1. Governing Equations II
by Clive Temperton (room 124) and Nils Wedi (room 128)

2. Overview Scale analysis of momentum equations
Geostrophic and hydrostatic relations
Eta – vertical coordinate
Putting it all together -- ECMWF’s equations
Map projections and alternative spherical coordinates
Shallow-water equations

3. Scale analysis
Typical observed values for mid-latitude synopic systems:
U ~ 10 ms-1
W ~ 10-2 ms-1
L ~ 106 m
~ 103 m2s-2
f0 ~ 10-4 s-1
a ~ 107 m
H ~ 104 m

4. Scale analysis (continued)
U2/L
f0U
f0W
U2/a
UW/a
10-4
10-3
10-6
10-8
10-5
UW/L
f0U
U2/a g
10-7
10-3
10-5 10

5. Scale analysis (continued)
Interpretation:
Geostrophic relationship, Accelerations du/dt, dv/dt are
small differences between two large terms!
Usually drop Coriolis and metric terms which depend on w.
Make the hydrostatic approximation.

6. Hydrostatic approximation
Ignore vertically propagating acoustic waves
w is obtained diagnostically from the continuity equation.
We discuss the validity of the approximation later.

7. vertical coordinate
with

8. Putting it all together --- ECMWF’s equations
Momentum equations
with
Verify by inserting as exercise!

9. Putting it all together --- ECMWF’s equations
Thermodynamic equation
Moisture equation
Given the invoked approximations these are also called
the primitive equations in a generalized vertical coordinate.
Note: virtual temperature Tv instead of T from the equation of state.

10. Map projections – Polar stereographic projection
Invented to have an angle preserving mapping from the sphere onto a plane for convenience of display.
Hence idea to perform computations already in transformed coordinates.
Map factor:

11. Rotated spherical coordinates
Move pole so that area of interest lies on the equator such that system gives more uniform resolution.
Limited-area gridpoint models: HIRLAM, Ireland, UK Met. Office….; Côté et al. MWR (1993)
Move pole to area of interest, then “stretch” in the new “north-south” direction to give highest resolution over the area of interest.
Global spectral models – Arpege/IFS: Courtier and Geleyn, QJRMS Part B (1988)
Ocean models have sometimes two poles in the continents to give uniform resolution over the ocean of interest.

12. Cote et al. (1993)

13. Courtier and Geleyn (1988)

14. Shallow water equations
Further reading: Gill (1982)
Shallow water equations
Useful for dynamical core test cases before full implementation.
Route to develop isentropic or isopycnic models.
eg. Williamson et. al., JCP Vol 102, p (1992)

15. Shallow water equations
(a) Assume constant density, and horizontal pressure force independent of height:
(b) Velocity field initially independent of height will remain so, therefore omit vertical advection terms:
(c) Assume incompressible motion:

16. Shallow water equations(continued)
Boundary conditions: w=0 at z=0 and free surface following the motion at the top (dh/dt=w). Integrating the continuity equation we obtain:

17. Shallow water equations(continued)
In component form in Cartesian geometry:
(1)
(2)
(3)

18. Shallow water equations(continued)
Deriving an alternative form:
Kinetic Energy
Vorticity:
Divergence:

19. Shallow water equations(continued)
The vector product of vorticity with velocity is called the Lamb vector and its transformation is therefore sometimes called ``Lamb's transformation'‘:
In spherical geometry:

20. Shallow water equations with topography+
Coriolis omitted!
Numerical implementation by transformation to a
Generalized transport form for the momentum flux:
Shallow water model accepted model for equatorial waves with d = H and second term in continuity d = scale height
This form can be solved by eg. MPDATA package
Smolarkiewicz and Margolin (1998)