A Harris theorem for enhanced dissipation, and an example of Pierrehumbert^*

Abstract In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known as enhanced dissipation . Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions . If κ denotes the strength of the diffusion, then we show that with probability at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>κ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> enhanced dissipation occurs on time scales of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>κ</mml:mi> <mml:mo fence="false" stretchy="false">|</mml:mo> </mml:mrow> </mml:math> , a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert’94.