ECMWF Governing Equations 3 Slide 1 Governing Equations III Thanks to Piotr Smolarkiewicz by Nils...
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Transcript of ECMWF Governing Equations 3 Slide 1 Governing Equations III Thanks to Piotr Smolarkiewicz by Nils...
ECMWFGoverning Equations 3 Slide 1
Governing Equations III
Thanks to Piotr Smolarkiewicz
by Nils Wedi (room 007; ext. 2657)
ECMWFGoverning Equations 3 Slide 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
ECMWFGoverning Equations 3 Slide 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
√
√
√
ECMWFGoverning Equations 3 Slide 4
Euler equations for isentropic inviscid motion
ECMWFGoverning Equations 3 Slide 5
Euler equations for isentropic inviscid motion
Speed of sound (in dry air 15ºC dry air ~ 340m/s)
ECMWFGoverning Equations 3 Slide 6
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
ECMWFGoverning Equations 3 Slide 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
Clark and Farley (1984)
ECMWFGoverning Equations 3 Slide 8
*NOT* approximated Euler perturbation equations
using:
eg. Durran (1999)
ECMWFGoverning Equations 3 Slide 9
Incompressible Euler equations
eg. Durran (1999); Casulli and Cheng (1992); Casulli (1998);
ECMWFGoverning Equations 3 Slide 10
"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
ECMWFGoverning Equations 3 Slide 11
Two-layer t=0.15
ECMWFGoverning Equations 3 Slide 12
Shallow water t=0.15
ECMWFGoverning Equations 3 Slide 13
Two-layer t=0.5
ECMWFGoverning Equations 3 Slide 14
Shallow water t=0.5
ECMWFGoverning Equations 3 Slide 15
Classical Boussinesq approximation
eg. Durran (1999)
ECMWFGoverning Equations 3 Slide 16
Projection method
Subject to boundary conditions !!!
ECMWFGoverning Equations 3 Slide 17
Integrability condition
With boundary condition:
ECMWFGoverning Equations 3 Slide 18
Solution Ap = fDue 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)
ECMWFGoverning Equations 3 Slide 19
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
ECMWFGoverning Equations 3 Slide 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);
ECMWFGoverning Equations 3 Slide 21
Anelastic approximationLipps and Hemler (1982);
ECMWFGoverning Equations 3 Slide 22
Numerical Approximation
Compact conservation-law form:
Lagrangian Form:
ECMWFGoverning Equations 3 Slide 23
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
Numerical Approximation
ECMWFGoverning Equations 3 Slide 24
time
Importance of implementation detail?
Example of translating oscillator (Smolarkiewicz, 2005):
ECMWFGoverning Equations 3 Slide 25
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)
ECMWFGoverning Equations 3 Slide 26
Compressible Euler equations
Davies et al. (2003)
ECMWFGoverning Equations 3 Slide 27
Compressible Euler equations
ECMWFGoverning Equations 3 Slide 28
Pressure based formulationsHydrostatic
Hydrostatic equations in pressure coordinates
ECMWFGoverning Equations 3 Slide 29
Pressure based formulationsHistorical NH
(Miller (1974); Miller and White (1984))
ECMWFGoverning Equations 3 Slide 30
Pressure based formulations
(Rõõm et. al (2001),
and references therein)
developed within the HIRLAM group
ECMWFGoverning Equations 3 Slide 31
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
ECMWFGoverning Equations 3 Slide 32
Laprise (1992)
Momentum equation
Thermodynamic equation
Continuity equation
Pressure based formulations
with
ECMWFGoverning Equations 3 Slide 33
Compressible vs. anelastic Davies et. al. (2003)
Hydrostatic
Lipps & Hemler approximation
ECMWFGoverning Equations 3 Slide 34
Compressible vs. anelastic
Equation set V A B C D E
Fully compressible 1 1 1 1 1 1
Hydrostatic 0 1 1 1 1 1
Pseudo-incompressible (Durran 1989) 1 0 1 1 1 1
Anelastic (Wilhelmson & Ogura 1972) 1 0 1 1 0 0
Anelastic (Lipps & Hemler 1982) 1 0 0 1 0 0
Boussinesq 1 0 1 0 0 0
ECMWFGoverning Equations 3 Slide 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. 10.8 in Dutton (1967))
ECMWFGoverning Equations 3 Slide 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.