2.1. Centroid theorem

The explicit expression for the propagation of the centre of mass anomaly is obtained by multiplying (2.1) by $(x,y)$ and integrating the result over the entire horizontal plane

(2.2)\begin{equation}\frac{{\textrm{d}{\kern 1pt} {x_c}}}{{\textrm{d}t}} ={-} \beta R_d^2,\quad \frac{{\textrm{d}{\kern 1pt} {y_c}}}{{\textrm{d}t}} = 0,\end{equation}

where $({x_c},{y_c})$ are the coordinates of the eddy centroid, defined as

(2.3)\begin{equation}{x_c} = \frac{{\displaystyle\iint {x\psi \,\textrm{d}x\,\textrm{d}y} }}{{\displaystyle\iint {\psi \,\textrm{d}x\,\textrm{d}y} }},\quad {y_c} = \frac{{\displaystyle\iint {y\psi \,\textrm{d}x\,\textrm{d}y} }}{{\displaystyle\iint {\psi \,\textrm{d}x\,\textrm{d}y} }}.\end{equation}

What makes the centroid theorem even more general and surprisingly robust, is that the centre-of-mass speed of coherent vortices embedded in steady large-scale flows is still given by (2.2). This ‘anti-Doppler’ effect (Nycander Reference Nycander1994) is caused by the variation in the upper layer thickness associated with the large-scale geostrophic current. The action of the resulting environmental potential vorticity gradient is similar to the beta-effect, causing the eddy to move at the same speed but in the direction opposite to the ambient current, thereby cancelling the advective tendency of the large-scale flow. It should be realized that the exact cancellation of these tendencies occurs only in the equivalent-barotropic model and uniform large-scale currents. The anti-Doppler effect takes on a less rigid and clear-cut form in more complicated models (Nycander Reference Nycander1994) and spatially non-uniform flows (Sutyrin & Carton Reference Sutyrin and Carton2006). Nevertheless, the fact that the advection by large-scale flows has a limited effect on the vortex speed makes it imperative to explain the wide range of eddy migration rates in the ocean (Cornillon et al. Reference Cornillon, Weyer and Flierl1989; Nof et al. Reference Nof, Jia, Chassignet and Bozec2011; Ni et al. Reference Ni, Zhai, Wang and Marshall2020). We should also mention that the analogous constraints on the propagation speed of coherent vortices do not arise in the purely barotropic model. In fact, the singular character of the barotropic model made it possible to develop several analytical solutions representing self-propagating vortices and demonstrate their ability to translate over large distances on the barotropic f-plane (e.g. Stern & Radko Reference Stern and Radko1998; Radko Reference Radko2008). The explicit constraint (2.2) on the centroid speed is ultimately caused by the effects of stretching/squeezing of water columns during the vortex motion, represented by the term $\psi /R_d^2$ in the potential vorticity (2.1).

Of course, it is possible that the velocity of the centroid simply differs from other – more local and precise – measures of the vortex movement, such as the displacement of potential vorticity extrema. The analysis of this difference is one of the key objectives of the present investigation. However, the corollary of the centroid theorem is that exact uniformly propagating solutions in the equivalent-barotropic model are possible only for the migration rate $(U,V) = ( - \beta R_d^2,0)$. An important exception, allowing vortices to move with a wide range of speeds, is the zero angular momentum configuration (1.2). In this case, (2.2) and (2.3) are inapplicable. However, the straightforward modification of the centroid theorem leads to

(2.4)\begin{equation}\frac{\textrm{d}}{{\textrm{d}t}}\,\iint {x\psi \,\textrm{d}x\,\textrm{d}y} = 0,\quad \frac{\textrm{d}}{{\textrm{d}t}}\,\iint {y\psi \,\textrm{d}x\,\textrm{d}y} = 0,\end{equation}

which is readily interpreted as a conservation principle for the first moment of the streamfunction. Equations (2.4) also imply that if a zero angular momentum vortex contains a substantial dipolar component initially, it will be maintained indefinitely. A well-known example of zero angular momentum vortices is the family of exact modon-with-the-rider solutions (e.g. Flierl et al. Reference Flierl, Larichev, McWilliams and Reznik1980). Free from the rigid constraints of the centroid theorem, such solutions can be found for any values of zonal propagation velocities outside of the Rossby wave range $( - \beta R_d^2 \lt {U_{Rossby}} \lt 0)$.

