The Dispersion Analysis of Taylor

Lagrangian Covariance Analysis of β -Plane Turbulence 3

2. Problem Statement and Background

Inviscid unforced barotropic flow on a β-plane satisfies the following evolution equation: D Dt ζx, t = − βu 2 x, t, 1 where u 2 x, t is the Eulerian meridional velocity. There is no mean flow and the flow is nondivergent as corresponds to a fluid of uniform depth. Expression 1 is the equation for the relative vorticity, ζ, in the presence of a mean potential vorticity gradient induced solely by the meridional northward gradient of the vertical component of the planetary rotation rate, β. The streamfunction, velocity and relative vorticity are related: ζ = ∇ 2 ψ, ζ = u 2 , 1 − u 1 , 2 , where u 1 = −ψ, 2 and u 2 = ψ, 1 . In the Lagrangian frame the substantial derivative is DDt = ddt. The Eulerian, u i , and Lagrangian v i velocity components are related in the usual way: for the meridional component u 2 x, t = v 2 xt, t|x , t = v 2 t|x , t = vt|x , t where d dtx i t|x , t = v i t|x , t . Without loss of generality, the particle identification, by initial condition, is often assumed. The Lagrangian form of the vorticity equation, d d t ζt = −βvt, 2 is used to infer a number of properties of the statistics of the solution and the mixing by the velocity field determined by 2. An equation for the enstrophy, hζζi, is easily derived from 2: d d t hζζi = 0, 3 indicating that the the vorticity variance is stationary, hζζi = hζ ζ i = hζζi . The angle brackets represent an average over an ensemble of Lagrangian particle trajectories. The fact that for stationary stochastic processes the process and its derivative are uncorrelated, hζtvti ∼ hζtddtζti = 0, has been used.

2.1. The Dispersion Analysis of Taylor

The dispersion analysis of Taylor [12] is well known and its generalization to the anisotropic homogeneous problem is summarized in [24]: hx i tx j ti = hx i x j i + Z t Z t 1 [ hv i t 1 v j t 2 i + hv i t 2 v j t 1 i] dt 1 d t 2 . 4 Taylor’s dispersion result is the formal consequence of the kinematic relationship, d dtxt|x , t = vt|x , t , describing a specific fluid particles trajectory whose solution is given by the first integral. Omit- ting the dependence on the initial conditions, the short and long time results are hx α x β i ∼ hv α v β iR αβ t 2 as t → 0 and hx α x β i ∼ 2hv α v β iT αβ t as t → ∞ where no sum is implied on the Greek subscripts. Several things should be noted: in the short time, ballistic limit, the dispersion scales with t 2 . This is very much in accord with the short time numerical diffusion studies on a β-plane of [4]. In the long time limit the dispersion scales with t. As the long time scaling is that of a Brownian diffusion, ∼ t, the long time limit is also called the diffusion limit. The long time limit depends, crucially, on the integral time scale T αβ . In the studies of [4] the diffusion limit is approached in about about one eddy turnover time ∼ 1ζ rms . The variance of the meridional dispersion is hx 2 tx 2 ti = 2 Z t Z t 1 hv 2 t 1 v 2 t 2 i dt 2 d t 1 = 2 hvvi Z t t − ηR vv η dη. 5 The definition of the autocorrelation, for a stationary process, has been used: R vv τ = hvtvt + τi hvvi . Defining the first and second integral time scales as the zeroth- and first-order moments of the autocorrelation 4 J.R. Ristorcelli and A.C. Poje T vv = Z ∞ R vv η dη, 6 T 2 1 vv = − Z ∞ η R vv η dη, 7 the meridional dispersion becomes, in the diffusion limit [ t → ∞], hx 2 tx 2 ti ∞ = 2 hvvi[T vv t + T 2 1 vv ] . 8 This kinematic result indicates that asymptotic meridional dispersion is determined by the zeroth and first moments of the meridional velocity autocorrelation. As one of the objectives of this article is to understand the role that β plays on the dispersal of fluid particles it is necessary to understand how the imposition of a mean vorticity gradient influences the integral time scales T vv and T 1 vv .

3. Moment Analysis of the Potential Vorticity Equation