adjoint diagnostics for the atmosphere and ocean · • why do we need an adjoint model and what is...
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ECMWF SEMINAR 20097-10 September
Adjoint diagnostics for theatmosphere and ocean
Jan BarkmeijerKNMI
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OUTLINE
• Why do we need an adjoint model and what is it?
• Easy non-trivial example
• Difficulties in developing adjoint models
• Applications for atmospheric/ocean models- ‘sensitivity calculation’- singular vectors- other use, i.e. not ‘initial condition’ related
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Contributions
Many thanks to:
• Bernard Bilodeau• Ron Errico• Ron Gelaro• Thomas Jung• Simon Lang• Martin Leutbecher• Andy Moore• Frank Selten• Florian Sevellec• Gerard van der Schrier
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Special guests: E. Lorenz (2002) and G.I. Marchuk (2004)
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Special guests: E. Lorenz (2002) and G.I. Marchuk (2004)
Application of adjoint equations to virus infection modelling(Marchuk et al., 2005).
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Why is an adjoint model useful?
Suppose we are dealing with a nonlinear model M of the form:
y=M(x)
and a differentiable scalar J defined for model output fields y :
J=J(y)=J(M(x))
Dependence of J on y is often straightforward,
but determining seems impossible for high-dimensional models.
It would require perturbed model runs for every (~108 ) entry of x.
xJ ∂∂ /
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Example 0: Sensitivity calculation(method to improve a forecast retrospectively)
y=M(x)
x=analysis1 analysis2
perturbed analysis1
J?
J(x)=[y-analysis2,y-analysis2], with [.,.] a suitable inner product
See: Rabier et al. (1996), Klinker et al. (1998), ……,…, Isaksen et al. (2005), Caron et al. (2006),…
(J=0)
How to minimize J ?
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j
k
k
M
kj xy
yJ
xJ
∂∂
∂∂
=∂∂ ∑
=1
JJ yT
x ∇=∇ M
Application of the chain rule learns that
Assume that a small perturbation δyj of yj is associated to a smallperturbation δxk of xk through:
and consequently
jdefkk k
jj x
xy
y )x( δδδ M=∂
∂= ∑
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How to determine MT ?
Assume that the linear model describing the evolution of initial time perturbations has the form
(1) dε/dt = Lε
with propagator M: ε(t2)=M(t1,t2) ε(t1)
Define the adjoint model by
(2) dε/dt = -LTε , with [La,b] = [a,LTb],
with propagator S and where [.,.] is a suitable inner product.
N.B. Adjoint model depends on chosen inner product [.,.].
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Solutions a(t) and b(t) of (1) and (2) respectively satisfy the property:
d/dt [a(t),b(t)]=[La(t),b(t)]+[a(t),-LTb(t)]=0
and consequently
[M(t1,t2)a(t1),b(t2)]=[a(t1),S(t2,t1)b(t2)]
M(t1,t2)T= S(t2,t1)
YES!
a(t1) M(t1,t2)a(t1)
S(t2,t1)b(t2) b(t2)time
How to determine MT ? (2)
Gradient J can be determined efficiently by running the adjoint model (2) backwards in time!
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Adjoint of barotropic vorticity equation (BVE)
∫∫ Σ= d.),( baba
),(Jac 1ζζζ −Δ+=∂∂ ftλμμλ ∂
∂∂∂
−∂∂
∂∂
=hghghg ),(Jac
),(Jac ψεε Δ=L
),(d),Jac(.d),(Jac.
d),(Jac.d).,(Jac),(*bababa
bababa
L
L
∫∫∫∫∫∫∫∫
=ΣΔ=Σ∇∇−
=ΣΔ=ΣΔ=
ψψ
ψψ
One of the components of the linearised BVE reads as:
Use inner product defined by:
and thus (N.B. )
, with
),~(Jac~* ψεε Δ−=L -1* LL ≠
!
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While developing the SL2TL adjoint …..
THE ADJOINT BLUES
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Linear vs. nonlinear model
• More elaborate physics in linear models, as nonlinear modelis growing increasingly complex/realistic.
(see Marta Janiskova’s contribution to 2003 ECMWF seminar)
- ensemble forecasting (e.g. in the tropics, tropical cyclones)- investigate physics driven instability mechanisms- data-assimilation of variety of data types (e.g. radar)
• Not straightforward to check the correctness of linear models.(e.g., by comparing difference between two nonlinear runs and
the outcome of a linear run)
• Possibility to trigger unwanted instability mechanisms.
