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Data Assimilation and Numerical Models
Richard Swinbank
UTLS International School, Cargese
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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.
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Contents
Numerical modelsModel gridsDynamical CorePhysical parametrizationsThe Met Office Unified Model
Interfacing with data assimilationObservationsAssimilation methods
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Introduction
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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
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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
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Global Model Grids (2)
Skipped gridReduced (Kurihara) grid
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Global Model Grids (3)
(Smoothed) Cubed sphere (Conformal) Icosahedral
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Global Model Grids (4)
Yin-Yang (overset) grid Fibonacci grid(Swinbank and Purser, 1999)
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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)
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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
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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
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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
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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
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Physics
Represent non-dynamical, e.g.RadiationLarge-scale rainfall
and sub-grid scale processes, e.g.ConvectionBoundary-layer turbulenceGravity-wave drag
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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.
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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
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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
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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)
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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
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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
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Current ND levels
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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.
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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.
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Effect of Methane Oxidation
10 year mean Jan
UARS observed
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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.
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Equatorial Zonal Wind for L50
VN 5.4
Assim Obs
Observations
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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
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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
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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?)
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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
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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
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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
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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
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Assimilation methods (2)
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
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Assimilation methods (3)
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
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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
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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
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Any Questions?
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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.