Cross-correlations:

The following are cross-correlations defined as follows:

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

Note: The following plot are the cross- correlations of velocity, acceleration, and velocity-acceleration. Some of which (most correlations that involve the homogeneous component z, for example) are expected to be uncorrelated at all times, but are included here anyway for completeness.

The data used to make these plots can be found here and the script used to plot these results can be found here

The following correlations are organized as:

Velocity

source:trunk/Figures/Correlations/Normalized_correlations/v_x(t)v_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_y(t)v_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_x(t)v_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)v_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_y(t)v_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)v_y(t+\tau).png

Acceleration

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)a_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_y(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)a_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_z(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_y(t)a_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_z(t)a_y(t+\tau).png

Velocity-Acceleration (same-component)

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)v_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_x(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_y(t)v_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_y(t)a_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_z(t)v_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)a_z(t+\tau).png

Velocity-Acceleration (cross-component)

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)v_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_y(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)v_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_y(t)v_z(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)a_y(t+\tau).png