Archive for the ‘Ocean Physics’ Category

In Part One we looked at how solar radiation and DLR (or “back radiation”) were absorbed by the ocean. And we had a brief look at how little heat would move by conduction into the deeper ocean if the ocean was “still”.

There were some excellent comments in part one from Nick Stokes, Arthur Smith and Willis Eschenbach – probably others as well – take a look if you didn’t see them first time around.

We will shortly look at mixing and convection, but first we will consider some absolute basics.

The First Law of Thermodynamics

How does the ocean sustain its (high) temperature? Every second, every square meter of the ocean is radiating energy. The Stefan-Boltzmann relationship tells us the value:

j = εσT4, where ε is the emissivity of the ocean (0.99), σ = 5.67 x 10-8 and T is the temperature in K

For example:

  • if T = 20°C (293K), j = 415 W/m²
  • if T = 10°C (283K), j = 361 W/m²

Now, many people are confused about how temperatures change with heat imbalances. If, for some reason, more heat is absorbed by a system than is radiated/conducted/convected away – what happens?

More heat absorbed than lost = heat gained. Heat gained leads to an increase of temperature (see note 1). When the temperature of a body increases, it radiates, conducts and convects more heat away (see note 2). Eventually a new equilibrium is reached at a higher temperature. It is important to grasp this concept. Read it again if it isn’t quite clear. Ask a question for clarification..

Questions are welcome.

A Simple Model

Evaluating a very simple energy balance model might help to set the scene.

Here is the radiative input – solar radiation and “back radiation” from the atmosphere, with typical values for a tropical region:

The primary question – the raison d’êtra for this article –  is what happens if only the solar radiation heats the ocean? And compared with if the back-radiation also heats the ocean?

It’s easy to find the basic equilibrium point using the first law of thermodynamics. All you need to know is the energy in, and the equation which links energy radiated with temperature.

For radiation, this is the Stefan-Boltzmann law cited earlier. The starting temperature for the ocean surface in this example was set to 300K (27°C). Depending on whether solar and back-radiation or just solar is heating the surface, here is the surface temperature change:

Notice the difference in the temperature trends for the two cases.

Now the model doesn’t yet include convective heat transfer from the ocean to the atmosphere (or movement of heat from the tropics to the poles), which is why in the first graph the temperature gets so high. Convection will reduce this temperature to a more “real world” value.

The second graph has only solar radiation heating the ocean. Notice that the temperature drops to a very low value (-15°C) in just a few years. Clearly the climate would be very different if this was the case, and the people who advocate this model need to explain exactly how the ocean temperature manages to stay so much higher.

By the way, if we made the “well-mixed layer”, dmixed, of the ocean deeper it would increase the time for the temperature to change by any given amount. That’s because more ocean has more heat capacity. But it doesn’t change the fact of the energy imbalance, or the final equilibrium temperature.

The model is a very simplistic one. That’s all you need to demonstrate that DLR, or “back radiation” must be absorbed by the ocean and contributing to the ocean heat content.

Turbulence and the Mixed Layer

Let’s take a look at a slightly more complex model to demonstrate an important point. This simulation has four main elements:

  • radiation absorbed in the ocean at various depths, according to the results in Part One
  • conduction between layers in the ocean
  • convective heating from the ocean surface to the atmosphere, according to a simple model with a fixed air temperature

This model is not going to revolutionize climate models as it has many simplifications. The important factor – there is no convection between different ocean layers in this model.

Now conductivity in still water is very low (as explained in Part One).

The starting condition – the “boundary condition” – was for the temperature to start at 300K (27°C) for the first 100m, with the ocean depths below to be a constant 1°C.

The model is for illumination. Let’s see what happens:

The wide bars of blue and green are because the day/night variation is significant but squashed horizontally. If we expand one part of the graph to look at the first few days:

You can see that the day/night variation of the top 1mm and 10cm are significant.

Look back at the first graph which covers four years. Notice the purple line, 10m depth, the blue line, 3m depth; and the red line, 1m depth.

Why is the ocean 1-10m depth increasing to such a high temperature?

The reason is simple. This model is flawed– these results don’t occur in practice. (And yes, the ocean would boil from within..)

