Melvin E. Stern

Ageostrophic instability of ocean currents

We investigate the stability of gravity currents, in a rotating system, that are infinitely long and uniform in the direction of flow and for which the current depth vanishes on both sides of the flow. Thus, owing to the role of the Earths rotation in restraining horizontal motions, the currents are bounded on both sides by free streamlines, or sharp density fronts. A model is used in which only one layer of fluid is dynamically important, with a second layer being infinitely deep and passive. The analysis includes the influence of vanishing layer depth and large inertial effects near the edges of the current, and shows that such currents are always unstable to linearized perturbations (except possibly in very special cases), even when there is no extremum (or gradient) in the potential vorticity profile. Hence the established Rayleigh condition for instability in quasi-geostrophic models, where inertial effects are assumed to be vanishingly small relative to Coriolis effects, does not apply. The instability does not depend upon the vorticity profile but instead relies upon a coupling of the two free streamlines. The waves permit the release of both kinetic and potential energy from the mean flow. They can have rapid growth rates, the e -folding time for waves on a current with zero potential vorticity, for example, being close to one-half of a rotation period. Though they are not discussed here, there are other unstable solutions to this same model when the potential vorticity varies monotonically across the stream, verifying that flows involving a sharp density front are much more likely to be unstable than flows with a small ratio of inertial to Coriolis forces. Experiments with a current of buoyant fluid at the free surface of a lower layer are described, and the observations are compared with the computed mode of maximum growth rate for a flow with a uniform potential vorticity. The current is observed to be always unstable, but, contrary to the predicted behaviour of the one-layer coupled mode, the dominant length scale of growing disturbances is independent of current width. On the other hand, the structure of the observed disturbances does vary: when the current is sufficiently narrow compared with the Rossby deformation radius (and the lower layer is deep) disturbances have the structure predicted by our one-layer model. The flow then breaks up into a chain of anticyclonic eddies. When the current is wide, unstable waves appear to grow independently on each edge of the current and, at large amplitude, form both anticyclonic and cyclonic eddies in the two-layer fluid. This behaviour is attributed to another unstable mode.

The physical significance of modons: Laboratory experiments and general integral constraints

A barotropic jet emerging from a point source in a rotating fluid is deflected to the right (northern hemisphere) and starts to accumulate in an anticyclonic vortex. This gives rise to a cyclonic neighbor, and the dipole (modon) then propagates away from the source in a circular path. A modification of Batchelors (1967) solution, which takes into account the different strenghts of the anticyclonic-cyclonic pair, is able to account for the path curvature. The experiment shows that highly organized modons can be realized in the laboratory with rather nondescript forcing. The s-effect (not noticeably present in the experiment) should enhance the realizability of these structures in geophysical flows. Therefore, it is suggested that the modon model captures certain essential features of geophysical eddies. This is based on a derived theorem which shows that any slowly varying (not necessarily uniformly propagating) and isolated disturbance on the beta plane must have zero net relative angular momentum, so that the dipole is the simplest dynamically consistent representation of such a disturbance. Some interesting aspects of two-dimensional turbulence in a rotating fluid are also indicated by the laboratory esperiments and by the general integral theorems presented.

Collective instability of salt fingers

We first consider a steady laminar model of salt fingers and show that the latter become unstable with respect to internal gravity waves when the finger Reynolds number exceeds a critical value. The criterion is then used in speculations about the statistically steady state in a fully developed similarity model where horizontally averaged temperature and salinity gradients are constant at all depths. Dimensional reasoning is used to obtain the asymptotic dependence of the turbulent flux on the molecular salt diffusivity. From this and other relationships order-of-magnitude estimates are obtained and compared with laboratory experiments and ocean observations.

Lateral mixing of water masses

The vertical transport of heat and salt in the “salt-finger” convective regime can be enhanced by the vertical shear of a “medium” scale motion. A weak shear converts compensating temperature-salinity variations on isopycnal surfaces into strong vertical T-S gradients. The cumulative and selective modification of the density field amplifies the medium scale motion. The generation of geostrophic shear by this mechanism is discussed using a basic state with uniform horizontal and vertical mean T-S gradients. It is suggested that the main halocline of the oceanic central waters tends to a marginally stable and statistically steady state which has no T-S variations on isopycnal surfaces. Dimensional considerations lead to an order of magnitude estimate of the r.m.s. geostrophic velocity.

