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Register a new userEnergy Conservation and Gravity Waves in Sound-proof Treatments of Stellar Interiors: Part I Anelastic Approximations
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Typical flows in stellar interiors are much slower than the speed of sound. To follow the slow evolution of subsonic motions, various sound-proof equations are in wide use, particularly in stellar astrophysical fluid dynamics. These low-Mach number equations include the anelastic equations. Generally, these equations are valid in nearly adiabatically stratified regions like stellar convection zones, but may not be valid in the sub-adiabatic, stably stratified stellar radiative interiors. Understanding the coupling between the convection zone and the radiative interior is a problem of crucial interest and may have strong implications for solar and stellar dynamo theories as the interface between the two, called the tachocline in the Sun, plays a crucial role in many solar dynamo theories. Here we study the properties of gravity waves in stably-stratified atmospheres. In particular, we explore how gravity waves are handled in various sound-proof equations. We find that some anelastic treatments fail to conserve energy in stably-stratified atmospheres, instead conserving pseudo-energies that depend on the stratification, and we demonstrate this numerically. One anelastic equation set does conserve energy in all atmospheres and we provide recommendations for converting low-Mach number anelastic codes to this set of equations.
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Recent photometric observations of massive stars show ubiquitous low-frequency red-noise variability, which has been interpreted as internal gravity waves (IGWs). Simulations of IGWs generated by convection show smooth surface wave spectra, qualitatively matching the observed red-noise. On the other hand, theoretical calculations by Shiode et al (2013) and Lecoanet et al (2019) predict IGWs should manifest at the surface as regularly-spaced peaks associated with standing g-modes. In this work, we compare these theoretical approaches to simplified 2D numerical simulations. The simulations show g-mode peaks at their surface, and are in good agreement with Lecoanet et al (2019). The amplitude estimates of Shiode et al (2013) did not take into account the finite width of the g-mode peaks; after correcting for this finite width, we find good agreement with simulations. However, simulations need to be run for hundreds of convection turnover times for the peaks to become visible; this is a long time to run a simulation, but a short time in the life of a star. The final spectrum can be predicted by calculating the wave energy flux spectrum in much shorter simulations, and then either applying the theory of Shiode et al (2013) or Lecoanet et al (2019).
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We show that polarization singularities, generic for any complex vector field but so far mostly studied for electromagnetic fields, appear naturally in inhomogeneous yet monochromatic sound and water-surface (e.g., gravity or capillary) wave fields in fluids or gases. The vector properties of these waves are described by the velocity or displacement fields characterizing the local oscillatory motion of the medium particles. We consider a number of examples revealing C-points of purely circular polarization and polarization M{o}bius strips (formed by major axes of polarization ellipses) around the C-points in sound and gravity wave fields. Our results (i) offer a new readily accessible platform for studies of polarization singularities and topological features of complex vector wavefields and (ii) can play an important role in characterizing vector (e.g., dipole) wave-matter interactions in acoustics and fluid mechanics.
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