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

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<p><strong>Abstract.</strong> 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. <br><br> 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. <br><br> The time stepping in Eulerian representation is achieved in two steps: first a mass-conserving Eulerian or semi-Lagrangian 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. <br><br> The HEL scheme has been designed to be accurate, multi-tracer 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. <br><br> The properties of HEL are here verified in two-dimensional tests. These include deformational passive transport on the sphere, and simulations with a semi-implicit shallow water model including topography.</p>




Kaas, E., Sørensen, B., Lauritzen, P. H., & Hansen, A. B. (2013). A hybrid Eulerian-Lagrangian numerical scheme for solving prognostic equations in fluid dynamics. Geoscientific Model Development, 6(6), 2023–2047. https://doi.org/10.5194/gmd-6-2023-2013

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