Governing Equations I: reference equations at the scale of the continuum

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. Mean free path Newton’s second law From molecules to continuum Navier-Stokes equations Boltzmann equations kinematic viscosity ~1.x10-6 m2s-1, water ~1.5x10-5 m2s-1, air number of particles Euler equations individual particles statistical distribution 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 or or

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 moveswith the fluid and always contains the same particles => Lagrangian viewpoint

7. Eulerian versus Lagrangian 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., then Let 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 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 Eulerianbudget :

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

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. 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!

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.