The effect of changing the Coriolis force gradient parameter on the escape probability and mean residence time☆

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Abstract

We investigate a quasigeostrophic vortex flow with random perturbations. The system we will look at is rotating and has a gradient in the Coriolis force (the force due to the earth's rotation). When these two characteristics are present, the conserved quantity is the potential vorticity q=∇2ψ+βy. Thus, the dynamics are determined by the quasigeostrophic equation (∂t+v→·∇)q=0. In this, the parameter β is proportional to the Coriolis force gradient. Whenever we have a system with random perturbations, we may talk of the escape probability of fluid particles crossing a portion of the boundary for a fluid domain, and the mean residence time of fluid particles inside a fluid domain. The goal of this paper is to determine the relationship between the parameter β and the escape probability and the mean residence time. We will look at a vortex of the flow and determine the escape probability crossing the upper and lower boundaries of the vortex of a particle (we assume that the particles are uniformly distributed in the vortex). We will also calculate the mean residence time for a particle inside a vortex. We find that as β increases, the escape probability for a particle crossing the lower boundary decreases. We also find that as β increases the mean residence time of a particle inside a vortex decreases.

Introduction

In recent years, many models have been proposed to model the movement of particles in oceanic flows. Tracking the movement of the particles within the flows is an area that has generated much interest. We would like to be able to determine where the particles will end up in the long run. In order to do this, we must look at the flow from a probabilistic point of view. We are then able to generate partial differential equations that will enable us to track the movement of the particles. In this paper, we look at a specific model of a flow in the presence of vortices that has a parameter, β. We would like to determine the relationship between the parameter and the movement of the particles.

Section snippets

Preliminary material

Before we begin, we must introduce some definitions and equations that we will use. The quantities that we are concerned with calculating are the escape probability, average escape probability, and mean exit time. We will look at a stochastic dynamical system of the general formẋ=a1(x,y)+b1(x,y)ẇ1,ẏ=a2(x,y)+b2(x,y)ẇ2,where w1(t),w2(t) are two real independent Wiener processes (then ẇ1 and ẇ2 are white noises) and a1,a2,b1,b2 are given deterministic (non-random) functions. In these

The parameter β

We will look at rotating fluid flow. Assume that the flow is two-dimensional and incompressible. Under these conditions, we know that the Lagrangian trajectories of the particles are given by the solution of the equations of motiondxdt=−∂ψ∂y,dydt=∂ψ∂x,where ψ(x,y) is the streamfunction defined by v→=(−∂yψ,∂xψ), the velocity vector. We will take positive x to be pointing east and positive y to be pointing north. We will concentrate on a quasigeostrophic flow. This is a flow that is a

The streamfunction and the equations of motion

In order to get our equations of motion for a shear flow with a chain of vortices, we need to define our streamfunction ψ(x,y). Since the streamfunction is a composition of simpler functions (and variables), we define them separately to use their name in ψ(x,y). First define the variable χ by the relationcos(χ)=334|β|.Now define c and kc=s23cosχ+π3,k=1−c2.For this model to make sense, 0<β<4/33≈0.7698. The variable s corresponds to which way the fluid is flowing. We choose s=−1, which

Escape probability

Now we are concerned with the probability that a particle will escape a vortex. Thus, we take our domain Ω to be the vortex (as shown in Fig. 1). We (arbitrarily) take our subboundary Γ to be either the top or bottom boundary of the vortex. We choose β to take on the values: 0.0, 0.15, 0.30, 0.45, 0.60 and 0.75 . When the value of β was specified, we calculated our equations of motion on a Maple worksheet. With the equations of motion, we were then able to use the program dstool in order to see

Mean residence time

We would also like to compute the mean residence time of a particle starting at (x,y) inside a vortex. This is the average time until the particle escapes through any portion of the boundary. From the equations of motion, we know that our mean residence time u(x,y) satisfies (see Section 2)12ϵ2Δu+sinh(y)cosh(y)−ϵφ′(y)cos(kx)−cux−ϵφ(y)ksin(kx)uy=−1subject to:u|∂Ω=0.As with the escape probability, we need to numerically approximate the solution to this partial differential equation. The same

Discussion

The goal of this paper was to determine and quantify the relationship between the parameter β and both the escape probability and mean residence time. We see that as β increases the escape probability for the lower boundary, and the maximum mean residence time decreases. The values of β were chosen so that they were equally spaced in the interval [0,0.75]. The only reason for this is that the system is stable. As was previously mentioned, for β>4/33, the system is unstable.

Acknowledgements

We would like to thank Dr. Jim Brannan (Clemson), Dr. Vincent Ervin (Clemson) and Dr. Diego del-Castillo-Negrete (Los Alamos) for helpful discussions and comments.

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☆

This work was supported by the National Science Foundation Grant DMS-9704345.

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