Toward a PV-Based Algorithm for the Dynamical Core of Hydrostatic Global Models

Abstract

The diabatic contour-advective semi-Lagrangian (DCASL) algorithms previously constructed for the shallow-water and multilayer Boussinesq primitive equations are extended to multilayer non-Boussinesq equations on the sphere using a hybrid terrain-following-isentropic (s-u) vertical coordinate. It is shown that the DCASL algorithms face challenges beyond more conventional algorithms in that various types of damping, filtering, and regularization are required for computational stability, and the nonlinearity of the hydrostatic equation in the s-u coordinate causes convergence problems with setting up a semi-implicit timestepping scheme to reduce computational cost. The prognostic variables are an approximation to the RossbyErtel potential vorticity Q, a scaled pressure thickness, the horizontal divergence, and the surface potential temperature. Results from the DCASL algorithm in two formulations of the s-u coordinate, differing only in the rate at which the vertical coordinate tends to u with increasing height, are assessed using the baroclinic instability test case introduced by Jablonowski and Williamson in 2006. The assessment is based on comparisons with available reference solutions as well as results from two other algorithms derived from the DCASL algorithm: one with a semi-Lagrangian solution for Q and another with an Eulerian grid-based solution procedure with relative vorticity replacing Q as the prognostic variable. It is shown that at intermediate resolutions, results comparable to the reference solutions can be obtained.

The diabatic contour‐advective semi‐Lagrangian algorithms for the dynamical core of global models in terrain‐following isentropic and pressure vertical coordinates

“…The behaviour of the – solution is nearly the same as that due to the corresponding DCASL algorithm developed in Mohebalhojeh et al . (2016), with substantial deviations from the reference solution over the first 10 days of the simulation. As noted in Mohebalhojeh et al .…”

“…As noted in Mohebalhojeh et al . (2016), the deviation mainly results from the initialisation problem in the zonal mode of the solution which is largely reduced in norm. With the use of exact initialisation, the two – p solutions closely follow each other in both norms and fall in the range of reference values reported in Jablonowski and Williamson (2006a).…”

“…Φ denotes geopotential, which is obtained at full levels by the proper discrete form of the hydrostatic equation in each vertical coordinate layers (half-levels in Figure 1). Equation 17is the same as equation (3.18) of Mohebalhojeh et al (2016) except for the addition of two terms in the right-hand side: the term due to the time tendency of F which comes from the implementation of ALE, and the term c for diabatic heating. When F∕ = 0, (18) for vertical velocity becomes singular.…”

“…Among the promising methods for representation of PV is the contour‐advective semi‐Lagrangian (CASL) algorithm (Dritschel and Ambaum, 1997; Dritschel et al ., 1999; Mohebalhojeh and Dritschel, 2004) and its extension to include diabatic processes called DCASL (Dritschel and Ambaum, 2006). The development of a dynamical core for a hydrostatic atmospheric general circulation model based on the DCASL algorithm has followed successful implementation of DCASL for the single‐layer shallow‐water, multi‐layer Boussinesq and non‐Boussinesq primitive equations, and the hybrid isentropic, terrain‐following vertical coordinate on the Charney–Phillips (CP) grid (Mohebalhojeh and Dritschel, 2007; 2009; Mirzaei et al ., 2012; Mohebalhojeh et al ., 2016). In the DCASL algorithm, both contour and grid representations are used for PV or a variant of PV.…”

An assessment of radiative impacts of CO2 on baroclinic instability using idealized life cycles

“…Given that the baroclinic life‐cycle problem is inherently related to the dynamics of storm tracks, the experiments and analyses provided may also help us in understanding the response of the extratropical storm tracks to climate change. To carry out the experiments, the Rapid Radiative Transfer Model for General Circulation Models (RRTMG) parametrization scheme has been coupled with the Diabatic Contour‐Advective Semi‐Lagrangian dynamical core called DCASL (Mohebalhojeh et al ., 2016; Laghaeizadeh et al ., 2020). To complete the coupling in terms of radiative interaction with the ground, a representation of the surface processes adopted from ECHAM6 has been used.…”