Abstract:We examine the equilibrium forms, linear stability and nonlinear evolution of two patches having oppositesigned, uniform potential vorticity anomalies in a single-layer shallow-water flow, under the quasi-geostrophic approximation. We widely vary the vortex area ratio, the potential vorticity anomaly ratio, as well as the Rossby deformation length to unravel the full complexity of possible interactions in this system. Oppositesigned vortex interactions turn out to be far richer than their like-signed counterpa… Show more
“…Supplementary movie 1 available at shows the full evolution of the modon for the case depicted in figure 5. A similar breakdown due to asymmetric instability is observed by Makarov & Kizner (2011) and Jalali & Dritschel (2020) for the case of oppositely signed vortex patches. Figure 5.Plots of the surface buoyancy for and ( a ) , ( b ) , ( c ) , ( d ) , ( e ) , and ( f ) .…”
Section: Vortex Breakdownsupporting
confidence: 82%
“…Supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.607 shows the full evolution of the modon for the case λ = 0.8 depicted in figure 5. A similar breakdown due to asymmetric instability is observed by Makarov & Kizner (2011) and Jalali & Dritschel (2020) for the case of oppositely signed vortex patches. Snyder et al (2007) considered the evolution of SQG modons in a primitive equation model and observed that waves could form within the modon.…”
This work discusses modons, or dipolar vortices, propagating along sloping topography. Two different regimes exist, which are studied separately using the surface quasi-geostrophic equations. First, when the modon propagates in the direction opposite to topographic Rossby waves, steady solutions exist and a semi-analytical method is presented for calculating these solutions. Second, when the modon propagates in the same direction as the Rossby waves, a wave wake is generated. This wake removes energy from the modon, causing it to decay slowly. Asymptotic predictions are presented for this decay and found to agree closely with numerical simulations. Over long times, decaying vortices are found to break down due to an asymmetry resulting from the generation of waves inside the vortex. A monopolar vortex moving along a wall is shown to behave in a similar way to a dipole, though the presence of the wall is found to stabilise the vortex and prevent the long-time breakdown. The problem is equivalent mathematically to a dipolar vortex moving along a density front, hence our results apply directly to this case.
“…Supplementary movie 1 available at shows the full evolution of the modon for the case depicted in figure 5. A similar breakdown due to asymmetric instability is observed by Makarov & Kizner (2011) and Jalali & Dritschel (2020) for the case of oppositely signed vortex patches. Figure 5.Plots of the surface buoyancy for and ( a ) , ( b ) , ( c ) , ( d ) , ( e ) , and ( f ) .…”
Section: Vortex Breakdownsupporting
confidence: 82%
“…Supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.607 shows the full evolution of the modon for the case λ = 0.8 depicted in figure 5. A similar breakdown due to asymmetric instability is observed by Makarov & Kizner (2011) and Jalali & Dritschel (2020) for the case of oppositely signed vortex patches. Snyder et al (2007) considered the evolution of SQG modons in a primitive equation model and observed that waves could form within the modon.…”
This work discusses modons, or dipolar vortices, propagating along sloping topography. Two different regimes exist, which are studied separately using the surface quasi-geostrophic equations. First, when the modon propagates in the direction opposite to topographic Rossby waves, steady solutions exist and a semi-analytical method is presented for calculating these solutions. Second, when the modon propagates in the same direction as the Rossby waves, a wave wake is generated. This wake removes energy from the modon, causing it to decay slowly. Asymptotic predictions are presented for this decay and found to agree closely with numerical simulations. Over long times, decaying vortices are found to break down due to an asymmetry resulting from the generation of waves inside the vortex. A monopolar vortex moving along a wall is shown to behave in a similar way to a dipole, though the presence of the wall is found to stabilise the vortex and prevent the long-time breakdown. The problem is equivalent mathematically to a dipolar vortex moving along a density front, hence our results apply directly to this case.
“…For QGSW equations, there are some numerical stability results [30,23], but there seem to be very few results on mathematical stability. However, due to the special energy characteristics of our constructed solution, we can prove its orbital stability.…”
In this paper, we construct a family of traveling wave vortex pairs with specific forms like Lamb Dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The solutions are obtained by maximization of a penalized energy with multiple constraints. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Based on the vorticity method, we prove the orbital stability of the counter-rotating vortex pairs for the QGSW equations.
“…Most previous studies have focused on symmetric merging of two identical vortices or, to a lesser extent, asymmetric merging with a few examples of different strength ratios, see, among others, Melander, Zabusky & McWilliams (1988), Meunier et al (2002) and Dritschel (1995). In two recent studies by Jalali & Dritschel (2018, 2020 the general inviscid interactions of vortex patches are studied with many examples over a wide parameter space, including the ratio of sizes and vorticity. Our study is not the first that attempts to describe all interaction scenarios in terms of different flow regimes.…”
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