Numerical solution of a vectorial model problem of solar oscillations

Physical background

The solar convection zone (outer 28% of the Sun) is the seat of complex dynamical processes that transport heat to the surface, maintain differential rotation and meridional circulation, and sustain the solar magnetic field. Fully developed stellar convection, however, is very difficult to model and not yet fully understood. Helioseismology -- the analysis and interpretation of the solar seismic waves -- is our only hope to probe the subsurface layers. The pressure (p) and surface-gravity (f) waves are observed in Dopplergrams, images of the line-of-sight component of velocity in the solar photosphere measured by Doppler shifts of absorption lines. Such Dopplergrams have been continuously collected since 1995 by ground and satellite based instruments as detailed in the data management plan. Mapping the Sun in three dimensions is only possible with local helioseismology (see, e.g., [3]). Unlike global-mode helioseismology (the study of the frequencies of the normal modes of oscillations, see e.g. [1]) local helioseismology uses information about the phases, frequencies, and amplitudes of the solar seismic waves.

Let us describe the equations of solar and stellar oscillations without magnetic fields in the frequency domain as derived in [5]. Consider the velocity \(\mathbf{u}\), the density \(\rho\), the pressure \(p\) and the gravitational potential \(\phi\) of a stationary equilibrium solution of the conservation equations of mass and momentum, given by the Euler equations and a potential equation for the gravitational potential. Then the displacement perturbations \(\boldsymbol{\xi}\) of Lagrangian particles and the Eulerian perturbation \(\varphi\) of the gravitational background potential \(\phi\) satisfies the following system of differential equations in the frequency domain:

\begin{align} \rho ( - i \omega + \mathbf{u} \cdot \nabla + \Omega \times)^2 \boldsymbol{\xi} -\nabla (\rho c^2 \nabla \cdot \boldsymbol{\xi} ) +(\nabla \cdot \boldsymbol{\xi}) \nabla p -\nabla(\nabla p \cdot \boldsymbol{\xi} ) &\\ +(\operatorname{Hess}(p) \boldsymbol{\xi} - \rho \operatorname{Hess}(\phi)) \boldsymbol{\xi} - i \gamma \rho \omega \boldsymbol{\xi} + \rho \nabla \varphi &= \mathbf{s} \quad \text{in} \quad D, \label{eq:vector_pde}\\ -\frac{1}{4 \pi G} \Delta \varphi + \nabla \cdot ( \rho \boldsymbol{\xi}) &= 0 \quad \text{in} \quad \mathbb{R}^3, \nonumber \end{align}

Here, \(G\) is the gravitational constant, \(\mathbf{s}\) are source terms caused by turbulent convection, \(\Omega\) is the uniform angular velocity of the frame of reference, and \(\omega\) the angular frequency of the waves. Correspondingly \(\times\) denotes the cross product in \(\mathbb{R}^3\). %Further, we note that \(\rho\) and \(p\) coincide with their versions of the equilibrium solution up to first order. A well-posedness result for the model has recently been shown by Halla & Hohage [6], essentially under the assumption that background flows \(\mathbf{u}\) are subsonic, which is the case in the Sun.

Scope of the thesis

In this thesis a compatible discretization for this set of equations shall be investigated. Key for the well-posedness result in Halla & Hohage [6] is a Helmholtz-type space decomposition that we want to transfer to a finite element discretization.

For the thesis we will initially (and mostly) consider the case \(\varphi=0\), \((\operatorname{Hess}(p) \boldsymbol{\xi} - \rho \operatorname{Hess}(\phi))=0\) and \(\nabla p = 0\).

In this case the sesquilinear form \(a(\cdot,\cdot)\) of a corresponding variational formulation simplifies to \begin{align} a(\boldsymbol{\xi}, \boldsymbol{\xi}') = \langle \rho c^2 ~ \nabla \cdot \boldsymbol{\xi}, \nabla \cdot \boldsymbol{\xi}' \rangle -\langle \rho \mathcal{D} \boldsymbol{\xi}, \mathcal{D} \boldsymbol{\xi}'\rangle -i \rho \gamma \omega \langle \boldsymbol{\xi}, \boldsymbol{\xi}'\rangle, \quad \boldsymbol{\xi}, \boldsymbol{\xi}' \in \boldsymbol{X}. \end{align} where \(\mathcal{D} := - i \omega + \mathbf{u} \cdot \nabla + \Omega \times\) is the differential operator corresponding to transport. In the discretization a (possibly only implicit) decomposition of a finite element space into an exactly divergence-free subspace and gradient fields is crucial. The \emph{compatible} treatment of divergence-free subspaces is also an important aspect in finite element discretizations for incompressible flows, cf. for instance [7], and successful, i.e. efficient and provably robust, discretization schemes have been developed based on \(\mathbf{H}_0(\operatorname{div},D)\)-conforming finite element spaces in recent years, cf. [8].

In this theses these approaches (combined with Discontinuous Galerkin techniques for the treatment of non-conformities) shall be considered for discretization.

References

[1] Jørgen Christensen-Dalsgaard. Lecture notes on stellar oscillations. Technical report, Institut for Fysik og Astronomi, Aarhus Universitet, Denmark, 2003.

[2] L. Gizon and A. C. Birch. Time-distance helioseismology: The forward problem for random distributed sources. The Astrophys. J., 571(2):966, 2002.

[3] Laurent Gizon, Aaron C. Birch, and Henk C. Spruit. Local helioseismology: Three- dimensional imaging of the solar interior. Ann. Rev. Astron. Astrophys., 48(1):289–338, 2010.

[4] Linus Hägg and Martin Berggren. arXiv:1912.04364, 2019.

[5] D Lynden-Bell and JP Ostriker. On the stability of differentially rotating bodies. Mon. Not. R. Astr. Soc., 136(3):293–310, 1967.

[6] Martin Halla and Thorsten Hohage. On the well-posedness of the damped time-harmonic Galbrun equation and the equations of stellar oscillations. arXiv:2006.07658, 2020.

[7] Volker John, Alexander Linke, Christian Merdon, Michael Neilan, and Leo G. Rebholz. On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Review, 59(3):492–544, 2017.

[8] Christoph Lehrenfeld and Joachim Schöberl. High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Computer Meth. Applied Mech. Engin., 307:339 – 361, 2016.