On caustics associated with the linearized vorticity equation

Ivanova, Petya N.; Gorman, Arthur D.

doi:10.1023/a:1023265821269

Cited by 2 publications

(3 citation statements)

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“…2020). Moreover, the emergence of finite-time singularities in hydrodynamic flows is a problem in its own right in mathematical fluid dynamics (Ivanova & Gorman 1998).…”

Section: Introductionmentioning

confidence: 99%

“…In addition to their role in infection spreading they are expected to play a role in the formation of rain droplets (Falkovich, Fouxon & Stepanov 2002;Shaw 2003;Wilkinson, Mehlig & Bezuglyy 2006;Bec et al 2016;Ravichandran et al 2022) due to enhanced collision rates (Ravichandran & Govindarajan 2015), and hence for the modelling of precipitation in modern climate models (Ashwin et al 2012;Schneider et al 2017;Prabhakaran et al 2020). Moreover, the emergence of finite-time singularities in hydrodynamic flows is a problem in its own right in mathematical fluid dynamics (Ivanova & Gorman 1998).…”

Section: Introductionmentioning

confidence: 99%

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Caustics in turbulent aerosols: an excitable system approach

Barta

Vollmer²

2022

J. Fluid Mech.

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Aerosol particles are inertial particles. They cannot follow the surrounding fluid in a region of high vorticity. These encounters render the particle velocity field ${{\boldsymbol {v}}}$ locally compressible. Caustics occur if the trace of the gradient matrix $\textsf{$\boldsymbol{\sigma}$}$ of the field ${{\boldsymbol {v}}}$ locally diverges. For three-dimensional isotropic and homogeneous turbulence, the dynamics of the gradient matrix can be expressed in terms of three geometric invariants. In the present paper we establish a parametrisation of this problem where the dynamics takes the form of an excitable stochastic dynamical system with a three-dimensional phase space. The deterministic part of the dynamics is solved analytically. We show that the deterministic system has a globally attractive stable fixed point. Small noise induces excursions from the fixed point that typically relax straight back towards the fixed point. Caustics emerge as non-trivial return to a global fixed point when noise excites a trajectory across a stability threshold. The relaxation to the global fixed point will then involve at least one, and it may involve two or even three divergences of $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$ . Based on a combination of analytical insights and numerical analysis, we determine the rate of occurrence, duration and relative observation probability of caustic events in turbulent aerosols. Our analysis reveals that each approach towards a divergence proceeds along a straight line in the phase space of the dynamical system, which can help identify caustics. Moreover, there are infinite ways in which caustics can arise, namely whenever $\boldsymbol {Tr}\,\textsf{$\boldsymbol{\sigma}$}$ tends to $-\infty$ , so that no two caustics look the same.

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“…2020). Moreover, the emergence of finite-time singularities in hydrodynamic flows is a problem in its own right in mathematical fluid dynamics (Ivanova & Gorman 1998).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Caustics in turbulent aerosols: an excitable system approach

Barta

Vollmer²

2022

J. Fluid Mech.

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“…the Lagrange manifold formalism, as adapted to geophysical models [2,3], to study wave phenomena near caustics for those environments considered by Yang. Our purpose is to complement his results and to illustrate how the approach may be used to study the caustic curve itself.…”

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confidence: 99%

Caustic consideration of long planetary wave packet analysis in the continuously stratified ocean

Gorman

Yang

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The wave packet method, one form of the WKB technique, recently has been employed to investigate the evolution of long planetary wave packets in relation to the complex climate variability in the world oceans. However, such a method becomes invalid near the caustics. Here, the Lagrange manifold formalism is used to extend this analysis to include the caustic regions. We conclude that even though the wave packet method fails near the caustics, the equations derived from this method away from caustics are identical to the ones from the Lagrange manifold formalism near caustics.2000 Mathematics Subject Classification. Primary 76B65, 76M45, 86A05. Introduction.Recently, wave packet theory has been used by Yang [10,11] in two studies to analyze the evolution of long planetary wave packets. In the first study, the evolution of long planetary wave packets in a continuously stratified ocean with exponentially decaying stratification and mean zonal current was considered. The analysis successfully explained all major features of long planetary Rossby waves recently observed in the Topex/Poseidon satellite data in the global ocean. In the second study, the evolution of three-dimensional wave packets in a subtropical gyre was studied. This analysis led to a rudimentary theory for ocean climate variability on the inter-annual to decadal time scale observed in the sea surface height changes and the ocean temperature changes and provided insight of dynamical process of complex ocean climate variability. The ocean response may undergo transition between different regimes. The transition of regime may be one more reason that the observed climate variability in the ocean is so complicated with a variety of time scales between interannual and decadal. These predictions are consistent with observations in the North Pacific and other analytic and numerical model results. These results were also further discussed in [12]. While the focus of each investigation was the development of analytical solutions, wave packet theory enabled an analysis of the structural evolution of the wave packet associated with each setting [9].Wave packet theory is based on the WKB formalism, originally developed for water waves but primarily associated with quantum mechanics [4]. Near caustic or turning points the classical WKB approach is not valid [6], for example, physically in regimes, where the phase velocity of the wave packet coincides with the velocity of the large-scale current. A related approach that does apply in such regimes is the Lagrange manifold formalism developed by Maslov [7] and Arnol'd [1]. Here, we use

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On caustics associated with the linearized vorticity equation

Cited by 2 publications

References 8 publications

Caustics in turbulent aerosols: an excitable system approach

Caustics in turbulent aerosols: an excitable system approach

Caustic consideration of long planetary wave packet analysis in the continuously stratified ocean

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