**Introduction and benchmarking**

The convection of the Earth’s mantle is usually modelled as an incompressible process, referred to as the Boussinesq approximation. However, in the Earth’s mantle, the pressure increase associated with depth also increases the density due to self-compression (King et al. 2010). In some applications, this compressibility may be non-negligible and modelling it may be desirable. Over the years, several mantle convection formulations have been devised to take this compressibility into account. One such formulation is the Truncated Anelastic Liquid Approximation (TALA) (Ita and King, 1994; Jarvis and Mckenzie, 1980). In this approximation, a depth-depended reference profile is defined which dictates how the density varies with depth. This is subsequently used to calculate the volumetric changes as a result of pressurization and heating (Gassmöller et al., 2020). In addition, the buoyancy term in the momentum equation precludes the contributions of the density changes due to pressure variations (van Zelst et al., 2022). Taking compressibility into account adds temperature gradients and adiabatic density which increases the complexity of mantle dynamics (Tan and Gurnis, 2007). Here, we easily implement an isoviscous compressible convection model with the Truncated Anelastic Liquid Approximation (TALA) formulation in cartesian coordinates using Underworld3.

This is a companion discussion topic for the original entry at https://www.underworldcode.org/articles/compressible-convection-in-cartesian-coordinates-in-underworld3