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Joint Spatio-temporal Modes of Wind and Sea Surface Parameters Variability in
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the North Indian Ocean during 1993-2005
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Thaned Rojsiraphisala, Balaji Rajagopalanb,c, and Lakshmi Kanthac
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Department of Mathematic, Faculty of Science, Burapha University, Thailand
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b
Department of Civil, Environmental, and Architectural Engineering, University of
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Colorado, Boulder, Colorado, USA
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c
Co-operative Institute for Research in Environmental Sciences, versity of Colorado,
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Boulder, Colorado, USA
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Colorado, USA
Department of Aerospace Engineering Sciences, University of Colorado, Boulder,
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Submitted to
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…………………………………..
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Journal of Geophysical Research
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2008
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Corresponding author:
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Thaned Rojsiraphisal
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Department of Mathematics, Faculty of Science, Burapha University
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169 Long-Hard Bangsaen Road, T. Saensook, A. Muang, Chonburi 20131, Thailand
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Tel: (66)-89-174-6747 Email: thaned@buu.ac.th
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Abstract
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Sea surface height (SSH) and sea surface temperature (SST) in the North Indian
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Ocean are mostly affected by the monsoon. Their variability and dynamics are of interest to
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surrounding population. In this study we use a set of data generated from a data-assimilative
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model to examine coherent spatio-temporal modes of winds and surface parameters via
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Multiple Taper Method with Singular Value Decomposition (MTM-SVD). Our analysis
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shows significant variability at annual and semi-annual, while only joint variability of winds
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and SSH is significant at an inter-annual mode (2-3 yr timescale) which is related to ENSO
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mode and its pattern exhibits a situation of “dipole” mode. The winds seem to be the driver of
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variability in SSH and SST at the annual cycle, semi-annual and interannual frequency bands.
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Furthermore, the forcing patterns in the winds are consistent with the large scale monsoon,
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especially the Indian summer monsoon feature in the basin.
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Keyword: MTM-SVD, joint spatiotemporal variability, ENSO
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1. Introduction
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North Indian Ocean (NIO hereafter) is the least explored among the Oceans in the
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world. It is forced by seasonally reversing monsoons, which play an important role in its
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variability. In particular, the monsoonal winds in the NIO greatly influence the variability of
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Sea surface height (SSH) and sea surface temperature (SST) which have an impact on the
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regional rainfall and consequently, the socio-economic well being of large population in the
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surrounding regions. Thus, better understanding of the nature of this variability is critical for
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improved resources planning and management. .
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The relationship of ENSO-monsoon during 1997-98 created climate and oceanic
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anomalies (Saji et al., 1999; Webster et al., 1999). SST anomaly affected from ENSO line
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condition yields SSH anomalies piling up along the South Asia coast. Sea level rises up and
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retards the outflow of estuaries resulting flood in the deltaic area especially in Bangladesh
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and India (University of Colorado at boulder, 2001; Singh, 2002).
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The SST anomaly is also an important key to climate change. One of the strong SST
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variations shows an out-of-phase SST in the tropic in the Indian Ocean, called Indian dipole
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Ocean (Saji et al., 1999; Webster et al., 1999; Yu and Rienecker 1999 and 2000) or the Indian
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Ocean zonal mode (IOZM). Situation of warm SST anomaly in the Indian Ocean, usually in
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order of 0.5oC, leads too strong monsoon (Hastenrath, 1987; Harzallah and Sadourny, 1997;
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Clark et al., 2000). Not only the monsoon affects the South Asia area but also the East
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African. A few studies confirmed that the SST variation in the Indian Ocean is a major
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contribution to the East African precipitation variation (Mutai et al., 1998; Goddard and
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Graham, 1999; Clark et al., 2003).
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In the past few decades, several mathematical and statistical techniques have been
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developed and used to identify variability from signal data. Techniques such as Singular
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Value Decomposition (SVD) (Wallace et al., 1992; Bretheron et al., 1992) related orthogonal
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multivariate spatiotemporal decompositions (Bretherton et al., 1992), Spectral Analysis
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(Emery and Thomson, 1998), Principal Oscillation Patterns (von Storch and Zwiers, 2003),
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Principal Component Analysis (PCA) (Preisendorfer, 1988; Jolliffe 2002), Wavelet Spectrum
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(WS) Analysis (Torrence and Compo, 1998) as well as Multiple-Taper Method-Singular
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Value Decomposition (MTM-SVD) (Mann and Park, 1994) have been applied in the analysis
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of atmospheric and oceanic data. Some methods only limit to identify either frequency
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domain or spatial domain. Some methods can only identify standing mode without
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information of dynamics and some methods can only identify localized variation.
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Unlike other univariate or multivariate, MTM-SVD is a powerful multivariate tool to
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simultaneously detect an oscillatory signal in both spatial and temporal data that can be used
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to identify frequencies where an unusual concentration of narrowband variance occurs. It can
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also be used to reconstruct the time history and spatial pattern associated with a frequency of
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interest (Mann and Park, 1999). Mann et al. (1996) applied the MTM-SVD to investigate
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decadal climate variation in the Northern America. Tourre et al. (1999) identified joint
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variability between SST and sea level pressure with MTM-SVD. Rajagopalan et al. (1998)
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and Mann and Park (1999) used the MTM-SVD to show how reliable the method was by
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testing through a synthetic data set with colored noise added and applied the method to
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climate data.
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In this study we intend to investigate joint variability and its dynamics of wind and
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two surface parameters; SSH and SST using MTM-SVD. The paper is organized as follows.
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Data description used in this study is first discussed and follows by brief detail of the MTM-
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SVD on joint variability. Results of MTM-SVD analysis is presented next. Discussions of the
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results conclude the paper.
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2. Data
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Wind and surface parameters are derived from satellite observations and hence, have
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coarse resolution and sometimes contain missing values. Thus, to have data on a finer scale
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and devoid of missing values, we used assimilation numerical ocean model applied to the
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NIO on a hindcast mode for the period 1993-2005 (Rojsiraphisal, 2007). The numerical ocean
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model used is the University of Colorado version Princeton Ocean Model (CUPOM) based
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on a primitive equation model using topographically conformal coordinate in the vertical and
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orthogonal curvilinear coordinates in the horizontal. Details of the basic features of CUPOM
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can be found in Kantha and Clayson (2000) and Mellor (1996).
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The NIO hindcast model has 1/4o resolution in the horizontal and 38 sigma levels in the
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vertical, with the levels closely spaced in the upper 300 m. The model is forced by 6-hourly
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ECMWF winds with Smith (1980) formulation for the drag coefficient CD to convert the
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ECMWF 10-meter wind speed to wind stress at the surface:
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1.110-3 ,
U10  6 m / s

Cd  
.
3
(0.61  0.063U10 ) 10 , U10  6m / s
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It assimilates altimetric sea surface height anomalies and weekly composite MCSST
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using a simple optimal interpolation-based assimilation technique. The model and the
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assimilation methodology based on conversion of SSH anomalies into pseudo-BT anomalies
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for adjusting the model temperature field via optimal interpolation, have been described in
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great detail in Lopez and Kantha (2000a and b). Details of CUPOM applied to the NIO along
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with validation of model’s results can be found in Kantha et al. (2008) and Rojsiraphisal
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(2007). The SSH is calculated explicitly using the split-mode technique.
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This coupled assimilation resulted in a continuous space-time data of wind and all the
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surface parameters (i.e., SST, SSH) for the 1993-2005 period. Weekly values are computed
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from this data for use in this research. To keep the data size reasonable and maintain all the
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spatial information data at every 1.5o×1.5o degree resolution north of 10oS in the Indian
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Ocean is used in the analysis. In all the data consists of 13-years (1993 to 2005) of weekly
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and monthly SSH (N = 676 weeks assuming 52 weeks a year) covering 10oS-26oN, 39o-
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120oE; total of 888 locations (M = 888 grid points) in the spatial domain (Figure 1).
