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The article of record as published may be found at http://dx.doi.org/10.1175/2010JPO4459.1 This study examines dynamics of finescale instabilities in thermohaline–shear flows. It is shown that the presence of the background diapycnal temperature and salinity fluxes due to double diffusion has a destabilizing effect on the basic current. Using linear stability analysis based on the Floquet theory for the sinusoidal basic velocity profile, the authors demonstrate that the well-known Richardson number criterion (Ri , 1/ 4) cannot be directly applied to doubly diffusive fluids. Rigorous instabilities are predicted to occur for Richardson numbers as high as—or even exceeding—unity. The inferences from the linear theory are supported by the fully nonlinear numerical simulations. Since the Richardson number in the main thermocline rarely drops below 1/ 4, whereas the observations of turbulent patches are common, the authors hypothesize that some turbulent mixing events can be attributed to the finescale instabilities associated with double diffusive processes.

Two-dimensional direct numerical simulations (DNS) are used to investigate the growth and nonlinear equilibration of spatially periodic double-diffusive intrusion for negative vertical temperature Tz < 0 and salinity Sz < 0 gradients, which are initially stable to small-scale double diffusion. The horizontal temperature Tx and salinity Sx gradients are assumed to be uniform, density compensated, and unbounded. The weakly sloping intrusion is represented as a mean lateral flow in a square computational box tilted with a slope equal to that of the fastest-growing linear theory mode; the vertical (η) domain size of the box L*η is a multiple of the fastest-growing wavelength. Solutions for the fastest-growing wavelength show that the intrusion growth is disrupted by salt fingers that develop when the rotation of the isotherms and isohalines by the intrusion shear results in temperature and salinity inversions; the thick inversion regions are separated by a thin interface supporting diffusive convection. These equilibrium solutions were always unstable to longer vertical wavelengths arising because of the merging of the inversion layers. The DNS predicts the following testable results for the maximum lateral velocity U* max = 0.13NSL*η, the lateral heat flux F* = 0.008ρCP(Sx/Sz)1/2(NS/KT)1/4NSL*η2.5(βSz/α), and the interface thickness hρ = 0.12L*η, where NS = , g is the gravity acceleration, ρ is the density, β/α is the haline contraction/heat expansion coefficient, and CP is the specific heat capacity. The results are compared with observations in the Arctic Ocean.

The slow downstream (x) variation of a dense and inviscid bottom current (u) in a parabolic strait with a sill at y = 0 is investigated. Vanishing potential vorticity is assumed and the density interface in the 1 1/2-layer model intersects the bottom at y = y1 and y = y2 < y1, where the vanishing layer thickness (h) provides the free dynamical boundary condition. For time-dependent finite-amplitude waves, the nonlinear hyperbolic equations obtained here give the wave velocity and indicate the sense in which lateral wave steepening occurs. The long-wave perturbations of y1(x,t), y2(x,t) are stationary if where g′ is the reduced gravity, μ = ∂2M/∂y2 is the parabolic curvature of the bottom elevation (M), and f is the Coriolis parameter. This controls the upstream–downstream flow, and the downstream nonlinearity generates ‘short’ waves which may initiate lateral mixing with the adjacent (less dense) water mass.It is also shown that short waves are exponentially amplified with a maximum growth rate (about 1/day) depending only on g′μ/f2. When g′μ/f2 = 1 (a narrow strait) the instability is suppressed, but for small g′μ/f2≪1 the growth rate is comparable to the flat bottom case μ = 0, studied by Griffiths, Killworth & Stern (J. Fluid Mech. Vol. 117, 1982, p. 343.).

This paper considers the equilibration of lateral intrusions in a doubly diffusive fluid with uniform unbounded basic-state gradients in temperature and salinity. These are density compensated in the horizontal direction and finger favorable in the vertical direction. Previous nonlinear studies of this effect have qualitative and quantitative limitations because of their fictitious parameterizations of the weak “turbulence” that arises. Here, two-dimensional direct numerical simulations (DNS) that resolve scales from the smallest to the intrusive are used to predict the equilibrium state. This is achieved by numerically tilting the x–z computational box so that the mean intrusion is represented by a mode with no lateral variation, but smaller-scale 2D eddies comparable to the intrusion thickness are resolved. The DNS show that the initial plane wave intrusion evolves to an equilibrium state containing both a salt finger interface and a diffusive interface, surrounded by well-mixed layers. The inversion of the horizontally averaged density in the mixed layer is negligibly small, but the salt finger buoyancy flux produces large transient density inversions that drive the mixed layer convection. For the considered values of horizontal/vertical gradients, the calculations yield small Cox numbers and buoyancy Reynolds numbers [comparable to those measured in staircases during the Caribbean-Sheets and Layers Transects (C-SALT) program]. An important testable result is the time-averaged maximum velocity of the fastest-growing intrusion Umax = 18.0 (Σ*z/Σ*x)+1/2KT(gΘ*z/νKT)1/4. Here Θ*z is the undisturbed vertical temperature gradient in buoyancy units, Σ*z and Σ*x are the corresponding vertical and horizontal salinity gradients, g is the gravity acceleration, and ν and KT are the respective values of the molecular viscosity and heat diffusivity. The paradoxical inverse dependence on the horizontal gradient results from the assumption that the latter is unbounded.

