by Donald Rapp
This paper describes a model that uses the basics of heat transfer to demonstrate than an increase in downwelling infrared radiation associated with increased CO2 reduces heat loss from the mixed layer of the ocean, causing the ocean to warm.
This Posting is a brief summary of the full paper (18 pages) that can be obtained athttp://home.earthlink.net/~drdrapp/ocean.heating.v3.pdf
Climate models indicate that when the CO2 concentration increases above the pre-industrial level of about 280 ppm, there is a consequent increase in downward back IR radiation impinging on the surface of the Earth, including the world oceans. A doubling of the CO2 concentration results in a downward radiative forcing at the surface of about 1 W/m2. Over the past half century the forcing averaged about 0.4 W/m2. Questions arise as to whether, how, and how much does this heat the oceans? The oceans have a heat capacity about 1,000 times greater than the atmosphere and land surface. Although air temperatures may change much more rapidly than ocean temperatures, it is the ocean temperature distribution that will ultimately determine the climate of the Earth.
There is considerable evidence that sea surface temperatures (SST) have warmed significantly over the course of the 20th century. There is also evidence that on average, the bulk oceans have warmed over this time period but the temperature gains were smaller. Nevertheless, the oceans are so vast that this represents a very large amount of heat. While the data on ocean warming leaves much to be desired, there seems to be little doubt that the oceans have acquired a significant amount of heat over the last five decades. Aside from the still unsettled issue of specifying the ocean warming to higher accuracy, the question arises as to what factors caused the ocean warming. Had there been long-term systematic increases in solar intensity and/or decreases in cloud cover, that would certainly have contributed significantly to ocean warming. There are several models that attempt to estimate how the solar intensity has varied during the 20th century, but they all make assumptions that cannot be validated and therefore they remain highly speculative. Estimates of global variation of cloud cover have been made by a number of investigators. Dai et al. (2006) admit “large inadequacies in monitoring long-term changes in global cloudiness with surface and satellite observations” and the sub-title of their paper is “A Tale of Monitoring Inadequacies”.
A number of websites and blogs currently claim that downwelling infrared radiation from greenhouse gases cannot heat the oceans. For example, hockeyschtick asserts: “since the LWIR re-radiation from increasing ‘greenhouse gases’ is only capable of penetrating a minuscule few microns (millionths of a meter) past the surface and no further, it could therefore only cause evaporation (and thus cooling) of the surface ‘skin’ of the oceans”. Hockeyschtick further asserts: “It is impossible for a 1.7 W/m2 increase [predicted by the IPCC due to man-made greenhouse gases] in downward ‘clear sky’ atmospheric LWIR flux to heat the oceans.” Steven Goddard ridicules Livermore scientists for claiming that rising greenhouse gases warm the oceans. Finally, Tallbloke asks: “Since the ocean is on average warmer than the atmosphere, the energy flux across the ocean/atmosphere interface is on average carrying heat from the ocean to the air…. So given the general direction of the motion of the energy, how can infrared energy be pushed into the ocean, when it can’t penetrate the surface further than its own wavelength?”
If these claims were correct, then any warming of the oceans would have to be attributed to increases in solar intensity or decreases in cloud cover. This paper describes a model that uses the basics of heat transfer to demonstrate than an increase in downwelling infrared radiation associated with increased CO2 reduces heat loss from the mixed layer of the ocean, causing the ocean to warm.
Models were developed in the 1970s and 1980s to estimate the magnitude of this effect. In my paper, I develop a simple model based on these earlier studies to estimate the rate of heat gain by the oceans due to an increase in back IR radiation. This model is based on taking an energy balance around the ocean surface. Unfortunately, it is difficult to estimate the rate of heat gain by the oceans for any given level of increased back radiation, because (i) the calculation is extremely sensitive to how the air above the ocean reacts as the ocean warms, and (ii) it is very difficult to estimate how the air above the ocean reacts to a warming ocean.
The profile of temperatures below the surface at a tropical location is shown schematically in Figure 3.
Figure 3. Schematic temperature profile in a tropical ocean. The initial curve (A) refers to equilibrium before applying a forcing to the surface. The short-term curve (B) shows the initial response to the forcing. This graph is not to scale. The difference in temperatures between the mixed layer and the ocean surface was exaggerated to make the graphic clearer.
