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									Adjoint Atmospheric Models and a Proposal for Adaptive Resolution Refinement
Martin Ehrendorfer
Institut für Meteorologie und Geophysik Universität Innsbruck
Presentation at

Very High Resolution Environmental Modeling (VHREM) European Science Foundation (ESF) Exploratory Workshop
University of Hohenheim – 21/23 September 2006 22 September 2006 Univ. Hohenheim, Germany

http://www2.uibk.ac.at/meteo

Outline • Adjoint models

• • • •

theory mathematical examples simple illustrations meteorological examples

• A proposal on ARR within VHREM

• strenghten network/collaboration within „VHREM community“ on topic of joint interest
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Adjoint Models
• sensitivity and/or gradient of model output

• with respect to model input
• adjoint of nonlinear model linearized around state „TLM“

• linearization „along“ model trajectory
• applications

• • • • •

data assimilation: gradients of cost function sensitivity studies synoptic – meteorology studies optimal growth: SVs and predictability parameter estimation
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Bulletin AMS, 1997

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Definitions – Adjoint Models

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Example 1 – Fortran Code

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Example 2 – Adjoint zonal derivative

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Spectral QG Model for illustration

T106L9 and T45L6
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QG Model (nonlinear spectra)

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The Adjoint FFT

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Checking: QGTN and ADQGTN

• (Ax)^T (Ax) = x^T (A^T A x) ……..? „Norm test“

b1ho
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Checking: „Matrix Test“

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Use of adjoint models: Gradients

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Adjoint Models: Gradients

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Example: Sensitivity field

b1ho; 24h sensitivity; field at model level indicating sensitivity of J (=integral of psi^2 at final time) with respect to initial condition; Initial condition: 200501011200 (irec=4); model including low-level damping; level 5 = 700 hPa, T45L6.
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initial condition relevant for sensitivity

streamfunction 400 hPa, T45L6
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Gradients through adjoints: testing

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Adjoint gradient test

b1ho, 24h integration, T45L6
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Singular Vectors through Adjoints

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SVs – TE norm (spectrally formulated)
• Solve through Lanczos iteration (avoid large matrices):

• A^(–1)^T . M^T . C^T . C . M . A^(–1) z = λ z

initial condition 200501010000 (irec=1), first 24 h SV, λ=111.80 Model includes low-level damping, 400 hPa, streamfunction,
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Basic state for SV

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Full QG TE SV spectrum

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Comments on adjoints
• Adjoints

• derived for linear routines
• routine by routine, statement by statement • eventually available for entire tangent-linear model TLM

• testing individual routines (norm and matrix test, gradient tests)
• Applications – wide range

• • • •
• •

sensitivities data assimilation SVs analysis error estimates
• sensitivity of forecast error w. r. t. initial condition

Further meteorological examples → …
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First variational assimilation: 4DVAR

1987

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Controlling gravity waves – 1990

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Adjoint literature – 1993

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What is „wrong“ in initial condition?

1996

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Key analysis errors – 1998 (JAS)

temperature 1.5 K

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Balance issues – 2000

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Sensitivity wrt observations – 2004

adjoint of assimilation system

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NRL NOGAPS fcst error sensitivity

operational

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Synoptic – dynamic interpretations

… of forecast sensitivities

2005

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Atmosphere – Simulation and Sensitivity [observed] input

nonlinear model
HPC – high-performance computing

linear sensitivity

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Lorenz (1984) – nonlinearity

NONLINEARITY

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Tangent-Linear Model

• linearised evolution of differences

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TLM (solid) versus NLM (broken)

0.2

0.4

0.8
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Proposal within VHREM
• central topic

• Adaptive Resolution Refinement (ARR) in Atmospheric and Climate Modeling
• submitted to funding agencies/programs

• special funding at Univ. Innsbruck (under review) • Austrian Science Fund – individual project, June 2006 • total requested funding, 220.000,- Euros, 3 years (under review) • SFB in planning phase at FWF/ASF – December 2006
• • • • primary idea collaborations mentioned scientific questions working steps planned and extensions
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ARR Proposal within VHREM
• primary idea

• strenghten network within „VHREM community“ on topic of joint interest
• collaborations

• • • • • • • •

ESF VHREM initiative (A. Gadian, S. Mobbs) IACETH Zürich (C. Schär) U. Michigan (C. Jablonowski) ECMWF (T. Palmer) RAMS (W. Cotton) Austrian A–GRID Institute of Computer Science, U. Innsbruck (T. Fahringer) High-Performance Computing Consortium, U. Innsbruck

• RAMS/OLAM (Ocean-Land-Atmosphere Model) (Walko/Avissar)
• • scientific questions working steps planned and extensions

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Jablonowski

spherical mesh – with block-structured approach

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background
• uniform resolution increase

•
•

ECMWF presently T799L91 (50km)

tradeoff

•
• •

specific features more detailed than remaining flow

nested grids stretched grids

• • •
• •

non-uniform but constant number of points Continuous Dynamic Grid Adaption (CDGA) reduction of resolution in other parts of domain

OMEGA model – adaptive unstructured triangular prism grid block-structured approach (Jablonowski):

• • • •
•

local refinement ahead of specific feature – adding additional points coarsening with disappearance of feature adaptive nature of approach grid-point based model

spectral methods – and – adaptive methods

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Scientific questions

• A-1) use of ARR methods in spectral models • A-2) adaptive refinement in the vertical and horizontal directions

• B-1) adaptive grid refinement in existing highresolution limited-area model • B-2) adaptive grid refinement vs. nested-grid approach in limited-area model

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approach – and – workplan

• questions A • use AMR/ARR in grid-point SWM • global spectral SWM • global spectral PE – PEAK (Primitive Equation Atmospheric Research Kernel) • questions B • RAMS/OLAM • European Microscale Model EMM

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schematic workplan

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WORKFLOW
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Summary

• Adjoint models

• powerful but underutilized tool • nonlinear code development always with a view towards adjoints • Adjoint Workshop
• VHREM and EMM in context of ARR

• joint initiative proposed • one question: Adaptive Resolution Refinement
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Adjoint Workshop, Obergurgl, Tyrol

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References

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