ECMWF Governing Equations 3 Slide 1 Governing Equations III Thanks to Piotr Smolarkiewicz by Nils Wedi (room 007; ext. 2657)

ECMWFGoverning Equations 3 Slide 1

Governing Equations III

Thanks to Piotr Smolarkiewicz

by Nils Wedi (room 007; ext. 2657)


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


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

√

√

√


Euler equations for isentropic inviscid motion


Euler equations for isentropic inviscid motion

Speed of sound (in dry air 15ºC dry air ~ 340m/s)


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


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

using:

eg. Durran (1999)


Incompressible Euler equations

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


"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


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 = 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)


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


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 approximationLipps and Hemler (1982);


Numerical Approximation

Compact conservation-law form:

Lagrangian Form:


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


time

Importance of implementation detail?

Example of translating oscillator (Smolarkiewicz, 2005):


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)



Davies et al. (2003)




Pressure based formulationsHydrostatic

Hydrostatic equations in pressure coordinates


Pressure based formulationsHistorical NH

(Miller (1974); Miller and White (1984))


Pressure based formulations

(Rõõm et. al (2001),

and references therein)

developed within the HIRLAM group


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


Laprise (1992)

Momentum equation

Thermodynamic equation

Continuity equation

Pressure based formulations

with


Compressible vs. anelastic Davies et. al. (2003)

Hydrostatic

Lipps & Hemler approximation


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



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))



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.

Documents

ECMWF Governing Equations 3 Slide 1 Governing Equations III Thanks to Piotr Smolarkiewicz by Nils Wedi (room 007; ext. 2657)