The equations of motion and their numerical solutions II
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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. MPDATASmolarkiewicz and Margolin (1998)
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Isopycnic/isentropic equations
eg. Bleck (1974); Hsu and Arakawa (1990);
" "d m
1" "d
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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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
Reference and environmental profiles
e
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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
using:
eg. Durran (1999)
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Incompressible Euler equations
eg. Durran (1999); Casulli and Cheng (1992); Casulli (1998);
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"two-layer" simulation of a critical flow past a gentle mountain
reduced domain simulation with H prescribed by an explicit shallow water model
Animation:
Compare to shallow water:
Example of simulation with sharp density gradient
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Two-layer t=0.15
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Shallow water t=0.15
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Two-layer t=0.5
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Shallow water t=0.5
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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
Incompressible Euler two-layer fluid flow past obstacle
Two layer flow animation with density ratio 1:1000 Equivalent to air-water
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
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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
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
with
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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
Davies et al. (1998,2005)
(not as implemented, Davies et al. (2005) for details)
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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
Define ‘mass-based coordinate’ coordinate: Laprise (1992)
Relates to Rõõm et. Al (2001):
By definition monotonic with respect to geometrical height
‘hydrostatic pressure’ in a vertically unbounded shallow atmosphere
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Pressure based formulations
Laprise (1992)
with
Momentum equation
Thermodynamic equation
Continuity equation
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Pressure based formulationsECMWF/Arpege/Aladin NH model
Bubnova et al. (1995); Benard et al. (2004), Benard (2004)
hybrid vertical coordinate
coordinate transformation coefficient
scaled pressure departure
‘vertical divergence’
with
Simmons and Burridge (1981)
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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-hydrostaticNon-hydrostatic flow past a mountain without wind shear
Hydrostatic flow past a mountain without wind shear
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Hydrostatic vs. Non-hydrostaticNon-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 hillIdealized 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)
Hydrostatic
Lipps & Hemler approximation
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Compressible vs. anelastic
Equation set V A B C D E
Fully compressible 1 1 1 1 1 1Hydrostatic 0 1 1 1 1 1Pseudo-incompressible (Durran 1989) 1 0 1 1 1 1Anelastic (Wilhelmson & Ogura 1972) 1 0 1 1 0 0Anelastic (Lipps & Hemler 1982) 1 0 0 1 0 0Boussinesq 1 0 1 0 0 0
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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 …