A density-compensated thermohaline front with a finite width L∗ and total lateral salinity variation ΔS superimposed on a finger-favorable thermocline is studied by means of linear theory and nonlinear numerical calculations using parameterized finger fluxes. By retaining the relatively small molecular heat diffusion and the dependence of the finger Nusselt number Nu on the density ratio, it is shown that marginal stability with critical ΔS is possible. The lateral fluxes induce a mean vertical velocity w0(x), which can sharpen the front, but when ΔS exceeds a certain value the effect of the lateral fluxes exceeds that of w0(x) and the front weakens. Nonlinear 2D spectral calculations showed that the modification of the mean horizontal gradients is small and that the intrusions continue to grow exponentially even after they have produced overturns. By extrapolating the linear theory results beyond the overturning stage, the intrusion velocity was estimated to be proportional to the Brunt–Vaisala ...

The free discharge of a layer of bottom water in a wide strait (e.g., the Denmark Strait) differs from the classical control problem because of the strong geostrophic turbulence. As a consequence, the cross-stream (x) variation of the time-averaged downstream velocity υ(x) is “underdetermined” and depends on more parameters than available conditions. To resolve this “degeneracy” the classical control condition is generalized; the result requires the discharge Q to be extremized with respect to the degeneracy parameters and with respect to the constraints. One of these constraints is that a branch point or a local stationary wave can be supported at some section of the long channel. By either maximizing Q or by requiring a stationary wave, useful approximations are obtained. Future work should consider the joint variational problem. Specific calculations are made for nonuniform potential vorticity in a rectangular channel and also for variable cross-stream bottom topography. In the latter case, th...

The mechanism by which amplifying salt fingers in an unbounded uniform T/S gradient are equilibrated is determined, starting with a time-dependent asymptotic field equation for (Rτ) -1 -1=e→0, where R > 1 is the T/S density ratio and τ = K S /K T < 1 is the molecular diffusivity ratio. A mode truncation of that equation yields an ODE which shows that the fastest growing finger mode transfers energy to two 'slave' modes with relatively small vertical scale; the finger mode thereby attains a statistically steady amplitude. The results for τ = 1/3 are compared with spectral solutions of the non-truncated equations in two and three dimensions; the predicted fluxes are testable in sugar (S), salt (T) laboratory experiments.

A vertically stable density stratification with temperature gradient T z < 0, salinity gradient S z < 0, and with density compensating horizontal T/S gradients is unstable to lateral intrusions because the molecular heat diffusivity is much larger than the salt diffusivity. Previous marginal instability theory is extended to the supercritical regime. The fastest growing vertical wavelength (δ z ) is obtained in a model of a T/S front with finite lateral width (L * ) and lateral salinity variation (ΔS). The value of δ z ΔS/|S z | varies only slightly with the parameters, and this result is applied to the smallest step size observed in the main thermocline of the Weddell Sea. Nonlinear considerations show that the laterally divergent heat/salt flux produced by the instability forces a mean vertical circulation which enhances the compensating lateral T/S gradients, thereby accelerating the intrusive instability, and eventually producing local density ratios sufficient to initiate strong vertical convection. This suggests that weak isopycnal T/S gradients are necessary to initiate the steps which subsequently merge into larger layers.

The salt finger fluxes obtained in small-domain direct numerical simulations (DNSs) are used to parameterize the fluxes in a larger domain that resolves internal gravity waves. For the case in which the molecular diffusivity ratio τ = KS/KT < 1 is not excessively small, the wave amplification and overturning in a large-domain DNS that resolves both fingers and waves agrees with the result of a parameterized model. Since such a DNS is not feasible for heat–salt (τ = 1/80), the parametric model is used for 2D spectral calculations with periodic boundary conditions. For density ratios R = 1.25 and 1.5 an initialized low-frequency internal wave amplifies because of the wave strain; the irreversible (finger) fluxes and the wave potential energy increase until the isopycnals overturn. The maximum value of heat–salt flux produced by the wave is comparable to the finger fluxes in the absence of the wave. Even larger total fluxes should occur in the subsequent turbulent overturning stage (not computed), i...

