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

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## Governing Equations II

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**Governing Equations II**by Clive Temperton (room 124) and Nils Wedi (room 128)**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**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**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)**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.**Hydrostatic approximation**Ignore vertically propagating acoustic waves w is obtained diagnostically from the continuity equation. We discuss the validity of the approximation later.**vertical coordinate**with**Putting it all together --- ECMWF’s equations**Momentum equations with Verify by inserting as exercise!**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.**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:**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.**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)**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:**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**Shallow water equations(continued)**• In component form in Cartesian geometry: (1) (2) (3)**Shallow water equations(continued)**• Deriving an alternative form: Kinetic Energy Vorticity: Divergence:**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:**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)

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