Ocean Dynamics (2010) 60:155–170DOI 10.1007/s10236-009-0255-9

On non-Gaussian SST variability in the Gulf Streamand other strong currents

Philip Sura

Received: 8 May 2009 / Accepted: 26 November 2009 / Published online: 22 December 2009© Springer-Verlag 2009

Abstract This paper examines the physics of observednon-Gaussian sea surface temperature (SST) anomalyvariability in the Gulf Stream system in a recentlydeveloped stochastic framework. It is first shown froma new high-resolution observational data set that theGulf Stream system is very clearly visible as a band ofnegative skewness all the way from Florida, over CapeHatteras, to the central North Atlantic. To get an ideaabout the detailed non-Gaussian variability along theGulf Stream, probability density functions are calcu-lated at several locations. One important observationalresult of this study is that the non-Gaussian tails ofSST variability in the Gulf Stream system follow apower-law distribution. The study then shows that theobserved non-Gaussianity is consistent with stochas-tic advection of SST anomalies in an idealized zonalcurrent. In addition, stochastic advection is compatiblewith the observed northward eddy heat flux in the GulfStream, providing a new dynamical view at the heatbalance in strong currents.

Keywords Non-Gaussian sea surface temperaturevariability · Gulf Stream · Power-law ·Stochastic advection

Responsible Editor: Pierre Lermusiaux

P. Sura (B)Department of Meteorology, Florida State University,1017 Academic Way, Tallahassee, FL 32306-4520, USAe-mail: psura@fsu.edu

1 Introduction

Since the very early days of physical oceanography,the Gulf Stream system plays a central role in thedynamical description of the general circulation of theocean. One reason for that is, of course, that the NorthAtlantic has been the major shipping route connectingthe Old with the New World for about 500 years now.The Gulf Stream is a warm western boundary currentthat transports large amounts of heat northward, andconsequently is a major part of the global climate sys-tem. Therefore, it is important to study and understandthe physics behind its temperature fluctuations. Here,we will study the physics of non-Gaussian sea surfacetemperature (SST) variability in the Gulf Stream andother strong currents in a recently developed stochasticframework.

The study of non-Gaussian variability and dynamicsin the ocean and atmosphere became more popularonly recently. That is because we now have the requiredhigh-resolution data and models available to meaning-fully explore non-Gaussian statistics from observationsand models. For a review of non-Gaussian statistics inthe atmosphere, see, for example, Stephenson et al.(2004) or Sura et al. (2005) and references therein.Here, we focus on oceanic variability, keeping in mindthat non-Gaussianity in the atmospheric forcing fields(e.g., Monahan 2006a, b; Gille 2005) may account forsome fraction of the oceanic non-Gaussianity.

The research of non-Gaussian oceanic variabilityincludes many relevant subtopics. It is obvious that ex-treme value (non-Gaussian) statistics of extreme waveheights and coastal sea levels are of utmost importance(e.g., Jha and Winterstein 2000; Coles 2001). More rel-evant to the present study are recent efforts to quantify

156 Ocean Dynamics (2010) 60:155–170

forecast errors in coastal and shelf sea prediction mod-els (e.g., Lermusiaux 2006; Lermusiaux et al. 2006).By performing ensemble simulations of the models,the non-Gaussian distribution of forecast errors canbe determined. Assuming ergodicity, the non-Gaussiandistribution of the ensemble members equals the localtemporal non-Gaussianity. As it turns out, the spatialpatterns of skewness and kurtosis of SST and sea sur-face height forecast errors resemble the spatial fieldsof non-Gaussian SST variability (e.g. Lermusiaux et al.2002; Auclair et al. 2003). In particular, Lermusiauxet al. (2002) showed that the skewness of sound speed(which is proportional to SST) variability changes signat a temperature front for a shelfbreak applicationoff Cape Cod, and Auclair et al. (2003) showed thatthe skewness of sea surface height variability in theGulf of Lions is negative (positive) at the northern(southern) edge of a strong alongshore current. Theseregional applications are very useful to elucidate theeffects of very complex and specific local conditions onnon-Gaussian ocean variability. Here, we are taking adifferent approach, neglecting local conditions as muchas possible to provide a more general view of non-Gaussian SST variability.

The earliest work on observed non-Gaussian SSTvariability using Ocean Weather Station data datesback two to three decades (Blaauboer et al. 1982;Müller 1987). More recently, Burgers and Stephenson(1999) analyzed the non-Gaussianity of observed ElNiño SSTs. While these studies are important, theydid not provide a global view of non-Gaussian SSTvariability. In a recent paper, Sura and Sardeshmukh(2008) closed this gap by providing a global stochastic–dynamical view of non-Gaussian SST variability. There,the starting point is the observation that daily SSTanomalies are highly non-Gaussian all over the globefor observed skewness and kurtosis, respectively. Mostimportantly, the skewness and kurtosis of SST anom-alies are found to be strongly linked at most loca-tions around the globe. The link, or constraint, is thatthe kurtosis (kurt) is everywhere equal to or larger thanone-and-a-half times the squared skewness (skew):kurt ≥ (3/2)skew2. This constraint is then analyticallyexplained with a simple multiplicative noise model,which can be directly derived from basic mixed-layerdynamics. That means that a simple linear stochas-tic differential equation (SDE) with multiplicativenoise, derived from first principles, captures the over-all dynamics of global SST variability remarkablywell. In particular, this shows that the observed non-Gaussianity of SST anomalies is due to multiplicativenoise rather than to nonlinearities in the deterministicpart of the SST equation.

The theory presented in Sura and Sardeshmukh(2008) does not predict the parameters of the stochas-tic model. In fact, all specific parameters drop out ofthe general constraint. That is, Sura and Sardeshmukh(2008) provide observational evidence that the dynam-ics of non-Gaussian SST variability is governed by alinear SDE with linear multiplicative noise. They donot provide specific local parameters based on specificphysical processes. These parameters, however, deter-mine the sign and magnitude of the observed skewnessand kurtosis at a given location beyond the generalconstraint. Therefore, the approach taken by Sura andSardeshmukh (2008) applies the ideas of statisticalmechanics to SST variability, neglecting detailed localdynamics as much as possible.

This study looks into the more detailed SST dynam-ics within a specific flow regime. It has been notedbefore that strong currents such as the Gulf Streamshow a very pronounced negative SST anomaly skew-ness. Thus, this study takes a closer look at non-Gaussian SST variability in the Gulf Stream systemwithin the stochastic framework provided by Sura andSardeshmukh (2008). One new observational result isthat the non-Gaussian tails of SST variability in theGulf Stream system follow a power-law distribution,consistent with multiplicative noise dynamics of, so far,unspecified physical origin. This study then shows thatthe negative SST skewness in the Gulf Stream mayoriginate from stochastic advection of SST anomalies.Stochastic advection is also shown to be consistentwith the observed northward eddy heat flux in theGulf Stream, providing a novel dynamical look at theheat balance in strong currents. While this approachmight look like a purely academic endeavor, there areimportant applications closely related to this study. Oneis connected to the recent interest in quantifying un-certainties in ocean predictions (e.g., Lermusiaux 2006;Lermusiaux et al. 2006). To parameterize the effect ofsub-grid scale variability in ocean models, stochasticparameterizations are becoming more popular. As wewill see, this study gives some guidance on the physicsof stochastic model components by showing that ob-served non-Gaussian SST variability is consistent withstochastic advection of SSTs in an idealized zonal cur-rent. In addition, the knowledge of the detailed non-Gaussian statistics of an oceanic variable such as SSTallows to better quantify sampling uncertainties anderrors in observed or modeled data.