Numerical solutions for salt e ngers in an unbounded thermocline with uniform overall vertical temperature-salinity gradients are obtained from the Navier-Stokes-Boussinesq equations in a e nite computational domain with periodic boundary conditions on the velocity. First we extend previous two-dimensional (2D) heat-salt calculations [Prandtl number Pr 5 n/k T 5 7 and molecular diffusivity ratio t 5 k S/k T 5 0.01] for density ratio R 5 2; as R decreases we show that the average heat and salt e uxes increase rapidly. Then three-dimensional (3D) calculations for R 5 2.0, Pr 5 7, and the numerically “ accessible” values of t 5 1/6, 1/12 show that the ratio of these 3D e uxes to the corresponding 2D values [at the same ( t, R, Pr)] is approximately two. This ratio is then extrapolated to t 5 0.01 and multiplied by the directly computed 2D e uxes to obtain a e rst estimate for the 3D heat-salt e uxes, and for the eddy salt diffusivity (dee ned in terms of the overall vertical salinity gradient). Since these calculations are for relatively “ small domains” [ O(10) e nger pairs], we then consider much larger scales, such as will include a slowly varying internal gravity wave. An analytic theory which assumes that the e nger e ux is given parametrically by the small domain e ux laws shows that if a critical number A is exceeded, the wave-strain modulates the e nger e ux divergence in a way which amplie es the wave. This linear theoretical result is cone rmed, and the e nite amplitude of the wave is obtained, in a 2D numerical calculation which resolves both waves and e ngers. For highly supercritical A (small R) it is shown that the temporally increasing wave shear does not reduce the e uxes until the wave Richardson number drops to ;0.5, whereupon the wave starts to overturn. The onset of density inversions suggests that at later time (not calculated), and in a sufe ciently large 3D domain, strong convective turbulence will occur in patches.

We compute numerically the amplitude of long thin fingers that form in a liquid stratified with sugar S * and salt T' (measured in buoyancy units), for which τ = k S /k T = 1/3 is the ratio of the two diffusivities and the Prandtl number is Pr = v/k T ∼ 10 3 , where V is the viscosity. The finger layer in our model is bounded by rigid and slippery horizontal surfaces with constant T, S * (the setup is similar to the classical Rayleigh convection problem). The numerically computed steady fluxes compare well with laboratory experiments in which the fingers are sandwiched between two deep (convectively mixed) reservoirs with given concentration differences ΔT * , ΔS * . The model results, discussed in terms of a combination of asymptotic analysis and numerical simulations over a range of density ratio R = ΔT * /ΔS * , are consistent with the (ΔS * ) 4/3 similarity law for the fluxes. The dimensional interfacial height (H * ) in the reservoir experiments (unlike that in our rigid lid model) is not an independent parameter, but it adjusts to a statistically steady value proportional to (ΔS * ) -1/3 . This similarity law is also explained by our model when it is supplemented by a consideration of the stability of the very thin horizontal boundary layers with large gradients (∂S * /∂z) which form near the rigid surfaces. The preference for three-dimensional salt fingers is also explained by a combination of analytical and numerical considerations.

Analytical solutions of the shallow-water system of equations in a 1½-layer f-plane model are obtained for stable quasi-monopolar vortices propagating due to a dipolar (azimuthal mode m = 1) component whose amplitude is small compared to the circularly symmetric (m = 0) part. The ability of the f-plane eddies with O(1) Rossby number to propagate distances greater than their size is demonstrated by finite amplitude numerical calculations. Numerical and analytical considerations also indicate that significant departures may occur between the “centroid of mass anomaly” (commonly used in theoretical estimates of the speed of oceanic vortices) and other measures of the eddy center.

A series of multi-layer numerical experiments show that classical e nite amplitude instabilities in boundary currents are not sufficient to account for the pinched-off eddies observed in the ocean and in laboratory experiments. These instabilities (barotropic or baroclinic) are shown to lead to an entrainment of offshore e uid into the boundary currents. Eddy separation, on the other hand, requires an additional process, such as a larger scale of motion containing a downstream velocity convergence of e nite amplitude; this might be produced by long period e uctuations in the discharge from an upstream source region which controls the boundary current, or by topographic features. In our spatially idealized model, we numerically computed the temporal evolution of an assumed initial state consisting of a fast moving upstream region separated by a potential vorticity front from a slow moving downstream region. We verify long-wave theories which show that this initial state indeed leads to frontal steepening and to a blocking wave. This eventually produces large transverse velocities followed by complete detrainment of eddies without any entrainment into the residual boundary current.

