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Archive for the ‘Ocean Physics’ Category

In Thirteen – Terminator II we had a cursory look at the different “proxies” for temperature and ice volume/sea level. And we’ve considered some issues around dating of proxies.

There are two main proxies we have used so far to take a look back into the ice ages:

  • δ18O in deep ocean cores in the shells of foraminifera – to measure ice volume
  • δ18O in the ice in ice cores (Greenland and Antarctica) – to measure temperature

Now we want to take a closer look at the proxies themselves. It’s a necessary subject if we want to understand ice ages, because the proxies don’t actually measure what they might be assumed to measure. This is a separate issue from the dating: of ice; of gas trapped in ice; and of sediments in deep ocean cores.

If we take samples of ocean water, H2O, and measure the proportion of the oxygen isotopes, we find (Ferronsky & Polyakov 2012):

  • 16O – 99.757 %
  • 17O –   0.038%
  • 18O –   0.205%

There is another significant water isotope, Deuterium – aka, “heavy hydrogen” – where the water molecule is HDO, also written as 1H2HO – instead of H2O.

The processes that affect ratios of HDO are similar to the processes that affect the ratios of H218O, and consequently either isotope ratio can provide a temperature proxy for ice cores. A value of δD equates, very roughly, to 10x a value of δ18O, so mentally you can use this ratio to convert from δ18O to δD (see note 1).

In Note 2 I’ve included some comments on the Dole effect, which is the relationship between the ocean isotopic composition and the atmospheric oxygen isotopic composition. It isn’t directly relevant to the discussion of proxies here, because the ocean is the massive reservoir of 18O and the amount in the atmosphere is very small in comparison (1/1000). However, it might be of interest to some readers and we will return to the atmospheric value later when looking at dating of Antarctic ice cores.

Terminology and Definitions

The isotope ratio, δ18O, of ocean water = 2.005 ‰, that is, 0.205 %. This is turned into a reference, known as Vienna Standard Mean Ocean Water. So with respect to VSMOW, δ18O, of ocean water = 0. It’s just a definition. The change is shown as δ, the Greek symbol for delta, very commonly used in maths and physics to mean “change”.

The values of isotopes are usually expressed in terms of changes from the norm, that is, from the absolute standard. And because the changes are quite small they are expressed as parts per thousand = per mil = ‰, instead of percent, %.

So as δ18O changes from 0 (ocean water) to -50‰ (typically the lowest value of ice in Antarctica), the proportion of 18O goes from 0.20% (2.0‰) to 0.19% (1.9‰).

If the terminology is confusing think of the above example as a 5% change. What is 5% of 20? Answer is 1; and 20 – 1 = 19. So the above example just says if we reduce the small amount, 2 parts per thousand of 18O by 5% we end up with 1.9 parts per thousand.

Here is a graph that links the values together:

From Hoef 2009

From Hoefs 2009

Figure 1

Fractionation, or Why Ice Sheets are So Light

We’ve seen this graph before – the δ18O (of ice) in Greenland (NGRIP) and Antarctica (EDML) ice sheets against time:

From EPICA 2006

From EPICA 2006

Figure 2

Note that the values of δ18O from Antarctica (EDML – top line) through the last 150 kyrs are from about -40 to -52 ‰. And the values from Greenland (NGRIP – black line in middle section) are from about -32 to -44 ‰.

There are some standard explanations around – like this link – but the I’m not sure the graphic alone quite explains it – unless you understand the subject already..

If we measure the 18O concentration of a body of water, then we measure the 18O concentration of the water vapor above it, we find that the water vapor value has 18O at about -10 ‰ compared with the body of water. We write this as δ18O = -10 ‰. That is, the water vapor is a little lighter, isotopically speaking, than the ocean water.

The processes (fractionation) that cause this are easy to reproduce in the lab:

  • during evaporation, the lighter isotopes evaporate preferentially
  • during precipitation, the heavier isotopes precipitate preferentially

(See note 3).

So let’s consider the journey of a parcel of water vapor evaporated somewhere near the equator. The water vapor is a little reduced in 18O (compared with the ocean) due to the evaporation process. As the parcel of air travels away from the equator it rises and cools and some of the water vapor condenses. The initial rain takes proportionately more 18O than is in the parcel – so the parcel of air gets depleted in 18O. It keeps moving away from the equator, the air gets progressively colder, it keeps raining out, and the further it goes the less the proportion of 18O remains in the parcel of air. By the time precipitation forms in polar regions the water or ice is very light isotopically, that is, δ18O is the most negative it can get.

As a very simplistic idea of water vapor transport, this explains why the ice sheets in Greenland and Antarctica have isotopic values that are very low in 18O. Let’s take a look at some data to see how well such a simplistic idea holds up..

The isotopic composition of precipitation:

From Gat 2010

From Gat 2010

Figure 3 – Click to Enlarge

We can see the broad result represented quite well – the further we are in the direction of the poles the lower the isotopic composition of precipitation.

In contrast, when we look at local results in some detail we don’t see such a tidy picture. Here are some results from Rindsberger et al (1990) from central and northern Israel:

From Rindsberger et al 1990

From Rindsberger et al 1990

Figure 4

From Rindsberger et al 1990

From Rindsberger et al 1990

Figure 5

The authors comment:

It is quite surprising that the seasonally averaged isotopic composition of precipitation converges to a rather well-defined value, in spite of the large differences in the δ value of the individual precipitation events which show a range of 12‰ in δ18O.. At Bet-Dagan.. from which we have a long history.. the amount weighted annual average is δ18O = 5.07 ‰ ± 0.62 ‰ for the 19 year period of 1965-86. Indeed the scatter of ± 0.6‰ in the 19-year long series is to a significant degree the result of a 4-year period with lower δ values, namely the years 1971-75 when the averaged values were δ18O = 5.7 ‰ ± 0.2 ‰. That period was one of worldwide climate anomalies. Evidently the synoptic pattern associated with the precipitation events controls both the mean isotopic values of the precipitation and its variability.

The seminal 1964 paper by Willi Dansgaard is well worth a read for a good overview of the subject:

As pointed out.. one cannot use the composition of the individual rain as a direct measure of the condensation temperature. Nevertheless, it has been possible to show a simple linear correlation between the annual mean values of the surface temperature and the δ18O content in high latitude, non-continental precipitation. The main reason is that the scattering of the individual precipitation compositions, caused by the influence of numerous meteorological parameters, is smoothed out when comparing average compositions at various locations over a sufficiently long period of time (a whole number of years).

The somewhat revised and extended correlation is shown in fig. 3..

From Dansgaard 1964

From Dansgaard 1964

Figure 6

So we appear to have a nice tidy picture when looking at annual means, a little bit like the (article) figure 3 from Gat’s 2010 textbook.

Before “muddying the waters” a little, let’s have a quick look at ocean values.

Ocean δ18O

We can see that the ocean, as we might expect, is much more homogenous, especially the deep ocean. Note that these results are δD (think, about 10x the value of δ18O):

From Ferronsky & Polyakov (2012)

From Ferronsky & Polyakov (2012)

Figure 7 – Click to enlarge

And some surface water values of δD (and also salinity), where we see a lot more variation, again as might expect:

From Ferronsky & Polyakov 2012

From Ferronsky & Polyakov 2012

Figure 8

If we do a quick back of the envelope calculation, using the fact that the sea level change between the last glacial maximum (LGM) and the current interglacial was about 120m, the average ocean depth is 3680m we expect a glacial-interglacial change in the ocean of about 1.5 ‰.

This is why the foraminifera near the bottom of the ocean, capturing 18O from the ocean, are recording ice volume, whereas the ice cores are recording atmospheric temperatures.

Note as well that during the glacial, with more ice locked up in ice sheets, the value of ocean δ18O will be higher. So colder atmospheric temperatures relate to lower values of δ18O in precipitation, but – due to the increase in ice, depleted in 18O – higher values of ocean δ18O.

Muddying the Waters

Hoefs 2009, gives a good summary of the different factors in isotopic precipitation:

The first detailed evaluation of the equilibrium and nonequilibrium factors that determine the isotopic composition of precipitation was published by Dansgaard (1964). He demonstrated that the observed geographic distribution in isotope composition is related to a number of environmental parameters that characterize a given sampling site, such as latitude, altitude, distance to the coast, amount of precipitation, and surface air temperature.

Out of these, two factors are of special significance: temperature and the amount of precipitation. The best temperature correlation is observed in continental regions nearer to the poles, whereas the correlation with amount of rainfall is most pronounced in tropical regions as shown in Fig. 3.15.

The apparent link between local surface air temperature and the isotope composition of precipitation is of special interest mainly because of the potential importance of stable isotopes as palaeoclimatic indicators. The amount effect is ascribed to gradual saturation of air below the cloud, which diminishes any shift to higher δ18O-values caused by evaporation during precipitation.

[Emphasis added]

From Hoefs 2009

From Hoefs 2009

Figure 9

The points that Hoefs make indicate some of the problems relating to using δ18O as the temperature proxy. We have competing influences that depend on the source and journey of the air parcel responsible for the precipitation. What if circulation changes?

For readers who have followed the past discussions here on water vapor (e.g., see Clouds & Water Vapor – Part Two) this is a similar kind of story. With water vapor, there is a very clear relationship between ocean temperature and absolute humidity, so long as we consider the boundary layer. But what happens when the air rises high above that – then the amount of water vapor at any location in the atmosphere is dependent on the past journey of air, and as a result the amount of water vapor in the atmosphere depends on large scale circulation and large scale circulation changes.

The same question arises with isotopes and precipitation.

The ubiquitous Jean Jouzel and his colleagues (including Willi Dansgaard) from their 1997 paper:

In Greenland there are significant differences between temperature records from the East coast and the West coast which are still evident in 30 yr smoothed records. The isotopic records from the interior of Greenland do not appear to follow consistently the temperature variations recorded at either the east coast or the west coast..

This behavior may reflect the alternating modes of the North Atlantic Oscillation..

They [simple models] are, however, limited to the study of idealized clouds and cannot account for the complexity of large convective systems, such as those occurring in tropical and equatorial regions. Despite such limitations, simple isotopic models are appropriate to explain the main characteristics of dD and d18O in precipitation, at least in middle and high latitudes where the precipitation is not predominantly produced by large convective systems.

Indeed, their ability to correctly simulate the present-day temperature-isotope relationships in those regions has been the main justification of the standard practice of using the present day spatial slope to interpret the isotopic data in terms of records of past temperature changes.

Notice that, at least for Antarctica, data and simple models agree only with respect to the temperature of formation of the precipitation, estimated by the temperature just above the inversion layer, and not with respect to the surface temperature, which owing to a strong inversion is much lower..

Thus one can easily see that using the spatial slope as a surrogate of the temporal slope strictly holds true only if the characteristics of the source have remained constant through time.

[Emphases added]

If all the precipitation occurs during warm summer months, for example, the “annual δ18O” will naturally reflect a temperature warmer than Ts [annual mean]..

If major changes in seasonality occur between climates, such as a shift from summer-dominated to winter- dominated precipitation, the impact on the isotope signal could be large..it is the temperature during the precipitation events that is imprinted in the isotopic signal.

Second, the formation of an inversion layer of cold air up to several hundred meters thick over polar ice sheets makes the temperature of formation of precipitation warmer than the temperature at the surface of the ice sheet. Inversion forms under a clear sky.. but even in winter it is destroyed rapidly if thick cloud moves over a site..

As a consequence of precipitation intermittancy and of the existence of an inversion layer, the isotope record is only a discrete and biased sampling of the surface temperature and even of the temperature at the atmospheric level where the precipitation forms. Current interpretation of paleodata implicitly assumes that this bias is not affected by climate change itself.

Now onto the oceans, surely much simpler, given the massive well-mixed reservoir of 18O?

Mix & Ruddiman (1984):

The oxygen-isotopic composition of calcite is dependent on both the temperature and the isotopic composition of the water in which it is precipitated

..Because he [Shackleton] analyzed benthonic, instead of planktonic, species he could assume minimal temperature change (limited by the freezing point of deep-ocean water). Using this constraint, he inferred that most of the oxygen-isotope signal in foraminifera must be caused by changes in the isotopic composition of seawater related to changing ice volume, that temperature changes are a secondary effect, and that the isotopic composition of mean glacier ice must have been about -30 ‰.

This estimate has generally been accepted, although other estimates of the isotopic composition have been made by Craig (-17‰); Eriksson (-25‰), Weyl (-40‰) and Dansgaard & Tauber (≤30‰)

..Although Shackleton’s interpretation of the benthonic isotope record as an ice-volume/sea- level proxy is widely quoted, there is considerable disagreement between ice-volume and sea- level estimates based on δ18O and those based on direct indicators of local sea level. A change in δ18O of 1.6‰ at δ(ice) = – 35‰ suggests a sea-level change of 165 m.

..In addition, the effect of deep-ocean temperature changes on benthonic isotope records is not well constrained. Benthonic δ18O curves with amplitudes up to 2.2 ‰ exist (Shackleton, 1977; Duplessy et al., 1980; Ruddiman and McIntyre, 1981) which seem to require both large ice- volume and temperature effects for their explanation.

Many other heavyweights in the field have explained similar problems.

We will return to both of these questions in the next article.

Conclusion

Understanding the basics of isotopic changes in water and water vapor is essential to understand the main proxies for past temperatures and past ice volumes. Previously we have looked at problems relating to dating of the proxies, in this article we have looked at the proxies themselves.

