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Gillian Baxter
University of ReadingDepartments of Mathematics and Meteorology
Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. BallardSponsored by NERC and CASE studentship with the Met Office
Multiscale Data Assimilation for High
Resolution Nested Models
Contents
A brief introduction to Data Assimilation High resolution models The “long wave” problem A few numerical results Conclusions
In order to run a weather forecasting model we need initial conditions.
We want initial conditions which most accurately describe the observed reality.
Data Assimilation allows us to combine observational data with a previous model forecast (background).
The initial conditions are known as the “analysis”.
What is data assimilation?
t
kkkkk
Tkkkb
Tb hhJ
0
11 ))(())((21
)()(21
)( xyRxyxxBxxx
time
ax
bx
4D-Var
constrained by the numerical model kk Mxx 1
Accurate forecasting of convective storms is important because these storms can lead to dangerous flooding events, such as the Boscastle flood in 2004.
High resolution models
Pictures from BBC website
Boscastle Flood: Comparison of 00 UTC 12, 4 and 1 km forecasts
12-18 from 00 UTC12km
12-18 from 00 UTC4km
12-18 from 00 UTC1km
Actual peak accumulations reached about 200 mm (gauge), 130 mm (2 km radar)
Plots from the Met Office, slide provided by S. L. Dance
If we want a model with high resolution it can only have a limited domain size due to restrictions in computer power.
This can create some problems;
• The “long wave problem”.
• A limited area model (LAM) needs boundary conditions.
Met Office HRTM domains
The “long wave” problem
The 1D heat equationxxt uu
2
1,11,1,11,,
/
)2(
xt
uuuuu kikikikiki
Discretisation
where
with homogeneous boundary conditions
Some numerical results Model set up
buffer zone buffer zone
Truth
LAM
Parent model
Boundary conditions provided by the parent model
0625.0x 0125.0t
the LAM has 4 times the spatial resolution and 16 times the temporal resolution of the parent model
The truth has twice the spatial resolution and 4 times the temporal resolution of the LAM
domain is [0,1]
the LAM starts at the first internal grid point of the parent model and covers exactly half of its domain
The 1D heat equationxxt uu
2
1,11,1,11,,
/
)2(
xt
uuuuu kikikikiki
Discretisation
Davies Relaxation scheme
Gi
Li
newi uuu )1(
]/)1[(1 bi where b is the width of the buffer
where
Model set up
with homogeneous boundary conditions
Aim:
To consider and compare the treatment in the parent model and the LAM of wavelengths which are
1.Shorter than the resolution of the parent model
Parent model outputs
)8sin()4sin()2sin(2 iii zzztruth
--- truth--- parent background__ parent analysis + observations
ii xz 2where
LAM model outputs
--- truth--- parent analysis__ LAM analysis+ observations
)8sin()4sin()2sin(2 iii zzztruth
ii xz 2where
LAM domain
__ LAM__ parent model__ truth
The Discrete Fourier Transform of the function is
wavenumber the is and1,,0 ,2
where kNjN
jj
kiN
jjkj
jefffDFT
)(
1
0
~
jf
Wavenumber k
2~
|| kf
Aim:
To consider and compare the treatment in the parent model and the LAM of wavelengths which are
1. Shorter than the resolution of the parent model
2.Longer than the domain of the LAM
__ LAM__ parent model__ truth
10 2 3 4 5 6 7 8Wavenumber k
2~
|| kf
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
Summary
The LAM can represent wavelengths that are missed by the parent model.
However, the LAM may actually be worse at reproducing the longer wavelengths.
When the wavelength is longer than the domain of the LAM it cannot be correctly reproduced.
Future Work
The “long wave” information is important and the challenge is how to assimilate observations of these “long waves” in order to feed information from the large scales into the analysis for the limited area model.
One option may be to split the scales. Take longer wavenumbers from the parent model and shorter wavenumbers from the LAM.
Plot taken from “Development of 1-4km Resolution Data Assimilation for Nowcasting at the Met Office”. S. Ballard, Z. Li, M. Dixon, S. Swarbrick, O. Stiller and H. Lean. WMO Nowcasting Meeting Abstract 3.02
Parent model outputs
--- truth--- parent background__ parent analysis + observations
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
ii xz 2where
LAM model outputs
--- truth--- parent analysis__ LAM analysis+ observations
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
ii xz 2where
