1. Governing Equations III
by Nils Wedi (room 007; ext. 2657)
Thanks to Piotr Smolarkiewicz

2. Introduction
Continue to review and compare a few distinct modelling approaches for atmospheric and oceanic flows
Highlight the modelling assumptions, advantages and disadvantages inherent in the different modelling approaches

3. Dry “dynamical core” equations
Shallow water equations
Isopycnic/isentropic equations
Compressible Euler equations
Incompressible Euler equations
Boussinesq-type approximations
Anelastic equations
Primitive equations
Pressure or mass coordinate equations √ √
All variants are or have been used in atmospheric or oceanic predictions, Note on wave breaking: waves can break but invalidate
The hydrostatic assumption as they do, there is no verification as to what wave breaking does in hydrostatic vs non-hydrostatic
Formulations nor is there global scale proof for the advantage of non-hydrostatic formulations, however, I’ll show some smaller scale
Differences that may matter √

4. Euler equations for isentropic inviscid motion
Not sure if those have been used for atmospheric predictions, but always used for academic exercises

5. Euler equations for isentropic inviscid motion
Speed of sound (in dry air 15ºC dry air ~ 340m/s)

6. Reference and environmental profiles
Distinguish between
(only vertically varying) static reference or basic state profile (used to facilitate comprehension of the full equations)
Environmental or balanced state profile (used in general procedures to stabilize or increase the accuracy of numerical integrations; satisfies all or a subset of the full equations, more recently attempts to have a locally reconstructed hydrostatic balanced state or use a previous time step as the balanced state
Overbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subset

7. The use of reference and environmental/balanced profiles
For reasons of numerical accuracy and/or stability an environmental/balanced state is often subtracted from the governing equations
Overbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subset
Clark and Farley (1984) 

8. *NOT* approximated Euler perturbation equations
eg. Durran (1999)
Always possible, hope is however, that if the basic state is chosen well and the perturbations are small
If not the perturbations may be as large as the basic state !
Now, we introduce various approximations, all introduced in such a way that the resulting equations
Satisfy some important invariants like energy and potential vorticity, however, note that these properties while
Invariant in their respective system are not necessarily the same when compared to each other !!!
using:

9. Incompressible Euler equations
eg. Durran (1999); Casulli and Cheng (1992); Casulli (1998);
Note, that incompressible does not necessarily mean \rho=constant !!! Example later.

10. Example of simulation with sharp density gradient
Animation:
"two-layer" simulation of a critical flow
past a gentle mountain
Fig5 as movie
Compare to shallow water:
reduced domain simulation with H prescribed
by an explicit shallow water model

11. Two-layer t=0.15

12. Shallow water t=0.15

13. Two-layer t=0.5

14. Shallow water t=0.5

15. Classical Boussinesq approximation
eg. Durran (1999)
System closes via diagnostic equation for pressure

16. Projection method
Subject to boundary conditions !!!

17. Integrability condition
Ensure that continuity is fulfilled. The numerics and all approaches in CFD are to avoid the creation of spurious forces
that alter flow magnitude through discretization, formulation or truncation errors.
With boundary condition:

18. Ap = f Solution Due to the discretization in space a banded matrix
A arises with size (N x L)2 N=number of gridpoints, L=number of levels
Classical schemes include Gauss-elimination for
small problems, iterative methods such as Gauss-Seidel and over-relaxation methods. Most commonly used techniques for the iterative solution of sparse linear-algebraic systems that arise in fluid dynamics are the preconditioned conjugate gradient method, e.g. GMRES, and the multigrid method (Durran,1999). More recently, direct methods are proposed based on matrix-compression techniques (e.g. Martinsson,2009)