There is good evidence that current values of isotopes in precipitation and ocean values give us a consistent picture that we can largely understand. The question about the past is more problematic.

I started looking seriously at proxies as a means to perhaps understand the discrepancies for key dates of ice age terminations between radiometric dating and ocean cores (see Thirteen – Terminator II). Sometimes the more you know, the less you understand..

Articles in the Series

Part One – An introduction

Part Two – Lorenz – one point of view from the exceptional E.N. Lorenz

Part Three – Hays, Imbrie & Shackleton – how everyone got onto the Milankovitch theory

Part Four – Understanding Orbits, Seasons and Stuff – how the wobbles and movements of the earth’s orbit affect incoming solar radiation

Part Five – Obliquity & Precession Changes – and in a bit more detail

Part Six – “Hypotheses Abound” – lots of different theories that confusingly go by the same name

Part Seven – GCM I – early work with climate models to try and get “perennial snow cover” at high latitudes to start an ice age around 116,000 years ago

Part Seven and a Half – Mindmap – my mind map at that time, with many of the papers I have been reviewing and categorizing plus key extracts from those papers

Part Eight – GCM II – more recent work from the “noughties” – GCM results plus EMIC (earth models of intermediate complexity) again trying to produce perennial snow cover

Part Nine – GCM III – very recent work from 2012, a full GCM, with reduced spatial resolution and speeding up external forcings by a factors of 10, modeling the last 120 kyrs

Part Ten – GCM IV – very recent work from 2012, a high resolution GCM called CCSM4, producing glacial inception at 115 kyrs

Pop Quiz: End of An Ice Age – a chance for people to test their ideas about whether solar insolation is the factor that ended the last ice age

Eleven – End of the Last Ice age – latest data showing relationship between Southern Hemisphere temperatures, global temperatures and CO2

Twelve – GCM V – Ice Age Termination – very recent work from He et al 2013, using a high resolution GCM (CCSM3) to analyze the end of the last ice age and the complex link between Antarctic and Greenland

Thirteen – Terminator II – looking at the date of Termination II, the end of the penultimate ice age – and implications for the cause of Termination II

Fourteen – Concepts & HD Data – getting a conceptual feel for the impacts of obliquity and precession, and some ice age datasets in high resolution

Fifteen – Roe vs Huybers – reviewing In Defence of Milankovitch, by Gerard Roe

Sixteen – Roe vs Huybers II – comparing the results if we take the Huybers dataset and tie the last termination to the date implied by various radiometric dating

Eighteen – “Probably Nonlinearity” of Unknown Origin – what is believed and what is put forward as evidence for the theory that ice age terminations were caused by orbital changes

Nineteen – Ice Sheet Models I – looking at the state of ice sheet models

References

Isotopes of the Earth’s Hydrosphere, VI Ferronsky & VA Polyakov, Springer (2012)

Isotope Hydrology – A Study of the Water Cycle, Joel R Gat, Imperial College Press (2010)

Stable Isotope Geochemistry, Jochen Hoefs, Springer (2009)

Patterns of the isotopic composition of precipitation in time and space: data from the Israeli storm water collection program, M Rindsberger, Sh Jaffe, Sh Rahamim and JR Gat, Tellus (1990) – free paper

Stable isotopes in precipitation, Willi Dansgaard, Tellus (1964) – free paper

Validity of the temperature reconstruction from water isotopes in ice cores, J Jouzel, RB Alley, KM Cuffey, W Dansgaard, P Grootes, G Hoffmann, SJ Johnsen, RD Koster, D Peel, CA Shuman, M Stievenard, M Stuiver, J White, Journal of Geophysical Research (1997) – free paper

Oxygen Isotope Analyses and Pleistocene Ice Volumes, Mix & Ruddiman, Quaternary Research (1984)  – free paper

- and on the Dole effect, only covered in Note 2:

The Dole effect and its variations during the last 130,000 years as measured in the Vostok ice core, Michael Bender, Todd Sowers, Laurent Labeyrie, Global Biogeochemical Cycles (1994) – free paper

A model of the Earth’s Dole effect, Georg Hoffmann, Matthias Cuntz, Christine Weber, Philippe Ciais, Pierre Friedlingstein, Martin Heimann, Jean Jouzel, Jörg Kaduk, Ernst Maier-Reimer, Ulrike Seibt & Katharina Six, Global Biogeochemical Cycles (2004) – free paper

The isotopic composition of atmospheric oxygen Boaz Luz & Eugeni Barkan, Global Biogeochemical Cycles (2011) – free paper

Notes

Note 1: There is a relationship between δ18O and δD which is linked to the difference in vapor pressures between H2O and HDO in one case and H216O and H218O in the other case.

δD = 8 δ18O + 10 – known as the Global Meteoric Water Line.

The equation is more of a guide and real values vary sufficiently that I’m not really clear about its value. There are lengthy discussions of it and the variations from it in Ferronsky & Polyakov.

Note 2: The Dole effect

When we measure atmospheric oxygen, we find that the δ18O = 23.5 ‰ with respect to the oceans (VSMOW) – this is the Dole effect

So, oxygen in the atmosphere has a greater proportion of 18O than the ocean

Why?

How do the atmosphere and ocean exchange oxygen? In essence, photosynthesis turns sunlight + water (H2O) + carbon dioxide (CO2) –> sugar + oxygen (O2).

Respiration turns sugar + oxygen –> water + carbon dioxide + energy

The isotopic composition of the water in photosynthesis affects the resulting isotopic composition in the atmospheric oxygen.

The reason the Dole effect exists is well understood, but the reason why the value comes out at 23.5‰ is still under investigation. This is because the result is the global aggregate of lots of different processes. So we might understand the individual processes quite well, but that doesn’t mean the global value can be calculated accurately.

It is also the case that δ18O of atmospheric O2 has varied in the past – as revealed first of all in the Vostok ice core from Antarctica.

Michael Bender and his colleagues had a go at calculating the value from first principles in 1994. As they explain (see below), although it might seem as though their result is quite close to the actual number it is not a very successful result at all. Basically due to the essential process you start at 20‰ and should get to 23.5‰, but they o to 20.8‰.

Bender et al 1994:

The δ18O of O2.. reflects the global responses of the land and marine biospheres to climate change, albeit in a complex manner.. The magnitude of the Dole effect mainly reflects the isotopic composition of O2 produced by marine and terrestrial photosynthesis, as well as the extent to while the heavy isotope is discriminated against during respiration..

..Over the time period of interest here, photosynthesis and respiration are the most important reactions producing and consuming O2. The isotopic composition of O2 in air must therefore be understood in terms of isotope fractionation associated with these reactions.

The δ18O of O2 produced by photosynthesis is similar to that of the source water. The δ18O of O2 produced by marine plants is thus 0‰. The δ18O of O2 produced on the continents has been estimated to lie between +4 and +8‰. These elevated δ18O values are the result of elevated leaf water δ18O values resulting from evapotranspiration.

..The calculated value for the Dole effect is then the productivity-weighted values of the terrestrial and marine Dole effects minus the stratospheric diminution: +20.8‰. This value is considerably less than observed (23.5‰). The difference between the expected value and the observed value reflects errors in our estimates and, conceivably, unrecognized processes.

Then they assess the Vostok record, where the main question is less about why the Dole effect varies apparently with precession (period of about 20 kyrs), than why the variation is so small. After all, if marine and terrestrial biosphere changes are significant from interglacial to glacial then surely those changes would reflect more strongly in the Dole effect:

Why has the Dole effect been so constant? Answering this question is impossible at the present time, but we can probably recognize the key influences..

They conclude:

Our ability to explain the magnitude of the contemporary Dole effect is a measure of our understanding of the global cycles of oxygen and water. A variety of recent studies have improved our understanding of many of the principles governing oxygen isotope fractionation during photosynthesis and respiration.. However, our attempt to quantitively account for the Dole effect in terms of these principles was not very successful.. The agreement is considerably worse than it might appear given the fact that respiratory isotope fractionation alone must account for ~20‰ of the stationary enrichment of the 18O of O2 compared with seawater..

..[On the Vostok record] Our results show that variation in the Dole effect have been relatively small during most of the last glacial-interglacial cycle. These small changes are not consistent with large glacial increases in global oceanic productivity.

[Emphasis added]

Georg Hoffmann and his colleagues had another bash 10 years later and did a fair bit better:

The Earth’s Dole effect describes the isotopic 18O/16O-enrichment of atmospheric oxygen with respect to ocean water, amounting under today’s conditions to 23.5‰. We have developed a model of the Earth’s Dole effect by combining the results of three- dimensional models of the oceanic and terrestrial carbon and oxygen cycles with results of atmospheric general circulation models (AGCMs) with built-in water isotope diagnostics.

We obtain a range from 22.4‰ to 23.3‰ for the isotopic enrichment of atmospheric oxygen. We estimate a stronger contribution to the global Dole effect by the terrestrial relative to the marine biosphere in contrast to previous studies. This is primarily caused by a modeled high leaf water enrichment of 5–6‰. Leaf water enrichment rises by ~1‰ to 6–7‰ when we use it to fit the observed 23.5‰ of the global Dole effect.

Very recently Luz & Barkan (2011), backed up by lots of new experimental work produced a slightly closer estimate with some revisions of the Hoffman et al results:

Based on the new information on the biogeochemical mechanisms involved in the global oxygen cycle, as well as new and more precise experimental data on oxygen isotopic fractionation in various processes obtained over the last 15 years, we have reevaluated the components of the Dole effect.Our new observations on marine oxygen isotope effects, as well as, new findings on photosynthetic fractionation by marine organisms lead to the important conclusion that the marine, terrestrial and the global Dole effects are of similar magnitudes.

This result allows answering a long‐standing unresolved question on why the magnitude of the Dole effect of the last glacial maximum is so similar to the present value despite enormous environmental differences between the two periods. The answer is simple: if DEmar [marine Dole effect] and DEterr [terrestrial Dole effect] are similar, there is no reason to expect considerable variations in the magnitude of the Dole effect as the result of variations in the ratio terrestrial to marine O2 production.

Finally, the widely accepted view that the magnitude of the Dole effect is controlled by the ratio of land‐to‐sea productivity must be changed. Instead of the land‐sea control, past variations in the Dole effect are more likely the result of changes in low‐latitude hydrology and, perhaps, in structure of marine phytoplankton communities.

[Emphasis added]

Note 3:

Jochen Hoefs (2009):

Under equilibrium conditions at 25ºC, the fractionation factors for evaporating water are 1.0092 for 18O and 1.074 for D. However under natural conditions, the actual isotopic composition of water is more negative than the predicted equilibrium values due to kinetic effects.

The discussion of kinetic effects gets a little involved and I don’t think is really necessary to understand – the values of isotopic fractionation during evaporation and condensation are well understood. The confounding factors around what the proxies really measure relate to the journey (i.e. temperature history) and mixing of the various air parcels as well as the temperature of air relating to the precipitation event – is the surface temperature, the inversion temperature, both?

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In the last article we had a look at the depth of the “mixed ocean layer” (MLD) and its implications for the successful measurement of climate sensitivity (assuming such a parameter exists as a constant).

In Part One I created a Matlab model which reproduced the same problems as Spencer & Braswell (2008) had found. This model had one layer  (an “ocean slab” model) to represent the MLD with a “noise” flux into the deeper ocean (and a radiative noise flux at top of atmosphere). Murphy & Forster claimed that longer time periods require an MLD of increased depth to “model” the extra heat flow into the deeper ocean over time:

Because heat slowly penetrates deeper into the ocean, an appropriate depth for heat capacity depends on the length of the period over which Eq. (1) is being applied (Watterson 2000; Held et al. 2010). For 80-yr global climate model runs, Gregory (2000) derived an optimum mixed layer depth of 150 m. Watterson (2000) found an initial global heat capacity equivalent to a mixed layer of 200 m and larger values for longer simulations.

This seems like it might make sense – if we wanted to keep a “zero dimensional model”. But it’s questionable whether the model retains any value with this “fudge”. So because heat actually moves from the mixed layer into the deeper ocean (rather than the mixed layer increasing in depth) I instead enhanced the model to create a heat flux from the MLD through a number of ocean layers with a parameter called the vertical eddy diffusivity to determine this heat flux.

So the model is now a 1D model with a parameterized approach to ocean convection.

Eddy Diffusivity

The concept here is the analogy of conductivity but when convection is instead the primary mover of heat.

Heat flow by conduction is governed by a material property called conductivity and by the temperature difference. Changes in temperature are governed by heat flow and by the heat capacity. The result is this equation for reference and interest – so don’t worry if you don’t understand it:

∂T / ∂tα∂²T / ∂z²  – the 1-d version (see note 1)

where T = temperature, t = time, α = thermal diffusivity and z = depth

What it says in almost plain English is that the change in temperature with respect to time is equal to the thermal diffusivity times the change in gradient of temperature with depth. Don’t worry if that’s not clear (and there is a explanation of the simple steps required to calculate this in note 1).

Now the thermal diffusivity, α:

α = k/cpρ, where k = conductivity, cp = heat capacity and ρ = density

So, an important bit to understand..

  • if the conductivity is high and the heat capacity is low then temperature can change quickly
  • if the conductivity is high and the heat capacity is high then it slows down temperature change, and
  • if the conductivity is low and the heat capacity is high then temperature takes much longer to change

Many researchers have attempted to measure an average value for eddy diffusivity in the ocean (and in lakes). The concept here, an explained in Part Two, is that turbulent motions of the ocean move heat much more effectively than conduction. The value can’t be calculated from first principles because that would mean solving the problem of turbulence, which is one of the toughest problems in physics. Instead it has to be estimated from measurements.

