1. ECMWF Governing Equations 2 Slide 1 Governing Equations II: classical approximations and other systems of equations Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz By Sylvie Malardel (room 10a; ext. 2414) after Nils Wedi (room 007; ext. 2657)

2. ECMWF Governing Equations 2 Slide 2 The continuous set of dry adiabatic equations

3. ECMWF Governing Equations 2 Slide 3 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? grid scale resolved scale (?) The scale of the grid is much bigger than the scale of the continuum

4. ECMWF Governing Equations 2 Slide 4 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.

5. ECMWF Governing Equations 2 Slide 5 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 T L 799, but will not be properly represented.

6. ECMWF Governing Equations 2 Slide 6 Overview Introduction Scale analysis of momentum equations Geostrophic and hydrostatic relations IFS hydrostatic equations Map projections and alternative spherical coordinates Shallow-water equations Isopycnic/isentropic equations

7. ECMWF Governing Equations 2 Slide 7 Introduction Classical (and educational) approach: Simplify the governing equations BEFORE numerics is introduced (e.g. by scale analysis, hydrostatic approximation, Boussinesq or Anelastic approximation) depending on the context. Solutions to the governing equations have three important propagation speeds: acoustic waves (speed of sound), gravity waves (gravity-wave speed), and advective motion (wind speed), which affect the time-step that can be used in numerical procedures (constraint : cdt<dx). The equations may be filtered through the choice of approximate governing equations BEFORE OR through appropriate numerical treatments AFTER. Ex: Explicit anelastic NH reasearch model vs. Fully compressible SI SL NH model.

8. ECMWF Governing Equations 2 Slide 8 Introduction Remove singularities (i.e. Pole problem) by choosing appropriate set of equations (e.g. in IFS we use the vector form of the equations for semi-Lagrangian advection; for limited-area applications the Pole is often rotated to another location). Add mapping transformations to the equations for convenience of presentation and accuracy in limited-area models. Choose a generalized vertical coordinate for a proper treatment of the boundaries or better treatment of conservative variables or easier interpretation of results etc. Change pronostic variables to make the numerics easier or more accurate or more stable.

9. ECMWF Governing Equations 2 Slide 9 Introduction Use simpler sets of equations as a first approach : Shallow water equations are a useful tool to test a new dynamical core, as they represent a single vertical mode but a comprehensive set of horizontal solutions. Adiabatic tests before introducing full physics Single column models, 2D vertical plane model, 3D cartesian models…. f-plane, beta-plane models

10. ECMWF Governing Equations 2 Slide 10 Scale analysis Example : Typical observed values for mid-latitude synoptic systems: U~ 10 ms -1 W~ 10 -2 ms -1 L~ 10 6 m ~ 10 3 m 2 s -2 f 0 ~ 10 -4 s -1 a~ 10 7 m H~ 10 4 m

11. ECMWF Governing Equations 2 Slide 11 Scale analysis (continued) UW/L f0Uf0UU 2 /a g 10 -7 10 -3 10 -5 10 U 2 /L f0Uf0U f0Wf0W U 2 /a UW/a 10 -4 10 -3 10 -6 10 -8 10 -5 10 -3

12. ECMWF Governing Equations 2 Slide 12 Scale analysis (continued) Consequences if you want to resolve synoptic motions in the mid-latitudes: Quasi-Geostrophic balance : accelerations du/dt, dv/dt are small differences between two large terms Allow to drop Coriolis and metric terms which depend on w Make the hydrostatic approximation Assume a shallow atmosphere with radius r = a + z ~ a 1 2 note

13. ECMWF Governing Equations 2 Slide 13 Scale analysis (continued) If you want to resolve smaller scales:, ex. If you want to cover other latitudes, ex: BUT A small scale circulation may be far from the geostrophic balance. The wind may be very ageostrophic. AND Near the equator, synoptic scale motion may be strongly ageostrophic

14. ECMWF Governing Equations 2 Slide 14 Hydrostatic balance The pressure gives the weight of the atmosphere above This is true for a very wide range of meteorological scales

15. ECMWF Governing Equations 2 Slide 15 Hydrostatic approximation UW/L 10 -7 10 -1 For synoptic, at mid-latitude The vertical acceleration is still very negligible compared with the residual force terms when the hydrostatic balance has been removed

16. ECMWF Governing Equations 2 Slide 16 Hydrostatic approximation : consequences Filter of isotropic acoustic waves : acoustic pressure perturbations are not related to the weight of the atmospheric column, then they are not described anymore. w is obtained diagnostically from the continuity equation, in agreement with an instantaneous mass reorganisation to fulfil the hydrostatic balance