Contour dynamics is used to compute the two-dimensional (f-plane) motion of an initially circularly symmetric barotropic eddy with piecewise-uniform vorticity as it is advected around a circular obstacle by a uniform upstream current. For grazing incidence of this ‘shielded’ eddy (compensating positive and negative vorticity) the main effect of the vortex images (inside the obstacle) is to change the speed of those particles in the outer portion of the eddy that are closest to the obstacle; a lesser velocity is induced on the oppositely signed vortices near the eddy centre. The result is a systematic separation of the centroids of the ± vortices in the eddy, and the eddy emerges far downstream with an invariant dipole moment (m = 1 azimuthal mode). This causes the eddy to move with a constant velocity V normal to the uniform basic flow. The ratio of the numerically computed V to the accompanying far-field dipole moment agrees with a previous analytical theory for a completely isolated eddy subjected to a small-amplitude m = 1 initial disturbance. The scattering effect might be realizable in a rotating homogeneous fluid by translating a cylinder relative to an otherwise stationary eddy. Application to a density-stratified model is suggested.

Three dimensional (3D) numerical calculations are made for a vertically unbounded fluid with initially uniform vertical gradients of sugar (S) and salt (T), where τ = κ s /κ T = 1/3 is the diffusivity ratio, and the molecular viscosity is ν >> κ T . The latter inequality allows us to neglect the nonlinear term in the momentum equation, while retaining such terms in the T-S equations. The discrete 3D Fourier spectrum resolves the fastest growing horizontal wavelength, as well as the depth independent Fourier component. Unlike previous calculations for the pure 2D case the finite amplitude equilibration in 3D is primarily due to the instability of the lateral S-gradients in the fingers, and the consequent transfer of energy to vertical scales comparable with the finger width. It is shown that finite amplitude two-dimensional disturbances are unstable and give way to three dimensional fingers with much larger fluxes. Calculations are also made for rigid boundary conditions at z = (0, L) in order to make a rough quantitative comparison with previous lab experiments wherein a finger layer of finite thickness is sandwiched between two well-mixed (T, S) reservoirs. The flux ratio is in good agreement, and the fluxes agree within a factor of two even though the thin interfacial boundary layer between the reservoir and the fingers is not quite rigid because sheared fingers pass through it. It is suggested that future experiments be directed toward the much simpler unbounded gradient model, for which flux and variance laws are given herein.

An analytical theory is developed for a class of stable quasi-geostrophic vortices propagating either in the westward direction with supercritical velocities (cR2d) or eastward, where Rd is the radius of deformation and β is the gradient of the Coriolis parameter. The numerical spectral calculations, initiated by the theoretical solutions, indicate that the supercritical vortices move initially with the predicted velocity, but later slow down to the speed of the long planetary waves. The period of time during which an eddy is propagating with its initial velocity is analysed as a function of its strength and (initial) speed. The theoretical solution has a distinctive streamline signature (‘alpha gyre’), which also appears in the final state of the numerical calculations, including those for an initially completely symmetric vortex placed on the β-plane.

A “wagon” rolls up a slightly inclined plane ( effective gravity acceleration = g ∗ ), and then it rolls down until it collides inelastically (coefficient of restitution = 1 − ∈) with a piston oscillating parallel to the plane between ξ = 0 and ξ = ξmax and with velocity νf(ωt), where ν is amplitude ω is frequency, and f is a specified waveform (e.g., sinusoidal). In the high frequency limit ( νω g ∗ → ∞ ) there are very many periodic solutions, but in the presence of a small noise level (e.g., the standard deviation in ∈) none of these are realizable because, when ∈ → ∞, this noise is assumed to introduce a random phase ωtn ≡ φn (mod 2π) of the piston each time (tn) the wagon arrives at ξ = ξmax moving down-hill with velocity V. For mathematical simplicity we confine our attention to the case of a square wave oscillator (f = ±1), and derive an evolution equation for the probability distribution function Pn+1(V). Numerical solutions reveal the approach to a unique statistically steady state (n → ∞) independent of the assumed initial distribution P1(V), and independent of the noise level.

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For O(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) offslope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important fo...