In Section 2, the observational results are presented.In particular, it is shown that the negatively skewednon-Gaussianity in the Gulf Stream system is consistentwith a power-law. Then, Section 3 provides a theoret-ical explanation of the negatively skewed power-law

Ocean Dynamics (2010) 60:155–170 157

behavior of SST anomalies in strong currents basedon stochastic advection. Finally, Section 4 provides asummary and discussion.

2 Observations

Probability density functions (PDFs) are useful diag-nostic measures of the dynamics of stochastic systems.In particular, higher moments beyond the variance(that is, deviations from Gaussianity) can shed lighton the underlying dynamics. For example, Sura andSardeshmukh (2008) provided a global view of non-Gaussian SST variability by presenting global maps ofskewness and kurtosis of SST anomalies to characterizethe overall shape of the PDF. They then used an ob-served skewness–kurtosis link to elucidate the underly-ing dynamics of global non-Gaussian SST variability.

2.1 Data

In this study, we use the same data as in Sura andSardeshmukh (2008) to study the non-Gaussian SSTvariability in the Gulf Stream system. To briefly reca-pitulate, this high-resolution observational data setcompiled and provided by Reynolds et al. (2007) con-sists of a blended analysis of daily SST fields basedon infrared satellite data from the Advanced VeryHigh Resolution Radiometer (AVHRR) and in situdata from ships and buoys. The analysis was performedusing optimum interpolation with a separate step tocorrect satellite biases relative to the in situ data. Thein situ data were obtained from the International Com-prehensive Ocean—Atmosphere Data Set (ICOADS;http://icoads.noaa.gov/). This daily SST analysis is avail-able on a 0.25-degree latitude/longitude grid fromJanuary 1985 to the present. A more detailed de-scription of the data set and analysis procedure canbe found in Reynolds et al. (2007). SST anomalieswere calculated by subtracting the daily climatologyand linear trend from the full daily values. Then, theextended summer (May–October) and extended winter(November–April) seasons are analyzed.

It should be noted that Reynolds et al. (2007) alsoprovided a more advanced data set combining AVHRRSSTs with the Advanced Microwave Scanning Ra-diometer (AMSR) data. The advantage of AMSR SSTsis that they have a near all-weather coverage; in con-trast, AVHRR SSTs are only available in cloud-freeconditions. That is, the combined AVHRR–AMSRdata set is less biased by missing data. So why didwe chose the AVHRR-based data set rather thanthe AVHRR–AMSR product? The answer is that to

reliably estimate higher moments such as skewnessand kurtosis (discussed below) requires timeseries aslong as possible. AMSR data, however, only becameavailable in June 2002. That is, the main rationale forusing the AVHRR SSTs is the length of the record: al-most 18 years more than the AVHRR–AMSR product.We are, however, aware of the potential biases of theAVHRR SSTs and discuss them accordingly in the nextsubsection, and also in the summary and discussion.

2.2 Non-Gaussian SST variability in the GulfStream system

First skewness (third moment) and kurtosis (fourthmoment) are used to characterize the overall shape ofa PDF. If the standard deviation of an anomaly x withzero mean (x = 0) is denoted by σ , the skewness (skew)and kurtosis (kurt) become

skew ≡ x3

σ 3 , kurt ≡ x4

σ 4 − 3 , (1)

where the overbar denotes a time or, assuming ergodic-ity, ensemble average. In the following discussion, wecan employ the notion of a time average in most ofthe cases. However, if we want to highlight the useof an ensemble average, we will use angle brackets〈...〉 (for example, in Appendix A, where we discusssome elements of SDEs). Skewness is a measure ofasymmetry of a PDF. For unimodal PDFs, the followinginterpretations are valid. If the left tail is heavier (morepronounced) than the right tail, the PDF has negativeskewness. If the reverse is true, it has positive skewness.If the PDF is symmetric, it has zero skewness. Kurtosis(or more accurately, “excess kurtosis,” since we sub-tract the kurtosis of 3 for a Gaussian distribution) mea-sures the excess probability (fatness) in the tails, whereexcess is defined in relation to a Gaussian distribution.In general, moments such as skewness and kurtosis aremeasures to obtain a first-order characterization of theoverall shape of a PDF. In principle, the knowledgeof all moments is almost equivalent to that of the fullPDF (see, e.g., Sornette 2006). That is, skewness andkurtosis are shown first, before moving on to presentthe detailed shape of the PDFs.

The skewness of SST anomalies in the NorthAtlantic in the extended summer (upper panel) andwinter (lower panel) is shown in Fig. 1. The related kur-tosis is shown in Fig. 2. One of the most striking featuresin the skewness fields is that the Gulf Stream (and itsextension, the North Atlantic Current) is very clearlyvisible as a band of negative skewness all the way

158 Ocean Dynamics (2010) 60:155–170

Fig. 1 Skewness of SSTanomalies for (top) extendedsummer and (bottom) winter

from Florida, over Cape Hatteras, to the central NorthAtlantic. It is also noteworthy that there is very littleseasonal dependence in the non-Gaussianity of theGulf Stream system. That is, full-year Gulf StreamPDFs will be studied subsequently to reduce the un-certainty of the power-law statistics discussed below.Other features such as the Loop Current in the Gulfof Mexico and the East Greenland Current are alsoclearly visible by negative skewness in at least oneof the extended seasons. However, in this paper, thefocus is on the year-round non-Gaussianity of the GulfStream system.

Is it possible that the non-Gaussianity in the GulfStream system is due to biases of the AVHRR SSTs?In particular, could the non-Gaussianity in the GulfStream system be biased due to cloud cover? Whilewe cannot completely rule out the possibility that thetails of the PDFs are somehow influenced by cloudcover, it seems very unlikely that a well defined andspatially coherent structure like the SST skewness ofthe Gulf Stream system (Fig. 1) is induced by a po-tential cloud cover bias. That is, there is little physicalreason to believe that the coherent skewness patternstrictly following the Gulf Stream can be due to a more

Ocean Dynamics (2010) 60:155–170 159

Fig. 2 Kurtosis of SSTanomalies for (top) extendedsummer and (bottom) winter

large-scale atmospheric cloudiness pattern. In fact,even on a global scale, the observed AVHRR SSTskewness and kurtosis patterns (Sura and Sardeshmukh2008) are congruent with known physical properties ofthe upper ocean such as currents and fronts. Therefore,we are confident that the moments and PDFs of theAVHRR SSTs are indeed reliable and not due to cloudbiases in the data set.