Initially, there is a temperature profile (A) for the equilibrium state before applying a forcing to the surface. The surface is cooler than the bulk ocean because of heat loss to the atmosphere. The short-term curve (B) shows the initial response to application of a forcing. The initial temperature rise at the surface is ΔT2 due to absorption of radiation. As a result, the temperature difference between the surface and the bulk mixed layer of the ocean decreases from ΔT1 to [ΔT1 – ΔT2]. This reduces the heat flow from the bulk ocean to the surface and the bulk ocean begins to warm. As it warms, there is a progression of temperature profiles until a new equilibrium is established at a new ΔT. Figure 4 shows the same information in greater detail, with curve C representing an intermediate stage in the passage from no forcing to a new equilibrium under forcing represented by curve D. The ultimate temperature rise of the mixed layer of the ocean is ΔT3 in this figure.
The ultimate new equilibrium established after passage of sufficient time is shown as (C). This graph is not to scale. The difference in temperatures between the mixed layer and the ocean surface was greatly exaggerated to make the graphic clearer.
Figure 4. Schematic temperature profile in a tropical ocean. The initial curve (A) refers to equilibrium before applying a forcing to the surface. The response (C) shows an intermediate stage in the process as time progresses. Curve (D) is the new equilibrium.
As these changes in the ocean take place, changes are likely to occur in the air above the ocean surface. In some zero’th order models, it has been assumed that the air remains unchanged in temperature and humidity, even as the ocean surface warms. At the other extreme, one can assume that the air temperature tracks the ocean surface temperature and the relative humidity in the air remains constant (the absolute humidity increases). Reality probably lies between these extremes.
The object is to estimate ΔT3 with a forcing due to doubling of CO2 from the pre-industrial era, F ~ 1 W/m2). On a transient basis, as the surface warms, while the bulk mixed layer temperature is unchanged, DT1 decreases. This reduces the rate of heat flow from the bulk mixed layer to the surface. Thus heat loss from the bulk mixed layer to the surface decreases, and the bulk mixed layer warms. Eventually, the bulk mixed layer warms enough that the temperature differential between the bulk ocean and the surface returns to approximately ΔT1, and a new equilibrium is established with the bulk mixed layer and the surface both at a higher temperature. The goal of the present study is to estimate ΔT3.
The model involves the following steps:
(1) Write an energy flux equation about the ocean-atmosphere interface, prior to application of the forcing introduced by increased CO2 concentration in the atmosphere. This is done for equilibrium night conditions when there is no solar input. In this equation, the equilibrium rate of heat loss by the ocean mixed layer per unit area is expressed as the sum of the rate of latent heat loss from surface to air per unit area, the rate of sensible heat loss from surface to air per unit area, and the net rate of back radiation from surface to air per unit area. These quantities were estimated by previous investigators in terms of the ocean surface temperature and properties of the atmosphere above it. The bulk mixed layer temperature is the surface temperature plus ΔT1, but ΔT1 need not be specified in this model.
(2) Write a new energy flux equation about the ocean-atmosphere interface, after application of the forcing introduced by increased CO2 concentration in the atmosphere. This defines the new equilibrium rate of heat loss by the bulk mixed layer in the presence of doubled CO2.
(3) Subtract the two energy flux equations, to obtain an equation that includes the change in ocean surface temperature (ΔT3) as well as the changes in properties of the atmosphere above the ocean. These properties are the air temperature and the relative humidity. If we could estimate the changes in the air temperature and the relative humidity in the atmosphere above the ocean, we could then determine (ΔT3).
(4) The problem is that we don’t know how the air above the ocean changes when we go from no forcing to forcing. Two extreme cases have been considered: (i) the air does not change as the ocean warms, and (ii) the air temperature above the ocean tracks the ocean surface temperature and the relative humidity remains constant as the air temperature rises. Unfortunately, the calculation turns out to be extremely sensitive to this assumption, and as a result it is impossible to make accurate quantitative estimates of ocean warming without much better data on the changes that occur in the air above the oceans.
Why the Ocean Warms
In this model, we start with the upper mixed layer of an ocean in equilibrium with the air above it. Next, we consider the same upper mixed layer of an ocean with a downward forcing on its surface and calculate a new equilibrium. We then compare the two calculations to determine how much warming occurs in the mixed layer as a consequence of this forcing.
The initial effect of turning on the forcing F, before the mixed ocean layer warms up, is to reduce the temperature differential between the mixed ocean layer and the surface as shown by curve B in Figure 3. The ocean is continuously losing heat to the surface, and this reduces the cooling rate of the mixed ocean layer by a flux slightly less than F. But that is equivalent to warming the mixed ocean layer by a flux slightly less than F. As the mixed ocean layer gradually warms up, this flux will increase, but nevertheless, in general the effect of adding the flux F is to warm the mixed ocean layer.
Clearly, the overwhelming effect of a rise in TS is to reduce the energy flux from the mixed layer to the ocean surface. Therefore, most of the effect of an increase in back radiation is to heat the mixed layer. This conclusion is independent of any assumptions regarding changes in the atmosphere that result from warming of the ocean surface. If we repeated the calculation with different assumptions about the air temperature and humidity, the importance of reduction of energy loss from the mixed layer to the ocean surface would not change.
As time progresses, the mixed layer continues to warm as shown in curve C in Figure 4. However, as the mixed layer warms, the rate of energy transfer from the mixed layer to the surface increases, and thus the rate of warming of the mixed layer decreases as time progresses. Eventually, the rate of energy transfer from the mixed layer increases enough to establish a new equilibrium at a higher mixed layer temperature (see curve D in Figure 4). When this new equilibrium is established we can treat the upper ocean as a mixed layer and ignore the small difference in temperature between the mixed layer and the surface. We then carry out an energy balance about the ocean surface.
The surface of the ocean is thus warmed slightly by IR absorption and this reduces the temperature differential from the bulk mixed layer to the surface. This reduces the rate of heat loss from the mixed layer of the ocean to the surface; thus the mixed layer does not cool as fast as it would without the IR forcing at the surface (i.e. it warms relative to the unforced state). It is important to understand that the IR absorbed at the surface does not flow down into the ocean. The energy flow is always upward.
In each model, one starts with the unforced ocean and the air above it. Then the forcing (~ 1 W/m2) is introduced into the energy balance equation in terms of the latent heat loss, the sensible heat loss, and the net back radiation, and the forcing.
In the so-called zero’th order calculation of Newell and Dopplick (1979), one assumes that the air temperature and the absolute humidity do not change after the forcing is applied. With this assumption, one finds that with a very small increase in surface temperature, the increase in sensible and latent heat flux loss to the atmosphere accounts for the energy flux added by the forcing. For a wind speed of 3 m/s one finds the ocean surface temperature rise is about 0.05°C. The ocean mixed layer temperature rises by the same amount. In the so-called first order calculation of Watts (1980), one assumes that the air temperature is equal to the ocean surface temperature and tracks the ocean surface temperature, and the relative humidity remains at 75% after the forcing is applied (i.e. the absolute humidity rises as the ocean warms). With this assumption, one finds that for a wind speed of 3 m/s, the ocean surface temperature increase is 0.6°C, and the ocean mixed layer temperature rises by the same amount. For a forcing of 0.4 W/m2 (average over last half-century) the temperature rise is 0.25°C.
Small changes in assumptions about the air above the ocean produce large changes in the temperature rise. For example, if the calculation is repeated with a wind speed of 5 m/s instead of 3 m/s, the temperature rise for 1 w/m2 forcing is reduced from 0.6°C to 0.25°C. If the wind speed is reduced to 2 m/s the calculation becomes unstable. As Watts (1980) emphasized, “The surface heat flux calculation is very, very sensitive to small changes in the components of the heat balance”.
He pointed out that a heat balance at the tropopause does not suffer from this problem. However, a heat balance at the tropopause does not provide us with direct insight as to the effect of back radiation on the oceans.