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3. Investigation Tool – Frequency Domain Multi Taper Method-Singular Value
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Decomposition (MTM-SVD) Approach
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Robust diagnosis of the key low-frequency modes of large-scale climate entails
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capturing the coherent space-time variations across multiple climate state variables.
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Traditional time-domain decomposition approaches for univariate and multivariate
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data provide useful details on the broad-scale patterns of variability. However, these
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approaches lack the ability to isolate narrow-band frequency domain structure (Mann
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and Park, 1994; 1996). Detailed methodology development and examples of the
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MTM-SVD methodology can be found in Thomson (1982); Mann and Park (1994,
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1996); Lees and Park (1995). The method relies on the assumption that climate modes
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are narrow band and evolve in a noise background that varies smoothly across the
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frequencies.
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computed based on the multi-taper spectral analysis (Thomson, 1982; Park et al.,
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1987).
Subsequently, spectral domain equivalents of each grid point are
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Here we applied the MTM-SVD approach to study the individual and joint
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spatiotemporal modes of variability of wind and sea surface parameters in the NIO region.
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The spatial time series of each parameter is standardized by removing the long-time mean at
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each grid point and normalized by dividing the long-term standard deviations by weekly (or
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monthly) basis so that the variability has a unit variance and also the weekly (seasonal) cycle
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is removed. This normalization eliminates the disparity in the units between the variables.
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The details of the technique can be found in the aforementioned references but below we
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provide a brief description of the method abstracted from the references for the benefit of the
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readers.
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Consider a standardized of spatiotemporal time series at site m-th of
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x  , n  1, 2,
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time series, M and L  M are numbers of sites of first and second variables, respectively.
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The first M term of x ( m ) are time series of the first variable, and the next L  M are time
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series of the second variable. The standardized xn( m ) is calculated from xn( m )  xn( m ) /  ( m ) ,
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where prime notation represents the anomalies data and  ( m ) is the standard deviation at site
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m-th. Note that the location of each variable need not be at the same site but both
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spatiotemporal time series must have same length in temporal space. Then we apply multiple-
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taper transform to the time series and obtain “eigenspectra” at each frequency as
(m)
n
, N and m  1, 2,
, M , M  1, M  2,
, L , where N is the length of the
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N
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(1)
Yk( m ) ( f )   wn( k ) xn( m ) ei 2 f n t ,
n 1
 
N
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where Δt is the sampling interval and wn( k )
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of K data tapers (also called Slepian tapers), k  1,
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have energy peaks within a narrow frequency bandwidth of f  pf R , where f is a given
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frequency and f R  1/( Nt ) is the Rayleigh frequency (the minimum resolvable frequency
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range for the time series). Note that the number of tapers K represents a compromise between
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the variance and frequency resolution of the Fourier transforms (Mann and park, 1996). Also,
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note that the Slepian tapers can considered as a sequence of weighting function, an example
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of the first three Slepain tapers can be found in Mann and Park (1999).
is the k-th member in an orthogonal sequence
, K . The set of K tapered eigenspectra
With the set of eigenspectra, we can form a (M  L)  K matrix
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n 1
(2)
 w Y (1)
1 1

 w2Y1(2)


 w Y (M )
M 1
A( f )  
 w Y (M 1)
 M 1 1
 w Y (M  2)
 M 2 1


( M  L)
 wM  LY1
w1Y2(1)
w2Y2(2)
wM Y2(M )
wM 1Y2(M 1)
wM  2Y2(M  2)
wM  LY2(M  L)


(2)

w2YK


wM YK(M ) 
,
wM 1YK( M 1) 

( M  2) 
wM  2YK



wM  LYK( M  L) 
w1YK(1)
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where each row is calculated from a different grid point time series. Each column uses a
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different data taper from equation (1) and wi represents specific weighting at each grid point.
