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Original Research Papers

A simple model of ocean temperature re-emergence and variability

Authors:

Peter Kowalski ,

Department of Mathematics, University College, London, GB
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Michael Davey

Met Office, Exeter; Centre for Mathematical Sciences, University of Cambridge, Cambridge, GB
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Abstract

A simple stochastic one-dimensional model of interannual mid-latitude sea surface temperature (SST) variability that can be solved analytically is developed. A novel two-season approach is adopted, with the annual cycle divided into two seasons denoted summer and winter. Within each season the mixed layer depth is constant, and the transition of the mixed layer from summer to winter and vice versa is discontinuous. SST anomalies are forced by random atmospheric heat fluxes, assumed to be constant within each season for simplicity, with linear damping to represent atmospheric feedback. At the start of summer the initial SST anomaly is set equal to that at the end of the previous winter, and at the start of winter the initial temperature anomaly is found by instantaneously mixing the summer mixed layer with the heat stored below in the deeper winter mixed layer, thereby explicitly taking into account the ‘re-emergence mechanism’. Two simple auto-regressive equations for the summer and winter SST anomalies are obtained that can be easily solved. Model parameters include seasonal damping coefficients, mixed layer depths and standard deviations of the atmospheric forcing. Analytic expressions for season-to-season correlation and variability and power spectra are used to explore and illustrate the effects of the parameters quantitatively. Among the results it is found that, with regard to winter-to-winter temperature correlation, the re-emergence pathway is more influential than persistence via the summer mixed layer when the winter layer is more than twice the depth of the summer layer. With regard to winter temperature variability, the effect of a deeper winter mixed layer is to decrease the sensitivity to surface forcing and thus decrease variability, but also to increase persistence via re-emergence and thus increase variance at multidecadal scales.

How to Cite: Kowalski, P. and Davey, M., 2015. A simple model of ocean temperature re-emergence and variability. Tellus A: Dynamic Meteorology and Oceanography, 67(1), p.28651. DOI: http://doi.org/10.3402/tellusa.v67.28651
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  Published on 01 Dec 2015
 Accepted on 15 Oct 2015            Submitted on 26 May 2015

1. Introduction

Namias and Born (1970, 1974) described a tendency for sea surface temperature (SST) anomalies to recur from one winter to the next without persisting in the intervening summer in the North Pacific and North Atlantic oceans. They hypothesised that the nature of this recurrence is closely tied to the seasonal mixed layer cycle. In the winter, upper ocean temperature anomalies are created in a deep mixed layer and then sequestered below the mixed layer as it shoals in the following spring and summer, sheltered from the summer surface heat fluxes. The summer SST anomalies are altered by the summer surface heat fluxes, subsequently losing their relationship with SST anomalies formed at the end of the previous winter. When the mixed layer deepens in the following late autumn and early winter, portions of these preceding winter temperature anomalies are re-entrained into the winter mixed layer, subsequently impacting the SST. Alexander and Deser (1995) investigated this theory of Namias and Born further using observational data taken from ocean weather ships in the North Atlantic and North Pacific oceans, and established a significant statistical link between subsurface temperature anomalies and SST anomalies from preceding and subsequent winter seasons. They termed the theory of Namias and Born ‘the re-emergence mechanism’. The type of re-emergence investigated by Namias and Born (1970,1974) and Alexander and Deser (1995) is termed ‘local’; re-emergence occurs at the same location where SST anomalies were formed in the previous winter. Since the work of Alexander and Deser (1995), further evidence for local re-emergence in the North Atlantic (Watanabe and Kimoto, 2000; Timlin et al., 2002; Deser et al., 2003; Hanawa and Sugimoto, 2004) and North Pacific (Alexander et al., 1999; Deser et al., 2003; Hanawa and Sugimoto, 2004) has been obtained. More recently, Ciasto and Thompson (2009) have presented observational evidence for re-emergence in the extratropical Southern Hemisphere. The focus of the present study is local re-emergence.

The influence of re-emergence on mid-latitude SSTs is highly relevant to seasonal prediction. Rodwell and Folland (2002) demonstrated that through re-emergence a pre-season North Atlantic SST pattern is a significant predictor for the winter North Atlantic Oscillation (NAO) index, and this work was extended by Folland et al. (2012). The relation of late winter 2009/10 North Atlantic SST to early winter 2010/11 SST through re-emergence, and hence on the NAO, is described in detail in Taws et al. (2011).

A commonly used measure of local re-emergence is the auto-correlation function (ACF) of the observed local SST (Alexander et al., 1999; Watanabe and Kimoto, 2000; Timlin et al., 2002; Deser et al., 2003; De Coëtlogon and Frankignoul, 2003; Hanawa and Sugimoto, 2004). If the winter-to-preceding-winter value of the ACF is larger than the winter-to-preceding-summer value, then re-emergence is likely to be influencing the winter SST. Key factors that influence the magnitude of winter-to-preceding winter and winter-to-preceding summer values of the ACF are:

  • The size of the mean winter mixed layer depth (e.g. Timlin et al., 2002; Deser et al., 2003); shallower mean winter mixed layers have a smaller heat capacity and thus subsurface temperature anomalies are less likely to have an influence on the SST in subsequent winter seasons through the entrainment process. The statistical signature of the re-emergence mechanism is therefore stronger in oceans associated with large mean winter mixed layers, such as the North Atlantic (e.g. Deser et al., 2003).
  • The difference between the mean summer and winter mixed layer depths; re-emergence dominates the winter temperature in regions where the mean winter mixed layer is much larger than the mean summer mixed layer (Timlin et al., 2002; Hanawa and Sugimoto, 2004).
  • Atmospheric feedback, which controls the rate at which SST anomalies are damped by the overlying atmosphere; stronger feedback reduces the persistence of SST anomalies (Ciasto et al., 2010).
  • The size of the winter net surface heat flux variations; if these are large then winter SST variability will be dominated by these, with less re-emergence effects (Zhao and Li, 2012).

These basic factors and processes can be represented by the following simple bulk mixed-layer model introduced by Deser et al. (2003):

(1 )
hdTdt= Fρ0cp - κρ0cpT - H(dhdt)dhdt(T-Tb),

where ρ0 is the characteristic density of the ocean, cp the specific heat capacity at constant pressure, h(t) is a fixed seasonal mixed layer depth cycle, T′(t) the temperature anomaly (constant throughout the mixed layer), Tb(t) the temperature anomaly just below the mixed layer, κ(t) the atmospheric damping coefficient with a fixed seasonal cycle, and F′ the stochastic atmospheric forcing typically modelled as Gaussian white noise that varies interannually as well as within the seasonal cycle. The Heaviside step function H term is zero if the mixed layer is steady or shoaling and 1 if the mixed layer is deepening. Equation (1) can be viewed as an extension to the classical climate noise paradigm of Frankignoul and Hasselmann (1977). Deser et al. (2003) demonstrated that the simulated ACFs of the North Pacific and North Atlantic, which were calculated using model SST data from eq. (1), were favourable fits to the corresponding observed ACFs, and subsequently proposed that eq. (1) forms the basis for understanding the persistence of mid-latitude SST anomalies.

In this paper, a version of eq. (1) is presented, simplified to the point that statistical relations such as the ACF can be obtained analytically. In Section 2, the simple two-season stochastic model of the re-emergence mechanism is derived. In Sections 3 and 4, we investigate and quantify the effects of varying model parameters on the winter-to-winter temperature correlation and the winter temperature variance. In Section 5, the power spectrum of the winter temperature is obtained analytically in terms of model parameters, and explored. Summer-to-winter statistics are described in Section 6, and some measures of re-emergence are discussed in Section 7. Summer-to-summer statistics are discussed briefly in Section 8.

2. The stochastic two-season auto-regressive model

The key simplification is to represent the seasonal cycle by two six-month seasons, summer and winter (denoted by subscripts S and W, respectively) in each year i. Thus the sequence is winter i-1summer iwinter i ...  …. The mixed layer depths hS and hW remain constant within each season, so from eq. (1) the mixed layer temperature variations within each season are governed by

(2 )
dTSdt = QSρ0cphS - κSρ0cphSTS,
(3 )
dTWdt = QWρ0cphW - κWρ0cphWTW.

The damping coefficients κS and κW are taken as constant in each season. The heat fluxes QS and QW are also taken to be constant within each season, and as such they represent the net effect of fluxes that fluctuate throughout each season on shorter ‘weather’ timescales. Interannual variations of QS and QW are modelled as uncorrelated random variables, so future atmospheric conditions are independent of those in the preceding seasons. Formally,

(4 )
QW   σQWN(0,1),
(5 )
QS   σQSN(0,1),

where N(0,1) is a normal random variable with mean zero and unit standard deviation, and σQ W and σQ S are the standard deviations of the summer and winter atmospheric forcing, respectively.

2.1. Transition relations

Denote the years by a subscript i, and the summer and subsequent winter of year i by the subscripts Si and Wi, respectively. At the start of summer in year i, the initial temperature anomaly Tsi0 is set equal to the anomaly TWi−1 at the end of the previous winter:

(6 )
TSi0 = TWi-1.

The temperature anomaly TWi0 at the start of winter in year i is found by instantaneously mixing the summer mixed layer and the sequestered winter layer heat content:

(7 )
ρ0cphWTWi0 = ρ0cphSTSi+ρ0cp(hW-hS)TWi-1    ,

where TSi denotes the end-of-summer temperature anomaly. Thus

(8 )
TWi0 = rTSi + (1-r)TWi-1,

where

(9 )
r = hS/hW

is the ratio of the summer and winter mixed layer depths, with r≤1.

The term (1−r)TWi−1 contains the re-emergence mechanism, and to help monitor its effect in various circumstances we introduce a ‘process flag’ parameter γ in eq. (8), so

(10 )
TWi0 = rTSi + γ(1-r)TWi-1,

where 0≤γ≤1. Effectively the layer sequestered below the summer mixed layer emerges with a temperature anomaly reduced by the factor γ, and by setting γ=0 in later expressions the effect of re-emergence via persistence of anomalies in the sequestered layer can be removed.

Similarly, we introduce another process flag η in eq. (6) to monitor the contribution of preceding winter temperature anomalies that influence the following summer and winter by persisting in the summer mixed layer:

(11 )
TSi0 = ηTWi-1.

The season-to-season evolution is summarised in the schematic diagram in Fig. 1. Note that Schneider and Cornuelle (2005) introduced a similar two-season model that was integrated numerically to explore some re-emergence effects.

Fig. 1  

Schematic of the two-season model. Note that TWi-1,0 represents the temperature anomaly at the start of winter in year i−1.

2.2. Season-to-season relations

The duration of each season is Δt=0.5 yr. Equation (2) can be integrated over this time interval, using the transition relation [eq. (11)], to relate the end-of-summer state to the end-of-previous-winter state:

(12 )
TSi = fSηTWi-1 + (1-fS)QSi/κS,

where

(13 )
fS = exp{-ΔtκS/ρ0cphS}

measures the fraction by which temperature anomalies are attenuated through the summer season. Similarly, from eqs. (3) and (10),

(14 )
TWi = fWrTSi + fWγ(1-r) TWi-1 + (1-fW)QWi/κW,

where

(15 )
fW = exp{-ΔtκW/ρ0cphW}.

Thus, using eq. (12), the relation between the end-of-winter state to the end-of-previous-winter state is

(16 )
TWi=fW[ rηfS+γ(1-r) ] TWi-1+fWr(1-fS)QSi/κS + (1-fW)QWi/κW  

The interpretation of the terms appearing in eq. (12) is as follows:

  • fSηTWi-1 is the influence of preceding winter temperature anomalies on those at the end of summer.
  • (1-fS)QSi/κS is the influence of the summer atmospheric forcing on the temperature at the end of summer.

The interpretation of the terms appearing in eq. (16) is as follows:

  • fWrηfSTWi-1 represents the influence of preceding winter temperature anomalies that persist in the summer mixed layer, which survive after the entrainment process ends, on temperature anomalies at the end of the following winter.
  • fWγ(1-r)TWi-1 is the influence of re-emergence on temperature anomalies at the end of the following winter.
  • fWr(1-fS)QSi/κS measures the influence of the portion of the summer atmospheric forcing that survives after the entrainment process ends on temperature anomalies at the end of the following winter.
  • (1-fW)QWi/κW measures the influence of the winter atmospheric forcing on the temperature at the end of winter.

For simplicity the model is derived in terms of end-of-season values, but note that as the thermal forcing Q is constant within each season then the end-of-season temperature is also indicative of the season-average temperature and the model could be formulated in terms of seasonal averages.

Effectively the model is an auto-regressive system. For later reference, the winter-to-winter relation, eq. (16), is written as

(17 )
TWi = CTWi-1 + Ri,

where

(18 )
C = fW[ rηfS+γ(1-r)],

with 0≤C≤1, and

(19 )
Ri = fWr(1-fS)QSi/κS +(1-fW)QWi/κW,

is a net stochastic temperature contribution.

In particular, when η=γ=0 then C=0, the previous winter has no influence, and TW evolution reduces to a white noise process.

Analytic expressions for the winter-to-winter and summer-to-winter correlations, the variance of the winter and summer temperature, and the power spectrum of the winter and summer temperature can be derived using eqs. (12) and (16), as described in the Appendix.

In exploring the effects of various parameters, departures from a set of standard values will be considered. Typical North Atlantic values of the damping parameters are κS=10Wm−2K−1 and κW=25Wm−2K−1 (e.g. Frankignoul et al., 1998; Deser et al., 2003). The summer mixed layer depth is fixed as hS=25 m. The selected value for the standard deviation of the winter atmospheric forcing is σQW=20Wm-2, and for summer σQS=10Wm-2. For reference, model variables, parameters and standard values are summarised in Table 1.

The fraction fS decreases as the damping κS increases. To quantify this effect, this dependence is shown in Fig. 2a: fS is below 0.1 when κS is above about 15Wm−2K−1. Likewise, as illustrated in Fig. 2b, fW decreases as κW decreases, but increases as hW increases.

Fig. 2  

(a) Dependence of the summer attenuation factor fS on the damping rate κS, with hS=25 m; (b) dependence of the winter attenuation factor fW on damping rate κW and depth hW.

3. Analysis of the winter-to-winter correlation

As derived in the Appendix, the winter-to-winter correlation C is

(20 )
C = Corr(TW,TW-1) = fW[rηfS + γ(1-r)],

where

  • ηfWrfS represents the influence of preceding winter temperature anomalies that persist in the summer mixed layer,
  • σR2 is the influence of re-emergence on the winter-to-winter persistence of temperature anomalies.

Note that Corr(TW, TW−1) is the correlation found for end-of-winter values and it is independent of σQW and σQS. For end-of-winter values this property that the correlation does not depend on the stochastic forcing can also be proven for eq. (1), by considering the history of sub-mixed-layer temperatures that are created and entrained each year.

In this section, we set η=γ=1 and investigate the dependence of Corr(TW, TW−1) on variations in κS, κW, and hW, with hS fixed to the standard value.

3.1. The impact of varying κS and κW on Corr(TW, TW−1)

Figure 3a shows C with κW fixed and varying hW and κS. (For reference, the black squares on this and subsequent diagrams indicate the standard values. Values of various statistics for standard values are provided in Table 2.) For large κSfS1) the preceding winter anomalies that influence the summer layer have negligible influence through to winter, and CfW(1−r). For small κS (fS≈1) the effect on C is weak. The winter depth hW has a much larger influence on C: although C is less than 0.1 for hW less than about 50 m, the correlation exceeds 0.5 for hW greater than about 150 m when the re-emergence mechanism has a dominant influence.

Fig. 3  

Winter-to-winter correlation Corr(TW, TW−1): (a) dependence on summer damping rate κS and winter depth hW, (b) dependence on winter damping rate κW and depth hW.

In Fig. 3b κS is fixed while κW and hW vary. Comparing the pattern of Fig. 3b with that of Fig. 2b, it is evident that C is strongly influenced by the attenuation factor fW. Correlations are high for large hW and small κW (e.g. larger than 0.8 when κW is less than about 10 Wm−2K−1 and hW larger than 250 m), when a relatively large heat content is sequestered for re-emergence.

It is interesting to compare the winter-to-winter correlation with the value when r=1. Let C1 denote the winter-to-winter correlation when r=1. From (20),

(21 )
C1 = ηfW1fS,

where fW1 is the value of fW when r=1. Then

(22 )
C - C1 = ηfS(rfW-fW1) + γfW(1-r),

where ηfS(rfW-fW1) represents the contribution of persistence via the summer mixed layer. When η=γ=1 it is straightforward to prove that C>C1 when hW>hS. Since fW1<fW for all hW>hS, and fS<1,

(23 )
C1 = fW1fS<fWfS = rfWfS+(1-r)fWfS < fW[ rfS+1-r ] = C,

which concludes the proof. The term rfW appears often in the properties of the model, and for reference it is illustrated in Fig. 4 for a range of values of hW and κW. As a function of hW this term has a maximum at a depth hW=ΔtκW/ρ0cp. When κW is less than about 7 Wm−2K−1 that depth is less than hS, and in Fig. 4rfW decreases as hW increases. For larger κW, rfW increases to a maximum and then decreases as hW increases. The line with rfW=fW1 is also included in Fig. 4. Below this line persistence increases CC1, but above the line persistence decreases CC1.

Fig. 4  

Dependence of rfW on the damping rate κW and depth hW. The thick line indicates where, for each κW, rfW=fW1.

This behaviour occurs due to the competing effects of hW: increasing the winter mixed layer depth reduces the relative contribution of preceding winter temperature anomalies via persistence, but also reduces the rate at which they are damped through winter.

The relative effects of re-emergence and persistence on the winter-to-winter correlation as hW varies can be compared. From eq. (20), with η=γ=1 , the former is larger than the latter when (1−r)>rfS. This condition (which is independent of κW) can be re-written as hW>(1+fS)hS, and as 0<fS<1, it follows that re-emergence always has the larger influence when hW>2hS.

To quantify the relative effects, the ratio rfS/(1−r) is shown in Fig. 5 for varying hW and κS. The ratio rapidly decreases as hW increases, the more so as κS increases. For the standard value κS=10Wm−2K−1 the ratio is 1 for hW≈30 m, but less than 0.2 when hW >52 m. Unless the seasonal range of mixed layer depth is small, re-emergence has a much larger influence on the winter-to-winter persistence of temperature anomalies than that of preceding winter temperature anomalies that persist through the summer mixed layer.

Fig. 5  

Dependence of rfS/(1−r) on summer damping rate κS and winter depth hW.

4. Analysis of the winter temperature variance

As derived in the Appendix, the winter temperature variance σTW2 is

(24 )
σTW2 = σR2/(1-C2),

where

(25 )
σR2 = σRS2 + σRW2,

is determined by the random stochastic forcing, with

(26 )
σRS2= r2fW2(1-fS)2(σQS2/κS2),σRW2= (1-fW)2(σQW2/κW2).

Overall the magnitude of σTW2 is determined by σR2, modified by the effect of C. (Note that σR does not depend on the process parameters η and γ.) When η=γ=0 then C=0, TW is a white noise process, and σTW=σR. When preceding winter has an influence, then C>0 and σTW is amplified above σR.

Both C and σR depend on several model parameters, and in this section the effect of parameter variations on σTW and its components is explored and quantified. For this purpose it is convenient to rewrite eq. (24) as

(27 )
σTW2 = σR2 + σP2,

where

(28 )
σP2 = σR2 C2/(1-C2),

contains the influence of preceding winters in the process. The fraction of variance associated with preceding winters is σP2/σTW2=C2, and is the fraction that would be predictable from preceding winter information using a linear regression approach based on eq. (17). Furthermore, the fraction of the variance due to random forcing alone is σR2/σTW2=1-C2, which is independent of the summer and winter atmospheric variability. When C2>0.5, σP2 makes a larger contribution to σTW2 than the random component σR2.

4.1. The impact of varying κW and hW on σTW2

Figure 6 illustrates the effect of varying hW and κW on the winter variance, with other parameters set to standard values. As shown in Fig. 6a, σTW2 is largest when hW=hS and κW=0. (Note that as κW0 then (1-fW)/κWΔt/ρcphW and thus remains finite.) As expected, σTW2 decreases as damping κW increases. For fixed κW, σTW2 decreases as hW increases, because the increased heat capacity of the deeper winter layer means less temperature change for the same heat input.

Fig. 6  

Dependence of the winter variance σTW2 on damping rate κW and depth hW.

The region with C=0.7 in Fig. 3b indicates approximately when the contribution to σTW2 from σP2 is greater than that of σR2 (i.e. when C2>0.5). For κW0 this occurs when hW is greater than about 70 m, and occurs at larger hW as κW increases. For all κW, when hW is very close to hS, σTW2σR2 and when hW is close to 500 m, σTW2σP2.

The winter and summer components σRS2 and σRW2 are plotted similarly in Fig. 7. (The ‘summer’ component depends on κW because the anomalies imposed in the summer season are attenuated through the following winter.) For the ranges of values shown σRW2 (Fig. 7a) decreases as κW and hW increase, and is much larger than σRS2 (Fig. 7b). σRS2 is negligible for all hW because when hW is close to hS anomalies forced in the preceding summer are relatively strongly damped in a shallow winter mixed layer, whereas for larger hW entrainment acts to significantly reduce their influence. Note that for κW above about 6 Wm−2K−1, σRS2 increases at first as hW increases from hS, then decreases: this is due to the effect of the factor rfW as described in Section 3.1.

Fig. 7  

Winter variance components associated with the random forcing. (a) dependence of σRW2 on winter damping rate κW and depth hW, (b) likewise for σRS2.

The relative effects on σTW of the near-surface and sequestered pathways for winter-to-winter connections are explored by plotting σP2 for γ=1, η=0 (Fig. 8a, sequestered path only) and for γ=0, η=1 (Fig. 8b, near-surface path only). Except for depths hW close to hS, the sequestered path has a much greater effect. Note that the behaviour of σP2 with hW when γ=0 and η=1 (Fig. 8b) is similar to that which was described for σRS2, with the effect of the term rfW again evident. It is also interesting to note that when γ=1, η=0, σP2 increases as hW increases from hS, and then decreases. This is linked to the effects of decreasing the winter mixed layer depth on the effects of the atmospheric forcing and re-emergence: decreasing (increasing) the winter mixed layer increases (decreases) the size of the temperature anomalies via the atmospheric forcing, which acts to increase (decrease) the effects of re-emergence on the temperature in the following winter.

Fig. 8  

Dependence of the predictable component of winter variance σP2 on winter damping rate κW and depth hW: (a) process flags γ=1, η=0, (b) γ=0, η=1.

4.2. The impact of varying κS and σQW on σTW2

As shown in Fig. 9a, σTW2 varies little as the summer damping coefficient κS varies. The apparent greater sensitivity for larger hW is due to the substantially reduced values of σTW2 for larger hW.

Fig. 9  

Winter variance σTW2 (a) dependence on summer damping κS and winter depth hW, (b) dependence on winter random forcing σQW2 and depth hW.

Figure 9b quantifies the response of σTW2 to σQW. As expected, increasing the winter forcing σQW increases σTW2 (roughly quadratically), by increasing present winter and previous winter temperature variances, with less sensitivity for larger hW.

5. The power spectrum of the winter temperature

From the winter-to-winter relation in eq. (17), the power spectrum of the winter temperature, PW(ω), can be derived. Equation (A25) gives

(29 )
PW(ω) = σR2 GW(ω),

where

(30 )
GW(ω) = 1/[ 1 - 2Ccos(2πω) + C2].

is the shape function that depends only on the winter-to-winter correlation C, and frequency ω[0,0.5] corresponds to periods from 2 yr upwards. Preceding winter conditions act to decrease power for short (interannual) periods, and increase power at long periods, with the crossover at GW=1 when cos(2πω)=C/2. For standard values, the crossover occurs at a period of 5 yr.

5.1. The effect of re-emergence and preceding winter temperature anomalies that persist in the summer mixed layer on PW

The expression for the power spectrum of the winter temperature enables us to establish the influence of re-emergence, of preceding winter temperature anomalies that persist in the summer mixed layer, and of summer atmospheric forcing, for a range of timescales. Various winter spectra are illustrated in Fig. 10, using standard values.

Fig. 10  

Power spectrum PW(ω) of winter temperature anomalies for standard parameter values and various combinations of process flags. Solid line γ=0, η=0; dashed line γ=0, η=1; thick line γ=1, η=0. Note that the thin solid and dashed lines nearly coincide.

When γ=η=0, GW(ω)=1 for all ω, and eq. (29) reduces to PW(ω) = σR2. When there are no effects of re-emergence and preceding winter temperatures that persist in summer the power spectrum is flat, as shown by the thin black line in Fig. 10.

When η=0 and γ=1, C=fW(1−r) in eq. (30) and only the effects of re-emergence influence PW. This case is shown by the thick line in Fig. 10. The shape factor has GW(0)=6.7, GW(0.5)=0.4.

For γ=0 and η=1, C=rfWf S , and PW is only influenced by preceding winter temperature anomalies that persist in the summer mixed layer. This case is illustrated by the broken line in Fig. 10: the effect of persistence on PW is much weaker than that of re-emergence, as evident in the shape factor values GW(0)=1.03, GW(0.5)=0.97.

With persistence and re-emergence processes included (η=γ=1), for standard values the spectrum is very similar to that with re-emergence only. The graph for this case is included in the parameter comparisons shown in Fig. 11. Note that re-emergence reddens the winter temperature spectrum.

Fig. 11  

Power spectrum PW(ω) of winter temperature anomalies. In each case the thin line is PW(ω) for standard values, the thick line for parameter variations. (a) winter damping κW increased to 40 Wm−2K−1, (b) summer damping κS increased to 40 Wm−2K−1, (c) winter depth hW doubled to 500 m, (d) winter random forcing σQW doubled to 40 Wm−2.

Schneider and Cornuelle (2005) described spectra from numerical integrations with a similar two-season model, in which re-emergence increased the spectral power at interannual timescales but not at longer timescales. One reason for the contrast with our result is the experimental design. They compare spectra from an integration with a constant deep (winter) mixed layer with that from an integration with deep winter and shallow summer layers, whereas in our experiments there is always a deep winter and shallow summer layer and spectral comparisons are made by varying the ‘process flags’ and parameters. In their comparison, decreasing the summer mixed layer depth increases the variability of the mixed layer temperature in summer, which results in an increase in the spectral power of the mixed layer temperature at interannual and shorter timescales. A further difference is the throughout-season data sampling in Schneider and Cornuelle (2005) versus the end-of-season sampling in our results. An increase in spectral power at decadal timescales was also found in the study with idealised models by De Coëtlogon and Frankignoul (2003), in which they compared spectra from an integration with a constant e-folding scale of 3 months and an integration with the addition of a simple re-emergence term in winter.

5.2. The effect of varying κS, κW and hW on PW

In this section, the effect of varying κS, κW and hW on PW is investigated. Throughout this section, we set γ=η=1, and the reference case (represented by the thin lines in Fig. 11) uses standard values.

The thick line in Fig. 11a shows PW when the winter atmospheric damping κW is increased to 40Wm−2K−1. Increasing κW reduces σR2, and also decreases C with the effect of flattening the shape of the spectrum. At interannual timescales these effects offset each other, and in this example the net result is very small [PW (0.5) reduces from 0.025 to 0.024], whereas at decadal timescales the effects reinforce and the power is more than halved for PW (0).

The thick line in Fig. 11b shows PW when the summer atmospheric damping is increased to 40Wm−2K−1. The system is less sensitive to κS, and in this case the power is reduced slightly.

The thick line in Fig. 11c shows PW when the winter mixed layer depth is doubled to 500m. The term σR2 is more than halved, but C is increased so the shape factor is steepened. The effects offset at long timescales, and the result in this case is a slight reduction of PW(0) from 0.49 to 0.48. The effects re-inforce at interannual scales, and PW(0.5) is reduced by about 75%.

Figure 11d shows how doubling σQW (thick line) acts to increase the winter temperature variability at all timescales, by increasing σR2 without affecting GW(ω).

6. Analysis of variances and the summer-to-winter correlation

A measure of the influence of re-emergence is the relative values of winter-to-winter correlation and summer-to-winter correlation. This involves in part the relative variances of summer and winter temperature anomalies, which are themselves of interest. Analytic expressions for these quantities are presented and analysed in this section.

The ratio of the summer and winter standard deviations of the temperature σTS/σTW is denoted α. Expressions for the variances σTS2 and σTW2 are derived in the Appendix. Note that these are related by

(31 )
σTS2 = fS2η2σTW2+(1-fS)2σQS2/κS2.

The expressions in the Appendix lead to

(32 )
α2= fS2η2 +(1-C2)r2fW2 + (σQW2/σQS2)(κS2/κW2)(1-fW)2/(1-fS)2

Note that when the ‘process flags’ η and γ are zero (so winter and summer are disconnected from the conditions in the previous winter, and C=0) the expression reduces to

(33 )
α2 = 1r2fW2 + (σQW2/σQS2)(κS2/κW2)(1-fW)2/(1-fS)2.

When re-emergence is activated by setting γ=1 then C increases and α decreases, so re-emergence decreases the ratio of σTS to σTW.

The covariance of summer and following winter anomalies (see Appendix A.2.3) can be written

(34 )
Cov(TW,TS)= fWfSrη2σTW2 + fWfSηγ(1-r)σTW2 +fW(1-fS)2rσQS2/κS2.

The first term is due to the previous winter influencing the summer which in turn influences the following winter; the second term is due to the previous winter influencing the following winter through re-emergence; and the third term is due to the summer forcing of summer anomalies that influence the following winter. The first and third terms can be combined to obtain

(35 )
Cov(TW,TS) = fWrσTS2 + fWfSηγ(1-r)σTW2,

from which it follows that the summer-to-following-winter correlation is

(36 )
Corr(TW,TS) = fWrα + fWfSηγ(1-r)/α.

The terms in eq. (36) are interpreted as follows:

  • fW represents the influence of summer temperature anomalies (due to both summer forcing and previous winter persistence) on those in the following winter.
  • fWfSηγ(1-r)/α is a contribution due to the influence of preceding winter temperature anomalies on TW through re-emergence. Note that the process flag η also appears here: when η=0 re-emergence still occurs, but the re-emerging anomalies have no correlation with TS as TS is determined only by QS when η=0.

Thus Corr(TW, TS) is not just a measure of the impact of summer temperature anomalies on those in the following winter.

6.1. The impact of varying κW and hW on Corr(TW, TS) and α

The effect of varying κW and hW, with other parameters set to standard values and η=γ=1, is described here. The effect on the summer-to-following-winter correlation Corr(TW, TS) is illustrated in Fig. 12a. As expected, for fixed hW the correlation decreases as the winter damping κW increases. The correlation is small for hW close to hS except when winter damping is small: when winter depths are small the anomalies induced by the random winter forcing dominate the influence of previous seasons. The correlation then increases as hW increases, then decreases: it is largest (over 0.4) for small κW and for hW about 75 m. For the standard value κW=25Wm−2K−1 correlation exceeds 0.2 for hW ranging from 100 to 400 m. Comparing the pattern in Fig. 3b with that of Fig. 12a, it is clear that Corr(TW, TS) is not as strongly influenced by variations in hW as Corr(TW, TW−1).

Fig. 12  

Dependence of summer and winter relations on the winter damping rate κW and depth hW (a) the summer-to-winter correlation Corr(TW, TS), (b) the ratio α=σTS/σTW, (c) the component fW of Corr(TW, TS), (d) the component fWfS(1−r)/α of Corr(TW, TS).

As shown in Fig. 12b, the ratio σTS/σTW increases as κW increases. As σTW decreases as winter damping increases, it is evident from eq. (31) that σTS also decreases but α increases as κW increases. Likewise the ratio also increases as hW increases, because σTW decreases. For the parameter values used, the ratio is larger than 1 when hW is larger than about 200 m when κW is small.

The contributions to the correlation from the two terms in eq. (35) are provided in Fig. 12c and d. In Fig. 12c the pattern is again linked to that of rfW described in Section 3.1. For the ‘re-emergence’ term in Fig. 12d, this contribution is largest for small κW, with a maximum at around hW=100 m for small κW. (The maximum is a result of the trade-off between increasing (1−r)fW and decreasing 1/α as hW increases. As hW increases, the amount of re-emerging water increases but the variance of its temperature decreases. This feature influences the occurrence and location of the maximum in correlation in Fig. 12a.) The two terms are similar in size: persistence of anomalies in surface layers and re-emergence of sub-surface information are both influential in the overall correlation between summer and following winter temperature anomalies. For the standard winter damping value κW=25Wm−2K−1 re-emergence is less influential than the other term.

6.2. The impact of varying κS and hW on Corr(TW, TS) and α

Similarly the effect of varying κS and hW is illustrated in Fig. 13. In Fig. 13a it can be seen that Corr(TW, TS) decreases as κS increases and summer anomalies are reduced. For hW close to hS the correlation is small (as in Fig. 12a). As hW increases from hS the correlation increases, and then weakly decreases for hW larger than about 200 m. For low κS the correlation exceeds 0.4 for hW between about 125 and 375 m. Comparing Fig. 3a with Fig. 13a, it is clear that Corr(TW, TS) is more sensitive to variations in κS than Corr(TW, TW−1).

Fig. 13  

Dependence of summer and winter relations on the summer damping rate κS and winter depth hW (a) the summer-to-winter correlation Corr(TW, TS), (b) the ratio α=σTS/σTW, (c) the component fW of Corr(TW, TS), (d) the component fWfS(1−r)/α of Corr(TW, TS).

The ratio α, shown in Fig. 13b, decreases as κS increases: while increasing the summer damping reduces both summer and winter variances, the more direct effect on the summer variance is greater. Similar to Fig. 12b, for fixed κS the ratio increases as hW increases and winter variances decrease. Small κS favours larger summer variance, and α is largest for low κS and large hW.

The components of the correlation are provided in Fig. 13c and d. For small fixed κS the term fW in Fig. 13c has a maximum at hW about 200 m. This contrast to the pattern in Fig. 12c occurs because α now increases as hW increases. Both terms have similar behaviour as κS and hW vary, with fW generally more than twice the re-emergence contribution.

6.3. The impact of varying σQW on Corr(TW, TS) and α

Changing the winter forcing standard deviation σQW changes the winter temperature variance correspondingly. The effect on Corr(TW, TS) and α is explored here by varying σQW and hW with other parameters set to their default values. (Note that the default for σQS is 10Wm−2, the default for σQW is 20 Wm−2, and σQW ranges from 5 to 90 Wm−2 in the results illustrated.)

Figure 14a shows Corr(TW, TS). For small σQW the random forcing of winter anomalies is weak and anomalies from the previous summer can have a stronger influence: thus the largest values in Fig. 14a occur with σQW at the low end of the range, reaching about 0.5 when hW is in the range 100–250 m. As σQW increases from 5 Wm−2 the correlations decrease at first, but then increase again for σQW larger than 40 Wm−2. The reason is that the re-emergence contribution to the correlation increases as σQW increases and winter variance increases. This is clear from the two contributions to the correlation mapped in Fig. 14c and d: for small σQWfW in Fig. 14c dominates, while for large σQWfWfS(1-r)/α dominates.

Fig. 14  

Dependence of summer and winter relations on the winter forcing σQW and depth hW (a) the summer-to-winter correlation Corr(TW, TS), (b) the ratio α=σTS/σTW , (c) the component fW of Corr(TW, TS), (d) the component fWfS(1−r)/α of Corr(TW, TS).

This behaviour is related to the effect of σQW on α shown in Fig. 14b. Decreasing σQW decreases both σTW and σTS, but the effect is relatively larger for σTW. Consequently the ratio α increases markedly as σQW decreases below about 20 Wm−2, particularly for larger hW. Increasing σQW above the default value of 20 Wm−2 has a weak decreasing effect on α.

For hW close to hS the correlation is weak for all σQW in the example.

7. Measures of the re-emergence signal

In previous studies, such as Timlin et al. (2002) and Deser et al. (2003), which show that the effect of summer SSTs on those in the following winter is weaker than that of preceding winter temperature anomalies, the winter-to-preceding winter value of the SST ACF is substantially larger than the winter-to-preceding summer value. The re-emergence signal can therefore be characterised by the ratio

R = Corr(TW,TW-1)/Corr(TW,TS),

which can be expressed analytically using eqs. (20) and (36):

(37 )
R = [ ηrfS+γ(1-r) ] / [ rα+γη(1-r)fSα-1 ].

Thus, summer temperature anomalies are having a relatively weak impact on the winter-to-winter persistence of temperature anomalies if R1 and vice versa if R is small.

As was shown in the previous section, Corr(TW, TS) includes a re-emergence component and overestimates the direct impact of summer temperature anomalies on those in the following winter. An alternative that can be assessed in the two-season formulation (but is more difficult to calculate from observations) is to use the correlation between winter temperature and the summer temperatures produced by the random atmospheric forcing, which is the same as the correlation Corr(TW, QS), as a measure of the direct summer-to-winter relation. The alternative ratio is

(38 )
R* = Corr(TW,TW-1)/Corr(TW,QS).

From eq. (16)

(39 )
Cov(TW,QS) = fWr(1-fS)σQS2/κS.

Making use of the expression for σTW2 in eq. (A6), the analytic expression for Corr(TW, QS) is

(40 )
Corr(TW,QS) = fWrα*,

where [cf. eq. (32)]

(41 )
α*2 = (1-C2)r2fW2 + (σQW2/σQS2)(κS2/κW2)(1-fW)2/(1-fS)2.

Thus

(42 )
R* = [ ηrfS+γ(1-r) ]/rα*.

For standard values, when r=1 and re-emergence has no role R and R* have similar values of about 0.2. R and R* both increase as hW increases, with R* larger than R: for standard values, when hW=500 m R is about 4, R* about 6.

7.1. The response of R and R* to varying κW, κS and σQW

Unless otherwise stated, parameters have their default values and η=γ=1. Figure 15a and b show R and R* when κW and hW are varied and other parameters have their default values. R and R* have similar relatively low values for hW close to hS, and increase as hW increases. R is not very sensitive to κW, whereas R* increases more rapidly with depth when κW is small. For small winter damping κW winter temperature variance is relatively large and re-emergence has a stronger effect, and this influence is emphasised in R*.

Fig. 15  

Parameter dependence of the correlation ratios R=Corr(TW,TW-1)/Corr(TW,TS) and R*=Corr(TW,TW-1)/Corr(TW,QS). (a) R and (b) R* dependence on κW and hW; (c) R and (d) R* dependence on κS and hW; (e) R and (f) R* dependence on σQW and hW.

As seen in Fig. 15c and d the effect of varying summer damping κS is very similar for R and R*. In this example the largest values are found for large hW and large κS, because summer temperature anomalies are strongly damped by large κS and re-emergence again has a stronger effect.

The effect of varying winter forcing σQW is illustrated in Fig. 15e and f. Differences between R and R* are most evident for larger σQW. For σQW larger than 40Wm−2, R decreases but R* increases markedly as σQW increases. This occurs because winter temperature variance increases as σQW increases: the re-emergence component maintains Corr(TW, TS) in R (cf. Fig. 14a), while Corr(TW, QS) decreases in R*.

8. Statistics for the summer temperature

8.1. The summer-to-summer correlation

As derived in the Appendix, the summer-to-summer correlation is

(43 )
Corr(TS,TS-1) = ηfSCorr(TW,TS)/α.

When η=0, Corr(TS, TS−1)=0, that is, preceding summer temperatures cannot influence the summer temperature if winter temperatures do not influence the summer temperature. (a), (b) and (c) of Fig. 16 show how Corr(TS, TS−1) varies with hW, κW, κS and σQW with other parameters in each figure set to their default values. It is clear that, as expected, preceding summer temperatures have little influence on those in the following summer for the ranges of parameters considered here.

Fig. 16  

Summer-to-summer correlation Corr(TS,TS−1). (a) dependence on winter damping rate κW and depth hW, (b) dependence on summer damping rate κS and winter depth hW, (c) dependence on winter forcing σQW and depth hW.

8.2. The power spectrum of the summer temperature

As derived in the Appendix, the power spectrum of the summer temperature is

(44 )
PS(ω) = σTS2GS(ω),

where

(45 )
GS(ω) = 1-A + A(1-C2)/[1-2Ccos(2πω)+C2],

and

(46 )
A = ηfSCorr(TW,TS)/αC.

When η=0 successive summers are uncorrelated and PS(ω)=σTS2 .

Figure 17 shows PS(ω) for standard values (thin line) and for some parameter variations (cf. the winter spectra in Fig. 11). For standard values the spectrum is weakly red. Increasing the winter damping rate κW to 40Wm−2K−1 reduces the winter temperature anomalies that persist into summer, flattening the spectrum (Fig. 17a). Increasing the summer damping rate κS to 40Wm−2K−1 reduces the summer variance considerably (Fig. 17b). Doubling the winter depth hW increases the power at interannual scales and reduces it at decadal scales (Fig. 17c). Doubling the winter forcing σQW increases the power slightly, more so at low frequencies (Fig. 17d).

Fig. 17  

Power spectrum PS(ω) of summer temperature anomalies. In each case the thin line is PS(ω) for standard values, the thick line for parameter variations. (a) winter damping kW increased to 40 Wm−2K−1, (b) summer damping κS increased to 40 Wm−2K−1, (c) winter depth hW doubled to 500 m, (d) winter random forcing σQW doubled to 40 Wm−2.

9. Discussion

In the mid to high latitude oceans the seasonal variability of SST is influenced by the re-emergence process, by which upper ocean temperature anomalies sequestered beneath the shallow summer mixed layer are mixed into the deeper winter mixed layer. The extent of this influence depends on factors such as the relative depth of the mixed layers and the strength of surface heat fluxes. The purpose of this article is to describe a novel idealised model aimed at exploring the effects of several factors. The main simplifying assumptions are the restriction to two seasons in the year, fixed mixed layer depths in the ‘summer’ and ‘winter’ seasons, and surface fluxes with a fixed forcing component within each season (varying stochastically from season to season) and a linear damping component. The strength of the model is that its simplicity allows analytic expressions to be derived for statistical properties such as seasonal temperature variance and season-to-season correlations. The main variables are end-of-season temperature anomalies: at the expense of extra algebraic complexity, the model could also be written in terms of seasonal-average anomalies, with similar qualitative behaviour.

The formulation of the model (Section 2) includes two ‘process flags’. The ‘re-emergence’ flag γ controls the subsurface temperature anomaly that influences the following winter, and the ‘persistence’ flag η controls the winter temperature anomaly that influences the following summer. These flags allow the roles of the respective processes to be traced in the derivation and interpretation of the analytic expressions. The parameters in the model are the summer and winter mixed layer depths hS and hW, the summer and winter damping rates κS and κW, the standard deviations of the summer and winter forcing σQS and σQW.

A set of standard values for the model parameters is provided in Table 1, representative of a mid-latitude ocean location, and select corresponding statistical values can be found in Table 2. The effects of parameter variations are described in Sections 3–8.

As derived in Section 2 and the Appendix, a particularly simple expression is obtained for the correlation C of end-of-winter temperature anomalies from one winter to the next:

(47 )
C = fW[ rηfS+γ(1-r) ],

where r is the depth ratio hS/hW, and fW and fS are expressions for the attenuation of anomalies through winter and summer, respectively, through damping effects (tending to zero for strong damping and 1 for weak damping). Note that C does not depend on the forcing terms. When flags η and γ are zero the anomalies each season are independent of those preceding, and C=0. When the flag γ is zero andη is unity then re-emergence is ‘off’, but C is positive due to persistence effects. When γ is also unity then re-emergence increases C. It can be deduced that the re-emergence contribution to C is larger when hW>(1+fS)hS, which is always true when hW>2hS. The dependence of C on damping and on hW is discussed in Section 3, with the tendency for larger C with larger hW being the dominant feature (see Fig. 3). Stronger winter damping and stronger re-emergence through deeper hW have competing effects, manifest in the parameter combination rfW illustrated in Fig. 4.

The equation for C also leads to a simple analytic expression for multi-year lag correlations and hence for the winter power spectrum, as described in Section 5. When C=0 (η=γ=0) the spectrum is white, with amplitude depending on a combination of the winter and summer forcing. For standard parameter values, activating persistence (η=1) has little effect, producing a slightly red spectrum, whereas activating re-emergence (γ=1) has a large effect, as shown in Fig. 10. Some effects of parameter variations on the winter spectrum are illustrated in Fig. 11.

The winter variance and its parameter dependence are discussed in Section 4. The variance decreases as hW increases, because winter surface forcing is spread over a large depth and resulting anomalies are smaller, and as damping increases. It can be regarded as having random and predictable components, with end-of-winter temperature anomaly as the predictor for the next winter and C2 as a measure of the predictable fraction. Re-emergence is the dominant process contributing to predictability, unless there is little difference between winter and summer depths. The amplitude of the predictable variance does not have a simple dependence on hW: there is an optimal depth, because increasing hW increases the influence of re-emergence but reduces the variance size.

Summer temperature variance and summer-to-winter correlations Corr(TW, TS) are described in Section 6. The ratio of summer to winter variance plays a role in the correlation. The ratio is increased by increasing hW (because winter variance is reduced), but decreased by re-emergence. The summer-to-winter correlation contains a contribution from conditions in the previous winter, because through persistence and re-emergence those conditions influence both following summer and winter conditions. Thus Corr(TW, TS) is not just a measure of direct summer influence on the following winter, but contains an indirect component, as illustrated in Fig. 14. The implications of this for defining a measure of the re-emergence signal in terms of winter-to-winter and summer-to-winter correlations are discussed in Section 7. Although season-to-season temperature correlations are relatively easy to estimate from temperature observations, some care is needed in interpreting the results.

To complete the description of the analytic properties of the simple two-season model, the summer-to-summer correlations and summer power spectrum are described in Section 8. The summer spectrum is relatively insensitive to parameter variations, with the exception of varying the summer forcing by which it is largely determined.

The model, however, neglects several important factors. As shown by Deser et al. (2003) and Frankignoul (1985), interannual mixed layer depth variability alters the entrainment rate, which influences the persistence of SST anomalies and the effects of re-emergence. Convective instability, which occurs when the temperature anomaly in the winter mixed layer is colder than that which resides just below the mixed layer can alter the upper ocean thermal structure, and subsequently the mixed layer depth. In the two-season model, entrainment occurs each year at the same depth. Similarly, the temperature anomaly at the start of winter can alter the mixed layer depth in the following winter. Interannual variability in the atmospheric damping may also impact re-emergence. Sura et al. (2006) showed that extending the model of Frankignoul and Hasselmann (1977) to include anomalous atmospheric feedback introduces an extra multiplicative noise term, which significantly enhances the overall stochastic forcing and produces a non-Gaussian probability density function of the winter SST similar to that which is found in observations. In the two-season model, the probability density function of the winter temperature is Gaussian. There are also vertical processes such as those associated with permanent thermocline variations induced by the first mode baroclinic Rossby wave (Zhang and Wu, 2010; Schneider and Miller, 2001); strong subduction (De Coëtlogon and Frankignoul, 2003); and non-local effects such as horizontal advection (Jin, 1997; Ostrovskii and Piterbarg, 2000) and remote ENSO forcing (Park et al., 2006) that influence mid-latitude temperature variability. The two-season model could be extended to include these factors and their effect together with re-emergence on mixed layer temperature investigated.

To summarise, the two-season approach provides a simple model of the effects of persistence and re-emergence, with parameters for layer depths, damping and forcing, in a stochastic forcing framework. The simplicity allows explicit analytic expressions to be obtained for the key properties of variance and correlation and power spectrum. Work is in progress on investigating the key results regarding for example temperature variance as a function of summer to winter mixed layer depth ratio, using ocean analysis datasets.

Acknowledgements

Part of this work was carried out while Peter Kowalski was at University College London, supported by a Co-operative Award in Science and Engineering studentship from the Engineering and Physical Sciences Research Council. Michael Davey was partly supported by the Joint UK DECC/Defra Met Office Hadley Centre Climate Programme (GA01101).

11. Appendix: Derivation of the analytic expressions

A.1. Notation

Let <x> denote the average of variable xi over a large sample. (Large means many times the damping timescale, which for standard parameters corresponds to several decades.) As <QS>=<QW>=0 in the damped two-season system, it follows from averaging the regression relations that <TS>=<TW>=0. The summer and winter temperature anomaly variances are σTS2=<TS2> and σTW2=<TW2>. The covariance between winter and previous summer is denoted

(A1 )
Cov(TW,TS)=<TWTS>,

and the lagged covariance between winter and winter j years previously is denoted

(A2 )
Cov(TW,TW-j)=<TWTW-j>.

The correlation is denoted, for example,

(A3 )
Corr(TW,TS)=Cov(TW,TS)/σTWσTS.

Note that as the stochastic atmospheric forcing Q is independent of preceding temperatures then, for example,

(A4 )
Cov(QS,TW-1)=0,Cov(QW,TS)=0.

A.2. Correlations and variances

A.2.1. Winter-to-winter correlation and winter temperature variance

From the winter-to-winter autoregression, eq. (17), it is straightforward to deduce that

(A5 )
Corr(TW,TW-1) = C,

and

(A6 )
σTW2 = σR2/(1-C2),

where

(A7 )
σR2 = r2fW2(1-fS)2(σQS2/κS2) + (1-fW)2(σQW2/κW2).

and C is defined in eq. (18).

A.2.2. Summer temperature variance

Multiplying eq. (12) by TSi and then taking the ensemble average yields

(A8 )
σTS2 = fSηCov(TS,TW-1) + (1-fS)Cov(TS,QS)/κS.

Multiplying eq. (12) by TWi-1, and using eq. (A4) gives

(A9 )
Cov(TS,TW-1) = fSησTW2.

Similarly, it can be shown using eq. (16) that

(A10 )
Cov(TS,QS) = (1-fS)σQS2/κS.

Substituting eq. (A10) and (A9) in eq. (A8) yields

(A11 )
σTS2 = fS2η2σTW2 + (1-fS)2σQS2/κS2.

A.2.3. Summer-to-winter correlation

Multiplying eq. (17) by TSi and using eqs. (A9) and (A10), leads to

(A12 )
Cov(TW,TS) = CfSησTW2 + fWr(1-fS)2σQS2/κS2.

Using eq. (A11) and the definition of C in eq. (18), this can be written as

(A13 )
Cov(TW,TS) = ηfSfWγ(1-r)σTW2 + fWrσTS2.

Dividing eq. (A13) by σTSσTW leads to an expression for the summer-to-winter correlation:

(A14 )
Corr(TW,TS) = fW[rα + γη(1-r)fS/α-1],

where α=σTS/σTW is the ratio of summer to winter standard deviation, which is known from eqs. (A6) and (A11).

A.2.4. Summer-to-summer covariance

Multiplying eq. (12) by TSi-1, and using <QSTS-1>=0 and <TW-1TS-1>=<TWTS>, leads to

(A15 )
Cov(TS,TS-1) = fSη Cov(TW,TS),

with Cov(TW, TS) known from eq. (A13). Similarly,

(A16 )
Cov(TS,TS-j) = fSη Cov(TW,TS-(j-1)).

Multiplying eq. (17) by TSi-1 and averaging, again using <TW-1TS-1>=<TWTS>leads to

(A17 )
Cov(TW,TS-1) = CCov(TW,TS).

Similarly,

(A18 )
Cov(TW,TS-k) = CkCov(TW,TS),

which can be substituted in eq. (A16) to give

(A19 )
Cov(TS,TS-j) = ηfSCj-1Cov(TW,TS).

It is convenient to write eq. (A19) as

(A20 )
Cov(TS,TS-j) = AσTS2Cj.

where

(A21 )
A = ηfSCov(TW,TS)/σTS2C.

Note that eq. (A20) is only valid when j≥1. When j=0, Cov(TS,TS)=σTS2 as defined in eq. (A11).

A.2.5. Summer-to-summer correlation

Using eqs. (A14), (A15), and (A20) it is straightforward to show that the summer-to-preceding summer correlation is

(A22 )
Corr(TS,TS-1)= AC= ηfSCorr(TW,TS)/α = fWfSη[r+γηfS(1-r)/α2]. 

A.3. Power spectra

A.3.1. Winter temperature

The power spectrum of the winter temperature, PW(ω), can be found by performing the discrete Fourier transform of the covariance function Cov(TW,TW-j):

(A23 )
PW(ω) = j=-j=Cov(TW,TW-j)e-i2πωj,

where ω[0,0.5], and the Nyquist frequency ω=0.5 corresponds to a period of 2 years in our model. From the winter-to-winter relations,

(A24 )
Cov(TW,TW-j) = CjσTW2.

and it follows that

(A25 )
PW(ω) = σR2GW(ω),

where the spectral shape is

(A26 )
GW(ω) = 1 / [ 1 - 2Ccos(2πω) + C2 ].

Note that GW(0)=1/(1-C)21, and GW(0.5)=1/(1+C)21.

A.3.2. Summer temperature

Similarly the power spectrum PS(ω) of the summer temperature can be found from

(A27 )
PS(ω) = j=-j=Cov(TS,TS-j)e-i2πωj,

which can be written as

(A28 )
PS(ω) = σTS2+2j=1j=Cov(TS,TS-j)cos(2πωj),

since Cov(TS,TSj) is an even function of j. Substituting eq. (A20) in (A27) yields

(A29 )
PS(ω) = σTS2[1+2Aj=1j=Cjcos(2πωj)],

which is

(A30 )
PS(ω) = σTS2[1+Aj=1j=(Cei2πω)j+(Ce-i2πω)j].

It straightforward to show that

(A31 )
PS(ω) = σTS2GS(ω),

where

(A32 )
GS(ω) = 1-A + A(1-C2)GW(ω).

References

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