Assessing Climate Forcing From the Sea Surface Temperature‐Surface Heat Flux Relation for SST‐Coupled Oscillatory Variability

The interaction between sea surface temperatures (SST) and surface heat flux (SHF) is vital for atmospheric and oceanic variabilities. This study investigates SST‐SHF relationship in the framework of a coupled oscillatory model, extending beyond previous research that predominantly used AR‐1 type simple stochastic climate models. In contrast to the AR‐1 type model, we reveal distinct features of SST‐SHF relationships in the oscillatory model: sign reversals occur in the imaginary part of SST‐SHF coherence and the low‐pass SST tendency‐SHF correlation. However, these sign reversals are absent in the real part of SST‐SHF coherence and in the low‐pass SST‐SHF correlation. We find these features are robust across both the twentieth Century Reanalysis and GFDL SPEAR model for El Niño‐Southern Oscillation (ENSO) variability. Furthermore, we develop a new scheme to assess ENSO's climate forcing magnitude and natural frequency. Our findings thus provide novel insights into understanding ENSO dynamics from the perspective of heat flux.


Introduction
The interaction between atmospheric and oceanic heat exchange is a critical process in generating climate variability (T.L. Delworth & Greatbatch, 2000;Frankignoul & Kestenare, 2002;Qiu & Kelly, 1993;Roberts et al., 2017;Stephens et al., 2012).Understanding the relationship between sea surface temperatures (SST) and surface heat flux (SHF), and its implication on the driving forcing, is thus one of the fundamental issues in climate science (Frankignoul et al., 1998;Large & Yeager, 2012;Liu et al., 2008;Park et al., 2005;Wills et al., 2019).Over half a century ago, Bjerknes (1964) hypothesized that the SST-SHF correlation should change from positive for interannual variability to negative for decadal variability, reflecting the change of the dominant driving from the atmospheric to oceanic forcing.Gulev et al. (2013) provided the first observational evidence supporting this hypothesis in the mid-latitude North Atlantic.Later studies have further investigated the low-pass correlation between SST and SHF in discerning the relative roles of atmospheric noise versus interior oceanic processes in driving SST variability (Bishop et al., 2017;Cane et al., 2017;Clement et al., 2016;Li et al., 2020;O'Reilly et al., 2016).
Recent studies have utilized Hasselmann's (1976) theoretical framework to investigate the SST-SHF relationship (O'Reilly et al., 2016;Zhang, 2017), confirming that SST-SHF correlation changes from positive to negative with low-pass filtering in the North Atlantic, suggesting an active role for oceanic processes (hereafter referred to as "oceanic forcing") on decadal variability.Liu et al. (2023) (hereafter Liu23) further developed a new stochastic model (hereafter LGD model) that includes red ocean forcing and showed that sign reversal of SST-SHF correlation is determined by both the magnitude and the timescale of the ocean forcing.The model is still based on the AR-1 process as in the classical Hasselmann model.In spite of its success for assessing oceanic variability in the North Atlantic, the LGD model has a limitation: it does not apply to quasi-oscillatory climate variability, or climate variability with preferred time scales, which have profound global impacts.An example is the El Niño-Southern Oscillation (ENSO), which has a preferred time scale of 2-7 years and is known to be driven by positive air-sea feedbacks and oceanic adjustment processes (Neelin et al., 1998;Wang, 2018;Wang et al., 2017).Therefore, how to assess the driving forcing from the SST-SHF relation for coupled oscillatory variability remains unclear.
In this study, we investigate this SST-SHF relationship in an oscillatory model that couples subsurface oceanic dynamics with SST variability: the recharge model for ENSO (Jin, 1996).It is found that the sign reversal in this coupled oscillatory model occurs in the SST tendency-SHF correlation (r T t H ) and the imaginary part of the SST-SHF coherence (Coh TH ), in contrast to the non-oscillatory LGD model, in which the sign reversal occurs in the SST-SHF correlation and real part of Coh TH .Additionally, the relative forcing magnitude of the atmosphere and ocean can be inferred by simultaneously low-pass SST-SHF correlation r T H (0) .We show that our theory can successfully explain the major features of the SST-SHF relationship over the ENSO region in the GFDL Seamless System for Prediction and Earth System Research (SPEAR) model and quasi-observational twentieth Century Reanalysis (20CR).Furthermore, we develop a new scheme to assess oceanic forcing and natural frequency from SST-SHF relations.Our estimation shows that oceanic forcing is significantly stronger than atmospheric forcing in influencing ENSO variability in both the 20CR and SPEAR model.

Theoretical Background
In the LGD model, SST variability exhibits largely the red noise behavior, as seen in the power spectrum (Figure 1b).To simulate SST oscillatory behavior in an idealized model, we use the following two-component coupled linear system: where T and h can be thought to represent the SST and the thermocline depth, respectively, thereby simulating the ENSO behavior as described the "Recharge paradigm" by Jin (1996).The ω is the coupling coefficient between SST and thermocline variability. Here, is the net SHF forcing, and is the net oceanic forcing on SST.The f a (t) and f o (t) are the stochastic climate forcing originating from the atmosphere and ocean, respectively, and, for simplicity, will be assumed independent of each other.The λ a and λ o are damping rates of the SST feedback associated via SHF and ocean interior processes, respectively, with the sum as the total SST damping rate Here, we set b 2 ) 2 < ω 2 such that the system is in the damped oscillation regime.The SST autocovariance function can be derived as: is the natural frequency of the oscillatory model.The β = b 2 is half the damping rate.σ a 2 and σ o 2 are the variance of atmospheric forcing f a and oceanic forcing f o .Unless otherwise specified, in the following, we will set b = 1, effectively nondimensionalizing all time scales by 1/b.

Observation and Model Data
Here, the observed SST are derived from the Hadley Centre's Sea Ice and Sea Surface Temperature (HadISST) data (Rayner, 2003).The net SHF (including sensible heat flux, latent heat flux, net shortwave radiation, and net longwave radiation) are sourced from the NOAA-CIRES twentieth Century Reanalysis V3 (20CR) (Slivinski et al., 2021).The ocean surface salinities are sourced from the ISHII data version 6.13 (Ishii et al., 2006) (Liu et al., 2023).linearly detrended, with monthly mean climatology subtracted to remove the seasonal cycle.All data are interpolated to 2°× 2°grid points before analyses.

SST-SHF Relationship in Spectral Analysis
Following Liu et al. (2023), we will study SST-SHF relationship first in the spectral analysis and then the correlation analysis (Section 4).The coherence and correlation analyses, fundamentally, yield consistent outcomes but from distinct perspective.Coherence analysis evaluates how consistently two signals maintain their phase relationship across different frequencies, offering insights into frequency-dependent relationships.In contrast, correlation analysis measures the overall strength and direction of the linear relationship between data sets, focusing on overall trends rather than the frequency-specific dynamics.Notably, the sign change of correlation with time scale is seen most easily in the mathematical derivation of the coherence analysis.
Denote the Fourier transform of the time series T(t), h(t), f a (t), f o (t) in the frequency space as [Τ(σ), h(σ), F a (σ), F o (σ)]e iσt , respectively.After applying the Fourier transform to Equations 1 and 2, the SST response can be derived as The coherence between SHF H(σ) = λ a Τ(σ) + F a (σ) and SST is therefore where |X(σ)| 2 = <X*(σ)X(σ)> can be considered as proportional to the power level at the frequency σ, X* is the complex conjugate of X, and the "< >" represents ensemble average.
Given that the correlation at lag zero is proportional to the real part of the coherence, we have: This shows clearly that the sign of r TH (0) is determined by the relative strength of effective atmospheric and oceanic forcing, while oceanic forcing |F o ⃒ ⃒ 2 contributing to negative correlation.In our oscillatory model, there is no sign change in r TH (0) for all frequencies.Rather, the sign reverses in the imaginary part of the coherence: The sign reversal of Im(Coh TH ) = 0 occurs at the reversal frequency: Thus, from Equation 7, the intrinsic oscillation frequency can be derived as Geophysical Research Letters 10.1029/2024GL108552 Figure 1 shows some basic features of the coherence analysis for an example.The parameters are chosen as λ a = 0.2, λ o = 0.8, couple coefficient ω = 2, atmospheric forcing variance σ a 2 = 0.1 and ocean forcing variance σ o 2 = 10.These parameters are estimated from the observational SST, SHF and SSS, closely relevant to the SST-SHF relations of 20CR (Appendix B).As shown in Figure 1a, the SST and SHF both have a spectral peak at σ 0 = 2.The spectral peak of SST originates from the coupling process near ω = 2, whereas the spectral peak of SHF is attributed to the SST feedback H = λ a Τ + F a .Figure 1c shows the coherence between SST and SHF, which has a coherence peak at σ 0 = 2 consistent with the spectral peak frequency.As expected, Re(Coh TH ) remains negative across all frequencies, indicating a stronger effective oceanic forcing than atmospheric forcing in this example ( 1e, blue curve).While Im(Coh TH ) changes sign from negative to positive with decreasing frequency, and the reversal frequency is identified as σ 0 = ω = 2 (Figure 1e, orange curve).This suggests that the relative forcing magnitude can be inferred from the sign of Re(Coh TH ) while the sign of the lowpass couple coefficient can be discerned from the sign reversal of Im(Coh TH ).Thus, the natural frequency Ω can be estimated from Equation 13.Furthermore, these characteristics significantly differ from those in the LGD model, which does not involve the coupling of SST with subsurface variability (Liu et al., 2023).In the LGD model, SST exhibits a red noise feature (Figure 1b) and lacks a distinct spectral peak in the coherence (Figure 1d).A noteworthy distinction is that in the LGD model, the sign reverses in Re(Coh TH ), but not in Im(Coh TH ) (Figure 1f), in contrast with the sign reversal in the oscillatory model.This difference of sign reversal in real part mainly occur at high frequencies where Re (Coh TH ) is negative in the oscillatory model but positive in the LGD model.This indicates that atmospheric forcing plays a dominant role in the LGD model, particularly in mid-to high-latitudes, whereas it is relatively weak in the coupled model for tropical ocean, as will be discussed in Section 5.This likely stems from the strong atmospheric forcing at mid-latitudes, which originates from large-scale storm tracks, compared to the weaker atmospheric forcing in the tropical ocean, arising from small-scale convections.

The Correlations Between SST and SHF
This section further study the SST-SHF relation in the "low-pass filtered correlation" approach, following many traditional studies in Coupled Global Climate Model and observations (Bishop et al., 2017;Cane et al., 2017;O'Reilly et al., 2016;Zhang, 2017).We first study it in theory and then apply it to both the 20CR and the SPEAR model (detailed in next section).
We use the running mean as the low-pass filter, which can be defined for, say, SST, as where L is the window length.After substantial algebra, low-pass covariances and correlations can be derived analytically (Appendix A).The simultaneous low-pass covariance and correction between SST and SHF can be derived as: where E = Ω e βL [β sin(ΩL) + Ω cos(ΩL)] > 0, σ a 2 , and σ o 2 are the variance of atmospheric forcing f a and oceanic forcing f o .C T T (0) and C H H (0) are the variance of SST and SHF, respectively (Equations A12 and A13 in Appendix A).This shows clearly that the sign of r T H (0) is determined by the relative strength of effective atmospheric and oceanic forcing, σ a 2 λ a and σ o 2 λ o , which is consistent with the result in spectrum analysis Equation 10.
Since the imaginary part of the coherence is related to the correlation between SST tendency and heat flux, it is conceivable that the correlation may reverses its sign with low-pass filter in the simultaneous correlation between SST tendency and SHF.These correlation and covariance can be derived as: Geophysical Research Letters where C T t T t (0) is the variance of SST tendency (Equation A14 in Appendix A).The C T t H (0) and r T t H (0) exhibits damped harmonic behaviors with several sign reversal points at L n = nπ Ω (n = 1,2…) .Thus, given the running mean window of the first reversal L 1 = π Ω , we can estimate the nature frequency as follows: This method provides an alternative way to determine the intrinsic oscillation frequency, supplementing the estimation from reversals in the imaginary part of coherence as outlined in Equation 13.

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Figure 2 shows the low-pass correlations between SST and SHF, as well as between SST tendency and SHF, across lag and window length (τ, L).Notably, for instantaneous correlation along τ = 0, the low-pass correlation r TH (0) remains negative for all running mean windows L (Figures 2a and 2e).This feature aligns with the negative real part of the coherence, suggesting that the effective oceanic forcing is stronger than atmospheric forcing ).In the cases with strong atmospheric forcing ), the low-pass correlations r T H (0) remains positive across all running mean windows L (Figures S1c and S1d in Supporting Information S1), and in the neutral case ), it remains zero (Figures S1b, S1e, and S1g-S1i in Supporting Information S1).Thus, in the coupled oscillatory model, r T H (0) never changes the sign, regardless of the low-pass filter.This significantly differs from LGD model where r T H (0) tends to change its sign with low-pass filter (Figures 2b and 2f).
In contrast, the low-pass correlations between SST tendency and SHF (r T t H (0)) exhibits sign reversals, transitioning from positive to negative at L 1 = 1.6 and then back to positive (Figures 2c and 2e), when normalized by the total SST damping time 1/b.The second and subsequent reversal points are not very clear, as the amplitude of r T t H (0) diminishes with lag as e β|τ| (Equation 6).These reversals in r T t H (0) are robust across all cases, regardless of the forcing magnitude (Figures S1a-S1f in Supporting Information S1).This contrasts with the LGD model which shows no sign reversal in r T t H (0) (Figures 2d and 2f).Furthermore, using the running mean window of the first reversal, L 1 = 1.6, we can estimate the natural frequency as Ω est = π L 1 = 1.9 (Equation 17).This estimation is consistent with the theoretical frequency Ω = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ ω 2 β 2 √ = 1.9 in this example, where ω = 2 and β = b 2 = 0.5.

The SST-SHF Relationship of ENSO
We now apply analysis to the ENSO variability in both the 20CR and the SPEAR model.We conduct the spectrum analysis and correlation analysis for ENSO variability within the Nino3.4region (170-120°W, 5°N-5°S) using monthly SST and SHF from 20CR for 1870-2015 and the SPEAR control simulation of 1000 years.
In the 20CR, the correlation analysis reveals that along τ = 0, r T H (0) remains negative across all running mean windows L (Figures 3a and 3c), suggesting the effective oceanic forcing is stronger than the effective atmospheric forcing for ENSO variability.Additionally, using our novel method (described in Appendix B) to estimate the magnitudes of oceanic and atmospheric forcings, we found that the oceanic forcing (σ o ) is approximately 9 times stronger than atmospheric forcing (σ a ).The estimated difference in effective forcings ( σ a 2 λ a σ o 2 λ o ) is about 1.28 K 2 month 1 , which contributes to the negative r T H (0) .This is consistent with our understanding of ENSO dynamics, which depends critical on equatorial ocean dynamics.Notably, r T t H (0) exhibits a damped oscillatory behavior (Figure 3c) with a sign reversal at L 1 = 15 months.This allow us to estimate the natural frequency of ENSO as Ω est = π L 1 = 0.21 rad/month, corresponding to a period of approximately 30 months.The spectral analysis indicates that the imaginary part of the coherence reverses sign at σ 0 = 0.22 rad/month (Figure 3f), aligning with the coherence peak (Figure 3e) and SST spectrum peak (Figure 3f).Equation 12enables us to estimate the coupling coefficient as ω = σ 0 = 0.22 rad/month.Similar patterns are also identified in the SPEAR model (Figure 4).Here, r T H (0) remains negative across all lowpass windows (Figures 4a and 4c) and r T t H (0) has a sign reversal point at L 1 = 21 months (Figures 4b and 4c).The oceanic forcing (σ o ) is estimated to be about 19 times stronger than atmospheric forcing (σ a ) and the effective forcing difference ( σ a 2 λ a σ o 2 λ o ) is about 1.07 K 2 month 1 , suggesting the dominant role of ocean dynamics over equatorial regions.Furthermore, the imaginary part of coherence also reverses the sign at σ 0 = 0.15 rad/month (Figure 4f), corresponding to the coherence peak (Figure 4d) and SST spectrum peak (Figure 4e).Consequently, the estimated ENSO frequency in the SPEAR model is Ω est = π L 1 = 0.15 (rad/month) and corresponding oscillatory period is 42 months.
Overall, key characteristics of ENSO, such as the sign reversal in r T t H and Im(Coh TH ) , are in alignment with our theory (Figures 1 and 2).This consistency suggests that the coupled oscillatory model not only reproduces the Geophysical Research Letters 10.1029/2024GL108552 oscillatory behavior of ENSO but also accurately captures the SST-SHF relationship features both in correlation and coherence analyses.

Conclusion and Discussion
In this study, we explore the SST-SHF relationship in the framework of a coupled oscillatory model, employing both spectrum and correlation analyses.Our findings indicate that the SST-SHF relation is dramatically different between the oscillatory model and the AR1-type of model such as the LGD model.In the oscillatory model, the sign reverses in the imaginary part of the SST-SHF coherence and the low-pass SST tendency-SHF correlation, as opposed to in the real part of the SST-SHT coherence and the low-pass SST-SHF correlation as in the LGD model.These reversals, stemming from the coupling process, are instrumental in estimating the natural frequency and coupling coefficient.Furthermore, in the oscillatory model, the sign of the SST-SHF correlation remains unchanged with time scale, positive for stronger atmospheric forcing and negative for stronger oceanic forcing.Thus, the SST-SHF correlation can be utilized to assess the relative forcing magnitudes of the atmosphere and ocean.These fundamental characteristics of the SST-SHF relationship markedly differ from those in the LGD model.
Our theory is further applied to the ENSO variability in 20CR and the SPEAR model.In both SPEAR and 20CR, the sign changes in both the low-pass SST tendency-SHF correlation and the imaginary part of the SST-SHF

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10.1029/2024GL108552 coherence, consistent with our theory.This suggests a significant role of coupling process in ENSO variability.The coupling coefficient and natural frequency can be estimated through these sign reversals using two distinct methods (Equations 13 and 17).Additionally, the consistently negative low-pass SST-SHF correlation across all running mean windows implies predominant oceanic forcing in ENSO variability, aligning with our quantitative estimates.We further found that oceanic forcing is 9 times stronger than atmospheric forcing in 20CR and 19 times stronger in the SPEAR model.In some sense, our analysis is not surprising, because the recharge model has been suggested to apply to ENSO very well (Jin, 1996).Nevertheless, our analysis sheds new light on the dynamics of ENSO, and especially from the perspective of SHF.
Our study also proposes innovative methods to estimate the natural frequency and forcing magnitude in coupled oscillatory models, with applications to ENSO variability.Previous studies, utilizing the residue method with SST tendency (Equation B8), estimated oceanic forcing from monthly (Patrizio & Thompson, 2022) timescale to decadal timescale (Gu et al., 2024;Liu et al., 2023).The residue method may have significant biases on monthly timescale due to discretization errors from using the centered finite difference scheme for SST tendency estimation.These biases can be reduced on decadal timescales by filtering highfrequency signals from the SST tendency.In this study, we applied and compared our new method (Equation B6) with residue methods in a coupled oscillatory system to estimate forcing magnitude using a Monte Carlo test.We found that our new method accurately estimates oceanic forcing, whereas residue methods significantly underestimate it by over 30% (Figure S2 in Supporting Information S1).Furthermore, while many previous studies have identified preferred time scales and periods from power spectral analysis (Hope et al., 2017;Kestin et al., 1998;Santoso et al., 2017), with inherent limitations such as power leakage due to discrete Fourier transform and poor resolution resulting from inadequate data length (Yiou

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10. 1029/2024GL108552 et al., 1996)).Our method enhances these traditional approaches by providing a novel perspective on estimating natural frequencies and preferred time scales through SST-SHF relations, improving accuracy and reliability through cross-examination with previous methods.
Finally, we should point out that, although our recharge model analysis applies well to ENSO, as it should be, it does not apply to multidecadal variability over the mid-latitude North Atlantic.Figures S3a and S3b in Supporting Information S1 show that the low-pass SST-SHF correlation in the mid-latitude North Atlantic exhibits multi-decadal parallel bands that are largely independent of the low-pass window length, in both the 20CR and the SPEAR model.This multi-decadal variability corresponds to the multi-decadal AMV in the 20CR and, especially, in the SPEAR model.This feature, however, can be reproduced by neither the AR-1 LGD model nor our current recharge oscillatory model.As shown in Figures S3c and S3d of Supporting Information S1, in these two models, the correlation bands always expands with the low-pass window.Our further preliminary analysis suggests a potential difference between the oscillation in the recharge model and the multi-decadal variability: the former is produced by the active coupling between SST and subsurface oceanic dynamics, while the latter is likely produced by the subsurface dynamics itself, potentially via subsurface ocean dynamic process associated with ocean circulation and baroclinic Rossby waves (T.Delworth et al., 1993;T. L. Delworth et al., 2017;Goodman & Marshall, 1999;Johnson & Marshall, 2002).The multi-decadal signal is only entrained "passively" into the mixed layer to affect SST.This part will be presented in a separate work.

Appendix A: Low-Pass Covariances and Correlations
The low-pass covariance can be derived, following the formula of Frankignoul et al. (1998), analytically as follows.
where σ a 2 and σ o 2 are the variance of atmospheric forcing f a and oceanic forcing f o , respectively.
Thus, we obtain the low-pass covariance between SST and SHF: The low-pass covariance between SST tendency and SHF: Here where The corresponding correlations can be derived as Where the simultaneously low-pass covariance can be derived as: where

Appendix B: Forcing Estimation Scheme for Coupled Oscillatory Model
Our forcing estimation mostly based on the LGD scheme (Liu et al., 2023) for parameters estimation.This scheme has been used to quantify the decadal forcing magnitude over extra-tropical regions over global ocean (Gu et al., 2024).Here, we develop a new scheme for oceanic forcing estimation for coupled oscillatory model.
Step 1: Estimate damping rates b, λ o , λ a (Figures S3a-S3i in Supporting Information S1): The total SST damping rate b is estimated from the SST lag-1 autocorrelation coefficient, while the ocean forcing damping on SST λ o is estimated from the sea surface salinity lag-1 autocorrelation coefficient (Hall & Manabe, 1997;Zhang, 2017) such Step 2: Estimate the heat flux H in the dimension of temperature tendency (°C s 1 ): H is estimated from the dimensional heat flux H (in W m 2 ) as H = H/(ρC p h e ) .Here, h e is an effective mixed layer depth estimated indirectly from the air-sea feedback as follows.In the original heat flux dimension (W m 2 ), the equivalent form of Equation 3 is H = μ a T + f a , where the feedback coefficient μ a is related to the atmospheric damping rate on SST as λ a = μ a /(ρC p h e ), with ρ = 1,024 kg m 3 and C p = 3,850 Jkg 1 K 1 as the density and specific heat of seawater, respectively.Thus, h e is estimated as: Here, μ a is estimated using the Equilibrium Feedback Analysis (Frankignoul et al., 1998;Liu et al., 2008;Park et al., 2005)

Figure 1 .
Figure 1.(a, c, and e) Analytical solutions of coupled oscillatory model and (b, d, and f) LGD model.(a and b) Power spectrum of SST and SHF.(c and d) Magnitude-squared coherence between SST and SHF.(e and f) The real and imaginary parts of coherence Coh TH .For coupled oscillatory model, b = 1, λ a = 0.2, λ o = 0.8, ω = 2, σ a 2 = 0.1, σ o 2 = 10.All parameters are nondimensionalized by 1/b.The parameters of LGD model is as same as Liu23(Liu et al., 2023).

Figure 2 .
Figure 2. (a, c, and e) Low-pass SST-SHF and SST tendency-SHF correlations of coupled oscillatory model and (b, d, and f) LGD model.(a and b) Low-pass SST-SHF correlations r T H as a function of window length L and lag τ. (c and d) Low-pass SST tendency-SHF correlations r T t H . Positive lag τ represents SHF leads SST or SST tendency.(e and f) Simultaneously low-pass correlations r T H (0) and r T t H (0) as a function of window L. The heavy black dotes indicates the reversal windows.For coupled oscillatory model, b = 1, λ a = 0.2, λ o = 0.8, ω = 2, σ a 2 = 0.1, σ o 2 = 10.All parameters are nondimensionalized by 1/b.The parameters of LGD model is as same as Liu23 (Liu et al., 2023).

Figure 3 .
Figure 3. Correlations, power spectral and coherence analysis for ENSO (170-120 o W, 5 o N-5 o S) using monthly 20CR data from 1870 to 2015.(a) Low-pass SST-SHF correlations r T H .(b) Low-pass SST tendency-SHF correlations r T t H . Positive lag τ represents SHF leads SST or SST tendency.(c) Simultaneously low-pass correlations r T H (0) and r T t H (0). (d) Normalized power spectrum of SST and SHF.(e) Magnitude-squared coherence between SST and SHF.(f) The real and imaginary parts of coherence.

Figure 4 .
Figure 4.As in Figure 3, but for the preindustrial control run of SPEAR model.