If an azimuthally symmetric barotropic eddy on the f plane is subject to a relatively small amplitude disturbance of unit azimuthal wavenumber (m 5 1), it can propagate very many diameters away from its origin, as shown by a weak nonlinear theory for a piecewise uniform vorticity eddy, and also for one with continuous vorticity inside a finite area. In the former case an initial value contour dynamical calculation shows that the analytical solution is realizable over long distances; the same is true in the latter case, as shown by spectral calculations using the full two-dimensional vorticity equation (with small dissipation). The oceanographic significance of this effect lies in the ability of almost symmetric eddies to self-propagate over large distances and collide with other eddies, currents, and continents; this produces important mixing effects, as illustrated herein. It is also shown how the analysis and the effect is generalizeable to a 1‰-layer density model on the b plane.

Finite amplitude numerical calculations are made for a completely unbounded salt e nger domain whose overall vertical ‘ ‘ property’ ’ gradients ( Tz andSz) are uniform and remain unaltered in time. For diffusivity ratio t 5 k S/k T 5 O(1), Prandtl number n /k T ae 1, and density ratio R 5 Tz/Sz . 1 this regime corresponds to a ‘ ‘ double gradient’ ’ sugar ( S) 2 salt (T) experiment. Two-dimensional pseudo-spectral calculations are made in the vicinity of the minimum critical condition for salt e nger instability, viz., small e ; (Rt ) 2 1 2 1 . 0; the allowed spectrumincludes the fastest growing wave of linear theory. When the vertical wavelength of the fundamental Fourier component is systematically increased the solution changes from a single steady vertical mode to a multi-modal statistically steady chaotic state. Each of the long vertical modes can be amplie ed by the (unchanging overall) gradient Sz, and can be stabilized by the induced vertical T, S gradients on the same scale as the modes; nonlinear triad interactions in the T 2 S equations can also lead to amplitude equilibration even though e , k T/n , and the Reynolds number are extremely small. When subharmonics of the horizonal wavelength of maximum growth are introduced into the numerical calculations the new wave amplie es (via Sz) and produces a quantitative change in the time average e uxes. Experimentally testable values of heat e ux and rms horizontal T-e uctuations are computed in the range2.8 . R . 1.6 for t 5 1 A3.Asymptotic similarity laws e ® 0 are also presented.

It is suggested that the large inshore shear of Western Boundary Currents, such as is observed on the continental slope of the Florida Current, can be maintained by instability waves which produce a downgradient potential vorticity flux and an upgradient flux of relative vorticity. This is shown for a non-inflected barotropic shear layer above continuously varying bottom topography, and also for a ‘ 1 1 2 layer’ quasi-geostrophic current bounded below by a density interface. The linear eigenfunctions are substituted into the nonlinear potential vorticity equation to obtain the mean vorticity tendency, and then the quantitative increase (in the case of the barotropic model) is computed by a pseudo-spectral calculation of the nonlinear equation.

If a steady jet with two free streamlines ascends a sill while entirely remaining on one sloping side, then the rising motions must be compensated by transverse sinking motions in order that the cross-stream integral of relative vorticity vanishes at all downstream sections. The maximum sill height which allows such a flow is computed assuming slow downstream changes in a quasi-geostrophic barotropic model; a 1½ layer baroclinic model is also considered. If the critical sill height is exceeded, it is suggested that the jet will split with only part of it crossing the sill on the original side. This might explain why the Antarctic Circumpolar Current bifurcates at the Falkland Plateau, where one branch forms the Malvinas Current and the other branch turns eastward across the Atlantic. The theory is related to previous laboratory observations in which bifurcation occurs.

The previously observed spatial evolution of the two-dimensional turbulent flow from a source on the vertical wall of a shallow layer of rapidly rotating fluid is strikingly different from the non-rotating three-dimensional counterpart, insofar as the instability eddies generated in the former case cause the flow to separate completely from the wall at a finite downstream distance. In seeking an explanation of this, we first compute the temporal evolution of two-dimensional finite-amplitude waves on an unstable laminar jet using a finite difference calculation at large Reynolds number. This yields a dipolar vorticity pattern which propagates normal to the wall, while leaving some of the near-wall vorticity (negative) of the basic flow behind. The residual far-field eddy therefore contains a net positive circulation and this property is incorporated in a heuristic point-vortex model of the spatial evolution of the instability eddies observed in a laboratory experiment of a flow emerging from a source on a vertical wall in a rotating tank. The model parameterizes the effect of Ekman bottom friction in decreasing the circulation of eddies which are periodically emitted from the source flow on the wall. Further downstream, the point vortices of the model merge and separate abruptly from the wall; the statistics suggest that the downstream separation distance scales with the Ekman spin-up time (inversely proportional to the square root of the Coriolis parameter f) and with the mean source velocity. When the latter is small and f is large, qualitative support is obtained from laboratory experiments.