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Note that each row is computed from a different series (grid point), and each column
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corresponds to using a different taper. Subsequently, a complex singular value
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decomposition (CSVD)is performed through,
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K
A( f )   k ( f ) u k ( f )  vk ( f )
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k 1
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where u k is complex spatial pattern and v k is spectral domain containing information of both
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variables (i.e., the first M components contain information of the first variable and the next
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L  M components contain information of the second variable). These eigenvectors can be
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inverted to obtain the smoothly varying envelope of the kth mode of variability at
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frequency f (Mann and Park, 1996). The localized fractional variance (LFV) provides
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a measure of the distribution of variance by frequency, and above a select confidence
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level threshold (e.g., 90%, 95%), represents a dominant narrow band mode. The
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confidence levels are computed based on the locally white noise assumption, and are
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constant outside the secular band. Mann and Park (1996) describe a bootstrap method
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used to obtain the confidence bands for this study. In general, the computed principal
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eigen-spectrum (described above) yields a number of narrowband peaks. The MTM-
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SVD technique has been effectively applied to the analysis of global SSTs and SLPs
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(Mann and Park, 1994 and 1996), identification of dominant modes of variability in
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the Atlantic basin (Tourre et al., 1999), and also for forecasting (Rajagopalan et al.,
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1998).
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The LFV spectrum was used to identify significant frequencies, and temporal and
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spatial reconstructions were carried out to understand the joint variability of the
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climate fields in the NIO region. In this study, we used the choice of bandwidth parameter
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p = 2 and K = 3 tapers, which provide enough spectral degrees of freedom for signal-noise
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decomposition, and allow reasonably good frequency resolution as well as stability of
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spectral estimates according to Mann and Park, (1994) and Mann and Lees, (1996). The
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monthly data allows for a maximum frequency of f = 6 cycle yr-1 (2 months period) to be
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resolved efficiently.
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In this section we will isolate the dominant frequencies of each individual surface
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parameters; SSH and SST then we will discuss its individual dynamic. Later, we will analyze
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spatiotemporal variability jointly between a pair of zonal and meridional wind stresses and
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SSH and SST. This allows us to obtain insight information of possible dynamical processes
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governing such signals.
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Results
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The MTM-SVD method is applied to the fields individually and also jointly. First we
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present results from the individual analysis of SSH and SST and then jointly between winds
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and SSH and winds and SST.
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Sea Surface Heights
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The LFV spectrum of SSH based on the analysis of monthly data is shown in Figure
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2a (blue). The dominant frequencies can be seen in, 1.6-4 years period band (f ~ 0.3-0.6 cycle
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yr-1), annual cycle (f ~ 1.0 cycle yr-1), semi-annual cycle (f ~ 2.0 cycle yr-1), and seasonal
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cycle (f ~ 4.0 cycle yr-1). All these peaks appear to breach the 99% confidence level. The LFV
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spectrum from the weekly data (green curve in the figure) is similar to the the one from
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monthly data.
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Sea Surface Temperature
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The LFV spectrum of SST (Figure 2b—Plan to omit) reveals two significant
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frequencies (>99% confidence limit) at annual and semi-annual periods (f = 1.0 and 2.0 cycle
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yr-1) in both weekly and monthly data. The frequencies in the interannual bands at f = 0.25-
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0.4 and 3.0 cycle yr-1 are barely significant at 90% confidence level – this a difference from
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the SSH results. A linear trend is evident between f = 0 and 0.75. cycle yr-1 , this reddening of
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the spectrum and a secular trend is consistent with prior results in the Indian Ocean (xxxx).
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Joint Wind and SSH Variability
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NIO, as mentioned earlier, is most influenced by the Indian summer monsoon, thus,
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the variability of SSH and SST are likely to be tied firmly to the monsoonal winds. To
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identify the joint variability, we performed joint analysis of the winds (zonal, WSX and
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meridional, WZY, winds separately) with SSH. Figure 4 shows the LFV of this joint analysis.