For both finite and zero angular momentum vortices, the question arises regarding the impact of the dipolar component on their movement and long-term evolution. This question is addressed by superimposing relatively weak dipoles on stable circularly symmetric riders and examining the evolution of the resulting systems numerically. The dipolar components producing the initial motion with a given velocity $(U,V)$ are computed using the following optimization procedure.

2.2. Optimal non-axisymmetric perturbations

To reduce the number of controlling parameters, we non-dimensionalize (2.1) using ${R_d}$ as the unit of length, $\beta R_d^2$ as the unit of velocity and ${(\beta {R_d})^{ - 1}}$ as the unit of time. As a result, (2.1) reduces to

(2.5)\begin{equation}\frac{\partial }{{\partial t}}({\nabla ^2}\psi - \psi ) + J(\psi ,{\nabla ^2}\psi ) + \frac{{\partial \psi }}{{\partial x}} = {\nu _{nd}}{\nabla ^4}\psi ,\end{equation}

where ${\nu _{nd}} = \nu /(\beta R_d^3)$. Introducing the fluid potential vorticity (PV)

(2.6)\begin{equation}q = {\nabla ^2}\psi - \psi ,\end{equation}

further reduces (2.5) to

(2.7)\begin{equation}\frac{{\partial q}}{{\partial t}} + J(\psi ,q) + \frac{{\partial \psi }}{{\partial x}} = {\nu _{nd}}{\nabla ^4}\psi .\end{equation}

Since the existence of exact stable solutions of (2.5) representing quasi-monopolar vortices uniformly propagating in space is not guaranteed, we now attempt to construct some accurate approximations. For that, we consider a circularly symmetric and preferably stable rider with the PV distribution ${q_0} = {q_0}(r)$, where $r = \sqrt {{x^2} + {y^2}} $, and search for a non-axisymmetric perturbation $q^{\prime}$ that would cause the vortex to move with a given speed $(U,V)$. The entire PV field is expressed in terms of its azimuthal Fourier components as follows:

(2.8)\begin{equation}q = {q_0}(r) + \sum\limits_{i = 1}^{{n_\theta }} {[{q_{ci}}(r)\,\textrm{cos}(i\theta ) + {q_{si}}(r)\,\textrm{sin}(i\theta )]} ,\end{equation}

where ${n_\theta }$ is the number of azimuthal harmonics and $\theta $ is the polar angle defined by

(2.9)\begin{equation}\textrm{cos}\,\theta = \frac{x}{r},\quad \textrm{sin}\,\theta = \frac{y}{r}.\end{equation}

To find uniformly translating patterns, we rewrite the governing equation (2.7) in the coordinate system moving with the target velocity $(U,V)$

(2.10)\begin{equation}\frac{{\partial q}}{{\partial t^{\prime}}} - U\frac{{\partial q}}{{\partial x^{\prime}}} - V\frac{{\partial q}}{{\partial y^{\prime}}} + J(\psi ,\,q) + \frac{{\partial \psi }}{{\partial x^{\prime}}} = {\nu _{nd}}{\nabla ^4}\psi ,\end{equation}

where the new independent variables $(t^{\prime},x^{\prime},y^{\prime})$ are related to the original ones through

(2.11)\begin{equation}t^{\prime} = t,\quad x^{\prime} = x - Ut,\quad y^{\prime} = y - Vt.\end{equation}

Equation (2.10) can also be expressed in terms of the streamfunction tendency

(2.12)\begin{equation}\frac{{\partial \psi }}{{\partial t^{\prime}}} = F(q),\end{equation}

where F is defined as

(2.13)\begin{equation}F \equiv {({\nabla ^2} - 1)^{ - 1}}\left[ {U\frac{{\partial q}}{{\partial x^{\prime}}} + V\frac{{\partial q}}{{\partial y^{\prime}}} - J(\psi ,\,q) - \frac{{\partial \psi }}{{\partial x^{\prime}}} + {\nu_{nd}}{\nabla^4}\psi } \right]\end{equation}

and $\psi $ is obtained from q by inverting the Helmholtz operator (2.6).

The requirement for the vortex to move almost uniformly with speed $(U,V)$ is equivalent to the statement that it is nearly steady in the moving coordinate system (2.11). To obtain such solutions, we introduce the global quadratic norm of the streamfunction tendency

(2.14)\begin{equation}N = \overline {{F^2}} ,\end{equation}

where the overbar denotes the spatial average. The norm (2.14) is then minimized by varying non-axisymmetric PV components ${q_{ci}}(r)$ and ${q_{si}}(r)$ in (2.8). Our approach can be viewed as the nonlinear generalization of the model developed by Nycander (Reference Nycander1988), who obtained asymptotic quasi-monopolar solutions by combining the dominant axisymmetric flow and a weak dipolar correction.

