Thermal Convection in a Central Force Field Mediated by Sound

Convection in radial force fields is a fundamental process behind weather on Earth and the Sun, as well as magnetic dynamo action in both. Until now, benchtop experiments have been unable to study convection in radial force fields due to the inability to generate radial forces of sufficient strength. Recently, it has been appreciated that sound, when averaged over many cycles, exerts a force on density gradients in the gas it travels through. The acoustic radiation pressure on thermal gradients draws cooler gas to regions with large time-averaged acoustic velocity and can be modeled as an effective acoustic gravity. We have constructed a system which generates a high amplitude, spherically symmetric acoustic wave in a rotating spherical bulb containing weakly ionized sulfur gas. Without sound, the gas stratifies itself into an initial state with the warmest gas near the center of the bulb, and the coolest gas near the bulb surface. When the sound is initiated, the acoustic radiation pressure is not balanced and a convective instability is triggered. With high speed videography, we observe the initial shape and growth rate of the most unstable mode at various acoustic amplitudes. Acoustic and rotational forces both contribute to the detailed mode shape, which changes qualitatively at low amplitudes where acoustic forces no longer surpass rotational ones everywhere in the bulb.

Figure 1

(a) Convection cells driven by the radial “acoustic gravity” of a spherical standing sound wave in a rotating sulfur plasma bulb. Direction of flow indicated by the streamlines has been extracted from high speed video available in the Supplemental Material. Cells are axisymmetric about the rotation axis (vertical line centered in figure). The hexagonal pattern visible in the image is the shadow of the metal mesh which forms the microwave cavity. (b) The acoustic velocity and effective gravity as a function of radius. Note that the acoustic gravity changes sign at the velocity antinode, which occurs at nearly half the bulb radius (dashed white circle). Since the plasma is volumetrically heated internally and is hottest at the center, the region inside of the antinode is stable, while the region outside of the velocity antinode is unstable. Convection in the outer zone is similar to Rayleigh-Bénard convection in a time-varying, spherical, acoustic gravity field.

Figure 2

Block diagram of microwave system which heats the sulfur and generates sound. Amplitude modulation of the microwave power generates sound waves inside the bulb. The acoustic gravity of the standing sound wave destabilizes the stratification of the quiescent bulb and the growth of the most unstable mode is captured on high-speed video as a function of microwave modulation depth.

Figure 3

Top view of instability evolution indicates the instability is axially symmetric. The feature visible at the center of the bulb in every frame is a pontil mark on the bulb from glassblowing. Panels (a)–(c) and (g)–(i) are frames of high speed video while panels (d)–(f) and (j)–(l) are false color images of the difference in brightness from panel a. Red (blue) coloring indicates the pixel value is lower (higher) than it was in panel (a).

Figure 4

Side view of instability evolution. Panels (a)–(d) and (i)–(l) are frames of high speed video while panels (e)–(h) and (m)–(p) are false color images of the difference in brightness from panel (a). Red (blue) coloring indicates the pixel value is lower (higher) than it was in panel (a).

Figure 5

Timescales of convection onset are extracted from the top view high frame rate videos by finding the radius of circles where the brightness drops below a threshold value over time and fitting exponentials (dashed black curves). Each color is a different threshold, with inner contours corresponding to higher thresholds (brighter pixels). The growth time constant of the instability at 100% (a), 80% (b), and 40% (c) modulation depth is 1.9 ms, 2.8 ms, and 3.0 ms respectively.

Figure 6

Although the growth times of the convective modes at 40% and 80% modulation depth are similar, their shape is qualitatively different. This difference is due to a change in the balance of rotational and acoustic forces.