Appendix A: Eulerian Statistics

The five years of daily sub-sampled North Atlantic drifter velocities were spatially averaged into 2°x2° bins. Spatially and temporally co-located model fields were similarly averaged for the five years period. The was achieved by first applying a binary mask to the full horizontal model grid each day; at the real drifters’ co-ordinates model velocities were active; elsewhere, the model velocities were "turned off" The masked fields were then spatially averaged. The model velocities are taken from the model level whose depth most closely matches the 15 m drifter drogue depth. For the 0.1° model that is the second upper-most level (15 m) and for the 0.28° model it is the uppermost level (12.5 m).

If the zonal velocity (cms^-1) is denoted as u, and the meridional velocity (cms^-1) as v, then the kinetic energy of the mean flow (per unit mass), also called the mean kinetic energy (MKE, cm² s^-2) is

where the operator <>_E denotes a simple average taken over all the data falling into a given bin. The mean eddy kinetic energy (per unit mass), also called eddy kinetic energy (EKE, cm²s^-2) is defined by

where l₁ and l₂ (l₁ l₂) are the principal components of variance, computed (Freeland et al., 1975; Emery and Thompson, 1998) as the roots of

where the prime indicates the residual about the Eulerian mean. The roots l₁ and l₂ are therefore the eigenvalues of the Eulerian sample covariance matrix. The two eigenvalues and their corresponding eigenvectors define an ellipse describing the distribution of velocity variance for the given bin; equivalently, a standard deviation ellipse is defined. The rotation angle q for the ellipse is calculated (Freeland et al., 1975) via

A James test (Seber, 1984, pp. 114-117) can be used to compare two sets of vectors with different variances; in this case the 2°x2° mean velocity fields obtained by binning the drifter trajectories and either the 0.28° or 0.1° numerical trajectories. Following Garraffo et al. (2001) the null hypothesis is that the drifter velocities, U_m, and the Lagrangian model velocities, U_d, are equal:

A test statistic for the null hypothesis is:

where T is the matrix transpose, S_m and S_d are the covariance matrices of the drifter and model velocities, respectively, and n_d and n_m are the number of independent points for the data and the model respectively. The number of independent data points are obtained by dividing the number of data points per bin by twice the characteristic integral time scale. In this case, we used 4 days based on the Lagrangian time scale analysis in Section 6. Using Taylor expansions, James (1954) showed that the upper critical value a critical value, k_a, correct to order n^-2, is

where

and i = 1, 2 refer to the model and drifter fields, respectively. Finally, is the upper quantile value of , that is, ; here a is 0.05. In other words, when the test statistic is greater than the modified value, the model and the data are said to differ significantly. The null hypothesis of equal means is rejected for those bins with a possibility of being incorrect 5% of the time.