On the determination of solenoidal or compressible velocity fields from measurements of passive or reactive scalars

Several techniques have been proposed for determining two- or three-dimensional velocity fields from measurements of one passive scalar. It is shown that measurements of one scalar and knowledge of the equation governing its transport determine a velocity field, only up to an additive vector field locally perpendicular to the gradient of the scalar field but otherwise arbitrary. Three previously proposed procedures for selecting a unique velocity field from among the uncountable infinity consistent with the scalar transport data and equation are then discussed, and it is shown that a recent ''iterative inversion'' procedure for ''solution'' of a singular linear equation system (obtained using only measurements of one scalar and the equation governing its transport) cannot converge as claimed. A method for determining the correct n-dimensional (n=2 or 3) divergence-free velocity field from measurements of n-1 passive or reactive scalars is then developed. Finally, it is shown how the velocity field in an n-dimensional compressible flow can be determined from measurements of density and n-1 passive or reactive scalars