Autocorrelations

The following are autocorrelations defined as follows:

\[ \rho(\tau)= \frac{\langle v(t) v(t+\tau)\rangle}{\sqrt{\langle v ^2(t) \rangle  \langle  v ^2(t+\tau) \rangle  } } \quad \quad \text{Where} \quad v(t) = V(t) - \langle V(t) \rangle \quad \text{i.e. the velocity less the Lagrangian mean velocity at time = t} \]

The following figure shows the autocorrelation of each component of the acceleration, the magnitude of acceleration, and the two spherical angles that are defined as follows:

\[ \phi = atan2 \left( \frac{a_y}{a_z} \right) \quad \quad \theta = acos \left( \frac{a_z}{|a|} \right) \]

Where $atan2$ is a function that avoids the singularity, and has a range of $-\pi/2, \pi/2$, more  here

The following figure shows the autocorrelations from all tracks that start at $y^+=0-38$, i.e. $y^+(t=0) = 0-38$

source:trunk/Figures/autocorrelation_all.png

The data shown in the above figure can be found here, with reasonably intuitive variable names. A simple script to plot this data is also in the repository here.

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