The equations of motion and their
numerical solutions II
by Nils Wedi (2006)
contributions by Mike Cullen and Piotr Smolarkiewicz
Dry “dynamical core” equations
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Shallow water equations
Isopycnic/isentropic equations
Compressible Euler equations
Incompressible Euler equations
Boussinesq-type approximations
Anelastic equations
Primitive equations
Pressure or mass coordinate equations
Shallow water equations
eg. Gill (1982)
Numerical implementation by transformation to a
Generalized transport form for the momentum flux:
This form can be solved by eg. MPDATA
Smolarkiewicz and Margolin (1998)
Isopycnic/isentropic equations
eg. Bleck (1974); Hsu and Arakawa (1990);
isentropic
" d "  m
isopycnic
1
" d "      

shallow water
defines depth between “shallow water layers”
More general isentropic-sigma
equations
Konor and Arakawa (1997);
Euler equations for isentropic
inviscid motion
Euler equations for isentropic
inviscid motion
Speed of sound (in dry air
15ºC dry air ~ 340m/s)
Reference and environmental
profiles
Distinguish between
•  (only vertically varying) static reference
or basic state profile (used to facilitate
comprehension of the full equations)
•  e 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
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
Clark and Farley (1984)

*NOT* approximated
Euler perturbation equations
eg. Durran (1999)
using:
Incompressible Euler equations
eg. Durran (1999); Casulli and Cheng (1992);
Casulli (1998);
Example of simulation with sharp
density gradient
Animation:
"two-layer" simulation of a critical flow
past a gentle mountain
Compare to shallow water:
reduced domain simulation with H prescribed
by an explicit shallow water model
Two-layer t=0.15
Shallow water t=0.15
Two-layer t=0.5
Shallow water t=0.5
Classical Boussinesq
approximation
eg. Durran (1999)
Projection method
Subject to boundary conditions !!!
Integrability condition
With boundary condition:
Solution
Ap = f
Since there is a discretization in space !!!
Most commonly used techniques for the iterative
solution of sparse linear-algebraic systems that arise
in fluid dynamics are the preconditioned conjugate
gradient method and the multigrid method. Durran (1999)
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
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
Anelastic approximation
Batchelor (1953); Ogura and Philipps (1962);
Wilhelmson and Ogura (1972); Lipps and Hemler
(1982); Bacmeister and Schoeberl (1989); Durran
(1989); Bannon (1996);
Anelastic approximation
Lipps and Hemler (1982);
Numerical Approximation
Compact conservation-law form:
Lagrangian Form:

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
Importance of implementation
detail?
Example of translating oscillator (Smolarkiewicz, 2005):
time
Example
”Naive” centered-in-space-and-time discretization:
Non-oscillatory forward in time (NFT) discretization:
paraphrase of so called “Strang splitting”,
Smolarkiewicz and Margolin (1993)
Compressible Euler equations
Davies et al. (2003)
Compressible Euler equations
A semi-Lagrangian semi-implicit
solution procedure
(not as implemented, Davies et al. (2005) for details)
Davies et al. (1998,2005)
A semi-Lagrangian semi-implicit
solution procedure
A semi-Lagrangian semi-implicit
solution procedure
Non-constantcoefficient approach!
Pressure based formulations
Hydrostatic
Hydrostatic equations in pressure coordinates
Pressure based formulations
Historical NH
Miller (1974); Miller
and White (1984);
Pressure based formulations
Hirlam NH
Rõõm et. Al (2001),
and references therein;
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):
Pressure based formulations
Laprise (1992)
Momentum equation
Thermodynamic equation
Continuity equation
with
Pressure based formulations
ECMWF/Arpege/Aladin NH model
Bubnova et al. (1995); Benard et al. (2004), Benard (2004)
hybrid vertical coordinate
Simmons and Burridge (1981)
coordinate transformation coefficient
scaled pressure departure
‘vertical divergence’
with
Pressure based formulations
ECMWF/Arpege/Aladin NH model
Hydrostatic vs. Non-hydrostatic
eg. Keller (1994)
• Estimation of the validity
Hydrostaticity
Hydrostaticity
Hydrostatic vs. Non-hydrostatic
Hydrostatic flow past a
mountain without wind shear
Non-hydrostatic flow past a
mountain without wind shear
Hydrostatic vs. Non-hydrostatic
Hydrostatic flow past a mountain
Non-hydrostatic flow past a
with vertical wind shear
mountain with vertical wind shear
But still fairly high resolution L ~ 30-100 km
Hydrostatic vs. Non-hydrostatic
hill
hill
Idealized T159L91 IFS simulation with parameters [g,T,U,L]
chosen to have marginally hydrostatic conditions NL/U ~ 5
Compressible vs. anelastic
Davies et. Al. (2003)
Lipps & Hemler approximation
Hydrostatic
Compressible vs. anelastic
Equation set
V A B C D E
Fully compressible
1
0
1
1
1
1
Hydrostatic
Pseudo-incompressible (Durran 1989)
Anelastic (Wilhelmson & Ogura 1972)
Anelastic (Lipps & Hemler 1982)
Boussinesq
1
1
0
0
0
0
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
0
0
0
1
1
1
0
0
0
Normal mode analysis of the
“switch” equations Davies et. Al. (2003)
• Normal mode analysis done on linearized
equations noting distortion of Rossby modes if
equations are (sound-)filtered
• Differences found with respect to gravity modes
between different equation sets. However,
conclusions on gravity modes are subject to
simplifications made on boundaries, shear/nonshear effects, assumed reference state, increased
importance of the neglected non-linear effects …