Asymptotic diffusion coefficients and anomalous diffusion in a meandering jet flow under environmental fluctuations

The nontrivial dependence of the asymptotic diffusion on noise intensity has been studied for a Hamiltonian flow mimicking the Gulf Jet Stream. Three different diffusion regimes have been observed depending on the noise intensity. For intermediate noise the asymptotic diffusion decreases with noise intensity at a rate which is linearly dependent to the flow's meander amplitude. Increasing the noise the fluid transport passes through a superdiffusive regime and finally becomes diffusive again at large noise intensities. The presence of inner circulation regimes in the flow has been found to be determinant to explain the observed behavior.

Figure 1

Stream function of the jet flow (4). Arrows indicate the direction of motion and dots correspond to the different regimes in the flow for the jet amplitude $B<B_{\text{crit}}$. The high density of contour lines indicates the position of the central eastward jet. The cross indicates the position of the stationary point inside the inner circulation core. The domain shown in the figure is extended periodically in the $x$ direction. Set of parameters as in Table .

Figure 2

(a) Relative dispersion $\langle R^2\left(t\right)\rangle$ for three values of noise intensity. Solid lines correspond to the best fitting to Eq. (1) to extract $\gamma$. Noise intensities: $\xi ={10}^{-9}$ (blue circles), $\xi ={10}^{-5}$ (red squares), and $\xi ={10}^{-2}$ (green triangles). (b) Exponent $\gamma$ as a function of noise intensity for two intervals of time, $t\in [7\times {10}^5,1\times {10}^7]$ (blue circles) and $t\in [2\times {10}^5,6\times {10}^5]$ (red squares). Lines indicate normal (- -) and ballistic ($·$ -) dispersion. Rest of parameters as in Table .

Figure 3

Asymptotic diffusion coefficient $D_A$ (2) at time $t={10}^7$ as a function of noise intensity for different values of the jet amplitude. From left to right, solid lines correspond to $D_A\sim \xi$ and $D_A\sim {\xi }^{-\alpha }$ with $\alpha =0.90\pm 0.03$. Dashed vertical lines separate regions for high (HN), intermediate (IN), and low (LN) noise. Set of parameters: $B=1$ (blue circles), $B=3$ (red squares), and $B=5$ (black stars). Rest of parameters as in Table .

Figure 4

Logarithmic plot of the maximum value of $D_A$ at two instants of time [$t=7\times {10}^5$ (red squares) and $t=1\times {10}^7$ (blue circles)] (a) and the scaling coefficient $\alpha$, Eq. (3) (b) as a function of the jet amplitude $B$. Parameters as in Table .