SRNWP Mini-Workshop on Numerical Techniques

Meteo-France, Toulouse, 12-13 December 2002

I) OVERVIEW

-----------

The areas of interest for the Working Group have been
divided into five topics, on which active research is being done

1) Semi-Lagrangian

2) Time and Space
Numerical Schemes

3) Numerics of
Physics-Dynamics Interface Schemes

4) Orography

5) Shared
academics experiments

II) REPORT OF TALKS

-------------------

1) Semi-Lagrangian

Despite the renaming of the working group (from "Semi-Lagrangian"
to "Numerical Techniques"), a significant level of activity is still
taking place in the Semi-Lagrangian. However, most of the basic numerical
problems linked to the operational use of SL schemes for NWP seem to be solved,
and the main focus is now set on refinements of the properties of the scheme,
as e.g. conservation, definite positiveness, monotonicity, etc.

Among the Groups participating to the SRNWP Network, UKMO and HIRLAM Groups have an active position for these topics, since they examined formulations trying to achieve the above desirable properties.

While HIRLAM CISL (Cell-Integrated Semi-Lagrangian) scheme is monotonous, mass-conserving, definite positive, and is implemented successfully in a shallow-water model with semi-implicit treatment of the inertia-gravity waves, it results in a significant increase of computation and storage, and the required spatial truncation could result in exaggerated diffusive properties. A strategy for the 3D formulation has already been defined.

The SLICE (Semi-Lagrangian Inherently Conserving and Efficient) scheme of UKMO, based on a cascade interpolation is conservative (but not monotonous) and is implemented in 2D. However it seems to be cost-effective and could possibly be extended to 3D with no major difficulty, leading to a possible application for NWP in short terms, a perspective which will be followed with interest by the other partners of the Network. The authors also claim that their scheme achieves competitive or better accuracy than that of standard non-conservative interpolating semi-Lagrangian schemes.

ECMWF expressed their current interest in also studying
the possibility of using more conservative schemes (possibly based on cascade
interpolations).

The Institute of Numerical Mathematics of the Russian
Academy of Sciences is implementing a finite-difference 2D SL model with a
variable resolution unstaggered grid, which requires choosing the vorticity and
divergence as prognostic variables. This minimises the number of trajectories
and interpolations and results in a cost-effective model. The extension to 3D
is straightforward.

ECMWF reported a case for which the so-called SETTLS (Stable Extrapolation Two-Time-Level Scheme) used for
the 2-Time-Level SL was not enough to prevent the growth of an instability over
Antarctica. In the absence of a better proposal so far, the problem, identified
in the vertical part of the trajectory computations, was solved by using
selectively a first-order vertical trajectory scheme instead of the usual
second-order scheme at places and time where the instability is suspected to
occur.

2) Time and Space Numerical Schemes

A significant level of activity is arising in this field,
especially in application to nonhydrostatic systems. RPN de Dorval (Canada),
although not present at the workshop, is also participating to developing this
area of interest.

UKMO Group analysed the numerical stability and accuracy
of the 1D and 2D versions of the New Dynamics (ND), in particular top and
bottom boundary conditions of potential temperature (ND uses the
Charney-Phillips staggering in the vertical) for isentropic and isothermal
basic state. They identified pathological combinations of the BC, which were
subsequently removed from the operational use.

ALADIN Group used a very similar approach to show the
beneficial impact that can be expected from using an Iterative Centred Implicit
scheme to solve their Euler Equations system with an enhanced robustness. Using
this scheme, Aladin-NH was shown to be able to perform stable integrations of a
real case at mesoscale resolution with a 2-TL scheme free of any artificial
stabilising feature as decentring or others.

3) Numerics of Physico-Dynamical Interface Schemes

HIRLAM Group is still continuing investigations on the
best way to include physical tendencies in the time-loop of their dynamical
model-core (2TLSLSI). They found better results and a significant decrease in
the sensitivity to the time-step length when using tendencies averaged along
the SL trajectories, as done at ECMWF. Some tendencies can also be optionally
averaged in time.

UKMO Group is building an analytical framework to study
this issue from the theoretical point of view. Their aim is to compare the
various possible physico-dynamical interfaces in terms of stability, accuracy,
steady-state response of the forced solution and resonant response. The
approach takes into account various formulations of the time-discretisation of
dynamical or physical forcings, but ignores the details linked to the different
individual physical parameterizations.

4) Orography

COSMO Group is still studying alternative solutions to terrain-following vertical coordinates which, they claim, could become problematic with very steep orography. The main problems are

- the generation of spurious local circulations over very steep slopes (probably due, at least partially, to the inaccuracy in the horizontal pressure double term)

- the exaggerated intensities of the precipitations, very
probably due to too strong a vertical component of the wind.

