Presentation on theme: "Governing Equations III"— Presentation transcript:
1 Governing Equations III by Nils Wedi (room 007; ext. 2657)Thanks to Piotr Smolarkiewicz
2 IntroductionContinue to review and compare a few distinct modelling approaches for atmospheric and oceanic flowsHighlight the modelling assumptions, advantages and disadvantages inherent in the different modelling approaches
3 Dry “dynamical core” equations Shallow water equationsIsopycnic/isentropic equationsCompressible Euler equationsIncompressible Euler equationsBoussinesq-type approximationsAnelastic equationsPrimitive equationsPressure or mass coordinate equations√√All variants are or have been used in atmospheric or oceanic predictions, Note on wave breaking: waves can break but invalidateThe hydrostatic assumption as they do, there is no verification as to what wave breaking does in hydrostatic vs non-hydrostaticFormulations nor is there global scale proof for the advantage of non-hydrostatic formulations, however, I’ll show some smaller scaleDifferences that may matter√
4 Euler equations for isentropic inviscid motion Not sure if those have been used for atmospheric predictions, but always used for academic exercises
5 Euler equations for isentropic inviscid motion Speed of sound (in dry air 15ºC dry air ~ 340m/s)
6 Reference and environmental profiles 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 stateOverbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subset
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 equationsOverbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subsetClark and Farley (1984)
8 *NOT* approximated Euler perturbation equations eg. Durran (1999)Always possible, hope is however, that if the basic state is chosen well and the perturbations are smallIf not the perturbations may be as large as the basic state !Now, we introduce various approximations, all introduced in such a way that the resulting equationsSatisfy some important invariants like energy and potential vorticity, however, note that these properties whileInvariant in their respective system are not necessarily the same when compared to each other !!!using:
9 Incompressible Euler equations eg. Durran (1999); Casulli and Cheng (1992); Casulli (1998);Note, that incompressible does not necessarily mean \rho=constant !!! Example later.
10 Example of simulation with sharp density gradient Animation:"two-layer" simulation of a critical flowpast a gentle mountainFig5 as movieCompare to shallow water:reduced domain simulation with H prescribedby an explicit shallow water model
15 Classical Boussinesq approximation eg. Durran (1999)System closes via diagnostic equation for pressure
16 Projection methodSubject to boundary conditions !!!
17 Integrability condition Ensure that continuity is fulfilled. The numerics and all approaches in CFD are to avoid the creation of spurious forcesthat alter flow magnitude through discretization, formulation or truncation errors.With boundary condition:
18 Ap = f Solution Due to the discretization in space a banded matrix A arises with size (N x L)2 N=number of gridpoints, L=number of levelsClassical schemes include Gauss-elimination forsmall 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)
19 Importance of the Boussinesq linearization in the momentum equation Two layer flow animation with density ratio 1:1000Equivalent to air-waterIncompressible Euler two-layer fluid flow past obstacleHowever, in the atmospheric context it is not so clear what influence this has on other processes.Test the formulation of Durran, the pseudo compressible equations!Incompressible Boussinesq two-layer fluid flow past obstacleTwo layer flow animation with density ratio 297:300Equivalent to moist air [~ 17g/kg] - dry airIncompressible Euler two-layer fluid flow past obstacleIncompressible Boussinesq two-layer fluid flow past obstacle
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);
21 Anelastic approximation Lipps and Hemler (1982);
23 Numerical Approximation withLE, flux-form Eulerian or Semi-Lagrangianformulation using MPDATA advection schemesSmolarkiewicz and Margolin (JCP, 1998)withPrusa and Smolarkiewicz (JCP, 2003)specified and/or periodic boundaries
24 Importance of implementation detail? Example of translating oscillator (Smolarkiewicz, 2005):time
25 Example ”Naive” centered-in-space-and-time discretization: However, in the atmospheric context it is not so clear what influence this has on other processes.Test the formulation of Durran, the pseudo compressible equations!Non-oscillatory forward in time (NFT) discretization:paraphrase of so called “Strang splitting”,Smolarkiewicz and Margolin (1993)
26 Compressible Euler equations Davies et al. (2003)
28 Pressure based formulations Hydrostatic Hydrostatic equations in pressure coordinatesNote, that in particular popular in meteorology due to the good approximation of the hydrostatic assumptionLike shallow water and isentropic models, success of ECMWF based on these equations !
29 Pressure based formulations Historical NH (Miller (1974); Miller and White (1984))
30 Pressure based formulations (Rõõm et. al (2001),and references therein)developed within theHIRLAM groupSame is true for these formulations since it requires that \partial p/\partial z and \rho is always well defined.This is not the case in fluids with density discontinuities !!!
31 Pressure based formulations Mass-coordinate Define ‘mass-based coordinate’ coordinate:Laprise (1992)‘hydrostatic pressure’ in a verticallyunbounded shallow atmosphereBy definitionmonotonic withrespect to geometricalheightrelates to Rõõm et. al (2001):
32 Pressure based formulations Laprise (1992)Momentum equationThermodynamic equationNote the easy form of the continuity equation and the prognostic equation for pressure absent in anelastic modelsContinuity equationwith
35 Compressible vs. anelastic Normal mode analysis done on linearized equations noting distortion of Rossby modes if equations are (sound-)filteredDifferences 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 in Dutton (1967))
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.