We systematically study the evolution of Larichev–Reznik dipoles in an equivalentbarotropic quasigeostrophic betaplane model in highresolution numerical simulations. Our results shed light on the selforganization and rich dynamics of dipolar vortices, which are ubiquitous in geophysical flows. By varying both dipole strength and initial angle α_{0} of dipole tilt to the zonal direction, we discover new breakdown mechanisms of the dipole evolution. The dipoles are quickly destroyed by Rossby wave radiation, if initial tilt is too large or the dipole is too weak; otherwise, via damped oscillations, the dipoles tend to adjust themselves to different states drifting eastward. Two competing physical mechanisms that govern dipole transformations are found: (1) spontaneous dipole instability due to a growing critical linear mode and (2) meridional separation of dipole partners that accumulates over the adjustment period and prevents the above instability. Which mechanism prevails depends on the initial tilt and dipole strength, and the details of this are discussed.
I. INTRODUCTION
The dynamics of isolated and coherent vortices has attracted the attention of many researchers trying to understand longlived geophysical vortices, which are ubiquitous in the ocean^{1–3} and atmospheres.^{4–6} Within this broad class of phenomena, we focus on vortex dipoles (i.e., pairs of oppositesign vortices) and their longterm evolution. Dipolar couples transporting fluid inside their cores across large distances are widespread at the ocean surface,^{7,8} stimulating growing interest in the underlying vortex dynamics.^{9–11}
A classical example exhibiting steady propagation is the Lamb–Chaplygin dipole (hereafter, LCD),^{12,13} which is a solution of the twodimensional (2D) Euler equations. As indicated in Ref. 14, Sec. 4.3, an LCD with a circular core tends to slowly evolve toward a state with smoother vorticity distribution in a slightly elliptical core. While no instability was detected in the inviscid discretized version of the LCD with circular separatrix,^{15} the authors noted that the lowest vorticity levels (not represented in their isovortical discretization) might be rapidly stripped from the configuration through the neighborhood of the rear stagnation point. Indeed, linearly unstable modes were found when explicit viscosity was included in the system.^{16}
Building from this, a 2D stationary solution (socalled modon) on the βplane (taking into account planetary rotation and sphericity) was obtained because Stern^{17} was found to be unstable.^{18} Furthermore, a more general solution of the classical quasigeostrophic (QG) equivalentbarotropic model (given by rotating toplayer shallowwater dynamics with deformable lower interface above the motionless deep layer) for propagating dipoles with a circular separatrix was obtained by Larichev and Reznik (hereafter, LRD).^{19}
When a finite internal Rossby radius of deformation is assumed, the LRD solutions are capable of zonal drift in both eastward and westward directions. The eastward propagation speed is unbounded, whereas the propagation speed to the west must exceed the maximum Rossby wave phase speed^{20} to prevent the vortex losing energy to generated waves.^{21} In the past, westward propagating LRDs were suggested as a paradigm for atmospheric blocking,^{22,23} but were later found unstable,^{24} hence not suitable for this due to rapid destruction. On the other hand, eastward propagating LRDs were found to be remarkably robust in numerical simulations, even in the presence of weak friction,^{25} shortwave perturbations,^{26} and topographic perturbations.^{27} Proof of their stability, however, remains evasive.^{24,28,29}
Recently, the phenomenon of spontaneous symmetry breaking was discovered in eastward weakintensity LRDs leading to their ultimate destruction.^{30} The corresponding unstable growing perturbation has even symmetry about the zonal axis (similar to the unstable modes obtained for the evolving viscous LCD^{16}) despite the odd flow symmetry of the LRD. A normal mode representation of this growing perturbation accurately replicates the time evolution, confirming that the eastward LRD is linearly unstable. This growing mode was initially referred to as the Amode,^{30} in accord with some earlier notation for a similar mode; however, in this study, we refer to it as the Davies mode (Dmode), to emphasize its discovery and differentiate it from the Amode for the LCD when viscosity is imposed in the system.
The effect of dipole tilt relative to the zonal direction was examined in Ref. 31, where two regimes were reported. For weak dipoles with a large tilt, the dipole rapidly disintegrated. However, for most of the cases studied, the dipole experienced damped oscillations along the zonal axis before adjusting to new eastward propagating steady states. Here, we include the effects of dipole tilt using the numerical framework of Ref. 30. We argue that the instabilities were not captured in Ref. 31 due to computational constraints on the spatial resolution of the simulations.
