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  • View in gallery

    (a) Annual global values of land–ocean surface temperature anomalies from Hansen et al. (2010). (b) Global mean sea level anomalies from Jevrejeva et al. (2009) (dashed line) and from Church and White (2006) (solid line). (c) Total radiative forcing anomalies from Crowley (2000) (solid line) and Myhre at al. (2001) (dashed line).

  • View in gallery

    Time differences of (a) global land–ocean surface temperature anomalies from Hansen et al. (2010) and (b) global mean sea level from Church and White (2006) as function of year.

  • View in gallery

    Examples of realizations of bivariate VAR processes of length 100defined as in Eq. (2) with different impact matrices: (a) , (b) , and (c) . In all three cases . The variable T is shown with a solid curve and h is shown with a dashed curve.

  • View in gallery

    Disequilibrium error given by (10) as function of year.

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    Partial scatterplot of (a) and (b) , both corrected for linear dependence on and , against . Associated partial correlation coefficients are −0.31 and 0.06, respectively.

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    Disequilibrium error for the conditional model (solid line) and the original model (dashed line).

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    Histograms of adjustment coefficients from MC experiments for (a) the equation and (b) the equation. Solid lines are for short time series and dashed are for long time series.

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Statistical Analysis of Global Surface Temperature and Sea Level Using Cointegration Methods

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  • 1 Danish Meteorological Institute, Copenhagen, Denmark
  • 2 University of Copenhagen, Copenhagen, Denmark
  • 3 Danish Meteorological Institute, Copenhagen, Denmark
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Abstract

Global sea level rise is widely understood as a consequence of thermal expansion and the melting of glaciers and land-based ice caps. Because of the lack of representation of ice-sheet dynamics in present-day physically based climate models, semiempirical models have been applied as an alternative for projecting future sea levels. There are, however, potential pitfalls in this because of the trending nature of the time series. A statistical method called cointegration analysis that is capable of handling such peculiarities is applied to observed global sea level and land–ocean surface temperature. The authors find a relationship between sea level and temperature and find that temperature causally depends on the sea level, which can be understood as a consequence of the large heat capacity of the ocean. They further find that the warming episode in the 1940s is exceptional in the sense that sea level and warming deviate from the expected relationship. This suggests that this warming episode is mainly due to internal dynamics of the ocean rather than external radiative forcing. On the other hand, the present warming follows the expected relationship, suggesting that it is mainly due to radiative forcing. In a second step, the total radiative forcing is used as an explanatory variable, but it is unexpectedly found that the sea level does not depend on the forcing. The authors hypothesize that this is due to a long adjustment time scale of the ocean and show that the number of years of data needed to build statistical models that have the relationship expected from physics exceeds what is currently available by a factor of almost 10.

Corresponding author address: Torben Schmith, Danish Meteorological Institute, Lyngbyvej 100, DK-2100 Copenhagen Ø, Denmark. E-mail: ts@dmi.dk

Abstract

Global sea level rise is widely understood as a consequence of thermal expansion and the melting of glaciers and land-based ice caps. Because of the lack of representation of ice-sheet dynamics in present-day physically based climate models, semiempirical models have been applied as an alternative for projecting future sea levels. There are, however, potential pitfalls in this because of the trending nature of the time series. A statistical method called cointegration analysis that is capable of handling such peculiarities is applied to observed global sea level and land–ocean surface temperature. The authors find a relationship between sea level and temperature and find that temperature causally depends on the sea level, which can be understood as a consequence of the large heat capacity of the ocean. They further find that the warming episode in the 1940s is exceptional in the sense that sea level and warming deviate from the expected relationship. This suggests that this warming episode is mainly due to internal dynamics of the ocean rather than external radiative forcing. On the other hand, the present warming follows the expected relationship, suggesting that it is mainly due to radiative forcing. In a second step, the total radiative forcing is used as an explanatory variable, but it is unexpectedly found that the sea level does not depend on the forcing. The authors hypothesize that this is due to a long adjustment time scale of the ocean and show that the number of years of data needed to build statistical models that have the relationship expected from physics exceeds what is currently available by a factor of almost 10.

