page 1© crown copyright 2005 data assimilation and numerical models richard swinbank utls...

© Crown copyright 2005

Data Assimilation and Numerical Models

Richard Swinbank

UTLS International School, Cargese


Aims of lecture

The aim of the lecture is to give an overview of atmospheric models used for Numerical Weather Prediction.

I will discuss some of the practicalities of assimilating data into those models with reference to the UTLS and stratosphere.


Contents

Numerical modelsModel gridsDynamical CorePhysical parametrizationsThe Met Office Unified Model

Interfacing with data assimilationObservationsAssimilation methods


Introduction


Numerical models

Dynamical Core Horizontal grid – staggering, grid types Vertical – staggering, coordinates Numerics - spatial & temporal differencing

Physical parametrizations Focused on the stratosphere

Sub models (not addressed here) Oceans, Land Surface, Chemistry.…

Met Office Unified Model, as an example


Global Model Grids (1)

The conventional latitude-longitude grid suffers from converging meridians, so a variety of different approaches have been proposed:

Reduced (Kurihara) grid Skipped grid (Smoothed) Cubed sphere (Conformal) Icosahedral Yin-Yang (overset) grid Fibonacci grid

With thanks to Jim Purser



Skipped gridReduced (Kurihara) grid



(Smoothed) Cubed sphere (Conformal) Icosahedral



Yin-Yang (overset) grid Fibonacci grid(Swinbank and Purser, 1999)


Space discretization methods

Simple finite difference - values defined at grid points

Galerkin methods – model fields defined as sum of basis functions

finite elements - local basis functions

global spectral expansion (analytical, hence highly accurate spatial derivatives)


Grid staggering - horizontal

Arakawa and Lamb (1977) defined several types of staggered grid.

A (unstaggered) is simple, differences calculated over distances 2d. Not so good for conservation.

C is better for conservation. Convergence and pressure gradient terms calculated over shorter distances (d). But Coriolis terms require horizontal averaging.

B is sometimes used as a compromise.

D has little merit. E is a B-grid, rotated by 45

degrees.

uvΦ

A

Φ

v

u

v

uC

Φ

uv

uvuv

uvB

Φ

u

v

u

vD


Vertical Grid

Staggering: Lorenz: only vertical

velocity is staggered – allows a spurious computational mode;

Charney-Phillips: more consistent with hydrostatic equation.

Vertical coordinate: pressure; sigma, p/ps; height; isentropic; hybrid / terrain-following.

u,v,T,Φw

u,v,T,Φw

w=0

Lorenz grid

u,v,Tw,Φ

u,v,Tw,Φ

w=0

Charney-Phillips grid


Time stepping

Forward Good for diffusive terms, but

unstable for hyperbolic equations Leapfrog

Prone to time-splitting; often used with Asselin filter

Predictor-Corrector No computational mode Heun, as an example

Implicit Stable – but damping

Semi-implicit α=1 fully implicit α=0.5 Crank-Nicholson α>0.5 stable

1n n

nx x F xt

1 1

1 1

22

n nn

n n n n n

x x F xt

x x x x x

1

1n n

nx x F xt

1

1 1n n

n nx x F x xt

*

1*1

2

nn

n nn

x x F xt

x x F x xt


Model resolution and time-step

Resolution A grid-point model can only resolve features of bigger scale than the grid length (rule of thumb: λ>4d)

CFL (Courant-Friedrichs-Levy) limitIn general, simple finite difference schemes cannot move information more than 1 grid-length in a time-step.Courant number μ must be less than 1; μ=cΔt/Δd<1Accuracy diminishes as μ approaches 1

Semi-Lagrangian methodInstead of using local values, the semi-Lagrangian method uses values around a calculated departure point.Because there is no extrapolation, S-L schemes are absolutely stable


Physics

Represent non-dynamical, e.g.RadiationLarge-scale rainfall

and sub-grid scale processes, e.g.ConvectionBoundary-layer turbulenceGravity-wave drag


General issues relevant to stratospheric modelling

Model resolution There should be consistency between vertical and horizontal

resolution. Aspect ratio of grid for representing large-scale flow should

ideally be around N/f (~1:100 in troposphere, 1:200 in stratosphere, Lindzen & Fox-Rabinovitz, 1989)

Often much poorer than ideal vertical resolution in the stratosphere;

Difficult to resolve the tropopause.Transport:

conservation; monotonicity constraints; tracer correlations.


Other issues

Numerical diffusion to control instabilitiesNoise from neutral/unstable parts of dynamics, plus

physics needs to be controlled. Produce realistic power spectrum

Physics-dynamics couplingParallel – each scheme unaware of each other; sum

total tendenciesSequential – order of processes is important Ideally do “slow” physics in parallel, followed by

“fast” in sequence (fastest process last)

An example - the Met Office Unified Model


New Dynamics v Old Dynamics

New Dynamics

Semi-LagrangianSemi-implicit (predictor-

corrector)Arakawa C-gridHeight based: hybrid

terrain-following gridCharney-Phillips Full 3D Helmholtz solver

Old Dynamics

Explicit HeunSplit-explicit (2 time-

level)Arakawa B-gridPressure based: hybrid

sigma-pressure gridLorenzReference state profile


Equation Set Options

Deep Shallow(r→a,

neglect boxed terms)

Non-hydrostatic

Complete equations

(new dynamics)

Non-hydrostatic primitive(Robert)

Hydrostatic(neglect dw/dt)

Quasi-hydrostatic(old dynamics)

Hydrostatic primitive

(e.g. ECMWF)


New Dynamics Equation Set

tan 2 sinco

coss

2r pd v uuw wr r

cD u v SDt r

uv

2 tan 2 sin pd vrc vw

r ru

rD v u SDt

2 2

2 cos wpd

rvcr r

g

u vu SD w

Dt r

2 2cos cos 0cos

ry y

D u v wDt r

r rr r

rD SDt


New Dynamics – Global model configurations

38-level, N216 (0.55o x 0.83o) Top at 39km Operational (NWP) in August 2002

50-level, N48 (2.5o x 3.75o) Methane oxidation and spectral GWD Top at 64 km Operational (NWP) in October 2003

Improved resolution due for implementation around end 2005: 50-level, N320


Current ND levels


Physics Package (HadGAM)

New Dynamics (Davies et al, 2005) 2-stream Radiation (Edwards & Slingo,1996) Mass flux convection (Gregory & Rowntree, 1990) Non-local Boundary Layer (Lock et al., 2000) Sub-grid Orography and GWD (Webster et al., 2003) Statistical cloud scheme (Smith, 1990) Prognostic Ice Microphysics (Wilson & Ballard, 1999) Met Office Surface Exchange (Cox et al., 1999) Cubic Monotonic Tracer Advection.


Methane Oxidation

Conservation of ( 2CH4 + H2O ) = 6 ppmvIdealised oxidation rate as a function of height based on ECMWF’s scheme

Only operates in middle atmosphere Photolysis of water vapour at higher levels.


Effect of Methane Oxidation

10 year mean Jan

UARS observed


Gravity wave drag

• Typical model errors are alleviated using a parametrization of drag due to breaking gravity waves

• A version of the USSP scheme (Warner and McIntyre, 2000) has been implemented in the UM (Scaife et al., 2000)

• Isotropic and homogeneous source of gravity waves in the lower atmosphere

• Launch spectrum proportional to m-3 at large m• Hydrostatic, non-rotating dispersion relation: /k=N/m• “Transparent” upper boundary.


Equatorial Zonal Wind for L50

VN 5.4

Assim Obs

Observations


Observations

When assimilating data into a GCM, it is important to take account of the general characteristics of the observations.

where they are (horizontally and vertically)when they are (synoptic or asynoptic)how accurate they are (observation errors)

measurement errorserrors of representativenessexplicitly included in variational formulation

Observation data coverage plots


AMSU nadir soundings – vertical resolution

AMSU is the primary sounding instrument in the ATOVS package.

AMSU-A is for temperature soundings and AMSU-B for water vapour.

Although the horizontal density of ATOVS is very high, the vertical resolution is rather poor


Satellite data - some issues

Measurement geometry :nadir soundings (e.g. AMSU; high horizontal

resolution, poor vertical) limb soundings (good vertical and poor horizontal

resolution – more consistent with typical model grid, bigger problems with clouds, harder to do radiance assimilation)

Radiances or retrievals? retrievals (simpler, but hard to characterise errors) radiances (more fundamental - get the best out of

the data, better characterised errors; need higher model lid for radiances?)


Contrasting nature of observations

IN SITU REMOTE SENSING

Conventional Satellite

"Point" measurements

Average measurements

Simple interpolation More complex observation operator

Synoptic Asynoptic

Suits Analysis-Forecast Cycle

Suits continuous assimilation process


OSEs and OSSEs

A technique often used to evaluate components of an existing observing system is the “Observing System Experiment” (OSE)

An OSE studies the impact of one observation type by removing it from the system under study

An Observing System Simulation Experiment (OSSE) applies the same idea to evaluate future observations. However, in that case the observations need to be simulated.

