Verification of the Meridional Particle Dispersion Result

Lagrangian Covariance Analysis of β -Plane Turbulence 11 0.5 1 1.5 2 2.5 3 −0.2 0.2 0.4 0.6 0.8 1 t−t R Beta = 20 R uu R vv R zz 0.5 1 1.5 2 2.5 3 −0.2 0.2 0.4 0.6 0.8 1 t−t R Beta = 40 R uu R vv R zz 0.5 1 1.5 2 2.5 3 −0.2 0.2 0.4 0.6 0.8 1 t−t R Beta = 60 R uu R vv R zz 0.5 1 1.5 2 2.5 3 −0.2 0.2 0.4 0.6 0.8 1 t−t R Beta = 80 R uu R vv R zz Figure 2 . The Lagrangian correlation functions, R ζζ , R uu , R vv , for four values of β . like nature of the R vv process is apparent in the computations for larger β values. Due to the increasingly important effects of nonstationarity over the larger sampling times required for higher order statistics, moments such as T 2 1 vv did not converge satisfactorily for the given resolution and particle sample size. Such experimental limitations indicate the utility of 41 which provides an exact relation between such higher-order statistics and lower-order ones.

6.2. Verification of the Meridional Particle Dispersion Result

It has been deduced that the meridional particle dispersion was bounded 33. This key result is verified in Figure 5. The dispersion curves in Figure 5 have been normalized by β 2 ζζ Due to the viscous decay of enstrophy and to variations in the enstrophy decay rate with β, the enstrophy scale, ζζ , in 46 was chosen as the average enstrophy, at each β, over the time taken for the particle dispersion to asymptote. The results, for the fixed value of Lagrangian-averaged enstrophy, unquestionably validate the analysis. The early time departures from the asymptotic bound are due to the fact that the meridional dispersion asymptotes as quickly as the vorticity autocorrelation. R ζζ is nonzero at short times and the relation 1 − R ζζ t = 1 2 ˆ β 2 h ˆx 2 t ˆx 2 ti 48 holds. 12 J.R. Ristorcelli and A.C. Poje 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t−t T Beta = 20 T uu T vv T zz 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 t−t T Beta = 40 T uu T vv T zz 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 t−t T Beta = 60 T uu T vv T zz 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 t−t T Beta = 80 T uu T vv T zz Figure 3 . The Lagrangian time scales, T ζζ , T uu , T vv , for four values of β . The effects of finite viscosity on the derived dispersion bound enter almost exclusively through the time dependence of the decaying enstrophy. The direct effects of viscous drain on the temporal behavior of hx 2 x 2 i are apparently quite small since the long-time value of the unscaled meridional dispersion is practically constant in all finite β simulations. To test the pointwise temporal applicability of 46, we have conducted similar experiments at a higher 256 2 resolution and a lower ν p = 10 − 15 viscosity. The resulting temporal dependence of β 2 hx 2 x 2 it 2 hζζit is shown in Figure 6 for the same four values of β. As before, 46 provides the proper scaling of the meridional particle dispersion with both β and the local time value of the Lagrangian averaged enstrophy. All four curves indicate an increase with time in the normalized dispersion due to the viscous decay of the particle enstrophy in the numerical simulations. Increasing β slows the enstrophy decay rate decreasing the exponential decay constant. To comprehend Figure 6 it is important to understand that all four simulations were begun at the same initial time. The initial time was determined by half the enstrophy e-folding time of the β = 20 flow. Thus the increase in the scaled meridional dispersion is more pronounced for the highest β value since this flow sees the largest proportional change in enstrophy over the ten time units shown. While the mathematical results have been verified it is helpful to obtain a physical picture of the particle dispersion. To provide some idea of the Lagrangian particle behavior for which the above mathematical Lagrangian Covariance Analysis of β -Plane Turbulence 13 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 t−t R zz Beta = 20 Beta = 40 Beta = 80 0.5 1 1.5 2 2.5 3 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 t−t T vv Beta = 20 Beta = 40 Beta = 80 Figure 4 . Comparison of the Lagrangian vorticity correlation function, R zz and the meridional Lagrangian timescales, T vv , for four values of β . results are relevant the flow was specially seeded with only a few particles. The particle trajectories are shown in Figure 7 over a period of a few eddy turnovers. The meridional boundedness is clearly seen. Our primary result, the magnitude of the meridional particle dispersion bound, has been verified. We now ask whether there is a phenomenological connection between the scale of the rectified Eulerian flow and the meridional length scale imposed by the Lagrangian particle dispersion bound.

6.3. Length Scales of Zonally Averaged Eulerian Structure