The intrusion of a density current along the coast of a rotating fluid

When light rotating fluid spreads over heavier fluid in the vicinity of a vertical wall (coast) a boundary jet of width Λ forms, the leading edge or nose of which propagates with speed ĉ along the coast. A certain fraction 8 of the boundary transport is not carried by the nose but is deflected backwards (detrained) and left behind the propagating nose. Theoretical and experimental results for Λ,ĉ, and δ are given for a quasi-equilibrium (constant-ĉ) regime. Over longer time intervals the laboratory observations suggest that the nose slows down and stagnates, whereupon the trailing flow separates from the coast and an intermittent boundary current forms. These processes may be relevant to the mixing of oceanic coastal currents and the maintenance of the mean current.

Salt fingers and convecting layers

Abstract It is shown that convective ‘fingers’ can form not only in the salt-heat system, but also in a fluid containing two solutes with much closer diffusivities (for example sugar above a salt solution). This experimental device is used to explore various phenomena which cannot easily be produced in the laboratory with salt and heat. We have investigated: (a) the behaviour of an interface containing fingers and separating two convecting layers, (b) the production of a series of steps from a smooth gradients by imposing a flux at the top, (c) and the mechanism of formation of layers from two opossing gradient bof the solutes. In each case the related theoretical ideas are summarized, and it is suggested how these can be extended to describe the new features of the observations.

Geostrophic fronts, bores, breaking and blocking waves

An undisturbed geostrophic density current flows along a vertical wall (the coast) with the free streamline (the front) located at a distance L from the wall which is comparable to the Rossby radius of deformation. Finite amplitude perturbations with downstream wavelengths much larger than L are discussed, and it is shown that the slope of the front in the horizontal plane increases with time. Some perturbations tend to ‘break’ seaward by developing large transverse velocities away from the coast. The temporal evolution of some perturbations is such as to completely ‘block’ the upstream flow, but the subsequent behaviour is beyond the scope of the theory. We also discuss the propagation of the nose of the intrusion when a density current debouches from a coastal source and then flows along the coastal boundary.

Interaction of a uniform wind stress with a geostrophic vortex

Abstract We consider the weak (small Rossby numbers) non-linear interaction between a quasi-geostrophic vortex and the Ekman flow which is produced by a uniform wind stress acting at the top of the mixed layer. The differential advection of geostrophic vorticity by the undisturbed Ekman components tends to tilt the axis of the vortex away from the vertical but, because of the strong rotational constraint, the vortex maintians vertical ‘rigidity’ by developing vertical velocities which balance the advective tendency (and frictional rotation). In this approximate vorticity balance for a ‘frictional’ component of the total motion, the vertical integral implies a net vertical velocity (‘Ekman suction’) at the free surface. To maintain the proper boundary condition an equal and opposite velocity must be applied to a second (geostrophic) component of the motion. This coupling condition has been determined as a function of the surface wind stress, the gradient of geostrophic vorticity, and the depth of the homogeneous mixed layer. If the base of the mixed layer were rigid the vortex would move as a unit with the velocity of the mean Ekman drift. The suction velocity formula is then generalized to include the special case ( Charney 1955) of a global wind variation and variable Coriolis parameter ( Sverdrup 1947). The effect of the local wind in supplying kinetic energy to the deeper ocean is also discussed, and suggestions are made for including the baroclinicity of the friction layer into the geostrophic dynamics.

Dynamics of Potential Vorticity Fronts and Eddy Detachment

Abstract The formation and detachment of quasi-geostrophic eddies in a 1½ layer jet is studied using a piecewise uniform potential vorticity model. A vorticity front separates the two pieces, and thus the jet has cusplike character. The evolution of large amplitude initial disturbance (whose origin may be attributed to barotropic-baroclinic instability mechanisms not explicit in our model) is computed by the method of contour dynamics. Certain numerical results such as the steepening of the front prior to eddy detachment can be physically explained in terms of differential mean field advection and vortex induction. Computations are made for a variety of initial conditions and we indicate the amplitude/scale conditions necessary for the detachment of an eddy. The discussion is directed to the problem of the formation of warm/cold rings in the Gulf Stream. The effect of a coast on large perturbations of a jet is also briefly discussed.

Coherent baroclinic eddies on a sloping bottom

A coherent and stable baroclinic eddy in a rotating fluid was produced on a sloping bottom by releasing a dome of salt water into the ambient fresh water. A strong cyclonic vortex is produced above the heavy dome. The entire eddy system moves ‘north-westward’ (with the up-slope direction designated ‘north’) as a ‘Taylor column’. The eddy system displays long lifetimes, but it is shown that a theory of isolated systems cannot account for the experimental observations. Instead, it is demonstrated that the vortex flow above the lens is along the lines of constant depth, producing a net pressure force on the lens, which approximately balances the buoyancy force. When Ekman friction is also included, it accounts for the northward motion of the dome.