Non-local, time-dependent convection models have been used to explain the location of double-mode pulsations in Cepheids in the HR diagram as well as the existence and location of the red edge of the instability strip. These properties are highly sen
sitive to model parameters. We use 2D radiation hydrodynamical simulations with realistic microphysics and grey radiative-transfer to model a short period Cepheid. The simulations show that the strength of the convection zone varies significantly over the pulsation period and exhibits a phase shift relative to the variations in radius. We evaluate the convective flux and the work integral as predicted by the most common convection models. It turns out that over one pulsation cycle the model parameter $alpha_{rm c}$, has to be varied by up to a factor of beyond 2 to match the convective flux obtained from the simulations. To bring convective fluxes integrated over the He II convection zone and the overshoot zone below into agreement, this parameter has to be varied by a factor of up to $sim 7.5$ (Kuhfu{ss}). We then present results on the energetics of the convection and overshoot zone by radially symmetric and fluctuating quantities. To successfully model this scenario by a static, one dimensional or even by a simple time-dependent model appears extremely challenging. We conclude that significant improvements are needed to make predictions based on 1D models more robust and to improve the reliability of conclusions on the convection-pulsation coupling drawn from them. Multidimensional simulations can provide guidelines for developing descriptions of convection then applied in traditional 1D modelling.
We have extended the ANTARES code to simulate the coupling of pulsation with convection in Cepheid-like variables in an increasingly realistic way, in particular in multidimensions, 2D at this stage. Present days models of radially pulsating stars as
sume radial symmetry and have the pulsation-convection interaction included via model equations containing ad hoc closures and moreover parameters whose values are barely known. We intend to construct ever more realistic multidimensional models of Cepheids. In the present paper, the first of a series, we describe the basic numerical approach and how it is motivated by physical properties of these objects which are sometimes more, sometimes less obvious. - For the construction of appropriate models a polar grid co-moving with the mean radial velocity has been introduced to optimize radial resolution throughout the different pulsation phases. The grid is radially stretched to account for the change of spatial scales due to vertical stratification and a new grid refinement scheme is introduced to resolve the upper, hydrogen ionisation zone where the gradient of temperature is steepest. We demonstrate that the simulations are not conservative when the original weighted essentially non-oscillatory method implemented in ANTARES is used and derive a new scheme which allows a conservative time evolution. The numerical approximation of diffusion follows the same principles. Moreover, the radiative transfer solver has been modified to improve the efficiency of calculations on parallel computers. We show that with these improvements the ANTARES code can be used for realistic simulations of the convection-pulsation interaction in Cepheids. We discuss the properties of several models which include the upper 42% of a Cepheid along its radial coordinate, assume different opening angles, and are suitable for an in-depth study of convection and pulsation.
The ANTARES code has been designed for simulation of astrophysical flows in a variety of situations, in particular in the context of stellar physics. Here, we describe extensions as necessary to model the interaction of pulsation and convection in cl
assical pulsating stars. These extensions encomprise the introduction of a spherical grid, movable in the radial direction, specific forms of grid-refinement and considerations regarding radiative transfer. We then present the basic parameters of the cepheid we study more closely. For that star we provide a short discussion of patterns of the H+HeI and the HeII convection zones and the interaction with pulsation seen in the pdV work or atmospheric structures.
