The interaction of acoustic radiation with turbulence

Abstract

We derive expressions for the spectral emissivity and absorptivity of acoustic radiation by low Mach number (M ≪ 1) turbulent fluids. The emissivity and absorptivity depend on the manner in which the turbulence is excited. We consider three types of turbulence. The first is free turbulence, that is, turbulence which is not subject to external forces. The second and third examples are special cases of forced turbulence, turbulence maintained by stirring with spoons and turbulent pseudoconvection. Acoustic quadrupoles are the lowest order acoustic multipoles present in free turbulence, and they control both its emissivity and absorptivity. Acoustic dipoles are created in forced turbulence, and they enhance the acoustic emissivity by M^(-2) compared to that of free turbulence. The acoustic absorptivity of forced turbulence is quite subtle. The absorptivity of turbulence which is maintained by stirring is dominated by acoustic dipoles and exceeds that of free turbulence by M^(-2). The dipole absorptivity of turbulent pseudoconvection is reduced by M^2 below that of turbulence maintained by stirring. Thus, the absorptivity of turbulent pseudoconvection is no larger than that of free turbulence. We apply our results to estimate the equilibrium energies of the acoustic modes in a box filled with fluid some of which is turbulent. For both free turbulence and turbulence maintained by stirring, the most highly excited acoustic modes attain energies E ~ Mv^2, where M and v are the typical mass and velocity of an energy bearing eddy. The quality factors, or Q's, of the modes are larger by M^(-2) in the former case than in the latter. For turbulent pseudoconvection, the most energetic acoustic modes have equilibrium energies E ~ Mc^2 , where c is the sound speed. Their Q's are comparable to those of modes in equilibrium with free turbulence. We evaluate the scattering of acoustic radiation by turbulent fluids. For all types of turbulence, the scattering opacity is smaller by M^3 than the absorptive opacity for frequencies near the peak of the acoustic spectrum. Radiation scattered by free turbulence and turbulent pseudoconvection suffers frequency shifts Δω ~ ω. The frequency shifts are much smaller, Δω ~ Mω, for radiation scattered by turbulence maintained by stirring. We investigate the rate at which nonlinear interactions transfer energy among the acoustic modes. If all of the fluid in the box is turbulent, this rate is slower, by M^3 for free turbulence, by M^5 for turbulence maintained by stirring, and by M for turbulent pseudoconvection, than the rate at which the individual acoustic modes exchange energy with the turbulence. If only a small portion of the fluid is turbulent, the nonlinear mode interactions can be significant, especially for modes in equilibrium with turbulent pseudoconvection. Our results have potential applications to the acoustic radiation in regions of extended turbulence which often arise in nature. In particular, they should prove useful in understanding the excitation of solar oscillations.

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