Answers to physics quiz

[JasonC]

Sorry this is so off topic. Rather than burying it in the frankly already moribund "Fanboy" thread, here are the answers to my warming debate entry level quiz -

a column of water 2 km long covered by a black body is in thermal equilibrium with its exterior environment at 291K. The measured power at its surface is 300 watts per square meter. A new power source is then added of 2 watts per square meter at one end of the water column.

(a) what is the heat capacity of the water column?

The volume of the water per square meter of surface area is 1 m^2 x 2000 m = 2000 m^3. A m^3 of water has a mass of 1 metric ton, 1000 kg. So the mass is 2 million kg per m^2 of surface. The heat capacity of water is 1 calorie per gram per degree or 4.2 joules per gram per degree. There are 2x10^9 grams, so the capacity is 8.4x10^9 joules per degree.

( what is the instantaneous rate of change in the average temperature of the water the moment the new power is applied?

2 watts of new power means 2 joules per second of energy are being added to the water. Since the heat capacity is 8.4x10^9 joules per degree, the added energy is raising the temperature by 1 / 4.2x10^9 degrees, per second.

© how long would it take to raise the average water temperature 1K at that rate, ignoring restoring forces for now?

There are 3x10^7 seconds in a year, so the rate can also be expressed as 0.0075 degrees per year or 1 degree every 133 years.

(d) if no other changes are made and assuming the water is thermally well mixed, which of the following describes the overall future trajectory of the water's average temperature -

1. it will go to infinity in finite time.

2. it will rise indefintely at a constant rate.

3. it will rapidly fall to absolute zero.

4. it will remain unchanged.

5. it will rise at a slowing rate to a new equilibrium level, approaching that level asymptotically

5. The new power will raise the temperature only until the hotter body re-radiates an equivalent higher power. Just like a burner on an electric stove, turning up the power to a higher setting and leaving it there, raises the equilibrium temperature of the burner. It gets hotter and glows as a result. That glow exports power as radiation, balancing the increased input power. When the two new power terms match, the temperature stops changing.

(e) What physical law describes the additional effect that will operate on the water's average temperature?

1. law of entropy increase

2. conservation of energy

3. Stefan-Boltzman radiation law

4. adiabatic expansion

5. the "waxy buildup" theory

3. The Stefan-Boltzmann law describes the power spectrum radiated by a black body at a given temperature. As the temperature increases, so does the radiated power, as described by the SB law.

(f) According to the S-B radiation law, the power output of a glowing black body is what function of its absolute temperature in degrees K?

1. third root

2. exponential

3. fourth power

4. inverse square

5. linear

3. The SB law states that the total power output of a black body goes as the fourth power of its absolute temperature, P = constant * T ^ 4.

(g) Calculate the average water temperature at the new, elevated power level at which its re-radiation equals the total power input.

The new power of 302 watts divided by the old power of 300 watts gives a proportion of 302/300 = 1.00667 times the power. To find the temperature change that will compensate, take the fourth root. That gives 1.001667. If the temperature rises 1.001667 times in absolute terms (degrees K), then the new radiated power will be 1.00667 times the old radiated power. And since the old radiated power was 300 watts, the new one will by 302 watts. The old temperature was 291, so the new temperature is 291 * 1.001667 = 291.484 degrees K. Up half of a degree.

Notice, this answer is independent of the heat capacity issue, which only effects the time it takes to reach the new level. The level itself is set by power balancing (energy rates of change), not stored heat energy (which is a stock, not a flow).

(h) At the temperature calculated in (g), the future course of the water's mean temperature would be which of the following?

1. steady state, equilibrium

2. rise continually

3. fall continually

4. go to infinity in finite time

5. go to absolute zero in finite time

1. At 291.484 K, the water body will reradiate 302 watts, while receiving 302 watts as input. Net joules entering the water per unit time are therefore zero, and the temperature hold steady at the new higher level. The full ongoing power input is necessary to maintain the marginally higher new temperature.

(i) If the instantaneous rate of change when the power is first applied is as calculated in (, and the that rate at the new higher temperature is as implied by (h), a decent approximation of the average rate of change over the period between them is -

1. answer to ( plus answer in (h) divided by 2.

2. answer to ( time answer in (h)

3. zero

4. infinity, the temperature change is instantaneous

1. The initial rate was one degree every 133 years. Once the temperature rises 0.484 degrees, however, the rate would be zero, as the whole balancing power from higher re-radiation is "on line" at that point. The actual rate will fall throughout the period of temperature rise, as first a little, and then more, extra re-radiated power partially balances the added 2 watts.

An exact answer can be obtained by a simple integral of the net power. But a decent approximation is the diagonal line, full at start and zero at end, area half that of the rectangle or equal to the simple average of the two endpoints. So the average rate of change is approximately 1 degree every 266 years - faster at the start, slower at the end.

(j) at the estimated average rate of change in (i), what is the period of time before the temperature rises (most of the way) to its new level?

To go up 0.484 degrees at a rate of 1 degree every 266 years would take 128 years and 9 months, about.

(k) someone alleges that the average temperature of the water column will rise from 291 to 296K within 100 years. A reasonable response to this claim would be

1. where are you going to get the power?

2. it can't possibly warm that fast with so little power input

3. someone in a red hat appointed by a pope says it, so it must be true

4. regression lines through short periods of noisy data are more trustworthy than such calculations, so it is almost certainly true

5. everything is connected to everything else, and some systems are sensitive and non-linear, so anything might happen and probably will

6. we must pray to Gaia to forgive our rape of sacred mother earth

1, or 2.

The new power needed to get 5K warming from a 291 base can be calculated from the SB law directly. 296/291 = 1.0172, to the fourth power to get a power term, means the total power into the system would have to rise by 7%.

On a base of 300 watts per square meter, that means you'd need 21 watts per square meter of new power. The 2 watts given in the problem are an order of magnitude too low to cause a permanent temperature change of that scale.

If there were another 19 watts per square meter of power, the prediction would be perfectly reasonable. So it is reasonable to ask where the extra 19 watts of power are supposed to come from.

Or one can calculate the power needed to push the temperature up at that rate, from heat capacity issues. 5K in 100 years means 1K every 20 years as an average rate. Since at the higher equilibrium, the rate of change will be zero, to average that over the whole period you'd need an initial rate more like 1K every 10 years. Which is 13.3 times the instantaneous rate given by the first calculation above. Again the rate of change alleged is an order of magnitude too large for the stated power.

(l) someone acknowledges (k) but alleges that as yet unspecified new power terms caused by hidden feedback mechanisms might conjure up the missing power. A reasonable response to this claim might be which of the following?

1. Ok. Where? What are the new power terms? What are their magnitudes and signs?

2. If most of the power is to come from some other cause, in what sense is the original 2 watts the cause of the predicted change?

3. If a complicated nonlinear system with such massive feedbacks is a correct model, then why don't its ongoing fluctuations, larger than the 2 watts actually seen, set off equally large sustained fluctuations, randomly? If they do, why is the predicted direction any more likely than the opposite?

4. Well then it must be so. Let's do everything possible to reduce that 2W to 1.97 W within 20 years, lest catastrophy ensue.

5. All but 4.

5. 1 is reasonable because if there were another 19 watts, the temperature response could be as predicted. That is a 6% change in the total power coursing through the system, and not obviously easy to get. But if some other modelers has an actual source in mind, it is perfectly reasonable to consider the hypothesis and measure that power term. (Aside, half a dozen have been measured since this was all pointed out, and their signs are randomly plus or minus, and all are 1-2 orders of magnitude too small).

2 is reasonable because if 21 watts of power are causing the warming, then the unstated 19 watts are causing 90% of the warming change. Watts turn into temperature in an entirely fungible fashion by well known physics. There aren't magic A watts that cause 10 times the temperature increase of unimportant B watts - a watt is a watt is a joule per second.

If one wanted to control the temperature change, one would look for the easiest watts to "turn off", not for the first 2 watts you noticed. Anything supposedly sensitively triggered as a feedback by smaller changes, is going to be easier, not harder, to tweak.

3 is reasonable because non-linear systems with positive feedbacks ("amplifiers") that magnify any input signal 10 fold, do not show long run stability. They jump all over the map. Strictly, such feedbacks are inherently unsustainable and go off in either direction until new forces begin to operate. Systems with negative feedbacks are broadly stable.

