Rotation Prevents Finite-Time Breakdown

Abstract

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, $U_t+U\cdot {\nabla }_{x}U=2k{U}^{\perp }$, with a fixed $2k$ being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, $U^{\perp }$, prevents the generic finite time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite time breakdown of solutions may and actually does occur. Thus, global regularity depends on whether the initial configuration crosses an intrinsic, $\mathcal{O}(1)$ critical threshold, which is quantified in terms of the initial vorticity, ${\omega }_0=\nabla \times U_0$, and the initial spectral gap associated with the $2\times 2$ initial velocity gradient, ${\eta }_0:={\lambda }_2(0)-{\lambda }_1(0),{\lambda }_j(0)={\lambda }_j(\nabla U_0)$. Specifically, global regularity of the rotational Euler equation is ensured if and only if $4k{\omega }_0(\alpha )+{\eta }_0^2(\alpha )<4k^2,\forall \alpha \in {\mathbb{R}}^2$ . We also prove that the velocity field remains smooth if and only if it is periodic. We observe yet another remarkable periodic behavior exhibited by the {\em gradient} of the velocity field. The spectral dynamics of the Eulerian formulation reveals that the vorticity and the eigenvalues (and hence the divergence) of the flow evolve with their own path-dependent period. We conclude with a kinetic formulation of the rotating Euler equation.

References in corpus (4)

• Spectral Dynamics of the Velocity Gradient Field in Restricted Flows
• Near identity transformations for the Navier-Stokes equations
• Critical Thresholds in 2D Restricted Euler-Poisson Equations
• Transport in Rotating Fluids

Cited by in corpus (15)

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• Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty
• The influence of an external magnetic field on cold plasma oscillations
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• Existence of radial global smooth solutions to the pressureless Euler-Poisson equations with quadratic confinement
• Development of singularities for the compressible Euler equations with external force in several dimensions
• Singularity formation for rotational gas dynamics
• Well-posedness Properties for a Stochastic Rotating Shallow Water Model
• Eulerian dynamics in multi-dimensions with radial symmetry
• On plane oscillations of the cold plasma in a constant magnetic field
• Critical thresholds in the Euler-Poisson-alignment system
• Formation of singularity for the rotating shallow water system
• Nonlinear stability of two-dimensional axisymmetric vortices in compressible inviscid medium in a rotating reference frame
• Sharp critical thresholds in a hyperbolic system with relaxation
• Critical thresholds in a nonlocal Euler system with relaxation