Time-space dependent Downscaling of Wind Stress Data

For Ocean modellings

byJun-Ichi Yano

Hiromi Kobayashi Pascale Bouruet-Aubertot

Downscaling:

Downscaling:

?Principal Approach

Oceanographic Application

Downscaling:

?

Spatial scale: X x (< X )?originaloutput/

observationalscale

(e.g., atmosphere)= 50 – 200 km

requiredinformation/model-input

scale(e.g., oceans)= 5 – 10 km

Alternatively:

Temporal scale: T ?

t (< T )

Downscaling:

Principal Approach

Power Law in Tubulence (Scaling Law)

Wavelets

Downscaling

+

(Spatial-Temporal Heterogeneity)

Oceanographic Application

OceanographicContext:Wind Stress

Data:X=60km

Data:X=125km

Model:X=80km

Zonal Component(Alexandra Bozec)

Response to the 3 types of Wind StressInputsin 4 regions(rows):Depth of MixedLayer

Data:X=60km

Data:X=120km

Model:X=80km

(Alexandra Bozec)

log P(k)

log k

X x

?Extrapolation

Question:(t)?:Modificationof variability with time& space:Heterogeneity

Downscaling Principle: Fully-Nonlinear Turbulent System:Power-law Spectra :

Methods:Observational Data Analysis:

Data Set: Buoy Time Series (off coast Nice): Wind Speed:Mar 1999-Dec 2001, t=1h

x

Methods:Observational Data Analysis:

Data Set: Buoy Time Series (off coast Nice): Wind Speed:Mar 1999-Dec 2001, t=1h

Methodology: Taylor’s frozen hypothesis:k=/u: k

Seek a power law: Pt) ~(t) as a function of time (Morlet wavelet spectra)

How to estimate: =(t) ?

Spe ctre à t = 999h

(instantaneous)

Wavelet Transform of the Wind Speed Time series: 6500:6000)( jpourTSetU ijj

Power Law•Time sereis (segment)

•Wavelet Spectrum

Strategy for the Downscaling

log P()

log t T

?

Strategy for the Downscaling

log P()

log t

Tc=?

Strategy for the Downscaling

•Estimation of wavelet coefficients:

•Reconstruction of a time series

i) |<U, u,s>| ~ s(u)/2

<U, u,s> = |<U, u,s>| ei(u,s)

ii) (u,s) = ?

NB: stochastic : probability

:present work

:future work

NB: u t

Strategy for the Downscaling

log P()

log t

Tc=?

Determination of the Power Exponent:Seek a spectrum of the form : P(T) ~T

for a limited period

Mean over 2n+1 spectra, n=24Spectrum at t=15787h

Seek a spectrum of the form : P(T) ~T

for Ti < Tc:

= ?

Tc = ?

Determination of the Power Exponent:

Spectre en loi de puissanceOn cherche un spectre de la forme : P(T) ~T

pour Ti < Tc:

= ?

Tc = ?

Probability Distribution of the Exponent:t: exponent of the spectrum

Probability Density :

p(

Tot

al e

ner

gy

How to Estimate the Exponent from the

Other Conditions?: Joint-Probabilities

?

ConclusionsBuoy Data: Surface Wind-Speed time serieswith t = 1 h:Power-Law Spectra in Wavelet Space:

Most likely exponent (14% chance): Two Regimes:

2nd Regime with

Preliminary anlyses for the joint-probabilites

Future Work: Statistics for the Phase: A 1st-order Markov Model?

http://www.ipsl.jussieu.fr/CLIMSTAT/CARGESE/TALKS/JUNICHI/downscale.pptftp://ftp.lmd.jussieu.fr/pub/yano/cargese/review.ftp://ftp.lmd.jussieu.fr/pub/yano/cargese/hiromi.psftp://ftp.lmd.jussieu.fr/pub/yano/cargese/poster_bozec.ppt

Final Remark: Two Schools in Downscaling:

ClimatologicalPlanetary-Scale

SynotpicScale

Meso-Convective Scale

104 km 103 km 10 - 102 km

I. X* x

« Regionalization » (CL12, HS9, …)

