Joseph Pedlosky

The Ventilated Thermocline

Abstract A simple theoretical model for the oceanic thermocline and the associated field of current is presented. The model consists of a finite but arbitarily large number of inviscid, homogeneous fluid layers each with a different density. The dynamical balances everywhere are Sverdrupian. IN regions where the Ekman pumping is negative (downward) the surface density is specified, i.e., the position of the outcrop of density interfaces is specified. This outcropping of density layers allows deep motion to be excited by the ventilation provided by Ekman pumping even in latitudes far south of the outcrop where the layer is shielded from direct influence of the wind. Analytical solutions are presented in the case where the density-outcrop lines are coincident with latitude circles. The solutions are not self-similar and important sub-domains of the solution are defined by critical potential vorticity trajectories which separate the ventilated from the unventilated regions in the lower thermocline. These cri...

Finite-Amplitude Baroclinic Waves

Abstract A theory is presented for the finite-amplitude behavior of unstable baroclinic waves in a quasi-geostrophic two-layer model. It is shown that in the absence of dissipation the equilibrated finite-amplitude state exhibits an oscillation, both of the mean flow and the baroclinic wave. On the other hand, if sufficient dissipation is present, the final state is a wave whose amplitude is steady and the approach to that state is non-oscillatory.

Circulation around islands and ridges

The circulation in an ocean basin containing an island is studied under nearly geostrophic, beta plane dynamics. The model is a fluid of uniform density driven by wind forcing or sources and sinks of mass at the upper boundary of the flow. The circulation is studied analytically, numerically, as well as in the laboratory through the device of the “sliced cylinder” model for the ocean circulation. Of particular interest is the estimate of the transport between the island and the oceanic basin’s boundary. The model is conceived as relevant to both the wind-driven circulation as well as the circulation of abyssal waters around deep topographic features such as mid-ocean ridge segments. Godfrey’s Island Rule for the transport is rederived in general form and the validity of the original approximation of Godfrey (1989) is examined in a variety of circumstances. In particular, the role of dissipative boundary layers and inertial effects such as vortex shedding are scrutinized to determine their role in determining the net transport around the island. Linear theory in many cases predicts a recirculation on the eastern side of the island, provided the meridional extent of the island is large enough. The existence of the recirculation, containing trapped fluid, is confirmed in both laboratory and numerical experiments and the evolution of the structure of the recirculation is examined as a function of the boundary layer Reynolds number. In both the laboratory and numerical studies, the recirculation predicted by linear theory is joined and then superseded by an inertial recirculation springing from boundary layer separation as the Reynolds number increases past a critical value. Even in the linear limit it is shown that the recirculation region, which is closed in quasigeostrophic theory, is subject to a small leak due to planetary geostrophic effects, which prediction is confirmed in the laboratory. The original island rule of Godfrey yields an estimate of the transport which is surprisingly robust and generally within 75% of the values measured in our numerical experiments. Agreement is moderately good when island western boundary layer transport is used as a basis for comparison. Several cases are discussed, however, in which the assumptions made by Godfrey are violated. One occurs when the frictional boundary layers of the island and the basin boundary overlap. We derive a threshold width for the gap for the case where the island is close to a northern or southern boundary of the basin and show how the transport is increasingly blocked as the gap is reduced. A second case occurs when the island is thin and zonally elongated so that the dissipative effects on the northern and southern boundaries of the island become important. Here the vorticity balance assumed in the simple Island Rule is fundamentally altered, and we extend the Island Rule to account for the new dissipation.

A unified linear theory of homogeneous and stratified rotating fluids

A unified picture of the linear dynamics of rotating fluids with given arbitrary stratification is presented. The range of stratification which lies outside the region of validity of both the theories of homogeneous fluids,

