Particle transport in horizontal convection: Implications for the "Sandström theorem"

The “Sandstr¨om theorem” as interpreted by Jeffreys is that in a flow maintained by a temperature difference, the pathline from the cold region to the warm region must lie below the return path. A formal demonstration of the argument for a rotating fluid requires three assumptions about the relative circulation around a closed material line: (1) the flowis steady, (2) the closed material line is a closed streamline and (3) the work done by friction along the streamline is negative. The argument extends to unsteady flows, thereby relaxing (1–2), if the absolute circulation along the material line is a bounded function of time—a condition that is met for flows with small Rossby number. Its validity for time-periodic two-dimensional flows of horizontal convection is verified numerically. Poincar´e sections reveal the presence of chaotic particle transport in these flows, even though the Eulerian velocity fields have a simple time dependence. In spite of chaotic advection, particle motion is in general downwards in the cold region and upwards in the warm region of the fluid, which is consistent with the flow shape envisioned by Jeffreys. This paper gives support to the validity of his argument for the unsteady case and enhances its relevance for the dynamical interpretation of the basic structure of geophysical flows