The structure of this paper is the following: in Sec. II, we formulate the nondimensional model, derive the tilted dipole state, and briefly introduce an asymptotic theory for strong dipoles; in Sec. III, we detail our findings for both strong and weak dipoles; and in Sec. IV, we summarize and discuss the results.
II. MODEL DESCRIPTION
A. Equivalentbarotropic framework
Explicit Newtonian viscosity can be incorporated into the equivalentbarotropic model by setting the lefthand side of Eq. (7) equal to $ R e \u2212 1 \u2207 4 \psi $, where $ R e = ( L \u0302 U \u0302 ) \u2212 1 \nu \u0302$ is the Reynolds number, and $ \nu \u0302$ denotes eddy viscosity. However, since we are interested in solutions to the inviscid problem (7), we make use of very small implicit numerical viscosity in our simulations, which we discuss later. This decision is also motivated by the results of the sensitivity analysis carried out in Ref. 30, where consistency is found between simulations using explicit Newtonian viscosity and implicit numerical viscosity.
B. LRD initialization
In this work, we investigate tilted dipoles for S = 1, i.e., L = R_{d}, launched at angles $ 0 < \alpha 0 \u2264 90 \xb0$ to the zonal axis. As a consequence of this, q_{A} is nonzero and the tilted LRDs do not satisfy (7), i.e., the dipoles we consider are no longer steady states. Note that in this case, the A and Scomponents are computed about the zonal coordinate axis, rather than the dipole translation axis, which are equivalent only for the eastward LRD. Figure 1 contains visualizations of the A and Scomponents and will help the readers to understand the story, as it evolves. Hereafter, the Acomponent as depicted in Fig. 1(a) is referred to as the tilted mode (Tmode).
In the (eastward propagating) case $ \alpha 0 = 0 \xb0$ and for dipoles with $ \beta / c \u2265 2$, the component q_{A} (i.e., Dmode) was initially zero but appeared due to spontaneous symmetry breaking and shown to oscillate and grow exponentially over time.^{30} This growth caused the dipole to develop asymmetries through elongation and compression of the vortex pair (Fig. 2). The ultimate result was the destruction of the dipole, with the vortex partners decelerating and drifting apart, before propagating in the (opposite) westward direction. At later times, they continued drifting apart and eventually disintegrated into the background flow as in Ref. 34. A combination of dynamical evolution associated with Tmode depicted in Fig. 1(a) and Dmode (Fig. 2) is expected to be seen in the case of $ \alpha 0 \u2260 0 \xb0$.
C. Kinematics of tilted dipoles
III. NUMERICAL EXPERIMENTS
A. Methodology
To verify our results, we made use of two distinctly different numerical models. For consistency with Ref. 30, we use finite differences with the CABARET advection scheme; the algorithm is described in detail in Ref. 37, and the numerical convergence properties of the model in relevant flow regimes with rich mesoscale dynamics are discussed in Refs. 38 and 39.
For consistency with Ref. 31 and to validate our results, we also performed simulations using the Dedalus Python package, which combines a pseudospectral approach in space with a fourthorder implicit–explicit Runge–Kutta scheme for time integration.^{40} This approach necessarily includes explicit dissipation represented through a hyperdiffusion term, whereas the CABARET approach has implicit numerical diffusion, which is minimized at each time step through the advection scheme.
We estimated a numerical viscosity value corresponding to dissipation in our CABARET simulations and used it to set equivalent hyperdiffusion parameter in the pseudospectral model. We found that both methods yield similar solutions (see the supplementary material); therefore, we primarily present CABARET results, though supplementary animations in Fig. 3 are obtained with the Dedalus package.
B. Overview of results
The dipole evolution is analyzed by considering both the trajectory of the dipole center (x_{c}, y_{c}) and changes in the internal dipole structure, as characterized by the propagation speed, V, and the distance between extrema, D, defined in Sec. II C. The coordinates of the positive and negative dipole extrema (X_{1}, Y _{1}) and (X_{2}, Y _{2}) are evaluated using local 2Dparabolic interpolation.
Comparing with the kinematic theory, we found new behavior in the evolution of dipoles with $ \beta / c = 1$ (strong dipoles) and with $ \beta / c \u2265 2$ (weak dipoles). These modifications to the dipole dynamics are mostly related to the growing Dmode on top of decaying Tmode, which become clearly visible when the vortex crosses the zonal axis.
Strong dipoles adjust themselves along the zonal axis and appear to continue eastward propagation. Nevertheless, over the course of this adjustment phase, one can see moderate increase in partner separation and associated deceleration in the upper panels in Fig. 4 and Table I. Note that the originally Dmode was described only for weak eastward LRD solution.^{30} Here, growing Dmode is revealed also for the strong dipoles though they are not destroyed during the time of integration and their structure still resembles the initial LRD.
α_{0} .  $ D ( t ) / D 0 \u2212 1$ .  $ V ( t ) / c \u2212 1$ . 

