Deducing kinetic energy spectra from pair dispersion

An important application of relative (particle pair) dispersion is its use in diagnosing the kinetic energy spectrum. Pairs of GPS-tracked drifters can sample down to the 10 m scale, well below the scales resolved by satellite altimetry, so relative dispersion can be used advantageously to study submesoscale dynamics. However it is often observed that different statistics yield qualitatively different results. In particular, statistics based on averaging with respect to distance (structure functions and relative diffusivity) frequently differ from those involving time averages (dispersion and pair separation PDFs). We demonstrate theoretically that this is to be expected. We then examine how the energy spectrum can be inferred in such cases.