There is an inherent problem with eddy diffusivity for vertical heat transfer that we will come back to shortly.

There is also a minor problem of notation that is “solved” here by changing the notation. Usually conductivity is written as “k”. However, most papers on eddy diffusivity write diffusivity as “k”, sometimes “K”, sometimes “κ” (Greek ‘kappa’) – creating potential confusion so I revert back to “α”. And to make it clear that it is the convective value rather than the conductive value, I use αeddy. And for the equivalent parameter to conductivity, keddy.

keddy = αeddycpρ

because cp= 4200 J/K.kg and ρ ≈ 1000 kg/m³:

keddy =4.2 x 106  αeddy – it’s useful to be able to see what the diffusivity means in terms of an equivalent “conductivity” type parameter

Measurements of Eddy Diffusivity

Oeschger et al (1975):

α is an apparent global eddy diffusion coefficient which helps to reproduce an average transport phenomenon consisting of a series of distinct and overlapping mechanisms.

Oeschger and his co-workers studied the problem via the diffusion into the ocean of 14C from nuclear weapons testing.

The range they calculated for αeddy = 1 x 10-4 – 1.8 x 10-4 m²/s.

This equates to keddy = 420 – 760 W/m.K, and by comparison, the conductivity of still water, k = 0.6 W/m.K – making convection around 1,000 times more effective at moving heat vertically through the ocean.

Broecker et al (1980) took a similar approach to estimating this value and commented:

We do not mean to imply that the process of vertical eddy mixing actually occurs within the body of the main oceanic thermocline. Indeed, the values we require are an order of magnitude greater than those permitted by conventional oceanographic wisdom (see Garrett, 1979, for summary).

The vertical eddy coefficients used here should rather be thought of as parameters that take into account all the processes that transfer tracers across density horizons. In addition to vertical mixing by eddies, these include mixing induced by sediment friction at the ocean margins and mixing along the surface in the regions where density horizons outcrop.

Their calculation, like Oeschger’s, used a simple model with the observed values plugged in to estimate the parameter:

Anyone familiar with the water mass structure and ventilation dynamics of the ocean will quickly realize that the box-diffusion model is by no means a realistic representation. No simple modification to the model will substantially improve the situation.

To do so we must take a giant step in complexity to a new generation of models that attempt to account for the actual geometry of ventilation of the sea. We are as yet not in a position to do this in a serious way. At least a decade will pass before a realistic ocean model can be developed.

The values they calculated for eddy diffusivity were broken up into different regions:

  • αeddy(equatorial) = 3.5 x 10-5 m²/s
  • αeddy(temperate) = 2.0 x 10-4 m²/s
  • αeddy(polar) = 3.0 x 10-4 m²/s

We will use these values from Broecker to see what happens to the measurement problems of climate sensitivity when used in my simple model.

These two papers were cited by Hansen et al in their 1985 paper with the values for vertical eddy diffusivity used to develop the value of the “effective mixed depth” of the ocean.

In reviewing these papers and searching for more recent work in the field, I tapped into a rich vein of research that will be the subject of another day.

First, Ledwell et al (1998) who measured eddy diffusivity via SF6 that they injected into the ocean:

The diapycnal eddy diffusivity K estimated for the first 6 months was 0.12 ± 0.02 x10-4 m²/s, while for the subsequent 24 months it was 0.17 ± 0.02 x10-4 m²/s.

[Note: units changed from cm²/s into m²/s for consistency]

It is worth reading their comment on this aspect of ocean dynamics. (Note that isopycnal = contact density surfaces and diapycnal = across isopycnal):

The circulation of the ocean is severely constrained by density stratification. A water parcel cannot move from one surface of constant potential density to another without changing its salinity or its potential temperature. There are virtually no sources of heat outside the sunlit zone and away from the bottom where heat diffuses from the lithosphere, except for the interesting hydrothermal vents in special regions. The sources of salinity changes are similarly confined to the boundaries of the ocean. If water in the interior is to change potential density at all, it must be by mixing across density surfaces (diapycnal mixing) or by stirring and mixing of water of different potential temperature and salinity along isopycnal surfaces (isopycnal mixing).

Most inferences of dispersion parameters have been made from observations of the large-scale fields or from measurements of dissipation rates at very small scales. Unambiguously direct measurements of the mixing have been rare. Because of the stratification of the ocean, isopycnal mixing involves very different processes than diapycnal mixing, extending to much greater length scales. A direct approach to the study of both isopycnal and diapycnal mixing is to release a tracer and measure its subsequent dispersal. Such an experiment, lasting 30 months and involving more than 105 km² of ocean, is the subject of this paper.

From Jayne (2009):

For example, the Community Climate Simulation Model (CCSM) ocean component model uses a form similar to Eq. (1), but with an upper-ocean value of 0.1 x 10-4 m²/s and a deep-ocean value of 1.0 x 10-4 m²/s, with the transition depth at 1000 m.

However, there is no observational evidence to suggest that the mixing in the ocean is horizontally uniform, and indeed there is significant evidence that it is heterogeneous with spatial variations of several orders of magnitude in its intensity (Polzin et al. 1997; Ganachaud 2003).

More on eddy diffusivity measurements in another article – the parameter has a significant impact on modeling of the ocean in GCMs and there is a lot of current research into this subject.

Eddy Diffusivity and Buoyancy Gradient

Sarmiento et al (1976) measured isotopes near the ocean floor:

Two naturally occurring isotopes can be applied to the determination of the rate of vertical turbulent mixing in the deep sea: 222Rn (half-life 3.824 days) and 228Ra (half-life 5.75 years). In this paper we discuss the results from fourteen 222Rn and two 228Ra profiles obtained as part of the GEOSECS program.

From these results we conclude that the most important factor influencing the vertical eddy diffusivity is the buoyancy gradient [(g/p)(∂ρpot/∂z)]. The vertical diffusivity shows an inverse proportionality to the buoyancy gradient.

Their paper is very much about the measurements and calculations of the deeper ocean, but is relevant for anywhere in the ocean, and helps explain why the different values for different regions were obtained by Broecker that we saw earlier. (Prof. Wallace S. Broecker was a co-author on this paper as well, and has authored/co-authored 100’s of papers on the ocean).

What is the buoyancy gradient and why does it matter?

Cold fluids sink and hot fluids rise. This is because cold substances contract and so are more dense. So in general, in the ocean, the colder water is below and the warmer water above. Probably everyone knows this.

The buoyancy gradient is a measure of how strong this effect is. The change in density with depth determines how resistant the ocean is to being overturned. If the ocean was totally stable no heat would ever penetrate below the mixed layer. But it does. And if the ocean was totally stable then the measurements of 14C from nuclear testing would be zero below the mixed layer.

But it is not surprising that the more stable the ocean is due to the buoyancy gradient the less heat diffuses down by turbulent motion.

And this is why the estimates by Broecker shown earlier have a much lower value of diffusivity for the tropics than for the poles. In general the poles are where deep convection takes place – lots of cold water sinks, mixing the ocean – and the tropics are where much weaker upwelling takes place – because the ocean surface is strongly heated. This is part of the large scale motion of the ocean, known as the thermohaline circulation. More on this another day.

Now water is largely incompressible which means that the density gradient is only determined by temperature and salinity. This creates the problem that eddy diffusivity is a value which is not only parameterized, but also dependent on the vertical temperature difference in the ocean.

Heat flow also depends on temperature difference, but with the opposite relationship. This is not something to untangle today. Today we will just see what happens to our simple model when we use the best estimates of vertical eddy diffusivity.

Modeling, Non-Linearity and Climate Sensitivity Measurement Problems

Murphy & Forster agreed in part with Spencer & Braswell about the variation in radiative noise from CERES measurements. I quote at length, because the Murphy & Forster paper is not freely available:

For the parameter N, SB08 use a random daily shortwave flux scaled so that the standard deviation of monthly averages of outgoing radiation (N – λT) is 1.3 W/m².

They base this on the standard deviation of CERES shortwave data between March 2000 and December 2005 for the oceans between 20 °Nand 20 °S.

We have analyzed the same dataset and find that, after the seasonal cycle and slow changes in forcing are removed, the standard deviation of monthly means of the shortwave radiation is 1.24 W/m², close to the 1.3 W/m² specified by SB08. However, longwave (infrared) radiation changes the energy budget just as effectively from the earth as shortwave radiation (reflected sunlight). Cloud systems that might induce random fluctuations in reflected sunlight also change outgoing longwave radiation. In addition, the feedback parameter λ is due to both longwave and shortwave radiation.

Modeled total outgoing radiation should therefore be compared with the observed sum of longwave and shortwave outgoing radiation, not just the shortwave component. The standard deviation of the sum of longwave and shortwave radiation in the same CERES dataset is 0.94 W/m². Even this is an upper limit, since imperfect spatial sampling and instrument noise contribute to the standard deviation.

[Note I change their α (climate feedback) to λ for consistency with previous articles].

And they continue:

We therefore use 0.94 W/m² as an upper limit to the standard deviation of outgoing radiation over the tropical oceans. For comparison, the standard deviation of the global CERES outgoing radiation is about 0.55 W/m².

All of these points seem valid (however, I am still in the process of examining CERES data, and can’t comment on their actual values of standard deviation. Apart from the minor challenge of extracting the data from the netCDF format there is a lot to examine. A lot of data and a lot of issues surrounding data quality).

However, it raised an interesting idea about non-linearity. Readers who remember on Part One will know that as radiative noise increases and ocean MLD decreases the measurement problem gets worse. And as the radiative noise decreases and ocean MLD increases the measurement problem goes away.

If we average global radiative noise and global MLD, plug these values into a zero-dimensional model and get minimal measurement problem what does this mean?

Due to non-linearity, it tells us nothing.

Averaging the inputs, applying them to a global model (i.e., a zero-dimensional model) and calculating λest (from the regression) gets very different results from applying the inputs separately to each region, averaging the results and calculating λest

I tested this with a simple model – created two regions, one 10% of the surface area, the other 90%. In the larger region the MLD was 200m and the radiative noise was zero; and in the smaller region the MLD was 20m and the (standard deviation of) radiative noise was varied from 0 – 2. The temperature and radiative flux were converted into an area weighted time series and the regression produced large deviations from the real value of λ.

A similar run on a global model with an MLD of 180m and radiative noise of 0-0.2 shows an accurate assessment of λ.

This is to be expected of course.

So with this in mind I tested the new 1D model with different values of ocean depth eddy diffusivity,  radiative noise, and an AR(1) model for the radiative noise. I used values for the tropical region as this is clearly the area most likely to upset the measurement – shallow MLD, higher radiative noise and weaker eddy diffusivity.

As best as I could determine from de Boyer Montegut’s paper, the average MLD for the 20°N – 20°S region is approximately 30m.

Here are the results using Oeschger’s value of eddy diffusivity for the tropics and the tropical value of radiative noise from MF2010 – varying ocean depth around 30m and the value of the AR(1) model for radiative noise:

Figure 1

For reference, as it’s hard to read off the graph, the value at 30m and φ=0.5 is λest = 2.3.

Using the current CCSM value of eddy diffusivity for the upper ocean:

Figure 2

For reference,  the value at 30m and φ=0.5 is λest = 0.2. (Compared with the real value of 3.0)

Note that these values are only for one region, not for the whole globe.

Another important point is that I have used the radiative noise value as the standard deviation of daily radiative noise. I have started to dig into CERES data to see whether such a value can be calculated, and also what typical value of autoregressive parameter should be used (and what kind of ARMA model), but this might take some time.

Yet smaller values of eddy diffusivity are possible for smaller regions, according to Jochum (2009). This would likely cause the problems of estimating climate sensitivity to become worse.

Simple Models

Murphy & Forster comment:

Although highly simplified, a single box model of the earth has some pedagogic value. One must remember that the heat capacity c and feedback parameter λ are not really constants, since heat penetrates more deeply into the ocean on long time scales and there are fast and slow climate feedbacks (Knutti et al. 2008).

It is tempting to add a few more boxes to account for land, ocean, different latitudes, and so forth. Adding more boxes to an energy balancemodel can be problematic because one must ensure that the boxes are connected in a physically consistent way. A good option is to instead consider a global climate model that has many boxes connected in a physically consistent manner.

The point being that no one believes a slab model of the ocean to be a model that gives really useful results. Spencer & Braswell likewise don’t believe that the slab model is in any way an accurate model of the climate.

They used such a model just to demonstrate a possible problem. Murphy & Forster’s criticism doesn’t seem to have solved the problem of “can we measure climate sensitivity?

Or at least, it appears easy to show that slightly different enhancements of the simple model demonstrate continued problems in measuring climate sensitivity – due to the impact of radiative noise in the climate system.

Conclusion

I have produced a simple model and apparently demonstrated continued climate sensitivity measurement problems. This is in contrast to Murphy & Forster who took a different approach and found that the problem went away. However, my model has a more realistic approach to moving heat from the mixed layer into the ocean depths than theirs.

My model does have the drawback that the massive army of Science of Doom model testers and quality control champions are all away on their Xmas break. So the model might be incorrectly coded.

It’s also likely that someone else can come along and take a slightly enhanced version of this model and make the problem vanish.

