Observational Data

Many remote sensing techniques actually measure the Lagrangian mean flow

It was recently shown that remote sensing techniques that are based on observing current-induced shifts in wave dispersion actually measure the Lagrangian, rather than Eulerian, mean current (Pizzo et al. 2023). Recall the Lagrangian mean velocity (written in terms of the usual Eulerian spatial coordinates) can be written as

\[ \overline{\mathbf{u}}_L = \overline{\mathbf{u}}_E + \mathbf{u}_S, \]

where $\overline{\mathbf{u}}_E$ is the Eulerian mean velocity and $\mathbf{u}_S$ is the Stokes drift. Doppler-based techniques such as HF radar, Doppler marine radar, and along-track SAR, DoppVis, etc., which measure the phase speed of the waves, actually yield $\overline{\mathbf{u}}_L$ rather than $\overline{\mathbf{u}}_E$. Now we know that $\overline{\mathbf{u}}_L$ rather than $\overline{\mathbf{u}}_E$ plays the key role in the generation of waves by wind. This role of the Lagrangian drift should be indicative of that in other related processes as well. The nontrivial role of the Lagrangian drift should therefore be considered when using the data collected via these methods, or when considering whether to use these data versus truly Eulerian measurements.

Coastal HF radar array measuring surface currents — Coastal high-frequency (HF) radar arrays infer surface currents from the Doppler shift of Bragg-resonant short waves, providing near–real-time maps of the upper-ocean flow. Photo credit: IOOS/NOAA.

Why it is the Lagrangian mean?

In Pizzo et al. (2023), the phase speed $c$ is written as

\[ c = \frac{ 2 g y_0 - U\,\gamma + 2 \displaystyle\int_{-\infty}^{0} U\langle \Gamma J\rangle\,\mathrm{d}\beta }{ U + \gamma }\Bigg|_{b=0} \]

where $U$ is the Lagrangian drift. Importantly, the current-induced shift in $c$ obtained from this expression is controlled by the Lagrangian mean flow: remote-sensing techniques that invert dispersion shifts are therefore measuring the Lagrangian, not Eulerian, surface current.