solution for the sum of the two layers converges, via an incorrect limit, to the one-layer baroclinic model.
3. As the thickness of the upper layer becomes small while the density difference across it remains proportional to the thickness (a constant density "gradient" in the upper layer), the range of latitude over which there exists a correct solution tends to zero. The incorrect solution goes to the one-layer model via an incorrect limit. This result suggests that continuously stratified, purely inertial models of the Gulf Stream are impossible for finite density gradients.
4. In the limit as the interface between the lower moving layer and the resting layer becomes horizontal, the lower layer velocity goes to zero. No solution exists as the limit is approached. It is an impossible limit.
5. As the density contrast between the upper and lower layer becomes small in the barotropic-baroclinic model, the solution goes via a correct limit to the homogeneous barotropic model.

In an attempt to model the actual Gulf Stream, parameters are selected for a model of two moving layers, the upper about 600 meters thick and the lower about 400 meters. This model is close to the impossible limit of 4. above, and no solution exists. The physical reason for this is that because of the small transport in the lower layer, the velocity in the lower layer must be small, which is incompatible with the large velocity gradient needed for conservation of potential vorticity as required in an inertial model. It therefore seems questionable that the deeper waters of the Gulf Stream can be modelled by a purely inertial theory.

No off-shore countercurrents can be found, despite fairly accurate modelling of boundary conditions which might be expected to give them.

The general implication of this work is that steady, purely inertial models are inadequate to describe even the lower latitude growth region of Gulf Stream if density stratification is taken into account, and that viscosity or unsteadiness must therefore be introduced.

Thesis Files