To find optimal non-axisymmetric perturbations, functions ${q_{ci}}(r)$ and ${q_{si}}(r)$ in the series (2.8) are discretized using a non-uniform (exponential) grid. The grid points $r = {r_j}$, $j = 0,1, \ldots ,{n_r}$, ${n_r} + 1$, are closely (sparsely) distributed in the near (far) field of the vortex $r \lesssim 1$ $(r \gg 1)$. Extensive experimentation with the proposed algorithm indicates that the increase in the number of azimuthal harmonics beyond ${n_\theta } = 3$ and of radial grid points beyond ${n_r} = 200$ leads to insignificant improvements in the model performance. Therefore, these values are used in all experiments reported here. In the subsequent calculations, the radius of the optimization region is set to $R = 15$, which greatly exceeds the radius of deformation and effective radii of vortices considered in this study. We assume that ${q_{ci}}(r)$ and ${q_{si}}(r)$ vanish at the edge points ${r_0} = 0$ and ${r_{{n_r} + 1}} = R$. The PV values at the interior points ${q_{ci}}({r_j})$ and ${q_{si}}({r_j})$, where $j = 1,2, \ldots ,{n_r}$, are used as the input variables into the iterative algorithm, which minimizes the quadratic norm N of trial solutions. The minimization module is available in the Matlab library as the function ‘fminunc.m’. The resulting patterns are then interpolated onto the Cartesian grid and used as initial conditions for numerical integrations of (2.5), which are described next.

3.1. Finite angular momentum vortices

The first series of experiments explores the dynamics of finite angular momentum eddies, which is represented in our study by Gaussian streamfunction patterns – a popular choice in theoretical and numerical models of coherent vortices (e.g. Early, Samelson & Chelton Reference Early, Samelson and Chelton2011; Sutyrin & Radko Reference Sutyrin and Radko2019)

(3.2)\begin{equation}{\psi _0} ={-} A\,\textrm{exp}( - a{r^2}).\end{equation}

Without loss of generality, we shall consider only cyclonic vortices $(A \gt 0)$. Because of the invariance of the governing equation (2.5) to the transformation $(\psi ,y) \to ( - \psi , - y)$, the evolution of anticyclones in the equivalent-barotropic model mirrors that of corresponding cyclonic eddies. While the Gaussian vortex (3.2) is known to be formally unstable, this instability is relatively benign and does not lead to its fragmentation or significant reorganization (e.g. Sutyrin & Radko Reference Sutyrin and Radko2019). Since the centroid theorem (2.2) is directly applicable to the Gaussian vortex, we can expect to find solutions uniformly propagating with the non-dimensional velocity

(3.3)\begin{equation}(U,V) = ( - 1,0).\end{equation}

In the following simulation, we use $a = 1$ and the amplitude of streamfunction $(A = 29.1)$ that corresponds to the maximal azimuthal velocity of

(3.4)\begin{equation}{v_{max}} = \mathop {\max }\limits_r ({v_0}) = 25,\end{equation}

where

(3.5)\begin{equation}{v_0}(r) = \frac{{\partial {\psi _0}}}{{\partial r}}.\end{equation}

The solution conforming to (3.2)–(3.4) was obtained using the optimization procedure (§ 2.2) and is shown in figure 1. The optimization resulted in the minimal tendency norm of $N = 1.6 \times {10^{ - 5}}$, which is much less than the tendency norm of the corresponding axisymmetric vortex $({N_0} = 0.828)$. Such a low value of N implies that the resulting structure is expected to move almost steadily with target velocity (3.3). The perturbation components $q^{\prime}$ and $\psi ^{\prime}$ of the obtained solution are shown in panels (a) and (b) of figure 1, respectively. The total PV and streamfunction fields $q = {q_0} + q^{\prime}$ and $\psi = {\psi _0} + \psi ^{\prime}$ are presented in (c) and (d), respectively. The perturbation is predominantly dipolar and is much weaker than the prescribed axisymmetric component.

Figure 1.

The propagating solution obtained using the optimization algorithm (§ 2.2) for the Gaussian basic state and the target propagation velocity of $(U,V) = ( - 1,0)$. The perturbation components of the solution $q^{\prime}$ and $\psi ^{\prime}$ are shown in panels (a) and (b), respectively. The total PV and streamfunction fields $q = {q_0} + q^{\prime}$and $\psi = {\psi _0} + \psi ^{\prime}$ are presented in (c) and (d), respectively.

