1. Governing Equations I: reference equations at the scale of the continuum
by Sylvie Malardel (room 10a; ext. 2414)
after Nils Wedi (room 007; ext. 2657)
Recommended reading:
An Introduction to Dynamic Meteorology, Holton (1992)
An Introduction to Fluid Dynamics, Batchelor (1967)
Atmosphere-Ocean Dynamics, Gill (1982)
Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Durran (1999)
Illustrations from “Fondamentaux de Meteorologie”, Malardel (Cepadues Ed., 2005)
Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz

2. Overview Introduction : from molecules to “continuum”
Eulerian vs. Lagrangian derivatives
Perfect gas law
Continuity equation
Momentum equation (in a rotating reference frame)
Thermodynamic equation
Spherical coordinates
“Averaged” equations in numerical weather prediction models
(Note : focus on “dry” (no water phase change) equations)

3. From molecules to continuum
Mean free path
Newton’s second law
Navier-Stokes equations
Boltzmann equations
number of particles
kinematic viscosity
~1.x10-6 m2s-1, water
~1.5x10-5 m2s-1, air
Euler equations
individual particles
statistical distribution
continuum
From molecules to continuum
Mean value of the parameter
Molecular
fluctuations
Continuous variations
Mean value at the scale of the continuum
Note: Simplified view !

4. Perfect gas law The pressure force on any surface element containing M
The temperature is defined as
The link between the pressure, the temperature and the density of molecules in a perfect gas :

5. Perfect gas law : other forms

6. Fundamental physical principles
Conservation of mass
Conservation of momentum
Conservation of energy
Consider budgets of these quantities for a control volume
(a) Control volume fixed relative to coordinate axes
=> Eulerian viewpoint
(b) Control volume moves with the fluid and always contains the same particles
=> Lagrangian viewpoint
Conservation in Eulerian depends on fluxes through the control volume
Conservation assured since always the same particles contained in control volume

7. Eulerian versus Lagrangian
Conservation in Eulerian depends on fluxes through the control volume
Conservation assured since always the same particles contained in control volume
Lagrangian : evolution of a quantity following the particules in their motion
Eulerian : Evolution of a quantity inside a fixed box

8. Eulerian vs. Lagrangian derivatives
Particle at temperature T at position at time moves to in time .
Temperature change given by Taylor series:
i.e.,
Let
then
local rate of change
is the rate of change
following the motion.
is the rate of change
at a fixed point.
advection
total derivative

9. Sources vs. advection

10. Mass conservation Mass flux Mass flux on left face on right face
Inflow at left face is Outflow at right face is
Difference between inflow and outflow is per unit volume.
Similarly for y- and z-directions.
Thus net rate of inflow/outflow per unit volume is
= rate of increase in mass per unit volume
= rate of change of density
=> Continuity equation (Eulerian point of view)
Mass flux
on left face
Mass flux
on right face
Eulerian budget :

11. Mass conservation (continued)
Continuity equation, mathematical transformation
Continuity equation : Lagrangian point of view
By definition, mass is conserved in a Lagrangian volume:

12. Momentum equation : frames
Newton’s Second Law in absolute frame of reference:
N.B. use D/Dt to distinguish the total derivative in the absolute frame of reference.
(1)
We want to express this in a reference frame which rotates with the earth
Rotating frame
“fixed” star

13. Momentum equation : velocities
= angular velocity of earth .
= position vector relative to earth’s centre
For any vector
Evolution in absolute frame or
Evolution in rotating frame

14. Momentum equation : accelerations
Then from (1):
In practice:

15. Momentum equation : Forces
Forces pressure gradient, gravitation, and friction
Where = specific volume (= ), = pressure,
= sum of gravitational and centrifugal force,
= molecular friction,
= vertical unit vector
Magnitude of varies by ~0.5% from pole to equator and by ~3% with altitude
(up to 100km).

16. Spherical polar coordinates (in the same rotating frame)
: = longitude, = latitude, = radial distance
Orthogonal unit vectors: eastwards, northwards, upwards.
As we move around on the earth, the orientation of the coordinate system changes:
Extra terms due to
the spherical curvature

17. Components of momentum equation
with

18. The “shallow atmosphere” approximation
“Shallowness approximation” –
For consistency with the energy conservation and angular momentum conservation, some terms have to be neglected in the full momentum equation. This approximation is then valid only if these terms are negligible.
with

19. Energy conservation : Thermodynamic equation (total energy conservation – macroscopic kinetic energy equation)
First Law of Thermodynamics:
where I = internal energy,
Q = rate of heat exchange with the surroundings
W = work done by gas on its surroundings by compression/expansion.
For a perfect gas, ( = specific heat at constant volume),

20. Thermodynamic equation (continued)
Alternative forms:
Eq. of state:
R=gas constant and

21. Thermodynamic equation (continued)
Alternative forms:
where
is the potential temperature
The potential temperature is conserved in a Lagrangian “dry” and adiabatic motion

22. The scale of the grid is much bigger than the scale of the continuum
Where do we go from here ?
So far : We derived a set of evolution equations based on 3 basic conservation principles valid at the scale of the continuum : continuity equation, momentum equation and thermodynamic equation.
What do we want to (re-)solve in models based on these equations?
The scale of the grid is much bigger than the scale of the continuum
resolved scale (?)
grid scale

23. Observed spectra of motions in the atmosphere
Spectral slope near k-3 for wavelengths >500km. Near k-5/3 for shorter wavelengths.
Possible difference in larger-scale dynamics.
No spectral gap!
Possibly investigate the Fourier representation of the heavy side function, example that single point can transfer
Energy to all scales

24. Scales of atmospheric phenomena
Practical averaging scales do not correspond to a physical scale separation.
If equations are averaged, there may be strong interactions between resolved and unresolved scales.

25. “Averaged” equations : from the scale of the continuum to the mean grid size scale
The equations as used in an operational NWP model represent the evolution of a space-time average of the true solution.
The equations become empirical once averaged, we cannot claim we are solving the fundamental equations.
Possibly we do not have to use the full form of the exact equations to represent an averaged flow, e.g. hydrostatic approximation OK for large enough averaging scales in the horizontal.

26. “Averaged” equations
The sub-grid model represents the effect of the unresolved scales on the averaged flow expressed in terms of the input data which represents an averaged state.
The mean effects of the subgrid scales has to be parametrised.
The average of the exact solution may *not* look like what we expect, e.g. since vertical motions over land may contain averages of very large local values.
The averaging scale does not correspond to a subset of observed phenomena, e.g. gravity waves are partly included at TL799, but will not be properly represented.

27. Demonstration of the averaging effect
Use high resolution (10km in horizontal) simulation of flow over Scandinavia
Average the results to a scale of 80km
Compare with solution of model with 40km and 80km resolution
The hope is that, allowing for numerical errors, the solution will be accurate on a scale of 80km
Compare low-level flows and vertical velocity cross- section, reasonable agreement
Cullen et al. (2000) and references therein

28. High resolution numerical solution
Test problem is a flow at 10 ms-1 impinging on Scandinavian orography
Resolution 10km, 91 levels, level spacing 300m
No turbulence model or viscosity, free-slip lower boundary
Semi-Lagrangian, semi-implicit integration scheme with 5 minute timestep

29. Low-level flow
40km resolution
10km resolution

30. Low-level flow
40km resolution
10km resolution averaged to 80km

31. Cross-section of potential temperature

32. x-z vertical velocity
40km resolution
10km resolution averaged to 80km

33. Conclusion
Averaged high resolution contains more information than lower resolution runs.
The better ratio of comparison was found approximately as
dx (averaged high resol) ~ ·dx (lower resol) with  ~1.5-2
The idealised integrations suggest that the predictions represent the averaged state well (despite hydrostatic assumption for example).
The real solution is much more localised and more intense.
Piotr mentions sqrt(3), theory unknown.