To get an idea about the detailed shape of the PDF,we picked several locations along the Gulf Stream andNorth Atlantic Current and calculated the full PDFsthere. As expected from the minor seasonal depen-

dence in SST skewness and kurtosis of the Gulf Streamsystem, the PDFs of extended winter and summerseasons have the same non-Gaussian structure. Thus,for the sake of clarity and considerable statistical sig-nificance, only the PDFs of full-year SST anomaliesare presented. First, a parametric method is used tocalculate the PDFs of observed SST anomalies, inwhich the parameters of a “skew-t” distribution aredetermined by a maximum likelihood estimate. Theskew-t distribution, a skewed and kurtotic alternativeto the normal distribution, is used because it is capableof adapting very closely to skewed and heavy-tailed

160 Ocean Dynamics (2010) 60:155–170

data (Azzalini and Capitanio 2003; Jones and Faddy2003; Azzalini 2005; R Development Core Team 2004).Two representative PDFs are presented in Fig. 3. Forthe examples, two locations with strong skewness werechosen. In Fig. 3a, the PDF of full-year SST anomaliesoff Cape Hatteras (35◦ N, 75◦ W) is shown, whereas,in Fig. 3b, the PDF of full-year SST anomalies at a

a) SST − Cape Hatteras

−4 −2 2 4

1e−0

41e

−03

1e−0

21e

−01

1e+

00

T'

PD

F

b) SST − North Atlantic Current

−4 −2 2 4

1e−0

41e

−03

1e−0

21e

−01

1e+

00

T'

PD

F

0

0

Fig. 3 a PDF of full-year SST anomalies off Cape Hatteras (35◦N, 75◦ W). b PDF of full year SST anomalies at a location in theNorth Atlantic Current (40◦ N, 50◦ W). In both plots, the skew-t distribution is denoted by the thick solid black line, a simplehistogram by the thin red line. The corresponding Gaussian dis-tribution is included as a dashed line. Note the logarithmic scaleon the ordinates

location in the North Atlantic Current (40◦ N, 50◦ W)are presented. The skew-t distribution is denoted bythe thick solid black line, a simple histogram by thethin red line. The corresponding Gaussian distributionis also included in both figures as a dashed line. Notethe logarithmic scale on the ordinates.

To examine the non-Gaussian structure of SST vari-ability in the Gulf Stream system in even more detail,the raw PDFs (histograms) are also presented on a log–log scale to highlight the scaling properties of the data(Figs. 4 and 5). For example, a highly non-Gaussianpower-law tail (that is, a PDF P(x) ∝ x−α with theexponent α) will appear as a straight line on a log–log plot. As shown below, PDFs of anomalous SSTvariability in the Gulf Stream system indeed obey apower-law, as tested by a Monte-Carlo approach usingKolmogorov–Smirnov statistics. Note that the power-law characteristics are estimated from the histogram(and not the skew-t fit) to avoid nested fits (fitting a

a)

-5

-4

-3

-2

-1

0

1 10

log[

P(|T

’|)]

|T’|

Positive SST Anomalies - Cape Hatteras

power-law exponent: 6.9

b)

-5

-4

-3

-2

-1

0

1 10

log[

P(|T

’|)]

|T’|

Negative SST Anomalies - Cape Hatteras

power-law exponent: 5.5

Fig. 4 Log-log scale PDFs (circles) of full-year a positive andb negative SST anomalies off Cape Hatteras (35◦ N, 75◦ W).The straight dashed lines are maximum likelihood estimates ofthe power-law behavior and the solid lines denote Gaussiandistributions. The lower bounds of the power-law scaling |T ′|minare shown by the dotted-dashed vertical lines

Ocean Dynamics (2010) 60:155–170 161

a)

-5

-4

-3

-2

-1

0

1 10

log[

P(|T

’|)]

|T’|

Positive SST Anomalies - North Atlantic Current

power-law exponent: 6.0

b)

-5

-4

-3

-2

-1

0

1 10

log[

P(|T

’|)]

|T’|

Negative SST Anomalies - North Atlantic Current

power-law exponent: 4.1

Fig. 5 Log–log scale PDFs (circles) of full-year a positive and bnegative SST anomalies in the North Atlantic Current (40◦ N, 50◦W). The straight dashed lines are maximum likelihood estimatesof the power-law behavior and the solid lines denote Gaussiandistributions. The lower bounds of the power-law scaling |T ′|minare shown by the dotted-dashed vertical lines

power-law to a skew-t fit). Because of the log scale onthe x-axis, absolute values |T ′| are shown. In addition,because the emphasis is on extreme events, the centerof the PDFs (± one standard deviation) are not shown.The PDFs of positive/negative SST anomalies at CapeHatteras are shown in Fig. 4a and b, and the PDFs ofpositive/negative SST anomalies in the North AtlanticCurrent are shown in Fig. 5a and b. In all plots, the solidline denotes a Gaussian distribution. Note the distinctheavy-tail power-law behavior for negative anomaliesgiving rise to negative skewness. However, even thepositive anomalies follow a power-law (weaker thanGaussian, though). The straight dashed lines are max-imum likelihood estimates of the power-law behavior(given by the power-law exponent α also includedin each plot) above a systematically estimated lowerbound |T ′|min. The lower bounds |T ′|min are shownby the dotted–dashed vertical lines. The procedure toestimate the power-law parameters and the related

goodness-of-fit test are described in the following para-graph. The numerical values of α, |T ′|min, and the re-sults of a goodness-of-fit test (p values) are summarizedin Table 1. In fact, in all cases, the power-law hypothesisis statistically significant (cannot be rejected) as testedby Kolmogorov–Smirnov statistics.

As shown in many publications (e.g., Sornette 2006;Newman 2005; Clauset et al. 2009), the best-fit (thatis, maximum likelihood) power-law exponent α for ageneral positive timeseries xi of length n is

α = 1 + n

[n∑

i=1

ln(

xi

xmin

)]−1

, (2)

where xmin is the lower bound of the power-law be-havior. Note that the general description in this sectionrequires xi and xmin to be positive and, therefore, notdirectly applicable to T ′. Hence, Eq. 2 has to be appliedseparately to the absolute values of SST anomalies |T ′|for positive and negative perturbations. The approxi-mate standard error ε of α can be derived from thewidth of the likelihood maximum as

ε = α − 1√Nin

, (3)

where Nin is the effective number of independent ob-servations. That is, the interval α ± 1.96ε contains thetrue power-law exponent with approximately 95% cer-tainty. In our case, Nin is estimated by dividing the totalnumber of observations n by the observed decorrela-tion timescale of SST anomalies. xmin is chosen to makethe PDF of the observed data and the best-fit power-law model as similar as possible above xmin (Clausetet al. 2009). As a measure for quantifying the distancebetween two PDFs, the Kolmogorov–Smirnov statisticD (e.g., Wilks 2006; Press et al. 1992) is used (there aremore sophisticated test statistics available, but as thenon-Gaussianity is rather strong in our example, a basictest is appropriate here). D is simply the maximum

Table 1 Best-fit power-law and goodness-of-fit test parameters:power-law exponent α, lower power-law bound |T ′|min, and pvalue of Kolmogorov–Smirnov statistics

Location α Tmin p

Cape Hatteras (35◦ N, 75◦ W)Positive anomalies 6.9 ± 0.4 1.9 0.13Negative anomalies 5.5 ± 0.3 2.7 0.87

North Atlantic Current (40◦ N, 50◦ W)Positive anomalies 6.0 ± 0.3 1.7 0.12Negative anomalies 4.1 ± 0.2 1.5 0.73

162 Ocean Dynamics (2010) 60:155–170

distance between the cumulative density function(CDF) of the data and the best-fit model:

