Governing Equations II

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

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. 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. 211-224 (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: 0

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: This form can be solved by eg. MPDATA package Smolarkiewicz and Margolin (1998)