Atlantic Oceanographic and Meteorological Laboratory

National Oceanographic and Atmospheric Administration

and

Hugo F. Bezdek

Cooperative Institute for Marine and Atmospheric Science

University of Miami

Submitted to:

*Journal of Climate*

12 December 1997

# Abstract

Using summaries drawn from the Comprehensive Ocean Atmosphere Data Set
(COADS) for the period 1947-1990 in the North Atlantic Ocean, we explore
the geographical variability of the structure of the cross correlation
function between anomalies of sea surface temperature (SSTA) and
anomalies of the sum of the latent and sensible heat fluxes (LSHFA). Two
distinct cross correlation structures emerge, one of which dominates in
the open ocean at midlatitudes and the other of which dominates near the
western boundary current and in the tropics. The former is characterized
by a strong antisymmetry, while the latter is one-sided with a peak at
zero-lag. A discrete stochastic model (vector first-order
auto-regressive model) is proposed to account for the observed
structure. The model, which reproduces these structures with fidelity,
implies that the SSTA and LSHFA must be mutually coupled in order to
reproduce the characteristics of the observations, especially in the
open ocean at midlatitudes. The low-frequency, decadal-scale propagating
signals that have been the subject of several recent studies are not
captured by this model as seen in the residuals that remain from a fit
of the model to the observed anomalies. This suggests that the
development of SSTA has a significant contribution from processes that
cannot be characterized as an integral response to atmospheric
white-noise forcing.

# I. Introduction

## A. Ocean/Atmosphere Interactions in the North Atlantic

Many authors have suggested that coupled interactions between the ocean
and atmosphere may influence climate variability on seasonal to decadal
timescales \[Bjerknes, 1964; Frankignoul, 1985; Deser and Blackmon,
1993; Kushnir, 1994; Frankignoul, 1995\]. One fundamental coupling
between these fluids is through the transfer of heat across the air-sea
interface. This consideration has motivated a number of observational
and numerical studies to support the hypothesis that anomalies of the
atmospheric circulation are mainly responsible for the appearance of
anomalous wintertime SST \[see Delworth (1996) for an overview\]. In
these studies, the atmospheric circulation is often parameterized and
the choice of the parameter used to characterize the atmosphere varies.
Davis (1976) chose sea level pressure to represent the atmospheric
circulation for a study of seasonal to annual variability over the North
Pacific. He found that a significant correlation between sea level
pressure anomalies (SLPA) and SSTA occurs only when SLPA leads SSTA by a
month or so. He pointed out that this suggests the ocean is being driven
by the atmosphere. Davis (1978) also found that a seasonal
stratification of the data indicates that autumn and winter SLPA are
predictable from prior observations of either SLPA or SSTA.

Frankignoul and Hasselmann (1977) explored a stochastic model
(univariate first-order auto-regressive model) of climate variability in
which the SST tendency is driven by random fluxes of latent plus
sensible heat. They recognized that the form of the cross correlation
function between the SSTA tendency and the chosen atmospheric parameter
provides a critical test of the basic premise of the model. To
facilitate a comparison with the results of Davis (1976), they
parameterized the atmospheric circulation using anomalous surface
pressure to represent the anomalous surface heat flux. They found that
the cross correlation between SLPA and SST tendency anomalies is zero
for negative lags, and is a decaying exponential for positive lags with
its maximum at zero lag. They attribute the observed peak at a one-month
lag to the 1-month averaging performed on the observed time series prior
to performing the correlation and they conclude that the stochastic
model explains most of the features of mid-latitude SST anomalies. As
they explicitly state: \"We suggest here that the development of SST
anomalies can be understood\..., without invoking feedback mechanisms,
as the integral response of the ocean to random short-time-scale forcing
by the atmosphere.\"

Frankignoul and Reynolds (1983) recognized that cross-correlations
between dominant empirical orthogonal functions of SST variability and
the variability of net surface heat flux have an "antisymmetric aspect"
(Frankignoul, 1985) that cannot be successfully modeled by the
development proposed by Frankignoul and Hasselmann (1977). They
hypothesized an atmospheric feedback, which implies that the atmospheric
parameter (net surface heat flux in this case) consists of both the
random fluxes of heat and a term proportional to SSTA. No feedback from
the ocean to the atmosphere is included. Although purely antisymmetric
cross correlations cannot be obtained via this model, significant
antisymmetric character can be introduced this way. Again, the peak
values tend to occur at zero lag, but two factors contribute to moving
the peaks away from zero: (1) the introduction of colored noise and (2)
the 1-month averaging performed on the observed time series prior to
calculating the correlation.

