Contour Dynamics Research Articles

Interaction of two unequal corotating vortices

Previous high-resolution contour dynamics calculations [Dritschel and Waugh, Phys. Fluids A 4, 1737 (1992)] have shown that in two-dimensional inviscid flow the interaction of two unequal corotating vortices with uniform vorticity is not always associated with vortex growth and may lead to vortices smaller than the original vortices. In the present study, we investigate whether these results also hold for two-dimensional vortices with continuous vorticity distributions. Similar flow regimes are found as for uniform vorticity patches, but the variation of the flow regimes with the initial vortex radii and peak vorticities is more complicated and strongly dependent on the initial shape of the vorticity profile. It is found that the “halo” of low-value vorticity, which surrounds the cores of continuous vortices, significantly increases the critical distance at which the weaker vortex is destroyed. The halo also promotes the vortex cores to merge more efficiently, since it accounts for a substantial part of the loss of circulation into filaments. Simple transformation rules and merger criteria are derived for the inviscid interaction between two Gaussian vortices. The strong dependence of the flow regimes on the initial vorticity distribution partly explains why previous laboratory experiments in an electron plasma [Mitchell and Driscoll, Phys. Fluids 8, 1828 (1996)] show complete merger of two unequal vortices in a range of parameter space where contour dynamics simulations with uniform vorticity patches predict partial merger or partial straining-out of the smaller vortex. It is shown that the measured times for complete merger are in reasonable agreement with inviscid dynamics when the vortices are very similar. For more distinct vortices the weaker vortex is often observed to be destroyed on a time scale much smaller than expected from inviscid numerical simulations. An explanation for this discrepancy is given by the combined effects of vortex stripping and viscous diffusion, which leads to an enhanced erosion of the weaker vortex. These results are verified by laboratory experiments in a conventional (rotating) fluid.

Watershed-based multiscale segmentation method for color images using automated scale selection

An automated multiscale segmentation approach for color images is presented. The scale-space stack is generated us- ing the Perona-Malik diffusion approach and the watershed algo- rithm is employed to produce the regions at each scale. A minima- linking process by downward projection is carried out over the successive scales, and a region dissimilarity measure—combining scale, contrast, and homogeneity—is subsequently estimated on the finer scale (localization scale). The dissimilarity measure is es- timated as a function of two different features, i.e., the dynamics of contours and the relative entropy of color region distributions, com- bined by means of a fuzzy-rule-based system. A region-merging process is also applied to the localization scale to produce the final regions. To validate the performance of the proposed multiscale segmentation, qualitative and quantitative results are provided in comparison to its single-scale counterpart. We also deal with the topic of localization scale selection. This stage is critical for the final segmentation results and can be used as a preprocessing step for higher level computer vision applications as well. A preliminary study of localization scale selection techniques is carried out. A scale se- lection method that originates from the evolution of the probability distribution of a region homogeneity measure across the generated scales is proposed next. The proposed algorithm is finally compared to a previously reported approach to indicate its efficiency. © 2005

Coastal currents and eddies and their interaction with topography

The nonlinear interaction of vorticity driven coastal currents and eddies with topography is studied. The topography is either a semi-infinite escarpment perpendicular to the coast (such that topographic waves propagate toward the coast) or a semi-circular canyon or seamount attached to the coast. Assuming a piecewise constant potential vorticity distribution, the quasigeostrophic equations are solved using contour dynamics. Offshore propagating dipole eddies occur, whenever a coastal current or eddy interacts with escarpment and canyon topographies. The size and frequency at which dipoles form are found to depend on the vorticity of the current and amplitude of the topography. However, for a seamount, little eddy shedding is observed and the coastal current or eddy skirts around topography.

Universality in the merging dynamics of parametric active contours: a study in MRI based lung segmentation

Measurement of lung ventilation is one of the most reliable techniques in diagnosing pulmonary diseases. The time-consuming and bias-prone traditional methods using hyperpolarized H3He and 1H magnetic resonance imageries have recently been improved by an automated technique based on ‘multiple active contour evolution’. This method involves a simultaneous evolution of multiple initial conditions, called ‘snakes’, eventually leading to their ‘merging’ and is entirely independent of the shapes and sizes of snakes or other parametric details. The objective of this paper is to show, through a theoretical analysis, that the functional dynamics of merging as depicted in the active contour method has a direct analogue in statistical physics and this explains its ‘universality’. We show that the multiple active contour method has an universal scaling behaviour akin to that of classical nucleation in two spatial dimensions. We prove our point by comparing the numerically evaluated exponents with an equivalent thermodynamic model.