The equations that make up this model have used:

  • the radiation absorbed from the sun and the atmosphere (as described in part one)
  • the radiation emitted from the surface layer (the Stefan-Boltzmann equation)
  • conductivity transferring heat between layers

If these were the only mechanisms for transferring heat, the ocean 1m – 10m deep would be extremely hot in the tropics. This is because the ocean where the radiation is absorbed cannot radiate back out.

For a mental picture think of a large thick slab of PVC which is heated from electrical elements within the PVC. Because it is such a poor conductor of heat, the inner temperature will rise much higher than the surface temperature, so long as the heating continues..

The reason this doesn’t happen in practice in the ocean is due to convection.

If you heat a gas or liquid from below it heats up and expands. Because it is now less dense than the layer above it will rise. This is what happens in the atmosphere, and it also happens in the ocean. The ocean under the very surface layer heats up, expands and rises – overturning the top layer of the ocean. This is natural convection.

The other effect that takes place is forced convection as the wind speed “stirs” the top few meters of the ocean. Convection is the transfer of heat by bulk motion of a fluid. Essentially, the gas or liquid moves, taking heat with it.

Price & Weller (1986) commented:

Under summer heating conditions with vanishing wind, the trapping depth of the thermal response is only about 1m (mean depth value), and the surface amplitude is as large as 2ºC or 3ºC. But, more commonly, when light or moderate winds are present, solar heating is wind mixed vertically to a considerably greater depth than is reached directly by radiation: the trapping depth is typically 10m, and the surface amplitude is reduced in inverse proportion to typically 0.2ºC. Given that the surface heating and wind stress are known, then the key to understanding and forecasting the diurnal cycle of the ocean is to learn how the trapping depth is set by the competing effects of a stabilizing surface heat flux and a destabilizing surface stress.

Here are the results from a model with another slight improvement. This includes natural convection. The mechanism is very rudimentary at this stage. It simply analyzes the temperature profile at each time step and if the temperature is inverted from normal buoyancy a much higher value of thermal conductivity is used to simulate convection.

The “bumpiness” you see in the temperature profile is because the model has multiple “slabs”, each with an average temperature. This could be reduced by a finer vertical grid.

During the early afternoon with peak solar radiation, the ocean becomes stratified. Why?

Because lots of heat is being absorbed in the first few meters with some then transported upwards to the surface via convection – but while the solar radiation value is high this heat keeps “pouring in” lower down. However, once the sun sets the surface will cool via radiation to the atmosphere and so become less buoyant. With no solar radiation now being absorbed lower down, the top few meters completely mix – from natural convection.

I did have a paper with a perfect set of measurements to illustrate these points. It showed day/night and seasonal variation. Sadly I put it down somewhere. Many hours of hunting for the physical paper and for the file on my PC but it is still lost..

Note that the large variation of surface temperature (4-5°C) is just a result of the convective mixing element in the model being too simplistic and moving heat much faster than happens in reality.

Kondo and Sasano (1979) said:

In the upper part of the ocean, a mixed layer with homogeneous density (or nearly homogeneous temperature) distribution is formed during the night due to free convection associated with heat loss from the sea surface and to forced convection by wind mixing.

During the daytime, the absorption of solar radiation which occurs mostly near the sea surface causes the temperature to rise, and a stable layer is formed there; as a consequence, turbulent transport is reduced.

Daily mean depth of the mixed layer increases with the wind speed. When the wind speed is lower than about 7-8 m/s, the mixed layer disappears about noon but it develops again in the later afternoon. A mixed layer can be sustained all day under high wind speeds..


The subject of convection and oceans is a fascinating one and I hope to cover much more. However, convection is a complex subject, the most complex mechanism of heat transfer “by a mile”.

There are also some complexities with the skin layer of the ocean which are worth taking a closer look at in a future article.

This article uses some very simple models to demonstrate that energy radiated from the atmosphere is being absorbed in the ocean surface and affecting its temperature. If it wasn’t the ocean surface would freeze. Therefore, if atmospheric radiation increases (for example, from an increase in “greenhouse” gases), then, all other things being equal, this will increase the ocean temperature.

The models also demonstrate that conduction of heat on its own cannot explain the temperature profiles we see in the ocean. Natural convection and wind speed both create convection, which is a much more effective heat transport mechanism in gases and liquids than conduction.

Updates: Does Back Radiation “Heat” the Ocean? – Part Three

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


Diurnal Cycling: Observations and Models of the Upper Ocean Response to Diurnal Heating, Cooling and Wind Mixing, James Price & Robert Weller, Journal of Geophysical Research (1986)

On Wind Driven Current and Temperature Profiles with Diurnal Period in the Oceanic Planetary Boundary Layer, Kondo and Sasano, Journal of Physical Oceanography (1979)


Note 1 – For the purists, heat retained can go into chemical energy, it can go into mechanisms like melting ice, or evaporating water which don’t immediately increase temperature.

Note 2 – For the purists, the actual heat transfer mechanism depends on the physical circumstances. For example, in a vacuum, only radiation can transfer heat.

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In the three part series on DLR (also known as “back radiation”, also known as atmospheric radiation), Part One looked at the network of stations that measured DLR and some of the measurements, Part Two reviewed the spectra of this radiation, and Part Three asked whether this radiation changed the temperature of the surface.

Very recently, on another blog, someone asked whether I thought “back radiation” heated the ocean. I know from a prominent blog that a very popular idea in blog-land is that the atmospheric radiation doesn’t heat the ocean. I have never seen any evidence for the idea. That doesn’t mean there isn’t any..

See note 1 on “heat”.

The Basic Idea

From what I’ve seen people write about this idea, including the link above, the rough argument goes like this:

  • solar radiation penetrates tens of meters into the ocean
  • atmospheric radiation – much longer wavelengths – penetrates only 1μm into the ocean

Therefore, solar radiation heats the ocean, but atmospheric radiation only heats the top few molecules. So DLR is unable to transfer any heat into the bulk of the ocean, instead the energy goes into evaporating the top layer into water vapor. This water vapor then goes to make clouds which act as a negative feedback. And so, more back-radiation from more CO2 can only have a cooling effect.

There are a few assumptions in there. Perhaps someone has some evidence of the assumptions, but at least, I can see why it is popular.

Solar Radiation

As regular readers of this blog know, plus anyone else with a passing knowledge of atmospheric physics, solar radiation is centered around a wavelength of 0.5μm. The energy in wavelengths greater than 4μm is less than 1% of the total solar energy and conventionally, we call solar radiation shortwave.

99% of the energy in atmospheric radiation has longer wavelengths than 4μm and along with terrestrial radiation we call this longwave.

Most surfaces, liquids and gases have a strong wavelength dependence for the absorption or reflection of radiation.

Here is the best one I could find for the ocean. It’s from Wikipedia, not necessarily a reliable source, but I checked the graph against a few papers and it matched up. The papers didn’t provide such a nice graph..

Absorption coefficient for the ocean - Wikipedia

Absorption coefficient for the ocean - Wikipedia

Figure 1

Note the logarithmic axes.

The first obvious point is that absorption varies hugely with the wavelength of incident radiation.

I’ll explain a few basics here, but if the maths is confusing, don’t worry, the graphs and explanation will attempt to put it all together. The basic equation of transmission relies on the Beer-Lambert law:

I = I0.exp(-kd)

where I is the radiation transmitted, I0 is the incident radiation at that wavelength, d is the depth, and k is the property of the ocean at this wavelength

It’s not easy to visualize if you haven’t seen this kind of equation before. So imagine 100 units of radiation incident at the surface at one wavelength where the absorption coefficient, k = 1:

Figure 2

So at 1m, 37% of the original radiation is transmitted (and therefore 63% is absorbed).

At 2m, 14% of the radiation is transmitted.

At 3m, 5% is transmitted

At 10m, 0.005% is transmitted, so 99.995% has been absorbed.

(Note for the detail-oriented people, I have used the case where k=1/m).

Hopefully, this concept is reasonably easy to grasp. Now let’s look at the results of the whole picture using the absorption coefficient vs wavelength from earlier.

Figure 3

The top graph shows the amount of radiation making it to various depths, vs wavelength. As you can see, the longer (and UV) wavelengths drop off very quickly. Wavelengths around 500nm make it the furthest into the ocean depths.

The bottom graph shows the total energy making it through to each depth. You can see that even at 1mm (10-3m) around 13% has been absorbed and by 1m more than 50% has been absorbed. By 10m, 80% of solar radiation has been absorbed.

The graph was constructed using an idealized scenario – solar radiation less reflection at the top of atmosphere (average around 30% reflected), no absorption in the atmosphere and the sun directly overhead. The reason for using “no atmospheric absorption” is just to make it possible to construct a simple model, it doesn’t have much effect on any of the main results.

If we considered the sun at say 45° from the zenith, it would make some difference because the sun’s rays would now be coming into the ocean at an angle. So a depth of 1m would correspond to the solar radiation travelling through 1.4m of water (1 / cos(45°)).

For comparison here is more accurate data:

From "Light Absorption in Sea Water", Wozniak (2007)

From "Light Absorption in Sea Water", Wozniak (2007)

Figure 4

On the left the “surface” line represents the real solar spectrum at the surface – after absorption of the solar radiation by various trace gases (water vapor, CO2, methane, etc). On the right, the amount of energy measured at various depths in one location. Note the log scale on the vertical axis for the right hand graph. (Note as well that the irradiance in these graphs is in W/m².nm, whereas the calculated graphs earlier are in W/m².μm).

From "Light Absorption in Sea Water", Wozniak (2007)

From "Light Absorption in Sea Water", Wozniak (2007)

Figure 5

And two more locations measured. Note that the Black Sea is much more absorbing – solar absorption varies with sediment as well as other ocean properties.

DLR or “Back radiation”

The radiation from the atmosphere doesn’t look too much like a “Planck curve”. Different heights in the atmosphere are responsible for radiating at different wavelengths – dependent on the concentration of water vapor, CO2, methane, and other trace gases.

Here is a typical DLR spectrum (note that the horizontal axis needs to be mentally reversed to match other graphs):

Pacific, Lubin (1995)

Pacific, Lubin (1995)

Figure 6

You can see more of these in The Amazing Case of Back Radiation – Part Two.

But for interest I took the case of an ideal blackbody at 0°C radiating to the surface and used the absorption coefficients from figure 1 to see how much radiation was transmitted through to different depths:

Figure 7

As you can see, most of the “back radiation” is absorbed in the first 10μm, and 20% is absorbed even in the first 1μm.

I could produce a more accurate calculation by using a spectrum like the Pacific spectrum in fig 6 and running that through the same calculations, but it wouldn’t change the results in any significant way.

So we can see that while around half the solar radiation is absorbed in the first meter and 80% in the first 10 meters, 90% of the DLR is absorbed in the first 10μm.

So now we need to ask what kind of result this implies.

Heating Surfaces and Conduction

When you heat the surface of a body that has a colder bulk temperature (or a colder temperature on the “other side” of the body) then heat flows through the body.

Conduction is driven by temperature differences. Once you establish a temperature difference you inevitably get heat transfer by conduction – for example, see Heat Transfer Basics – Part Zero.

The equation for heat transfer by conduction:

q = kA . ΔT/Δx

where k is the material property called conductivity, ΔT is the temperature difference, Δx is the distance between the two temperatures, and q is the heat transferred.

However, conduction is a very inefficient heat transfer mechanism through still water.

For still water, k ≈ 0.6 W/m.K (the ≈ symbol means “is approximately equal to”).

So, as a rough guide, if you had a temperature difference of 20°C across 50m, you would get heat conduction of 0.24 W/m². And with 20°C across 10m of water, you would only get heat conduction of 1.2 W/m².

However, the ocean surface is also turbulent for a variety of reasons, and in Part Two we will look at how that affects heat transfer via some simulations and a few papers. We will also look at the very important first law of thermodynamics and see what that implies for absorption of back radiation.

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


Light Absorption in Sea Water, Wozniak & Dera, Atmospheric and Oceanographic Sciences Library (2007)


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.

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