## B. Geographical Variability

Using a simple thermodynamic model to relate the flux anomalies to the
tendency of the SST anomalies, Cayan (1992) sought to test the
hypothesis that the atmosphere forces the ocean. In this model, the
upper-ocean heat balance is driven by the latent and sensible heat flux.
The signature of this process is expected to be a negative correlation
between anomalies of latent plus sensible heat flux and anomalies of SST
tendency. This signature was found nearly everywhere in the
extratropics, where the flux and the tendency are well correlated. In
the tropics, however, Cayan found that the correlations are smaller and
the flux anomalies tend to be in phase with the SST (not tendency)
anomalies. This suggested to him that the flux anomalies are driven by
the ocean thermal field. Although he did not directly comment about
this, Figure 2 of Cayan (1992) reveals a small, positive correlation
between flux and tendency anomalies in both the tropics and a region
centered around 40˚N off the coast of North America.

Battisti *et al.* (1995) conducted some numerical experiments to test
the hypothesis that North Atlantic SST variability is due to the
variability in the surface heat flux (which is dominated by latent plus
sensible heat flux). They found the hypothesis to be significantly in
error only along the U.S. coast, in a small region extending from Cape
Hatteras to Nova Scotia. Here the observed flux anomalies are
anticorrelated with the SSTA, which they suggested implicates ocean
advection in generating SSTA. They did not examine relationships in the
tropics.

As noted above, the nature of the cross correlations between the SSTA
and LSHFA can provide a critical test of the nature of air-sea coupling,
especially in midlatitude regions. Evidence for geographical variability
of the cross correlations between SSTA and LSHFA suggest that this
property could delineate regions where the nature of air-sea
interactions are significantly different. In this paper, we will explore
this hypothesis, using an analysis of observations from the North
Atlantic Ocean.

# II. Data Processing

The data set for this study was drawn from the Comprehensive
Ocean-Atmosphere Data Set (COADS) trimmed file, for which individual
observations were averaged both temporally and geographically into
monthly means for 2˚ x 2˚ bins (Woodruff *et al.*, 1987). Our study
region covers the North Atlantic Ocean from 8˚N to 60˚N and from 94˚W to
6˚W. We eliminate from further consideration all bins which have more
than 9 years of missing data for any given month. Our analysis uses the
remaining 737 2˚ x 2˚ bins (gray, red and blue squares in Figure 1a),
each with a time series of 528 monthly values for the period from
January 1947 to December 1990. In this study, we concentrate on three
variables: sea-surface temperature (SST), latent heat flux (LHF) and
sensible heat flux (SHF). Flux estimates in COADS were calculated via
bulk formulae from measurements of SST, air temperature, surface winds
and humidity. See Cayan (1992) for both a discussion of the methods used
by COADS and an analysis of the errors in that database. Anomalous
values of SST, LHF and SHF were calculated as departures from the
monthly means of the detrended time series at each location; missing
values were replaced with climatology. LHFA and SHFA were summed to
provide a time series of LSHFA at each location. Our sign convention has
positive flux anomalies leaving the ocean.

# III. Results

## A. Observations

At each of the 737 locations, we computed the cross correlation between
SSTA and LSHFA. Figure 1a shows the locations of the bins associated
with the lowest (red) and highest (blue) deciles based on the value of
the cross correlation at zero lag. Notice that the lowest decile (red)
corresponds to the midlatitude region away from the Gulf Stream, which
has been characterized as a place where "\...wintertime interannual to
subdecadal variability in SST is mainly due to local anomalies in the
air-sea flux of sensible and latent heat" (Battisti *et al.,* 1995). The
highest decile (blue) corresponds to the regions noticed by Cayan (1992)
and Battisti *et al.* (1995) as being exceptions to this
characterization. Henceforth, we will refer to these as the red and blue
regions, respectively. The median cross correlation functions (CCF) for
these two regions (solid lines in Figure 1b,c) emphasize the difference
inherent in the relationship between SSTA and LSHFA. The CCF in the blue
region is virtually zero for negative lags, peaks at zero lag and decays
for positive lags. For the red region, however, the CCF has a strong
antisymmetric component. The difference between the two regions is
further emphasized by the cross spectrum and phase spectrum between SSTA
and LSHFA (Figure 2a,b respectively). Both regions have cross spectral
densities which decrease with increasing frequency. More importantly,
the nearly constant phase spectra are at distinctly different phases for
the two regions with SSTA leading LSHFA by about 90˚ and 20˚ in the red
and blue regions, respectively. Of course, this is equivalent to saying
that LSHFA leads SSTA by 270˚ and 340˚, respectively.

