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

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 √ √ All variants are or have been used in atmospheric or oceanic predictions, Note on wave breaking: waves can break but invalidate The hydrostatic assumption as they do, there is no verification as to what wave breaking does in hydrostatic vs non-hydrostatic Formulations nor is there global scale proof for the advantage of non-hydrostatic formulations, however, I’ll show some smaller scale Differences that may matter √

Euler equations for isentropic inviscid motion Not sure if those have been used for atmospheric predictions, but always used for academic exercises

Euler equations for isentropic inviscid motion Speed of sound (in dry air 15ºC dry air ~ 340m/s)

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 state Overbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subset

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 Overbar denotes a (vertically varying) basic state; _e denotes any environmental balanced state, that satisfies the equations or a subset Clark and Farley (1984) 

*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 small If 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 equations Satisfy some important invariants like energy and potential vorticity, however, note that these properties while Invariant in their respective system are not necessarily the same when compared to each other !!! using:

Incompressible Euler equations eg. Durran (1999); Casulli and Cheng (1992); Casulli (1998); Note, that incompressible does not necessarily mean \rho=constant !!! Example later.

Example of simulation with sharp density gradient Animation: "two-layer" simulation of a critical flow past a gentle mountain Fig5 as movie Compare to shallow water: reduced domain simulation with H prescribed by an explicit shallow water model

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) System closes via diagnostic equation for pressure

Projection method Subject to boundary conditions !!!

Integrability condition Ensure that continuity is fulfilled. The numerics and all approaches in CFD are to avoid the creation of spurious forces that alter flow magnitude through discretization, formulation or truncation errors. With boundary condition:

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 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 Two layer flow animation with density ratio 1:1000 Equivalent to air-water Incompressible Euler two-layer fluid flow past obstacle 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! 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 with 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

Importance of implementation detail? Example of translating oscillator (Smolarkiewicz, 2005): time

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)

Compressible Euler equations Davies et al. (2003)

Compressible Euler equations

Pressure based formulations Hydrostatic Hydrostatic equations in pressure coordinates Note, that in particular popular in meteorology due to the good approximation of the hydrostatic assumption Like shallow water and isentropic models, success of ECMWF based on these equations !

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

Pressure based formulations (Rõõm et. al (2001), and references therein) developed within the HIRLAM group Same 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 !!!

Pressure based formulations Mass-coordinate Define ‘mass-based coordinate’ coordinate: Laprise (1992) ‘hydrostatic pressure’ in a vertically unbounded shallow atmosphere By definition monotonic with respect to geometrical height relates to Rõõm et. al (2001):

Pressure based formulations Laprise (1992) Momentum equation Thermodynamic equation Note the easy form of the continuity equation and the prognostic equation for pressure absent in anelastic models Continuity equation with

Compressible vs. anelastic Davies et. al. (2003) Lipps & Hemler approximation Hydrostatic

Compressible vs. anelastic Equation set V A B C D E Fully compressible 1 Hydrostatic Pseudo-incompressible (Durran 1989) Anelastic (Wilhelmson & Ogura 1972) Anelastic (Lipps & Hemler 1982) Boussinesq

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))

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.