I have used values for MLD and eddy diffusivity that seem to represent real-world values but I have no idea as to the correct values for standard deviation and auto-correlation of daily radiative noise (or appropriate ARMA model). These values have a big impact on the climate sensitivity measurement problem for reasons explained in Part One.

A useful approach to determining the effect of radiative noise on climate sensitivity measurement might be to use a coupled atmosphere-ocean GCM with a known climate sensitivity and an innovative way of removing radiative noise. These kind of experiments are done all the time to isolate one effect or one parameter.

Perhaps someone has already done this specific test?

I see other potential problems in measuring climate sensitivity. Here is one obvious problem – as the temperature of the mixed layer increases with continued increases in radiative forcing the buoyancy gradient increases and the eddy diffusivity reduces. We can calculate radiative forcing due to “greenhouse” gases quite accurately and therefore remove it from the regression analysis (see Spencer & Braswell 2008 for more on this). But we can’t calculate the change in eddy diffusivity and heat loss to the deeper ocean. This adds another “correlated” term that seems impossible to disentangle from the climate sensitivity calculation.

An alternative way of looking at this is that climate sensitivity might not be a constant – as already noted in Part One.

Articles in this Series

Measuring Climate Sensitivity – Part One

Measuring Climate Sensitivity – Part Two – Mixed Layer Depths

References

Potential Biases in Feedback Diagnosis from Observational Data: A Simple Model Demonstration, Spencer & Braswell, Journal of Climate (2008) – FREE

On the accuracy of deriving climate feedback parameters from correlations between surface temperature and outgoing radiation, Murphy & Forster, Journal of Climate (2010)

A box diffusion model to study the carbon dioxide exchange in nature, Oeschger et al, Tellus (1975)

Modeling the carbon system, Broecker et al, Radiocarbon (1980) – FREE

Climate response times: dependence on climate sensitivity and ocean mixing, Hansen et al, Science (1985)

The study of mixing in the ocean: A brief history, MC Gregg, Oceanography (1991) – FREE

Spatial Variability of Turbulent Mixing in the Abyssal Ocean, Polzin et al, Science (1997) – FREE

The Impact of Abyssal Mixing Parameterizations in an Ocean General Circulation Model, Steven R. Jayne, Journal of Physical Oceanography (2009)

The relationship between vertical eddy diffusion and buoyancy gradient in the deep sea, Sarmiento et al, Earth & Planetary Science Letters (1976)

Mixing of a tracer in the pycnocline, Ledwell et al, JGR (1998)

Impact of latitudinal variations in vertical diffusivity on climate simulations, Jochum, JGR (2009) – FREE

Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology, de Boyer Montegut et al, JGR (2004)

Notes

Note 1: The 1D version is really:

∂T / ∂t = ∂/∂z (α.∂T/∂z)

due to the fact that α can be a function of z (and definitely is in the case of the ocean).

Although this looks tricky – and it is tricky to find analytical solutions – solving the 1D version numerically is very straightforward and anyone can do it.

In plain English is looks something like:

- Heat flow into cell X = temperature difference between cell X and cell X-1

- Heat flow out of cell X = temperature difference between cell X and cell X+1

- Change in temperature = (Heat flow into cell X – Heat flow out of cell X) x time / heat capacity

Note 2: I am in the process of examining CERES data. Apart from the challenge of extracting the data from the netCDF format there is a lot to examine. A lot of data and a lot of issues surrounding data quality.

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In Measuring Climate Sensitivity – Part One we saw that there can be potential problems in attempting to measure the parameter called “climate sensitivity”.

Using a simple model Spencer & Braswell (2008) had demonstrated that even when the value of “climate sensitivity” is constant and known, measurement of it can be obscured for a number of reasons.

The simple model was a “slab model” of the ocean with a top of atmosphere imbalance in radiation.

Murphy & Forster (2010) criticized Spencer & Braswell for a few reasons including the value chosen for the depth of this ocean mixed layer. As the mixed layer depth increases the climate sensitivity measurement problems are greatly reduced.

First, we will consider the mixed layer in the context of that simple model. Then we will consider what it means in real life.

The Simple Model of Climate Sensitivity

The simple model used by Spencer & Braswell has a “mixed ocean layer” of depth 50m.

Figure 1

In the model the mixed layer is where all of the imbalance in top of atmosphere radiation gets absorbed.

The idea in the simple model is that the energy absorbed from the top of atmosphere gets mixed into the top layer of the ocean very quickly. In reality, as we will see, there isn’t such a thing as one layer but it is a handy approximation.

Murphy & Forster commented:

For the heat capacity parameter c, SB08 use the heat capacity of a 50-m ocean mixed layer. This is too shallow to be realistic.

Because heat slowly penetrates deeper into the ocean, an appropriate depth for heat capacity depends on the length of the period over which Eq. (1) is being applied (Watterson 2000; Held et al. 2010).

For 80-yr global climate model runs, Gregory (2000) derived an optimum mixed layer depth of 150 m. Watterson (2000) found an initial global heat capacity equivalent to a mixed layer of 200 m and larger values for longer simulations.

Held et al. (2010) found an initial time constant τ = c/α of about four yr in the Geophysical Fluid Dynamics Laboratory global climate model. Schwartz (2007) used historical data to estimate a globally averaged mixed layer depth of 150 m, or 106 m if the earth were only ocean.

The idea is an attempt to keep the simplicity of one mixed layer for the model, but increase the depth of this mixed layer for longer time periods.

There is always a point where models – simplified versions of the real world – start to break down. This might be the case here.

The initial model was of a mixed layer of ocean, all at the same temperature because the layer is well-mixed – and with some random movement of heat between this mixed layer and the ocean depths. In a more realistic scenario, more heat flows into the deeper ocean as the length of time increases.

What Murphy & Forster are proposing is to keep the simple model and “account” for the ever increasing heat flow into the deeper ocean by using a depth of the mixed layer that is dependent on the time period.

If we do this perhaps the model will work, perhaps it won’t. By “work” we mean provide results that tell us something useful about the real world.

So I thought I would introduce some more realism (complexity) into the model and see what happened. This involves a bit of a journey.

Real Life Ocean Mixed Layer

Water is a very bad conductor of heat – as are plastic and other insulators. Good conductors of heat include metals.

However, in the ocean and the atmosphere conduction is not the primary heat transfer mechanism. It isn’t even significant. Instead, in the ocean it is convection – the bulk movement of fluids – that moves heat. Think of it like this – if you move a “parcel” of water, the heat in that parcel moves with it.

Let’s take a look at the temperature profile at the top of the ocean. Here the first graph shows temperature:

Soloviev & Lukas (1997)

Soloviev & Lukas (1997)

Figure 2

Note that the successive plots are not at higher and higher temperatures – they are just artificially separated to make the results easier to see. During the afternoon the sun heats the top of the ocean. As a result we get a temperature gradient where the surface is hotter than a few meters down. At night and early morning the temperature gradient disappears. (No temperature gradient means that the water is all at the same temperature)

Why is this?

Once the sun sets the ocean surface cools rapidly via radiation and convection to the atmosphere. The result is colder water, which is heavier. Heavier water sinks, so the ocean gets mixed. This same effect takes place on a larger scale for seasonal changes in temperature.

And the top of the ocean is also well mixed due to being stirred by the wind.

A comment from de Boyer Montegut and his coauthors (2004):

A striking and nearly universal feature of the open ocean is the surface mixed layer within which salinity, temperature, and density are almost vertically uniform. This oceanic mixed layer is the manifestation of the vigorous turbulent mixing processes which are active in the upper ocean.

Here is a summary graphic from the excellent Marshall & Plumb:

From Marshall & Plumb (2008)

Figure 3

There’s more on this subject in Does Back-Radiation “Heat” the Ocean? – Part Three.

How Deep is the Ocean Mixed Layer?

This is not a simple question. Partly it is a measurement problem, and partly there isn’t a sharp demarcation between the ocean mixed layer and the deeper ocean. Various researchers have made an effort to map it out.

Here is a global overview, again from Marshall & Plumb:

Figure 4

You can see that the deeper mixed layers occur in the higher latitudes.

Comment from de Boyer Montegut:

The main temporal variabilities of the MLD [mixed layer depth] are directly linked to the many processes occurring in the mixed layer (surface forcing, lateral advection, internal waves, etc), ranging from diurnal [Brainerd and Gregg, 1995] to interannual variability, including seasonal and intraseasonal variability [e.g., Kara et al., 2003a; McCreary et al., 2001]. The spatial variability of the MLD is also very large.

The MLD can be less than 20 m in the summer hemisphere, while reaching more than 500 m in the winter hemisphere in subpolar latitudes [Monterey and Levitus, 1997].

Here is a more complete map by month. Readers probably have many questions about methodology and I recommend reading the free paper:

From de Boyer Montegut et al (2004)

Figure 5 – Click for a larger image

Seeing this map definitely had me wondering about the challenge of measuring climate sensitivity. Spencer & Braswell had used 50m MLD to identify some climate sensitivity measurement problems. Murphy & Forster had reproduced their results with a much deeper MLD to demonstrate that the problems went away.

But what happens if instead we retest the basic model using the actual MLD which varies significantly by month and by latitude?

So instead of “one slab of ocean” at MLD = choose your value, we break up the globe into regions, have different values in each region each month and see what happens to climate sensitivity problems.

By the way, I also attempted to calculate the global annual (area weighted) average of MLD from the maps above, by eye. I also emailed the author of the paper to get some measurement details but no response.

My estimate of the data in this paper was a global annual area weighted average of 62 meters.

Trying Simple Models with Varying MLD

I updated the Matlab program from Measuring Climate Sensitivity – Part One. The globe is now broken up into 30º latitude bands, with the potential for a different value of mixed layer depth for each month of the year.

I created a number of different profiles:

Depth Type 0 – constant with month and latitude, as in the original article

Type 1 – using the values from de Boyer’s paper, as best as can be estimated from looking at the monthly maps.

Type 2 – no change each month, with scaling of 60ºN-90ºN = 100x the value for 0ºN – 30ºN, and 30ºN – 60ºN = 10x the value for 0ºN – 30ºN – similarly for the southern hemisphere.

Type 3 – alternating each month between Type 2 and its inverse, i.e., scaling of 0ºN – 30ºN = 100x the value for 60ºN-90ºN and 30ºN – 60ºN = 10x the value for 60ºN-90ºN.

Type 4 – no variation by latitude, but  month 1 = 1000x month 4, month 2 = 100x month 4, month 3 = 10x month 4, repeating 3 times  per year.

In each case the global annual (area weighted) average = 62m.

Essentially types 2-4 are aimed at creating extreme situations.

Here are some results (review the original article for some of the notation), recalling that the actual climate sensitivity, λ = 3.0:

Figure 6

Figure 7 – as figure 6 without 30-day averaging

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

What’s the message from these results?

In essence, type 0 (the original) and type 1 (using actual MLDs vs latitude and month from de Boyer’s paper) are quite similar – but not exactly the same.

However, if we start varying the MLD by latitude and month in a more extreme way the results come out very differently – even though the global average MLD is the same in each case.

This demonstrates that the temporal and area variation of MLD can have a significant effect and modeling the ocean as one slab – for the purposes of this enterprise – may be risky.

Non-Linearity

We haven’t considered the effect of non-linearity in these simple models. That is, what about interactions between different regions and months. If we created a yet more complex model where heat flowed between regions dependent on the relative depths of the mixed layers what would we find?

Losing the Plot?

Now, in case anyone has lost the plot by this stage – and it’s possible that I have – don’t get confused into thinking that we are evaluating GCM’s and gosh aren’t they simplistic.. No, GCM’s have very sophisticated modeling.

What we have been doing is tracing a path that started with a paper by Spencer & Braswell. This paper used a very simple model to show that with some random daily fluctuations in top of atmosphere radiative flux, perhaps due to clouds, the measurement of climate sensitivity doesn’t match the actual climate sensitivity.

We can do this in a model – prescribe a value and then test whether we can measure it. This is where this simple model came in. It isn’t a GCM.

However, Murphy & Forster came along and said if you use a deeper mixed ocean layer (which they claim is justified) then the measurement of climate sensitivity does more or less match the actual climate sensitivity (they also had comment on the values chosen for radiative flux anomalies, a subject for another day).

What struck me was that the test model needs some significant improvement to be able to assess whether or not climate sensitivity can be measured. And this is with the caveat – if climate sensitivity is a constant.

The Next Phase – More Realistic Ocean Model

As Murphy & Forster have pointed out, the longer the time period, the more heat is “injected” into the deeper ocean from the mixed layer.

So a better model would capture this better than just creating a deeper mixed layer for a longer time. Modeling true global ocean convection is an impossible task.

As a recap, conducted heat flow:

q” = k.ΔT/d

where q” = heat flow per unit area, k = conductivity, ΔT = temperature difference, and d = depth of layer

Take a look at Heat Transfer Basics – Part Zero for more on these basics.

For water, k = 0.6 W/m².K. So, as an example, if we have a 10ºC temperature difference across 1 km depth of water, q” = 0.006 W/m². This is tiny. Heat flow via conduction is insignificant. Convection is what moves heat in the ocean.

Many researchers have measured and estimated vertical heat flow in the ocean to come up with a value for vertical eddy diffusivity. This allows us to make some rough estimates of vertical heat flow via convection.

In the next version of the Matlab program (“in press”) the ocean is modeled with different eddy diffusivities below the mixed ocean layer to see what happens to the measurement of climate sensitivity. So far, the model comes up with wildly varying results when the eddy diffusivity is low, i.e., heat cannot easily move into the ocean depths. And it comes up with normal results when the eddy diffusivity is high, i.e., heat moves relatively quickly into the ocean depths.

Due to shortness of time, this problem has not yet been resolved. More in due course.

This article is already long enough, so the next part will cover the estimated values for eddy diffusivity because it’s an interesting subject

Conclusion

Regular readers of this blog understand that navigating to any kind of conclusion takes some time on my part. And that’s when the subject is well understood. I’m finding that the signposts on the journey to measuring climate sensitivity are confusing and hard to read.

And that said, this article hasn’t shed any more light on the measurement of climate sensitivity. Instead, we have reviewed more ways in which measurements of it might be wrong. But not conclusively.

Next up we will take a detour into eddy diffusivity, hoping in the meantime that the Matlab model problems can be resolved. Finally a more accurate model incorporating eddy diffusivity to model vertical heat flow in the ocean will show us whether or not climate sensitivity can be accurately measured.

Possibly.

Articles in this Series

Measuring Climate Sensitivity – Part One

Measuring Climate Sensitivity – Part Three – Eddy Diffusivity

References

Potential Biases in Feedback Diagnosis from Observational Data: A Simple Model Demonstration, Spencer & Braswell, Journal of Climate (2008)

On the accuracy of deriving climate feedback parameters from correlations between surface temperature and outgoing radiation, Murphy & Forster, Journal of Climate (2010)

Observation of large diurnal warming events in the near-surface layer of the western equatorial Pacific warm pool, Soloviev & Lukas, Deep Sea Research Part I: Oceanographic Research Papers (1997)

Atmosphere, Ocean and Climate Dynamics: An Introductory Text, Marshall & Plumb, Elsevier Academic Press (2008)

Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology, de Boyer Montegut et al, JGR (2004)

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The Cool Skin of the Ocean

In the ensuing discussion on Does Back Radiation “Heat” the Ocean? – Part Four, the subject of the cool skin of the ocean surface came up a number of times.

It’s not a simple subject, but it’s an interesting one so I’m going to plough on with it anyway.

Introduction

The ocean surface is typically something like 0.1°C – 0.6°C cooler than the temperature just below the surface. And this “skin”, or ultra-thin region, is less than a 1mm thick.

Here’s a diagram I posted in the comments of Does Back Radiation “Heat” the Ocean? – Part Three:

Kawai & Wada (2007)

from Kawai & Wada (2007)

Figure 1

There is a lot of interest in this subject because of the question: “When we say ‘sea surface temperature’ what do we actually mean?“.

As many climate scientists note in their papers, the relevant sea surface temperature for heat transfer between ocean and atmosphere is the very surface, the skin temperature.

In figure 1 you can see that during the day the temperature increases up to the surface and then, in the skin layer, reduces again. Note that the vertical axis is a logarithmic scale.

Then at night the temperature below the skin layer is mostly all at the same temperature (isothermal). This is because the surface cools rapidly at night, and therefore becomes cooler than the water below, so sinks. This diurnal mixing can also be seen in some graphs I posted in the comments of Does Back Radiation “Heat” the Ocean? – Part Four.

Before we look at the causes, here are a series of detailed measurements from Near-surface ocean temperature by Ward (2006):

Figure 2

Note: The red text and arrow is mine, to draw attention to the lower skin temperature. The measurements on the right were taken just before midday “local solar time”. I.e., just before the sun was highest in the sky.

And in the measurements below I’ve made it a bit easier to pick out the skin temperature difference with blue text “Skin temp“. The blue value in each graph is what is identified as ΔTc in the schematic above. The time is shown as local solar time.

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

The measurements of the skin surface temperature were made by MAERI, a passive infrared radiometric interferometer. The accuracy of the derived SSTs from M-AERI is better than 0.05 K.

Below the skin, the high-resolution temperature measurements were measured by SkinDeEP, an autonomous vertical profiler. This includes the “sub-skin” measurement, from which the sea surface temperature was subtracted to calculate ΔTc (see figure 1).

The Theory

The existence of the temperature gradient is explained by the way heat is transferred: within the bulk waters, heat transfer occurs due to turbulence, but as the surface is approached, viscous forces dominate and molecular processes prevail. Because heat transfer by molecular conduction is less efficient than by turbulence, a strong temperature gradient is established across the boundary layer.

Ward & Minnett (2001)

Away from the interface the temperature gradient is quickly destroyed by turbulent mixing. Thus the cool-skin temperature change is confined to a region of thickness, which is referred to as the molecular sublayer.

Fairall et al (1996)

What do they mean?

Here’s an insight into what happens at fluid boundaries from an online textbook (thanks to Dan Hughes for letting me know about it) – this textbook is freely available online:

From "A Heat Transfer Textbook", by Prof Lienhard & Prof Lienhard (2008)

From "A Heat Transfer Textbook", by Prof Lienhard & Prof Lienhard (2008)

Figure 8

The idea behind turbulent mixing in fluids is that larger eddies “spawn” smaller eddies, which in turn spawn yet smaller eddies until you are up against an interface for that fluid (or until energy is dissipated by other effects).

In the atmosphere, for example, large scale turbulence moves energy across many 100’s of kilometers. A few tens of meters above the ground you might measure eddies of a few hundreds of meters in size, and in the last meter above the ground, eddies might be measured in cms or meters, if they exist at all. And by the time we measure the fluid flow 1mm from the ground there is almost no turbulence.

For some basic background over related terms, check out Heat Transfer Basics – Convection – Part One, with some examples of fluid flowing over flat plates, boundary layers, laminar flow and turbulent flow.

Therefore, very close to a boundary the turbulent effects effectively disappear, and heat transfer is carried out via conduction. Generally conduction is less effective than turbulence movement of fluids at heat transfer.

A Note on Very Basic Theory

The less effectively heat can move through a body, the higher the temperature differential needed to “drive” that heat through.

This is described by the equation for conductive heat transfer, which in (relatively) plain English says:

The heat flow in W/m² is proportional to the temperature difference across the body and the “conductivity” of the body, and is inversely proportional to the distance across the body

Now during the day a significant amount of heat moves up through the ocean to the surface. This is the solar radiation absorbed below the surface. Near the surface where turbulent mixing reduces in effectiveness we should expect to see a larger temperature gradient.

Taking the example of 1m down, if for some reason heat was not able to move effectively from 1m to the surface, then the absorbed solar radiation would keep heating the 1m depth and its temperature would keep rising. Eventually this temperature gradient would cause greater heat flow.

An example of a flawed model where heat was not able to move effectively was given in Does Back-Radiation “Heat” the Ocean? – Part Two:

A Flawed Model

Note how the 1m & 3m depth keep increasing in temperature. See that article for more explanation.

The Skin Layer in Detail

If the temperature increases closer to the surface, why does it “change direction” in the last millimeter?

In brief, the temperature generally rises in the last few meters as you get closer to the surface because hotter fluids rise. They rise because they are less dense.

So why doesn’t that continue to the very last micron?

The surface is where (almost) all of absorbed ocean energy is transferred to the atmosphere.

  • Radiation from the surface takes place from the top few microns.
  • Latent heat – evaporation of water into water vapor – is taken from the very top layer of the ocean.
  • Sensible heat is moved by conduction from the very surface into the atmosphere

And in general the ocean is moving heat into the atmosphere, rather than the reverse. The atmosphere is usually a few degrees cooler than the ocean surface.

Because turbulent motion is reduced the closer we get to the boundary with the atmosphere, this means that conduction is needed to transfer heat. This needs a temperature differential.

I could write it another way – because “needing a temperature differential” isn’t the same as “getting a temperature differential”.

If the heat flow up from below cannot get through to the surface, the energy will keep “piling up” and, therefore, keep increasing the temperature. Eventually the temperature will be high enough to “drive the heat” out to the surface.

The Simple 1-d Model

We saw a simple 1-d model in Does Back Radiation “Heat” the Ocean? – Part Four.

Just for the purposes of checking the theory relating to skin layers here is what I did to improve on it:

1. Increased the granularity of the model – with depths for each layer of: 100μm, 300μm, 1mm, 5mm, 20mm, 50mm, 200mm, 1m, 10m, 100m (note values are the lower edge of each layer).

2. Reduced the “turbulent conductivity” values as the surface was reached – instead of one “turbulent conductivity” value (used when the layer below was warmer than the layer above), these values were reduced closer to the surface, e.g. for the 100μm layer, kt=10; for the 300μm layer, kt=10; for the 1mm layer, kt=100; for the 5mm layer, kt=1000; for the 20mm layer, kt=100,000. Then the rest were 200,000 = 2×105 – the standard value used in the earlier models.

3. Reduced the time step to 5ms. This is necessary to make the model work and of course does reduce the length of run significantly.

The results for a 30 day run showed the beginnings of a cooler skin. And the starting temperatures for the top layer down to the 20mm layer were the same. The values of kt were not “tuned” to make the model work, I just threw some values in to see what happened.

As a side note for those following the discussion from Part Four, the ocean temperature also increased for DLR increases with these changes.

Now I can run it for longer but the real issue is that the model is not anywhere near complex enough.

Further Reading on Complexity

There are some papers for people who want to follow this subject further. This is not a “literature review”, just some papers I found on the journey. The subject is not simple.

Free

Saunders, Peter M. (1967), The Temperature at the Ocean-Air InterfaceJ. Atmos. Sci.

Tu and Tsuang (2005), Cool-skin simulation by a one-column ocean model, Geophys. Res. Letters

Paywall

McAlister, E. D., and W. McLeish (1969), Heat Transfer in the Top Millimeter of the OceanJ. Geophys. Res.

Fairall et al, reference below

GA Wick, WJ Emery, LH Kantha & P Schlussel (1996), The behavior of the bulk-skin sea surface temperature difference under varying wind speed and heat fluxJournal of Physical Oceanography

Hartmut Grassl, (1976), The dependence of the measured cool skin of the ocean on wind stress and total heat flux, Boundary Layer Meteorology

Conclusion

The temperature profile of the top mm of the ocean is a challenging subject. Tu & Tsuang say:

Generally speaking, the structure of the viscous layer is known to be related to the molecular viscosity, surface winds, and air-sea flux exchanges. Both Saunders’ formulation [Saunders, 1967; Grassl, 1976; Fairall et al.,1996] and the renewal theory [Liu et al., 1979; Wick et al.,1996; Castro et al., 2003; Horrocks et al., 2003] have been developed and applied to study the cool-skin effect.

But the exact factors and processes determining the structure is still not well known.

However, despite the complexity, an understanding of the basics helps to give some insight into why the temperature profile is like it is.

I welcome commenters who can make the subject easier to understand. And also commenters who can explain the more complex elements of this subject.

References

A Heat Transfer Textbook, by Prof Lienhard & Prof Lienhard, Phlogiston Press, 3rd edition (2008)

Cool-skin and warm-layer effects on sea surface temperature, Fairall, Bradley, Godfrey, Wick, Edson & Young, Journal of Geophysical Research (1996)

Near-surface ocean temperature, Ward, Journal of Geophysical Research (2006)

An Autonomous Profiler for Near Surface Temperature Measurements, Ward & Minnett, Accepted for the Proceedings Gas Transfer at Water Surfaces 4th International Symposium (2000)

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In Part One we saw how the ocean absorbed different wavelengths of radiation:

  • 50% of solar radiation is absorbed in the first meter, and 80% within 10 meters
  • 50% of “back radiation” (atmospheric radiation) is absorbed  in the first few microns (μm).

This is because absorption is a strong function of wavelength and atmospheric radiation is centered around 10μm, while solar radiation is centered around 0.5μm.

In Part Two we considered what would happen if back radiation only caused evaporation and removal of energy from the ocean surface via the latent heat. The ocean surface would become much colder than it obviously is. That is a very simple “first law of thermodynamics” problem. Then we looked at another model with only conductive heat transfer between different “layers” in the ocean. This caused various levels below the surface to heat to unphysical values. It is clear that turbulent heat transport takes place from lower in the ocean. Solar energy reaches down many meters heating the ocean from within – hotter water expands and so rises – moving heat by convection.

In Part Three we reviewed various experimental results showing how the temperature profile (vs depth) changes during the diurnal cycle (day-night-day) and with wind speed. This demonstrates very clearly how much mixing goes on in the ocean.

The Different Theories

This series of articles was inspired by the many people who think that increases in back radiation from the atmosphere will have no effect (or an unnoticeable effect) on the temperature of the ocean depths.

So far, no evidence has so far been brought forward for the idea that back radiation can’t “heat” the ocean (see note 1 at the end), other than the “it’s obvious” evidence. At least, I am unaware of any stronger arguments. Hopefully as a result of this article advocates can put forward their ideas in more detail in response.

I’ll summarize the different theories as I’ve understood them. Apologies to anyone who feels misrepresented – it’s quite possible I just haven’t heard your particular theory or your excellent way of explaining it.

Hypothesis A – Because the atmospheric radiation is completely absorbed in the first few microns it will cause evaporation of the surface layer, which takes away the energy from the back radiation as latent heat into the atmosphere. Therefore, more back-radiation will have zero effect on the ocean temperature.

Hypothesis B – Because the atmospheric radiation is completely absorbed in the first few microns it will be immediately radiated or convected back out to the atmosphere. Heat can’t flow downwards due to the buoyancy of hotter water. Therefore, if an increase in back radiation occurs (perhaps due to increases in inappropriately-named “greenhouse” gases) it will not “heat” the ocean = increase the temperature of the ocean below the surface.

For other, more basic objections about back radiation, see Note 2 (at the end).

I believe that Part Two showed that Hypothesis A was flawed.

I would like to propose a different hypothesis:

Hypothesis C – Heat transfer is driven by temperature differences. For example, conduction of heat is proportional to the temperature difference across the body that the heat is conducted through.

Solar radiation is absorbed from the surface through many meters of the ocean. This heats the ocean below the surface which causes “natural convection” – heated bodies expand and therefore rise. So solar energy has a tendency to be moved back to the surface (this was demonstrated in Part Two).

The more the surface temperature increases, the less temperature difference there will be to drive this natural convection. And, therefore, increases in surface temperature can affect the amount of heat stored in the ocean.

Clarification from St.Google: HypothesisA supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation

An Excellent Question

In Part Three, one commenter asked an excellent question:

Some questions from an interested amateur.
Back radiation causes more immediate evaporation and quicker reemission of LWR than does a similar amount of solar radiation.

Does that mean that the earth’s temperature should be more sensitive to a given solar forcing than it would be to an equal CO2 forcing?

What percentage CO2 forcing transfers energy to the oceans compared to space and the atmosphere?

How does this compare with solar forcing?

Is there a difference between the effect of the sun and the back radiation when they are of equal magnitude? This, of course, pre-supposes that Hypothesis C is correct and that back radiation has any effect at all on the temperature of the ocean below the surface.

So the point is this - even if Hypothesis C is correct, there may still be a difference between the response of the ocean temperatures below the surface – for back radiation compared with solar radiation.

So I set out to try and evaluate these two questions:

  1. Can increases in back radiation affect the temperature of the ocean below the surface? I.e., is Hypothesis C supported against B?
  2. For a given amount of energy, is there a difference between solar forcing and back radiation forcing?

And my approach was to use a model:

Oh no, a model! Clearly wrong then, and a result that can’t fool anyone..

For a bit of background generally on models, take a look at the Introduction in Models On – and Off – the Catwalk.

Here is one way to think about a model

The idea of a model is to carry out some calculations when doing them in your head is too difficult

A model helps us see the world a bit more clearly. At these point I’m not claiming anything other than they help us see the effect of the well-known and undisputed laws of heat transfer on the ocean a little bit more clearly.

Ocean Model

The ocean model under consideration is about a billion times less complex than a GCM. It is a 1-d model with heat flows by radiation, conduction and, in a very limited form, convection.

Here is a schematic of the model. I thought it would be good to show the layers to scale but that means the thicker layers can’t be shown (not without taking up a ridiculous amount of blank screen space) – so the full model, to scale, is 100x deeper than this:

Figure 1

To clarify – the top layer is at temperature, T1, the second layer at T2, even though these values aren’t shown.

The red arrows show conducted or convected heat. They could be in either direction, but the upwards is positive (just as a convention). Obviously, only a few of these are shown in the schematic – there is a heat flux between each layer.

1. Solar and back radiation are modeled as sine waves with the peak at midday. See the graph “Solar and Back Radiation” in Part Two for an example.

2. Convected heat is modeled with a simple formula:

H=h(T1-Tair), where Tair = air temperature, T1 = “surface” temperature, h = convection coefficient = 25 W/m².K.

Convected heat can be in either direction, depending on the surface and air temperature. The air temperature is assumed constant at 300K, something we will return to.

3. Radiation from the surface:

E = εσT4 – the well-known Stefan-Boltzmann equation, and ε = emissivity

For the purposes of this simple model ε = 1. So is absorptivity for back radiation, and for solar radiation. More on these assumptions later.

4. Heat flux between layers (e.g. H54 in the schematic) is calculated using the temperature values for the previous time step for the two adjacent layers then using the conducted heat formula: q” = k.(T5-T4)/d54, where k= conductivity, and d54 = distance between center of each layer 5 to the center of layer 4.

For still water, k = 0.6 W/m.K – a very low value as water is a poor conductor of heat.

In this model at the end of each time step, the program checks the temperature of each layer. If T5 > T4 for example, then the conductivity between these layers for the next time step is set to a much higher value to simulate convection. I used a value for stirred water that I found in a textbook: kt = 2 x 105 W/m.K. What actually happens in practice is the hotter water rises taking the heat with it (convection). Using a high value of conductivity produces a similar result without any actual water motion.

For interest I did try lower values like 2 x 10³ W/m.K and the 1m layer, for example, ended up at a higher temperature than the layers above it. See the more detailed explanation in Part Two.

5. In Part Three I showed results from a number of field experiments which demonstrated that the ocean experiences mixing due to surface cooling at night, and due to high winds. The mixing due to surface cooling is automatically taken account of in this model (and we can see it in the results), but the mixing due to the winds “stirring” the ocean is not included. So we can consider the model as being “under light winds”. If we had a model which evaluated stronger winds it would only make any specific effects of back radiation less noticeable. So this is the “worst case” – or the “highlighting back radiation’s special nature” model.

Problems of Modeling

Some people will already know about the many issues with numerical models. A very common one is resolving small distances and short timescales.

If we want to know the result over many years we don’t really want to have the iterate the model through time steps of fractions of a second. In this model I do have to use very small time steps because the distance scales being considered range from extremely small to quite large – the ocean is divided into thin slabs of 5mm, 15mm.. through to a 70m slab.

If I use a time step which is too long then too much heat gets transferred from the layers below the surface to the 5mm surface layer in the one time step, the model starts oscillating - and finally “loses the plot”. This is easy to see, but painful to deal with.

But I thought it might be interesting for people to see the results of the model over five days with different time steps. Instead of having the model totally “lose the plot” (=surface temperature goes to infinity), I put a cap on the amount of heat that could move in each time step for the purposes of this demonstration.

You can see four results with these time steps (tstep = time step, is marked on the top left of each graph):

  • 3 secs
  • 1 sec
  • 0.2 sec
  • 0.05 sec

Figure 2 – Click for a larger image

I played around with many other variables in the model to see what problems they caused..

The Tools

The model is written in Matlab and runs on a normal PC (Dell Vostro 1320 laptop).

To begin with there were 5 layers in the model (values are depth from the surface to the bottom edge of each layer):

  • 5 mm
  • 50 mm
  • 1 m
  • 10 m
  • 100 m

I ran this with a time step of 0.2 secs and ended up doing up to 15-year runs.

In the model runs I wanted to ensure that I had found a steady-state value, and also that the model conserved energy (first law of thermodynamics) once steady state was reached. So the model included a number of “house-keeping” tests so I could satisfy myself that the model didn’t have any obvious errors and that equilibrium temperatures were reached for each layer.

For 15 year runs, 5 layers and 0.2s time step the run would take about two and a half hours on the laptop.

I find that quite amazing – showing how good Matlab is. There are 31 million seconds in a year, so 15 years at 0.2 secs per step = 2.4 billion iterations. And each iteration involves looking up the solar and DLR value, calculating 7 heat flow calculations and 5 new temperatures. All in a couple of hours on a laptop.

Well, as we will see, because of the results I got I thought I would check for any changes if there were more layers in my model. So that’s why the 9-layer model (see the first diagram) was created. For this model I need an even shorter time step – 0.1 secs and so long model runs start to get painfully long..

Results

Case 1: The standard case was a peak solar radiation, S, of 600 W/m² and back radiation, DLR of 340 with a 50 W/m² variation day to night (i.e., max of 390 W/m², min of 290 W/m²).

Case 2a: Add 10 W/m² to the peak solar radiation, keep DLR the same. Case 2b – Add 31.41 W/m² to solar.

Case 3a: Keep solar radiation the same, add 3.14 W/m² to DLR. This is an equivalent amount of energy per day to case 2, see note 3. Case 3b – Add 10 W/m² to DLR.

Many people are probably asking, “Why isn’t case 3a – Add 10 W/m² to DLR?”

Solar radiation only occurs for 12 out of the 24 hours, while DLR occurs 24 hours of the day. And the solar value is the peak, while the DLR value is the average. It is a mathematical reason explained further in Note 3.

The important point is that for total energy absorbed in a day, case 2a and 3a are the same, and case 2b and 3b are the same.

Let’s compare the average daily temperature in the top layer, 1m, 10m and 100m layer for the three cases (note: depths are from the surface to the bottom of each layer; and only 4 layers of the 5 were recorded):

Figure 3

The time step (tstep) = 0.2s.

The starting temperatures for each layer were the same in all cases.

Now because the 4 year runs recorded almost identical values for solar vs DLR forcing, and because the results had not quite stabilized, I then did the 15 year run and also recorded the temperature to the 4 decimal places shown. This isn’t because the results are this accurate – this is to see what differences, if any, exist between the two different scenarios.

The important results are:

  1. DLR increases cause temperature increases at all levels in the ocean
  2. Equivalent amounts of daily energy into the ocean from solar and DLR cause almost exactly the same temperature increase at each level of the ocean – even though the DLR is absorbed in the first few microns and the solar energy in the first few meters
  3. The slight difference in temperature may be a result of “real physics” or may be an artifact of the model

And perhaps 5 layers is not enough?

Therefore, I generated the 9-layer model, as shown in the first diagram in this article. The 15-year model runs on the 9-layer model produced these results:

Figure 4

The general results are similar to the 5-layer model.

The temperature changes have clearly stabilized, as the heat unaccounted for (inputs – outputs) on the last day = 41 J/m². Note that this is Joules, not Watts, and is over a 24 hour period. This small “unaccounted” heat is going into temperature increases of the top 100m of the ocean. (“Inputs – outputs” includes the heat being transferred from the model layers down into the ocean depths below 100m).

If we examine the difference in temperature for the bottom 30-100m deep level for case 2b vs 3b we see that the temperature difference after 15 years = 0.011°C. For a 70m thick layer, this equates to an energy difference = 3.2 x 106 J, which, over 15 years, = 591 J/m².day = 0.0068 W/m². This is spectacularly tiny. It might be a model issue, or it might be a real “physics difference”.

In any case, the model has demonstrated that DLR increases vs solar increases cause almost exactly the same temperature changes in each layer being considered.

For interest here are the last 5 days of the model (average hourly temperatures for each level) for case 3b:

Figure 5

and for case 2b:

Figure 6

Pretty similar..

Results – Convection and Air Temperature

In the model results up until now the air temperature has been at 300K (27°C) and the surface temperature of the ocean has been only a few degrees higher.

The model doesn’t attempt to change the air temperature. And in the real world the atmosphere at the ocean surface and the surface temperature are usually within a few degrees.

But what happens in our model if real world situations cool the ocean surface more? For example, higher temperatures locally create large convective currents of rising hot air which “sucks in” cooler air from another area.

What would be the result? A higher “instantaneous” surface temperature from higher back radiation might be “swept away” into the atmosphere and “lost” from the model.. This might create a different final answer for back radiation compared with solar radiation.

It seemed to be worth checking out, so I reduced the air temperature to 285K (from 300K) and ran the model for one year from the original starting temperatures (just over 300K). The result was that the ocean temperature dropped significantly, demonstrating how closely the ocean surface and the atmosphere (at the ocean surface) are coupled.

Using the end of the first year as a starting temperature, I ran the model for 5 years for case 1, 2a and 3a (each with the same starting temperature):

Figure 7

Once again we see that back radiation increases do change the temperatures of the ocean depths – and at almost identical values to the solar radiation changes.

Here is a set of graphs for one of the 5-year model runs for this lower air temperature, also demonstrating how the lower air temperature pulls down the ocean surface temperature:

Figure 8 – Click for a larger image

The first graph shows how the average daily temperature changes over the full time period – making it easy to see equilibrium being reached. The second graph shows the hourly average temperature change for the last 5 days. The last graph shows the heat which is either absorbed or released within the ocean in temperature changes. As zero is reached it means the ocean is not heating up or cooling down.

Inaccuracies in the Model

We can write a lot on the all the inaccuracies in the model. It’s a very rudimentary model. In the real world the hotter tropical / sub-tropical oceans transfer heat to higher latitudes and to the poles. So does the atmosphere. A 1-d model is very unrealistic.

The emissivity and absorptivity of the ocean are set to 1, there are no ocean currents, the atmosphere doesn’t heat up and cool down with the ocean surface, the solar radiation value doesn’t change through the year, the top layer was 5mm not 1μm, the cooler skin layer was not modeled, a number of isothermal layers is unphysical compared with the real ocean of continuously varying temperatures..

However, what a nice simple model tells us is how energy only absorbed in the top few microns of the ocean can affect the temperature of the ocean much lower down.

It’s obvious“, I could say.

Conclusion

My model could be wrong – for example, just a mistake which means it doesn’t operate how I have described it. The many simplifications of the model might hide some real world physics effect which means that Hypothesis C is actually less likely than Hypothesis B.

However, if the model doesn’t contain mistakes, at least I have provided more support for Hypothesis C – that the back radiation absorbed in the very surface of the ocean can change the temperature of the ocean below, and demonstrated that Hypothesis B is less likely.

I look forward to advocates of Hypothesis B putting forward their best arguments.

Update - Code files saved here

Notes

Note 1 – To avoid upsetting the purists, when we say “does back-radiation heat the ocean?” what we mean is, “does back-radiation affect the temperature of the ocean?”

Some people get upset if we use the term heat, and object that heat is the net of the two way process of energy exchange. It’s not too important for most of us. I only mention it to make it clear that if the colder atmosphere transfers energy to the ocean then more energy goes in the reverse direction.

