The onset of filamentation on vorticity interfaces in two-dimensional Euler flows
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| Název: | The onset of filamentation on vorticity interfaces in two-dimensional Euler flows |
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| Autoři: | David G. Dritschel, Adrian Constantin, Pierre M. Germain |
| Přispěvatelé: | University of St Andrews.Marine Alliance for Science & Technology Scotland, University of St Andrews.Applied Mathematics |
| Zdroj: | Journal of Fluid Mechanics. 1008 |
| Publication Status: | Preprint |
| Informace o vydavateli: | Cambridge University Press (CUP), 2025. |
| Rok vydání: | 2025 |
| Témata: | MCC, Wave breaking, QC Physics, Contour dynamics, T-NDAS, Vortex dynamics, Physics - Fluid Dynamics, Mathematics - Dynamical Systems, QC |
| Popis: | Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation. |
| Druh dokumentu: | Article |
| Popis souboru: | application/pdf |
| Jazyk: | English |
| ISSN: | 1469-7645 0022-1120 |
| DOI: | 10.1017/jfm.2025.190 |
| Přístupová URL adresa: | http://arxiv.org/abs/2410.09610 https://hdl.handle.net/10023/31805 |
| Rights: | CC BY |
| Přístupové číslo: | edsair.doi.dedup.....f073784c63519b3492e7be131fcbadd9 |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation. |
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| ISSN: | 14697645 00221120 |
| DOI: | 10.1017/jfm.2025.190 |