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The equations of motion and their numerical

solutions I

by Nils Wedi (2006) contributions by Mike Cullen

and Piotr Smolarkiewicz

Why this presentation ?

- There is NO perfect model description for either

atmosphere or ocean (so far?)

Introduction

- Observations
- What do we solve in our models ?
- Some properties of the equations we solve
- What do we (re-)solve in our models ?
- How do we treat artificial and/or natural

boundaries and how do these influence the

solution ?

Introduction

- Review and comparison of a few distinct modelling

approaches for atmospheric and oceanic flows - Highlighting the modelling assumptions,

advantages and disadvantages inherent in the

different modelling approaches - Highlighting issues with respect to upper and

lower boundary conditions

Observations

- Much information about the solution of the

equations can be deduced from observations. - Satellite pictures show large-scale structures,

but closer examination reveals more and more

detail. - High resolution in-situ observations show large

small-scale variability of comparable amplitude

to underlying larger scale variations

Satellite picture of N. Atlantic

Large-Scale Frontal systems on scale of

N.Atlantic. Cellular convection S. of Iceland. 2d

vortex shedding S. of Greenland.

Enlarged picture over U.K.

More detail Gravity-wave train running SW-NE

over UK.

Balloon measurements of static stability

Stratification S Brunt-Väisällä frequency

N Potential temperature ?

Idealized reference or initial state in

numerical models

Time series of wind speed from anemograph

High resolution observations are still above the

viscous scale! However, only spatially and

temporally averaged values suitable for

numerical solution.

Observed spectra of motions in the atmosphere

Spectral slope near k-3 for wavelengths gt500km.

Near k-5/3 for shorter wavelengths. Possible

difference in larger-scale dynamics.

No spectral gap!

Scales of atmospheric phenomena

Practical averaging scales do not correspond to a

physical scale separation. If equations are

averaged, there may be strong interactions

between resolved and unresolved scales.

Observations of boundary layers the tropical

thermocline

M. Balmaseda

Observations of boundary layers EPIC - PBL over

oceans

M. Koehler

Averaged equations

- The equations as used in an operational NWP model

represent the evolution of a space-time average

of the true solution. - The equations become empirical once averaged, we

cannot claim we are solving the fundamental

equations. - Possibly we do not have to use the full form of

the exact equations to represent an averaged

flow, e.g. hydrostatic approximation OK for large

enough averaging scales in the horizontal. - The sub-grid model represents the effect of the

unresolved scales on the averaged flow expressed

in terms of the input data which represents an

averaged state. - The average of the exact solution may not look

like what we expect, e.g. since vertical motions

over land may contain averages of very large

local values. - Averaging scale does not correspond to a subset

of observed phenomena, e.g. gravity waves are

partly included at T511, but will not be properly

represented.

Demonstration of averaging

- Use high resolution (10km in horizontal)

simulation of flow over Scandinavia - Average the results to a scale of 80km
- Compare with solution of model with 40km and 80km

resolution - The hope is that, allowing for numerical errors,

the solution will be accurate on a scale of 80km - Compare low-level flows and vertical velocity

cross-section, reasonable agreement

Cullen et al. (2000) and references therein

Mean-sea-level pressure

Satellite picture

VIS 18/03/1998 1300

High resolution numerical solution

- Test problem is flow at 10 ms-1 impinging on

Scandinavian orography - Resolution 10km, 91 levels, level spacing 300m
- No turbulence model or viscosity, free-slip lower

boundary - Semi-Lagrangian, semi-implicit integration scheme

with 5 minute timestep - Errors in flow Jacobian are the errors of the

Lagrangian continuity equation integrated over a

timestep contour interval is 3.

Low-level flow 10km resolution

Cross-section of potential temperature

x-y errors in the inverse flow Jacobian

In IFS

Flow Jacobian in T799 L91

Remember Contour line 1 is the correct

answer!

y-z errors in the inverse flow Jacobian

Low level flow- 40km resolution

Low level flow-10km resolution averaged to 80km

x-z vertical velocity - 40km resolution

Vertical velocity - 10km resolution averaged to

80km

Conclusion

- Averaged high resolution contains more

information than lower resolution runs - The ratio of comparison was found approximately

as - dx (averaged resol) ?dx (lower resol) with ?

1.5-2

Vertical velocity cross-section ECMWF operational

run

Observations show mountain wave activity limited

to neighbourhood of mountains. Operational

forecast (T511L60) shows unrealistically large

extent of mountain wave activity. Other tests

show that the simulations are only slightly

affected by numerical error. The idealised

integrations suggest that the predictions

represent the averaged state well (despite

hydrostatic assumption). The real solution is

much more localised and more intense.

What is the basis for a stable numerical

implementation ?

- A Removal of fast - supposedly insignificant -

external and/or internal acoustic modes (relaxed

or eliminated), making use of infinite sound

speed (cs??) from the governing equations BEFORE

numerics is introduced. - B Use of the full equations WITH a semi-implicit

numerical framework, reducing the propagation

speed (cs ?0) of fast acoustic and buoyancy

disturbances, retaining the slow

convective-advective component (ideally)

undistorted. - C Split-explicit integration of the full

equations, since explicit NOT practical (100

times slower) - ? Determines the choice of the numerical scheme

Rõõm (2001)

Choices for numerical implementation

- Avoiding the solution of an elliptic equation
- fractional step methods Skamrock and Klemp

(1992) Durran (1999) - Solving an elliptic equation
- Projection method Durran (1999)
- Semi-implicit Durran (1999) Cullen et.

Al.(1994) - Preconditioned conjugate-residual solvers or

multigrid methods for solving the resulting

Poisson or Helmholtz equations Skamarock et. al.

(1997)

Split-explicit integration

Skamarock and Klemp (1992) Durran (1999) Doms

and Schättler (1999)

Semi-implicit schemes

- (i) coefficients constant in time and

horizontally (hydrostatic models Robert et al.

(1972), ECMWF/Arpege/Aladin NH) - (ii) coefficients constant in time Thomas (1998)

Qian, (1998) see references in Bénard (2004) - (iii) non-constant coefficients Skamarock et. al.

(1997), (UK Met Office NH model)