The equations of motion and their numerical solutions II

1
The equations of motion and their numerical
solutions II
by Nils Wedi (2006) contributions by Mike Cullen
and Piotr Smolarkiewicz
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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

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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 Smolarkiewic
z and Margolin (1998)
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Isopycnic/isentropic equations

• eg. Bleck (1974) Hsu and Arakawa (1990)

isentropic
isopycnic
shallow water
defines depth between shallow water layers
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More general isentropic-sigma equations
Konor and Arakawa (1997)
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Euler equations for isentropic inviscid motion
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Euler equations for isentropic inviscid motion
Speed of sound (in dry air 15ºC dry air 340m/s)
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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

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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)
?
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NOT approximated Euler perturbation equations

• eg. Durran (1999)

using
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Incompressible Euler equations

• eg. Durran (1999) Casulli and Cheng (1992)
  Casulli (1998)

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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
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Two-layer t0.15
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Shallow water t0.15
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Two-layer t0.5
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Shallow water t0.5
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Classical Boussinesq approximation

• eg. Durran (1999)

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Projection method
Subject to boundary conditions !!!
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Integrability condition
With boundary condition
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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)
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Importance of the Boussinesq linearization in the
momentum equation
Two layer flow animation with density ratio
11000 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
297300 Equivalent to moist air 17g/kg - dry
air
Incompressible Euler two-layer fluid flow past
obstacle
Incompressible Boussinesq two-layer fluid flow
past obstacle
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Anelastic approximation

• Batchelor (1953) Ogura and Philipps (1962)
  Wilhelmson and Ogura (1972) Lipps and Hemler
  (1982) Bacmeister and Schoeberl (1989) Durran
  (1989) Bannon (1996)

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Anelastic approximation
Lipps and Hemler (1982)
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Numerical Approximation
Compact conservation-law form
Lagrangian Form
?
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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
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Importance of implementation detail?
Example of translating oscillator (Smolarkiewicz,
2005)
time
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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)
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Compressible Euler equations

• Davies et al. (2003)

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Compressible Euler equations
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A semi-Lagrangian semi-implicit solution procedure
(not as implemented, Davies et al. (2005) for
details)
Davies et al. (1998,2005)
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A semi-Lagrangian semi-implicit solution procedure
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A semi-Lagrangian semi-implicit solution procedure
Non-constant- coefficient approach!
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Pressure based formulationsHydrostatic
Hydrostatic equations in pressure coordinates
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Pressure based formulationsHistorical NH
Miller (1974) Miller and White (1984)
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Pressure based formulationsHirlam NH
Rõõm et. Al (2001), and references therein
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Pressure based formulationsMass-coordinate
Laprise (1992)
Define mass-based coordinate coordinate
hydrostatic pressure in a vertically
unbounded shallow atmosphere
By definition monotonic with respect to
geometrical height
Relates to Rõõm et. Al (2001)
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Pressure based formulations
Laprise (1992)
Momentum equation
Thermodynamic equation
Continuity equation
with
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Pressure based formulationsECMWF/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
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Pressure based formulations ECMWF/Arpege/Aladin
NH model
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Hydrostatic vs. Non-hydrostatic

• eg. Keller (1994)
• Estimation of the validity

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Hydrostaticity
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Hydrostaticity
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Hydrostatic vs. Non-hydrostatic
Non-hydrostatic flow past a mountain without wind
shear
Hydrostatic flow past a mountain without wind
shear
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Hydrostatic vs. Non-hydrostatic
Non-hydrostatic flow past a mountain with
vertical wind shear
Hydrostatic flow past a mountain with vertical
wind shear
But still fairly high resolution L 30-100 km
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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
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Compressible vs. anelastic

• Davies et. Al. (2003)

Lipps Hemler approximation
Hydrostatic
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Compressible vs. anelastic
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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/non-shear effects, assumed reference state,
  increased importance of the neglected non-linear
  effects