Transcript Model talk for Cargese

Data Assimilation and Numerical Models
Richard Swinbank
UTLS International School, Cargese
© Crown copyright 2005
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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.
© Crown copyright 2005
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Contents
Numerical models
 Model grids
 Dynamical Core
 Physical parametrizations
 The Met Office Unified Model
Interfacing with data assimilation
 Observations
 Assimilation 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 latitudelongitude 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)
Reduced (Kurihara) grid
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Skipped grid
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Global Model Grids (3)
(Smoothed) Cubed sphere
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(Conformal) Icosahedral
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Global Model Grids (4)
Yin-Yang (overset) grid
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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.
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A
uvΦ
uv
uv
B
C
u
Φ
v
Φ
u
uv
v
uv
D
v
u
Φ
v
u
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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.
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Lorenz grid
w
u,v,T,Φ
w
u,v,T,Φ
w=0
Charney-Phillips grid
w,Φ
u,v,T
w,Φ
u,v,T
w=0
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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
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x n1  x n
 F  xn 
t
x n 1  x n 1
 F  xn 
2 t
x n  x n    x n 1  2 x n  x n 1 
x*  x n
 F  xn 
t
x n 1  x n
1

 F   x n  x*  
t
2

x n 1  x n
 F  x n 1 
t
x n1  x n
 F  x n1  1    x n 
t
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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) limit
In 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<1
Accuracy diminishes as μ approaches 1
 Semi-Lagrangian method
Instead of using local values, the semiLagrangian 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.
 Radiation
 Large-scale rainfall
and sub-grid scale processes, e.g.
 Convection
 Boundary-layer turbulence
 Gravity-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 instabilities
 Noise from neutral/unstable parts of dynamics, plus
physics needs to be controlled.
 Produce realistic power spectrum
Physics-dynamics coupling
 Parallel – each scheme unaware of each other; sum
total tendencies
 Sequential – order of processes is important
 Ideally do “slow” physics in parallel, followed by
“fast” in sequence (fastest process last)
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An example the Met Office Unified Model
New Dynamics v Old Dynamics
New Dynamics
Old Dynamics
 Semi-Lagrangian
 Semi-implicit (predictorcorrector)
 Arakawa C-grid
 Height based: hybrid
terrain-following grid
 Charney-Phillips
 Full 3D Helmholtz solver
 Explicit Heun
 Split-explicit (2 timelevel)
 Arakawa B-grid
 Pressure based: hybrid
sigma-pressure grid
 Lorenz
 Reference state profile
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Equation Set Options
Deep
Shallow
(r→a,
neglect boxed
terms)
Nonhydrostatic
Hydrostatic
(neglect dw/dt)
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Complete
equations
Non-hydrostatic
primitive
(new dynamics)
(Robert)
Quasi-hydrostatic
Hydrostatic
primitive
(old dynamics)
(e.g. ECMWF)
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New Dynamics Equation Set
c pd v 
Dr u uv tan 
 uw


 2 sin  v 
    2 cos  w   S u
Dt
r
r cos  
 r

c pd 
Dr v u 2 tan 
 vw 
v

 2 sin  u 
 

S

Dt
r
r 
r


 
Dr w
 
 c pd v


Dt
r
r
g
u
2
 v2 
r
 2 cos  u  S w
   u    v  w 
Dr
2
2
 y r cos    y r cos  
   0


Dt
   r cos    r  r 
Dr 
S
Dt
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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 ppmv
Idealised 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: w/k=N/m
• “Transparent” upper boundary.
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Equatorial Zonal Wind for L50
VN 5.4
Assim
Obs
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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 errors
 errors of representativeness
 explicitly included in variational formulation
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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
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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
Simple interpolation
Synoptic
Average
measurements
More complex
observation operator
Asynoptic
Suits AnalysisForecast 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 locally
 3D-Var – use variational methods to get global
solution (minimising cost function J)

T
1
b T
1
b
o
J (x)  (x  x ) B (x  x )   y  H (x)  R 1  y o  H (x) 
2

 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.
x  x  BH
a
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b
T
 HBH  R 
T
1

y o  H  xb 

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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-Var
 As discussed in my previous lecture, ensembles are
a good way of estimating error covariances
 More on ensemble forecasts in my third lecture
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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 QBO
 Observations
 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 has
 Provided a rich resource for improving our
understanding of stratospheric dynamics
 Helped 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 Reading
Atmospheric 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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