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The LFV of joint variability of both pairs (WSX-SSH and WSY-SSH) yield
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significant peaks (over 99% confidence limit) at inter-annual, annual, and semi-annual
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periods seasonal and intra-seasonal (at 90% confidence level). The significant peaks observed
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in this joint analysis are same as the frequencies identified in the individual analysis of SSH.
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Here too, the LFV spectra from the weekly and monthly data are similar.
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To understand the joint variability in space and their evolution at these dominant
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frequencies, we performed spatial reconstructions for the two fields at the frequencies
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identified above. Spatial patterns of the zonal and meridional wind fields and the
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corresponding SSH at the annual cycle (f = 1.0 cycle yr-1) frequency, from the respective joint
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analyses are shown in Figure 5. The vector lengths in this (and subsequent spatial
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reconstructions) figure are indicative of the magnitude of the signal and the phase (i.e.,
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directions) are with respect to a reference location (grid 529th at 7.25oN, 76.75oE) – the vector
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at the reference location is always horizontal pointing right (i.e., at the 3 o’clock position).
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The direction of vectors at other locations corresponds to temporal difference (time lead/lag)
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relative to the reference vector. In the spatial pattern, clockwise vector rotation represents
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negative relative lag (or “lead”), and counter-clockwise rotation represents positive lag (or
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“lag”). A complete rotation represents a periodicity of the mode; e.g. 1 year for f = 1.0 cycle
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yr-1, and a grid point vector at 12 o’clock (90o) experiences peaks at 4 months lag relative to
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the reference grid point.
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Annual frequency
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The zonal winds (Figure 5a) exhibit an out of phase relationship between the NIO
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region and the Southern Indian Ocean region – this is indicative of the annual cycle in that the
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winds in the northern hemisphere are always out of phase with the southern. The meridional
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winds (Figure 5c) show a strong signal in the Arabian Sea region of the NIO the Bay of
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Bengal region and the South China Sea – all active regions of the Asian monsoon, and Indian
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monsoon in particular. In fact, these wind patterns are almost identical to the monsoon wind
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features (xxx reference xxx). The vectors all in the same direction indicative of the feature
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happening in the same period (i.e. the summer period), the slight changes in vector directions
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between the South China Sea region and NIO shows the delay in the South China monsoon
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relative to the Indian.
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The spatial pattern of SSH variability at this mode (Figures 5b and 5d) from the joint
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analysis of the two wind components is similar. The SSH signal is strong over most of the
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basin except in the southern hemisphere where the winds are quite a bit weaker as seen
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above. variability is dominant in most area of the NIO except area of 50o-80oE and 5oS-0oN.
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From these plots, we see that the strong magnitude of SSH in the southwest region is not
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locally affected by the zonal wind since the zonal wind in that region is weak during that
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time. It may be affected from strong easterly wind in the south leading by about 2 months.
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The propagation of the SSH signal at the annual cycle frequency is apparent and can
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be explained as follows. Starting in the black-vector region in the southern Indian Ocean it
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takes about 1-2 months (~30o-45o) to propagate westward into the southwest (green-vector)
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area. The large magnitude along the west coast of India propagates westward into the red-
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vector area in the southern Arabian Sea. Similarly, the signal in the southwest propagates
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northward along the Somali coast into red/blue-vector areas. Then the red area near the
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Socotra Island propagates eastward into the middle of Arabian Sea; while the blue area
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moves southward into the Equatorial region which next propagates eastward along the
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equatorial waveguide into the Sumatra Island area and completes the cycle with a stronger
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magnitude along the east coast of Bay of Bengal. The propagation of the SSH signal lagged
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by a few months to the wind signal – which is consistent, in that the SSH anomalies are
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driven in large part by the strong wind forcing in the basin (xxx refs xxx).