$ 5 \xb0$  (0.5, 1.1)%  −(1.0, 3.0)% 
$ 30 \xb0$  (1.0, 6.0)%  −(5.2, 20.1)% 
$ 45 \xb0$  (0.71, 5.1)%  −(7.1, 21.0)% 
$ 60 \xb0$  (0.14, 1.5)%  −(6.7, 12.2)% 
$ 90 \xb0$  (1.7, 3.2)%  −(16.7, 25.5)% 
α_{0} .  $ D ( t ) / D 0 \u2212 1$ .  $ V ( t ) / c \u2212 1$ . 

$ 5 \xb0$  (0.5, 1.1)%  −(1.0, 3.0)% 
$ 30 \xb0$  (1.0, 6.0)%  −(5.2, 20.1)% 
$ 45 \xb0$  (0.71, 5.1)%  −(7.1, 21.0)% 
$ 60 \xb0$  (0.14, 1.5)%  −(6.7, 12.2)% 
$ 90 \xb0$  (1.7, 3.2)%  −(16.7, 25.5)% 
Weak dipoles for $ \alpha \u2264 45 \xb0$ decelerate with partners separating by more than 30% and a reversal in the direction of zonal propagation (lower panels in Fig. 4), similar to the effects of a growing Dmode in the eastward LRD.^{30} For strong and weak dipoles with a range of tilts, our findings are summarized in Fig. 5, where there are both similarities with and notable differences from Ref. 31.
C. Strong dipole evolution
Figure 6 shows the evolution of the meridional position of the dipole center, y_{c}, for the case of strong dipoles. Decaying oscillations can clearly be seen. It should be noted that these oscillations persist for all cases studied—see the supplementary material for the case of weak dipoles.
In the case of strong dipoles with $ \alpha 0 = 5 \xb0$, the modeled dipole trajectory is compared with the kinematic theory trajectory (Fig. 7), where we used $ \lambda = 0.2$ and normalized $ \Delta X \u223c D \u2009 sin \u2009 \alpha $ and y_{c} with $ D 0 \u2009 sin \u2009 \alpha $ and Y_{m} [as defined in Eq. (17)], respectively. The similarity between numerical and predicted trajectories suggests that the theory is valid within the strong dipole regime for small initial tilts as the vortex structure remains approximately unchanged throughout the evolution. However, significant deviations in the numerical trajectory in Fig. 7 become visible when $ t \u2265 300$, suggesting that a transition in dynamics results in a discrepancy with the kinematic theory to be investigated further.
By t = 400, the vortex has undergone a small deceleration and an increased partner separation of $ V / c \u2212 1 \u223c \u2212 3 %$ and $ D / D 0 \u2212 1 \u223c 1.1 %$, respectively. These alterations are consistent with the dynamics of symmetric eastward drifting LRDs studied in Ref. 30, which continued translating as a steady state, with deceleration and small partner separation resulting from implicit numerical viscosity.
To better understand the spiral deviations in Fig. 7, we consider times $ t \u2113$ corresponding to intersections of the dipole trajectory with the zonal axis [i.e., $ y c ( t \u2113 ) = 0$, with $ \u2113 = 1 , 2 , 3 \u2026$]. Eight intersections are observed in Fig. 6 for $ \alpha 0 = 5 \xb0 , 30 \xb0 , 60 \xb0$ over the time of integration $ 0 \u2264 t \u2264 400$ and seven intersections for $ \alpha 0 = 90 \xb0$. The simple tilted mode (or Tmode) structure of the