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

Abstract

Abstract.A new hybrid Eulerian-Lagrangian numerical scheme (HEL) for solving prognostic equations in fluid dynamics is proposed. The basic idea is to use an Eulerian as well as a fully Lagrangian representation of all prognostic variables.The time step in Lagrangian space is obtained as a translation of irregularly spaced Lagrangian parcels along downstream trajectories. Tendencies due to other physical processes than advection are calculated in Eulerian space, interpolated, and added to the Lagrangian parcel values. A directionally biased mixing amongst neighboring Lagrangian parcels is introduced. The rate of mixing is proportional to the local deformation rate of the flow.The time stepping in Eulerian representation is achieved in two steps: first a mass-conserving Eulerian or semiLagrangian scheme is used to obtain a provisional forecast. This forecast is then nudged towards target values defined from the irregularly spaced Lagrangian parcel values. The nudging procedure is defined in such a way that mass conservation and shape preservation is ensured in Eulerian space.The HEL scheme has been designed to be accurate, multitracer efficient, mass conserving, and shape preserving. In Lagrangian space only physically based mixing takes place; i.e., the problem of artificial numerical mixing is avoided. This property is desirable in atmospheric chemical transport models since spurious numerical mixing can impact chemical concentrations severely.The properties of HEL are here verified in twodimensional tests. These include deformational passive transport on the sphere, and simulations with a semi-implicit shallow water model including topography.

Construction of the SILAM Eulerian atmospheric dispersion model based on the advection algorithm of Michael Galperin

“…The application of the scheme to the highly challenging tests of Lauritzen et al (2012) allowed its evaluation in a global 2-D case and comparison with the state-of-the-art schemes evaluated by L14 and Kaas et al (2013).…”

“…In particular, the run with 0.75 • resolution and 120 time steps can be related to the performance of the HEL and CSLAM schemes, which were tested against the same test collection by Kaas et al (2013). Extrapolating the charts of Fig.…”

“…Extrapolating the charts of Fig. 13 of Kaas et al (2013) to one species (the range given there is 2-20 species), the test takes about 190 s for HEL and 300 s for CSLAM, but only 47 s for SILAM; i.e., the difference was about 4 and 6 times, respectively.…”

“…Formal benchmarks of the computers, the main uncertainty in this comparison, are essentially the same: Kaas et al (2013) with a mobile Intel Core i5-540M Duo (Intel Linpack 18.5 GFlops). These CPUs were also compared in http://www.…”

“…The memory bandwidth of our notebook, as always for compact computers, was modest: 7.2 GB s −1 (STREAM test, http://www.cs.virginia.edu/stream/ref.html accessed 5 October 2015). We used a GNU compiler with -O3 optimisation without parallelisation, similar to Kaas et al (2013).…”