To further illustrate the extent to which this optimal solution represents a steadily propagating pattern, we present (figure 2) the scatterplot of the total PV

(3.6)\begin{equation}{q_{tot}} = q + y = {\nabla ^2}\psi - \psi + y\end{equation}

and the streamfunction evaluated in the moving coordinate system

(3.7)\begin{equation}{\psi _{mov}} = \psi + Uy - Vx.\end{equation}

For steadily propagating non-dissipative solutions, (2.5) reduces to

(3.8)\begin{equation}J({q_{tot}},{\psi _{mov}}) = 0,\end{equation}

which implies that ${q_{tot}}$ and ${\psi _{mov}}$ are functionally related. The scatterplot in figure 2 confirms the existence of a relatively tight PV–streamfunction relation for our optimal solution. This relation is largely bi-modal. The non-monotonic curve in figure 2 represents the ${q_{tot}}({\psi _{mov}})$ pattern realized in the vortex interior $(r \lt 2.5)$ and the nearly horizontal low-PV line describes the exterior flow field $(r \gt 2.5)$. It is interesting that the interior pattern closely follows the relation between ${q_0}$ and ${\psi _0}$, which is also shown (red curve) in figure 2. This correspondence implies that the optimization, whilst adding new non-axisymmetric components, largely retains the PV–streamfunction relation of the basic axisymmetric circulation.

Figure 2.

The scatterplot of ${q_{tot}}$ vs. ${\psi _{mov}}$ for the optimal solution in figure 1 (black dots). The red curve represents the corresponding relation between ${q_0}$ and ${\psi _0}$.

The state in figure 1 was used as an initial condition for the simulation shown in figure 3. As expected, the vortex moved linearly in space with the prescribed velocity and exhibits no visible signs of instability. In this regard, the solution in figure 3 can be considered as an interesting example of a stable modon-with-the-rider vortex. This experiment demonstrates that the instability (Swenson Reference Swenson1987) of the classical exact solution (Flierl et al. Reference Flierl, Larichev, McWilliams and Reznik1980) does not imply that all uniformly propagating vortices are inherently unstable. The calculation in figure 3 was extended for 200 units of time. During this time interval, the vortex crossed the periodic computational domain six times, with the net westward displacement of $\Delta X = 182$. This displacement is dimensionally equivalent to approximately $5000\;\textrm{km}$ – roughly the width of the Atlantic Ocean – and it occurred over two and a half years. The vortex maintained its motion pattern, exhibiting only a modest reduction of its propagation speed in the late stages of the experiment. The streamfunction amplitude ${A_{fit}}(t)$ was evaluated by fitting the Gaussian pattern (3.2) to instantaneous $\psi $ fields and recorded throughout the simulation. During the entire journey, the effective radius of the vortex ${r_{fit}} = {a_{fit}}{(t)^{ - 0.5}}$ remained largely unchanged, barely increasing from ${r_{fit}} = 1$ to ${r_{fit}} = 1.048$. However, the vortex amplitude reduced from ${A_{fit}} = 29.1$ initially to ${A_{fit}} = 18.1$ at $t = 200$, which could be attributed to the combination of Rossby wave radiation, northward displacement, and frictional decay. Among these processes, frictional effects are the most significant. This was established by reproducing the calculation in figure 3 with the reduced viscosity of ${\nu _{nd}} = {10^{ - 4}}$, which resulted in the final amplitude of ${A_{fit}} = 24.27$. The meridional drift, on the other hand, plays a largely negligible role in the vortex spin-down, which was ascertained by the scaling analysis based on the Lagrangian conservation of the total PV (3.6).

Figure 3.

Numerical simulation initiated by the solution in figure 1. The instantaneous PV patterns are shown at $t = 0,\;5,\;15,\;25$ and 35 in panels (a)–(e), respectively. Only the central part of the computational domain $( - 5 \lt y \lt 5)$ is shown.

