Contour Dynamics Model Research Articles

Three-component contour dynamics model to simulate and analyze amoeboid cell motility in two dimensions.

Amoeboid cell motility is relevant in a wide variety of biomedical processes such as wound healing, cancer metastasis, and embryonic morphogenesis. It is characterized by pronounced changes of the cell shape associated with expansions and retractions of the cell membrane, which result in a crawling kind of locomotion. Despite existing computational models of amoeboid motion, the inference of expansion and retraction components of individual cells, the corresponding classification of cells, and the a priori specification of the parameter regime to achieve a specific motility behavior remain challenging open problems. We propose a novel model of the spatio-temporal evolution of two-dimensional cell contours comprising three biophysiologically motivated components: a stochastic term accounting for membrane protrusions and two deterministic terms accounting for membrane retractions by regularizing the shape and area of the contour. Mathematically, these correspond to the intensity of a self-exciting Poisson point process, the area-preserving curve-shortening flow, and an area adjustment flow. The model is used to generate contour data for a variety of qualitatively different, e.g., polarized and non-polarized, cell tracks that visually resemble experimental data very closely. In application to experimental cell tracks, we inferred the protrusion component and examined its correlation to common biomarkers: the F-actin density close to the membrane and its local motion. Due to the low model complexity, parameter estimation is fast, straightforward, and offers a simple way to classify contour dynamics based on two locomotion types: the amoeboid and a so-called fan-shaped type. For both types, we use cell tracks segmented from fluorescence imaging data of the model organism Dictyostelium discoideum. An implementation of the model is provided within the open-source software package AmoePy, a Python-based toolbox for analyzing and simulating amoeboid cell motility.

Stratospheric vacillations in a minimal contour dynamics model

AbstractA low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow water flow on a polar f-plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave/mean flow interaction. The approach can be generalized to other geometries.The resulting three-component system depends on four non-dimensional parameters, and the structure of the steady state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton-Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton-Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations.

Nested contour dynamics models for axisymmetric vortex rings and vortex wakes

AbstractInviscid models for vortex rings and dipoles are constructed using nested patches of vorticity. These models constitute more realistic approximations to experimental vortex rings and dipoles than the single-contour models of Norbury and Pierrehumbert, and nested contour dynamics algorithms allow their simulation with low computational cost. In two dimensions, nested-contour models for the analytical Lamb dipole are constructed. In the axisymmetric case, a family of models for vortex rings generated by a piston–cylinder apparatus at different stroke ratios is constructed from experimental data. The perturbation response of this family is considered by the introduction of a small region of vorticity at the rear of the vortex, which mimics the addition of circulation to a growing vortex ring by a feeding shear layer. Model vortex rings are found to either accept the additional circulation or shed vorticity into a tail, depending on the perturbation size. A change in the behaviour of the model vortex rings is identified at a stroke ratio of three, when it is found that the maximum relative perturbation size vortex rings can accept becomes approximately constant. We hypothesise that this change in response is related to pinch-off, and that pinch-off might be understood and predicted based on the perturbation responses of model vortex rings. In particular, we suggest that a perturbation response-based framework can be useful in understanding vortex formation in biological flows.

Minimization of instabilities in growing interfaces: A variational approach

The Mullins-Sekerka and the electric breakdown instabilities are well known to lead to the spontaneous formation of a variety of complex spatial structures, among them dendritic crystal shapes, and treelike electric discharge patterns. Controlling such systems by suppressing predominantly excited solutions offers the opportunity to manipulate and stabilize these patterns in a defined way for a wide range of technological applications. In this work, we employ a variational approach which enables one to systematically search for the ideal conditions under which the patterns grow, but where interfacial deformations are efficiently minimized. The effectiveness of our variational control method is illustrated via linear stability calculations on both two-dimensional and three-dimensional contour-dynamics models for crystal growth and electric discharge phenomena.

Jupiter’s Great Red Spot: Fine-scale matches of model vorticity patterns to prevailing cloud patterns

We report on a set of six new matches between fine-scale features in the vorticity field of a three-dimensional (3D), primitive-equation, finite-difference model of Jupiter’s Great Red Spot that includes no clouds or cloud physics, and quasi-permanent structures in reflected visible-band images of the clouds. These add to similar success by Cho et al. (Cho, J., de la Torre Juárez, M., Ingersoll, A.P., Dritschel, D.G. [2001]. J. Geophys. Res. 106, 5099–5106), who earlier captured four characteristic features of the GRS, also reproduced here, using a 3D quasi-geostrophic, cloud-free contour-dynamics model. In that study and this, the key enabling model attribute is sufficient horizontal resolution, rather than the moist-convective and cloud-microphysics processes often required to match the patterns of clouds in terrestrial hurricanes. The only significant feature that these dry models do not capture is the episodic moist-convective plumes seen in the northwest quadrant adjacent to the GRS. We initialize with Jupiter’s averaged zonal winds plus an approximately balanced, smooth 3D ellipsoidal anticyclone. The threshold horizontal grid-resolution to obtain the fine-scale matches is approximately Δy/Ld≲0.15, where Δy≲300km is the meridional grid spacing and Ld∼2000km the Rossby deformation length. For models with this or finer horizontal resolution, the best correspondence with observations is reached after about six vortex turnaround times from initialization (∼30 Earth days), but good facsimiles of nearly all the studied features appear after only 1.5 turnaround times (∼7–8days). We conclude that in images of Jupiter, it is not accurate to associate clouds with upward motion, since these dry models reproduce the observed cloud patterns without this association, and indeed the synoptic-scale vertical motions in the model, as well as those deduced from observations, do not at all correspond to the observed cloud patterns. Instead, Jupiter’s cloud-top patterns indicate the effects of local shear in the manner of passive-tracer fields. As a corollary, the water clouds on Jupiter, which lie unseen below its visible clouds, are the only ones on the planet likely to correlate with upwelling in the manner that clouds do on Earth. The next step is to extend studies such as this past the reflected visible band, for example to include the GRS’s 5-μm emission bright collar, which may require the inclusion of cloud physics to enable the successful simulation of large voids.

Stirring Vortices with Vorticity Holes

A vorticity hole is a region with, in absolute value, significantly lower vorticity than its surroundings. Here we discuss the dynamics of a Rankine vortex with two equal circular holes. Since a symmetric initial condition is assumed, the evolution depends on three parameters only: the vorticity drop, the hole radius and the distance between the holes. We computed the evolution with a contour-dynamics model and analysed the stirring of fluid particles using the Lagrangian flow geometry, i.e., the set of hyperbolic trajectories and associated manifolds of the time-dependent velocity field. The vorticity holes evolve similarly to a pair of vortices in an otherwise quiescent fluid; their interaction with the boundary of the Rankine vortex being relevant only when they are close to it. We found that the strongest stirring occurs when the holes interact elastically and that it always takes place in the centre of the Rankine vortex. This result contradicts the generally accepted notion that vortices are regions of null to weak stirring.

Two-layer quasi-geostrophic model of contour dynamics for a round basin

A quasi-geostrophic contour dynamics model permitting one to study flows induced by a system of vortex patches in a two-layer ocean with round shore boundaries in the presence of specified background flows caused by the bottom relief, the β effect, and sources and sinks at the boundaries is proposed. The principal relations of the model are presented and the algorithm of its numerical realization is described. Some experimental results of the study of the evolution of unstable two-layer vortices are demonstrated.

Electric-discharge contour-dynamics model: The effects of curvature and finite conductivity

In this paper we present the complete derivation of the effective contour model for electrical discharges which appears as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, when the electron diffusion is small. It consists of two integro-differential equations defined at the boundary of the plasma region: one for the motion and a second equation for the net charge density at the interface. We have computed explicit solutions with cylindrical symmetry and found the dispersion relation for small symmetry-breaking perturbations in the case of finite resistivity. We implement a numerical procedure to solve our model in general situations. As a result we compute the dispersion relation for the cylindrical case and compare it with the analytical predictions. Comparisons with experimental data for a 2D positive streamers discharge are provided and predictions confirmed.

Instability of Shielded Surface Temperature Vortices

Abstract The stability characteristics of the surface quasigeostrophic shielded Rankine vortex are found using a linearized contour dynamics model. Both the normal modes and nonmodal evolution of the system are analyzed and the results are compared with two previous studies. One is a numerical study of the instability of smooth surface quasigeostrophic vortices with which qualitative similarities are found and the other is a corresponding study for the two-dimensional Euler system with which several notable differences are highlighted.

Interaction of an Asymmetric Double Vortex and Trochoidal Motion of a Tropical Cyclone with the Concentric Eyewall Structure

Abstract Radar echo images demonstrate that mature tropical cyclones frequently have a concentric eyewall structure, which consists of the inner eyewall, echo-free moat, and outer eyewall regions. Near the inner and outer eyewalls, well-defined wind maxima are generally observed. This indicates that two large vertical vorticity regions exist just inside radii of the two wind maxima near the inner and outer eyewalls. Therefore, the concentric eyewall structure can be considered to be a double vortex composed of the inner vortex and the outer vortex ring. In this study, the contour dynamics model is used on the f plane to analyze the characteristics of flows with either a symmetric double vortex or an asymmetric one, and examined the relationship between the movement of the inner vortex in an asymmetric double vortex and a trochoidal motion of a tropical cyclone with an asymmetric concentric eyewall structure. Results show that, depending on the degree of an interaction of a double vortex, the evolution of the inner vortex is classified into three patterns: the first is that the center of the inner vortex is stationary, which is seen only for the symmetric double vortex; the second is that the track of the center of the inner vortex draws a circle; and the third is that it draws a spiral. A numerical experiment based on an observed flow around Typhoon Herb was also performed. The time evolution of the double vortex is very similar to that of radar echo intensity of Typhoon Herb. Also the rotation period and amplitude of the inner vortex in the numerical experiment were comparable with those of the trochoidal motion in the observation. These suggest that, in tropical cyclones with the concentric eyewall structure, the interaction of an asymmetric double vortex can become a cause of trochoidal motion.

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.

Synoptic Development in a Two-and-One-Half-Layer Contour Dynamics System

Synoptic development is studied using a two-and-one-half-layer contour dynamics model in full spherical geometry. The model has three isentropic layers: two lower layers that are dynamically active and one upper layer that is kept motionless. The isentropic layers cover the whole sphere and are confined in the vertical direction by two rigid horizontal boundaries. The model is assumed to be in hydrostatic and geostrophic equilibrium; forcing/friction and diabatic heating/cooling are neglected. In both active layers, the horizontal structure is represented by a piecewise-uniform distribution of potential vorticity with a single front in each layer. The potential vorticity front in the upper active layer can be associated with the tropopause or, to be more specific, with the sudden change in height of the tropopause at the jet stream. The potential vorticity front in the lower active layer enhances the baroclinicity of the system and may be associated with the polar front. Because of the assumption of hydrostatic and geostrophic equilibrium, the model atmosphere is completely defined by the instantaneous positions of the contours and can be integrated in time using the technique of contour dynamics. It is shown that realistic zonal flows can be obtained by a suitable choice of parameters. A linear stability analysis reveals that small-amplitude perturbations of given planetary wavenumber may grow for only specific latitudinal positions of the potential vorticity fronts. The maximum growth rates and highest planetary wavenumbers are found for potential vorticity fronts that are located at approximately the same latitude. Because of the conceptual simplicity of the model, in which the potential vorticity structure is represented by only two contours, the instability mechanism expresses itself in a clear way. The contour dynamics model also captures the nonlinear stages of cyclogenesis remarkably well, as is evident from the time evolution of the fronts and the time evolution of a passive tracer in a numerical simulation of a cutoff cyclone.

Thin jet and contour dynamics models of Gulf Stream meandering

This paper reexamines the theory of the meandering of the Gulf Stream and other inertial jets. We develop a hybrid model (with piecewise constant potential vorticity in the upper layer and a deep layer initially at rest) which allows us to clarify the relationships among thin jet, contour dynamics, and instability models. Approximating the hybrid model leads to a simple two-contour model which can be analyzed easily and can be integrated numerically for large amplitude disturbances. The jet evolution predicted by the approximate model is quite similar to the meander development under the full dynamics, except that the time scales are shorter. The model shows that baroclinic processes clearly play a significant role in the growth of meanders, while upper-layer interactions drive the final pinch-off of eddies. In addition to such process studies, the approximate model provides a simple dynamical system for further investigations.

Interacting Vortex and Vortex Layer: How Length Scale Affects Entrainment and Ejection

A contour dynamics model of the interaction between a point vortex and a layer of constant vorticity, bounded below by a slip wall and above by irrotational e ow, isstudied. Theheight of thevortex above the layerestablishes a length scale that is found to have a strong ine uence on both the evolution of the layer and the vertical displacement of the vortex. A vortex far above the boundary layer generates long-wavelength disturbances. These exhibit earlytime growth, which saturates, taking on the shape of a vortex-like structure, suggesting that a vortex far above a boundary layer can create another one within the layer. Concurrently, entrainment of irrotational e ow deep into the layer occurs within a narrow crevice. Alternatively, for a vortex initially placed close to the layer, a short-scale disturbance occurs that exhibits rapid, continuous growth that eventually rolls up around the vortex. Here, accounting for the vertical displacement of the vortex is necessary to accurately determine the interface evolution. The results support the contention that a vortex can induce ejection and roll up of the boundary layer and entrainment of irrotational e ow into the layer.

Interface proliferation and the growth of labyrinths in a reaction-diffusion system.

In the bistable regime of the FitzHugh-Nagumo model of reaction-diffusion systems, spatially homogeneous patterns may be nonlinearly unstable to the formation of compact ``localized states." The formation of space-filling patterns from instabilities of such structures is studied in the context of a nonlocal contour dynamics model for the evolution of boundaries between high and low concentrations of the activator. An earlier heuristic derivation {[}D.M. Petrich and R.E. Goldstein, Phys. Rev. Lett. {\bf 72}, 1120 (1994){]} is made more systematic by an asymptotic analysis appropriate to the limits of fast inhibition, sharp activator interfaces and small asymmetry in the bistable minima. The resulting contour dynamics is temporally local, with the normal component of the velocity involving a local contribution linear in the interface curvature and a nonlocal component having the form of a screened Biot-Savart interaction. The amplitude of the nonlocal interaction is set by the activator-inhibitor coupling and controls the ``lateral inhibition" responsible for the destabilization of localized structures such as spots and stripes, and the repulsion of nearby interfaces in the later stages of those instabilities. The phenomenology of pattern formation exhibited by the contour dynamics is consistent with that seen by Lee, McCormick, Ouyang, and Swinney in experiments on the iodide-ferrocyanide-sulfite reaction in a gel reactor. Extensive numerical studies of the underlying partial differential equations are presented and compared in detail with the contour dynamics. The similarity of these phenomena (and their mathematical description) with those observed in amphiphilic monolayers, Type-I superconductors in the intermediate state, and magnetic fluids in Hele-Shaw geometry are emphasized.

Orography in a Contour Dynamics Model of Large-Scale Atmospheric Flow

The influence of orography on the structure of stationary planetary Rossby waves is studied in the context of a contour dynamics model of the large-scale atmospheric flow. Orography of infinitesimal and finite amplitude is studied using analytical and numerical techniques. Three different types of orography are considered: idealized orography in the form of a global wave, idealized orography in the form of a local table mountain, and the earth's orography. The study confirms the importance of resonances, both in the infinitesimal orography and in the finite orography cases. With finite orography the stationary waves organize themselves into a one-dimensional set of solutions, which due to the resonances, is piecewise connected. It is pointed out that these stationary waves could be relevant for atmospheric regimes.

Nonlocal contour dynamics model for chemical front motion.

Pattern formation exhibited by a two-dimensional reaction-diffusion system in the fast inhibitor limit is considered from the point of view of interface motion. A dissipative nonlocal equation of motion for the boundary between high and low concentrations of the slow species is derived heuristically. Under these dynamics, a compact domain of high concentration may develop into a space-filling labyrinthine structure in which nearby fronts repel. Similar patterns have been observed recently by Lee, McCormick, Ouyang, and Swinney in a reacting chemical system.

The splitting of eddies along boundaries

This paper addresses the question of what happens to an eddy that is forced “violently” against a boundary by, say, an advective current or another vortex. The detailed temporal evolution of such a collision on an f-plane is examined using a barotropic, as well as a one-and-a-half-layer contour dynamics model with surgery procedures. Both the barotropic eddy and the one-and-a-half-layer eddy initially have two circular potential vorticity fronts: an inner front at which the velocity increases to a maximum from zero at its center, and an outer front (e.g., the edge of the eddy) at which the velocity reduces to zero. At t = 0, the circular eddy is conceptually cut off by the wall. It is demonstrated that such a cut corresponds to the violent forcing of the eddy against the wall. One intuitively expects that, as a result of the collision, the eddy would simply leak fluid along the wall (forming a thin jet) until it shrinks to such a size that it is merely “kissing” the wall (Nof, 1988a). In contrast to this intuition, however, it is found that, after the eddy is cut by the wall (i.e., t > 0), the annulus fluid (i.e., the fluid between two fronts) is gradually advected along the wall forming a new eddy next to the interior (i.e., the region inside the inner front). After formation, both the off-spring eddy and the parent eddy migrate along the wall away from each other. This migration is mainly due to the image effect that is created by the wall and separates the eddies even farther. As time goes on, the migration intensifies because the mutual advection of the eddies forces them farther into the wall so that the image effect increases. These results of our contour dynamics study are in good agreement with additional experiments of an isopycnic, primitive equation model. Namely, both of these studies illustrate that an eddy-wall collision causes the parent eddy to split into two migrating eddies, one that contains the core of the parent and the other that contains fluid from the rim. Possible applications of these models to eddies pushed against the shelf in the Gulf of Mexico are discussed.

Vorticity and Instability during Flow Reversals on the Continental Shelf

Abstract Flow reversals during relaxation of the equatorward wind on the northern California shelf are studied with observations and a simple numerical model. Data from the CODE experiment are used to document the changes in the cross-shelf profiles of alongshelf currents and potential vorticity when the wind forcing changes from upwelling-favorable to near-zero. Upwelling vorticity dynamics are considered using a simple equation, where also the importance of stress curls is discussed. During the wind relaxation events, the usually equatorward flow reveres to poleward on the inner shelf, resulting in a region of high shear and occasionally a localized maximum in potential vorticity on the shelf. In the course of such an episode, a single eddylike structure on the shelf was observed with satellite imagery. In order to simulate the basic dynamics of the flow reversals and instability, a contour dynamics model is developed for barotropic flow on an exponentially sloping bottom, in which potential vorticity i...

On the instability of geostrophic vortices

The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases, the other two kinds of instability are nonlinearly destabilized.