It is a dull point.

Note 2 - Some people think that back radiation can’t occur at all, and others think that it can’t affect the temperature of the surface for reasons that are a confused mangle of the second law of thermodynamics. See Science Roads Less Travelled and especially Amazing Things we Find in Textbooks – The Real Second Law The Real Second Law of Thermodynamics and The Three Body Problem. And for real measurements of back radiation, see The Amazing Case of “Back Radiation” -Part One.

Note 3 – If we change the peak solar radiation from 600 to 610, this is the peak value and only provides an increase for 12 out of 24 hours. By contrast, back radiation is a 24 hour a day value. How much do we have to change the average DLR value to provide an equivalent amount of energy over 24 hours?

If we integrate the solar radiation for the before and after cases we find the relationship between the value for the peak of the solar radiation and the average of the back radiation = π (3.14159). So if the DLR increase = 10, the peak solar increase to match = 10 x π = 31.4159; and if the solar peak increase = 10, the DLR increase to match = 10/π = 3.1831.

If anyone would like this demonstrated further please ask and I will update in the comments. I’m sure I could have made this easier to understand than I actually have (haven’t).

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A long time ago I wrote the article The Dull Case of Emissivity and Average Temperatures and expected that would be the end of the interest in emissivity. But it is a gift that keeps on giving, with various people concerned that no one has really been interested in measuring surface emissivity properly.

Background

All solid and liquid surfaces emit thermal radiation according to the Stefan-Boltzmann formula:

E = εσT4

where ε=emissivity, a material property; σ = 5.67×10-8 ; T = temperature in Kelvin (absolute temperature)

and E is the flux in W/m²

More about this formula and background on the material properties in Planck, Stefan-Boltzmann, Kirchhoff and LTE.

The parameter called emissivity is the focus of this article. It is of special interest because to calculate the radiation from the earth’s surface we need to know only temperature and emissivity.

Emissivity is a value between 0 and 1. And is also depends on the wavelength of radiation (and in some surfaces like metals, also the direction). Because the wavelengths of radiation depend on temperature, emissivity also depends on temperature.

When emissivity = 1, the body is called a “blackbody”. It’s just the theoretical maximum that can be radiated. Some surfaces are very close to a blackbody and others are a long way off.

Note: I have seen many articles by keen budding science writers who have some strange ideas about “blackbodies”. The only difference between a blackbody and a non-blackbody is that the emissivity of a blackbody = 1, and the emissivity of a non-blackbody is less than 1. That’s it. Nothing else.

The wavelength dependence of emissivity is very important. If we take snow for example, it is highly reflective to solar (shortwave) radiation with as much as 80% of solar radiation being reflected. Solar radiation is centered around a wavelength of 0.5μm.

Yet snow is highly absorbing to terrestrial (longwave) radiation, which is centered around a wavelength of 10μm. The absorptivity and emissivity around freezing point is 0.99 – meaning that only 1% of incident longwave radiation would be reflected.

Let’s take a look at the Planck curve – the blackbody radiation curve – for surfaces at a few slightly different temperatures:

The emissivity (as a function of wavelength) simply modifies these curves.

Suppose, for example, that the emissivity of a surface was 0.99 across this entire wavelength range. In that case, a surface at 30°C would radiate like the light blue curve but at 99% of the values shown. If the emissivity varies across the wavelength range then you simply multiply the emissivity by the intensity at each wavelength to get the expected radiation.

Sometimes emissivity is quoted as an average for a given temperature – this takes into account the shape of the Planck curve shown in the graphs above.

Often, when emissivity is quoted as an overall value, the total flux has been measured for a given temperature and the emissivity is simply:

ε =  actual radiation measured / blackbody theoretical radiation at that temperature

[Fixed, thanks to DeWitt Payne for pointing out the mistake]

In practice the calculation is slightly more involved, see note 1.

It turns out that the emissivity of water and of the ocean surface is an involved subject.

And because of the importance of calculating the sea surface temperature from satellite measurements, the emissivity of the ocean in the “atmospheric window” (8-14 μm) has been the subject of many 100’s of papers (perhaps 1000’s). These somewhat overwhelm the papers on the less important subject of “general ocean emissivity”.

Measurements

Aside from climate, water itself is an obvious subject of study for spectroscopy.

For example, 29 years ago Miriam Sidran writing Broadband reflectance and emissivity of specular and rough water surfaces, begins:

The optical constants of water have been extensively studied because of their importance in science and technology. Applications include a) remote sensing of natural water surfaces, b) radiant energy transfer by atmospheric water droplets, and c) optical properties of diverse materials containing water, such as soils, leaves and aqueous solutions.

In this study, values of the complex index of refraction from six recent articles were averaged by visual inspection of the graphs, and the most representative values in the wavelength range of 0.200 μm to 5 cm were determined. These were used to find the directional polarized reflectance and emissivity of a specular surface and the Brewster or pseudo-Brewster angle as functions of wavelength.

The directional polarized reflectance and emissivity of wind-generated water waves were studied using the facet slope distribution function for a rough sea due to Cox and Munk [1954].

Applications to remote sensing of sea surface temperature and wave state are discussed, including effects of salinity.

Emphasis added. She also comments in her paper:

For any wavelength, the total emissivity, ε, is constant for all θ [angles] < 45° [from vertical]; this follows from Fig. 8 and Eq. (6a). It is important in remote sensing of thermal radiation from space, as discussed later..

The polarized emissivities are independent of surface roughness for θ < 25°, while for θ > 25°, the thermal radiation is partly depolarized by the roughness.

This means that when you look at the emission radiation from directly above (and close to directly above) the sea surface roughness doesn’t have an effect.

I thought some other comments might also be interesting:

The 8-14-μm spectral band is chosen for discussion here because (a) it is used in remote sensing and (b) the atmospheric transmittance, τ, in this band is a fairly well-known function of atmospheric moisture content. Water vapor is the chief radiation absorber in this band.

In Eqs. (2)-(4), n and k (and therefore A and B) are functions of salinity. However, the emissivity value, ε, computed for pure water differs from that of seawater by <0.5%.

When used in Eqs. (10), it causes an error of <0.20°C in retrieved Ts [surface temperature]. Since ε in this band lies between 0.96 and 0.995, approximation ε= 1 is routinely used in sea surface temperature retrieval. However, this has been shown to cause an error of -0.5 to -1.0°C for very dry atmospheres. For very moist atmospheres, the error is only ≈0.2°C.

One of the important graphs from her paper:

Click to view a larger image

Emissivity = 1 – Reflectance. The graph shows Reflectance vs Wavelength vs Angle of measurement.

I took the graph (coarse as it is) and extracted the emissivity vs wavelength function (using numerical techniques). I then calculated the blackbody radiation for a 15°C surface and the radiation from a water surface using the emissivity from the graph above for the same 15°C surface. Both were calculated from 1 μm to 100 μm:

The “unofficial” result, calculating the average emissivity from the ratio: ε = 0.96.

This result is valid for 0-30°C. But I suspect the actual value will be modified slightly by the solid angle calculations. That is, the total flux from the surface (the Stefan-Boltzmann equation) is the spectral intensity integrated over all wavelengths, and integrated over all solid angles. So the reduced emissivity closer to the horizon will affect this measurement.

Niclòs et al – 2005

One of the most interesting recent papers is In situ angular measurements of thermal infrared sea surface emissivity—validation of models, Niclòs et al (2005). Here is the abstract:

In this paper, sea surface emissivity (SSE) measurements obtained from thermal infrared radiance data are presented. These measurements were carried out from a fixed oilrig under open sea conditions in the Mediterranean Sea during the WInd and Salinity Experiment 2000 (WISE 2000).

The SSE retrieval methodology uses quasi-simultaneous measurements of the radiance coming from the sea surface and the downwelling sky radiance, in addition to the sea surface temperature (SST). The radiometric data were acquired by a CIMEL ELECTRONIQUE CE 312 radiometer, with four channels placed in the 8–14 μm region. The sea temperature was measured with high-precision thermal probes located on oceanographic buoys, which is not exactly equal to the required SST. A study of the skin effect during the radiometric measurements used in this work showed that a constant bulk–skin temperature difference of 0.05±0.06 K was present for wind speeds larger than 5 m/s. Our study is limited to these conditions.

Thus, SST used as a reference for SSE retrieval was obtained as the temperature measured by the contact thermometers placed on the buoys at 20-cm depth minus this bulk–skin temperature difference.

SSE was obtained under several observation angles and surface wind speed conditions, allowing us to study both the angular and the sea surface roughness dependence. Our results were compared with SSE models..

The introduction explains why specifically they are studying the dependence of emissivity on the angle of measurement – for reasons of accurate calculation of sea surface temperature:

The requirement of a maximum uncertainty of ±0.3 K in sea surface temperature (SST) as input to climate models and the use of high observation angles in the current space missions, such as the 55° for the forward view of the Advanced Along Track Scanning Radiometer (AATSR) (Llewellyn-Jones et al., 2001) on board ENVISAT, need a precise and reliable determination of sea surface emissivity (SSE) in the thermal infrared region (TIR), as well as analyses of its angular and spectral dependences.

The emission of a rough sea surface has been studied over the last years due to the importance of the SSE for accurate SST retrieval. A reference work for many subsequent studies has been the paper written by Cox and Munk (1954)..

The experimental setup:

From Niclos (2004)

From Niclos (2004)

The results (compared with one important model from Masuda et al 1988):

From Niclos (2004)

From Niclos (2004)

Click on the image for a larger graphic

This paper also goes on to compare the results with the model of Wu & Smith (1997) and indicates the Wu & Smith’s model is a little better.

The tabulated results, note that you can avoid the “eye chart effect” by clicking on the table:

Click on the image for a larger view

Note that the emissivities are in the 8-14μm range.

You can see that the emissivity when measured from close to vertical is 0.98 – 0.99 at two different wind speeds.

Konda et al – 1994

A slightly older paper which is not concerned with angular dependence of sea surface emissivity is by Konda, Imasato, Nishi and Toda (1994).

They comment on a few older papers:

Buettner and Kern (1965) estimated the sea surface emissivity to be 0.993 from an experiment using an emissivity box, but they disregarded the temperature difference across the cool skin.

Saunders (1967b, 1968) observed the plane sea surface irradiance from an  airplane and determined the reflectance. By determining the reflectance as the ratio of the differences in energy between the clear and the cloudy sky at different places, he calculated the emissivity to be 0.986. The process of separating the reflection from the surface irradiance, however, is not precise.

Mikhaylov and Zolotarev (1970) calculated the emissivity from the optical constant of the water and found the average in the infrared region was 0.9875.

The observation of Davies et al. (1971) was performed on Lake Ontario with a wave height less than 25 cm. They measured the surface emission isolated from sky radiation by an aluminum cone, and estimated the emissivity to be 0.972. The aluminum was assumed to act as a mirror in infrared region. In fact,aluminum does not work as a perfect mirror.

Masuda et al. (1988) computed the surface emissivity as a function of the zenith angle of observed radiation and wind speed. They computed the emissivity from the reflectance of a model sea surface consisting of many facets, and changed their slopes according to Gaussian distribution with respect to surface wind. The computed emissivity in 11 μm was 0.992 under no wind.

Each of these studies in trying to determine the value of emissivity, failed to distinguish surface emission from reflection and to evaluate the temperature difference across the cool skin. The summary of these studies are tabulated in Table 1.

The table summarizing some earlier work:

Konda (1994)

Konda (1994)

Konda and his co-workers took measurements over a one year period from a tower in Tanabe Bay, Japan.

They calculated from their results that the ocean emissivity was 0.984±0.004.

One of the challenges for Konda’s research and for Niclòs is the issue of sea surface temperature measurement itself. Here is a temperature profile which was shown in the comments of Does Back Radiation “Heat” the Ocean? – Part Three:

Kawai & Wada (2007)

Kawai & Wada (2007)

The point is the actual surface from which the radiation is emitted will usually be at a slightly different temperature from the bulk temperature (note the logarithmic scale of depth). This is the “cool skin” effect. This surface temperature effect is also moderated by winds and is very difficult to measure accurately in field conditions.

Smith et al – 1996

Another excellent paper which measured the emissivity of the ocean is by Smith et al (1996):

An important objective in satellite remote sensing is the global determination of sea surface temperature (SST). For such measurements to be useful to global climate research, an accuracy of ±0.3K or better over a length of 100km and a timescale of days to weeks must be attained. This criterion is determined by the size of the SST anomalies (≈1K) that can cause significant disturbance to the global atmospheric circulation patterns and the anticipated size of SST perturbations resulting from global climate change. This level of uncertainty is close to the theoretical limits of the atmospheric corrections..

It is also a challenge to demonstrate that such accuracies are being achieved, and conventional approaches, which compare the SST derived from drifting or moored buoys, generally produce results with a scatter of ±0.5 to 0.7K. This scatter cannot be explained solely by uncertainties in the buoy thermometers or the noise equivalent temperature difference of the AVHRR, as these are both on the order of 0.2K or less but are likely to be surface emissivity/reflectivity uncertainties, residual atmospheric effects, or result from the methods of comparison

Note that the primary focus of this research was to have accurate SST measurements from satellites.

From Smith et al (1996)

From Smith et al (1996)

The experimental work on the research vessel Pelican included a high spectral resolution Atmospheric Emitted Radiance Interferometer (AERI) which was configured to make spectral observations of the sea surface radiance at several view angles. Any measurement from the surface of course, is the sum of the emitted radiance from the surface as well as the reflected sky radiance.