Acomponent at initialization [see Fig. 1(a)] is assumed to remain at each intersection according to the kinematic theory, while the amplitude of the Acomponent, $ q A \u223c \u2009 sin \u2009 \alpha $, is expected to decay at subsequent crossings together with the angle $ \alpha ( t \u2113 )$, similar to $  \Delta X ( t \u2113 )  / D \u2009 sin \u2009 \alpha 0$ in Fig. 7. To evaluate the predictions of the kinematic dipole theory for strong dipoles, we compare the predicted amplitude decay with the values of $ q m ( t \u2113 )$ in our solutions for different values of α_{0}.
Values of $ q m ( t \u2113 )$ are plotted in Fig. 8 for different initial angles and correspond well to the theoretical Tmode prediction of $ \alpha ( t \u2113 )$ for the earlier crossings. However, a significant increase in the value of q_{m} can be seen in Fig. 8 at $ \u2113 = 7 , 8$ (t > 300) for $ \alpha 0 = 5 \xb0$ and $ \alpha 0 = 30 \xb0$. Such increases indicate the dominance of a growing Dmode (compare with Fig. 2) that becomes clearly visible at the final two crossings (see the supplementary material and Fig. 9). This earlier Dmode appearance in comparison with strong nontilted eastward LRDs in Ref. 30 indicates that the dipole perturbations due to tilt play a catalytic role and favor the instability, which appears only in weak eastward LRDs over the time of integration.
For an initial launch of $ \alpha 0 = 60 \xb0$, the growing Dmode is not seen; instead, we observed a monotonic and approximately linear decrease in $ q m ( t \u2113 )$ at consecutive values of $\u2113$ (see Fig. 8) and find that the Acomponent evolves as a decaying Tmode of alternating sign (see Fig. 10). However, at later times, one can see meridional splitting of the Tmode [Fig. 10(d)], reflecting another mechanism to cause the separation between partners, though the apparent partner separation appears less than that caused by the growing Dmode for smaller tilt $ \alpha 0 = 45 \xb0$ [see Fig. 4(b)]. Such adjustment of the dipole to an elliptical (meridionally elongated) shape is accompanied by a decrease in the deceleration magnitude for t > 250. Further investigations in substantially longer domains, over much longer times, and with much higher resolution (to suppress implicit viscosity), go beyond the scope of this paper.
Similar Tmode persistence with alternating sign value is observed in the dynamics when $ \alpha 0 = 90 \xb0$ (see the supplementary material). However, despite approximately linear depreciation in q_{m} up until $ \u2113 = 5$, this decay becomes significantly weaker at $ \u2113 = 6 , 7$ (see Fig. 8). Even though this is still a monotonically decreasing pattern, the change in decay suggests that there might be some transition in dynamics to be investigated further.
In summary, our analysis for strong dipoles with $ \beta = c$ revealed two distinct scenarios, which are as follows:

