The equations of motion and their numerical solutions III

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

Boundaries

• We are used to assuming a particular boundary that fits the analytical or numerical framework but not necessarily physical free surface, non-reflecting boundary, etc.

• Often the chosen numerical framework favors a particular boundary condition where its influence on the solution remains unclear

Choice of vertical coordinate

• Ocean modellers claim that only isopycnic/isentropic frameworks maintain dynamic structures over many (life-)cycles ? .. Because coordinates are not subject to truncation errors in ordinary frameworks.

• There is a believe that terrain following coordinate transformations are problematic in higher resolution due to apparent effects of error spreading into regions far away from the boundary in particular PV distortion. However, in most cases problems could be traced back to implementation issues which are more demanding and possibly less robust to alterations.

• Note, that in higher resolutions the PV concept may loose some of its virtues since the fields are not smooth anymore, isentropes overturn and cross the surface.

Choice of vertical coordinate

• Hybridization or layering needed to exploit the strength of various coordinates in different regions (in particular boundary regions) of the model domain

• Unstructured meshes: CFD applications typically model only simple fluids in complex geometry, in contrast atmospheric flows are complicated fluid flows in relatively simple geometry (eg. gravity wave breaking at high altitude, trapped waves, shear flows etc.) Bacon et Al. (2000)

But why making the equations more difficult ?

• Choice: Use relatively simple equations with difficult boundaries or use complicated equations with simple boundaries

• The latter exploits the beauty of

“A metric structure determined by data”Freudenthal, Dictionary of Scientific Biography (Riemann),C.C. Gillispie, Scribner & Sons New York (1970-1980)

Examples of vertical boundary simplifications

• Radiative boundaries can also be difficult to implement, simple is perhaps relative

• Absorbing layers are easy to implement but their effect has to be evaluated/tuned for each problem at hand

Radiative Boundary Conditions for limited height models

eg. Klemp and Durran (1983); Bougeault (1983); Givoli (1991); Herzog (1995); Durran (1999)

Linearized BoussinesqEquations in x-z plane

Radiative Boundary Conditions

Inserting…

Phase speed:

Group velocity:

for hydrostatic waves

Dispersion relation:

Radiative Boundary Conditions

discretized Fourierseries coefficients:

Wave-absorbing layers

eg. Durran (1999)

Viscous damping:

Rayleigh damping:

Adding r.h.s. terms of the form …

Flow past Scandinavia 60h forecast 17/03/1998

divergence patterns in the operational configuration

Flow past Scandinavia 60h forecast 17/03/1998

divergence patterns with no absorbers aloft

Wave-absorbing layers: an engineering problem

Wave-absorbing layers: overlapping absorber regions

Finite difference example of an implicit absorber treatment including overlapping regions

The second choice …

• Create a unified framework to investigate the influence of upper boundary conditions on atmospheric and oceanic flows

• Observational evidence of time-dependent well-marked surfaces, characterized by strong vertical gradients, ‘interfacing’ stratified flows with well-mixed layers

Can we use the knowledge of the time evolution of the interface ?

Anelastic approximation

Generalized coordinate equations

Strong conservation formulation !

Explanations …Using:

Solenoidal velocity

Contravariant velocity

Transformation coefficients

Jacobian of the transformation

Physical velocity

Time-dependent curvilinear boundaries

• Extending Gal-Chen and Somerville terrain-following coordinate transformation ontime-dependent curvilinear boundaries

Wedi and Smolarkiewicz (2004)

The generalized time-dependent coordinate transformation

Gravity waves

Reduced domain simulation

Swinging membrane

Swinging membranes bounding a homogeneous Boussinesq fluid

Animation:

Another practical example

• Incorporate an approximate free-surface boundary into non-hydrostatic ocean models

• “Single layer” simulation with an auxiliary boundary model given by the solution of the shallow water equations

• Comparison to a “two-layer” simulation with density discontinuity 1/1000

Regime diagram

supercritical

subcriticalcritical, stationary lee jump

Critical, downstream propagating lee jump

Critical – “two-layer”

Critical – reduced domainflat

shallow water

Conclusions

• …collapses the relationship between auxiliary boundary models and the interior fluid domain to a single variable

• …aids the systematic investigation of influences of different boundary conditions on the same flow simulation

• …possibly simplifies the direct incorporation of observational data into flow simulations given the exact knowledge of H and it’s derivative

• Aids the inclusion of a free surface in ocean data assimilation and corresponding adjoint formulation

The stratospheric QBO

- westward+ eastward data taken from ERA40

Laboratory analogue of the quasi-biennial oscillation (QBO)

• The coordinate transformation allows for a time-dependent upper/lower boundary forcing without small amplitude approximation

• Demonstrates the importance of wave transience and their correct numerical realization when resulting in a dominant large scale oscillation

The laboratory experiment of Plumb and McEwan

• The principal mechanism of the QBO was demonstrated in the laboratory

• University of Kyoto

Plumb and McEwan, J. Atmos. Sci. 35 1827-1839 (1978)

http://www.gfd-dennou.org/library/gfd_exp/exp_e/index.htm

Animation:

Time – height cross section of the mean

flow Uin a 3D simulation

Animation

Wedi and Smolarkiewicz, J. Atmos. Sci., 2006 (accepted)

Eulerian semi-LagrangianViscous simulation

Wedi, Int. J. Numer. Meth. Fluids, 2006

Inviscid simulation

Eulerian St=0.25, n=384

semi-LagrangianSt=0.25, n=384

Eulerian St=0.36, n=640top absorber top rigid, freeslip top rigid, freeslip

Conclusions

• The flux-form Eulerian simulation is asymptotically close to “idealized inviscid”.

• The semi-Lagrangian simulation exhibits a more viscous behaviour due to enhanced dissipation, which contributes to the development of different bifurcation points in the flow evolution.

• The analysis is relevant to larger-scale geophysical flows driven by small scale fluctuations, where the interplay of reversible and irreversible fluxes determines the flow evolution.

A QBO in IFS ? Not really …

Numerically generated forcing !

Instantaneous horizontal velocity divergence at ~100hPa

Tiedke massflux scheme

No convection parameterization

T63 L91 IFS simulation over 4 years

Interaction of convection (parametrization) and the large-scale dynamics appears to play a key role in future advances of forecast ability!