D = maxx≥xmin

|S(x) − P(x)| , (4)

where S(x) is the CDF of the data above xmin, and P(x)

is the CDF of the best-fit power-law model above xmin.The best estimate of xmin is the value that minimizes D.It is also straightforward to use Kolmogorov–Smirnovstatistics to test if the observed data are consistent witha power-law distribution, that is, to perform a goodness-of-fit test. Here, this is done using a Monte-Carlo ap-proach. First, the power-law exponent α and the cutoffxmin is calculated from the data. The goodness-of-fitto the data is quantified by the Kolmogorov–Smirnovdistance D. Then, using the observed parameters, alarge number (here 10,000) of synthetic power-lawtimeseries of length Nin are generated and the power-law exponent α, lower-bound xmin, and Kolmogorov–Smirnov distance D (now from its know CDF) areestimated for each of them employing the tech-niques described above. Finally, the fraction of theKolmogorov–Smirnov statistics for the synthetic datasets whose value exceed the statistics of the real datayields the so called p value. Only if the p value issufficiently small can a power-law distribution be ruledout. Note that we use Kolmogorov–Smirnov statisticsand the related p value to reject (rather than confirm) ahypothesis for the observed data; hence, high p values,not low, are considered good. Here, a power-law isruled out if p ≤ 0.1. That is, if p ≤ 0.1, there is a 10% orless probability to draw a power-law sample merely bychance that agrees that poorly with power-law statistics,and the hypothesis that the data are drawn from apower-law is rejected. See Newman (2005) and Clausetet al. (2009) for more details on how to estimate power-law distributions, including goodness-of-fit tests, fromempirical data.

To summarize, the non-Gaussian structure with aheavy negative power-law tail, responsible for the neg-ative skewness, can be clearly seen at both locations.Even the positive SST anomalies show a weak power-law behavior. The PDFs at other locations along theGulf Stream and North Atlantic Current have the samestructure as both the shown PDFs. This non-Gaussianity is, therefore, representative for the non-Gaussian SST variability in the Gulf Stream systemand, perhaps, other strong currents. In particular, theobserved skewness along the Gulf Stream and theNorth Atlantic Current is indicative of a specific phys-ical process confined to the strong flow. In the nextsection, we will present a theory that explains the pro-

nounced negative skewness and the power-law behav-ior in strong currents.

3 Theory

In the previous section, we have seen that the GulfStream system shows a very pronounced negative skew-ness of SST anomalies. In this section, we present atheory that explains this interesting feature of SSTvariability in strong currents. To develop the theoryof non-Gaussian SST variability in strong currents, wehave to first briefly rehash the closely related studyof global non-Gaussian SST variability (see Sura andSardeshmukh 2008, for details).

3.1 A univariate theory of global non-GaussianSST variability

The starting point for the global view of non-GaussianSST variability is the heat budget equation for To,defined as an average temperature over the mixed-layerdepth h (e.g., Frankignoul 1985):

∂To

∂t=−vo · ∇To+ Q

qCh− we

h

(To − Tb

o

) + κ∇2To ≡ F ,

(5)

where the rate of change of To is governed by advectionthrough ocean currents, surface heat fluxes, verticalentrainment, and horizontal mixing. Here, vo is thehorizontal velocity in the mixed-layer, Q is the heat fluxthrough the sea surface, q and C are the density andheat capacity of sea water, we and Tb

o are the verticalvelocity and temperature just below the mixed-layer,and κ is the horizontal mixing coefficient. F denotesthe sum of all terms. Note that all of these terms aredirectly or indirectly affected by atmospheric quantitieslike winds and temperatures. For small temperatureanomalies, T ′

o, a Taylor expansion of the heat flux Fwith respect to To = To + T ′

o yields

∂T ′o

∂t= ∂ F

∂ToT ′

o + F ′ + R′ , (6)

where it is assumed that the evolution of the meantemperature To is balanced by the mean heat flux F(using F = F + F ′), and that the derivative is evaluatedat To. We also added a residual term R′ to representhigher-order terms of the Taylor expansion and otherprocesses not included in the mixed-layer Eq. 5. Be-cause the heat flux anomaly F ′ is, to a large degree,due to rapid transient atmospheric fluctuations, it maybe represented as temporal noise ηF ′ . For the sake of

Ocean Dynamics (2010) 60:155–170 163

simplicity (and the lack of detailed knowledge), theresidual R′ is dealt with in the same way and repre-sented as another (uncorrelated to ηF ′) noise term ηR′ .The derivative ∂ F/∂To, on the other hand, is usuallyrepresented by a constant parameter −λ. That is, theeffect of atmospheric forcing on SST anomalies T ′

o isoften represented by a simple stochastic driving of theoceanic mixed layer,

∂T ′o

∂t= −λT ′

o + η , (7)

where λ is a rate coefficient representing the damp-ing of the slowly evolving mixed-layer heat anomaly,and η = ηF ′ + ηR′ is Gaussian white-noise represent-ing the heat flux due to rapidly varying weather fluc-tuations and other residual processes. Equation 7 is,of course, the classic stochastic model of SST vari-ability (Frankignoul and Hasselmann 1977) in partic-ular, and low-frequency climate variability in general(Hasselmann 1976).

The stochastic view in Eq. 7 implies that T ′o has a

Gaussian PDF. Indeed, temporally (monthly, season-ally, or even yearly) or spatially (e.g., several degrees)averaged SST anomalies are nearly Gaussian. We ex-pect this partly from the Central Limit Theorem (e.g.,Gardiner 2004; Paul and Baschnagel 1999) to the extentthat it is applicable to time-averaged quantities. How-ever, we have already seen in Figs. 1–5 that, on dailyscales, observed SSTs are significantly non-Gaussian. Inparticular, we showed in Sura and Sardeshmukh (2008)that there is a striking parabolic relationship betweenskewness and kurtosis, kurt ≥ (3/2)skew2. Sura andSardeshmukh (2008) used a multiplicative noise modelto explain this observed constraint. The derivation issketched in the following.

We already noticed (Eq. 6) that a Taylor expansionof the heat flux with respect to To = To + T ′

o yields

∂T ′o

∂t= ∂ F

∂ToT ′

o + ∂ F ′

∂ToT ′

o + F ′ + R′ , (8)

where, in contrast to Eq. 6, we just replaced the fullheat flux F with its mean and perturbation, F = F + F ′.As before in Eq. 6, the derivatives are evaluated atTo. This, at first glance, trivial replacement is done tohighlight the fact that the derivative ∂ F/∂To actuallyconsists of two terms: the constant term ∂ F/∂To and therapidly varying term ∂ F ′/∂To. As a relevant exampleclosely related to Frankignoul and Hasselmann (1977),consider the form of a common bulk heat flux para-meterization F ≡ FQ = β(Ta − To)|U|, which dependson the SST To, air temperature Ta, wind speed |U|,

and a constant β. We set F Q = β(Ta − To)|U| andF ′

Q = β(Ta − To)|U|′, allowing for rapidly varying windspeed fluctuations |U|′. Note that the mean SST To

is still allowed to vary very slowly. We then obtain∂ F Q/∂To = −β|U| and ∂ F ′

Q/∂To = −β|U|′, or, equiv-alently, ∂ FQ/∂To = −β|U| = −β(|U| + |U|′). It is theconstant term that justifies the introduction of the con-stant feedback parameter −λ in the general formula-tion Eq. 7. However, the rapidly varying term ∂ F ′/∂To

cannot be neglected as is done in many studies (e.g., inFrankignoul and Hasselmann 1977). Why has this termbeen neglected in previous studies?