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Spurious perturbation growth in the tropics
linear run nonlinear (difference of 2 runs)
Excessive growthin the lowerstratosphere
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Example 0: Sensitivity calculation(how to improve a forecast retrospectively)
y=M(x)
x=analysis1 analysis2
perturbed analysis1
J?
Minimize cost function J :
J(x)=[y-analysis2,y-analysis2], with [.,.] a suitable inner product
(J=0)
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(Isaksen et al., 2005)
Example 0: Sensitivity calculation (contnd)
Be careful with theinterpretation of‘key analysis errors’
For example, seeECMWF 2003 Seminar
Isaksen: Realism of sensitivity patterns.
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Example 1: Periodic weather.Unstable Periodic Orbits (UPO) can be used to describe certain characteristics of the ‘climate attractor’ efficiently.
Define a cost function J by: J (x(0))=1/2 || x(0)-x(T) ||2 ,with ||.|| the Euclidean norm.
The gradient of J is given by:
(0))-(T)]([ T xxJ IdM −=∇
UPO.x(0)x(T)
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original initial condition
and after minimization
head and tail of UPOUnstable Periodic Orbit(period is 10 days)
Streamfunction at 500 hPain a T21L3 Quasi-Geostrophic model
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Periodic weather
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Almost all data points lie close to a small set of UPO’s
Points on the attractor not close to a UPO
(Gritsun and Branstator, 2007)UPO
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Example 2: Upwelling
• Decadal variations in California Current Upwelling Cells(Chhak & Di Lorenzo, 2007)
• Model: Regional Ocean Modeling System (ROMS)(Moore et al., 2004)
• Potential mechanisms for the sharp decline in zooplanktonbiomass off the coast of California after the mid ‘70s
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Pacific Decadal Oscillation (PDO)
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Cold phase: deepersource of upwelled,nutrient rich water.
Supported by observations ofphytoplankton &zooplankton
Passive tracer introduced mid-April each year (55 yrs); adjoint run for 1 yr
Origin of upwelling water (%) 1 yr prior to following year upwelling max.
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Example 3: Sensitivity in surface precipitationfor 2 convection schemes
(Mahfouf and Bilodeau, 2007)
• costfunction J is mean surface precipitation over a selected domain• relative importance of u,v,T,q and ps depends on convection scheme
start of adjoint model integration
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SINGULAR VECTORS
If more realistic models with many thousands of variables also have the property that a few of the eigenvalues of ATA are much larger than the remaining, a study based upon a small ensemble of initial errors should give a reasonable estimate of the growth rate of random errors … It would appear then, that the best use could be made of computational time by choosing only a small number of error fields for superposition upon a particular initial state …(Lorenz 1965)
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Perturbations ε of the initial condition that maximize the ratio
),(),(
),(),(
εεεε
εεεε
CEMM
CMEM T
=
where M is the propagator of the tangent linear model and C and Edefine a perturbation norm at initial and final time respectively.
- Popular choice: C=E= ‘total energy’ norm
- Other choice for C: Hessian norm
-1-14DVAR AHRHBC -1T =+=∇∇= J
A : estimate of the analysis error covariance matrix B/R :background/observational error covariance matrix
Σ+∂∂
Σ++= ∫∫∫∫∫ d]ln[dpd][ 2'r2'2'2's
r
p
r
p pTc
RTTc
vu ηη
E
final norm
begin norm
Orr mechanism (1907)
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Equivalently, Hessian singular vectors (HSV) satisfy:
MTEM ε = λ A-1ε
Solvers exist, even when M and A are not known explicitly
Note that (E1/2MA)MTEM ε = λ(E1/2 MA)A-1ε
orE1/2 (MAMT) E1/2 E1/2 M ε = λE1/2M ε
so
time-evolved HSVs E1/2M ε are eigenvectors of E1/2(MAMT) E1/2,
which is the forecast error covariance matrix in the E-norm(Observe MAMT= M δδΤ MT= Mδ (Mδ)T )
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Analysis error variance field for temperature at 500 hPa
Hessian singular vectors know about this:
higher analysisuncertainty
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(Barkmeijer et al., 2000)
• reduced growth (50%) for HSV’s in terms of total energy.• potential (kinetic) energy dominant for initial TE (Hessian) SVs• no energy for wave number >25 in case of HSV without observations
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(Lawrence et al., 2009)
TE HSV no OBS HSV
T=0h
T=48h
T=48h
pe keke pe
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The Observing System Research and Predictability Experiment (THORPEX)
To improve weather forecasts by collecting observations in data-sensitive locations where analysis errors would have the largest impact on the forecast for a specific event or region of interest
One of the objectives:
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data-sensitive areas
(Gelaro et al., 2002)
TESV
VARSV
case 1 case 2
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Example 2: Sea Surface Salinity SVs
• Optimal surface salinitiy perturbations for the Meridional OverturningCirculation (Sévellec et al., 2008)
• Final norm measures (northward) mass transport at 48oN in the Atlantic
• Model: OPA and OPA Tangent Adjoint Model (OPATAM, Weaver et al., 2003)
• Estimate the influence of sea surface salinity (SSS) perturbations on the North-Atlantic circulation as suggested by observations and modeling studies.
