1,763 research outputs found
Gap-Leaping Vortical Currents
A one-parameter family of exact solutions describing the bifurcation of a steady two-dimensional current with uniform vorticity near a gap in a thin barrier is found. The unsteady evolution of source-driven flows toward these steady states is studied using a version of contour dynamics, appropriately modified to take into account the presence of a barrier with a single gap. It is shown that some of the steady solutions are realizable as large-time limits of the source-driven flows, although some are not owing to persistent eddy-shedding events in the vicinity of the gap. For the special case when there is zero net flux through the gap, numerical experiments show that the through-gap flux of vortical fluid increases with the width of the gap and that this flux approaches a steady limit with time
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 Kelvin's 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
Surf-zone vortices over stepped topography
The problem of vortical motions in the surf zone is simplified by taking the bottom topography to be piecewise flat while allowing finite-height jumps in depth between flat regions. The motion of an arbitrary number of singular vortices is cast into Hamiltonian form and the rule for relating Hamiltonians in conformally equivalent domains derived. Examples are given of a singular vortex pair colliding head-on with a step, of a vortex propagating along a curved coast to cross a step, and of a vortex being swept past a circular island straddling a step. Surf-zone vortices are then modelled as finite-area vortex patches and their motion followed by contour dynamics. It is shown that the paths of singular vortices can yield highly accurate explicit predictions of the paths of the centroids of vortex patches. Possible applications to surf-zone rip currents are noted
Vortex scattering by step topography
The scattering at a rectilinear step change in depth of a shallow-water vortex pair consisting of two patches of equal but opposite-signed vorticity is studied. Using the constants of motion, an explicit relationship is derived relating the angle of incidence to the refracted angle after crossing. A pair colliding with a step from deep water crosses the escarpment and subsequently propagates in shallow water refracted towards the normal to the escarpment. A pair colliding with a step from shallow water either crosses and propagates in deep water refracted away from the normal or, does not cross the step and is instead totally internally reflected by the escarpment. For large depth changes, numerical computations show that the coherence of the vortex pair is lost on encountering the escarpment
The longwave vorticity dynamics of rotating buoyant outflows
This paper discusses the evolution of coastal currents by considering, relative to a
rotating frame, the flow development when buoyant fluid is injected into a quiescent
fluid bounded by a solid wall. The initial rapid response is determined by the
Coriolis force–pressure gradient balance with a Kelvin wave propagating rapidly,
at the long-wave speed, with the bounding wall to its right (for positive rotation).
However fluid columns can stretch or squash on ejection from coastal outflows so
that the ejected fluid gains positive or negative relative vorticity. Depending on its
sign, the image in the solid wall of this vorticity can reinforce or oppose the zero
potential-vorticity-anomaly (PVa) current set up by the Kelvin wave (KW). This paper
presents a simple, fully nonlinear, dispersive, quasi-geostrophic model to discuss the
form of coastal outflows as the relative strength of vortex to KW driving is varied.
The model retains sufficient physics to capture both effects at finite amplitude and
thus the essential nonlinearity of the flow, but is sufficiently simple so as to allow
highly accurate numerical integration of the full problem and also explicit, fully
nonlinear solutions for the evolution of a uniform PVa outflow in the hydraulic limit.