While quasi-steady motion with the preferred migration rate (3.3) is a plausible consequence of the centroid theorem, the question arises whether vortices with other target speeds can also maintain their motion patterns for extended periods. To examine this possibility, we turn to a series of experiments in which we systematically vary both the launch angle $\varphi $ and the magnitude W of the target drift rate

(3.9)\begin{equation}(U,V) = (W\,\textrm{cos}\,\varphi ,\;W\,\textrm{sin}\,\varphi ).\end{equation}

The resulting displacement patterns are shown in figure 4. Panel (a) presents simulations performed with $W = 2$, ${v_{max}} = 25$ and $\varphi = 0,{\rm \pi} \textrm{/}6,2{\rm \pi} \textrm{/}6, \ldots ,11{\rm \pi} \textrm{/}6$. Each experiment in figure 4 was extended in time until the absolute displacement $\Delta R = \sqrt {\Delta {X^2} + \Delta {Y^2}} $ exceeded $\varDelta {R_{max}} = 50$. All vortices eventually evolve to the predominantly westward propagation with speeds close to the preferred migration rate (3.3). During the final stages of experiments, defined here as the last 20% of the integration periods, the meridional/zonal speed ratios $|{V/U} |$ range from 0.0032 to 0.0647, and $|{1 + U} |$ from 0.07 to 0.20. However, the initial conditions clearly affect the evolution of self-propagating vortices for extended periods. This influence becomes evident when their trajectories are compared to the track of the corresponding axisymmetric vortex, indicated by the dashed curve in figure 4(a). Of particular interest are the instances of eastward displacement, which can be as large as $\Delta X\sim 30$ – the length scale that is dimensionally equivalent to ${\sim} 1000\;\textrm{km}$.

Figure 4.

Trajectories of vortices constructed using the Gaussian basic state for (a) $(W,{v_{max}}) = (2,25)$, (b) $(W,{v_{max}}) = (1,25)$, (c) $(W,{v_{max}}) = (4,25)$ and (d) $(W,{v_{max}}) = (2,100)$. In all simulation sets, the launch angles are $\varphi = 0,{\rm \pi} \textrm{/}6,2{\rm \pi} \textrm{/}6, \ldots ,11{\rm \pi} \textrm{/}6$. Dashed curves represent tracks of the initially axisymmetric vortices. Different colours are used to help distinguish individual trajectories.

Panel (b) shows the analogous series of simulations performed with a relatively low target velocity $(W = 1)$. As expected, this reduction substantially limited the long-term effect of initial non-axisymmetric perturbations. The increase in the target speed to $W = 4$ in panel (c), on the other hand, dramatically increases the ability of vortices to self-propagate. Six out of twelve vortices maintained their predominantly eastward movement throughout the entire experiment, thereby overcoming the opposing tendency of the centre-of-mass propagation (2.2).

Finally, panel (d) examines the ramifications of increasing the vortex strength. These simulations were performed for ${v_{max}} = 100$ and $W = 2$. Despite the fourfold difference in ${v_{max}}$, the patterns of trajectories in (a) and (d) are qualitatively similar. This indicates that the ability of vortices to self-propagate is not particularly sensitive to the strength of their axisymmetric components. In this regard, it should be noted that the values of ${v_{max}}$ used for the calculations in figure 4 are broadly representative of coherent eddies in the ocean. For instance, the historical census of long-lived oceanic vortices (Olson Reference Olson1991) lists 35 well-studied examples with azimuthal velocities ranging from 0.22 to 2 m s⁻¹. For typical oceanic parameters (3.1), this range translates to the non-dimensional velocities of $22 \lt {v_{max}} \lt 200$.

3.2. Zero angular momentum vortices

The second type of solutions considered in this study is that of zero angular momentum vortices. Such solutions are particularly interesting because they do not satisfy the conditions of the centroid theorem (2.2). Thus, a priori, one expects to find a wider range of migration rates and enhanced self-propagation tendencies amongst zero angular momentum eddies. These vortices are modelled in our study using the double-Gaussian streamfunction

(3.10)\begin{equation}{\psi _0} ={-} A{a_1}\,\textrm{exp}( - {a_1}{r^2}) + A{a_2}\,\textrm{exp}( - {a_2}{r^2}),\end{equation}

which satisfies the zero angular momentum condition (2). However, to use the double-Gaussian pattern, we must first identify relatively stable solutions, which do not disintegrate in simulations. Extensive exploration of the parameter space revealed that the double-Gaussian vortices tend to become more stable when ${a_1}$ and ${a_2}$ are assigned dissimilar values. Since no instances of vortex fragmentation occurred in simulations performed with $({a_1},{a_2}) = (2,0.4)$, these parameters are used in all following experiments. The azimuthal velocity profile of the double-Gaussian vortex is shown in figure 5, along with its Gaussian counterpart. While qualitatively similar, the double-Gaussian profile is characterized by a slight reversal of the velocity in the vortex periphery $(r \gt 1.4)$.

