Direct multi-scale reconstruction of velocity fields from measurements of particle tracks

We present a method for reconstructing two-dimensional velocity fields at specified length scales using observational data from tracer particles in a flow, without the need for interpolation or smoothing. The algorithm, adapted from techniques proposed for oceanography, involves a least-squares projection of the measurements onto a set of two-dimensional, incompressible basis modes with known length scales. Those modes are constructed from components of the velocity potential function, which accounts for inflow and outflow at the open boundaries of the measurement region; and components of the streamfunction, which accounts for the remainder of the flow. All calculations are evaluated at particle locations, without interpolation onto an arbitrary grid. Since the modes have a well-defined length scales, scale-local flow properties are available directly. The technique eliminates outlier particles automatically and reduces the apparent compressibility of the data. Moreover the technique can be used to produce spatial power spectra and to evaluate the spatial effects of open boundaries; it also holds promise for direct calculation of scale-to-scale transfer of enstrophy and energy.