On approximations of the basic equations of terrestrial mantle convection used in published literature

Previous studies have suggested some approximations for the basic equations of terrestrial mantle convection. The approximations are based on five dimensionless parameters-M (Much number), Pr (Prandtl number), Di (dissipation number), Co (Compressibility number; the ratio of dissipation number to the Grüneisen number), and υ (fraction of density change due to thermal expansion). These approximations are given by: (i) M2Pr → 0 for the anelastic liquid approximation (ALA), (ii) M2Pr → 0 and υ → 0 for the truncated anelastic liquid approximation (TALA), (iii) M2Pr → 0, υ → 0, and Co → 0 for the extended Boussinesq approximation (EBA), (iv) M2Pr → 0, υ → 0, and Di/Ra → 0 for the superadiabatic Boussinesq approximation (SBA), and (v) M2Pr → 0, υ → 0, Co → 0, and Di → 0 for the Boussinesq approximation (BA). This study suggests the use of five dimensionless parameters, namely, M, Pr, Di, Co, and Ra (Rayleigh number), to reduce the number of approximations to four: (I) M2Pr → 0 for the ALA, (II) M2Pr → 0 and Co → 0 for the EBA, (III) M2Pr → 0 and Di/Ra → 0 for the SBA, and (IV) M2Pr → 0, Co → 0, and Di → 0 for the BA. This is because υ is simply defined by υ = M2PrRa/Co and is automatically approximated to 0 when M2Pr → 0. In other words, approximations of ALA and TALA can be unified because they represent the same sense physically. This conclusion is valid for mantle convection in the present Earth whose Ra ∼ O(107) is approximately one order smaller than the threshold Rayleigh number, Rathr = Co/(M2Pr) ∼ O(108).