In their shaved-element method, a merge of adjacent
elements is used to avoid over-restrictive CFL stability conditions. The method
is shown to be effective, and leads to reasonable results.

However the other SRNWP Groups do not seem to face severe
problems with the use of terrain-following coordinates so-far. It must be nevertheless
said that they avoid the problem by smoothing the orography.

From the physical point of view, the COSMO Group also stressed that more attention should be paid to the definition of the orography and its links with the precipitations. They are now testing a precipitation scheme with explicit horizontal advection of the precipitation. This scheme should transport precipitation to the lee side of the Alps where a systematically deficit occurs. The COSMO Group has still no solution for the "roughness length dilemma" over mountains. The problem is that the large roughness lengths requested in order to have correct momentum fluxes and momentum budgets reduce far too strongly the surface winds when compared with observations. This problem has also been recognised and studied by the UKMO Group which also has no "universal" solution (i.e. a parameterization satisfying both aspects).

5) Shared academics experiments

The usefulness of this field of activity is usually
strongly advocated by UKMO group. This year, a "demanding" academic
experiment has been submitted to 4 groups (HIRLAM, UKMO, ALADIN, and the French
Meso-NH which was at the origin of this test). The settings and the results
have been presented by the ALADIN group. The case is demanding since it
involves a non-stationary non-linear trapped lee-wave which needs a good
numerical scheme in order to be simulated satisfactorily. The French groups
ALADIN-NH and Meso-NH did extensive experiments in order to get a good
knowledge of the behaviour of the system, and obtained quite similar results.
UKMO performed a preliminary experiment which exhibited some slight
differences, which should be explained by some further experiments. The HIRLAM
NH model gave significantly different results.

II) COMMENTS

------------

1) Semi-Lagrangian

The problems of conservation, monotonicity and
definite-positiveness are still of actuality, but only if they can be solved
with computationally efficient schemes. Simple "corrective" schemes
as the quasi-monotonous option in the ARPEGE/IFS model are not necessarily the
best solution (this option is abandoned for the vertical part of the moisture transport
in IFS). However, simple algorithm such as cascade interpolations will be
considered with interest for NWP if they are confirmed to provide a good
solution at a reasonable cost.

The 2-time-level SLSI (Semi-Lagrangian with Semi-Implicit) scheme is still at the limit of stability when large time-steps are used. Despite the SETTLS extrapolation, the SLSI scheme seems to exhibit unstable behaviours under certain conditions in the IFS.

It must be noticed that the interest for local mass conservative advection schemes does not come from the short-range NWP community, but primarily from the climate simulation scientists. For their GCM, they need mass conservation, particularly conservation of the water vapour.

There is definitely a revival of the cascade interpolation. But it must not be forgotten that this technique cannot be applied with a reduced grid.

2) Time and Space Numerical Schemes

The need of a deeper understanding of the behaviour of efficient numerical schemes especially in emerging NH models is a clear tendency. UKMO and Aladin-NH Groups achieved in some sense a remarkable convergence on this issue. To be well accepted, numerical schemes should ideally prove their suitability with theoretical arguments. Analytical studies often lead also to a greater potential for innovative ideas. UKMO and Aladin-NH Groups agreed to share more activity in this field in the next future.

3) Numerics of Physics-Dynamics Interface Schemes

The coupling of the physics to the dynamics seems to be
still a burning question in spite of the small number of talks on the subject.
Each Group is more or less working on the subject even if it is not the main
activity pushed in the flood-light. The question of the detrimental effect of only
first-order accuracy in time for the inclusion of physical parameterizations is
apparently concerning more than one Group.

4) Orography

The COSMO Group still seems isolated inside the Network,
when pointing to the unsuitability of terrain-following vertical coordinates
for horizontal resolutions of a few km over the Alps. A possible explanation is
that even if the problem actually exists, current NWP applications are maybe not
approaching the dangerous steepness. It would be interesting to better quantify
when terrain-following coordinates will be so bad that they must be abandoned.
This makes this topic a perfect candidate for the proposal developed in item
(5) below.

5) Shared academics experiments

Shared experiments can be viewed from two points of view:

- (i) Comparison
of results (and models) on a given fixed experiment

- (ii) Tool for trying to compare options which are not
under switch for any available model.

The first approach has been used for the experiments described above for the non-linear non-stationary trapped lee-wave experiment. However, this exercise has its own limits which are easily reached. We claim that the second approach could be more fructuous, and we propose to promote it in the future instead of the first one. A formal proposal in this way will be sent to all SRNWP-NT participants, with more details on the motivations.

Pierre Bénard (with some complements of J. Quiby