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Linear toroidal-inertial waves on a differentially rotating sphere with application to helioseismology: Modeling, forward and inverse problems

Tram Thi Ngoc Nguyen, Damien Fournier, Laurent Gizon, Thorsten Hohage

TL;DR

The paper develops a rigorous framework for modeling solar inertial waves in a purely toroidal, linear regime on a differentially rotating sphere, deriving a time-harmonic fourth-order Orr-Sommerfeld-type equation for the stream function and its separated-m$^m$ counterparts. It proves well-posedness under explicit smallness or regularity conditions on the differential rotation and viscosity, and formulates an inverse problem to jointly identify $\gamma$ and $\Omega$ from surface observations, establishing a lifted regularity approach and tangential cone condition for convergence of iterative regularization methods. Local identifiability results are shown (with viscosity known or unknown) under full measurements, and partial-data results are obtained under additional Cauchy-data assumptions. Numerical experiments with an accelerated Nesterov-Landweber scheme demonstrate robustness of reconstructions across observation strategies and noise levels, validating the practical potential for helioseismic inferences of internal solar rotation and viscosity.

Abstract

This paper develops a mathematical framework for interpreting observations of solar inertial waves in an idealized setting. Under the assumption of purely toroidal linear waves on the sphere, the stream function of the flow satisfies a fourth-order scalar equation. We prove well-posedness of wave solutions under explicit conditions on differential rotation. Moreover, we study the inverse problem of simultaneously reconstructing viscosity and differential rotation parameters from either complete or partial surface data. We establish convergence guarantee of iterative regularization methods by verifying the tangential cone condition, and prove local unique identifiability of the unknown parameters. Numerical experiments with Nesterov-Landweber iteration confirm reconstruction robustness across different observation strategies and noise levels.

Linear toroidal-inertial waves on a differentially rotating sphere with application to helioseismology: Modeling, forward and inverse problems

TL;DR

The paper develops a rigorous framework for modeling solar inertial waves in a purely toroidal, linear regime on a differentially rotating sphere, deriving a time-harmonic fourth-order Orr-Sommerfeld-type equation for the stream function and its separated-m counterparts. It proves well-posedness under explicit smallness or regularity conditions on the differential rotation and viscosity, and formulates an inverse problem to jointly identify and from surface observations, establishing a lifted regularity approach and tangential cone condition for convergence of iterative regularization methods. Local identifiability results are shown (with viscosity known or unknown) under full measurements, and partial-data results are obtained under additional Cauchy-data assumptions. Numerical experiments with an accelerated Nesterov-Landweber scheme demonstrate robustness of reconstructions across observation strategies and noise levels, validating the practical potential for helioseismic inferences of internal solar rotation and viscosity.

Abstract

This paper develops a mathematical framework for interpreting observations of solar inertial waves in an idealized setting. Under the assumption of purely toroidal linear waves on the sphere, the stream function of the flow satisfies a fourth-order scalar equation. We prove well-posedness of wave solutions under explicit conditions on differential rotation. Moreover, we study the inverse problem of simultaneously reconstructing viscosity and differential rotation parameters from either complete or partial surface data. We establish convergence guarantee of iterative regularization methods by verifying the tangential cone condition, and prove local unique identifiability of the unknown parameters. Numerical experiments with Nesterov-Landweber iteration confirm reconstruction robustness across different observation strategies and noise levels.

Paper Structure

This paper contains 26 sections, 17 theorems, 102 equations, 2 figures.

Table of Contents

  1. Keywords.
  2. Acknowledgement.
  3. Introduction
  4. Contributions.
  5. Structure.
  6. Model and observation of inertial waves
  7. Reduced-order modeling of inertial waves
  8. Separated equation as inversion model
  9. Observation of solar inertial modes
  10. Inertial waves on the manifold ${r{\mathbb{S}^2}}$
  11. Boundedness
  12. Fredholm property and well-posedness
  13. Separated equations
  14. Additional regularity
  15. Well-posedness on the restricted domain
  16. ...and 11 more sections

Key Result

Lemma 1

If $\Omega\in C^2([0,\pi])$, then for any $m\in\mathbb{Z}$ and $\Psi_m,\psi_m\in V_m\cap C^{\infty}({r{\mathbb{S}^2}})$ we have where $\tilde{\alpha}_{\Omega}(\theta):= \Omega'(\theta)\sin\theta+2\Omega(\theta)\cos\theta$. Moreover, where $C:=3C_{H_{\diamond}^1\to L^6}C_{H_{\diamond}^{1/2}\to L^3}$. Under the regularity assumption eq:Omega_regularity, the right hand side of eq:alt_defi_Aalpha ha

Figures (2)

  • Figure 1: 1% data noise. Top to bottom: full and leaked data (filled area) at different levels: 20%, 50%, missing imaginary part.
  • Figure 2: Leaked data. Top to bottom: 1%, 5%, 10%, 20% relative noise level.

Theorems & Definitions (37)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof : Proof of Corollary \ref{['coro:analyticFredholm']}
  • Corollary 5
  • proof
  • ...and 27 more