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. MPDATA Smolarkiewicz and Margolin (1998)

Isopycnic/isentropic equations eg. Bleck (1974); Hsu and Arakawa (1990); 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

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

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

Non-constant- coefficient approach!

Pressure based formulations Hydrostatic Hydrostatic equations in pressure coordinates

Pressure based formulations Historical NH Miller (1974); Miller and White (1984);

Pressure based formulations Hirlam NH Rõõm et. Al (2001), and references therein;

Pressure based formulations Mass-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 formulations ECMWF/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

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

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 ~ km

Hydrostatic vs. Non-hydrostatic hill Idealized 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V AA BB CC DD EE Fully compressible Hydrostatic Pseudo-incompressible (Durran 1989) Anelastic (Wilhelmson & Ogura 1972) Anelastic (Lipps & Hemler 1982) Boussinesq

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 …