Autocorrelation of ζt and the Second Integral Time Scale of vt

Lagrangian Covariance Analysis of β -Plane Turbulence 5

3.2. Integral Time Scale of vt

The Lagrangian solution for the vorticity, 2, can be formally written as ζt|x , t = ζ − β Z t vt 1 |x , t d t 1 . 14 The meridional velocity that drives 2 is related to the vorticity through a Poisson equation. Thus expression 14, while not immediately useful as a solution per se, allows several rigorous results for the statistical properties of the solution of 2 to be deduced. The quantity ζ is a random variable denoting the vorticity at the tagging time and location, t , x i . Multiplying 14 by ζt and averaging produces hζtζti = hζtζ i − β Z t hζtvt 1 i dt 1 . 15 The fluid particle identification has, for convenience, and without loss of generality, been dropped. As t → ∞, hζtζ i → 0 and one finds that hζ ζ i = −β Z ∞ hζtvt − ηi dη; 16 the integral time scale of the joint process, {ζ, v}, is finite and known in terms of the initial enstrophy and β. The fact that the variance, hζζi = hζ ζ i, is stationary for the inviscid problem has been used. The moment of 14 with the meridional velocity produces hvtζ i − β Z t hvtvt 1 i dt 1 = 0 , 17 where again stationarity implies hvtζti = 0. This can be rewritten as hvtζ i = βhvvi Z t R vv η dη, 18 which indicates that, in the diffusion limit, the first integral time scale of the meridional velocity is given by T vv = Z ∞ R vv η dη = 0, 19 since hvtζ i → 0 as t → ∞. This result and its consequences are the central result of this article. The vanishing of the first integral time scale for finite β implies that the meridional particle dispersion is bounded see [18]. The above results are a rigorous consequence of 2, hζvi = 0, stationarity, and that the statistics of the flow forget their initial condition, i.e., hζtζ i → 0 and hvtζ i → 0 as t → ∞.

3.3. Autocorrelation of ζt and the Second Integral Time Scale of vt

The formal solution, 14, can also be written as ζt + τ |x , t = ζt|x , t − β Z t + τ t vt 1 |x , t d t 1 . 20 The moment with ζt is now used to obtain an equation for the autocorrelation. Dropping the particle labels and substituting t 1 = t + η produces hζt + τζti = hζtζti − β Z τ hζtvt + ηi dη. 21 The solution for hζt + τζti requires the two-time correlation hζtvt + ηi. Taking the moment with respect to velocity in 14 produces 6 J.R. Ristorcelli and A.C. Poje hvt + ηζti = hvt + ηζ i − β Z t hvt + ηvt 1 i dt 1 . 22 Use of the result 18 and defining a two-point meridional velocity autocorrelation allows the equation to be written as hvt + ηζti = βhvvi Z t + η R vv η 1 d η 1 − βhvvi Z t + η η R vv η 1 d η 1 23 = βhvvi Z η R vv η 1 d η 1 , 24 which has the form, as might be expected, of a stationary process. Substitution into 21 produces the following relationship between the autocorrelations of the v and ζ processes: R ζζ τ = 1 − β 2 hvvi hζζi Z τ Z η R vv η 1 d η 1 d η. 25 The above result is a consequence of the vorticity equation, 2, and nothing else. Note, as β → 0, R ζζ τ → 1 as would occur in statistically homogeneous two-dimensional flows for which vorticity is materially conserved. Integrating by parts, the equation becomes R ζζ τ = 1 − β 2 hvvi hζζi Z τ τ − ηR vv η dη. 26 The result for the autocorrelation, 26, has several consequences. The second integral time scale can be obtained by taking the limit τ → ∞; with R ζζ τ → 0 and as T vv = 0 one finds T 2 1 vv = − Z ∞ ηR vv η dη = hζζi β 2 hvvi . 27 Thus while the first integral time scale of the v process is zero, the second integral time scale, T 1 vv , is related to β and two single-point second-order moments, hvvi and hζζi. Additionally the second integral time scale of v is related to the micro-time-scale of ζ: T 2 1 vv = − Z ∞ ηR vv η dη = hζζi β 2 hvvi = 1 2 τ 2 ζ . 28 The result 26 also leads to the differential equation d 2 d τ 2 R ζζ τ = − β 2 hvvi hζζi R vv τ 29 relating the vorticity autocorrelation to the autocorrelation of the meridional velocity. Evaluating the differ- ential equation at τ = 0 reproduces the definition of the temporal microscale of ζt.

3.4. Integral Time Scale of ζt