Also measured:

  • ocean salinity
  • intake water temperature
  • surface air temperature
  • humidity
  • wind velocity
  • SST within the top 15cm of depth

There was also independent measurement of the radiative temperature of the sea surface at 10μm with a Heimann broadband radiation thermometer “window” radiometer. And radiosondes were launched from the ship roughly every 3 hours.

Additionally, various other instruments took measurements from a flight altitude of 20km. Satellite readings were also compared.

The AERI measured the spectral distribution of radiance from 3.3μm to 20μm at 4 angles. Upwards at 11.5° from zenith, and downwards at 36.5°, 56.5° and 73.5°.

There’s a lot of interesting discussion of the calculations in their paper. Remember that the primary aim is to enable satellite measurements to have the most accurate measurements of SST and satellites can only really “see” the surface through the “atmospheric window” from 8-12μm.

Here are the wavelength dependent emissivity results shown for the 3 viewing angles. You can see that at the lowest viewing angle of 36.5° the emissivity is 0.98 – 0.99 in the 8-12μm range.

From Smith et al (1996)

From Smith et al (1996)

Note that the wind speed doesn’t have any effect on emissivity at the more direct angle, but as the viewing angle moves to 73.5° the emissivity has dropped and high wind speeds change the emissivity considerably.

Henderson et al – 2003

Henderson et al (2003) is one of the many papers which consider the theoretical basis of how viewing angles change the emissivity and derive a model.

Just as an introduction, here is the theoretical variation in emissivity with measurement angle, versus “refractive index” as computed by the Fresnel equations:

The legend is refractive index from 1.20 to 1.35. Water, at visible wavelengths, has a refractive index of 1.33. This shows how the emissivity reduces once the viewing angle increases above 50° from the vertical.

The essence of the problem of sea surface roughness for large viewing angles is shown in the diagram below, where multiple reflections take place:

Henderson (2003)

Henderson (2003)

Henderson and his co-workers compare their results with the measured results of Smith et al (1996) and also comment that at zenith viewing angles the emissivity does not depend on the wind speed, but at larger angles from vertical it does.

A quick summary of their model:

We have developed a Monte Carlo ray-tracing model to compute the emissivity of computer-rendered, wind-roughened sea surfaces. The use of a ray-tracing method allows us to include both the reflected emission and shadowing and, furthermore, permits us to examine more closely how these processes control the radiative properties of the surface. The intensity of the radiation along a given ray path is quantified using Stokes vectors, and thus, polarization is explicitly included in the calculations as well.

Their model results compare well with the experimental results. Note that the approach of generating a mathematical model to calculate how emissivity changes with wind speed and, therefore, wave shape is not at all new.

Water retains its inherent properties of emissivity regardless of how it is moving or what shape it is. The theoretical challenge is handling the multiple reflections, absorptions, re-emissions that take place when the radiance from the water is measured at some angle from the vertical.

Conclusion

The best up to date measurements of ocean emissivity in the 8-14 μm range are 0.98 – 0.99. The 8-14 μm range is well-known because of the intense focus on sea surface temperature measurements from satellite.

From quite ancient data, the average emissivity of water across a very wide broadband range (1-100 μm) is 0.96 for water temperatures from 0-30°C.

The values from the ocean when measured close to the vertical are independent of wind speed and sea surface roughness. As the angle of measurement moves from the vertical around to the horizon the measured emissivity drops and the wind speed affects the measurement significantly.

These values have been extensively researched because the calculation of sea surface temperature from satellite measurements in the 8-14μm “atmospheric window” relies on the accurate knowledge of emissivity and any factors which affect it.

For climate models – I haven’t checked what values they use. I assume they use the best experimental values from the field. That’s an assumption. I’ve already read enough on ocean emissivity.

For energy balance models, like the Trenberth and Kiehl update, an emissivity of 1 doesn’t really affect their calculations. The reason, stated simply, is that the upwards surface radiation and the downward atmospheric radiation are quite close in magnitude. For example, the globally annually averaged values of both are 396 W/m² (upward surface) vs 340 W/m² (downward atmospheric).

Suppose the emissivity drops from 0.98 to 0.97 – what is the effect on upwards radiation through the atmosphere?

The upwards radiation has dropped by 4W/m², but the reflected atmospheric radiation has increased by 3.4W/m². The net upwards radiation through the atmosphere has reduced by only 0.6 W/m².

One of our commenters asked what value the IPCC uses. The answer is they don’t use a value at all because they summarize research from papers in the field.

Whether they do it well or badly is a subject of much controversy, but what is most important to understand is that the IPCC does not write papers, or perform GCM model runs, or do experiments – and that is why you see almost no equations in their many 1000’s of pages of discussion on climate science.

For those who don’t believe the “greenhouse” effect exists, take a look at Understanding Atmospheric Radiation and the “Greenhouse” Effect – Part One in the light of all the measured results for ocean emissivity.

On Another Note

It’s common to find claims on various blogs and in comments on blogs that climate science doesn’t do much actual research.

I haven’t found that to be true. I have found the opposite.

Whenever I have gone digging for a particular subject, whether it is the diurnal temperature variation in the sea surface, diapycnal & isopycnal eddy diffusivity, ocean emissivity, or the possible direction and magnitude of water vapor feedback, I have found a huge swathe of original research, of research building on other research, of research challenging other research, and detailed accounts of experimental methods, results and comparison with theory and models.

Just as an example, in the case of emissivity of sea surface, at the end of the article you can see the first 30 or so results pulled up from one journal – Remote Sensing of the Environment for the search phrase “emissivity sea surface”. The journal search engine found 348 articles (of course, not every one of them is actually about ocean emissivity measurements).

Perhaps it might turn out to be the best journal for this subject, but it’s still just one journal.

References

Broadband reflectance and emissivity of specular and rough water surfaces, Sidran, Applied Optics (1981)

In situ angular measurements of thermal infrared sea surface emissivity—validation of models, Niclòs, Valor, Caselles, Coll & Sànchez, Remote Sensing of Environment (2005)

Measurement of the Sea Surface Emissivity, Konda, Imasato, Nishi and Toda, Journal of Oceanography (1994)

Observations of the Infrared Radiative Properties of the Ocean—Implications for the Measurement of Sea Surface Temperature via Satellite Remote Sensing, Smith, Knuteson, Revercomb, Feltz, Nalli, Howell, Menzel, Brown, Brown, Minnett & McKeown, Bulletin of the American Meteorological Society (1996)

The polarized emissivity of a wind-roughened sea surface: A Monte Carlo model, Henderson, Theiler & Villeneuve, Remote Sensing of Environment (2003)

Notes

Note 1: The upward radiation from the surface is the sum of three contributions: (i) direct emission of the sea surface, which is attenuated by the absorption of the atmospheric layer between the sea surface and the instrument; (ii) reflection of the downwelling sky radiance on the sea, attenuated by the atmosphere; and (iii) the upwelling atmospheric radiance emitted in the observing direction.

So the measured radiance can be expressed as:

where the three terms on the right are each of the three contributions noted in the same order.

Note 2: 1/10th of the search results returned from one journal for the search term “emissivity sea surface”:

Remote Sensing of Environment - search results

Remote Sensing of Environment - search results

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In Part One we looked at the absorption of solar and atmospheric radiation (aka “back radiation”) in the ocean.

About half of the solar energy is absorbed in the first meter, but most of the DLR (downward longwave radiation), or “back radiation”, from the atmosphere is absorbed in the first few microns of the ocean surface. And all of the DLR is absorbed in the first 0.1mm of the ocean.

Warmer water is less dense (more buoyant) than cooler water. Less dense water rises to the top over more dense water and so a surface being heated at the very top is “stratified”. This really means that there is no tendency for water from below to displace water above. How then can heat from the top layer of the atmosphere make it into the ocean?

In Part Two we looked first at what would happen if the “back radiation” was absorbed and immediately consumed in the process of evaporation. The ocean would be much colder.

Then we looked at simple models of heating from solar radiation and DLR if there was no convection (within the ocean). Because the conductivity of water is quite low the results are strange – the water boils around 1m down. Of course, this doesn’t happen – the water heats up, expands and rises. This is “natural convection”. The article touched on this and also on “forced convection” via the effects of the wind “stirring” the water.

The ocean is a fascinating place, very complex, but of course the basics of heat transfer still stay the same.

How Does Heat Move Through the Ocean?

There are competing forces, as you might expect in a dynamic system.

In “broadbrush”:

  • the sun heats the ocean, mostly in the first few meters
  • the ocean heats the boundary layer of the atmosphere via convection
  • the ocean heats the lower atmosphere via radiation

Huge amounts of energy also move from the equator to the poles, approximately 50:50 via the ocean and the atmosphere.

To consider the complete picture of the energy transfer – the atmosphere also radiates to the ocean (see The Amazing Case of “Back Radiation” -Part One and the following parts). The issue that was probably implied in the original question was “if back radiation increases due to more CO2 (or other trace gas increases) will that energy be mixed into the ocean?”.

If we “dive in” to a little more detail, as in Part Two, then we see that solar radiation is absorbed in the top few meters of the ocean and as a consequence, this ocean heat rises to the surface.

In this article we will try to understand more about how the upper layer of the ocean mixes.

Temperature – Depth Profile vs Wind Speed and SST Variation

There is a huge wealth of experimental results in ocean “heat budgets” and temperature profiles.

First, 35 vertical temperature profiles during the warming and cooling phases of the diurnal cycle at low wind speed from Soloviev & Lukas (1997):

Soloviev & Lukas (1997)

Soloviev & Lukas (1997)

Note: Each successive temperature profile is artificially shifted in temperature for comparison purposes

From Price and Weller (1986) we see another example of how the vertical temperature profile goes from “well-mixed” under higher winds to a definite temperature profile (during the day) under light winds:

Price & Weller (1986)

Price & Weller (1986)

With these temperature profiles we also see the effect of the day-night variation. Regardless of wind speed, at night the temperature profile shows that the ocean has become well mixed.

Now temperature profiles under different wind speeds from Soloviev & Lukas (1997):

From Soloviev & Lukas (1997)

From Soloviev & Lukas (1997)

COARE is the coupled ocean-atmosphere response experiment, and the area of study is in the Western Pacific from the R.V. Moana Wave. What this diagram shows is very interesting, but not surprising:

  • in higher wind speeds (A) the ocean is very well mixed (the temperature is the same at the surface as 10m depth)
  • in very calm conditions (C) there is a very pronounced temperature profile with depth
  • in between (B), the ocean temperature profile shows a transition between these two results

Another dataset from Soloviev & Lukas (1997) – time-series of temperature, salinity and density on one day:

Soloviev & Lukas (1997)

Soloviev & Lukas (1997)

Note: there was a rain event at 19:00

The variation in surface temperature (SST) from day to night indicates the same effect taking place. During light winds in tropical waters very large temperature variations can be seen (up to 2-3°C and even 5°C), whereas more generally the SST diurnal variation is less than 0.5°C.

From Deschamps & Frouin (1984)

From Deschamps & Frouin (1984)

The SST measurements were done by the HCMR (Heat Capacity Mapping Radiometer) and the study area is East of Sardinia in the Mediterranean.

From Kawai & Wada (2007) an interesting time-based dataset of temperature vs depth against solar flux and wind speed:

Kawai & Wada (2007)

Kawai & Wada (2007)

Wind-Induced Convection and Diurnal Cycle

At night the surface of the ocean cools rapidly via radiation – but is not receiving any heat via solar radiation. Therefore, the top surface of the ocean cools – and so it sinks.

Therefore, every night, the ocean heat in the top few meters becomes very well mixed.

During the day (and the night), when the wind picks up over the surface two effects take place. One is that the top few meters of the ocean become well-mixed due to wind-induced stirring. The other effect is that the surface cools more rapidly than normal due to convective cooling (convective heat transfer to the atmosphere), and therefore is more likely to sink.

A Brief Digression on Cameos

Not only do climate scientists do a lot of experiments, but sometimes they even come together for a cameo role just because the stars align:

Let’s not pretend this paper was anything other than what it obviously appears to be..

Conclusion

The aim of this article is just to show some field research about how the top layer of the ocean mixes. Wind and diurnal cooling mix heat from the surface into the top few meters of the ocean.

In the next article we will analyze the possible effect of a little more “back radiation” in the light of these results.

Update – Does Back Radiation “Heat” the Ocean? – Part Four

References

Observation of large diurnal warming events in the near-surface layer of the western equatorial Pacific warm pool, Soloviev & Lukas, Deep Sea Research Part I: Oceanographic Research Papers (1997)

Large Diurnal Heating of the Sea Surface Observed by the HCMR Experiment, Deschamps & Frouin, Journal of Physical Oceanography (1984)

Diurnal Sea Surface Temperature Variation and Its Impact on the Atmosphere and Ocean: A Review, Kawai & Wada, Journal of Oceanography (2007)

Diurnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing, Price & Weller, Journal of Geophysical Research (1986)

Notes

Note 1 (added Dec 6th) – To avoid upsetting the purists, when we say “does back-radiation heat the ocean?” what we mean is, “does back-radiation affect the temperature of the ocean?”

Some people get upset if we use the term heat, and object that heat is the net of the two way process of energy exchange. It’s not too important for most of us. I only mention it to make it clear that if the colder atmosphere transfers energy to the ocean then more energy goes in the reverse direction.

It is a dull point.

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