Outflow evolutions are shown to depend strongly on the sign of the PVa of the
expelled fluid, which determines whether the vortex and KW driving are reinforcing
or opposing, and on the ratio of the internal Rossby radius to the vortex-source scale,
|V0/D
2Π0|
1/2
, of the flow (where D measures the outflow depth, Π0 the PVa of the
outflow and V0 the volume flux of the outflow), which measures the relative strengths
of the two drivers. Comparison of the explicit hydraulic solutions with the numerical
integrations shows that the analytical solutions predict the flow development well
with differences ascribable to dispersive Rossby waves on the current boundary and
changes in the source region captured by the full equations but not present in the
hydraulic solutions
Potential Vorticity Dynamics of Coastal Outflows
A simple quasigeostrophic model is used to examine the outflow from a river, estuary or strait into a coastal ocean. As shown by Johnson et al. (2016), these quasigeostrophic outflows are accurately described by analytical long wave solutions. This paper first uses these solutions and contour dynamics simulations to discuss the behaviour of coastal outflows. Second, it extends the model and the long wave theory to consider the effects of ambient currents, tides, winds or a variable source flux. Third, consideration of the momentum flux at the source is used to understand the turning of the current, showing that steady solutions conserve momentum, hence resolving the momentum imbalance paradox of Pichevin and Nof (1997). Finally, a new numerical scheme to compute steady outflow boundaries is developed. The model focuses on the key dynamics driven by the source velocity and the generation of vorticity as the buoyant fluid adjusts. The simplicity of the model, and insight given by the long wave solutions, enables a full understanding of the dynamics. The outflows display a range of behaviours including indefinitely growing near-source bulges, steady boundary profiles with varying offshore width, bidirectional currents and rarefying or eddy-like leading heads, all of which can be understood with the long wave theory. Despite the simplicity of the model, the results show good agreement in comparison with observations, experiments and numerical models
Finite Rossby radius effects on vortex motion near a gap
This work investigates the effect of the Rossby radius of deformation on the motion of a vortex near a gap in an infinitely long barrier. A key parameter determining the behaviour of the vortex is a, the ratio of the Rossby radius of deformation to the width of the gap. Assuming quasi-geostrophic dynamics for a single-layer, reduced-gravity fluid, an integral equation is derived whose solution gives the velocity at any point in the fluid. The integral equation is solved numerically and the velocity field is integrated to give the trajectories of point vortices. Combined with the method of contour dynamics, the method can be used to compute the evolution of finite area patches of constant vorticity. The trajectories of point vortices and vortex patches are compared. The patches are initially circular and the centroids of those vortex patches that remain close to circular follow the trajectory and speed of their equivalent point vortices when appropriately normalised. The critical point vortex trajectory (the separatrix) which divides vortices that leap across the gap and those that pass through, is computed for various a. Decreasing the Rossby radius of deformation increases the tendency of vortices to pass through the gap. The effect of various background flows on both point vortex and vortex patch motion is also described
Beach vortices near circular topography
Finite-area monopolar vortices which propagate around topography without change in shape are computed for circular seamounts and wells including the limiting cases of each: islands and infinitely-deep wells. The time-dependent behaviour of vortex pairs propagating toward circular topography is also examined. Trajectories of pointvortex pairs exterior to the topography are found and compared to trajectories of vortex patches computed using contour dynamics
A Simple Model for "Sheddies" – Ocean Eddies Formed from Shed Vorticity
Recent studies show that vertical eddy diffusivity is sufficient on its own to introduce intense horizontal shear layers at sloping ocean margins (Molemaker et al. 2015; Gula et al. 2015; Dewar et al. 2015). These layers influence mesoscale energy and potential vorticity budgets but cannot be fully represented in models without sloping boundaries, no slip boundary conditions and sufficiently high resolution. This paper investigates the detachment of these shear layers and their subsequent rolling up into concentrated eddies. These shed eddies, or “sheddies”, may have significant oceanographic impacts. Their growth is considered using a simple point vortex model that adapts the Brown–Michael model of vortex shedding to quasigeostrophic flow and allows detailed consideration of the vorticity fluxes. The model shows good qualitative agreement with observations and experimental and numerical results. It is applied to a number of examples of well known cases of sheddy formation, including the Agulhas Cyclones, California Undercurrent and Canary Eddy Corridor, and also to investigate the effects of shed vorticity in the growth of the Cook Strait Eddy and the interaction of the North Brazil Current Rings with the islands of the Lesser Antilles
A point vortex model for the formation of ocean eddies by flow separation
A simple model for the formation of ocean eddies by flow separation from sharply curved horizontal boundary topography is developed. This is based on the Brown–Michael model for two-dimensional vortex shedding, which is adapted to more realistically model mesoscale oceanic flow by including a deforming free surface. With a free surface, the streamfunction for the flow is not harmonic so the conformal mapping methods used in the standard Brown–Michael approach cannot be used and the problem must be solved numerically. A numerical scheme is developed based on a Chebyshev spectral method for the streamfunction partial differential equation and a second order implicit timestepping scheme for the vortex position ordinary differntial equations. This method is used to compute shed vortex trajectories for three background flows: (A) a steady flow around a semi-infinite plate, (B) a free vortex moving around a semi-infinite plate, and (C) a free vortex moving around a right-angled wedge. In (A), the inclusion of surface deformation dramatically slows the vortex and changes its trajectory from a straight path to a curved one. In (B) and (C), without the inclusion of flow separation, free vortices traverse fully around the tip along symmetrical trajectories. With the effects of flow separation included, very different trajectories are found: for all values of the model parameter—the Rossby radius—the free and shed vortices pair up and move off to infinity without passing around the tip. Their final propagation angle depends strongly and monotonically on the Rossby radius
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