At the northeast corner of Taiwan the direction of the continental slope isobaths changes rapidly relative to the oncoming Kuroshio, so that the inertia of a small inshore fraction of this current causes it to cross the slope, while the main branch follows the isobaths. It is suggested that the portion of the bifurcated current entering the shelf displaces ambient water of relatively high potential vorticity as a countercurrent, which flows across the slope. The vortex stretching and subsequent entrainment of this water into the main branch of the Kuroshio increases its maximum cyclonic vorticity and helps to maintain the inshore shear of the western boundary current. This Idea is supported by simple initial value and steady-state models, and also by dye observations of the flow from a source on the wall of a rotating tank.

An inviscid laminar boundary layer flow Û(ŷ) with vertical thickness λy, and free stream velocity U is disturbed at time $\tcirc$ = 0 by a normal velocity $\vcirc$ and by a spanwise velocity ŵ([xcirc ],ŷ, $\zcirc$, 0) of finite amplitude αU, with spanwise ($\zcirc$) scale λz, and streamwise ([xcirc ]) scale λx = λz/α; the streamwise velocity û([xcirc ],ŷ,$\zcirc$,$\tcirc$) is initially undisturbed. A long wave λy/λz → 0) expansion of the Euler equations for fixed α and time scale $\tcirc$s = U−1λz/α results in a hyperbolic equation for Lagrangian displacements ŷ. Within the interval $\tcirc$ > $\tcirc$s of asymptotic validity, finite parcel displacements (O(λy)) with finite (O(U)) û fluctuations occur, independent of α no matter how small; the basic flow Û is therefore said to be unstable to streaky (λx [Gt ] λz) spanwise perturbations. The temporal development of the ('spot’) region in the (x,z) plane wherein inflected û profiles appear is computed and qualitatively related to observations of ‘breakdown’ and transition to turbulence in the flow over a flat plate. The maximum $\vcirc$([xcirc ],ŷ,$\zcirc$,$\tcirc$) increases monotonically to infinity as $\tcirc$ → $\tcirc$s.

The use of a classical eddy parametrization in the analysis of continental boundary currents leads to the diffusion of momentum and relative vorticity and fails to recognize that the relevant eddies are dominated by the conservation of potential vorticity, which in turn may produce an increase in the mean relative vorticity. To illustrate this effect, we examine a non-inflected barotropic shear flow destabilized by the cross-steam variation in the bottom topògraphy of a continental slope. The finiteamplitude evolution of the waves is analysed in a simple model with a step-like bottom topography and with a piecewise-uniform potential vorticity distribution. The increase in maximum mean vorticity is computed for various values of the Rossby number and the topographic elevation, and it is suggested that a similar effect, taking into account the isopycnal topography as well as the isobaths, could maintain the large inshore shear of the Gulf Stream. Cross-shelf transport of different water ‘types’ (i.e. potential vorticity and passive tracers) are also computed and suggested to be pertinent to the more realistic oceanic problem involving baroclinic effects. The numerical calculation employs the well-known method of contour dynamics, and the Green's function appropriate for the step-like topography is derived.

A “gravitationally stable” stratification of salinity and temperature, such as is observed in the oceans, is actually unstable due to the fact that the molecular diffusivity of heat is much greater than the diffusivity of salt. We discuss this stability characteristic and the form of the convective motion in the laminar regime. Future studies of this model relative to the amplitude of the motion and the subsequent transition to turbulence should lead to the formulation of critical observational questions, which will determine whether the proposed mechanism is significant in the vertical mixing of the sea.DOI: 10.1111/j.2153-3490.1960.tb01295.x

When a low Rossby number barotropic flow accelerates in the laterally converging half of a strait, the local propagation speed of long topographic waves can be reduced to zero, thereby blocking or preventing the formation of a steady flow downstream from the strait. An inviscid longwave theory is presented for the new steady upstream and downstream states that evolve from the blocking wave. The enhanced inshore cyclonic vorticity extending far downstream suggests that topographic jetogenesis, rather than lateral eddy diffusion, in major ocean straits (e.g., Yucatan and Florida) may be important in generating or reforming boundary currents.

