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Properties of the equations
of motion
January 2001
By Mike
Cullen
European Centre for Medium-Range Weather
Forecasts
| NUMERICAL METHODS | |||||||
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Abstract
Some generic properties of the nonlinear equations
of fluid flow are demonstrated with simple illustrative problems. Properties
of the shallow water model are then described, and the solutions shown
to be close to that of a `balanced' approximation to them.
In three dimensions, the generalisation of the concept
of `balance' leads to models from which sound waves have been filtered,
in particular the incompressible and anelastic models. Properties of
their solutions are described. In particular, analytic solution of the
equations requires solution of an elliptic problem, which thus also
has to be solved in numerical models using the equations. The hydrostatic
equations can be solved without solving an elliptic problem, but it
is shown that this means that the solutions break down for weak stratification.
Use of the hydrostatic approximation in numerical models requires use
of a numerical equivalent of a non-hydrostatic pressure to ensure stability.
Operational models are more correctly viewed as solving
space-time averages of the equations. Both Eulerian and Lagrangian averaging
procedures are illustrated. In particular, both suggest that the averaged
variable representing the fluid trajectory is best treated as different
from that representing the momentum.
Averaged equations can be related to filtered models
in which all inertia-gravity waves are removed. While such models do
not give a complete description of the atmosphere, since they exclude
real waves, they can describe the motions that are well-resolved and
predictable by operational models. Their properties are thus useful
in designing models, particularly the way that the computation of the
resolved flow is related to the sub-grid models which parametrise the
unresolved motions.
1 . Introduction
2 . Observed behaviour
3 . Toy problems
| 3.1 Introductory remarks |
| 3.2 Examples |
| 3.3 Two-dimensional incompressible Euler equations |
4 . Shallow water equations
| 4.1 Basic properties |
| 4.2 Properties of `slow' solutions |
5 . Three dimensional equations
| 5.1 Basic equations, and filtering of fast waves |
| 5.2 Solution of the inviscid incompressible equations |
| 5.3 Practical implications of solution procedure |
| 5.4 The hydrostatic approximation |
6 . Averaged equations
| 6.1 Averaged equations, and their approximation by balanced models |
| 6.2 Computations of averaged solutions. |
| 6.3 Balanced models |
| 6.4 Structure of the solution of the semi-geostrophic model |
7 . Summary
REFERENCES
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13.06.2002 |
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