Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910150697156608 |
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| author | He, Jiyang Wang, Yan |
| author_facet | He, Jiyang Wang, Yan |
| contents | The final states of freely decaying two-dimensional (2D) topographic turbulence consist of a background flow and localized vortices. While the background flow satisfies a linear potential vorticity (PV)-streamfunction relation, the vortex structures remain poorly understood. To address this gap and ensure oceanic relevance, we examine quasi-stationary final states of 2D turbulence over a sinusoidal topography featuring a bump and a dip, where two oppositely signed vortices are locked to the topographic extrema. After subtracting the background flow, the vortices exhibit a "sinh"-like PV-streamfunction relation, as observed in flat-bottom turbulence. Motivated by Gaussian vortex profiles in flat-bottom turbulence, we propose an empirical model combining the background flow with Gaussian vortices centered at the topographic extrema. This model accurately reproduces quasi-stationary states and yields locally stationary solutions to the inviscid governing equation. We further test the model under complex topography and high-energy conditions, confirming that the "sinh"-like trend and Gaussian profiles are robust features of localized vortices. Linear stability analyses of these stationary vortex solutions reveal background flow-dependent stability: cyclone/elevation and anticyclone/depression configurations are stable at low background energy, while anticyclone/elevation and cyclone/depression configurations are stable at high background energy. These findings align with vortex-topography correlations observed in simulations across energy regimes. Our results provide explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence and elucidate the mechanism underlying vortex-topography correlations through stability analyses of vortices embedded in topographic background flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability He, Jiyang Wang, Yan Fluid Dynamics The final states of freely decaying two-dimensional (2D) topographic turbulence consist of a background flow and localized vortices. While the background flow satisfies a linear potential vorticity (PV)-streamfunction relation, the vortex structures remain poorly understood. To address this gap and ensure oceanic relevance, we examine quasi-stationary final states of 2D turbulence over a sinusoidal topography featuring a bump and a dip, where two oppositely signed vortices are locked to the topographic extrema. After subtracting the background flow, the vortices exhibit a "sinh"-like PV-streamfunction relation, as observed in flat-bottom turbulence. Motivated by Gaussian vortex profiles in flat-bottom turbulence, we propose an empirical model combining the background flow with Gaussian vortices centered at the topographic extrema. This model accurately reproduces quasi-stationary states and yields locally stationary solutions to the inviscid governing equation. We further test the model under complex topography and high-energy conditions, confirming that the "sinh"-like trend and Gaussian profiles are robust features of localized vortices. Linear stability analyses of these stationary vortex solutions reveal background flow-dependent stability: cyclone/elevation and anticyclone/depression configurations are stable at low background energy, while anticyclone/elevation and cyclone/depression configurations are stable at high background energy. These findings align with vortex-topography correlations observed in simulations across energy regimes. Our results provide explicit vortex solutions for quasi-stationary final states of 2D topographic turbulence and elucidate the mechanism underlying vortex-topography correlations through stability analyses of vortices embedded in topographic background flows. |
| title | Final states of two-dimensional turbulence above large-scale topography: stationary vortex solutions and barotropic stability |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2507.00691 |