# Continued Misconception Of The Concept of Heating In The Pipeline In The Paper Vermeer and Rahmstorf 2009 Titled “Global Sea Level Linked To Global Temperature”

There continue to be misunderstandings of the role of global warming and the concept of “heating in the pipeline” and “unrealized “heat”. I have posted on this in the past; e.g. see

Is There Climate Heating In “The Pipeline”?

Further Comments Regarding The Concept “Heating In The Pipeline”

My View Of The Terminology “Heating In The Pipeline”

This issue was recently brought to my attention by Leonard Ornstein associated with his posting of a comment on  Real Climate with respect to a guest post by Martin Vermeer.  Len’s comment was a response to the paper

Martin Vermeer and Stefan Rahmstorf, 2009: Global sea level linked to global temperature. PNAS http://www.pnas.org cgidoi 10.107 pnas.0907765106. December 22, 2009.

In their paper, they have the equations

d (sea level)/dt = a [T – T (equllibrium)]                             (equation 1)

d (sea level)/dt = a ([T – T (equllibrium)] + b d( T)/ dt   (equation 2)

where

“T (equllibrium) is a base temperature at which sea level is in equilibrium with climate, so that the rate of rise of sea level H, dH/dt, is proportional to the warming above this base temperature. T (equllibrium) and a are to be determined from data. For the ice-melt contribution (glaciers and ice sheets) this approach corresponds to one commonly used in ice modeling, where the rate of mass loss is assumed to be proportional to the temperature increase above a threshold value………Eq. 1 is based on the assumption that the response time scale of sea level is long compared with the time scale of interest (typically ~100 years).”

With respect to equation 2, they added a term. As they write

“However, some components of sea level adjust quickly to a temperature change, e.g., the heat content of the oceanic surface mixed layer.”

As they also write in their paper, these equations only involve

“the thermal expansion component of sea-level rise only”

Their equation 1 is a relationship between sea level rise (the dH/dt term) and the resultant change of T from To which is dependent on the response time that is represented by “a”.  If there was no mass to the system being studied, the change to T would be instantaneous. If the system has mass [as the real ocean does] and heat is added (in Joules), the temperature will continue to increase until it warms sufficiently so that the outgoing heat is equal to the incoming heat. If the heat is turned off, the temperature increase stops immediately.

The sea level response, however, is not lagged at all for heat (in Joules) that is input into the ocean at any level. This involves not just the temperatures in the ocean mixed layer, but the ocean temperature at all depths.  It is the mass weighted temperatures (i.e. the heat) that matter in terms of thermal expansion which results in sea level rise.

Their use of a new ¨dual model” does recognize that part of the sea level rise (that from heating in the ocean mixed layer is instantaneous) but also, by continuing to use temperature, rather than heat,  they retain a delay that does not recognize that heating causes expansion as soon as it is imposed.  They indicate a delay in sea level rise from this heating.

However, heating within the ocean at ALL levels will immediately be seen in the sea level. The only lags would involve the melting of glaciers and ice fields from warming, but since they state they are only evaluating the thermal expansion of the ocean, that lag is not be part of equations (1) or (2).

Moreover, since sea level and the diagnosed surface temperatures have increased over the last century, a strong correlation is guaranteed from their equation 2.  Indeed, I caution students in taking two curves which have similar shapes, but which are out of phase, and by lagging one of the curves in time, assume this provides a causal relationship.

The quantitative value of the regression they obtain and the extrapolation decades from now, therefore,  has to be viewed with significant caution since they have assumed the response of sea level rise to ocean heating can be delayed ( i.e. “in the pipeline”; “unrealized”). This certainly is true with  respect to the melting of glaciers and ice sheets, as well as for hydrologic responses over land, but not for heating within the ocean.

What is needed is a direct linkage between ocean heat content added in Joules and sea level rise rather than equation (1) or (2).

The papers

Leuliette, E. W., and L. Miller (2009), Closing the sea level rise budget with altimetry, Argo, and GRACE, Geophys. Res. Lett., 36, L04608, doi:10.1029/2008GL036010.

Willis J. K., D. P. Chambers, R. S. Nerem (2008), Assessing the globally averaged sea level budget on seasonal to interannual timescales, J. Geophys. Res., 113, C06015, doi:10.1029/2007JC004517

Cazenave et al. Sea level budget over 2003-2008: A reevaluation from GRACE space gravimetry, satellite altimetry and Argo. Global and Planetary Change, 2008; DOI:10.1016/j.gloplacha.2008.10.004

provide a more rigorous examination of the sea level issue, albeit only for very recent years. Nonetheless, we will have this data available from now forward and we should use this instead of T.

There is, thus, a fundamental error in assuming that if heat is added in the ocean but not yet communicated to the sea surface height means that heating is in the pipeline.  There is no heat in the pipeline that has not been realized in sea level change.

What they are assuming is the continued input of heat to the climate system. This is what could be referred to as “heat in the pipeline” (or “unrealized heat”) since it does not yet exist. The need for a continued radiative imbalance is needed for this heat to continue to be added. This can be illustrated from equation 1-3 in

National Research Council, 2005: Radiative forcing of climate change: Expanding the concept and addressing uncertainties. Committee on Radiative Forcing Effects on Climate Change, Climate Research Committee, Board on Atmospheric Sciences and Climate, Division on Earth and Life Studies, The National Academies Press, Washington, D.C., 208 pp.

where, using the notation of Vermeer and Rahmstorf

T – T (equillibium)  = f * lamda     (equation 3)

where f is radiative forcing (i.e. the radiative imbalance) at the tropopause and  lamda is a temperature feedback parameter.

Substituting 1-3 into 1-2 yields

d (sea level)/dt = a f * lamda  + b d( T)/ dt   (equation 4).

Since d(T)/dt is also zero if f is zero, there is no further change in d (sea level)/dt if f = zero.

If the phrase “heat in the pipeline” or “unrealized heat” are to be used, they are referring to f [radiative forcing] not to heat hidden somewhere within the climate system.

Clearly if f becomes zero, there is no further change in T.

This is an important issue. If you are boiling water and the temperature is initially 25C, you do not use the term “unrealized heat” or “heat in the pipeline” with respect to reaching 100C. The heat still needs to be added by the burners.

If the ocean stopped further heating today, the radiative imbalance of the climate system would be zero (if that metric accurately represented the other (smaller) heat reservoirs). There would be no further increase in the global average mean ocean heat content(i.e. there is nothing in the “pipeline”). nor a further rise in sea level from thermal expansion.

The concept of “heating in the pipeline” and of “unrealized heat” are misleading in terms of how this concept has been presented in the climate discussion, including in the Vermeer and  Rahmstorf paper.

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