4 is not reasonable because a change of 0.03 watts, in a system that is supposedly being powered to higher temperatures by 21 watts of new power all told, is only a 1/700 change. Instead of +5C in 100 years you'd get +4.964C in the same 100 years. Not that you can get either without 19 extra watts, source unspecified. But there would be no gain in reducing that, if it were happening, by a 30th of 1 degree over 100 years.

[BigDog944]

So I take it this proves that the role of greenhouse gasses in the supposed "global warming" is a non-issue?

Kyoto and other international environmental treaties are misguided and a waste of time?

I don't have strong opinions either way, but I am interested in hearing about the physics and background as you have posted them here.

[JasonC]

No. It is an entry level of understanding needed to appreciate the real debate at the level of the science, as opposed to the media war over policy.

Human activities can change atmospheric CO2 levels. Whether due mostly to human activities or not, the CO2 level is clearly and measurably changing. CO2 is only a trace gas, a tiny portion of the atmosphere, and the effects it can have on climate are therefore low order. But changing its concentration can produce a net new power term in the global energy equation.

Best estimates of that scale run 1-2 watts per square meter for past CO2 chances, and projections of possible added power from ongoing changes in the CO2 concentration run 1-4 watts.

It is therefore entirely reasonable to say, as the stock formula goes, that human activities can contribute to changes in global mean temperature. What is essentially never discussed at the media level is how much. Or rather, remarkably unsupported wild ass guesses are the sole form of how much discussed. (Those guesses have an interesting history, but that is a whole additional chapter in the piece).

I call it the entry level, because you can't understand the science debate unless you realize that it is physical nonsense to allege 3-5C changes in global mean temperature from either past or predicted changes in atmospheric CO2. The power simply is not there.

The best supporters of the global warming hypothesis acknowledge this. They are out hunting for additional power terms and feedback mechanisms.

This generally is subject to hand waving under the phrase "climate sensitivity". Is it low? Is it high? Can people just make it up? Physics says, for the power terms we have actually seen and can actually measure, it is low - way too low to justify the headline scare predictions of 3-5C warming.

If there are unknown factors at work in the climate that also strongly contribute to warming - the additional power I discussed - then climate might change by those amounts. Due to those other factors, whatever they are. This is a good reason to discuss it and to investigate it. But everybody should understand, the power from CO2 changes alone physically can't warm the whole earth appreciably. Ever.

Half a degree, easy and probably already happening. 5C, other causes have to be at work - or aren't, and you won't see that big a change. But CO2 won't get that done.

[ April 15, 2006, 07:05 PM: Message edited by: JasonC ]

[John Kettler]

JasonC,

Does anyone have even a halfway decent estimate of humankind's thermal output from vehicles, factories, power gen facilities, etc,, as compared to natural

internal sources and external ones, such as the sun?

Regards,

John Kettler

[LongLeftFlank]

I've taken the liberty of forwarding the above discussion to some friends of friends in the Princeton physics department. They may be able to either rebut or illuminate your argument. I'll let you know.

I generally think of myself as being among the better educated people I know, but you've definitely left me in the dust with this one....

[JasonC]

J.K. - the sun utterly dominates the global power budget. Every square meter at the earth's distance gets a continual flux of about 1360 watts. Since the earth re-radiates from the whole sphere while getting that flux only on one side (at a time, of course), and that with diminishing angle as you get to the ends, the effective power per unit surface area, average, is 340 watts per square meter.

The earth is almost "black" in radiative transfer terms, meaning most photons that hit it are absorbed. A typical estimate is an albedo of 0.85. Some of that incident sunlight is absorbed in the atmosphere before reaching the surface, some reaches the surface, some is reflected. Photons are continually re-emitted from the surface, mostly in the infrared, which re-exports the solar energy into space. Some are absorbed in the atmosphere on the way out, instead. The atmosphere is re-radiating too, some of it out and some down. There is some energy transfer from ground to sky from thermal convection.

You get a big system model tracking where the energy goes. Net, in from above the sky and out through the sky about balance. When the in is marginally higher there is net energy added to the system per unit time. That raises the temperature, and therefore it glows hotter, and more gets out as infrared - but only after you've transitioned to a marginally higher temperature.