The main reason is that, in older, coarse (in spaceand time) resolution data sets, SST variability is closeto being Gaussian, mainly because of the Central LimitTheorem. Then, the classic stochastic model of SSTvariability is sufficient to describe the Gaussian statis-tics of the data. Only recently, the new high-resolutiondata sets became available, allowing for a more detailedanalysis of non-Gaussian statistics [the exception beinginvestigators using Ocean Weather Station data, suchas Müller 1987; Blaauboer et al. 1982]. That meanswe are using the same very general assumption (slow–fast timescale separation) as in the classic papers, butinclude non-Gaussian effects through the inclusion ofthe so far neglected term ∂ F ′/∂To. In fact, the suc-cess of Sura and Sardeshmukh (2008) suggests thatthe timescale separation assumption is indeed valid fornon-Gaussian high-resolution SST dynamics, the sameway it is for Gaussian coarse-resolution SSTs. So howto model the rapidly varying derivative ∂ F ′/∂To?

The easiest way to model this term is to linear-ize it. That is, we assume that the derivative ∂ F ′/∂To

is proportional to the rapidly varying forcing F ′ it-self: ∂ F ′/∂To ≈ −φF ′, where φ is a locally constantparameter. Because φ can be positive or negative,the minus sign is arbitrary and just introduced forthe sake of convenience. As shown next, within theframework of small temperature anomalies T ′

o, thisapproximation means that F ′ depends weakly lin-early on To and, therefore, imposes no major con-straint on F ′ giving the arbitrary parameter φ. Theequation ∂ F ′/∂To = −φF ′, of course, means that F ′ =η(t) exp(−φTo), with the time-dependent, arbitraryamplitude white-noise term η(t). For small temper-ature anomalies, the exponential function can bevery well linearized as F ′ = η(t) exp[−φ(To + T ′

o)] ≈η(t) exp(−φTo)(1 − φT ′

o). Plugging this F ′ back into ourstarting Eq. 8, we obtain exactly the same functionalform we started with, namely, correlated additive andlinear multiplicative noise terms. That means that wemade a physically plausible and consistent approxima-tion by using ∂ F ′/∂To = −φF ′.

164 Ocean Dynamics (2010) 60:155–170

To summarize, we approximate the rapidly varyingderivative ∂ F ′/∂To as a constant times F ′, ∂ F ′/∂To =−φF ′, to get the following Stratonovich SDE for SSTanomalies T ′

o:

∂T ′o

∂t= −λeff T ′

o − φF ′T ′o + F ′ + R′ + φF ′T ′

o , (9)

with the locally constant parameters −λeff and −φ andthe rapidly varying (approximately Gaussian white-noise) forcing terms F ′ and R′. That is, we approxi-mated F ′ and R′ as independent, zero mean Gaussianwhite-noise processes with amplitudes σF ′ and σR′ :F ′(t)F ′(t′) = (σF ′)2δ(t − t′) and R′(t)R′(t′) = (σR′)2δ(t −t′). Recall that, in the Stratonovich calculus, we haveto include the noise-induced drift in the effective drift:−λeff = −λ + (1/2)(φσF ′)2. The noise-induced drift ap-pears in Stratonovich systems because then the timemean of the multiplicative noise term, −φF ′T ′

o, is notzero. This is also why we have to introduce the addi-tional mean forcing φF ′T ′

o to guarantee that the timemean of T ′

o is zero, T ′o = 0.

At this point, it is helpful to recall the scaling prop-erties of the noise terms in continuous SDEs such asEq. 9 to avoid confusion about dimensions. This scaling,of course, comes from the fact that SDEs are properlydefined only in the integral sense. A brief summary ofhow to interpret SDEs is given in Appendix A [for adetailed discussion, see, e.g., Gardiner 2004, Kloedenand Platen 1992, Paul and Baschnagel 1999, or anyother textbook on SDEs].

One basic and very important result is that the vari-ance of white-noise (the so called Wiener process W)scales with the time increment dt. Casually, we oftensay the that Wiener process effectively scales with

√dt

in contrast to the dt scaling of the determinist terms,when we actually mean the standard deviation of theWiener process (see below and Appendix A). In anutshell, it is this

√dt scaling that is responsible for so

called stochastic calculus, in contrast to the dt scalingof the ordinary deterministic calculus. For example, ifwe integrate over white-noise F ′ with amplitude σF ′

(that is, F ′ = σF ′η with the normalized, zero mean,and unit standard deviation, white-noise η), we obtain∫

F ′dt = σF ′∫

η dt = σF ′∫

dW, where dW ≡ η dt is theincrement of a Wiener process. However, as shownin Appendix A, dW has the dimensional variance dt:〈dW2〉 = dt (= dW2). That is, the Wiener process itselfscales with

√dt. That, in turn means that σF ′ does not

have the same units as F ′ itself, but has to be scaledby

√dt.

The multiplicative noise system Eq. 9 has one im-portant property of interest here. In general, Eq. 9

will produce non-Gaussian statistics. As shown in Suraand Sardeshmukh (2008), the second, third, and fourthmoments of the SDE Eq. 9 are (with T ′

o ≡ x)

x2 =(σ 2

F ′ + σ 2R′

)[2λeff − (φσF ′)2

] ,

x3 = −2φσ 2F ′ x2[

λeff − (φσF ′)2] ,

x4 =[−3φσ 2

F ′ x3 + (3/2)(σ 2

F ′ + σ 2R′

)x2

][λeff − (3/2)(φσF ′)2

] . (10)

These moment equations result in the general relationbetween skewness and kurtosis, kurt ≥ (3/2)skew2, inalmost perfect agreement with observations (Sura andSardeshmukh 2008). In the context of this study, itis important to note that T ′

o is positively (negatively)skewed if φ is negative (positive). This follows fromthe fact that, for the fourth moment to exist, there isan upper limit for the strength of the multiplicativenoise: (φσF ′)2 < (2/3)λeff . In such conditions, the term[λeff − (φσF ′)2] is always positive, and the sign of theskewness is solely determined by the sign of φ. Putdifferently, the skewness is positive if the additive andmultiplicative noises are positively correlated (that is,have the same sign), and negative if the noises arecorrelated negatively (that is, have opposite signs). Thisbehavior is closely connected to the following otherproperties (derived in Appendix B) we will use later.First, F ′T ′2

o φ ≤ 0. That is, if T ′o is positively skewed

(φ < 0), the third-order moment F ′T ′2o is also positive

(and vice versa). Second, in the pure additive noise caseφ = 0, we have F ′T ′2

o = 0 (see Appendix B). Third, thenoises and T ′

o are positively correlated, F ′T ′o > 0 and

R′T ′o > 0.

Besides the skewness and kurtosis link of the mul-tiplicative noise system, Eq. 9, there is one more im-portant non-Gaussian property of interest here. Asshown in Sardeshmukh and Sura (2009), Eq. 9 willproduce power-law tails. This can be easily seen fromthe Fokker–Planck equation for the stationary PDF pof SST anomalies T ′

o ≡ x governed by Eq. 9:

ddx

(λeff x p) + 12

d2

dx2

× [(σ 2

F ′ + σ 2R′ + φ2 σ 2

F ′ x2 − 2φ σ 2F ′ x

)p] = 0 , (11)

or, after one integration,

λeff x p + 12

ddx

[(σ 2

F ′ + σ 2R′ + φ2 σ 2

F ′ x2 − 2φ σ 2F ′ x

)p]

= const. ≡ 0 . (12)

Ocean Dynamics (2010) 60:155–170 165

Note that, in general, the constant has to be zero tofulfill the boundary conditions of vanishing probabilityat x = ±∞. For large x and small noise amplitudes, thisequation simplifies to

λeff x p + 12

ddx

(φ2 σ 2

F ′ x2 p) = 0 . (13)

Using the power-law ansatz p ∝ |x|−α , one easily findsthat p is a solution if

α = 2[

λeff

φ2 σ 2F ′

+ 1]

. (14)

That is, Eq. 9 will produce power-law tails in agree-ment with observations in the Gulf Stream system(Figs. 4 and 5). Note that there exists a closed solutionp(x) of the full Fokker–Planck Eq. 11 (see, e.g., vanKampen 1981; Müller 1987; Sardeshmukh and Sura2009) which has, of course, the same power-law tails.For the sake of simplicity, we omitted the full solutionhere to merely present the simple derivation of thepower-law exponent.