Dynamics of plasma vortices: The role of the electron skin depth

The drift-Alfvén equations for a quasineutral plasma that is permeated by a strong background magnetic field describe the plasma dynamics in the transverse plane. Three scalar fields—which are linear combinations of parallel current, fluid vorticity, and density—are incompressibly advected by three streaming potentials. These three “generalized vorticities” are point-wise conserved. Their dynamics is studied numerically using the method of Contour Dynamics, which for this purpose is generalized from one to three different types of vorticity. In the regime where the electron inertial skin depth de is smaller than the ion sound gyroradius, the interaction dynamics of patches of generalized vorticity shows a preference for the formation of structures with length scale O(de). In this regime quasiperiodic splitting-merging cycles of single vorticity patches occur. These vortex interactions are considered to be the paradigm for the interactions of separated vortical structures in a strongly magnetized turbulent plasma.

The temporal evolution of a geostrophic flow in a rotating stratified basin

A laboratory study in a rotating stratified basin examines the instability and long time evolution of the geostrophic double gyre introduced by the baroclinic adjustment to an initial basin-scale step height discontinuity Δ H in the density interface of a two-layer fluid. The dimensionless parameters that are important in determining the observed response are the Burger number S = R / R 0 (where R is the baroclinic Rossby radius of deformation and R 0 is the basin radius) and the initial forcing amplitude ɛ = Δ H / H 1 ( H 1 is the upper layer depth). Experimental observations and a numerical approach, using contour dynamics, are used to identify the mechanisms that result in the dominance of nonlinear behaviour in the long time evolution, τ > 2 ɛ − 1 (where τ is time scaled by the inertial period T I = 2 π / f ). When the influence of rotation is moderate ( 0.25 ≤ S ≤ 1 ), the instability mechanism is associated with the finite amplitude potential vorticity (PV) perturbation introduced when the double gyre is established. On the other hand, when the influence of rotation is strong ( S ≤ 0.1 ), baroclinic instability contributes to the nonlinear behaviour. Regardless of the mechanism, nonlinearity acts to transfer energy from the geostrophic double gyre to smaller scales associated with an eddy field. In the lower layer, Ekman damping is pronounced, resulting in the dissipation of the eddy field after only 40 T I . In the upper layer, where dissipative effects are weak, the eddy field evolves until it reaches a symmetric distribution of potential vorticity within the domain consisting of cyclonic and anticyclonic eddy pairs, after approximately 100 T I . The functional dependence of the characteristic eddy lengthscale L E on S is consistent with previous laboratory studies on continuously forced geostrophic turbulence. The cyclonic and anticyclonic eddy pairs are maintained until viscous effects eventually dissipate all motion in the upper layer after approximately 800 T I . The outcomes of this study are considered in terms of their contribution to the understanding of the energy pathways and transport processes associated with basin-scale motions in large stratified lakes.

Hamiltonian and action principle formulations of plasma physics

Hamiltonian and action principle formulations of the basic equations of plasma physics are reviewed. Various types of Lagrangian and Poisson bracket formulations for kinetic and fluid theories are discussed, and it is described how such formulations can be used to derive and approximate physical models. Additional uses are also described. Two applications are treated in greater detail: an algorithm based on Dirac brackets for the calculation of V-states of contour dynamics and the calculation of fluctuation spectra of Vlasov theory and shear flow dynamics by methods of statistical mechanics.

Evidence of singularities for a family of contour dynamics equations

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < alpha </= 1. The limiting case alpha --> 0 corresponds to 2D Euler equations, and alpha = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

Exact solutions of integrable 2D contour dynamics

A class of exact solutions of the dispersionless Toda hierarchy constrained by a string equation is obtained. These solutions represent deformations of analytic curves with a finite number of nonzero harmonic moments. The corresponding τ-functions are determined and the emergence of cusps is studied.

A parallel hierarchical-element method for contour dynamics simulations

Many two-dimensional incompressible inviscid vortex flows can be simulated very efficiently by means of the contour dynamics method. Several applications require the use of a hierarchical-element method (HEM), which is a modified version of the classical contour dynamics scheme based on the fast multipole method. The HEM can be used, for example, to study the large-scale motion of coherent structures in idealized geophysical fluid dynamics where the flow can be modelled as the motion in a thin layer of fluid in the presence of a non-uniform background rotation. Nevertheless, such simulations require a substantial computational effort, even when the HEM is used. In this article it is shown that the acceleration of contour dynamics simulations can be increased further by parallelizing the HEM algorithm. Speed-up, load balance and scalability are parallel performance features which are studied for several representative cases. The HEM has been parallelized using the OpenMP interface and is tested with up to 16 processors on an Origin 3800 CC-NUMA computer.