CO2 warms bu intercepting a percentage of the outward infrared photons and reradiating their energy in all directions, some outward again (so still gets out) but some downward, allowing that energy to make an additional "round trip" before leaving for space. The photons so absorbed are a few spectral lines of the molecule. Other trace elements absorb a few other lines.

By far the largest "greenhouse gas" this way is ordinary water vapor. If the earth had no atmosphere, it would be a ball of ice as cold as the air outside a jet or at the edge of space.

In order of magnitude terms, the biggest factors in earth mean temperature are (1) solar radiation hitting the earth (2) the earth's rotation (that 4 times reradiating area thing above) and (3) water vapor greenhouse.

But the earth's atmosphere is basically saturated in the water vapor wavelengths. The sky is opaque to those infrared photons, in other words. So adding a bit more water makes no difference really - it rains a little, that's all.

The sky is not saturated at the wavelength of CO2 because there is very little CO2 in the atmosphere. The atmosphere is almost all nitrogen and what isn't nitrogen is almost all oxygen. Water vapor is a percent or so amount below those two. And everything else is a trace type an order of magnitude lower again, including CO2. It is typically measured in parts per billion with a number in the mid 3 digits.

As for other forms of power for the earth system, there are radioactive elements throughout the crust and upper mantle. Mostly uranium, with naturally occurring 235 about a quarter the contribution of the most prevalent 238. (There is a lot less 235, but it has a significantly shorter half life. Not quite enough to catch the more abundant but more stable 238). A few other elements too, but none combine the abundance and radiactivity of uranium. This internal radiation power is orders of magnitude lower than solar, but still large.

There is also a term at the surface from the earth's core gradually cooling off. The interior is much hotter than the surface, creating a temperature gradient. But thick metal rich bodies don't exactly cool off on a dime. That term is a couple orders of magnitude lower again.

Plants capture less than 1% of the incident solar radiation and convert it to chemical energy in organic bonds, in all photosynthesis combine. Understand, 99 out of 100 of the energy in photons hitting the earth just warm things up a little then fly off into space (at higher entropy - a few high energy ultraviolet and visible photons coming in from the sun have much lower entropy than many infrared ones scattered in all directions).

Human power consumption of all kinds is approaching the same scale as photosynthesis, by the rough measures I've seen at least. Probably lower by a factor of 2, but same order of magnitude. Most of it is releasing energy photosynthesis captured a long time ago, and some is accelerating the release of crust radiation by getting uranium to "burn" in tens of years instead of hundreds of millions.

Among those who don't get the whole picture, I've seen a common misunderstanding that global warming is supposed to be simply entropy increase from human energy use. I've heard people say, "gosh, if we keep on polluting the world with entropy, we will end in heat death". I call it the "waxy build up theory" - like the old furniture polish commercial, we are doomed by waxy build-up from old energy use. Complete misunderstanding.

The earth re-exports gobs of entropy in its infrared glow. That is the warmer water downstream of the dam. The power source - the water flowing over the dam continually - is the sun's nuclear furnace converting matter into energy, an irreversible entropy increasing process, but one it has sufficient fuel to keep up for billions of years. The "pollution" exported is simply light beamed off into interstellar space.

The reason carbon based fuels are a concern is simply that their use involves releasing extra CO2 into the atmosphere, net. And CO2 is a non-saturated greenhouse gas. So increasing its concentration means intercepting a little bit of that outgoing light, for another round trip, and thus a minor new power term. The IPCC estimates it at 1-2 watts per square meter, which is based on direct measurements of the photon flux at the surface compared to that seen in a test airplane, and similar measurements.

There is enough intricacy and uncertainty in just the systems models of power moving from surface to atmosphere etc, that we don't really know if the power at the surface that 1-2 watts has to be used as an increment to, is the 300 watts in my quiz. But that is the right order of magnitude. It could easily be 200 or 500, the error bars are large. That won't make 1-2 into 21 watts, however.

To review, all this shows is that CO2 greenhouse alone will not cause 3-5C warming. That means either (a) modest warming on the order of 0.5C is happening, but nothing else or ( faster warming is happening, but has other significant power terms aka causes besides CO2, as yet unidentified.

If you look at the temperature measurements directly, they are ambiguous on this score. You don't take the temperature of the whole earth by sticking a thermometer under its tongue. It involves averages and proxy measures, all of which can introduce their own noise.