We considered another approach for choosing locations of the highest and
lowest deciles, based on the frequency with which the SSTA and the LSHFA
had the same or opposite signs. This approach changed some details, but
the conclusion that two distinct regions can be defined is still
correct, and the new regions are substantially the same as those found
previously (Figure 1d). Note the similarity of the median cross
correlations (Figure 1e,f) to those already presented (Figure 1b,c).

## B. Modeling

Inspired by the success of local linear stochastic theory in explaining
the character of climate variations (Hasselmann, 1976; Frankignoul,
1995), we sought the simplest stochastic model that would unify our
results. We particularly wanted the model to be able reproduce the forms
of the cross correlation functions derived from the data, including the
antisymmetric shape of the cross correlation function in the red region.
Hasselmann\'s model, as extended by Frankignoul and Reynolds (1983), is
unable to produce an antisymmetric cross correlation function (although
it can produce an "antisymmetric aspect" \[Frankignoul, 1985\]),
primarily because no feedback from ocean to atmosphere is included.
Accordingly, we turn our attention to a first order vector
auto-regressive model (Wei, 1993), which includes feedbacks in both
directions.

We consider the model

![](media/image1.pct){width="270.0in" height="16.0in"}

where **Z~t~** is a two dimensional column vector, Φ is a 2 x 2 matrix
of coefficients and **a~t~** is a two dimensional column vector of
innovations (random noise), with the property

![](media/image2.pct){width="284.0in" height="35.0in"}

where **Σ** is a 2 x 2 symmetric positive definite matrix. Following
Frankignoul and Reynolds (1983), we associate the ocean state, Z~1,t~
with standardized SSTA and the atmospheric state, Z~2,t~ with
standardized LSHFA, where the time series are standardized by
subtracting the monthly means and dividing by the monthly standard
deviations.

It is instructive to consider the impact that the parameters in this
model have on the character of the resultant CCF. There are seven
parameters required to fully specify Φ and **Σ.** We define the elements
of Φ and **Σ** by φ~ij~ and σ~ij~, respectively. The diagonal elements
of **Σ** represent the variances of the random noise that directly
forces the ocean and atmosphere, respectively, while the off-diagonal
term represents the correlation between the two noise processes. The
diagonal elements of Φ represent the memory or persistence of the ocean
and atmosphere states. The off-diagonal terms represent the feedback or
coupling whereby the ocean directly influences the atmosphere and visa
versa. We confine ourselves to the range 1 \> φ~11~, φ~22~ ≥ 0.

If Φ is identically zero, then correlations between the ocean and
atmosphere arise solely because of correlations between the two noise
processes. The CCF is zero everywhere except at zero lag where it takes
the value σ~12~, which may be positive or negative. Similarly, when the
diagonal elements of Φ are nonzero (memory), and the off-diagonal
elements of Φ are zero (no feedback), there is no correlation between
the ocean and atmosphere, unless the two noise processes are correlated
(σ~12~ ≠ 0). When correlations exist, the extreme value of the CCF
(positive or negative) is at zero lag and the magnitude of the
correlation decays exponentially on either side of the origin. The decay
rates depend on the values of φ~11~ and φ~22~. In general, these rates
are different on either side of the origin, but the CCF never crosses
zero. In summary, if there is no feedback, nonzero CCF can arise only
via correlations in the noise. If there is memory in addition to the
noise correlation, then the correlations are extended to nonzero lags,
but the CCF never crosses zero.

Provided that the system has memory (φ~11~ ≠ 0 or φ~22~ ≠ 0),
correlations at zero lag (and all other lags, too) may be introduced by
feedback; correlated noise is no longer necessary. If the noise
processes are uncorrelated (σ~12~ = 0) and there is only partial
feedback (φ~12~ ≠ 0), then the CCF never crosses zero. With the
combination of partial feedback (φ~12~ ≠ 0) and correlated noise (σ~12~
≠ 0), it becomes possible for the CCF to cross zero. This corresponds to
the "antisymmetric aspect" mentioned by Frankignoul (1985). In this case
the CCF can exhibit extrema for at most two lags, one at zero and
another at ±1. When both feedback terms are nonzero, a great variety of
behaviors for the CCF become possible. In particular, only with full
feedback (φ~12~ ≠ 0, φ~21~ ≠ 0) is it possible to obtain a CCF that
simultaneously exhibits extrema at both +1 and -1 lag and is small or
zero at zero lag.

How do these considerations relate to the observations? We seek
estimates for Φ and **Σ**, using a least-squares fit of the model to the
observations at each location (Ljung, 1987; Wei, 1993). Figure 3 depicts
the distribution functions of the parameters φ~ij~, along with the
formal uncertainties derived by the fit. φ~11~ (Figure 3a), which
represents the persistence of the SSTA, is clearly larger than the other
terms and has a median value of about 0.5. φ~22~ (Figure 3d), on the
other hand, indicates that the persistence of the LSHFA is relatively
short, with a median value of about 0.05. Median values of φ~12~ (Figure
3b) and φ~21~ (Figure 3c) are approximately of equal magnitude (about
0.15) and of opposite sign. Figure 3 (e,f,g) show the distributions
for σ~11~, σ~12~ and σ~22~ respectively. Clearly the noise input to the
LSHFA is more uniform and generally stronger than that for the SSTA and
a wide range of noise correlations of both signs exists.

A dramatic example of the organization of the observations by the fitted
parameters of the model is shown in Figure 4, where red/blue represents
the regions where the fitted values of σ~12~ are less/greater than zero.
Figure 4 captures very clearly the distinction between the two regions
discussed in the last section. The lowest and highest deciles of fitted
values for σ~12~ produce Figure 5. Comparison of Figure 5 with Figures
1a-c and 2 reveal that the extreme deciles based on the distribution of
fitted values for σ~12~ (Figure 5) are substantially the same as those
based on the distribution of the values of the CCF at zero lag (Figures
1a-c and 2). Using the former deciles (based on the sign of σ~12~), we
extract parameter values for the red/blue regions from the distributions
in Figure 3 to produce Figure 6. We see that the red region has
significantly higher SSTA persistence and that the feedback coefficient
φ~21~ is strongest in the red region. The blue region, on the other
hand, has a substantially reduced SSTA persistence and a small value for
φ~21~. Regional differences for φ~22~ and φ~12~ are much less
pronounced. φ~22~ is considerably smaller than φ~11~ in both regions, as
expected from the curves shown in Figure 3. This supports the idea that
the atmosphere has much less persistence (memory) the does the ocean.
Notice that the noise input into the SSTA in the blue region (σ~11~) is
considerably stronger than that for the red region.

We ran an ensemble (200) of realizations of the model using the median
values of Φ and **Σ** obtained from the two regions to produce
model-based estimates of the CCF, power spectra, cross spectrum and
phase spectrum. We find good agreement with the observed cross
correlation functions (compare Figure 7c,d with Figure 5d,e). We also
recover the basic structure of the phase spectra as well (compare Figure
7f with Figure 5g). The observations suggest that the cross spectral
amplitudes of the SSTA and the LSHFA are the same in the red and blue
regions (Figure 5f), while the model simulations indicate that the red
region has slightly lower cross spectral amplitudes at high frequencies.
Many realizations of the model were required to make these regional
distinctions apparent, so it may be that the uncertainty (noise) in the
observations is simply too large to allow the emergence of this subtle
feature. The model cross-spectra also have a distinct knee near a
frequency of about 1 cycle/year and are flat out to lower frequencies.
The observational cross-spectra, on the other hand, increase as the
frequency decreases. The autospectra are also well-modeled (compare
Figure 7a,b with Figure 5b,c) except at low-frequencies. These
discrepancies are related to the low-frequency structure in the
residuals (observations minus model fit).

Autocorrelations and autospectra of the residuals suggest that the model
is successful in capturing the organized temporal structure at each
location in the observations. Autocorrelation values are less than 0.2
and autospectra are approximately constant at all frequencies. Recall,
however, that we obtained the model fit at each location independently.
Spatial correlations in the observations are known to exist and
residuals from the fit display large scale spatial and temporal
structure. Figure 8a shows the annual-mean, zonally-averaged residuals
of SSTA and LSHFA together with an identically-processed field of
uncorrelated white noise. We see that the SSTA residuals contain the
large propagating features that have been the focus of recent studies
(Hansen and Bezdek, 1996; Sutton and Allen 1997). The LSHFA residuals,
although not as well-defined as in the SSTA residuals, also show
propagating features on scales much larger than those seen in the white
noise panel.

One characterization of the consequences of the spatial correlations can
be seen in power spectra of the basin-wide median time-series of the
residual SSTA and LSHFA (Figure 8b,c). The residual SSTA spectrum is
clearly dominated by low frequency energy, while that for the LSHFA
residuals is less evident.

In summary, the discrete first-order vector autoregressive model
captures the structure of the cross correlation functions, cross spectra
and phase spectra of the observations, including the geographical
variability of these functions. It does not, however, capture the
behavior of the observations at low frequencies and large space scales.

# IV. Discussion

A precise description of the interaction between the ocean and
atmosphere is bound to be a complicated matter. The development and
interpretation of the mixed-layer heat budget, from which SST may be
inferred, is complex (Frankignoul, 1985). This budget contains at least
5 important terms that are not directly a function of local air-sea heat
flux. A similar description for the coupled system (heat budget of the
oceanic mixed-layer/atmospheric surface boundary layer system) would be
even more complex. There is no hope of accurately measuring all of the
processes well enough to assess the nature of air-sea coupling on
climate time scales any time soon. In the face of these difficulties, it
is clear that progress on this critical issue will be possible only in
small steps. We propose here one such step by seeking to describe and
model statistical relationships that characterize the low-order behavior
of this intricate system.

In that spirit, we seek a simple parameterization of the ocean and
atmosphere that can be associated with observations. Of the oceanic
fields available, SST appears to be the one that most influences the
atmosphere; it is a natural choice. For the atmosphere, we choose the
local surface heat flux, derived from measurements of wind and
ocean/atmosphere differences of temperature and humidity, because it has
a direct link to the SST. The local surface heat flux is a forcing term
in the SST tendency equation. In other studies, some have chosen to
parameterize the atmosphere with sea-level pressure (SLP), or some other
measure of atmospheric mass distribution. The nature of the apparent
statistical connection between the ocean and the atmosphere can be quite
different depending on the choice of parameterization and further
consideration of the relationship between SLP and surface heat flux is
needed to properly interpret such studies.

For example, the antisymmetric character of the CCF between SSTA and
LSHFA is not apparent in the CCF between SLPA and SSTA. In the red
region of the North Atlantic, the CCF for SLP and SSTA is very similar
to that of Davis (1976). In this region, however, our model fits suggest
that both mutual feedback and correlated noise are required to explain
the observed structure. This is not be true for the CCF between SLPA and
SST because only partial feedback between SLPA and SSTA is required to
obtain the observed shape. This difference, which leads to a substantial
difference in interpretation, arises from the difference between surface
heat flux and surface pressure. Our result appears to be more basic
because the relationship between SSTA and LSHFA is more direct.

We propose a coupled model driven by two sets of random impulses
correlated only at zero lag. Nontrivial coupling (or feedback) is
essential for reproducing the observed CCF structure, so it appears
inappropriate to interpret the results as one medium forcing the other.
If either φ~12~ or φ~21~ are zero (partial feedback), one of the
variables does not have a direct dependence on the value of the other.
In this case, it is appropriate to say that one variable forces the
other. This might be thought of as the causal limit of the coupled
system. The success of the fully coupled model suggests that it is
better to think of SSTA and LSHFA as being mutually coupled, with the
nature of the coupling being different in different times and places.
Depending on the signs of the feedback terms, this means that the two
systems work together to reinforce or to oppose each other, but they
always work together. One of the systems never simply forces the other
in a causal sense.

The noise correlation σ~12~ may be interpreted as zero lag correlation
between SSTA and LSHFA due to unobserved physical mechanisms. For the
blue region, the noise correlation, σ~12~ (Figure 6f), is positive. With
our sign convention, this means that the random component of the forcing
of SSTA tends to be positive (negative) when the LSHFA are upward
(downward). Thus, the most important coupling between SSTA and LSHFA is
contemporaneous. φ~21~ is close to zero (Figure 6c), so the LSHFA is
only weakly affected by the previous month's SSTA. The significant, but
small, negative values for φ~12~ (Figure 6b) in this region indicate
that the LSHFA provides a negative feedback to the SSTA.

The contemporaneous coupling in the red region has the opposite sign,
with the random component of the forcing of SSTA tending to be positive
(negative) when the LSHFA are downward (upward). Furthermore, Φ tends to
be antisymmetric in the red region. This means that the coupling is
essential in the red region and it accounts for the antisymmetric shape
of the CCF in this region with extreme values at lags of 1 and -1. The
previous month\'s SSTA and LSHFA both provide feedback to influence the
results of both fields the following month. This conclusion differs from
those found in previous studies where the atmosphere is found to drive
the ocean (Delworth, 1996; Battisti *et al.*, 1995).

A comparison with the results of Frankignoul (1995, 1985) are in order.
A rigorous transformation between the development of Frankignoul (1995),
which uses a continuous model, and our development of the discrete model
is not transparent. Nonetheless, the spirit of the two models is very
much the same and the basic mechanisms of white noise forcing,
persistence and feedbacks are identical. Basically, the Frankignoul
(1995) approach is equivalent to our approach with partial feedback
(φ~21~ = 0). This produces SSTA spectra that are similar to those
observed and is able to produce CCF that have an "antisymmetric aspect"
(Frankignoul, 1985). This forms the basis of the success of the
Frankignoul approach, but colored noise and explicit inclusion of the
effect of using monthly-averaged data are required to move the extrema
in the CCF away from zero. In our model, ocean feedback to the
atmospheric state is explicitly included and shown to be nontrivial.
This added mechanism allows the possibility of a CCF with extrema at
lags of both ±1 with the value at zero lag small or zero, in accordance
with the observations. The model (with fitted parameters) produces
faithful estimates of the CCF, cross spectra and phase spectra.
Frankignoul (1995) did not explicitly consider the comparison of their
model cross and phase spectra with the observations, but we find that we
cannot get a good match with observations if we neglect the ocean to
atmosphere feedback (φ~21~ = 0).

The primary importance of these results is probably the guidance they
provide for the development of coupled general circulation models. Such
models must be able to reproduce the observed cross correlation
structure between LSHFA and SSTA before conclusions based on their
output can be made with confidence. These models, once verified, will
provide the best tool for exploring the potential connections between
the propagating low-frequency structures in SSTA and variability of the
earth's climate.

# V. Conclusions

Based on an analysis of COADS summaries from 1947-1990, we show that the
form of the cross-correlation between SSTA and LSHFA provides a clear
distinction between an open-ocean midlatitude region (red) and another
which dominates near the western boundary current and in the tropics
(blue). The red region is characterized by CCF which are strongly
antisymmetric and by phase spectra that are about 75° over a broad
frequency range. The blue region, on the other hand, has CCF which peak
at zero lag, are one sided, and have phase spectra that are about 25°
over a broad frequency range.

A discrete first-order vector auto-regressive model is used to interpret
and unify our findings . Correlations in such a model potentially arise
from two sources: (1) Noise that is correlated at zero lag only and
represents unobserved physical mechanisms and (2) Feedbacks between the
ocean and atmosphere. Fits of the observed time series show that the red
and blue regions have several distinctions: (1) The sign of the noise
correlation is positive in the red region and negative in the blue
region. (2) The persistence of SSTA is smaller in the blue region. (3)
The feedback of ocean to atmosphere is small in the blue region. (4) The
persistence of LSHFA is very small in the red region. The CCF in the red
region requires that the noise be correlated, the ocean have significant
persistence and the feedback between ocean and atmosphere be significant
in both directions. This is different than the model of Frankignoul and
Reynolds (1983) which included only feedback from atmosphere to ocean.
In the blue region, the CCF can be well-modeled with only correlated
noise and ocean persistence.

