The equations of motion and their numerical solutions II

by Nils Wedi (2006)contributions by Mike Cullen and Piotr Smolarkiewicz

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

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)

Isopycnic/isentropic equations

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

" "d m

1" "d

isentropic

isopycnic

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)

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 = 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

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 approximation

Lipps and Hemler (1982);

Numerical Approximation

Compact conservation-law form:

Lagrangian Form:

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

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

Davies et al. (1998,2005)

(not as implemented, Davies et al. (2005) for details)

A semi-Lagrangian semi-implicit solution procedure

A semi-Lagrangian semi-implicit solution procedure

Non-constant-coefficient approach!

Pressure based formulationsHydrostatic

Hydrostatic equations in pressure coordinates

Pressure based formulationsHistorical NH

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

Pressure based formulationsHirlam NH

Rõõm et. Al (2001), and references therein;

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

Pressure based formulations

Laprise (1992)

with

Momentum equation

Thermodynamic equation

Continuity equation

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)

Pressure based formulations ECMWF/Arpege/Aladin NH model

Hydrostatic vs. Non-hydrostatic

eg. Keller (1994)

• Estimation of the validity

Hydrostaticity

Hydrostaticity

Hydrostatic vs. Non-hydrostaticNon-hydrostatic flow past a mountain without wind shear

Hydrostatic flow past a mountain without wind shear

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

Hydrostatic vs. Non-hydrostatic

hill hillIdealized 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)

Hydrostatic

Lipps & Hemler approximation

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

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 …