# Relationship between sea level and temperature

### Global Temperature Increase of 1 Degree Caused Sea Level Rise of 6 Meters | IFLScience

The temperatures (relative to ) to go with the sea-level data from the second reference were taken from the . So, it appears that the sea-level rise by will be between and meters. The difference is a factor of about 7. Sea level rise is caused primarily by two factors related to global warming: the added water from melting ice sheets and glaciers and the expansion of seawater . The new record reveals a systematic equilibrium relationship between global temperature and CO2 concentrations and sea-level changes over.

Semi-empirical approaches have been implemented to estimate the statistical relationship between these two variables providing an alternative measure on which to base potentially disrupting impacts on coastal communities and ecosystems. However, only a few of these semi-empirical applications had addressed the spurious inference that is likely to be drawn when one nonstationary process is regressed on another. Furthermore, it has been shown that spurious effects are not eliminated by stationary processes when these possess strong long memory.

Our results indicate that both global temperature and sea level indeed present the characteristics of long memory processes. Nevertheless, we find that these variables are fractionally cointegrated when sea-ice extent is incorporated as an instrumental variable for temperature which in our estimations has a statistically significant positive impact on global sea level.

Introduction Coastal erosion, loss of coastal wetlands and increased risks of flooding are some of the negative impacts that increases in the sea-level would have on coastal communities and ecosystems [19].

Although the economic costs derived from some of these impacts have been found to be relatively small in terms of GDP losses [1]it is nevertheless important to improve the confidence on the estimates of the relationship between global average temperature and global sea level. Current physics based-models have been shown to be able only to partially replicate recent sea level rise observations based on global average temperature and semi-empirical approaches have been implemented with the intention of filling the predictive gaps in the former class of models [26].

However, most of these studies have not considered the potential spurious inference that is likely to be drawn from a regression of two series with strong temporal properties such as global sea level and global temperature [28]. Two noteworthy exceptions are [29] and [13]. The former performs a cointegration analysis to correct for potential spurious effects and the latter adjusts a nonstationary model through the Kalman filter.

However, while a statistically significant impact of sea level on temperature is found in [29]statistical significance for the converse causal relationship is not obtained, which has important implications for climate change adaptation policies.

Our study extends the work of Schmith et al. First, we obtain fractional orders of integration for the series and present evidence of a long-run relationship between sea level and temperature that is fractionally cointegrated. Second, through the use of instrumental variables we obtain a consistent estimate of the impact of global average temperature on global sea level.

When the distant past of a series affects its current levels, it is said that the process possess long memory. Granger and Joyeaux [11] formally introduced the related concepts of long memory and fractional integration into the field of econometrics.

Unit root tests are frequently used to detect the nonstationarity of a series, however, such tests are ill-suited in determining whether a series presents long memory see, for instance [8][9]. Tsay and Chung [35] show that the presence of long memory, even in stationary series, leads to spurious relationships. Therefore, determining the fractional order of integration is crucial if valid inferences are to be made regarding the statistical relationship between time series.

### Sea Level Versus Temperature

Although the approach taken in SJT12 has been long recognized as an appropriate mechanism to correct for the nonstationarity of the series in a regression context, the study does not incorporate more recent developments regarding the implications of long memory in series that seem to be nonstationary. As mentioned above, our study extends the work of SJT12 to the fractionally integrated case allowing us to make statements about the long memory properties of the series and their implications in terms of statistical inference.

The point at about The points chosen during the last Major Ice Ageybp to 20, ybp are for maxima and minima. In the interest of truthfulness, not all the other data points in the 2nd reference fit the curve so well. The shape of the fit curve cries out for a physical explanation. Try this one for size: At very low temperatures much or all of the Earth is covered with ice. Most of the ice is on land or prevented from displacing sea water by rigid attachment to land. Melting such ice adds to the ocean water, which adds to thermal expansion of the water to greatly increase sea level.

## Global Temperature Increase of 1 Degree Caused Sea Level Rise of 6 Meters

At very high temperatures most of the sea ice has already melted. So melting the remaining ice on the land adds to the ocean water, which adds to thermal expansion of the water to greatly increase sea level.

At intermediate temperatures, as is the case now, more melting is for sea ice than in the other two cases. Since sea ice is already floating in water, its melting adds very little to sea-level rise due to land-ice melting and thermal expansion of the water.

The inflection point is at That is, we are living during an Earth temperature that is near the minimum of change in equilibrium sea-level rise with changing temperature: Various references 1234 give the equilibrium sea-level rise for thermal expansion of 0. That rise is negligible relative to the equilibrium curve given in the figure above 7. So most of the equilibrium rise in sea level is due to melting ice.

Tanh Fit Another more physically realistic approach is to fit the sea-level-change data using two hyperbolic tangents: The best fit is shown in the following graph: This graph shows the fit projected to the asymptotic temperature-change regions.

The negative asymptote is Snowball Earth all land covered with ice and the positive asymptote is an ice-free Earth or at least no ice on land. Some papers indicate that the sea level was at least meters below now during Snowball Earth. Although the best fit, shown above, is about meters, a reasonably good fit can be obtained with meters.

## Long-Memory and the Sea Level-Temperature Relationship: A Fractional Cointegration Approach

More data at larger temperature changes are needed to have confidence in the values of the asymptotes. Back to the top Predicting Non-Equilibrium Sea Level Versus Temperature Temperatures calculated for global warming can be converted to approximate equilibrium-sea-level rises by using the equation given above.

However there is a time lag of perhaps years, so immediate-sea-level rises would be much smaller. Actually there may be different time lags for different Earth situations. Here I use only one time lag. Since that time lag is not known, I do the following calculation for five different hyperbolic time constants: