E. R. Johnson

The motion of a vortex near two circular cylinders

The motion of a vortex near two circular cylinders of arbitrary radii—a problem of geophysical significance—is studied. The fluid motion is governed by the two-dimensional Euler equations and the flow is irrotational exterior to the vortex. Two models are considered. First, the trajectories of a line vortex are obtained using conformal mapping techniques to construct the vortex Hamiltonian which respects the zero normal flow boundary condition on both cylinders. The vortex paths reveal a critical trajectory (i.e. separatrix) that divides trajectories into those that orbit both cylinders and those that orbit just one cylinder. Second, the motion of a patch of constant vorticity is computed using a combination of conformal mapping and the numerical method of contour surgery. Although the patch can deform, the results show that when the islands have comparable radii the patch remains remarkably coherent. Moreover, it is demonstrated that the trajectory of the centroid of the patch is well modelled by a line vortex. For the limiting case when one of the cylinders has infinite curvature (i.e. it becomes a straight line or wall) it is shown that the vortex patch, which propagates under the influence of its image in the wall, may undergo severe deformation as it collides with the smaller cylinder, with portions of the vortex passing around different sides of the cylinder.

Stratified taylor columns on a beta-plane

Abstract The form of quasigeostrophic flow past isolated topography on a beta-plane is obtained for both homogeneous and strongly stratified flows. The stratified flow patterns directly above the bottom boundary are qualitatively similar to the corresponding homogeneous patterns but do not exhibit as marked a deviation from uniform flow away from this boundary. Strong stratification limits the vertical extent of perturbations to western flows. causing them to decay within a distance of order fL/N of the bottom where f is the Coriolis parameter, L the horizontal length scale and N the buoyancy frequency. Perturbations to eastward flows are virtually unattenuated by this height due to the presence of a standing Rossby-wave wake behind the obstacle propagating energy vertically upwards. The minimum height required for an axisymmetric obstacle to cause blocking in a westward current is obtained as a function of the Burger number and the beta parameter. Expressions are given for the force exerted on axisymmetr...

NONLINEAR ROSSBY ADJUSTMENT IN A CHANNEL - BEYOND KELVIN WAVES

Nonlinear advective adjustment of a discontinuity in free-surface height under gravity and rotation is considered, using the method of contour dynamics. After linear wave-adjustment has set up an interior jet and boundary currents in a wide ([dbl greater-than sign] one Rossby radius) channel, fluid surges down-channel on both walls, rather than only that wall supporting a down-channel Kelvin wave. A wedgelike intrusion of low potential vorticity fluid on this wall, and a noselike intrusion of such fluid on the opposite wall, serve to reverse the sign of relative vorticity in the pre-existing currents. For narrower channels, a coherent boundary-trapped structure of low potential vorticity fluid is ejected at one wall, and shoots ahead of its parent fluid. The initial tendency for the current to concentrate on the ‘right-hand’ wall (the one supporting a down-channel Kelvin wave in the northern hemisphere) is defeated as vorticity advection shifts the maximum to the left-hand side. Ultimately fluid washes downstream everywhere across even wide channels, leaving the linearly adjusted upstream condition as the final state. The time necessary for this to occur grows exponentially with channel width. The width of small-amplitude boundary currents in linear theory is equal to Rossbys deformation radius, yet here we find that the width of the variation in velocity and potential vorticity fields deviates from this scale across a large region of space and time. Comparisons of the contour dynamics solutions, valid for small amplitude, and integration of the shallow-water equations at large amplitude, show great similarity. Boundary friction strongly modifies these results, producing fields more closely resembling the linear wave-adjusted state. Observed features include those suggestive of coastally trapped gravity currents. Analytical results for the evolution of vorticity fronts near boundaries are given in support of the numerical experiments.

Vortices near barriers with multiple gaps

Two models are presented for the motion of vortices near gaps in infinitely long barriers. The first model considers a line vortex for which the exact nonlinear trajectories satisfying the governing two-dimensional Euler equations are obtained analytically. The second model considers a finite-area patch of constant vorticity and is based on conformal mapping and the numerical method of contour surgery. The two models enable a comparison of the trajectories of line vortices and vortex patches. The case of a double gap formed by an island lying between two headlands is considered in detail. It is noted that Kelvins theorem constrains the circulation around the island to be a constant and thus forces a time-dependent volume flux between the islands and the headlands. When the gap between the island and a headland is small this flux requires arbitrarily large flow speeds through the gap. In most examples the centroid of the patch is constrained to follow closely the trajectory of a line vortex of the same circulation. Exceptions occur when the through-gap flow forces the vortex patch close to an edge of the island where it splits into two with only part of the vortex passing through the gap. In general the part squeezing through a narrow gap returns to near-circular to have a diameter significantly larger than the gap width.

Trapped vortices in rotating flow

A variational principle is presented which characterizes steady motions, at finite Rossby number, of rotating inviscid homogeneous fluids in which horizontal velocities are independent of depth. This is used to construct nonlinear solutions corresponding to stationary patches of distributed vorticity above topography of finite height in a uniform stream. Numerical results are presented for the specific case of a right circular cylinder and are interpreted using a series expansion, derived by analogy with a deformable self-gravitating body. The results show that below a critical free-stream velocity a trapped circular vortex is present above the cylinder and a smaller patch of more concentrated vorticity, of the opposite sign, maintains a position to the right (looking downstream) of the cylinder. An extension to finite Rossby number and finite obstacle height of Hupperts (1975) criterion for the formation of a Taylor column is presented in an appendix.

The motion of a vortex near a gap in a wall

Two models are presented for the motion of a vortex near a gap in an infinitely long straight barrier: a problem of geophysical significance. The first model considers a line vortex for which the trajectories are obtained analytically and are generalized to include simple ambient flows. The criterion determining whether in the absence of background flow a vortex originating far from the gap passes through or leaps across the gap is derived. The second model considers a finite area patch of constant vorticity and is based on conformal mapping and the numerical method of contour surgery. The two models enable a comparison of the trajectories of line vortices and vortex patches. In most examples the centroid of the patch is constrained to follow closely the trajectory of a line vortex of the same circulation. An exception occurs when flow through the gap forces the vortex patch close to one of the edges of the barrier where it splits into two with only one part of the vortex passing through the gap.

Baroclinic and Barotropic Instabilities of Coastal Currents

The two-layer baroclinic instability model of the California Undercurrent from Mysak (1977) is modified to investigate the effects of the lateral boundary conditions on the stability properties of the system. As is common in baroclinic instability calculations, Mysak (1977) assumes the mean flow along the continental rise to be bounded laterally by vertical rigid walls, thus allowing the cross-stream structure of the perturbation flow to be decomposed into simple normal modes. Instability then occurs when waves of the same cross-stream structure interact. The dominant instability is that associated with the gravest mode. In the first model presented here we consider the effect of replacing the rigid outer boundary with a quiescent, constant-depth ocean. Waves of short longshore wavelength are not greatly affected by the open seaward boundary. However, as consideration is turned to waves of longer longshore wavelength, the cross-stream wavenumber departs further from the integral values of the channel-flow problem and another class of baroclinic instabilities occurs due to interaction between waves of differing cross-stream structure. Nevertheless, the dominant baroclinic instability remains that associated with the gravest mode. A new barotropic instability is also present, drawing energy from the horizontal shear between the coastal current and the quiescent ocean. In the second model the rigid outer boundary is retained but the inner boundary is replaced by a shallow sloping region, modeling the effects of a sloping shelf adjoining the coastal current which flows along the continental rise. Topographic waves are present above the sloping inshore region. These waves are coupled with the channel waves. Once again the cross-stream wavenumber departs from the integral values of the channel problem and instabilities are present due to interaction between waves of differing cross-stream structure. As in the previous model the dominant baroclinic instability is that of the gravest mode and a new barotropic instability is present due to the lateral shear in the mean flow at the shelf break. For both models, a parameter study is presented in which we determine the effects of varying the shear, stratification and bottom slope.

Millennium Global Village-Net: bringing together Millennium Villages throughout sub-Saharan Africa.

The Millennium Villages Project (MVP), based at The Earth Institute at Columbia University, is a bottom-up, community led approach to show how villages in developing countries can get out of the poverty trap that afflicts more than a billion people worldwide. With well-targeted, practical inputs can help the community invest in a path leading to self-sustaining development. There are 80 Millennium Villages clustered in 10 countries throughout sub-Saharan Africa. MVP is an important development process for empowering communities to invest in a package of integrated interventions aiming to increase food production, improve access to safe water, health care, education and infrastructure. The process benefits from synergies of the integrated approach and relies on community leadership as empowered by proven technological inputs. MVP is committed to a science-based approach to assess and monitor the progress of the communities towards clear objectives; the Millennium Development Goals (MDGs) and to do so with mechanisms that are scalable and sustainable. This approach offers much more than simply collecting and analyzing data since the mechanism used for recording progress would provide a bridge over the divide which separates the haves and the have-nots (by facilitating the sharing of solutions from one community to another bidirectionally). By so doing, it allows people to enhance their own futures in a sustainable manner. Solutions found in one community are transferable to similar communities in other MVP villages. To achieve this goal, the MVP requires an information and communication system which can provide both necessary infrastructure for monitoring and evaluation, and tools for communicating among the villages, cities and countries. This system is called the Millennium Global Village-Net (MGV-Net). It takes advantage of the latest in open source software (OpenMRS), databases (MySQL), interface terminology, a centralized concept dictionary, and uses appropriate technology locally for data entry.

TOPOGRAPHIC WAVES AND THE EVOLUTION OF COASTAL CURRENTS

An initial-value problem is considered for the oceanographically relevant case of slow flow over obstacles of small height and horizontal scale of order the fluid depth or larger. Previous work on starting flow over obstacles whose contours are closed (Johnson 1984) is extended to flow forced by a source–sink pair to cross a step change in depth bounded by a vertical sidewall. Bottom contours thus end abruptly and the near-periodic solutions of the earlier work are no longer possible. The relevant timescale for the motion is again the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the step height. This time is taken to be long compared with the inertial period but short compared with the advection time. It is shown that if shallow water lies to the right (looking away from the wall) a wavefront moves outwards exponentially fast leaving behind a flow equivalent to that obtained by replacing the step by a rigid wall. If shallow water lies to the left the wavefront approaches the wall, forming at the wall–step junction an unsteady, exponentially thinning, singular region that transports the whole flux. The relevance of these solutions to experiments and steady solutions for free-surface and two-layer flows in Davey, Gill, Johnson & Linden (1984, 1985) is discussed.

Experimental study of the effect of rotation on nonlinear internal waves

Nonlinear internal waves are commonly observed in the coastal ocean. In the weakly nonlinear long wave regime, they are often modeled by the Korteweg-de Vries equation, which predicts that the long-time outcome of generic localised initial conditions is a train of internal solitary waves. However, when the effect of background rotation is taken into account, it is known from several theoretical and numerical studies that the formation of solitary waves is inhibited, and instead nonlinear wave packets form. In this paper, we report the results from a laboratory experiment at the LEGI-Coriolis Laboratory which describes this process.