Three-dimensional time-dependent convection model for asteroseismology: I. Mathematical model

Context. Due to an ill-depicting model of the convective layers below the photosphere in 1D stellar models (structural contribution) and/or a misrepresentation of the coupling between convection and oscillations (modal contribution), a well-known deviation appears between observed and theoretical frequencies, which grows towards high frequencies; the so-called surface effects. While satisfying solutions have been found regarding the structural contribution, the accurate modeling of the modal effect still represents a challenge. Alongside the frequency, the interaction between convection and oscillations also impacts the damping rate of the modes and forms an important part of the driving mechanism behind the stellar oscillations of low-mass stars. With increasing observational capabilities at our disposal with Kepler and TESS, shortcomings in modeling constitute the main limitation to accurate seismic probing of solar-like and red giant stars. Aims. We present the formalism of an approach that changes the current paradigm by addressing three-dimensional space. This new formalism consists in an original nonadiabatic 3D time-dependent convection model for asteroseismology. Methods. We aim to keep the entire 3D structure of the astrophysical flow in these superficial layers in order to fully account for the nature of turbulence in our model via the use of advanced hydrodynamic simulation. We use the perturbative approach and introduce a spectral decomposition approach that results in an entirely new formalism describing standing waves in 3D. This formalism is set to solve the quasi-radial global nonadiabatic oscillation equations in a full 3D framework. Results. Based on physical assumptions, we establish an eigenvalue problem describing the 3D quasi-radial global nonadiabatic stellar oscillation. We also provide a prescription for its numerical resolution alongside a proposed iteration method for our formalism. Finally, we derive the peculiar 3D work integral and establish the expression of the damping rate. We show how our formalism offers the possibility to probe the complex structure of stars and is able to precisely locate regions of the driving and damping of the modes as well as their physical origin.