A class of deformational flow test cases for linear transport problems on the sphere

Abstract

a b s t r a c tA class of new benchmark deformational flow test cases for the two-dimensional horizontal linear transport problems on the sphere is proposed. The scalar field follows complex trajectories and undergoes severe deformation during the simulation; however, the flow reverses its course at half-time and the scalar field returns to its initial position and shape. This process makes the exact solution available at the end of the simulation, and facilitates assessment of the accuracy of the underlying transport scheme. A procedure to eliminate possible cancellations of errors when the flow reverses is proposed.The test suite consists of four cases. Three are based on non-divergent flow fields and one on a divergent flow. The initial conditions are prescribed in terms of regular latitude-longitude spherical coordinates and are easy to implement. The divergent flow is specifically aimed for conservative global transport schemes to test for conservation, consistency and monotonicity (or positivity) of limiters/filters in a challenging flow environment. In the context of semi-Lagrangian schemes, the time-varying flow fields can be used to test trajectory algorithms where the exact trajectories do not follow great-circle arcs. The characteristics of the test cases are demonstrated with two different transport schemes.

A hybrid Eulerian–Lagrangian numerical scheme for solving prognostic equations in fluid dynamics

“…The two analytical flow fields used have originally been proposed by Nair and Lauritzen (2010), and they include a non-divergent as well as a divergent flow. In both cases the Lagrangian parcels follow relatively complex trajectories, and the flow is composed of a deformational deformation component, which is different in the two cases, and an overlaid translational flow.…”

Downscale cascades in tracer transport test cases: an intercomparison of the dynamical cores in the Community Atmosphere Model CAM5

“…The deformation test case number 4 given in Nair and Lauritzen (2010) for two-dimensional flow is a suitable example. The resolution that will be used for calculating the reference solution in our subgrid scale tests must be the same resolution that is used when calculating the accuracy of the reference solution on the deformation test of Nair and Lauritzen (2010). The deformation test of Nair and Lauritzen (2010) is designed so that the whole tracer is resolved during the simulation, and the flow is reversed so that error norms can be calculated using the initial conditions.…”

“…Mahlman and Sinclair, 1977;Rood, 1987;Zerroukat et al, 2005;Colella and Sekora, 2008), or solid-body rotation in two (Williamson et al, 1992) or three dimensions . More complex, deformational flow test cases have recently been developed (Nair and Machenhauer, 2002;Nair and Jablonowski, 2008;Nair and Lauritzen, 2010; to provide a more challenging test on the sphere. Each of these tests either returns the tracer to its starting position or has an analytical solution, which gives an exact solution that can be used to calculate error norms and convergence rates.…”

Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

“…In the following we consider the smooth deformational-flow test introduced in [26]. For the initial distribution we use two Gaussian hills which will be highly deformed through a non-divergent wind field.…”

“…For all test cases in this section we choose the time step in such a way that the maximal CFL is always ≤ 0.8 (fixed) for all resolutions. For a general description of this test case we refer to [26]. There, we can find an accurate trajectory computation algorithm (to find the departure points), which is a combination of a Taylor series expansion and splitting the trajectory into segments.…”