The model fits are performed on each location separately and the
residuals from each location do not show any remaining temporal
structure. Spatial-temporal correlations, however, are evident in the
residuals which propagate on decadal time scales. It appears that these
features cannot be modeled by local linear stochastic models. Some
method of incorporating spatial correlations must be developed to
account for these features. It therefore appears that it is premature to
conclude that "\...the development of SST anomalies can be
understood\..., without invoking feedback mechanisms, as the integral
response of the ocean to random short-time-scale forcing by the
atmosphere\" as suggested by Frankignoul and Hasselmann (1977) for two
reasons: (1) Full feedback is required to model the observations with a
linear, local stochastic model and (2) The residuals that remain after
fitting such a model to the observations contain important low-frequency
propagating signals that the model cannot account for.

# VI. Acknowledgments

We would like to thank Dr. M. O. Baringer for helpful discussions. We
would like to also acknowledge a number of useful suggestions from
Professors R. E. Davis and A. B. Baggeroer. This work was partially
supported by the NOAA/OGP/ACCP.

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# VIII. Figure Captions

Figure 1. Locations and structure of cross correlation functions between
observed values of SSTA and LSHFA in the red and blue regions (see
text). Locations are chosen from lowest (red) and highest (blue) deciles
of two distributions formed by: (1) Sorting the zero-lag value of the
cross correlation function (a-c), (2) Sorting the frequency with which
SSTA and LSHFA have the same sign (d-f). Red and blue in panel a (d)
show locations from which cross correlations shown in panels b and c (e
and f) were chosen. Black represents land, white depicts locations that
are considered data-poor and gray depicts the locations used in this
study that are not in the lowest or highest decile. Solid lines in
(b,c,e,f) are the medians of the individual cross correlation functions
shown in light gray. Solid lines are color coded to conform to locations
shown in (a,d). Positive/negative lags mean that SSTA leads/lags LSHFA.

Figure 2. Magnitude (a) and phase angle (b) of cross spectra between
SSTA and LSHFA observations for the red and blue regions as determined
in Figure 1a-c. Units of phase angle are degrees and negative angles
mean that SSTA leads LSHFA.

Figure 3. Distribution functions of fits for elements of Φ and **Σ**.
Dark line is the value and light lines show one standard error derived
from the fitting process.

Figure 4. Locations of bins with sign of the innovation (noise) cross
correlation (σ~12~) greater than (blue) or less than (red) zero.

Figure 5. (a) Locations, (b,c) auto spectra, (d,e) cross correlation
functions, (f) cross spectra and (g) phase spectra for observed values
of SSTA and LSHFA in the red and blue regions (see text). Locations are
chosen from lowest (red) and highest (blue) deciles of the distribution
of σ~12~. In (a), black represents land, white depicts locations that
are considered data-poor and gray depicts the locations used in this
study that are not in the lowest or highest decile. Solid lines in
(b,c,d,e,f,g) are the medians of the various individual functions.
Individual cross correlation functions (d,e) are shown in light gray.
Solid lines are color coded to conform to locations shown in (a).
Positive/negative lags mean that SSTA leads/lags LSHFA.

Figure 6. Distribution functions of fits for elements of Φ and
**Σ** based on the least and greatest deciles of the distribution of
σ~12~. Red/blue lines show values of the parameter and are color coded
to the red/blue regions. Light gray/dark gray denote one standard error
for values of Φ shown by red/blue curves.

Figure 7. Statistics derived from 200 realizations of the model using
median parameters from fits shown in Figure 6. All panels depict median
values derived from the 200 realizations. (a,b) Auto spectra of SSTA and
LSHFA for red and blue regions, (c,d) Cross correlation function of SSTA
and LSHFA, (e,f) Cross and phase spectra of SSTA and LSHFA. Figures are
color coded to denote appropriate regions.

Figure 8. (a) Latitude-time plots showing residuals remaining after the
fit to observations. Top: SSTA residuals, Middle: LSHFA residuals,
Bottom: white noise. Monthly SSTA and LSHFA were used to form annual
means and then averaged longitudinally over 2° latitude bands. Each of
the three data arrays were smoothed by a Hanning-weighted 3x3 window.
Values are coded in 10 gray scales (linear from -0.6 to 0.6), with
white/black being positive/negative, respectively. (b) Power spectrum of
median of SSTA residuals. (c) Power spectrum of median of LSHFA
residuals.