The decay of cyclonic eddies by Rossby wave radiation

It is argued that because shallow water cyclones on a β-plane drift westward at a speed equal to an available Rossby wave phase speed, they must radiate energy and cannot, therefore, be steady. The form of the Rossby wave wake accompanying a quasi-steady cyclone is calculated and the energy flux in the radiated waves determined. Further, an explicit expression for the radiation-induced northward drift of the cyclone is obtained. A general method for determining the effects of the radiation on the radius and amplitude of the vortex based on conservation of energy and potential vorticity is given. An example calculation for a cyclone with a ‘top-hat’ profile is presented, demonstrating that the primary effect of the radiation is to decrease the radius of the vortex. The dimensional timescale associated with the decay of oceanic vortices is of the order of several months to a year

The motion of an intense vortex near topography

The initial value problem for the motion of an intense, quasi-geostrophic, equivalent-barotropic, singular vortex near an infinitely long escarpment is studied in three parts. First, for times small compared to the topographic wave timescale the motion of the vortex is analysed by deriving an expression for the secondary circulation caused by the advection of fluid columns across the escarpment. The secondary circulation, in turn, advects the primary vortex and integral expressions are found for its velocity components. Analytical expressions in terms of integrals are found for the vortex drift velocity components. It is found that, initially, cyclones propagate away from the deep water region and anticyclones propagate away from the shallow water region. Asymptotic evaluation of the integrals shows that both cyclones and anticyclones eventually propagate parallel to the escarpment with shallow water on their right at a steady speed which decays exponentially with distance from the escarpment. Secondly, it is shown that for times comparable to, and larger than, the wave timescale, the vortex always resonates with the topographic wave field. The flux of energy in the topographic waves leads to a loss of energy in the vortex and global energy and momentum arguments are used to derive an equation for the distance (or, equivalently, the vortex velocity) of the vortex from the escarpment. It is shown that cyclones, provided they are initially within an O(1) distance (here a unit of distance is dimensionally equivalent to one Rossby radius of deformation) from the escarpment, drift further away from the deep water (i.e. toward higher ambient potential vorticity), possibly crossing the escarpment and accumulate at a distance of approximate to 1.2 on the shallow side of the escarpment. For distances larger than 1.2 there is essentially no drift of the vortex perpendicular to the escarpment. Anticyclones display similar behaviour except they drift in the opposite direction, i.e. away from the shallow water or toward lower ambient potential vorticity. Third, the method of contour dynamics is used to describe the evolution of the vortex and the interface representing the initial potential vorticity jump between the shallow and deep water regions. The contour dynamic results are in good quantitative agreement with the analytical results

Vortex equilibria in flow past a plate

Families of vortex equilibria, with constant vorticity, in steady flow past a flat plate are computed numerically. An equilibrium configuration, which can be thought of as a desingularized point vortex, involves a single symmetric vortex patch located wholly on one side of the plate. Given that the outermost edge of the vortex is unit distance from the plate, the equilibria depend on three parameters: the length of the plate, circulation about the plate, and the distance of the innermost edge of the vortex from the plate. Families in which there is zero circulation about the plate and for which the Kutta condition at the plate ends is satisfied are both considered. Properties such as vortex area, lift and free-stream speed are computed. Time-dependent numerical simulations are used to investigate the stability of the computed steady solutions

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

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

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

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 fundamental solutions of the curve shortening problem via the Schwarz function

Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy S_{t}S_{z} = S_{zz}, where S(z, t) is the Schwarz function of the curve. This equation is shown to have a parametric solution from which the known explicit solutions for curve shortening flow; the circle, grim reaper, paperclip and hairclip, can be recovered

Baroclinic geostrophic adjustment in a rotating circular basin

Baroclinic geostrophic adjustment in a rotating circular basin is investigated in a laboratory study. The adjustment process consists of a linear phase before advective and dissipative effects dominate the response for longer time. This work describes in detail the hydrodynamics and energetics of the linear phase of the adjustment process of a two-layer fluid from an initial step height discontinuity in the density interface DeltaH to a final response consisting of both geostrophic and fluctuating components. For a forcing lengthscale r(f) equal to the basin radius R-0, the geostrophic component takes the form of a basin-scale double gyre while the fluctuating component is composed of baroclinic Kelvin and Poincare waves. The Burger number S=R/r(f) (R is the baroclinic Rossby radius of deformation) and the dimensionless forcing amplitude epsilon = DeltaH/H-1 (H-1 is the upper-layer depth) characterize the response of the adjustment process. In particular, comparisons between analytical solutions and laboratory measurements indicate that for time tau: 1 < tau < S-1 (tau is time scaled by the inertial period 2pi/f), the basin-scale double gyre is established, followed by a period where the double gyre is sustained, given by S-1 < tau < 2epsilon(-1) for a moderate forcing and S-1 < tau < tau(D) for a weak forcing (tau(D) is the dimensionless dissipation timescale due to Ekman damping). The analytical solution is used to calculate the energetics of the baroclinic geostrophic adjustment. The results are found to compare well with previous studies with partitioning of energy between the geostrophic and fluctuating components exhibiting a strong dependence on S. Finally, the outcomes of this study are considered in terms of their application to lakes influenced by the rotation of the Earth

Fingering instability in wildfire fronts

A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number Pe as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that Pe increases in time, so all fire lines eventually become unstable. For periodic fire lines, Pe remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires