Simplified vortex models have been extremely useful in helping us understand the intricate behaviour of real coherent structures in the ocean and atmosphere. Such models are usually highly nonlinear, making it possible to gain insight into many phenomena that are difficult to predict within a geophysical setting . For instance, such vortex models can shed some light on the dynamics of interacting coherent mesoscale vortices , sustainability of such vortices against external flows , or topographic influence , and so on.

One of the most renowned vortex models is the model of an elliptic vortex subjected to linear deformation consisting of shear and rotational components . In the case of a stationary deformation, the elliptic vortex is able to perform three types of motion depending on the parameters of the deformation flow and the initial alignment of the ellipse against the exterior strain. There are two periodic states involving the vortex changing its eccentricity; these are (i) rotation and (ii) nutation. Moreover, there is one aperiodic state – infinite elongation. In this case, the vortex elongates continuously tending to be collinear with the strain axis. Moreover, for a specific initial alignment, the vortex can be stationary.

The model of an elliptic vortex embedded in a linear deformation field is the base model to assess the stability of elliptic vortex shapes occurring in nature. A large body of literature is devoted to the problem. Most of the papers consider spatial perturbations to the elliptic form in the case of constant linear deformation . A prominent result of this stability analysis is that an elliptic vortex is stable to linear perturbations of its form until its geometrical shape complies with the relation a/b≤3, where a and b are the major and minor semi-axes of the ellipse.

Another interesting aspect of the dynamics of an elliptic vortex is its response to time-dependent external deformation. In this case, the form of the vortex endures no changes but the vortex's aspect ratio and orientation can alter unpredictably demonstrating chaotic behavior. This problem has been addressed for small-amplitude strain oscillations by means of the Melnikov's integral technique in several works . The evolution of the vortex embedded in a time-dependent strain with a slowly varying frequency was addressed in . also assess the stability of the stationary configuration of the elliptic vortex; i.e., the vortex's aspect ratio and orientation do not change in time under a constant exterior deformation, given a time-dependent external perturbation. They show that such a stationary configuration of the vortex can be easily destabilized into the nutation or even rotation regimes because of a linear resonance effect. In this paper, we demonstrate that, in the case of relatively small amplitudes of the perturbation, the vortex loses its stationarity by means of parametric instability. In the case of finite amplitudes and optimal frequencies of the perturbation, the dynamics of the vortex is governed strictly by nonlinear effects.

It is also worth noting that fluid particle advection near an oscillating elliptic vortex embedded in a constant deformation flow manifests chaotic dynamics . This is because the oscillating elliptic vortex generates a time-periodic perturbation to the fluid particle motion. This results in the appearance of exponentially diverging trajectories in the unsteady velocity field governing the fluid particle advection. Similar chaotic dynamics is present in the case of the ellipsoid vortex model , which is a generalization of the elliptic vortex model taking into account a linear vertical stratification .

Let us consider an inviscid, incompressible, two-dimensional flow. In this flow, an elliptic patch of constant vorticity g, experiencing deformation from time-dependent strain e(t) and background rotation γ(t), is embedded. The patch conserves its elliptic form with a and b being the ellipse's semi-axes, ε=a/b being the aspect ratio, and φ being the angle between the ellipse's major semi-axis and the x axis of the Cartesian coordinate frame. The governing equations are ε˙=2eεcos⁡2φ,φ˙=γ+gεε+12-eε2+1ε2-1sin⁡2φ.

It is worth noticing that the motion of the ellipse's center (x0,y0) is governed by advection equations in the form dx0dt=u0+ex0-xd-γy0-yd,dy0dt=v0-ey0-yd+γx0-xd, where u0,v0 are the velocity's components of an arbitrary uniform flow; xd,yd are the coordinates of the constant deformation center. Equation () exactly coincides with the equations governing the motion of the vorticity center of an arbitrary number of point vortices embedded in such a deformation flow . Given periodic unequal dependencies of the strain and rotation components, these equations allow for parametric instability. This, in turn, may result in unbounded motion of the vorticity center depending on the strain and rotation oscillation parameters. However, the relative vortex motion is independent of the vortex's center motion. We further consider the evolution of the elliptic vortex when the vortex's center and the exterior deformation's center coincide at the coordinate's origin.

Without loss of generality, the ellipse's vorticity in Eq. () is further set to be g=1. Equation () represent a dynamical system with “one and a half” degrees of freedom given time-dependent strain and rotation rates e(t) and γ(t). This suggests that the dynamics of the phase variables may be chaotic for certain initial conditions. To start, it is informative to look into the stationary system given constant values of e(t)≡e0= const, γ(t)≡γ0= const. The critical points (φ0,ε0) corresponding to the stationary elliptic vortex ensue from the following relations : φ0=±π4,ε03+ε02γ0-e0sin⁡2φ0+1γ0-e0sin⁡2φ0-ε0γ0+e0sin⁡2φ0+1γ0-e0sin⁡2φ0-γ0+e0sin⁡2φ0γ0-e0sin⁡2φ0=0.

There are four qualitatively different phase portraits depending on the number and type of the critical points . Each phase portrait has at least one elliptic critical point. The corresponding phase portrait for the values e0=0.15, γ0=0.02 is shown in Fig. . The homoclinic separatrix delineates the regions of the initial conditions corresponding to the ellipse nutation (near the elliptic critical point) and infinite elongation (all the rest).

The elliptical critical point illustrated in Fig.  corresponds to the elliptic vortex, without performing any motion. Once slightly shifted from this stance, the vortex starts nutating, slightly changing the lengths of its semi-axes and the orientation angle. If an initial condition is taken outside of the nutation region near the elliptic critical point, the vortex starts elongating infinitely.

Now, let us examine the dynamics of the system near the elliptic critical point, given small time-dependent perturbations of the deformation flow e(t)=e0+e′(t)=e0+e0δcos⁡νt,γ(t)=γ0+γ′(t)=γ0+γ0δcos⁡νt. One can expand Eq. () into a Taylor series up to the first term over a small deviation from the elliptic critical point εt≈ε0+ε′t,φt≈φ0+φ′t=±π4+φ′t, where ε0,φ0 result from Eq. (); ε′t,φ′t are small time-dependent deviations. One then obtains the following for the deviations: dε′dt=-4etε0φ′sin⁡2φ0,dφ′dt=γt-etε02+1ε02-1sin⁡2φ0-ε′ε0-1ε0+13-4etε0ε02-12sin⁡2φ0. The linear system of Eq. () is an inhomogeneous one. However, in terms of determining whether the system has unbounded solutions, one can take advantage of the associated homogeneous system: dε′dt=-4etε0φ′sin⁡2φ0,dφ′dt=-ε′ε0-1ε0+13-4etε0ε02-12sin⁡2φ0.

It is worth noting that the term with the background rotation γ has vanished from the homogeneous system. This readily means that the existence of unbounded solutions about the stationary position is independent of the exterior rotation.

One can see that, if reduced to a second-order equation, the system of Eq. () represents a Hill equation . This, in turn, signifies that there is a possible manifestation of parametric instability near the vortex stationary position. In other words, given specific values of the perturbation's parameters, the phase trajectories become unbounded near the steady-state elliptic critical point. The values of the perturbation parameters resulting in parametric instability can be readily figured out using the Floquet analysis. An analytical estimate can be derived by means of averaging techniques . To do this, it is convenient to rewrite the system of Eq. () in the form dε′dt=-4etε0φ′sin⁡2φ0,dφ′dt=k24e0ε0sin⁡2φ0+4e0ε0sin⁡2φ0ε02-12δcos⁡νtε′, where k2=-4e0ε0sin⁡2φ0ε0+12ε0-1ε0+1-4e0ε0ε0-12sin⁡2φ0.

Let us introduce a new variable, ρ= -1+i4e0ε0sin⁡2φ0kφ′ε′e-iνt. Then, instead of Eq. (), one obtains

dρdt=i2k1+δeiνt+e-iνt2e0-νρ-ik16k2e02ε02ε02-12-1δ1+e-2iνt2e0+ikeiνt+δe2iνt+12e0ρ2. Then, taking into account that the fast-oscillating terms average to zero, one gets the following for the averaged value ρ‾: dρ‾dt=ikδ2e0ρ‾2+i2k-νρ‾-ik16k2e02ε02ε02-12-1δ2e0,ρ‾0=0.

The solution of Eq. () is ρ‾+e0δk2k-ν-Dρ‾+e0δk2k-ν+D=exp⁡ikDδe0, where D2=e0δk2k-ν2-16k2e02ε02ε0+12ε0-12-1.

The solution Eq. () grows unboundedly if the argument in the right-hand term exponential function is real. Therefore, a rough analytical estimate for the location of the primary parametric instability zone in the parametric space ensues 2k-ν=±2δsin⁡2φ0e0ε0ε0-1ε0+13.

Figure  depicts the precise parametric instability zones obtained by the Floquet analysis (the dark regions) and the analytically estimated values (the dashed line) in the (δ, ν) parametric space. Every parameter set from the dark regions leads to spiral-like unbounded trajectories of the linear system of Eq. (), while the parameters taken from the light areas result in periodic trajectories in bounded regions. The primary zone of the parametric instability is the widest zone located near ν=0.6. The secondary zone is the one located near ν=0.3.

Now, let us discuss possible implications of the parametric instability reported for the linear system of Eq. () that governs the dynamics of the elliptic vortex in the immediate vicinity of the elliptical critical point shown in Fig. . Since the original system of Eq. () is nonlinear, it is clear that the applicability range of the linear system of Eq. () is relatively limited.

First of all, in the case of parametric instability, the trajectories that originated near the steady-state elliptic critical point move unboundedly only up to the steady-state separatrix region where nonlinear effects prevail against the linear unbounded motion. This scenario occurs only if the nonlinear effects do not already drastically influence the dynamics in the immediate vicinity of the steady-state critical elliptic point. This is the case, for instance, for the primary parametric instability zone shown in Fig. . Given the corresponding perturbation values, the phase space near the steady-state critical elliptical point differs crucially from the unperturbed phase space.

To corroborate this effect, a Poincaré section shown in Fig. a is presented. To construct this and all the following Poincaré sections, we plot the position of a phase trajectory exactly in a perturbation period 2π/ν. Thus, a chaotic trajectory appears as a set of disorder points and a regular trajectory appears as a smooth, closed, linked orbit in the sections. Figure a illustrates that the dynamics near the steady-state elliptic critical point is dominated by a nonlinear resonance with winding number 1:2. Because of this effect, the phase trajectories that originated near the steady-state elliptic critical point do not demonstrate parametric instability. Figure b shows a trajectory starting at the steady-state elliptic critical point φ0=π/4, ε0≈2.09244 is clearly bounded to the region of the nonlinear resonance influence. Therefore, the linear system (Eq. ) cannot account for the dynamics in this case.

Nevertheless, the linear system is a good approximation for certain values of the perturbation's parameters. For instance, when one considers the second parametric instability zone, the perturbed phase space near the steady-state elliptic critical point features no nonlinear resonances (see the Poincaré section shown in Fig. a). Therefore, the dynamics near the steady-state critical elliptic point can be derived from the linear system (Eq. ). When this happens, the phase trajectory that starts near the steady-state critical elliptic point moves in a spiral-like divergent trajectory (see Fig. b). The trajectory spirals only until it reaches the region of high nonlinearity (a chaotic region in the place of the Poincaré section in Fig. a); then it spirals back. However, the parametric resonance results in a significant change of the ellipse characteristics contrary to the case shown in Fig. b.

In this work, we have considered a model of an isolated elliptic vortex embedded in a time-dependent deformation flow, consisting of strain and rotational components. The main focus is on the dynamics of the vortex near its steady-state stable position. In the linear approximation, the vortex dynamics is established to experience parametric instability. This instability entails the following motion of the vortex. The vortex starts nutating about the strain axis, continuously increasing the aspect ratio of its semi-axes. This evolution continues until the phase trajectory reaches the region of high nonlinearity near the steady-state separatrix. This regime proceeds, provided the nonlinear system governing the dynamics of the vortex remains relatively linear near the steady-state elliptic point under perturbation. We have shown that this, in fact, is possible for certain values of the perturbation parameters. Thus, our results suggest that the parametric instability, which is intrinsic to linear systems, can also be an important factor in the nonlinear system in question.

However, the influence of linear effects on the nonlinear systems should be generally interpreted with great care. Indeed, the primary parametric resonance zone produces the most effective unbounded motion of the phase trajectories in the linearized system, but the same driving parameters applied to the nonlinear system lead only to a bounded motion in a very small vicinity of the steady-state elliptic critical point. This is because a region of high nonlinearity appears near the steady-state elliptic critical point that completely suppresses the linear unbounded motion in the nonlinear system.

Considering the evolution of the elliptic vortex, the elliptic vortex nutates along the shear axis, becoming more oblong in time. This dynamics is regular (not chaotic), meaning that the vortex form returns to its initial shape after the locally unbounded motion is suppressed by the nonlinear effects near the steady-state separatrix. However, during this temporal elongation, the aspect ratio of the vortex may easily exceed the critical value of the vortex stability. Thus, in terms of the linear stability of the vortex to the perturbations of its elliptic form, the vortex turns unstable. Therefore, our results may suggest the following scenario of the evolution of an elliptic vortex in a time-dependent natural environment. First, it starts becoming more and more oblate due to the linear parametric instability. After attaining the critical aspect ratio, small-scale disturbances start transpiring on the boundary of the vortex. Finally, it may break into a number of smaller elliptic vortices .

No specific data are used. All the figures are obtained by numerically integrated equations (Eq. 1) using a standard Bulirsch–Stoer procedure, which uses the Richardson extrapolation and modified midpoint method.