It is suggested that the inshore shear of continental boundary flows like the Florida Current can be accounted for by a countergradient vorticity flux, rather than by lateral diffusion to the shore. Two simple barotropic models with cross-stream and downstream topographic variations illustrate the point. In the first case, a broad jet with piecewise uniform vorticity accelerates through a slowly converging strait, eventually becoming locally critical (in the hydraulic sense) and then undergoing a transition to a different downstream state in which the maximum inshore vorticity is increased. The topographic conditions for this to occur are determined by a nonlinear long-wave theory. In the second model, the computed flow around an idealized cape illustrates the role of lee waves in generating mean downstream vorticity. For large-amplitude capes a nonlinear long-wave theory shows that a downstream transition (similar to the strait problem) can occur as well as upstream “blocking.”

A semi-infinite jet flows along a vertical wall in a rotating fluid, with the nose of the intrusion approaching a corner where the wall turns through an obtuse angle θ + 180°. The jet separates at the corner and flows into the interior if θ exceeds a critical θc, otherwise part of the jet continues around the corner and flows along the downstream segment of the wall. The separation criterion is computed using an inviscid and piecewise-uniform-vorticity model, with s denoting the ratio of the maximum ‘offshore’ to ‘inshore’ vorticity. The separation effect is demonstrated by a laboratory experiment in which a two-dimensional jet flows along the wall from a source. Average velocities are used to estimate s, and to make semi-quantitative comparisons of experimental and theoretical θc. This suggests that the separation mechanism is independent of local viscous forces, although the cumulative effect of lateral eddy stresses in the jet is important in establishing the value of s immediately upstream from the corner. We suggest that our barotropic separation mechanism is relevant to mesoscale oceanic coastal currents.

GEOSECS measurements of excess radon 222 as a function of depth above the sediments in the Atlantic and Pacific are sufficient to resolve the median profile in each ocean, but not the mean profile. The transport process maintaining the oceanic means is highly intermittent, and is not consistent with mixing by small scale turbulence generated at the ocean bottom. We suggest that transport of radon above the benthic boundary layer is driven by isopycnal motions associated with baroclinic instabilities on benthic fronts. A method for generating benthic fronts together with the subsequent instabilities is demonstrated in the laboratory.

The primary instability of salt fingers in an unbounded fluid with uniform vertical salinity/temperature (S* z /T* z ) gradients consists of a vertically (z) uniform 'elevator' mode, amplifying exponentially in time. We compute the z-wavelength (h 0 ) of the fastest growing secondary instability by integrating a system of linear partial differential equations (PDEs) with time-dependent and horizontally periodic coefficients

[1] A single layer of fluid with thickness h(x) and geostrophic velocity u = (g′/f)∂(h + M)/∂x is conventionally used to model the flow of a density current over a sill/contraction, where M(x) is the cross-stream (x) bottom elevation, g′ = gΔρ/ρ is reduced gravity, and f is the Coriolis parameter. The interface [h] intersects the parabolic bottom at two points x = ξ,L, with M(L) = H given. A new extremum for the volume discharge Q is obtained which is approximately half that of the Killworth-Whitehead upper bound, and which agrees with measurements in the Denmark Strait.

A general variational principle for turbulence is applied to the large Reynolds number flow behind a cylinder, leading to the conclusion that the drag force must be extremized subject to the dynamical constraints. By using only two approximate versions of these, a drag coefficient equal to 3/8 is obtained. Comparison is made with experimental values in the ‘‘supercritical’’ (Re=106) and ‘‘transcritical’’ (Re=107) regimes.

An inviscid two-dimensional eddy surrounded by irrotational fluid is initially located near the edge of a jet, on the other side of which the vorticity increases. The interaction causes the eddy to move towards the edge and into the shear flow. Eventually the eddy and the ambient (irrotational) fluid are surrounded by the jet fluid

Baroclinic eddies in a rotating box with a sloping bottom were produced by squirting dense salt water up the sloping bottom and along the “eastern” wall. The jet stagnated in shallow water and was ejected normal to the wall. For certain parameters (volume flux of jet, etc.), a coherent lens of dense bottom water formed and propagated west with an overlying cyclonic vortex. The circulation in the bottom lens, on the other hand, was relatively weak. No such eddy forms when the depth of fresh water is relatively deep, and a regime diagram is given for the formation of the coherent eddies. Thus a relatively simple structure emerges despite the complexity of the generating process. The pressure field determined from density measurements is discussed in terms of an integral theorem for coherent eddies, and the westward propagation is also related to previous theories. Several other techniques for generating such eddies are discussed.

International audience; 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 c 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,c, and δ are given for a quasi-equilibrium (constant-c) 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.