This is more complicated, but still worthwhile for evaluating expensive future satellite missions


SWIFT OSSE

In a joint DARC / Met Office project, we evaluated the likely impact of the proposed SWIFT instrument (Lahoz et al, 2005)

SWIFT, Stratospheric Wind Interferometer for Transport Studies: 2-component line-of-sight winds using Doppler shift of thermal

emission (mid-IR) of ozone (1133 cm-1). Similar technology to UARS WINDII. Global measurements of wind and ozone profiles (~20-40 km)

Conclusions: SWIFT winds would have a significant impact in tropical stratosphere

(except lowermost levels) They could have significant impacts in the extra-tropics when flow

regime is variable (relatively fast changing) Improve information on tropical winds and wintertime variability


Assimilation methods (1)

Conventional Analysis-Forecast cycle OptimaI Interpolation – generally including

approximations to calculate local analysis locally3D-Var – use variational methods to get global

solution (minimising cost function J)

PSAS – “dual” of 3D-Var, solving same problem as 3D-Var, but in observation space. Less tied to model grid, but cost increases rapidly with number of observations.

1 11( ) ( ) ( ) ( ) ( )2

Tb T b o oJ H H x x x B x x y x R y x

1a b T T o bH

x x BH HBH R y x



4D-Var is an important extension of 3D-Var which treats observations distributed over a time window. (Increasingly important with the growth in satellite data.)

A (tangent linear) model and its adjoint are used to determine the misfit of the model to the observations, and make corrections to the initial conditions at the beginning of the time window.

The minimisation procedure in 4D-Var includes several iterations of an inner loop involving running the linear model and its adjoint.

To avoid problems such as physical processes switching on and off, the linear model generally uses a simplified version of the physical parametrizations.

Use of a simplified model is also more cost-effective



Some research groups are working on the development of Ensemble Kalman filter methods

Reasonably easy to implement quickly (given available assimilation infrastructure)

But currently does not do as well as 4D-VarAs discussed in my previous lecture, ensembles are

a good way of estimating error covariancesMore on ensemble forecasts in my third lecture

Final Comments


Some limitations of stratospheric data assimilation

Models The vertical resolution is often poor near and above the

tropopause Model biases are often much larger than in the troposphere Some models cannot simulate key stratospheric features – in

particular the QBOObservations

The stratospheric observing system is dominated by poor vertical resolution temperature soundings

The SWIFT study confirmed the potential usefulness of wind measurements

Error covariances As we saw in my previous lecture, it is difficult to get good

estimates of the background error covariances


But, on the other hand…

Stratospheric Data Assimilation hasProvided a rich resource for improving our

understanding of stratospheric dynamicsHelped put constituent measurements in a

dynamical context (e.g. NH ozone measurements from UARS-MLS)

Provided a basis for chemical data assimilation efforts, especially ozone

Contributed to improvements in weather forecast skill at many NWP centres


Any Questions?


Further reading

Recommended ReadingAtmospheric Modeling, Data Assimilation and Predictability by Eugenia Kalnay. Cambridge

University Press, 2003 Selected References

Arakawa, A. and V. Lamb, 1977: Computational design of the basic dynamical processes in the UCLA general circulation model. In General circulation models of the atmosphere, Methods in Computational Physics, Academic Press, pp 174-264.

Lindzen, R.S. and M. Fox-Rabinovitz, 1989: Consistent vertical and horizontal resolution. Mon. Wea. Rev., 117, 2575-2583.

Davies, T.D., M.J.P. Cullen, A. J. Malcolm, M.H. Mawson, A. Staniforth, A.A. White and N. Wood, 2005: A new dynamical core for the Met Office’s global and regional modeling of the atmosphere, Quart. J. Roy. Meteor. Soc., 131, 1759-1782.

Parrish, D.F. and J.C. Derber, 1992: The National Meteorological Center’s spectral statistical interpolation analysis scheme. Mon. Wea. Rev., 120, 1747-1763.

Scaife, A.A., N. Butchart, C.D. Warner, D. Stainforth, W.A. Norton and J. Austin, 2000: Realistic Quasi-Biennial Oscillations in a simulation of the global climate. Geophys. Res. Lett. 27, 3481-3484.

Swinbank, R. and R.J. Purser, 1999: Fibonacci grids. 13th Conference on Numerical Weather Prediction, AMS, 125-128.

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