The pulse tube and the pendulum.

An inverted pulse tube in which gravity-driven convection is suppressed by acoustic oscillations is analogous to an inverted pendulum that is stabilized by high-frequency vibration of its pivot point. Gravity acts on the gas density gradient arising from the end-to-end temperature gradient in the pulse tube, exerting a force proportional to that density gradient, tending to cause convection when the pulse tube is inverted. Meanwhile, a nonlinear effect exerts an opposing force proportional to the square of any part of the density gradient that is not parallel to the oscillation direction. Experiments show that convection is suppressed when the pulse-tube convection number N(ptc)=omega(2)a(2)DeltaT/T(avg)/[g(alphaD sin theta-L cos theta)] is greater than 1 in slender tubes, where omega is the radian frequency of the oscillations, a is their amplitude, DeltaT is the end-to-end temperature difference, T(avg) is the average absolute temperature, g is the acceleration of gravity, L is the length of the pulse tube and D is its diameter, alpha is about 1.5, and the tip angle theta ranges from 90 degrees for a horizontal tube to 180 degrees for an inverted tube. Theory suggests that the temperature dependence should be DeltaT/T(avg) instead of DeltaT/T(avg).