This is a summary of the paper published in the Fluids, January 2021. Code to reproduce the figures is separated into two parts, figures 2 and 12-17 and all the remaining figures. If applying this methodology to your own drifter trajectories, it is recommend that you use the spline based linear velocity field parameter estimation tools.

In the summer of 2011 an array of drifters were deployed in the Sargasso Sea as part of an ONR funded project to study submesoscale motions. After first smoothing and interpolating the noisy GPS data, the goal here is to develop a methodology for separating the mesoscale motions from the submesoscale motions.

The two movies below show the ~6 day deployments of nine drifter at Site 1 and Site 2. The key thing to notice is that, not only does the entire cluster of drifters move, but the drifters also separate (disperse) relative to one another. This relative dispersion between the drifters is assumed to have two sources: 1) the relatively large mesoscale feature that is also moving the entire cluster and 2) some unknown submesoscale process. The goal is then to parameterize the mesoscale feature, in order to isolate the submesoscale feature.

Mesoscale dispersion

The central idea with this methodology is that we will separate the total velocity \( \mathbf{u} \) into three parts,

\[ \mathbf{u}^\textrm{total} = \mathbf{u}^\textrm{bg} + \mathbf{u}^\textrm{meso} + \mathbf{u}^\textrm{sm} \]

background (bg), mesoscale (meso) and submesoscale (sm). Following Okubo and Ebbesmeyer, we assume a local Taylor expansion,

\[ \underbrace{ \frac{d}{dt} \begin{bmatrix} x_k(t) \\ y_k(t) \\ \end{bmatrix}}_{\mathbf{u}^\textrm{total}} = % \underbrace{\begin{bmatrix} u^\textrm{bg}(t) \\ v^\textrm{bg}(t) \\ \end{bmatrix}}_{\mathbf{u}^\textrm{bg}} + % \underbrace{ \begin{bmatrix} u_0 + u_1 t \\ v_0 + v_1 t \\ \end{bmatrix} + \frac{1}{2} \begin{bmatrix} \sigma_n+\delta & \sigma_s-\zeta \\ \sigma_s + \zeta & \delta-\sigma_n \\ \end{bmatrix} % \begin{bmatrix} x_k(t) – x_0 \\ y_k(t) – y_0 \\ \end{bmatrix}}_{\mathbf{u}^\textrm{meso}} + % \underbrace{\begin{bmatrix} u_k^\textrm{sm}(t) \\ v_k^\textrm{sm}(t) \\ \end{bmatrix}}_{\mathbf{u}^\textrm{sm}} \]

where \( (x_k,y_k \) are observations from drifter \( k \) at time \( t \). In this notation, \( \{u^\textrm{bg}(t),v^\textrm{bg}(t)\} \) is the spatially homogeneous time-varying background flow, \( \{u_0,v_0,u_1,v_1,\sigma_n,\sigma_s,\zeta,\delta\} \) are the model parameters for the mesoscale flow, \( \{x_0,y_0\} \) is the expansion location and has no consequence to the model, and \( \{u_k^\textrm{sm}(t), v_k^\textrm{sm}(t)\} \) are the residual `submesoscale’ velocities for each drifter. Note that the mesoscale parameters are just re-definitions of the usual spatial gradients, e.g., \( \zeta = \frac{\partial v}{\partial x} – \frac{\partial u}{\partial y} \).

Why is a Taylor expansion sensible thing to do? A key assumption in such a model is that the velocity gradients \( ( \sigma, \zeta, \delta ) \) across the drifters are the same (or similar anyway). This is certainly not guaranteed to be true, and it is easy to think of a situation where it is not (such as drifters on opposite sides of the earth). Using the terminology in turbulence theory, a structure that has a smoothly varying velocity field across the drifters has non-local dynamics. This concept is typically illustrated using the classic forced-dissipative flow.

The turbulence spin-up movie illustrates the difference between local and non-local flow. The panel on the left shows vorticity \( \zeta \) and the panel on the right shows the velocity spectrum. In the center panel you can see particles advected by the flow. The flow is generated by adding vorticity at a given length scale—the ‘forcing length scale’. The forcing length scale is easy to see in the first few frames of the movie as vorticity is added, and later as the typical size of the ‘eddies’. In the velocity spectrum panel on the right, the forcing length scale is shown as the dual green vertical lines. The key takeaway from this movie is that at length scales smaller than the forcing length scale, the velocity structure looks smooth, but at length scales larger than the forcing length scale, the velocity structure look far rougher.

This second movie shows forcing at a longer length scale, or, further “zoomed in”. In the context of the Taylor expansion then, the idea is that we are applying a the Taylor expansion to scales quite a bit less than the forcing length scale. If the drifters were separated at, or longer than, the forcing length scale, we would have no hope of capturing a smooth structure with a Taylor expansion.

Parameter estimation

This second movie shows forcing at a longer length scale, or, further “zoomed in”. In the context of the Taylor expansion then, the idea is that we are applying a the Taylor expansion to scales quite a bit less than the forcing length scale. If the drifters were separated at, or longer than, the forcing length scale, we would have no hope of capturing a smooth structure with a Taylor expansion.