The temporal evolution of large amplitude quasi-geostrophic disturbances in a piecewise uniform potential vorticity flow is elucidated by numerical solutions of the “contour dynamica” equations. Lateral wavebreaking occurs when the initial disturbance amplitude exceeds a certain value, and at later times tongues of the 1ower vorticity fluid are engulfed or entrained into the higher vorticity shear flow. The effect appears to be important for the evolution of “shingles” observed between the coastal water and the cyclonic side of the Gulf Stream. The effect may also be in important phase in initiating the mixing process at the perimeter of an eddy embedded in another water mass.

Laboratory experiments on the decay (spin-up) of fluid motion on the β-plane are compared with theory. Under weakly dissipative conditions, some particles conserve potential vorticity during the entire decay. We also study the rectified mean flow which is produced by the lateral Reynolds Stress when a low frequency force is applied to the planetary fluid. The possible connection of effects with oceanic phenomena is briefly discussed.

We study analytically and numerically small amplitude perturbations of a geostrophically balanced semi-infinite layer of light water having a surface front and lying above a heavier layer of finite vertical thickness which is at rest in the mean. In contrast with previous studies where the latter layer was infinitely deep we find that the equilibrium is always unstable regardless of the distribution of potential vorticity, and the maximum growth rates are generally much larger than in the "one-layer" case. The amplifying ageostrophic wave transfers kinetic energy from the basic shear flow as well as potential energy. Good quantitative agreement is found with the laboratory experiments of Griffiths and Linden (1982), and our model seems to be the simplest one for future investigations of cross frontal mixing processes by finite amplitude waves. The propagation speed of very low frequency and nondispersive frontal waves is also computed and is shown to decrease with increasing bottom layer depth.

The evolution of large-amplitude disturbances at the outer edge of a quasi-geostrophic shear layer depends on the sign of the outward gradient of potential vorticity. Entrainment of ambient water can occur when the gradient of relative vorticity dominates in the potential vorticity, and detrainment from the current can occur when gradient of isopycnal thickness dominates. In the latter case long, thin filaments of finite area are “pinched off” into the surrounding water mass. This is verified using a quasi-geostrophic model having piecewise uniform potential vorticity. Contour dynamical calculations for many initial conditions allow us to define and tabulate an entrainment/detrainment velocity. This is used for an order of magnitude estimate of the flux of heat or salt on an isopycnal surface in a warm core ring.

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.

We compute the evolution of disturbances on a circularly symmetric eddy having uniform vorticity in a central core, in a surrounding annulus, and in the irrotational exterior water mass. This vortex is known to be (Kelvin-Helmholtz) unstable when its annular width is less than the core radius. Our calculations for the nonlinear regime show that amplification of azimuthal wavenumber n = 2 causes the vortex to split into two dipoles, in agreement with previous numerical calculations for a smoothed version of our vorticity field. This paper concentrates on the evolution of large-amplitude disturbances on the outer edge of a stable and robust eddy. We show that lateral wave breaking of vorticity isopleths causes intrusions of the (irrotational) exterior water mass into the central core of the vortex, a physical process which is relevant to lateral diffusion and isopycnal mixing in baroclinic ocean eddies. Similar intrusive features occur for an n = 1 disturbance, which also causes a “self-propagation...

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.

It is shown that a coastal density current in a rotating system is unstable to downstream wave disturbances when the mean potential vorticity increases towards the (vertically-walled) coast and when the mean current vanishes there. Other new instability modes are also found which do not require the potential vorticity extremum of quasi-geostrophic theory. All the instabilities in our equivalent one-layer model release mean kinetic energy and most of them release mean potential energy, but an increase of the latter can occur under certain circumstances. Specifically, it is shown (a) that mean flows close to uniform potential vorticity can be unstable to disturbances of infinite wavelength (and hence also for finite wavelengths) even for monotonic potential vorticity distributions, and (b) that all mean flows which vanish at the wall and for which the potential vorticity has a maximum but not an extremum at the wall are unstable to waves of finite length. A logical extension shows that the second h...

Two inequalities are proposed for the purpose of bounding from below the mean shear U′o(z) in turbulent channel flow. The first of these inequalities pertains to the energetics of the boundary layer, and the second pertains to the logarithmic form of the asymptote to Uo. These inequalities imply a maximum value of von Kármán's constant, the numerical value of which lies between the measurements of Laufer (1951) and the value obtained from bulk discharge measurements in a pipe. The formalism, which contains no adjustable parameters, is then applied to the turbulent thermal convection problem, and a lower bound for the mean temperature $\overline{T}_o(z)$ is obtained. The minimum value of the latter at large distances from the boundary is in fair agreement with Townsend's (1959) measurements. Although the proposed inequalities have not been deduced from the equations of motion, they provide facts which may be useful in the search for new variational formulations of theturbulent transport problem.