The development of a slowly growing Dmode when initial tilt is small, i.e., $ 0 < \alpha 0 \u2264 45 \xb0$,

The gradual splitting of a decaying Tmode (without the appearance of a Dmode) when the initial tilt is large, i.e., $ 45 < \alpha 0 \u2264 90$.
Further investigations are needed over a much larger interval of time to clarify the fate of D and Tmodes in strong dipoles.
D. Slow destruction of weak dipoles
Our numerical simulations indicate that weak dipoles released at an angle $ \alpha 0 \u2208 [ 5 \xb0 , 90 \xb0 ]$ initially follow oscillatory paths that decay over time (see the supplementary material), comparable to the case of strong dipoles. In Ref. 31, weak dipoles only disintegrate at large α_{0} (corresponding to $ \alpha 0 = 90 \xb0$ when $ \beta = 0.2$ and $ \alpha 0 = 60 \xb0 , 90 \xb0$ when $ \beta = 0.35$), otherwise orienting themselves along the zonal axis, displaying approximate steady translation in the eastward direction. Given the destruction of purely eastward weak dipoles was attributed to the growth of the Dmode in Ref. 30, we anticipate this anomaly to develop on an initially tilted dipole after the adjustment phase.
For weak dipoles with $ ( \alpha 0 , \beta ) = ( [ 5 \xb0 , 30 \xb0 , 45 \xb0 ] , 2 c )$, we find a phase transition in q_{A} at intersections with the zonal axis (y_{c} = 0). In particular, when $ \alpha 0 = 5 \xb0$, the Tmode at zonal axis crossings rapidly evolves into a growing oscillatory Dmode (see Fig. 11). Consequently, the eddy partners are driven much further apart than for strong dipoles, with $ D / D 0 \u2212 1 \u223c 6.3 % , 36.6 %$ at t = 200, 400, respectively, [compare Figs. 4(a) and 4(c)]. This is also reflected in their deceleration, where $ V / c \u2212 1 \u223c \u2212 21.6 % , \u2212 89.2 %$ when t = 200, 400, which is consistent with the dynamics of eastward LRDs with the growing Dmode.^{30}
Observed destruction of tilted weak dipoles is consistent with previous numerical simulations using a barotropic model ( $ L \u226a R d$) (Ref. 14, Sec. 4.2.2). The similar weakintensity dipole ( $ \beta / c = 2$), launched at $ \alpha 0 = 5 \xb0$, relaxed along the zonal axis and proceeded to disintegrate. This disintegration was attributed to the filamentation process and the emission of Rossby waves, suppressed by the use of a cutting filter in the far field, which encouraged drastic deceleration and allowed the separation distance between partners to grow.
In our simulations, similar results are obtained when $ \alpha 0 = 30 , 45 \xb0$ (see Fig. 12), though with longer Tmode persistence before Dmode formation. The dipole partners experience increased separation and greater dipole deceleration [compare Figs. 4(c) and 4(d)], while the zonal dipole drift becomes westward at $ t S \u223c 400$. Therefore, even though such oppositesign couples remain selfadvecting eastward until t = 400, the amplifying Dmode will inevitably disintegrate the dipole in longer simulations.
For even weaker dipoles with $ \beta = 0.35$, a growing Dmode emerged for $ \alpha 0 = 5 \xb0 , 30 \xb0$, which results in the subsequent disintegration of the dipole for t < 200 (see the supplementary material and Fig. 3). Such disintegration is consistent with Ref. 30; however, our results throughout Sec. III D differ from those obtained in Ref. 31, where adjustment to a seemingly steady eastward propagating state was observed for $ 0 \u2264 t \u2264 400$. We believe that these discrepancies are due to the 16 times coarser numerical resolution adopted in their 30 years old study, which meant they were unable to capture the instability of weak eastward propagating dipoles.
To summarize our findings in this subsection, we have observed the following:

Weak dipoles developed a Dmode over time when $ 0 < \alpha 0 < 45 \xb0$ (and for $ \alpha 0 = 45 \xb0$ when $ \beta = 2 c )$.

A growing Dmode decelerates the weak dipoles and increases the core distance between the oppositesign pair, encouraging the spontaneous symmetry breaking phenomena, as found for symmetric weak eastward LRDs.^{30}