Equation 14 tells us to expect heavier tails (smallerα) for a weaker damping (or stronger multiplicativenoise forcing φ2 σ 2

F ′) and vice versa. It should be noted,though, that the limit of very small φ2 σ 2

F ′ (resultingin α → ∞) is not allowed within the approximationwe made: φ2 σ 2

F ′ → 0 counteracts the approximation oflarge x. The limit φ2 σ 2

F ′ → 0 simply means that themultiplicative noise is negligible, and the PDF p(x)

becomes Gaussian. It has already been shown that thepower-law exponent α has values between about fourand seven in the Gulf Stream system. Interestingly,negative anomalies exhibit heavier tails (smaller expo-nents α) than positives ones. In this stochastic frame-work, this points to a slightly weaker damping λeff (orstronger multiplicative noise forcing φ2 σ 2

F ′) of negativeanomalies. It is interesting that this is consistent withobservational studies showing a positive local (on scalessmaller than 1,000 km) correlation between SSTs andsea surface winds (Chelton et al. 2004; Small et al. 2005;Xie 2004; O’Neill et al. 2003). Negative SST anomaliesmean weaker mean winds and, therefore, a weakerdamping. The study of this phenomenon in the pres-ence of a strong current is part of our ongoing research.

Equation 14 also points to an interesting way to es-timate the strength of the multiplicative noise in Eq. 9.So far, we have not come up with a stable and reliablemethod to estimate all parameters of the multiplicativenoise SDE Eq. 9 from relatively short records in orderto directly compare modeled and observed power-lawexponents and PDFs. In general, it is non-trivial toestimate coefficients of SDEs from limited data (e.g.,Kloeden and Platen 1992; Sura and Barsugli 2002). One

method we are exploring is to use the observationalestimates of λeff and the moments x2, x3, and x4 tosolve the nonlinear set of Eq. 10 for the remaining noiseparameters σ 2

F ′ , σ 2R′ , and φ. Preliminary results show a

satisfactory agreement of the power-law exponent α

calculated from Eq. 14 with the direct estimate fromdata using Eq. 2. However, the method is fraught withthe potential for error and uncertainty due to the non-linearity of the equations involved. Therefore, we mightuse the stable estimate of α from Eq. 2 and the easy-to-determine effective damping parameter λeff to solveEq. 14 for φ2 σ 2

F ′ . The remaining additive noise couldthen be determined from Eq. 10. However, to exploreand discuss estimation techniques for SDEs is beyondthe scope of this paper, but also part of our currentresearch.

3.2 Non-Gaussian SST variability in strong currents

The general theory discussed above does not specify thevalue, magnitude, or even the sign of the parameter φ.Therefore, the general idea outlined above does allowfor positive and negative skewness, depending on thesign of φ. In order to closer determine φ, it is necessaryto take a more detailed look at the terms includedin the overall heat flux F. So far, we only used thefact that the timescale of the forcing F is faster thanthe SST response. Is is already known that, away fromstrong currents, the heat flux Q/ρCH through the seasurface is the dominating term over most parts of theocean. Aspects of the stochastic nature of this termhave already been discussed in Sura et al. (2006). Forexample, while the heat flux Q/ρCH is not entirelyrandom on a daily scale (large-scale pressure systemsmay have a decorrelation timescale of up to a week),it can be well modeled as a random process on a dailyscale because the “slow” ocean does not care about thespectral form of the “fast” atmospheric forcing belowits decorrelation scale of at least about 2 or 3 weeks.That is, as long there is some spectral gap between theocean and the atmosphere, a stochastic forcing of theocean is justified (see also Frankignoul and Hasselmann1977; Hasselmann 1976). However, most relevant hereis the fact that the heat flux Q/ρCH will induce positiveskewness, that is, a negative φ. This can be easily seenfrom the common form of the bulk formula for the fluxQ ∝ (Ta − To)|U|, which depends on the SST To, airtemperature Ta, and wind speed |U|. That is, the evo-lution equation for SST driven solely by fluxes throughthe sea surface is

∂To

∂t= β(Ta − To)|U| ≡ FQ , (15)

166 Ocean Dynamics (2010) 60:155–170

where all constants are included in β. Therefore,∂ FQ/∂To = −β|U| = −β(|U| + |U|′) and F ′

Q = β(Ta −To)|U|′. Because the annual mean temperature differ-ence Ta − To is negative (the ocean is warmer thanthe atmosphere) over large swaths of the ocean (withthe exception of regions with very cold currents, suchas areas south of 40◦, to the west off California, andthe cold tongue of the equatorial Pacific), the additiveand multiplicative noise terms due to the wind speedanomaly |U|′ are mostly positively correlated (that is,have the same sign), resulting in positively skewed SSTanomalies T ′

o. That is, SST anomalies driven by theheat flux Q are mostly positively skewed due to themean atmosphere–ocean temperature difference witha warmer ocean combined with the structure of thebulk heat flux parameterization. It has been shownthat this relation also holds for locally coupled air–seainteraction models (Sura and Sardeshmukh 2009). Tosummarize, the heat flux Q cannot explain the negativeskewness observed in strong currents such as the GulfStream. Thus, we have to look for another mechanism.The key candidate in a strong current is, of course, theadvection term −vo · ∇To. Therefore, let us study thedynamics of this term.

The corresponding evolution equation reads

∂To

∂t+ vo · ∇To = D , (16)

where D denotes a dissipative process such that DTo ≤0 (for example, the conventional harmonic diffusion hasthis property). Equation 16 is, of course, the advectionequation of a nearly conserved tracer (here SST). Anequation for the variance of SST anomalies can beobtained by decomposing the temperature and velocityfields into the usual time-mean and eddy components,and then multiplying the resulting SST anomaly equa-tion by the SST anomaly itself. After averaging, weobtain

12

∂

∂tT ′2

o + v′oT ′

o ·∇To + 12

vo ·∇T ′2o + 1

2∇

(v′

oT ′2o

)= D′T ′

o ,

(17)

where we again may assume for the dissipation D′T ′o ≤

0 (e.g., Vallis 2006). Very often, the time dependence,the advection by the mean current in a zonal flow, andthe third-order term in Eq. 17 are neglected to obtainthe balance v′

oT ′o · ∇To ≈ D′T ′

o ≤ 0, stating that, onaverage, the eddy temperature flux is downgradient inregions of dissipation (e.g., Vallis 2006). In the follow-ing, we apply Eq. 17, keeping the third-order term, to anidealized, baroclinically unstable, strictly zonal warm-core current driven by a meridional pressure and tem-

perature (linked through the thermal-wind balance)gradient. The condition of strict zonality ensures that allzonal gradients vanish in Eq. 17. While this is, of course,a simplification of the real world, it serves as a usefulapproximation to dynamically discuss the advection ofSST anomalies along-stream a strong current such asthe Gulf Stream. Therefore, with the velocity vectorvo = (uo, vo), the spatial terms become

v′oT ′

o · ∇To =(

u′oT ′

o, v′oT ′

o

)·(

0,∂To

∂y

)= v′

oT ′o

∂To

∂y,

(18)

vo · ∇T ′2o = (uo, 0) ·

(0,

∂T ′2o

∂y

)= 0 , (19)

and

∇(

v′oT ′2

o

)= ∇

(u′

oT ′2o , v′

oT ′2o

)= ∂

∂yv′

oT ′2o . (20)

Assuming a statistically steady state, the final form ofthe SST anomaly variance equation in a strictly zonalflow becomes

v′oT ′

o∂To

∂y+ 1

2∂

∂yv′

oT ′2o = D′T ′

o . (21)

In the following, we discuss Eq. 21, whereby we alsoneglect friction (D′T ′

o = 0) to highlight the dynamics ofthe third-order term in our simplified setting.

