A hybrid Eulerian–Lagrangian numerical scheme for solving prognostic equations in fluid dynamics 1Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
2Climate and Global Dynamics Division, Boulder, Colorado, USA
3Aarhus University, Department of Environmental Science, Roskilde, Denmark
4National Institute of Water and Atmospheric Research, Lauder, New Zealand
Received: 18 June 2013 – Published in Geosci. Model Dev. Discuss.: 18 July 2013 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
Revised: 30 September 2013 – Accepted: 15 October 2013 – Published: 22 November 2013
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 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.
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
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.
Citation: Kaas, E., Sørensen, B., Lauritzen, P. H., and Hansen, A. B.: A hybrid Eulerian–Lagrangian numerical scheme for solving prognostic equations in fluid dynamics, Geosci. Model Dev., 6, 2023-2047, doi:10.5194/gmd-6-2023-2013, 2013.