The Motion of a Point Vortex near Large-Amplitude Topography in a Two-Layer Fluid

Abstract This work examines the dynamics of point vortices in a two-layer fluid near large-amplitude, sharply varying topography like that which occurs in continental shelf regions. Topography takes the form of an infinitely long step change in depth, and the two-layer stratification is chosen such that the height of topography in the upper layer is a small fraction of the overall depth, enabling quasigeostrophic theory to be used in both layers. An analytic expression for the dispersion relation of free topographic waves in this system is found. Weak vortices are studied using linear theory and, if located in the lower layer, propagate mainly because of their image in the topography. Depending on their sign, they are able to produce significant topographic wave radiation in their wakes. Upper-layer vortices propagate much slower and produce relatively small amplitude topographic wave radiation. Contour dynamics results are used to investigate the nonlinear regions of parameter space. For lower-layer vortices, linear theory is a good approximation, but for upper-layer vortices complicated features evolve and linear theory is only valid for weak vortices.

Contour dynamics method for solving the Grad–Shafranov equation with applications to high beta equilibria

Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of a plasma confined by an axisymmetric magnetic field. Nevertheless, they are limited to low beta or small plasma pressure. Combining a nonconservative variational principle with a contour dynamics approach, the approach presented in this paper converges for extreme high beta configurations. By reducing the dimension of the problem from two to one, a compact and efficient numerical algorithm can be developed, and a wide range of boundary shapes can be utilized. Furthermore, the iterative nature of this technique greatly facilitates convergence at high beta while minimizing computation times.

Estudo teórico da dinâmica da confluência Brasil-Malvinas

The study of the mesoscale dynamics of the confluence of two western boundary currents was conducted in this work via a quasi-geostrophic inviscid f-plane two-layer ocean theoretical approach. The simplified vertical structure was calculated through the employment of a dynamic calibration scheme based on the dynamical mode structure of the Brazil-Malvinas Confluence Region. The Brazil-Malvinas mode structure was computed from climatological hydrography of the area delimited by 35o-40oS and 50o-60oW. Two contour dynamics models were constructed: a linear and a nonlinear version. The basic flow configuration consisted of two converging western boundary currents that form a zonal eastward current in the upper layer. The lower layer flow was essentially divergent as a result of a westward zonal jet impinging on the western border. This vertical shear choice assured that the system was baroclinically unstable. The results of the model experiments showed that the presence of the confluence, the western meridional boundary and the barotropic mode in the dynamical structure of the basic flow favored long wave patterns. The three experiments conducted with the nonlinear model exhibited the development of both a reflection pattern and vortical dipoles. The dipoles pinched off from either the retroflection lobe (i.e., the primary crest of the wave train) or the primary trough when the baroclinically unstable current system was perturbed at the boundary vicinities. It was verified that the nonlinear model simulations followed the instability properties predicted by the linear model in terms of meander growth rates, phase speeds and most unstable wavelengths. This suggested that while the baroclinic instability mechanism was responsible for the temporal growth of the meanders, the nonlinear effects caused the dipole isolation and pinch off of the finite amplitude meanders. These dipoles could leave and propagate away from the current axis.

Coastal currents generated by outflow and vorticity and their interaction with topography

The offshore spreading of coastal river outflows over shelf-like topography is inhibited by the demands of potential vorticity conservation. As a result, outflows tend to remain coastally trapped and flow parallel to the coast. Alongshore currents may also form when the outflow fluid has vorticity relative to the surrounding fluid, in which case the ‘image’ vorticity drives the outflow parallel to the coast. In the present work, the combined effects of topography and anomalous vorticity of the outflow in forming coastal currents are studied. Shelf-like topography is considered in the form of a sudden step change in depth, running parallel to a straight coast, some distance offshore. A simple model incorporating these effects is formulated using quasigeostrophic dynamics for a barotropic fluid with a rigid lid. The role of topographic wave radiation in establishing coastal currents is studied using the linearised barotropic potential vorticity equation for the case of a weak outflow of fluid with zero anomalous vorticity. The outflow is shown to be confined to the shelf and flows to the right of the source region. For stronger outflows, with relative vorticity, the method of contour dynamics is used to study the nonlinear evolution of the outflow plume and, in particular, its interaction with topography. The effects of outflow strength, magnitude and sign of the relative vorticity of the outflow fluid are considered in detail. In particular, the offshore penetration distance and rate of alongshore spreading of the outflow plume are studied. When the outflow comprises of fluid with positive (cyclonic) vorticity, the topographic and vorticity-driven image effects reinforce each other and increase the spreading rate of the outflow plume to the right of the source. For negative (anticyclonic) vorticity outflows the effects oppose each other leading to enhanced offshore spreading of the outflow. The numerical results demonstrate that the offshore transport is achieved by an efficient dipolar vortex mechanism.