Resonant Topographic Waves in Barotropic and Baroclinic Flows

\sigma S and the strongly stratified fluids, σ S > E ½ , is studied, where σ S = v α g Δ T /κΩ 2 L and E = v /Ω L 2 . The transition from one dynamics to the other is elucidated by a detailed study of the intermediate region E 2/3 S E ½ . It is shown that, within this intermediate stratification range, the dynamics differs from that of either extreme case, except in the neighbourhood of horizontal boundaries where Ekman layers are present. In particular the side wall boundary layer exhibits a triple structure and is made up of (i) a buoyancy sublayer of thickness (σ S ) −1/4 E ½ in which the viscous and buoyancy forces balance, (ii) an intermediate hydrostatic, baroclinic layer of thickness (σ S ) ½ and (iii) an outer E ¼ -layer which is analogous to the one occurring in a homogeneous fluid. In the interior, the dynamics is mainly controlled by Ekman-layer suction, but displays hybrid features; in particular the dynamical fields can be decomposed into a ‘homogeneous component’ which satisfies the Taylor-Proudman theorem, and into a ‘stratified component’ which is baroclinic and which satisfies a thermal wind relation. In all regions the structure of the flow is displayed in detail.

Finite-Amplitude Baroclinic Waves with Small Dissipation

Abstract The problem of resonant, topographic quasi-geostrophic waves is examined analytically by exploiting simplifications that arise when the flow is nearly resonant. The barotropic and (two-layer) baroclinic problems are studied. In each case the topographic linear stability problem is solved explicitly and analytic expressions are given for the growth rate. The bifurcation problem in finite amplitude also is described. Some differences with earlier treatment are noted. In particular, in the barotropic problem subresonant instability may occur if the zonal wavelength is long enough. In both the barotropic and baroclinic problems the critical point at which multiple equilibria occur does not correspond to the stability thresholds of the linear problem. In the baroclinic problem Reynolds stresses are found to be of equal importance with eddy heat fluxes in altering the zonal flow although only the latter can transfer energy to the wave field for the zonal velocity profile considered. Analysis of the mar...

A simple laboratory model for the oceanic circulation

Abstract A study of finite-amplitude baroclinic instability for a two-layer system with small but non-zero dissipation is presented. The presence of dissipation, however slight, allows the existence of steady finite-amplitude wave solutions. For sufficiently small friction, however, the steady wave may be unstable if a certain criterion, presented in this paper, is satisfied. Calculations indicate that in such cases a continuous, slow, periodic amplitude pulsation exists which is independent of the initial conditions.

On the steady motions produced by a stable stratification in a rapidly rotating fluid

A linear theory is developed for the motion of a viscous, incompressible fluid in a rotating cylinder with a sloping bottom. An analysis of the normal modes of oscillation reveals that the presence of the bottom slope introduces a new set of low frequency inertial oscillations to replace the purely geostrophic modes which are not allowed in this geometry. The new waves possess mean circulation and are the mechanism by which the fluid adjusts to changes in the rotation rate of the container, a process discussed in detail. The steady motion produced in the cylinder when the cylinders upper surface rotates at a different rate than the bottom surface is studied. It is shown that the presence of the bottom slope inhibits the steady fluid motion in the body of the cylinder and introduces a non-symmetric, high velocity side wall boundary layer. Experimental evidence, presented to validate the theory, reproduces certain important features of the oceanic circulation.

An Inertial Theory of the Equatorial Undercurrent

The equilibrium state of a rapidly rotating fluid, heated uniformly from above and cooled uniformly from below while contained in a cylinder with insulated side-walls is studied. The circulations which are produced by the resulting stratification are studied over a wide range of parameters and it is shown that many of the features of the linear theory of rotating stratified fluid flows found in earlier studies reappear in this non-linear problem. These include the gradual disappearance of Ekman layer suction and O (1) Ekman layers as the stratification increases, and the determination of the interior flow by the side-wall boundary layers in conjunction with the Ekman layers. It is suggested therefore, that studies of rotating stratified flows which are unbounded laterally may frequently be defective and lead to solutions which are not the limit of any physically realizable experiment.

A Study of the Time Dependent Ocean Circulation

Abstract An inertial nonlinear model of the equatorial undercurrent is presented. The model is a simple two-layer model whose lower layer represents the undercurrent. The flow in the lower layer preserves potential vorticity and Bernoulli function. The former includes the relative vorticity f the current and the latter includes the currents kinetic energy. The required relation between the potential vorticity and the Bernoulli function is determined by matching the solution far from the equator with the ventilated thermocline theory of Luyten et al. The model describes an eastward-accelerating undercurrent fed by a general wedge-shaped meridional circulation pattern. The general character of the meridional and zonal flow, as well as the magnitudes of the undercurrent velocity, the current width and thermocline depth agree reasonably well with observations.