To make further progress, we now make a commonapproximation used to study the statistics of a passivetracer: we consider the problem of stochastic advection.That is, we assume that the SST is advected by a ran-dom velocity field v′

o. This is the key step to link Eq. 21to our stochastic SST anomaly Eq. 9. Remember, thatwe have a very good stochastic model of non-GaussianSST anomaly variability in the SDE Eq. 9, where wehave not specified the detailed physical origin of themultiplicative noise F ′. However, from the generalform of the advection, Eq. 16, we know that a stochasticvelocity field v′

o will appear as a multiplicative noiseterm in a general SDE (e.g., Sardeshmukh and Sura2009; Majda et al. 2008). That is, for the stochasticadvection problem, we can set F ′ proportional to themeridional component v′

o of the random velocity vectorv′

o: F ′ ∝ v′o. That is, we obtain the inequalities (see

above)

v′oT ′2

o φ ≤ 0 and v′oT ′

o > 0 , (22)

linking the tracer variance Eq. 21 with the SDE Eq. 9.What do we learn from Eq. 22 in connection withEq. 21? The remainder of this section tries to answerthat question. To facilitate the discussion, a schematic

Ocean Dynamics (2010) 60:155–170 167

drawing of the underlying dynamical balance is pro-vided in Fig. 6.

Let us start by summarizing what we already know,including our assumptions. We derived Eq. 21 havinga strictly zonal, frictionless warm-core current (an ide-alized version of the Gulf Stream) in mind. The meantemperature is assumed to be warmest in the center(core) of the current; the boundaries are assumed tobe colder. That implies, of course, that the gradient∂To/∂y > 0 south of the center, and ∂To/∂y < 0 northof it. This behavior is depicted on the right-hand sideof Fig. 6. We also know that, if we observe nega-tive skewness (equivalent to φ > 0) of stochasticallyadvected SST anomalies (as in the Gulf Stream system),we have v′

oT ′2o < 0, and vice versa. Furthermore, on the

meridional boundaries of the current, the skewness ofSST anomalies vanishes, implying φ = 0 and v′

oT ′2o = 0.

This gives us the sign of the derivative: ∂v′oT ′2

o /∂y < 0south of the warm core, and ∂v′

oT ′2o /∂y > 0 north of the

warm core (also depicted in Fig. 6).We are now in a position the study the balance

v′oT ′

o∂To

∂y+ 1

2∂

∂yv′

oT ′2o = 0 (23)

in more detail:

• South of the center (warm core), we have ∂To/∂y >

0 and ∂v′oT ′2

o /∂y < 0. To balance the terms, we needa northward (up-gradient) eddy heat flux v′

oT ′o > 0,

consistent with our stochastic advection model.

• North of the center (warm core), we have ∂To/∂y <

0 and ∂v′oT ′2

o /∂y > 0. Again, to balance the terms,we need a northward (now down-gradient) eddyheat flux v′

oT ′o > 0, again consistent with our model.

Several interesting features emerge. First, in a sto-chastic non-Gaussian setting, friction is not explicitlynecessary to balance the eddy heat fluxes. The diver-gence of the third-order correlation v′

oT ′2o is sufficient

to balance the northward heat flux in a zonal warm-core current. While this is an idealization of the realworld (because randomly advected water parcels carryheat and momentum), it is enlightening to have a modelthat gives meaningful results without relying on poorlyunderstood frictional effects. Second, the non-Gaussianview is consistent with observations, provided that thestochastic advection model is applicable to tracer trans-ports in the Gulf Stream region. However, as discussedabove, we have clear observational evidence that thestochastic view is a very good model of non-GaussianSST variability, and that the assumptions made aboveare, therefore, approximately valid. In addition, thederived requirement that the eddy heat flux is solelynorthward (v′

oT ′o > 0) in the Gulf Stream System is

also consistent with observations and models (Stammer1998; Jayne and Marotzke 2002). Third, by intuition,one could expect the Gulf Stream SSTs to be negativelyskewed. In the warm Gulf Stream, the strongest coldanomalies result from rings and meanders that bringcold water southward from regions north of stream.Warm anomalies due to other processes, such asradiative effects and advection from downstream,

Fig. 6 Schematic drawing ofthe dynamical balance of theSST variance Eq. 21 in astrictly zonal, frictionlesswarm-core current. The meantemperature is assumed to bewarmest in the center (core)of the current; the boundariesare assumed to be colder.That implies that the gradient∂To/∂y > 0 south of thecenter, and ∂To/∂y < 0 northof it. The sign of thethird-order derivative canalso deduced fromobservations: ∂v′

oT ′2o /∂y < 0

south of the warm core, and∂v′

oT ′2o /∂y > 0 north of the

warm core. This balance thenrequires a northward eddyheat flux: v′

oT ′o > 0. See text

for more details

168 Ocean Dynamics (2010) 60:155–170

might well be weaker and less frequent. This is basicallythe intuition that the net eddy heat flux is northward inthe Gulf Stream. The fact that this comes out of ourindependent stochastic analysis is a satisfying result.

To summarize, stochastic advection of SST is a goodcandidate to account for the negative SST anomalyskewness observed in the Gulf Stream system. As wehave seen, the evolution equation for SST driven solelyby heat fluxes through the seas surface Eq. 15 wouldgive us positively skewed SST anomalies. That is, acombination of SST advection and heat flux forcingcould result in negative or positive skewness. It is therelative strength of each process responsible for thenet skewness. In very strong currents such as the GulfStream, the SST advection is strong enough to signif-icantly affect the mixed-layer heat budget, resultingin the pronounced negative SST skewness observedalong-stream strong currents. However, the heat fluxesthrough the seas surface are still important to providethe damping of SST anomalies without dominating thenet skewness.

4 Summary and discussion

This study has looked at non-Gaussian SST variabil-ity in the Gulf Stream system. It has been shownfrom observations that the Gulf Stream (and its ex-tension, the North Atlantic Current) is very clearlyvisible as a band of negative skewness all the wayfrom Florida, over Cape Hatteras, to the central NorthAtlantic. To get an idea about the detailed non-Gaussian variability along the Gulf Stream, PDFs werecalculated at several locations. One important observa-tional result of this study is that the non-Gaussianity inthe Gulf Stream system is consistent with a power-law,as tested by Kolmogorov–Smirnov statistics. The studythen provides a theoretical explanation of the nega-tively skewed power-law behavior of SST anomalies instrong currents.

The model that explains the power-law and the neg-ative skewness of SST anomalies in strong currents isbased on a recent stochastic theory of non-GaussianSST variability by Sura and Sardeshmukh (2008). Thattheory states that a stochastic approximation, includingmultiplicative noise, of the general mixed-layer SSTbudget is an excellent globally applicable model ofnon-Gaussian SST variability. The first candidate for astochastic forcing of SST anomalies is, of course, thewind-driven surface heat flux. However, it is known(Sura et al. 2006; Sura and Sardeshmukh 2009) thatthe surface heat flux alone cannot explain the coherentnegative skewness observed in swift currents such as

the Gulf Stream. Therefore, the starting hypothesisis that SST advection in the mixed layer significantlycontributes to the negatively skewed non-Gaussian SSTvariability in strong currents. It is then shown thatstochastic advection of SSTs is consistent with theobserved properties of the Gulf Stream system.

To summarize, SST advection in strong ocean cur-rents very likely contributes to the induction of ex-treme negative SST anomalies in such regions. Thediscussed results provide useful insight into the rela-tive importance of local air–sea coupling and oceanicheat transports in the dynamics of SST variability ingeneral, and of extreme SST anomalies in particular.Of course, other processes we neglected so far, such asvariable mixed-layer depth, entrainment, and skewedatmospheric wind contributions, may potentially beresponsible for negative skewness as well. A detailedanalysis of these factors is part of our ongoing research.

Acknowledgements The author thanks Dr. Pierre Lermusiauxand three anonymous reviewers whose comments greatly im-proved the manuscript. This work was supported by NSF GrantATM-0552047 “The impact of rapidly-varying heat fluxes on air-sea interaction and climate variability.”

Appendix A

Stochastic differential equations in a nutshell

This appendix heuristically reviews a few basic ideas ofSDEs used in this paper. More comprehensive treat-ments may be found in many textbooks (e.g., Gardiner2004; Horsthemke and Léfèver 1984; Paul andBaschnagel 1999; Kloeden and Platen 1992; vanKampen 1981). Note that we here use the notion of theensemble average 〈...〉 to be mathematically consistentwith textbooks on SDEs.

Consider the dynamics of a scalar x governed by theSDE

dxdt

= A(x) + B(x)η(t) , (24)

with the delta-correlated white-noise η: 〈η(t)〉 = 0 and〈η(t)η(t′)〉 = δ(t − t′). How do we interpret thisequation? The underlying problem is that, while thewhite-noise η(t) is defined for every time t, it is notcontinuous and, therefore, not differentiable. Thismeans that, mathematically speaking, the SDE Eq. 24is not well defined. An alternative interpretation iscalled for. The solution is to interpret Eq. 24 in itsintegral form and, hence, expect the noise term to be

Ocean Dynamics (2010) 60:155–170 169

integrable. That is, we integrate Eq. 24 to obtain theintegral equation

x(t) − x(t0) =∫ t

t0A[x(t′)]dt′ +

∫ t

t0B[x(t′)]η(t′)dt′ . (25)

which can be interpreted consistently. In addition,the discontinuous white-noise η(t) is usually replacedwith its continuous integral, the Wiener process W(t),given by

W(t) =∫ t

0η(t′)dt′ , (26)

or

dW(t)dt

= η(t) . (27)

Informally, we may write dW(t) = η(t)dt, keeping inmind that this expression is rigorously defined onlywithin an integral. From the definition of the Wienerprocess, its autocovariance can also be easily calcu-lated: 〈W(t)W(s)〉 = min(t, s). Using the definition ofthe Wiener process, the integral Eq. 25 becomes

x(t) − x(t0) =∫ t

t0A[x(t′)]dt′ +

∫ t

t0B[x(t′)]dW(t′) . (28)

To summarize, the SDE Eq. 24 has to be interpretedin its mathematically consistent integral form. As areminder of the integral definition, the SDE Eq. 24 isoften written as

dx = A(x)dt + B(x)dW(t) . (29)

The rigorous integral expression of the increment of theWiener process is

dW(t) = W(t + dt) − W(t)

=∫ t+dt

0η(t′)dt′ −

∫ t

0η(t′)dt′

=∫ t+dt

tη(t′)dt′ . (30)

Using this expression, we can calculate the autocovari-ance 〈dW(t)dW(s)〉 as

〈dW(t)dW(s)〉 =⟨∫ t+dt

t

∫ s+dt

sη(t′)η(s′) dt′ds′

⟩

=∫ t+dt

t

∫ s+dt

s〈η(t′)η(s′)〉 dt′ds′

=∫ t+dt

t

∫ s+dt

sδ(t′ − s′) dt′ds′

= dt δt,s , (31)

where δt,s denotes the Kronecker delta. For t = s, weobtain 〈dW(t)2〉 = dt, stating that the variance of white-

noise (the so called Wiener process W) scales with thetime increment dt. That is, the standard deviation of theWiener process effectively scales with

√dt in contrast

to the dt scaling of the determinist terms. It is this√

dtscaling of the stochastic terms that is responsible for socalled stochastic calculus, in contrast to the dt scaling ofthe ordinary deterministic calculus. Again, the readershould consult one of many textbooks on SDEs for amore comprehensive treatment (e.g., Gardiner 2004;Horsthemke and Léfèver 1984; Paul and Baschnagel1999; Kloeden and Platen 1992; van Kampen 1981).

Appendix B

More on third-order moments

The starting point is the Stratonovich SDE

∂T ′o

∂t= −λeff T ′

o − φF ′T ′o + F ′ + R′ + φF ′T ′

o , (32)

with the locally constant parameters −λeff and −φ andthe rapidly varying (approximately Gaussian white-noise) forcing terms F ′ and R′ with amplitudes σF ′

and σR′ : F ′(t)F ′(t′) = (σF ′)2δ(t − t′) and R′(t)R′(t′) =(σR′)2δ(t − t′). The effective drift −λeff equals the sumof the deterministic drift and the noise-induced drift:−λeff = −λ + (1/2)(φσF ′)2. Multiplying this equationwith T ′

o, averaging, and assuming steady-state statisticsyields

0 = −λeff T ′2o − φF ′T ′2

o + F ′T ′o + R′T ′

o , (33)

or

T ′2o = 1

λeff

(−φF ′T ′2

o + F ′T ′o + R′T ′

o

). (34)

The variance T ′2o is known from Eq. 10. Therefore, we

obtain

(σ 2

F ′ + σ 2R′

)[2λeff − (φσF ′)2

] = 1λeff

(−φF ′T ′2

o + F ′T ′o + R′T ′

o

).

(35)

We see that for no multiplicative noise (φ = 0) F ′T ′o >

0 and R′T ′o > 0. From Eq. 35, we can also derive the

following properties of F ′T ′2o :

• Solving Eq. 35 for F ′T ′2o and letting φ → 0, we ob-

tain the limit F ′T ′2o = 0. In short, we write F ′T ′2

o = 0for φ = 0.

• From Eq. 35, we can also derive the sign of F ′T ′2o φ.

We start by setting φ = 0. The remaining terms areall positive. If we now allow for a small perturbation

170 Ocean Dynamics (2010) 60:155–170

φ away from zero, keeping F ′T ′o and R′T ′

o constant,we obtain F ′T ′2

o φ < 0. Assuming continuous behav-ior of the moments, the general relations F ′T ′2

o φ ≤0 and F ′T ′

o > 0 follow.

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