Individual temperatures vary widely and wildly, showing nothing like the consistency of the *average* - because the local temp can always be higher here at the expense of being lower there, etc. And that "local" can be not Boston and Chicago, but the air a mile over Boston and the sea 500 meters below the surface 200 miles off the coast of Sri Lanka.

Measuring averages over very diverse places etc is an inherently messy and approximate business, that must be expected to introduce its own noise and purely statistical error terms. Different ways of taking such measurements have reported different results - unsurprisingly.

If you get some scatter plot of averages and you put a least squares regression line through it, you will often get a slope. Does that mean that slope is the real "signal" or mean rate of change, and can be confidently projected to other data points not yet measured, or to the indefinite future? Well, no. Particularly not when you know there is a constraint on the overall power.

But the reason people are concerned is, then do such a regression with some proxies and they see relatively high warming rates in recent times. They also see the CO2 level rising and know it can add a new power term.

Physics says, the CO2 power term can't account for a warming that rapid. Ergo, either some of that regression line slope is just plain wrong, noise, fooled by your proxies or outlier data points, and warming by CO2 is happening but 0.5C not 3-5.

Or, the regression line is right, and larger scale warming is already under way (though not all that far along yet, to be sure). CO2 can't be causing all of it. Perhaps something else is - some feedback mechanism or amplifier (slightly warmer somehow changing the albedo, or the water vapor power term, to give two popular if unconfirmed and frankly a bit hand-waving accounts).

That is the real scientific debate. It is over "climate sensitivity" and whether there are new unknown power terms already operating. People have been looking for them and many candidates have been proposed. They haven't (yet, at any rate) checked out, in the sense of finding a new term with the right sign and a large enough order of magnitude.

I hope this helps.

[ww2steel]

In the on topic realm of the East Front... who cares? I had hoped this was something about ballistics... oh well.

[Folbec]

Technical opinion : http://realclimate.org/

[Ike]

I'm not certain your audience is really paying attention, JasonC.

Guys, this is his point: There is only "X" amount of energy coming to the Earth in the form of light from the sun, heat rising to the surface from the still-molten core of the Earth, plant, animal and human activities, and the decay of radioactive elements near the surface of the Earth (in the ground). And perhaps other stuff, but we - the scientists - haven't discovered what that is, if there is anything.

In order for the Earth to heat up, the heat input from all these sources has to total up to an increase. The part he was outlining in his physics posts is the part of that heat which comes from the Sun and is - was? - usually re-radiated (sent out in infrared light form) into space, but is now being kept by the increase in CO2. That part is, according to his post, 2 parts in 300, less than 1%. A increase in energy the Earth gets and keeps. His calculations there are to show how much increase in temperature is possible with that extra two-thirds of one percent of energy.

The answer is: a lot less than what the "global warming by mankind" advocates are claiming, and a lot closer to the observed "average temperature" increases actually measured over the last 100 - 150 years. And keep in mind about measuring the Earth's temperature: there aren't good records for most of the Earth that go back more than about 100 years. There is the Central England - whatisit - that goes back longer, but for Indonesia what do we have? Now think about the rest of the world and temperature records.

You see, his point isn't about greehouse gases or other mechanizism for trapping the heat from the Sun. His point is, even if all that is 100% true, based on what the human race already knows about the sources of the Earth's warmth and how that energy is kept or lost, CO2 increases as actually measured cannot possibly result in an increase of temperature of 5 - 7 degree Celsius as is being predicted.

Sorry, JasonC; I don't mean to interfere, but sometimes a simpler wording gets the point across. And guys, I'm not implying that you're idiots, either. This is an area of science that is relatively complicated (climate) and JasonC's post was about another one, but one we know a lot more about, but it isn't something we all discuss over a beer or a wine at the local. Ya know?

[BigDog944]

This is an interesting topic, for sure. However, one thing I'd like to throw in is that perhaps an average temperature rise across the planet isn't the problem. The average temperature rise at the poles is what humanity should be concerned about.

The reason to focus on the poles is because there is a heck of a lot of water stored up and down there in the ice caps, and should that ice melt in any significant amount, ocean levels will rise and perhaps threaten some heavily populated, low lieing areas.