$ \beta = 3.5 c$ dipoles completely disintegrate within the time interval of $ 0 \u2264 t \u2264 400$, whereas $ \beta = 2 c$ dipoles begin to propagate westward much later and would completely disintegrate if simulated for longer time.
E. Adjustment or fast destruction of weak dipoles
Weak dipoles with $ ( \alpha 0 , \beta ) = ( 60 \xb0 , 2 c )$ have q_{A} evolve as a decaying Tmode (see Fig. 13) comparable to what we observed with strong dipoles (see Fig. 10). However, a notable difference is that $ D / D 0 \u2212 1 \u223c 62.2 %$ at t = 200, corresponding to a much greater separation than observed for $ \beta = c$. This deformation is similar to the meridional separatrix stretching captured in simulations for dipoles with initially circular separatrix in Ref. 14. Despite the large meridional separation at early times, this elliptical deformation does not change significantly in the range of $ 200 \u2264 t \u2264 400$ (see Table II), and these small alterations are likely due to implicit numerical viscosity.^{41} Furthermore, the much lower deceleration experienced compared to cases with smaller values of α_{0} suggests that the dipole transforms to a noncircular steady state. This agrees with the steady adjustment discussed in Ref. 31 and the theoretical predictions for strong dipoles in Ref. 36, though these studies did not discuss elliptical deformation.
α_{0} .  $ D ( t ) / D 0 \u2212 1$ .  $ V ( t ) / c \u2212 1$ . 

$ 5 \xb0$  (6.3, 36.6)%  −(21.6, 89.2)% 
$ 30 \xb0$  (18.1, 37.8)%  −(60.5, 100.1)% 
$ 45 \xb0$  (14.9, 39.8)%  −(52.4, 102.4)% 
$ 60 \xb0$  (62.2, 65.3)%  −(58.8, 67.0)% 
$ 90 \xb0$  (50.1, 300)%  −(−99.1, 57.4)% 
α_{0} .  $ D ( t ) / D 0 \u2212 1$ .  $ V ( t ) / c \u2212 1$ . 

$ 5 \xb0$  (6.3, 36.6)%  −(21.6, 89.2)% 
$ 30 \xb0$  (18.1, 37.8)%  −(60.5, 100.1)% 
$ 45 \xb0$  (14.9, 39.8)%  −(52.4, 102.4)% 
$ 60 \xb0$  (62.2, 65.3)%  −(58.8, 67.0)% 
$ 90 \xb0$  (50.1, 300)%  −(−99.1, 57.4)% 
Weak dipoles with $ ( \alpha 0 , \beta ) = ( 90 \xb0 , 2 c )$ disintegrate in the interval of $ 0 \u2264 t \u2264 280$ as observed in Ref. 31. Indeed, a decaying Tmode similar to that seen when $ \alpha 0 = 60 \xb0$ elucidates the dipole dynamics for this parameter regime where the values of D and V rapidly increase over time (see the supplementary material). This destruction occurs much earlier than that achieved by a Dmode and is due to significant Rossby wave radiation. More specifically, the trajectory of the dipole exposes the vortex to interaction with the trailing Rossby waves, which enhances the rapid disintegration.
Finally, when $ \alpha 0 \u2265 45 \xb0$ for the weaker $ \beta = 3.5 c$ dipole, a Dmode does not appear over the considered time interval, and instead, we see components of decaying Tmode decelerate and drift apart, before the disintegration of the dipole. Since a Dmode appears when $ \alpha 0 = 45 \xb0$ and $ \beta = 2 c$, this suggests the existence of some bifurcation point in $ ( \alpha 0 , \beta )$space, where there is a transition between Dmode and Tmode dominance. To confirm this, more research is needed. Finally, in the extreme case when $ \alpha 0 = 90 \xb0$, weak dipoles with $ \beta = 3.5 c$ disintegrate much faster than for other initial tilts, as in Refs. 31 and 34.
To summarize, we obtained the following results:

q_{A} evolves as a decaying Tmode rather than a growing Dmode for weak dipoles when $ \alpha 0 > 45 \xb0$ (and for $ \alpha 0 = 45 \xb0$ when $ \beta = 3.5 c$),

Weak dipoles with $ \beta = 3.5 c$ rapidly disintegrate when $ \alpha 0 \u2265 45 \xb0$, with the dipole lifespan shortening as α_{0} increases,

When $ \beta = 2 c$ and $ \alpha 0 = 90 \xb0$, the dipole disintegrates much faster than with a growing Dmode, as a consequence of Rossby wave radiation,

When $ \beta = 2 c$ and $ 45 \xb0 < \alpha 0 < 90 \xb0$, elliptic deformation of the separatrix and large enstrophy loss inhibit Dmode development and drive the adjustment to approximate steady propagation along the zonal axis.
IV. CONCLUSIONS AND DISCUSSION
Motivated by the ubiquity of isolated coherent vortices in geophysical (i.e., rotating and stratified) fluids, we investigated the dynamics of mesoscale vortex dipoles launched with different northeastward tilts to the zonal direction. We considered an idealized equivalentbarotropic QG model in an oceanic configuration and numerically simulated the evolution of individual LRDs^{19} for a physically relevant range of parameters, focusing both on transient and longtime behaviors. While motivated by the important work of our forerunner (Ref. 31), we were able to extend their analysis using more advanced numerical schemes and techniques. These tools allowed us to reexamine their results and discover new dynamical behaviors, thus enriching our understanding of the longterm behavior of coherent vortices. In particular, we found dynamical sensitivities to both the initialtilt angle, α_{0}, and the initial intensity of the dipole, $ \beta / c$.
Even strong dipoles (i.e., $ \beta = c$; here, nondimensional planetary vorticity gradient and dipole speed, respectively) for moderate initial tilts (i.e., $ \alpha 0 \u2264 45 \xb0$) eventually developed a critical Dmode instability,^{30} resulting in essential deviations from the predictions of the kinematic strongdipole theory in Ref. 35. This theory allows only for an alternating sign Tmode with the initial profile of q_{A}, as depicted in Fig. 1(a), and decaying with time as in Figs. 7 and 8 with damping suggested in Ref. 36. For large tilt (i.e., $ \alpha 0 > 45 \xb0$), our results are more aligned with the kinematic theory, while displaying meridional separation of dipole partners (illustrated by a splitting of the Tmode in Fig. 10) that accumulates over the adjustment period and prevents the above instability so that the Dmode did not develop for large tilt.
In the case of moderate initial tilt, weak dipoles experienced either a combination of significant partner separation with overall deceleration or complete destruction owing to fast developing of the Dmode. In the case of large initial tilts, more pronounced elliptical deformation of the dipole core—corresponding to meridional splitting of Tmode—was found to dominate over the Dmode, as the dipole underwent oscillations along the zonal axis until adjusting to steadystate eastward propagation. For extreme large tilts (e.g., $ \alpha 0 = 90 \xb0$) and/or very weak dipoles ( $ \beta = 3.5 c$), these structures disintegrated rapidly by radiating Rossby waves and proceeding to interact with them; leaving no time for the Dmode destruction mechanism.
Our numerical simulations employed very small implicit numerical viscosity in the background; however, the work presented in Ref. 41 showed that with explicit Newtonian viscosity (and β = 0), there exists an elliptical dipole structure that distinct circular dipole initializations converge toward as time progresses. These states are found to be unsteady, as characterized by decaying amplitude and increasing vortex size, which is broadly similar to our findings for large α_{0} and suggests that this adjustment would be stable in the inviscid limit (if implicit numerical viscosity was exactly zero).
In summary, we found and analyzed different dipole parameter regimes and reported evolution scenarios never previously discussed. All these cases may exist and coexist in nature, but they are likely to be influenced by other unaccounted physical processes, such as realistic largescale circulation, stratification, and topographic effects. Therefore, our results should act as a catalyst for further research.
SUPPLEMENTARY MATERIAL
See the supplementary material for animations of the dipole propagation and additional figures that support the results of this study.
ACKNOWLEDGMENTS
J.D. acknowledges support from the Roth scholarship, Imperial College London. G.S. acknowledges support from the U.S. National Science Foundation (No. NSFOCE1828843). P.B. was supported by the NERC Grant No. NE/T002220/1, the Leverhulme Trust Grant No. RPG2019024, and by the Moscow Centre for Fundamental and Applied Mathematics (supported by Agreement No. 0751520191624 with the Ministry of Education and Science of the Russian Federation).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Jack Davies: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead). Georgi Sutyrin: Conceptualization (equal); Methodology (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal). Matthew Crowe: Data curation (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Pavel Berloff: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this work are available from the corresponding author upon reasonable request.