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Semi-annual
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The spatial patterns of the wind and SSH fields at the semi-annual frequency (f = 2.0
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cycle yr-1) are shown in Figure 6. At this period, the strongest signal in the both the wind
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components are in the Arabian Sea region – which is the active Indian monsoon region. This
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is more so in the meridional component where the amplitudes are strongest in Western
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Arabian Sea region and also a bit in the South China Sea. The main aspect at the annual and
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semi-annual frequency is similar in terms of the winds in that the variability is strongest in
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the regions of monsoonal winds and the phase lags are consistent with spatial and temporal
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variability of monsoons in the region.
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The SSH signal is stronger with the meridional winds (Figure 6d). The large
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magnitude of SSH between 2o-10oN and 50o-77oE (referred as area A1 thereafter) is affected
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by strong WSY in the western Arabian Sea, which leads this by about a month or so. The
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strong magnitude of SSH in the southern region on the other hand is forced by zonal wind
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stress locally and is also affected by the strong magnitude of SSH in the area A1.
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Interannual band
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The dominant frequency in the inter-annual band is (f = 0.4 cycle yr-1). We show a
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snap-shot reconstruction for this frequency, in that snap shots of the magnitudes of the two
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fields at several times in the entire cycle are shown. Figure 7 shows the reconstruction of
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spatial snapshots of joint WSX (left panels) and SSH (right panels) variations of 2.5-year (f =
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0.4 cycle yr-1) period at various times, spanning one-half of a complete cycle (~1.25 years or
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15 months). Squares and triangles indicate sign differences and the sizes of these symbols
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represent magnitude of each variable that is relative to value of WSX at a reference grid point
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(grid 529th). The initial snapshots (at 0o) correspond to peak WSX anomalies in the east-
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central of the NIO and the corresponding SSH anomalies. Next snapshots reveals the
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evolution of WSX and SSH anomalies. The WSX anomalies are weakening in the next few
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months and reverse the signs in about 6-7 months (at about 90o); though the SSH anomalies
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are also weakening but at slower rate. While the final snapshots correspond to the opposite
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conditions reveals at about 15 months later. At the phase between 112.5o and 135o, large
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magnitude of SSH anomalies occur within 10oS - 10oN across the ocean with one sign
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covering most of the western side of equatorial region and the other sign occurs next to the
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Sumatra Island. These anomalies in the eastern side intrude into the middle of this region.
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This is reminiscent of the “dipole” feature (Saji et al., 1999; Webster et al., 1999; Yu and
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Rienecker 1999 and 2000) identified in the wind fields.
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To analyze the temporal variability, we reconstructed the time components of these
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fields at locations (see Figure 8) from regions exhibiting strong magnitudes in Figure 7. The
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temporal reconstructions are shown in Figure 9. Relationship between WSX and SSH at the
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western equatorial (top) locations are almost in-phase, while the WSX in this region leads
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SSH at Sumatra by 10-12 weeks. There is a rather sudden change in WSX during the late
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1997 that can be seen with an anomalous high in SSH at that time. This potentially could be
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driven by the strongest ENSO event in 1997-98.
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Wind and SST
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The LFV spectra of the joint analysis of WSX-SST and WSY-SST are shown in
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Figure 8. The spectra is very similar to that of joint wind and SSH analysis from before.
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Significant peaks are found at the annul, semi-annual and interannual bands.
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The spatial reconstructions of the SST fields (Figure not shown) shows anomalies of opposite
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sign in the Northern Arabian Sea, Northwestern Bay of Bengal and South China Sea and;
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near the Tanzanian coast. We observed that the wind fields lead the SST variations by a few
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months and that the strong SST variantion in the southwest region of the domain is affected
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by both local WSX and WSY variation. At the semi-annual (f = 2.0 cycle yr-1) frequency the
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SST anomalies are small everywhere except in Western Arabian Sea.
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5. Conclusion
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The MTM-SVD was applied to the joint fields of winds and variables at the surface.
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This analysis was able to identify frequencies where an unusual concentration of a
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narrowband variance occurs. In this study, we performed MTM-SVD along with bootstrap
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procedure on decimated weekly and monthly data. The results revealed higher energy of local
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fraction variance in weekly data at the high frequency as expected; while at other low
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frequency, the LFVs of both data are relatively comparable. The LFVs from joint MTM-SVD
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simultaneously revealed dominant cycles at annual and semi-annual frequencies which
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should be expected in the NIO because the variability in this region are strongly affected by
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seasonal reversing monsoons.
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The LFVs of winds and SSH also reveal dominant variability in the inter-annual
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frequency, which relates to the ENSO cycle; while variation of the wind components and SST
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at this mode are not statistically significant. This is not a surprise result since the NIO locates
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in the tropic region, thus SST is usually warm most of the time and the variability of SST in
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this region does not vary much from year to year.
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Reconstructed spatial patterns reveal the dynamics of joint variation at specific
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frequency. The phase and magnitude differences in each pair allow us to understand the
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relationship between the two fields. We observed that the anomalous wind during 1997-98
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strongly affected in the eastern side of NIO which can be seen from signature of SSH
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anomaly.
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One can extend the analysis of MTM-SVD to more than two fields but it requires
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higher computer resources. However, a disadvantage of MTM-SVD technique is that the
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dominant variability at any frequency requires strong variance in many spatial locations. If
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there are only few spatial locations with strong variability participating at that frequency, that
367
area will not be dominant.
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477
Figure Captions:
478
479
Figure 1
480
Locations of data used for MTM-SVD analysis in the North Indian Ocean. Data locations are
481
defaulted by  with reference location at the 529th grid point which is just below the tip of
482
Indian.
483
484
Figure 2
485
Comparing LFV spectra (relative variance explained by the first eigenspectra) of weekly
486
(green) and monthly (blue) SSH time series as a function of frequency. Black, red, cyan, and
487
magenta lines denote 99%, 95%, 90%, and 50% confident limits from bootstrap procedure,
488
respectively.
489
490
Figure 3 (not shown)
491
Same as Figure 2 but for SST.
492
493
Figure 4
494
LFVs for joint variability between WSX and SSH (a) and between WSY and SSH (b).
495
496
Figure 5
497
Spatial variations at annual (f = 1.0 cycle yr-1) cycle of (a) WSX on joint WSX-SSH; (b) SSH
498
on joint WSX-SSH; (c) WSY on joint WSY-SSH; and (d) SSH on joint WSY-SSH. Vector
499
lengths in (a) and (b) correspond to the magnitude of each variable relative to size of WSX at
500
a reference location (grid 529th at 7.25oN, 76.75oE). Phase of the vectors correspond to
501
temporal difference (time lead/lag) relative to the reference vector (at 3 o’clock). In the
502
spatial pattern, clockwise vector rotation represents negative relative lag (or “lead”), and
503
counter-clockwise rotation represents positive lag (or “lag”). A complete rotation represents a
504
periodicity of the mode; e.g. 1 year for f = 1.0 cycle yr-1, and a grid point vector at 12 o’clock
505
(90o) experiences peaks at 4 months lag relative to the reference grid point. Same analogous
506
is applied for a patterns in (c) and (d), where all vectors are related to vector at grid 529th of
507
WSY.
508
509
Figure 6
510
Same as Figure 5, but for semi-annual (f = 2.0 cycle yr-1) period.
Page 16 of 28
511
Figure 7
512
Spatial patterns shown at progressive intervals (~56 days or 22.5o from top to bottom),
513
spanning one-half of a complete cycle (~1.25 years or 15 months). Left and right panels show
514
variations of WSX and SSH, respectively. Sizes of square and triangle represents magnitude
515
of both variables are relative to value of WSX at a reference grid point (grid 529th). The
516
squares and triangles indicate different in signs. The initial snapshots correspond to peaks
517
WSX anomalies in the east-central of the North Indian Ocean and SSH anomalies. While the
518
final snapshot corresponds to the opposite conditions that obtain at one-half cycle later.
519
520
Figure 8 (Plan to omit)
521
LFVs for joint variability between WSX and SST (a) and between WSY and SST (b).
522
523
Figure 9
524
Locations of reconstructed time series, WSX at southeast of Sri Lanka (grid 450 th) in yellow
525
squares and SSH at west equatorial region (grid 346th), west Sumatra Island (grid 416th), and
526
west India (grid 484th) in green circles.
527
528
Figure 10
529
Time series reconstruction at inter-annual mode (f = 0.39 cycle yr-1) comparing WSX at
530
southeast of Sri Lanka and SSH at three other locations; western of equatorial region (top),
531
western Sumatra Island (middle), and western India (bottom).
532
Page 17 of 28
533
Page 18 of 28
Figures
Figure 1: Locations of data used for MTM-SVD analysis in the North Indian Ocean. Data
locations are defaulted by  with reference location at the 529th grid point which is just
below the tip of Indian.
Figure 2: Comparing LFV spectra (relative variance explained by the first eigenspectra) of
weekly (green) and monthly (blue) SSH time series as a function of frequency. Black, red,
cyan, and magenta lines denote 99%, 95%, 90%, and 50% confident limits from bootstrap
procedure, respectively.
Figure 3 (Plan to omit): Same as Figure 2 but for SST.
Figure 4: LFVs for joint variability between WSX and SSH (a) and between WSY and SSH
(b).
Page 2 of 28
Figure 5: Spatial variations at annual (f = 1.0 cycle yr-1) cycle of (a) WSX on joint WSXSSH; (b) SSH on joint WSX-SSH; (c) WSY on joint WSY-SSH; and (d) SSH on joint WSYSSH. Vector lengths in (a) and (b) correspond to the magnitude of each variable relative to
size of WSX at a reference location (grid 529th at 7.25oN, 76.75oE). Phase of the vectors
Page 3 of 28
correspond to temporal difference (time lead/lag) relative to the reference vector (at 3
o’clock). In the spatial pattern, clockwise vector rotation represents negative relative lag (or
“lead”), and counter-clockwise rotation represents positive lag (or “lag”). A complete rotation
represents a periodicity of the mode; e.g. 1 year for f = 1.0 cycle yr-1, and a grid point vector
at 12 o’clock (90o) experiences peaks at 4 months lag relative to the reference grid point.
Same analogous is applied for a patterns in (c) and (d), where all vectors are related to vector
at grid 529th of WSY.
Page 4 of 28
Figure 6: Same as Figure 5, but for semi-annual (f = 2.0 cycle yr-1) period.
Page 5 of 28
Figure 7: Spatial patterns shown at progressive intervals (~56 days or 22.5o from top to
bottom), spanning one-half of a complete cycle (~1.25 years or 15 months). Left and right
panels show variations of WSX and SSH, respectively. Sizes of square and triangle represents
magnitude of both variables are relative to value of WSX at a reference grid point (grid
529th). The squares and triangles indicate different in signs. The initial snapshots correspond
to peaks WSX anomalies in the east-central of the North Indian Ocean and SSH anomalies.
While the final snapshot corresponds to the opposite conditions that obtain at one-half cycle
later.
Page 6 of 28
Figure 7: (continue)
Page 7 of 28
Figure 7: (continue)
Page 8 of 28
Figure 8: (Plan to omit) LFVs for joint variability between WSX and SST (a) and between
WSY and SST (b).
Figure 9: Locations of reconstructed time series, WSX at southeast of Sri Lanka (grid 450 th)
in yellow squares and SSH at west equatorial region (grid 346th), west Sumatra Island (grid
416th), and west India (grid 484th) in green circles.
Page 9 of 28
Figure 10: Time series reconstruction at inter-annual mode (f = 0.39 cycle yr-1) comparing
WSX at southeast of Sri Lanka and SSH at three other locations; western of equatorial region
(top), western Sumatra Island (middle), and western India (bottom).
Page 10 of 28