Corresponding author address: Torben Schmith, Danish Meteorological Institute, Lyngbyvej 100, DK-2100 Copenhagen Ø, Denmark. E-mail: ts@dmi.dk

1. Introduction

Sea level rise is one of the important societal aspects of climate change but unfortunately there is large uncertainty about expected changes. The most recent assessment report from the Intergovernmental Panel on Climate Change (IPCC; Solomon et al. 2007) specifically mentions that the published projections cannot be taken as the upper limit because the full effect of Greenland ice streams and other effects are only partially included owing to a lack of understanding of the relevant processes.

As an alternative to process-based, physical models, Rahmstorf (2007a) suggested applying semiempirical models and he formulated a statistical relationship between global averages of surface temperature and the rate of sea level rise. The work caused some debate (Holgate et al. 2007; Schmith et al. 2007; von Storch et al. 2008; Rahmstorf 2007b), which led to a modification of the model (Vermeer and Rahmstorf 2009) debated in Taboada and Anadón (2010) and Vermeer and Rahmstorf (2010). Parallel to this, Grinsted et al. (2010) and Jevrejeva et al. (2009, 2010) formulated similar semiempirical models. Common to these semiempirical methods is that they yield larger projected global sea level changes by the end of this century than stated in Solomon et al. (2007) and therefore they have caused concern.

Our reservation about the statistical approach applied in the above work is that it does not take into account the possible nonstationarity of the time series. Such nonstationarity can be due to deterministic trends or stochastic trends. We consider that linear deterministic trends and any stochastic trend to be a consequence of the statistical model that we assume is used to generate the data. A typical stochastic trend is a realization of a so-called integrated process, consisting of a random walk generated by the accumulation of independent errors plus a stationary process. Neglecting such trends implies a risk of misleading and biased determination and estimation of statistical models (Yule 1926; Granger and Newbold 1974).

The presence of stochastic trends in climate series is well-documented (e.g., Kaufmann and Stern 1997, 2002; Kaufmann et al. 2006a,b; Richards 1993, 1998; Stern and Kaufmann 1999, 2000, and references therein). Within the field of econometrics there exists an extensive body of literature on analyses of time series containing a combination of stochastic and deterministic trends. We find it peculiar that these statistical tools, which are applied routinely in economical analyses, are rarely applied in the analysis of climate data, with Kaufmann and Stern (1997, 2002), Kaufmann et al. (2006a,b), Liu and Rodriguez (2005), Mills (2009), Stephenson et al. (2000), Stern (2006), and Stern and Kaufmann (1999, 2000) as notable exceptions. A central concept in these works is cointegration, which is also the backbone of the method we will apply. The concept of cointegration and common stochastic trends is thanks to Clive W. J. Granger, who in 2003 was awarded the Nobel Prize in economics for it. A popular explanation of cointegration can be found in Murray (1994).

2. Data

We use annual averages of the combined land–ocean surface temperature data described in Hansen et al. (2010) and shown in Fig. 1a. As the primary global sea level dataset we use that of Church and White (2006), constructed by combining satellite altimeter data from the beginning of the 1990s with conventional sea level records to obtain a record of global average sea level back to 1880. Additionally, we use the dataset of Jevrejeva et al. (2009), who use a “virtual stacking method” based solely on conventional sea level records (shown in Fig. 1b). Radiative forcing data on an annual basis from natural and anthropogenic sources, separated into volcanic, solar, greenhouse gases, and man-made tropospheric aerosols, were taken from Crowley (2000). As an alternative forcing dataset we used Myhre et al. (2001), with in total nine anthropogenic and natural forcing components. For each of these datasets, we calculate the total radiative forcing by simple addition of the individual components (shown in Fig. 1c). These data are analyzed in the overlapping period of 1880–1998 in what follows.

Fig. 1.
Fig. 1.

(a) Annual global values of land–ocean surface temperature anomalies from Hansen et al. (2010). (b) Global mean sea level anomalies from Jevrejeva et al. (2009) (dashed line) and from Church and White (2006) (solid line). (c) Total radiative forcing anomalies from Crowley (2000) (solid line) and Myhre at al. (2001) (dashed line).

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

Over this period, the global sea level (Fig. 1b) has been increasing almost steadily, while the temperature (Fig. 1a) has a more complicated development with an increase until around 1940, then a slight decrease until around 1980, followed by an increase. In particular, we note that the increase in sea level begins before the warming (and the increase in forcing). The forcing has also been steadily increasing throughout the period with intermittent negative spikes caused by major volcanic eruptions. None of these negative forcing spikes occur in the 1930s, 1940s, or 1950s.

The reasons for this complicated temperature behavior are still unclear: it could be due to variations in the external forcing, mainly anthropogenic and/or volcanic aerosols, and/or it could be due to internal variability in the climate system. Studies using coupled atmosphere–ocean GCMs arrive at the conclusion that the full range of anthropogenic and natural external forcings can account for the gross features in the observed global temperature changes over the twentieth century while natural external forcings alone cannot (e.g., Broccoli et al. 2003; Knutson et al. 2006; Stott et al. 2006). Other studies claim that a residual remains, which cannot be accounted for by external forcing (Andronova and Schlesinger 2000; Kravtsov and Spannagle 2008). It is our aim that the present statistical analysis will shed light on this problem.

The series described here possibly exhibit stochastic trends. This means that the series of time differences play a central role in this type of analysis and it is therefore illustrative to plot these two derived series (Fig. 2). They appear stationary and much more like “ordinary” time series when compared to the original series shown in Figs. 1a,b. The fact that the time difference series appear stationary is a sign that the original series are integrated. We also performed an augmented Dickey–Fuller test. This is a univariate test with the null hypothesis that the series is integrated. We specify two lags of the variable and constant term included and for the temperature series we get a p value of 0.83, while we get 0.99 for the sea level series. This means that the null hypothesis cannot be rejected for any of the series.

Fig. 2.
Fig. 2.

Time differences of (a) global land–ocean surface temperature anomalies from Hansen et al. (2010) and (b) global mean sea level from Church and White (2006) as function of year.

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

3. Method

Since the method we intend to apply is nonstandard within the climate community, we will explain it in some detail. More comprehensive descriptions can be found in, for example, Hendry and Juselius (1999, 2000) and Juselius (2006).

To begin with, we consider the bivariate time series (Tt, ht) of the annual global anomalies of surface temperature and sea level. We model it statistically as a vector-autoregressive (VAR) model, which in its simplest form (first order) is a dynamic linear stochastic model:
e1
where is a 2 × 2 matrix and is bivariate white noise. The model (1) is the multivariate analog of a univariate first-order autoregressive (AR) model. The properties of the matrix determine the properties of the VAR process, as will be demonstrated below. This is equivalent to the role that the lag-1 autocorrelation coefficient plays in a univariate AR model, . In the univariate case, whenever the parameter is strictly less than 1 in absolute value, the process adjusts to its mean value, with a longer adjustment time scale the larger the parameter. When the parameter equals unity the process becomes a random walk, , and the adjustment time scale becomes infinitely long [i.e., the process is a random walk without any affinity to approach its mean value (or any other value)]. Rather, such a process exhibits stochastic trends over long periods of time.
We reformulate our bivariate VAR model as
e2
where and are the time differences of global temperature and sea level, respectively. The matrix = 1 is called the impact matrix.

The rank of determines the properties of the process, as will be illustrated in the following: if the rank equals 2 (full rank), the model (2) describes two correlated and stationary processes (Fig. 3a). If the rank equals 0, all elements in are 0 and the process (2) describes two dynamically unrelated random-walk processes (Fig. 3b). Finally, when the rank equals 1 (Fig. 3c), the two time series exhibit nonstationarity and stochastic trends, like the rank-0 case. Unlike that case, however, the upward–downward stochastic trends of the two series appear concurrently. We say that the two series share a common stochastic trend—they are cointegrated. Cointegration can be thought of as the nonstationary analog to correlation.

Fig. 3.
Fig. 3.

Examples of realizations of bivariate VAR processes of length 100defined as in Eq. (2) with different impact matrices: (a) , (b) , and (c) . In all three cases . The variable T is shown with a solid curve and h is shown with a dashed curve.

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

This can be clarified further. If the impact matrix has rank 1, we can write
e3
and (2) can be rewritten as
e4
where
e5
(called the disequilibrium error) can be shown to be stationary (e.g., Juselius 2006), whereas Tt and ht themselves are both given as a random walk plus a stationary process, a so-called I(1) process.

The model (4)(5) is called the error correction form of the VAR (VECM), which we recall is only possible if the two series are cointegrated. By the transformation to the VECM description, the two coupled equations for the time differences in the VAR description separate into two equations relating the time differences of the original variables, Tt and ht, to the disequilibrium error zt and random and uncorrelated disturbances.

The disequilibrium error (5) defines a linear relation between temperature and sea level by
e6
and model (4) represents a feedback mechanism whereby the changes in temperature and sea level react to deviations from the line defined by (6) through the adjustment coefficients and , which therefore may give valuable information on the underlying physics of the system. These coefficients determine time scales with which each of the two variables adjusts to the cointegration line [see (6)], and therefore the adjustment time will in general be different for the two variables. In practice, only estimates of the adjustment coefficients are available and therefore inference on these becomes crucial. In particular, it could be that one coefficient is virtually 0, which means that this variable does not adjust to the cointegration line.
The VECM described by (4)(5) is the simplest one. We recall that it is developed from a VAR model where the present values depend on values from the previous time step only and therefore it is of order one. The VECM of any order could be formulated. For instance, the VECM of order two (i.e., with values from the two previous time steps) and with deterministic drift terms reads
e7

Here, is a 2 × 2 matrix of coefficients describing the direct response of the time differences to their values at the previous time step, called the short-run dynamics. The terms and are linear drift terms.

Certain choices must be made regarding the form of the drift terms and this requires some attention. With no restrictions (called “unrestricted constant”) on the drift terms in (7), deterministic linear trends may be present in the series and it can be shown that these trends are related to the drift terms. In the “restricted-constant” case we require that is proportional to , and with the constant of proportionality being the system (7) then reduces to
e8
This means that the series have no deterministic trends but the disequilibrium error
e9
can have an arbitrary nonzero mean level determined by the drift terms. As can be seen from (9), this is equivalent to taking arbitrary nonzero offsets of the variables Tt and ht into account.

In applications of VAR model fitting, the first step is to determine the order of the model. This is a balance between wanting a parsimonious model on one hand and retaining a sufficient number of lags to ensure normally distributed and uncorrelated residuals on the other hand. This is because having residuals with these properties ensures valid statistical inference.

The technique to ensure this is to begin by fitting a high-order model. The residuals and are then examined with respect to the occurrence of time trends and their histograms are examined with respect to being normal, and their autocorrelation functions are examined with respect to being 0 for non-0 lags. If all conditions are fulfilled, one repeats the procedure with lower-order models and the adequate order is the lowest order where all conditions are fulfilled.

Next we determine the rank of the impact matrix , since this determines the stationarity properties of the variables and whether or not the two series are cointegrated, and the VECM formalism therefore can be applied. Since only an estimate of is available, the rank of cannot be exactly determined yet, but will be determined by a statistical procedure described in Johansen (1988). The rank of is determined by formulating and testing a series of null hypotheses on . The first null hypothesis is taken to be no cointegration or rank equal to 0. If rank 0 is rejected at the significance level used, we assume that the rank is greater than 0 and test that it is 1, and so on. As with all statistical tests the possibility of finding the correct rank of depends on the amount of data available, through the power function that specifies the rejection probability of the null hypothesis when the null is not true.

If cointegration is found, our VAR model is reformulated as a VECM as described above and can be estimated using the maximum-likelihood procedure described in Johansen and Juselius (1990). This includes the estimation of all parameters in the model and their confidence intervals and significance test statistics.

Engle and Granger (1987) pointed out that once the parameters and have been estimated, one can calculate the disequilibrium error zt by (5) or (9) and use its lagged value as a regressor variable in the estimation of (7) or (8) by ordinary least squares regression. Here, ordinary test statistics, like a t test, can be used and make it easy to test the significance of each independent variable in (7) or (8) and to re-estimate restricted models with insignificant variables omitted. A backward elimination procedure is applied where in each iteration, we perform doubled-sided t-ratio tests on all coefficients and successively omit the variable with the highest p value. We stop the procedure when all coefficients have p values less than 0.05.

4. Application of the method

We apply the method described in the previous section to the global temperature series and the sea level series by Church and White (2006). We expect both series to exhibit an upward deterministic trend arising from the increase through time of the external radiative forcing and therefore we assume the unrestricted-constant case [see (7)] in the following analysis.

By inspecting the residuals from fitted VAR models of successively lower order, as described in the previous section, we find that VAR model of order two is a satisfactory description of the data. When applying the cointegration test we conclude that the impact matrix has rank 1, since we obtain a p value around 0.005 for the “rank-0” hypothesis, which we therefore reject, while we obtain a p value of 0.25 for the “rank-1” hypothesis, which we therefore cannot reject. Based on this, we regard the two series as cointegrated and describe their mutual time development as a VECM of order two.

Since the two series are cointegrated, we estimate the VECM using the maximum-likelihood procedure mentioned in the previous section and find the disequilibrium error:
e10
This relationship describes a line [see (6)] to which temperature and sea level adjust over time. We note that the disequilibrium error (10) implies that increasing temperatures and increasing sea level anomalies are positively connected as expected. Throughout this paper, we give the corresponding t ratio in parentheses below an estimated parameter.
Thermal expansion of the ocean is responsible for around half of the observed sea level rise over the period 1961–2003 (Domingues et al. 2008). Let us tentatively assume that only the ocean mixed layer of depth H, which is in direct thermal contact with the lower atmosphere on annual time scales, is involved in the thermal expansion. In that case, we have approximately , where is a typical value of the volumetric thermal expansion coefficient of seawater, and by comparing with the cointegration relation we see that , from which
e11

This value is obviously an order of magnitude larger than the generally agreed values for the thickness of the ocean mixed layer. Therefore, we can conclude that the thermal expansion is not confined to the mixed layer, but rather, the calculation indicates that the deeper-water masses are also heating. This is in accordance with Barnett et al. (2001).

As mentioned in section 2, the disequilibrium error should be a stationary series (i.e., without any random-walk character). This is confirmed by plotting zt, given by (10), against time (Fig. 4), although it admittedly contains some decadal-term deviations (e.g., around 1940). This means that the series cannot be described as a simple autoregressive-type relaxation. We will discuss the temporal evolution of the disequilibrium error later in the paper.

Fig. 4.
Fig. 4.

Disequilibrium error given by (10) as function of year.

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

Next we estimate the parameters in Eq. (7) for the time-differenced series. As explained in section 2, we do that by multiregression analysis and find that the statistical significance, evaluated by an ordinary t test, differs among the parameters. Retaining only significant terms, identified by the backward elimination procedure described above, the equations reduce to
e12
and
e13

In (12), the estimated significantly negative value for means that the temperature reacts to disequilibrium by adjusting to the cointegration line. The short-run dynamics is nonsignificant. We find in (13) that is not statistically different from zero, which means that sea level does not seem to react to disequilibrium. Variables that do not react to the disequilibrium error are called weakly exogenous. From (13) it is evident that the time differences have a positive mean value equivalent to a deterministic trend, described by the drift term. In addition to this there is a small antipersistence of the time differences.

The different properties of the adjustment coefficients in the two Eqs. (12) and (13) can be illustrated by the partial residual scatterplot of and , both corrected for any linear dependence on and , against , displayed in Fig. 5. Beginning with Fig. 5b, the estimated is evidently due to the lack of any relationship between the two variables and not due to, for example, a quadratic relationship. Continuing with Fig. 5a, the linear relationship appears convincing and not due to single outlying and therefore influential observations.

Fig. 5.
Fig. 5.

Partial scatterplot of (a) and (b) , both corrected for linear dependence on and , against . Associated partial correlation coefficients are −0.31 and 0.06, respectively.

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

In summary, our statistical analysis shows that the points (Tt, ht) tend to be clustered along the cointegration line in the (Tt, ht) plane. The two variables, however, behave very differently. The time differences in the global sea level exhibit a deterministic positive trend, representing the influence of the radiative forcing, while the time differences in the global temperature have the role of keeping (Tt, ht) near the cointegration line.

We can understand this result from a physical point of view, as follows: because of the much larger heat capacity of the ocean compared to that of the atmosphere, any change of the upper-ocean temperature, being due to external forcing or to internal climate variability, will spread to the atmospheric surface layer. On the other hand, if the surface temperature of the atmosphere changes, the change will vanish as soon as the associated heat anomaly spreads to the upper ocean. We can say that the surface air temperature adjusts to the average temperature of the upper ocean, which, as remarked earlier, is strongly related to the sea level because of thermal expansion (Domingues et al. (2008). The result is therefore in opposition to the physical thinking behind the model proposed by Rahmstorf (2007a), where adjustment of the sea level to temperature is assumed.

To investigate whether the reason for the above was due to the specific nature of the sea level data used, we redid the whole analysis using the sea level series by Jevrejeva et al. (2009), but this left all of the above results virtually unchanged.

5. Cointegration analysis conditioned on the external radiative forcing

In the analysis presented until now, the forcing of the sea level rise comes about through a deterministic trend, determined by the drift parameter . Since we expect the increasing level over time in the external radiative forcing to be responsible for this trend, it would be satisfactory if we were able to explain this trend in the sea level directly by the external forcing rather than just ad hoc as a parameter estimated in the analysis.

The cointegration methods described in section 3 can be extended to take weakly exogeneous variables into account. We then consider the VAR model for the time differences, and , conditional on the current value of the forcing Ft and lagged values of Ft, Tt, and ht. This model is fitted and cointegration is tested for, based on the rank of the impact matrix , which is now a 2 × 3 matrix. We will regard the total external radiative forcing Ft as a weakly exogeneous variable and can formulate a second-order VAR model as
e14
which in the case of cointegration can be transformed to
e15
As in (7), model (15) relates the time differences to a combination of adjustment terms, in which the disequilibrium error now includes the external forcing and is given by
e16
short-run dynamics determined by the 2 × 1 matrix , and the 2 × 3 matrix ; drift terms; and random disturbances.

We conduct an analysis with the total external forcing calculated from Crowley (2000) specified as a weakly exogenous variable. We thus assume no feedback from changes in global temperature forcing variables like the atmospheric carbon dioxide concentration.

As previously explained, our motivation for introducing the total radiative forcing as a weakly exogenous variable in the model was to replace the deterministic trend of sea level as a driver of the model. As a consequence of this, we expect no deterministic trends in our VAR model (14) and should test and estimate it under the restricted-constant assumption. However, this estimating yields residuals (not shown) with a nonzero average, while the residuals have a zero average. This is a sign that there is a trend in the sea level not explainable by the external forcing and that the model does not fit the data satisfactorily. We therefore continue our analysis using the unrestricted-constant assumption.

We conduct the cointegration test and find that the impact matrix has rank 1, since we obtain a p value of <0.03 for the rank-0 hypothesis, which we therefore reject, while we obtain a p value of 0.91 for the rank-1 hypothesis, which we therefore cannot reject. Thus, a reasonable description of the data is the cointegrated model (15)(16).

Estimating (16) under the unrestricted-constant assumption, we get the disequilibrium error
e17
For a fixed total forcing , (17) defines an equilibrium relationship between and and we note as in the unconditional model that increasing temperatures and increasing sea level anomalies are connected. The coefficient of ht has been reduced from −3.3 K m−1 in the unconditional case to −1.4 K m−1 in the conditional case.

We plot the disequilibrium error for both the unconditional and conditional model (Fig. 6) and see that this variable is almost unchanged, which tells us that the effect of introducing the external radiative forcing as an exogenous variable is small. There is still a large positive anomaly around the 1940s, which means that this anomaly cannot be explained by the external forcing.

Fig. 6.
Fig. 6.

Disequilibrium error for the conditional model (solid line) and the original model (dashed line).

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

Having determined the cointegration relation and the associated disequilibrium error, let us now turn to the equations for the time differences and . Estimating these using the backward elimination procedure described above yields
e18
and
e19

For the time differences in temperature, the first term in (18) is an adjustment to the equilibrium cointegrating relation, while the second term represents a short-run response in temperature to an increase in total forcing. This means that time differences in temperature are explained by either disequilibrium from the cointegration line or by immediate time differences in the external forcing.

For the time differences in sea level, we see from (19) that the sea level in this model also contains no term representing adjustment to the cointegration line; in fact, (19) is identical to (13). This unexpected result will be discussed in the next section.

We repeated the whole analysis using the total forcing calculated from Myhre et al. (2001) instead. This caused only minor changes to the estimation of (17) and the form of (18) and (19) was virtually unchanged as well as the estimation of the parameters. In particular, we note that here also the time differences of the sea level do not depend on the disequilibrium error.

6. Discussion

a. Robustness of the analysis

Whereas there is more or less agreement between different estimates of global surface temperature, this is not the case for estimates of global average mean sea level. This is because the basic data, which are historical sea level records, are sparsely distributed and mainly along the coastline, and the interior ocean completely lacks information on sea level prior to the satellite era. This leaves room for variations due to the methodology applied in calculating the global average (see Christiansen et al. 2010).

There is also much uncertainty about the historical radiative forcing. This is because both the series of historical concentrations of greenhouse gases, aerosols, and other radiative constituents as well as the model to transform these concentrations to net forcing differ.

We carried out the above analysis with two different global sea level time series and two different time series of external forcing and saw that the main conclusions remained unchanged, which gives confidence to our results.

b. Causes for the warming in the 1940s and the present warming

We have seen the disequilibrium error exhibit decadal-scale undulations from its average. The most prominent one is the period of above 0.1-K and up to 0.2-K positive deviations around 1940. In other words, in this period surface temperatures are relatively high or, equivalently, sea level is relatively low. This period, known as the early twentieth-century warming (E20CW), is known as a period with relatively high surface temperatures compared to surrounding decades, both averaged over the globe and in particular over the North Atlantic and the Arctic (e.g., Johannessen et al. 2004; Polyakov et al. 2003; Wood and Overland 2010). The 1960s and 1970s are characterized by moderately negative disequilibrium anomalies, while the period from 1980 onward has smaller positive values of the disequilibrium error. This last period is known as the present warming (PW). This intriguing difference in the disequilibrium error between the two warming periods of the twentieth century may tell us something about the different nature of these warming periods.

Whereas the PW is to a large extent ascribed to increased external forcing from greenhouse gases, it is still unclear whether the E20CW is entirely caused by changes in the external forcing due to changes in solar or volcanic activity or the emission of man-made aerosols (Stott et al. 2000) or can in part be attributed to a multidecadal variability mode in the thermohaline overturning circulation of the World Ocean (Delworth and Knutson 2000).

Polyakov et al. (2010) used hydrographic observations to identify such a variability mode in the North Atlantic with maximum amplitude around 1940 and with an out-of-phase relation between the temperature in the surface of the ocean and at depth. We would expect a quite modest signal in the sea surface height from such a mode, since expansion at the top would partly be compensated for by contraction at depth and vice versa. Therefore, we would see a high disequilibrium error in such a situation and thus our analysis supports the view that the E20CW is at least in parts attributable to an anomaly in the thermohaline overturning circulation.

If the PW is mainly caused by radiative anomalies and only to a minor degree by anomalies in the overturning circulation, we would not have the expansion/contraction described above and therefore we would see modest disequilibrium errors. This is indeed what we have found from our analysis, from which we conclude that there is a minor contribution from overturning anomalies but the main contribution is from radiative forcing.

c. Investigating the reasons behind the undetected adjustment of sea level: Monte Carlo experiments

In our last analysis with forcing included we would a priori expect that sea level changes were driven by the total external radiative forcing. Therefore, it is surprising that sea level does not react to the disequilibrium error given by (17).

Could this result be an effect of having too short a period of data at hand? To investigate this we performed Monte Carlo experiments based on the model described in Eq. (15), as follows: a greater than 1100-yr-long time series of external forcing was constructed from the forcing series by Crowley (2000) by repeating the series a sufficient number of times and in each repetition adding an offset to the series. The offset has a magnitude so that the final series is steadily increasing. We then use this long forcing series to construct, in a Monte Carlo simulation, artificial series of and using the Eqs. (17)(19) with the important difference that the term is retained in Eq. (19) with set equal to its central estimate. The white-noise series and are different in each Monte Carlo realization. The series constructed in this way are therefore by definition cointegrated. We then estimate a VECM model described in (15). This estimation is done based on both years 101–200 (“short” case) and 101–1100 (“long” case) of the artificial series. Based on the MC simulation, we finally make histograms of the estimated adjustment coefficients.

Looking at these histograms, shown in Fig. 7, we note that the adjustment coefficient is symmetrically distributed around its specified value of −0.49, and the histogram is clearly separated from zero both in the long and short cases. The adjustment coefficient is symmetrically distributed around its true value of 2.4 m K−1 but with a large spread, so that the distribution is not clearly separated from zero for the short case. For the long case, the distribution is almost separable from zero. From this we conclude that we need on the order of 1000 yr of data to detect any reaction of the sea level to disequilibrium.

Fig. 7.
Fig. 7.

Histograms of adjustment coefficients from MC experiments for (a) the equation and (b) the equation. Solid lines are for short time series and dashed are for long time series.

Citation: Journal of Climate 25, 22; 10.1175/JCLI-D-11-00598.1

7. Summary and concluding remarks

We have introduced a statistical tool, rather unknown within geophysics but widely used within the field of econometrics, and applied it to the problem of analyzing the relationship between global land–ocean surface temperature and sea level. We have found that the time series for global surface temperature and sea level both contain stochastic trends and therefore the risk of spurious regression is present, which in turn justifies the use of these methods. We found that the two time series shared a common stochastic trend, that is, they cointegrate. This is in accordance with our physical expectations, since we expect both series to be driven by the external radiative forcing in the long run. Accordingly, the common time development can be described by a VECM introduced in the beginning of the paper.

Three main conclusions can be drawn by fitting the VECM to data. First, we show that temperature adjusts to the cointegration equilibrium, while there is no detectable adjustment of sea level. We can understand this in physical terms. The ocean represents a much larger heat capacity than the atmosphere and therefore the surface air temperature adjusts to the average temperature of the upper ocean, which again is proportional to the sea level anomaly. This mechanism has also been shown in GCM modeling experiments (Hoerling et al. 2008; Dommenget 2009).

Second, by studying the disequilibrium error we have learned that there is an important difference between the two warming periods during the twentieth century in that the E20CW has a positive disequilibrium error, while the PW has close to zero disequilibrium error. We have suggested that the reason for this could be that while the PW is mainly caused by changes in the external radiative forcing, the E20CW seems to be at least partly connected to multidecadal variations internal to the climate system and to the thermohaline overturning circulation of the ocean.

In the second part of our analysis we introduced the external radiative forcing as an external, explanatory variable. Contrary to our expectations, this did not extensively change our overall conclusions. We expected the forcing to determine the development of sea level, but could not find any sign of this in our data material. A Monte Carlo experiment revealed that we need on the order of 1000 yr of data to detect this effect. This is in contrast to results from global climate model experiments, where a clear signal from expansion of the seawater is seen in twentieth-century integrations (e.g., Gregory et al. 2001) as a result of the increased forcing. At present, we can only suggest explanations for this discrepancy. As seen in Fig. 1b, the sea level has also been rising well before the forcing (Fig. 1c) increases in the mid-twentieth century. This secular sea level rise seems to have persisted for at least some centuries (Kearney 2001) and may dominate even today, which may be the reason why we cannot detect any influence from the external radiative forcing on sea level.

Another matter is data quality. There is an ongoing discussion on the discrepancy between the observed sea level rise and the contributions from the different components, when seen over the whole twentieth century (Munk 2002; Mitrovica et al. 2006; Woodworth 2006). One explanation for this “enigma” could be the poorer quality of the estimated historical sea level back in time, which would probably also affect our analysis. Besides that, the confidence we can have in the historical forcings is unknown. One meaningful next step would therefore be to do a similar analysis on a transient coupled GCM experiment, to see whether we arrive at similar conclusions when using such consistent data. This work will be described in a subsequent paper.

Acknowledgments

This work was supported by the European Science Foundation under Contract EW06-047 (LESC). The Gnu Regression, Econometrics and Time-series Library was widely used in the analyses.

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