Organising Numeracy Resources Thanks Gillian - it works Presentation prepared by S. Allen.
Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K....
-
date post
21-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K....
![Page 1: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/1.jpg)
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
![Page 2: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/2.jpg)
Contents
A brief introduction to Data Assimilation High resolution models The “long wave” problem A few numerical results Conclusions
![Page 3: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/3.jpg)
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?
![Page 4: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/4.jpg)
t
kkkkk
Tkkkb
Tb hhJ
0
11 ))(())((21
)()(21
)( xyRxyxxBxxx
time
ax
bx
4D-Var
constrained by the numerical model kk Mxx 1
![Page 5: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/5.jpg)
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
![Page 6: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/6.jpg)
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
![Page 7: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/7.jpg)
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
![Page 8: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/8.jpg)
The “long wave” problem
![Page 9: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/9.jpg)
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
![Page 10: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/10.jpg)
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
![Page 11: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/11.jpg)
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
![Page 12: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/12.jpg)
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
![Page 13: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/13.jpg)
Parent model outputs
)8sin()4sin()2sin(2 iii zzztruth
--- truth--- parent background__ parent analysis + observations
ii xz 2where
![Page 14: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/14.jpg)
LAM model outputs
--- truth--- parent analysis__ LAM analysis+ observations
)8sin()4sin()2sin(2 iii zzztruth
ii xz 2where
LAM domain
![Page 15: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/15.jpg)
__ 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
![Page 16: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/16.jpg)
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
![Page 17: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/17.jpg)
__ LAM__ parent model__ truth
10 2 3 4 5 6 7 8Wavenumber k
2~
|| kf
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
![Page 18: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/18.jpg)
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.
![Page 19: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/19.jpg)
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.
![Page 20: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/20.jpg)
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
![Page 21: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/21.jpg)
Parent model outputs
--- truth--- parent background__ parent analysis + observations
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
ii xz 2where
![Page 22: Gillian Baxter University of Reading Departments of Mathematics and Meteorology Supervisors: N. K. Nichols, S. L. Dance, A. S. Lawless, S. P. Ballard Sponsored.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d585503460f94a37fb3/html5/thumbnails/22.jpg)
LAM model outputs
--- truth--- parent analysis__ LAM analysis+ observations
)8sin()4sin()2sin(2)sin(2 iii zzzzitruth
ii xz 2where