Semi‐implicit integration of the unified equations in a mass‐based coordinate: model formulation and numerical testing

This paper derives the unified equations of Arakawa and Konor rigorously formulated in a suitable mass‐based sigma coordinate and develops an efficient semi‐implicit integration scheme. The unified equations accurately capture the non‐hydrostatic small‐scale effects and retain the hydrostatic compressibility of the flow at large scales. As with the classical quasi‐hydrostatic equations, the underlying approximations filter vertically propagating acoustic waves. In contrast to the quasi‐hydrostatic equations, however, the filtering property of the unified equations requires that the wind field satisfy a divergence constraint similar to anelastic and pseudo‐incompressible (small‐scale limit) soundproof systems. An efficient semi‐implicit integration scheme for the unified equation system is achieved by combining a constant‐coefficient linear partitioning approach with an iterative implicit treatment of the nonlinear residuals arising from the soundproof divergence constraint. The resulting linear implicit problem to be solved at each iteration may be reduced to a single Helmholtz equation with horizontally homogeneous coefficients, which is akin to the one typically solved in the semi‐implicit integration of the quasi‐hydrostatic equations. The stability and accuracy of the developed semi‐implicit scheme for the unified equations in the mass‐based coordinate is numerically assessed by means of standard vertical plane test cases in linear and nonlinear atmospheric flow regimes. Moreover, in order to ascertain the convergence of the iterative semi‐implicit scheme, the test cases also include a large‐scale 3D configuration that resembles the stiffness typically encountered in global atmospheric models. Horizontal gravity wave test: distributions of potential temperature at t = 3,000 s. The upper and lower panels, respectively, represent the Euler equations and the unified model numerical solutions using ∇t = 6 s, ∇x = 500 m and ∇z≈500 m. The solid lines represent positive values, while the dashed lines are strictly negative values. The contour interval is 5 × 10−4 K