A new locally mass conserving, monotonic and accurate method for
solving the continuityequations in Earth System Models
Eigil Kaas, University of Copenhagen (kaas@gfy.ku.dk)
Poster presented at the Second International Conference on Earth System
Modelling - Hamburg 2007.
Abstract
To correctly simulate the evolution
and interactions of chemical
constituents in Earth System Models it is important that the algorithms
of the underlying dynamical core fulfil a number of properties such as
high order of accuracy, inherent mass conservation, locality, positive
definiteness, monotonicity, consistency, and high order of numerical
stability and efficiency. Furthermore the so-called masswind
inconsistency problem should be minimised in a physically consistent
way. Aiming at fulfilling all these requirements simultaneously a new
scheme has been introduced. It is based on two new ingredients:
- a simple and cost effective method to ensure local, i.e.
inherent, mass conservation in traditional semi-implicit (SI),
semi-Lagrangian (SL)models
- an efficient locally mass conserving spatial filter.
The basic mass conserving SL method
is relatively simple to construct
from existing SL general circulation models. The trick is to introduce
weights at the upstream departure points ensuring that the total mass
given off by a given Eulerian grid point to the surrounding SL
departure points is equal to the volume represented by that grid point.
Based on the Eulerian grid point weights it is possible to obtain
departure point weights representing the divergence associated with the
SL trajectories as in the socalled cell-integrated semi-Lagrangian
(CISL) models. The filter is simple but it efficiently ensures
monotonicity and positive definiteness and surprisingly it enhances the
accuracy. Furthermore it can be constructed to generally maintain
constancy in non-divergent flows. So far the scheme has only been
tested in traditional two dimensional advection tests and in a
semi-Lagrangian, semi-implicit shallow water
model in plain geometry but with topography. Without the filter the
scheme is at least as accurate as the traditional SL schemes. The
filter enhances the accuracy for all tests carried out. However, it
becomes particularly impressive near sharp gradients and
discontinuities. The increase in numerical cost of the new scheme
relative to traditional SI-SL models is small, particularly when there
are several passive tracers