19. Importance of the Boussinesq linearization in the momentum equation
Two layer flow animation with density ratio 1:1000
Equivalent to air-water
Incompressible Euler two-layer fluid flow past obstacle
However, in the atmospheric context it is not so clear what influence this has on other processes.
Test the formulation of Durran, the pseudo compressible equations!
Incompressible Boussinesq two-layer fluid flow past obstacle
Two layer flow animation with density ratio 297:300
Equivalent to moist air [~ 17g/kg] - dry air
Incompressible Euler two-layer fluid flow past obstacle
Incompressible Boussinesq two-layer fluid flow past obstacle

20. Anelastic approximation
Batchelor (1953); Ogura and Philipps (1962); Wilhelmson and Ogura (1972); Lipps and Hemler (1982); Bacmeister and Schoeberl (1989); Durran (1989); Bannon (1996);

21. Anelastic approximation
Lipps and Hemler (1982);

22. Numerical Approximation
Compact conservation-law form:
Lagrangian Form: 

23. Numerical Approximation
with
LE, flux-form Eulerian or Semi-Lagrangian
formulation using MPDATA advection schemes
Smolarkiewicz and Margolin (JCP, 1998) 
with
Prusa and Smolarkiewicz (JCP, 2003)
specified and/or periodic boundaries

24. Importance of implementation detail?
Example of translating oscillator (Smolarkiewicz, 2005):
time

25. Example ”Naive” centered-in-space-and-time discretization:
However, in the atmospheric context it is not so clear what influence this has on other processes.
Test the formulation of Durran, the pseudo compressible equations!
Non-oscillatory forward in time (NFT) discretization:
paraphrase of so called “Strang splitting”,
Smolarkiewicz and Margolin (1993)

26. Compressible Euler equations
Davies et al. (2003)

27. Compressible Euler equations

28. Pressure based formulations Hydrostatic
Hydrostatic equations in pressure coordinates
Note, that in particular popular in meteorology due to the good approximation of the hydrostatic assumption
Like shallow water and isentropic models, success of ECMWF based on these equations !

29. Pressure based formulations Historical NH
(Miller (1974); Miller and White (1984))

30. Pressure based formulations
(Rõõm et. al (2001),
and references therein)
developed within the
HIRLAM group
Same is true for these formulations since it requires that \partial p/\partial z and \rho is always well defined.
This is not the case in fluids with density discontinuities !!!

31. Pressure based formulations Mass-coordinate
Define ‘mass-based coordinate’ coordinate:
Laprise (1992)
‘hydrostatic pressure’ in a vertically
unbounded shallow atmosphere
By definition
monotonic with
respect to geometrical
height
relates to Rõõm et. al (2001):

32. Pressure based formulations
Laprise (1992)
Momentum equation
Thermodynamic equation
Note the easy form of the continuity equation and the prognostic equation for pressure absent in anelastic models
Continuity equation
with

33. Compressible vs. anelastic
Davies et. al. (2003)
Lipps & Hemler approximation
Hydrostatic

34. Compressible vs. anelastic
Equation set V A B C D E
Fully compressible 1
Hydrostatic
Pseudo-incompressible (Durran 1989)
Anelastic (Wilhelmson & Ogura 1972)
Anelastic (Lipps & Hemler 1982)
Boussinesq

35. Compressible vs. anelastic
Normal mode analysis done on linearized equations noting distortion of Rossby modes if equations are (sound-)filtered
Differences found with respect to deep gravity modes between different equation sets. Conclusions on gravity modes are subject to simplifications made on boundaries, shear/non-shear effects, assumed reference state, increased importance of the neglected non-linear effects.
The Anelastic/Boussinesq simplification in the momentum equation (not when pseudo-incompressible) simplifies baroclinic production of vorticity, i.e. possible steepening effect of vortices missing (see also §10.4 and Fig in Dutton (1967))

36. Compressible vs. anelastic
Recent scale analysis suggests the validity of anelastic approximations for weakly compressible atmospheres, low Mach number flows and realistic atmospheric stratifications (Δ  30K) (Klein et al., 2010), well beyond previous estimates!
Recently, Arakawa and Konor (2009) combined the hydrostatic and anelastic equations into a quasi- hydrostatic system potentially suitable for cloud-resolving simulations.