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Effects of density stratification on Rossby waves in deep atmospheres

Catherine C. Blume, Bradley Hindman

TL;DR

This work develops a general framework for deep-fluid Rossby waves in the Sun by adopting a β-plane, hydrostatic, slow-rotation limit and using the Lagrangian pressure fluctuation $\delta P$ as the primary variable. The vertical structure reduces to a non-singular Sturm-Liouville problem with eigenvalues $h_n$, revealing two distinct radial cavities: one in the radiative interior and another in the convection zone, each with characteristic eigenfunctions and an associated dispersion relation $\omega_n(k_x,k_y) = -\frac{k_x \beta}{k_x^2 + k_y^2 + f_0^2/(g h_n)}$. Radiative interior modes show nearly constant vorticity through the convection zone and experience tiny frequency shifts relative to the 2-D limit, while convection-zone modes are surface-confined and can exhibit substantial negative shifts at higher azimuthal orders. These results imply potential seismic probes of solar stratification and highlight limitations due to curvature, differential rotation, and the finite observational resolution when interpreting solar $r$-modes.

Abstract

Though Rossby waves have been observed on the Sun, their radial eigenfunctions remain a mystery. The prior theoretical work either considers quasi-2D systems, which do not apply to the solar interior, or only considers fully radiative or fully convective atmospheres. This project calculates the radial eigenfunctions for Rossby waves in a deep atmosphere for a general stratification. Here, we use the $β$-plane approximation to derive a vertical equation in terms of the Lagrangian pressure fluctuation $δP$, and we then calculate radial eigenfunctions for Rossby waves in a standard solar model, Model S. We find that working in the Lagrangian pressure fluctuation results in cleaner wave equations that lack internal singularities that have been encountered in prior work. The resulting radial wave equation makes it abundantly clear that there are two wave cavities in the solar interior, one in the radiative interior and another in the convection zone. Surprisingly, our calculated radial vorticity eigenfunctions for the radiative interior modes are nearly constant throughout the convection zone, raising the possibility that they may be observable at the solar surface.

Effects of density stratification on Rossby waves in deep atmospheres

TL;DR

This work develops a general framework for deep-fluid Rossby waves in the Sun by adopting a β-plane, hydrostatic, slow-rotation limit and using the Lagrangian pressure fluctuation as the primary variable. The vertical structure reduces to a non-singular Sturm-Liouville problem with eigenvalues , revealing two distinct radial cavities: one in the radiative interior and another in the convection zone, each with characteristic eigenfunctions and an associated dispersion relation . Radiative interior modes show nearly constant vorticity through the convection zone and experience tiny frequency shifts relative to the 2-D limit, while convection-zone modes are surface-confined and can exhibit substantial negative shifts at higher azimuthal orders. These results imply potential seismic probes of solar stratification and highlight limitations due to curvature, differential rotation, and the finite observational resolution when interpreting solar -modes.

Abstract

Though Rossby waves have been observed on the Sun, their radial eigenfunctions remain a mystery. The prior theoretical work either considers quasi-2D systems, which do not apply to the solar interior, or only considers fully radiative or fully convective atmospheres. This project calculates the radial eigenfunctions for Rossby waves in a deep atmosphere for a general stratification. Here, we use the -plane approximation to derive a vertical equation in terms of the Lagrangian pressure fluctuation , and we then calculate radial eigenfunctions for Rossby waves in a standard solar model, Model S. We find that working in the Lagrangian pressure fluctuation results in cleaner wave equations that lack internal singularities that have been encountered in prior work. The resulting radial wave equation makes it abundantly clear that there are two wave cavities in the solar interior, one in the radiative interior and another in the convection zone. Surprisingly, our calculated radial vorticity eigenfunctions for the radiative interior modes are nearly constant throughout the convection zone, raising the possibility that they may be observable at the solar surface.

Paper Structure

This paper contains 12 sections, 18 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Wave Cavities and Vertical Eigenfunctions
  4. Eigenfrequencies
  5. Discussion
  6. $\delta P$ is a convenient variable
  7. Easily identifiable wave cavities
  8. Comparison to radial observations
  9. Radiative interior modes
  10. Convection zone modes
  11. Mode misidentification
  12. Distinguishing between mode families

Figures (4)

  • Figure 1: Propagation diagram for Model S---The orange (purple) region denotes propagation for Rossby waves in the radiative interior (convection zone). The blue (yellow) lines mark the possible values of the separation constant in each region respectively. The inset shows the very top of the convection zone, with $y$-axis values ranging from $-5 \times 10^{-10}$ to $0$.
  • Figure 2: Radial eigenfunctions for Rossby waves---Radial eigenfunctions for the radiative interior modes (top) and convection zone modes (bottom) in Lagrangian pressure fluctuation $\delta P$ (left column), reduced pressure $\delta P/\rho_0$ (middle column) and radial vorticity $\zeta_z$(right column). The radiative interior mode eigenfunctions are distributed across the region and are roughly constant in the convection zone. The convection zone mode eigenfunctions are confined near the surface and evanescent everywhere else.
  • Figure 3: Fractional frequency shift---The fractional frequency shift $(\omega_n - \omega_{2D})/\omega_{2D}$ with respect to the two-dimensional dispersion relation for modes of the (a) radiative interior and (b) convection zone. The figure was generated for a low-latitude $\beta$-plane located at $\theta$ = 10 degrees. The black dashed line represents the asymptotic behavior for large $m$. The frequency shift for the radiative interior modes is so small as to be undetectable. On the other hand, the convection zone modes can have quite a large frequency difference in comparison.
  • Figure 4: Artificial spectra---Artificial spectra calculated for the $\ell = m = 3$ mode at about (a) 0.03 nHz and (b) 3 nHz. Each panel displays the first 11 modes of increasing radial order for both the radiative interior and convection zone families given an arbitrary linewidth of 0.1 nHz and amplitude falling off like 1/$n^n$. The zeroth order mode for the radiative interior (convection zone) family is marked with an orange (purple) dashed line. With a more realistic resolution, it is impossible to distinguish between these modes.