II. X* x

Linear-Wave Dynamics

Nonlinear-Turbulent

How to Estimate the Exponent from the

Other Conditions?: Joint-Probabilities

Tc

How to Estimate the Exponent from the

Other Conditions?: Joint-Probabilities

Tc

Tot

al e

ner

gy

OutlineReview: what is downscaling?

scale dependence: types I & II

An explorative study for the downscaling of the wind stress over the Mediterranean Sea

approaches for the type I: linear approaches

limitations of linear approaches approaches for the type II: nonlinear

why necessary?: oceanographic context

Downscaling Type I

X* =104 km x =103 km

ClimatologicalState

SynotpicScale

determining factor « slaved »

e.g., ENSO, NAO e.g., European Rainfall

«  Weakly » Nonlinear Approaches

Methodologies: Linear•Classification methods (Pattern Recognition)•Linear statistical methods: CCA, SVD

•Artificial Neural Netwrok (ANN)•Kriging but with one-to-one correspondence

(i.e., almost linear)

Linear statistical methods: CCA, SVD

Singular Vector Decomposition,SVD: x(x’)>t = lllxlx’~ ~

where <l2>x= 1, <l

2>x= 1,> > >….>0~ ~

Canonical Correlation Analysis,CCA: x(x’)>t

-1/2x(x’)>t–1/2x(x’)>t

= lllxlx’~ ~

Limitations of the Downscaling Type I (Linear Statistical Approach)

Length of data required to establish a statistically-robust result could be huge & hard to estimate (van den Dool 1994 Tellus)

(Remeady X* -> +oo, Zorita&von Stroch 1999)

Synoptic-scale is not perfectly « slaved » to a climatological state « stochasticity  »

Physically-related two variables are not necessarily identified by this method (e.g., , Newman & Sardeshmakh 1995

JC)

Transfer from the Type-I Regime to Type II

increase of Nonlinearity~ Ro = U/L

Linear-WaveDynamicsTeleconnections (Hoskins&Karoly 1981 JAS)

Semi-Determinisitc

NonlinearDynamicsLocally-definedStochasitc

x

Previous Attempts for the Downscaling Type II (Synoptic Meso-Convective)

Weather generators: stochastic generation of local time series with a given climate state (Wilks, Wilby, ……)

Scaling-law based approaches: fractal, wavelets, etc. (e.g., Deidda 2000 WRS, Kuligowski

&Barros 2001 JAM, Croa&Wood 2002 JH)

Topographically-induced rains: local rainfall ~ vH.grad (h) (e.g., Sinclair 1994)

Generalization to a heterogeneous case

A Physical Basis for the Downscaling of the Fully-Nonlinear System (Type II)

log P(k)

log k

multiscalemode interactions

Linear MethodDoes not work

Data Set (Time series)

Bouée «Côte d’Azur» de Météo-France : vitesse du vent, à dt=1h, mars 1999 - décembre 2001

Correction des erreurs de chronologie de la série de mesures

Série temporelle Ui=1,M

avec M=24514, dont L=19522 termes qui seront analysés

m/s

Morlet Wavelet (continuous wavelet)

dts

ut

stffsufW

RLffonctionunedondeletteseneTransformé

ets

ut

stRsu

ondelettesdFamille

et

tMorletdeOndeletteEx

etdtt

RLmèreOndelette

su

susu

ti

)(1

)(,),)((

:)('

1)(1

)(;0,

:'

)2

exp()(

1)(:

10)(

)(:

*,

2

,,

2

2

412

2

Wavelet transform:

Théorème de reconstruction

20

2

0

2

2

)(1

),)((1

)(

)(

)(ˆ

:')(

s

dsdu

s

ut

ssufW

Ctf

RLfAlors

dC

itéadmissibildconditionlavérifiantRLSoit

NB: overcompleteness

Transformation en ondelettes de la série U

Spectre à t = 999h

2),)(()(

:

:1),)((

:

ijij

j

ij

TtUWTSi

tdatelaàUdeSpectre

MjavecTtUW

UdeondeletteseneTransformé

Time-mean

Fourier

Power Law

Methods:Observational Data Analysis:

Data Set: Buoy Time Series (off coast Nice): Wind Speed:Mar 1999-Dec 2001, t=1h

Methodology: Taylor’s frozen hypothesis:k=/u: k

Seek a power law: Pt) ~(t) as a function of time (wavelet spectra)

How to estimate: =(t) ?