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Optimal SSS Perturbation(scaled to 1 psu amp.)
• Upper bounds based on GSA:~0.8 Sv (11% of mean)
• SST perturbation less optimal:5oC required to achieve SSS impact
MOC Growth factor
ΔMOC
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Example 3: Stratosphere-Troposphere interaction
Look for structures ε that originate in the stratosphere and grow in thestratosphere or (lower) troposphere, by using appropriate projectionoperators Pini and Pevo.
><><
εε iniini
inievoinievo
, , PP
MPPMPP εε
100 hPa -------
500 hPa -------
Stratosphere SVs: S-SV
Stratosphere-Troposphere SVs: ST-SV
Pini
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ST-SVs
Downward propagating structuresare possible even during summerconditions
Energy distribution below 500 hPaof linearly evolved ST-SV at T=120h
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Example 4: Tropical singular vectors
• Case: Tropical cyclone Helene (September 2006)as seen from space shuttle Atlantis
resolution T42 T159
(Courtesy of S. Lang)
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• Target area is entire tropical strip 30oS – 30oN !• Tropical cyclone Helene shows up in the leading SVs
T=0h
T=48h
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ForcingThe regular SV and Sensitivity calculation can be exploited for studying model uncertainty. Assume error evolution is given by
)(f tdtd
+= εε L
then ∫+=t
0
d )(f),( (0))( sstst MMεε
ε(0)
f
timeand in case 0)0( =ε
f d )(f),()(t
0
N sstst == ∫Mε
Together with the corresponding adjoint N*
forcing singular vectors/sensitivity can be determined.
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The Reynolds system
Assume error dynamics is governed by a stable 2x2-matrix A
and further subject to a forcing f(s):
dε/dt = Aε + f(t)
Look for Forcing Singular Vectors FSV, which maximize
(Nf , Nf) for unit-sized f and Nf =
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2-0101-
A
∫t
0
d )(f),( sstsM
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(Farrell and Ioannou, 2005)
Components of the leading FSV for the Reynolds system usingan optimization time of 4 units.
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Size of the perturbation at optimization time
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0h 24h 48h 0h 24h 48h
Tendency perturbations in the sensitivity calculation
OR
Use N and its adjoint N* (instead of M and M*) to determine a constanttendency perturbation (forcing) f, which decreases the forecast error, orequivalently, minimizes the following cost function:
J=<fc-an+Nf , fc-an+Nf>
regular forcing
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Temperature perturbations at700 hPa
default sensitivity(T=0h)
forcing sensitivity(constant during 48-h forecast)
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T500 impact at forecast time T=48h
forcing
default
2-day forecast errortemperature 500 hPacontour interval 1K
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(Barkmeijer et al., 2003)
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Example 1: Increasing the Atlantic subtropical jet
• Determine every 72 hours an atmospheric forcing, which increases the Atlantic subtropical jet
• Apply this forcing in a coupled atmosphere-ocean model• Run the coupled model for 10 years
yearday
Jet
Str
eam
In
dex
FORCED
CONTROL
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(Van der Schrier et al., 2007)
control
forced
• Subtropical jet more zonalin the forced run
• Atmospheric meridional heat transport over the North Atlanticis reduced
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Resulting in a cooling over the Atlantic in the forced run.
Change in surface temperature for the forced run compared to the control run
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Adjoint models have been very useful and instructive in understanding ocean and atmospheric models!
Thank you for your attention
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Stratospheric forcing with the ECMWF model
• Apply a forcing F to the model tendency • Forcing F is constructed to change the strength of the
stratospheric polar vortex (18 sensitivity calculations). • Perform 60 forty-day T95L60 integrations during
DJF 1982-2001 with
dx/dt=EC(x), dx/dt=EC(x) + F and dx/dt=EC(x) - F
• Forcing F is small and zero below 150 hPa • and F is kept constant during the integration
Jung and Barkmeijer (2006)
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Stratospheric Response (50hPa) Weak-Ctl
Vortex Collapse
D+1-D+10 D+11-D+20
D+21-D+30 D+31-D+40
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Z1000 Response (Weak-CTL)
D+1-D+10 D+11-D+20
D+21-D+30 D+31-D+40