10 J.R. Ristorcelli and A.C. Poje
10 10
1
10
2
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
k Energy
Initial Conditions Beta = 10
Beta = 80
10 10
1
10
2
10
−5
10
−4
10
−3
10
−2
10
−1
10 10
1
k Enstrophy
Initial Conditions Beta = 10
Beta = 80
Figure 1 . The energy and enstrophy spectra for
β
= 10 and 80 at time
t
= 10. The initial spectra, used in all simulations, are shown by the bold line.
decrease in the enstrophy decay rate due to spectral cascade inhibition and to a reduction in Lagrangian time scales.
6.1. Autocorrelations and Integral Time Scales
The two-point Lagrangian correlation functions for the vorticity and the zonal and meridional velocity components are shown in Figure 2 for increasing values of
β. Of the three correlations, R
vv
is seen to go negative most quickly as might be expected from the analysis. Recall that one of the central deductions of
the analysis was that, unlike T
uu
and T
ζζ
, the integral time scale T
vv
is zero. The integral timescales are shown in Figure 3. Increasing
β appears to lead to a decrease in the vorticity timescale and in a more rapid approach to zero of the time scale of the meridional velocity. Relative to the time scale of the vorticity the
time scale of the fluctuating zonal velocity increases with β becoming commensurate with the time scale of
the vorticity suggesting an increased stability of zonal flow structures and a stronger coupling to the vorticity. A direct comparison of the enstrophy covariance for three different
β values, Figure 4, shows the decrease of the microscale increase of curvature with
β as seen in 39. Similarly, the decrease of the meridional microscale and the increase in the decay rate of the meridional macro-scale is shown. The increasingly wave-
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
