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

zg

NS

12

Stratification SBrunt-Väisällä frequency NPotential 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 >500km. 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 13:00

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

1*0

* J

X

XJ

01

10

1

pp

J

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)