, 2011). If bounded Galerkin projection is used the time required was found to increase to approximately two time steps. Simulation M2M2-mid was also profiled as a part of this investigation and the mesh adapt required a similar proportion of time to the simulations that use M∞M∞ (Hiester, 2011). In parallel, the overhead of adaptivity is relatively small with the overall cost of the adaptive step being dominated by the serial algorithm (Gorman et al., 2009). The background potential energy provides a measure of diapycnal mixing and is the main diagnostic used for analysis here, Section 4.1. The Froude number is also calculated providing

an additional diagnostic comparison, Section 4.2. The background potential energy is the potential energy Ibrutinib clinical trial of the minimum energy state (or reference state) that can be obtained by adiabatic redistribution of the system (Winters et al., 1995 and Winters and D’Asaro, 1996). Most crucially, for a closed system, changes to the reference state caused by diapycnal mixing correspond to increases in the find more background potential energy (Winters et al., 1995). Denoting the vertical

coordinate in the reference state z∗z∗, the background potential energy, EbEb, is given by equation(11) Eb=∫Ωρgz∗dV,where ΩΩ is the domain. z∗z∗ is calculated using the method of Tseng and Ferziger (2001), where a probability density function is constructed for the density (or here temperature) field and then integrated to give z∗z∗ (cf. Hiester, 2011). The background potential energy is decomposed further to account

Dapagliflozin for changes in EbEb that may occur due to non-conservation of the fields through the use of a non-conservative advection scheme and consistent interpolation. Following Ilıcak et al. (2012), ρρ and z∗z∗ are partitioned into a spatial mean and a perturbation: ρ=ρ‾+ρ′ and z∗=z∗‾+z∗′, where equation(12) ρ‾=1V∫ΩρdVandz∗‾=1V∫Ωz∗dV.EbEb then becomes equation(13) Eb=gρ‾z∗‾∫ΩdV︸Eb‾+g∫Ωρ′z∗′dV︸Eb′,where Eb‾ changes due to changes in mass and Eb′ changes due to diapycnal mixing (Ilıcak et al., 2012). The values will be presented as a change in Eb′, normalised by the initial value of EbEb: equation(14) ΔEb′(s)Eb0=Eb′(s)-Eb′(s=0)Eb(s=0),where s=t/Tbs=t/Tb or, for a closer analysis of the propagation stages, s=X/Hs=X/H with X   the position of the no-slip front. It is noted that whilst EbEb depends on density and hence ρ0ρ0, as the values are normalised, once again no value of ρ0ρ0 is required (cf. Section 2.1). The typical behaviour of the background potential energy is presented in Section 5.2. The Froude number, Fr=U/ubFr=U/ub, is the ratio of the front speed, U  , to the buoyancy velocity, ubub, Table 1. After an initial acceleration, the gravity current fronts travel at a constant speed until the end walls exert an influence or viscous forces begin to dominate ( Cantero et al., 2007, Härtel et al., 1999 and Huppert and Simpson, 1980).