Abstract
Recent studies of pseudo-plane ideal flow (PIF) reveal a ubiquitous presence of vortex alignment in both homogeneous and stratified fluids, and in both inertial and rotating reference frames as well. The exact solutions of a steady-state PIF model suggest that stagnation points tend to be vertically aligned and the concentric structure represents a fixed-point phenomenon of the Euler equations. Exception occurs in the rotating frame when a flow holds inertial period and skew center becomes possible. Properties of stagnation points based on Morse theory are obtained, leading to a topological explanation of vertical alignment via pressure Hessian. The study thus uncovers a new aspect of vortex behavior in ideal fluid that requires vortex center to align with the direction of gravity when vortex evolution reaches a laminar end state characterized by steady pseudo-plane velocities. Though the phenomenon arises from the constraint of the Euler equations, under specific conditions the topological theory is applicable to viscous fluid and explains the curvilinear tilting of von Kármán swirling vortex.
Highlights
Diagnoses of atmospheric and oceanic flows have revealed a basic state called Gravest Empirical Mode (GEM) in which a scalar property, such as temperature, has invariant vertical profile along a streamfunction contour (Sun and Watts 2001, Sun 2005)
The low-dimensional structure motivated Sun (2008) to develop a steady pseudo-plane ideal flow (PIF) model based on the notion that geostrophic flows are quasi-horizontal and stratified turbulence tends to collapse into a laminar end state with negligible vertical velocity
Numerical experiments show that geostrophic vortices in a freely evolving turbulence have an intriguing tendency toward vertical alignment (Polvani 1991, Nof and Dewar 1994, Viera 1995, McWilliams et al 1999)
Summary
Diagnoses of atmospheric and oceanic flows have revealed a basic state called Gravest Empirical Mode (GEM) in which a scalar property, such as temperature, has invariant vertical profile along a streamfunction contour (Sun and Watts 2001, Sun 2005). The low-dimensional structure motivated Sun (2008) to develop a steady pseudo-plane ideal flow (PIF) model based on the notion that geostrophic flows are quasi-horizontal and stratified turbulence tends to collapse into a laminar end state with negligible vertical velocity. Such pseudo-plane flows have vertically varying horizontal velocities but no vertical velocity (Saccomandi 1994). We intend to develop a topological theory for the concentric structure of the PIF solutions, and thereby provide a theoretical explanation for the vortex-alignment phenomenon As it turns out, the phenomenon stems from the Euler equations and has omnipresence beyond geophysical flows
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