Detecting contour crossings in contour dynamical and contour-advective semi-Lagrangian simulations

Contour dynamics and contour-advective methods are commonly used numerical techniques for simulating inviscid fluid motions. In these methods the vorticity or potential vorticity of a flow is represented by a series of contours which are advected according to the prevailing velocity field. In some circumstances the contours may cross, eroding the accuracy of the numerical solution and violating the equations of motion. This paper describes an automated method for explicitly revealing such crossings, first considering the case of determining if two contours cross and then later the more general case of determining if and where an arbitrary number of contours cross.

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.

Growing vortex patches

This paper demonstrates that two well-known equilibrium solutions of the Euler equations—the corotating point vortex pair and the Rankine vortex—are connected by a continuous branch of exact solutions. The central idea is to “grow” new vortex patches at two stagnation points that exist in the frame of reference of the corotating point vortex pair. This is done by generalizing a mathematical technique for constructing vortex equilibria first presented by Crowdy [D. G. Crowdy, “A class of exact multipolar vortices,” Phys. Fluids 11, 2556 (1999)]. The solutions exhibit several interesting features, including the merging of two separate vortex patches via the development of touching cusps. Numerical contour dynamics methods are used to verify the mathematical solutions and reveal them to be robust structures. The general issue of how simple vortex equilibria can be continued continuously to more complicated ones with very different vortical topologies is discussed. The solutions are examples of exact solutions of the Euler equations involving multiple interacting vortex patches.

Benjamin–Feir instability of Rossby waves on a jet

AbstractLarge‐scale waves on the extratropical tropopause have been widely observed to spontaneously organize into groups or wave packets. Here, a simple paradigm for this wave packet formation is presented. Firstly, a weakly nonlinear theory of Rossby wave propagation on a potential‐vorticity front, based on small non‐dimensional wave amplitude ε, is developed. As is typical for systems allowing conservative one‐dimensional wave propagation, the evolution of the wave envelope is governed by the nonlinear Schrödinger equation. The sense of the nonlinearity is consistent with Benjamin–Feir instability, where uniform wave trains are unstable to sideband modulations, leading to the formation of wave packets. Next, numerical results from contour dynamics integrations show that the weakly nonlinear predictions for sideband growth rates are quantitatively accurate up to ε∼0.5, and that unstable sideband growth is qualitatively similar, but slower than predicted, at higher values of ε. For ε≥0.6 the formation of wave packets leads to wave‐breaking, this occuring at much lower initial wave amplitudes than for unmodulated uniform wave trains previously studied. The numerical results reveal that the length and time‐scales of the Benjamin–Feir instability are broadly consistent with observed wave packet formation in the extratropics. Copyright © 2004 Royal Meteorological Society.

Enhancement of Rossby Wave Breaking by Steep Potential Vorticity Gradients in the Winter Stratosphere

This work investigates the extent to which potential vorticity gradients affect the vertical propagation of planetary-scale Rossby waves on the edge of a stratospheric polar vortex and their eventual nonlinear saturation and breaking. Using two different numerical modeling approaches, it is shown that wave propagation and wave breaking are significantly reduced when the potential vorticity gradients at the vortex edge are anomalously weak. The efficiency of the first model, based on high-resolution contour dynamics, permits a full exploration of the parameter space of wave forcing amplitude and edge steepness. A more realistic primitive equation model in spherical geometry both confirms the contour dynamics results and highlights some direct implications for stratospheric modeling in more comprehensive models. The results suggest that stratospheric models using horizontal resolutions of spectral T42 or less may significantly underestimate the vertical propagation and breaking of planetary waves, and may consequently misrepresent such important stratospheric processes as the mean meridional circulation, sudden warmings, and the mixing of chemically distinct polar and midlatitude air.

Local Approximation for Contour Dynamics in Effectively Two-Dimensional Ideal Electron-Magnetohydrodynamic Flows

The evolution of piecewise constant distributions of a conserved quantity related to the frozen-in canonical vorticity in effectively two-dimensional incompressible ideal EMHD flows is analytically investigated by the Hamiltonian method. The study includes the case of axisymmetric flows with zero azimuthal velocity component and also the case of flows with the helical symmetry of vortex lines. For sufficiently large size of such a patch of the conserved quantity, a local approximation in the dynamics of the patch boundary is suggested, based on the possibility to represent the total energy as the sum of area and boundary terms. Only the boundary energy produces deformation of the shape with time. Stationary moving configurations are described.