Escape Probability, Mean Residence Time and Geophysical Fluid Particle Dynamics

TL;DR

The paper addresses stochastic transport of fluid particles in a randomly forced quasigeostrophic meandering jet by formulating escape probability $p(x,y)$ and mean residence time $u(x,y)$ through elliptic PDEs derived from a two-dimensional drift-diffusion model. It constructs a random jet model with a baseline stream function $\Psi(x,y)$ parameterized by $\beta$ and adds white noise of strength $\epsilon$, yielding particle dynamics that balance advection and diffusion. Numerical solutions on domains representing eddies and unit jet cores reveal $\beta$-dependent bifurcations in escape routes (e.g., between exterior retrograde flow and jet-core leakage, and between northern and southern recirculation) and distinct trends in residence times, with eddies and jet cores showing different maxima as functions of $\beta$. The results provide a quantitative framework for cross-regime geophysical transport under stochastic forcing, highlighting how diffusion interacts with structured mean flows to shape mixing in jet-dominated regimes like the Gulf Stream. The approach leverages Fokker-Planck-type PDEs to quantify first-exit and residence-time statistics in complex, geophysically relevant flows.

Abstract

Stochastic dynamical systems arise as models for fluid particle motion in geophysical flows with random velocity fields. Escape probability (from a fluid domain) and mean residence time (in a fluid domain) quantify fluid transport between flow regimes of different characteristic motion. We consider a quasigeostrophic meandering jet model with random perturbations. This jet is parameterized by the parameter $β= (2Ω)/r \cos (θ)$, where $Ω$ is the rotation rate of the earth, $r$ the earth's radius and $θ$ the latitude. Note that $Ω$ and $r$ are fixed, so $β$ is a monotonic decreasing function of the latitude. The unperturbed jet (for $0 < β< 2/3$) consists of a basic flow with attached eddies. With random perturbations, there is fluid exchange between regimes of different characteristic motion. We quantify the exchange by escape probability and mean residence time.

Figures (13)

• Figure 1: Unperturbed jet: $\epsilon =0$ and $\beta=1/3$.
• Figure 2: An eddy: $\beta=1/3$
• Figure 3: A unit jet core near a trough: $\beta=1/3$
• Figure 4: Escape probability of fluid particles (initially in an eddy in Figure \ref{['eddy']}) exiting into the exterior retrograde region: $\beta=1/3$
• Figure 5: Escape probability of fluid particles (initially in an eddy in Figure \ref{['eddy']}) exiting into the jet core: $\beta=1/3$