17. ECMWF Governing Equations 2 Slide 17 Use p as a vertical coordinate or any other pressure type coordinate (terrain following : sigma, hybrid) Hydrostatic approximation : consequences Hypsometric equation : the thickness between 2 isobars is proportional to the mean temperature in the layer between these 2 isobars The geopotential of a layer is obtained thanks to the integration of the hydrostatic equation

18. ECMWF Governing Equations 2 Slide 18 Validity of hydrostatic approximation For internal gravity wave : Toward the smaller scales : Hydrostatic approximation if :

19. ECMWF Governing Equations 2 Slide 19 Hydrostatic vs. Non-hydrostatic Horizontal divergence for a flow past a 3D - mountain on the sphere ( r = a/100 ) with a T159L91 IFS simulation hydrostatic non-hydrostatic

20. ECMWF Governing Equations 2 Slide 20 Hydrostatic vs. Non-hydrostatic Hydrostatic waves only Hydrostatic + non-hydrostatic waves

21. ECMWF Governing Equations 2 Slide 21 Anelastic approximation What is anelastic approximation? Neglect the elasticity of the atmosphere which is responsible for the accoustic wave propagation. How to do that ? Modify the continuity equation in order to neglect the quick response of the density to compression The air is still compressible in the sense that its density may change, for exemple in a vertical motion, but it will change passively, without elastic reaction or oscillations.

22. ECMWF Governing Equations 2 Slide 22 Anelastic approximation : consequences Balance between horizontal and vertical mass fluxes The anelastic approximation may be useful if you need to take into account the NH effect and you dont have very sophisticated numerics to treat the sound waves.

23. ECMWF Governing Equations 2 Slide 23 Primitive (hydrostatic) equations in IFS for Momentum equations Sub-grid model : physics Numerical diffusion

24. ECMWF Governing Equations 2 Slide 24 IFS hydrostatic equations Thermodynamic equation Moisture equation Note: virtual temperature T v instead of T from the equation of state.

25. ECMWF Governing Equations 2 Slide 25 IFS hydrostatic equations Continuity equation Vertical integration of the continuity equation in hybrid coordinates

26. ECMWF Governing Equations 2 Slide 26 One word about water species…. Phase changes are treated inside the physics (P terms) But the pronostic water species have a weight. They are included in the full density of the moist air and in the definition of the specific variables. It does some tricky changes in the equations. For ex. : Pronostic water species should be advected. They are then also treated by the dynamics. Perfect gas equation

27. ECMWF Governing Equations 2 Slide 27 Map projections 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: Wind components in the model are then usually not the real zonal and meridional winds.

28. ECMWF Governing Equations 2 Slide 28 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.

29. ECMWF Governing Equations 2 Slide 29 Courtier and Geleyn (1988)

30. ECMWF Governing Equations 2 Slide 30 Shallow water equations Useful for (hydrostatic) dynamical core test cases before full implementation. Route to interpret isentropic or isopycnal models. eg. Williamson et. al., JCP Vol 102, p. 211-224 (1992) Further reading: Gill (1982) Free surface

31. ECMWF Governing Equations 2 Slide 31 Shallow water equations Assume constant density + free surface at z=h(x,y) horizontal pressure force independent of height: Horizontal wind independent of height as : Use only the horizontal motion equation at the ground, where w=0

32. ECMWF Governing Equations 2 Slide 32 Shallow water equations 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:

33. ECMWF Governing Equations 2 Slide 33 Shallow water equations In component form in Cartesian geometry: (1) (2) (3)

34. ECMWF Governing Equations 2 Slide 34 Shallow water equations Deriving an alternative form: Vorticity: Divergence: Kinetic Energy

35. ECMWF Governing Equations 2 Slide 35 Shallow water equations The advection term in the velocity equation may be transform into the Lamb form; the vector product of vorticity with velocity is called the Lamb vector (useful for generalised form of Bernoulli equation): In spherical geometry:

36. ECMWF Governing Equations 2 Slide 36 Isopycnal/isentropic coordinates : representation of a stratified fluid as a superposition of shallow water models (model levels = material surface) defines depth between shallow water layers (momentum) (continuity) (thermo) (hydrostatic)

37. ECMWF Governing Equations 2 Slide 37 More general isentropic-sigma equations Konor and Arakawa (1997); terrain-following