Figure 5.

The radial patterns of the azimuthal velocity ${v_0}(r)$ in the Gaussian (dashed curve) and double-Gaussian (solid curve) vortices.

As previously, we use the optimization procedure (§ 2.2) to construct non-axisymmetric perturbations for given target propagation velocity values. These perturbations are then superimposed on the symmetric state (3.10) and used as an initial condition for calculations shown in figure 6. Panel (a) shows a series of experiments performed with $W = 2$ and ${v_{max}} = 25$. The salient feature of these simulations is the ability of vortices to maintain the predominantly eastward course if their initial drift is eastward. Such persistence is seen even when the target speed is reduced to $W = 1$ in figure 6(b). An interesting aspect of high-speed experiments $(W = 4)$ is the nearly bi-modal pattern of vortex tracks. All trajectories in figure 6(c) are tightly grouped around two well-defined routes, westward and eastward. The increase in the eddy intensity to ${v_{max}} = 100$ (figure 6d) has a limited impact on the motion patterns (cf. figure 6a) and is reflected most notably in larger northward displacements of the eastward-propagating vortices.

Figure 6.

The same as figure 4 but for the double-Gaussian basic state.

Figure 7 examines in greater detail two representative examples of zero angular momentum vortices. The eastward-propagating case $(\varphi = 0)$ is illustrated in panels (a–e) and the westward example $(\varphi = {\rm \pi})$ is in (f–j). Although both experiments were performed using $W = 2$ and ${v_{max}} = 25$, they reveal considerably different dynamics and evolutionary patterns. The eastward-propagating vortex rapidly deviates from its initial course and by $t = 5$ starts moving to the north-east (figure 7b). At this stage, the vortex undergoes a substantial reorganization of its PV field. Negative PV accumulates in the south-east region of the vortex, forming a strongly asymmetric dipole. The resulting structure eventually settles into a predominantly eastward motion pattern (figure 7c,d) along the $y = 10$ latitude. Its zonal migration is accompanied by periodic meridional excursions of decreasing magnitude.

Figure 7.

Numerical simulations initiated by the zero angular momentum (double-Gaussian) vortices. Panels (a–e) present the evolution of the eastward-propagating vortex $(\varphi = 0)$, and the westward example $(\varphi = {\rm \pi})$ is in (f–j). The instantaneous PV patterns in each experiment are shown at $t = 0,\;5,\;15,\;25$ and 35. Both simulations were performed with $(W,{v_{max}}) = (2,25)$. Only the parts of computational domains containing the vortex are shown.

These oscillations and their eventual stabilization can be interpreted as a ramification of the Lagrangian conservation of the total PV (3.6). This principle implies that any northward displacement – and the associated increase in the planetary PV component – is necessarily accompanied by the decrease in the fluid PV (q). Thus, when an approximately balanced vortex that is travelling predominantly eastward drifts to the north, the decrease in q induces anticyclonic perturbation. As a result, the entire structure tends to rotate clockwise, and the vortex turns right, eventually reversing its northward drift. Applying similar arguments to the vortex moving south-east, we conclude that it should turn left and reverse its southward displacement. This negative feedback leads to the directional stabilization of the vortex trajectory and phase locking of its dipolar component into the orientation required for eastward propagation. It is interesting to note that directional stabilization was originally discovered in the context of dipolar eddy models (Zabusky & McWilliams Reference Zabusky and McWilliams1982). Subsequently, the tendency of approximately symmetric dipoles to evolve into eastward-propagating structures has been explored using simulations (Hesthaven, Lynov & Nycander Reference Hesthaven, Lynov and Nycander1993; Sutyrin et al. Reference Sutyrin, Hesthaven, Lynov and Rasmussen1994), laboratory experiments (Kloosterziel, Carnevale & Phillippe Reference Kloosterziel, Carnevale and Phillippe1993) and was recently identified in the ocean using satellite altimeter data (Hughes & Miller Reference Hughes and Miller2017). The present investigation, however, demonstrates that the directional stabilization of eastward propagation can also occur for quasi-monopolar vortices.

The motion pattern of the westward-propagating vortex in (f–j) is more regular and predictable. The vortex largely retains its initial structure and uniform motion pattern throughout the entire simulation. Perhaps the only qualitative difference between this experiment and its finite angular momentum counterpart in figure 3 is a larger northward drift in figure 7.