Abstract
The pseudo-incompressible approximation, which assumes small pressure perturbations from a one-dimensional reference state, has long been used to model large-scale dynamics in stellar and planetary atmospheres. However, existing implementations do not conserve energy when the reference state is time-dependent. We use a variational formulation to derive an energy-conserving pseudo-incompressible model in which the reference state evolves while remaining hydrostatic. We present an algorithm for solving these equations in the case of closed boundaries, for which the pseudo-incompressible velocity constraint is degenerate. We implement the model within the low-Mach-number code MAESTROeX, and validate it against a fully compressible model in several test cases, finding that our hybrid pseudo-incompressible–hydrostatic model generally shows better agreement with the compressible results than the existing MAESTROeX implementation.
Highlights
The time step t cannot exceed x/ max{c, u}, where x is the smallest scale resolved in the simulation, c is the sound speed and u is the magnitude of the fluid velocity
We note that MAESTROeX uses a slightly different definition for the quantity φ, which has been found to improve numerical stability (Almgren, Bell & Crutchfield 2000; Almgren et al 2008) but is mathematically equivalent to what we present here
In the case of closed boundaries, the velocity must satisfy a degenerate constraint equation, and this degeneracy is broken by requiring that the energy flux through the boundaries vanishes. We have implemented this model within the existing MAESTROeX framework and validated it in a number of test cases, including comparison with results from the fully compressible code CASTRO
Summary
In a wide variety of fluid dynamical systems sound waves play no significant role on the time and length scales of interest. (i) they become inaccurate on large horizontal scales (for which pressure perturbations are significant e.g. Davies et al 2003); and (ii) they imply a constraint on the velocity field that cannot generally be maintained in the presence of heat sources, at least for certain choices of the boundary conditions (e.g. Rehm & Baum 1978; Lecoanet et al 2014) For these reasons, and for the sake of greater generality, it is desirable to allow the background pressure field to evolve in time, whilst maintaining hydrostatic balance (Almgren 2000; Almgren et al 2008; O’Neill & Klein 2014). The purpose of this paper is to resolve this problem by generalising the method of Vasil et al (2013) to allow the background pressure P0 to vary in time, subject to the constraint that the horizontally averaged dynamics is hydrostatic To enforce this additional constraint, we introduce an additional Lagrange multiplier into the fluid action, which results in an additional force in the momentum equation. The pseudo-incompressible approximation is generally more accurate for modelling dynamics on small horizontal length scales, such as atmospheric convection and other buoyancy processes, which form the motivation for the present work
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