It is evident from these calculations that a crucial unknown is the change in the condition of the air above the ocean as the ocean surface warms. In the zeroth order model, it was assumed that the air did not change as the ocean surface warmed. As a result, when the ocean surface warms in this model, although the net back radiation increases extremely slowly, the latent heat loss and sensible heat loss increase at significant rates. This allows the ocean surface to rid itself of excess energy that would have accumulated due to a reduction in flux from the mixed layer to the ocean surface. Hence a new equilibrium is achieved with a rather small increase in ocean temperature. In the first-order model, the air temperature is set equal to the ocean temperature and the relative humidity is assumed to be constant. Thus the sensible heat loss is zero. The latent heat loss increases with temperature as before. However, the net back radiation now decreases sharply as the ocean and temperatures increase, due to the increased humidity in the air. Thus, the ocean surface is less able to lose energy to the air, and the temperature rise is greater. The problem is that the total heat loss from the ocean depends on three terms, one of which is assumed to be zero, a second decreases with temperature, and the third increases with temperature. The term that decreases with increasing temperature depends critically on the humidity in the air above the ocean. The term that increases with temperature depends critically on wind speed. Depending on assumptions made about these variables, the final equilibrium temperature of the mixed layer of the ocean can be almost any value. The experimental data on ocean warming have considerable variance from investigator to investigator, but the results suggest the oceans have received about 0.5 W/m2 of warming over the past half century.
In our model, the initial response of the ocean to an increase in back radiation flux is an increase in the ocean surface temperature, TS, while the mixed layer of ocean remains at its original temperature, TL. Although the increase in TS tends to increase heat loss from the surface, this increase is far less than the increase in back radiation flux. But the effect of an increase in TS is a decrease in TL – TS, resulting in a large reduction in energy flux transported from the mixed layer to the surface, so that the mixed layer loses heat at a lower rate (i.e. it warms). The new (increased) value of TS is obtained when the sum of increased heat losses upward and decreased ocean energy flux to the surface balances the increase in back radiation flux. As time progresses, the energy flux from the mixed layer to the surface remains smaller than it was before the forcing was applied, and therefore it warms (TL increases with time). As TL increases, the ocean energy flux to the surface gradually increases. After passage of sufficient time, a new equilibrium is eventually established in which TS and TL are both higher than before the forcing was applied. To model this new equilibrium it is sufficient to use a lumped ocean model for the mixed layer and surface by assuming TS ~ TL because the difference between TS and TL is expected to be smaller than the temperature change resulting from the forcing. One then takes an energy flux balance about the ocean surface, and varies TS ~ TL until the calculated heat loss from the surface equals the forcing. This is the new equilibrium value of TS ~ TL.
In order to utilize the model quantitatively, we need to estimate the increase in back radiation flux at the ocean surface due to increased CO2, and we also need to estimate the changes that occur in the air above the ocean (in order to estimate heat losses from the ocean to the air).
The forcing at the surface due to doubling of CO2 from the pre-industrial value is about 1 to 1.2 W/m2. What is not clear is to what extent this basic forcing is augmented by radiation from a warmer troposphere. Ramanathan (1981) concluded that this effect is very large but his results seem grossly exaggerated.
The effect of increased ocean surface temperature on the air above can only be conjectured. If radiation from a warmer troposphere is neglected, and one makes the zero’th order assumption that the atmosphere remains unchanged as the ocean warms, one finds a very small increase in ocean temperature (~0.05°C) due to an increase in back radiation of 1 W/m2. On the other hand, if radiation from a warmer troposphere is neglected, and one assumes that the air temperature remains equal to the ocean surface temperature and the relative humidity of the air remains constant, the calculated increase in ocean temperature due to an increase in back radiation of 1 W/m2 is about 0.6°C. Unfortunately, this calculation is very sensitive to assumptions made about the condition of the air, and even more sensitive to assumptions about the wind velocity. Hence it is not possible to obtain precise quantitative estimates of the ultimate equilibrium temperature of the mixed ocean layer when subjected to a forcing at the surface.
One thing we can assert however, is that warming of the oceans by an increase in downwelling infrared radiation to the surface is an efficient process, and the initial rate of heat loss by the mixed layer is only slightly less than the magnitude of the imposed forcing. The average surface forcing due to increased CO2 over the past 55 years was roughly 0.4 W/m2. It is therefore not unreasonable to expect that the upper mixed level of the ocean would have warmed by an input of roughly this amount over that time period. Experimental data on warming of the oceans indicate that over the past ~50 years, the average warming was due to a flux of about this magnitude.
JC note: This post was submitted via email. Here is Donald Rapp’s biosketch. I worked with Donald Rapp to shorten the post relative to his original paper. As with all guest posts, please keep your comments relevant and civil.