# Analytic Reconstruction of a Two-Dimensional Velocity Field from an Observed Diffusive Scalar

## TL;DR

This paper presents a method to uniquely reconstruct two-dimensional velocity fields from scalar measurements in incompressible flows, using high-resolution tracer data and PDE solving, with applications in oceanography.

## Contribution

It introduces a novel approach to invert scalar fields for velocity reconstruction in 2D flows, extending to shallow-water equations and spatially varying diffusivity.

## Key findings

- Unique velocity reconstruction possible with high-res scalar and transverse velocity measurements.
- Method applicable to Navier-Stokes flows and ocean circulation models.
- Potential for extending local velocity data to larger areas using scalar fields.

## Abstract

Inverting an evolving diffusive scalar field to reconstruct the underlying velocity field is an underdetermined problem. Here we show, however, that for two-dimensional incompressible flows, this inverse problem can still be uniquely solved if high-resolution tracer measurements, as well as velocity measurements along a curve transverse to the instantaneous scalar contours, are available. Such measurements enable solving a system of partial differential equations for the velocity components by the method of characteristics. If the value of the scalar diffusivity is known, then knowledge of just one velocity component along a transverse initial curve is sufficient. These conclusions extend to the shallow-water equations and to flows with spatially dependent diffusivity. We illustrate our results on velocity reconstruction from tracer fields for planar Navier- Stokes flows and for a barotropic ocean circulation model. We also discuss the use of the proposed velocity reconstruction in oceanographic applications to extend localised velocity measurements to larger spatial domains with the help of remotely sensed scalar fields.