My understanding of the physics of the issue isn't very deep (but I can follow it when it's laid out in front of me like in this post), but an average *planetary* temperature rise of 1 degree C may be manifested as a 0.25 degree rise near the equator, and a 3 degree rise at the poles. That would still be something we should want to avoid, I think.

Can you shed any light on that, Jason? I've not seen any mention of polar average temperatures in your discussion so far. I'm genuinely interested in your opinion on that part of this topic.

[JasonC]

General circulation models are even less trustworthy than regression lines through noise. Nobody has any real idea where warming would show up, and anyone who tells you otherwise is selling something. Some empirical stuff suggests more than its fair share happens in continental northern hemisphere locations in winter - milder Siberian Februaries, that sort of thing. But the data are so noisy (it happens where it happens, who knows) it is really just a guess.

The reason people worry about the poles is not primarily sea level stuff, though that figures in the popular scare stories in the media, every time somebody makes an allegation with a "could" in it. It is really the possible albedo feedback, that is the real science reason to care about it.

The earth is mostly a black body but not quite, and one of the contributors to the "not quite" bit is the reflectivity of ice and snow. If the whole earth were white, it would keep less of the incident solar radiation, reradiating more of it in the visible spectrum, instead of downshifted to infrared after heating things up. Effectively the power at the surface would be lower.

White at the poles doesn't make that much difference, though, because the solar radiation is much more intense at the equator and diminishes toward the poles, so you are reflecting some but a modest portion of the "incoming". But the feedback created by lots of white farther south can be significant, and is thought to be part of the ice age mechanism. Ice retreat works the other way, though with diminishing returns if the only stuff getting darker is in places with little incoming solar.

So, delta ice coverage can give you a delta power term, and probably has in the long term past, in climatically meaningful amounts. That puts it in everybody's models. But with large uncertainties. And it is much more obvious that big continental ice sheets well down into the temperate zone, will act as large net coolers, than that marginally thinner ice in greenland or antartica would make any difference.

[BigDog944]

As always, Jason, thanks for sharing your knowledge with the forum!

[Ike]

And here are the two grand prize questions in the whole climate change issue: (1) Do we genuinely need to do something about a warming change in the climate? (2) Are we able to do something significant about it, without reducing our industrial output or power useage or whatever but do something that reduces our wealth creation to an extent that causes deaths?

It is my opinion that the answer to #1 is "No, we don't need to do anything except buy more shorts and short-sleeved shirts." That makes the answer to #2 moot. But ...

In case you believe #2 to be an exaggeration, suppose that it were possible to return to a 19th Century level of industrial production and assume that level is low enough to affect the climate "favorably" (whatever that means): consider how many people lived in the world in the 19th Century and how many live now and who gets to die from a lack of industrial output. Your children? My grandchildren? China's? India's? Whose?

Any plan that actually reduces human produced CO2 without either massive nuclear energy production facilities being built or the invention or (and?) the introduction of some entirely new energy source necessarily reduces the production of wealth - not money, useable goods and services that people use to live. Renewable energy sources don't work; if they did, it wouldn't need subsidies from the government because the naughty nasty capitalists would be using it already and the CO2 production would (arguably)be lower already.

Science aside: why are there people who insist that we have to curtail human freedom and wealth production right now for some maybe-could be-might be-if only-perhaps bad event in the next 100 years? They use the word catastrophe, but that word implies a sudden event; what is being described is the gradual increase in the "average global temperature". Do you ever wonder why? Not altruism, I bet.

[Guest Mike]

Originally posted by JasonC:

J.K. - the sun utterly dominates the global power budget. Every square meter at the earth's distance gets a continual flux of about 1360 watts.

Is a flux measured in Watts?

And is that continuous during day or night? And which season at which parallel?

[JasonC]

A watt is a joule of energy per second, which is a energy flow aka flux. The solar radiation figure of 1360 watts per square meter is the absolute power from sunlight at the earth's orbital distance. A square meter plane perpendicular to the sun's rays would intercept that much energy from all wavelengths of photons combined. If you turn the plane you get a cosine term as the effective intercepting area decline. An infinitely flat sheet oriented parallel to the photons of course present zero area.

Since the earth intercepts as a disk and re-radiates as a sphere, there is a factor of 4 difference between its *average* power received per unit of re-radiating area, and the max solar flux itself. Thus the 340 watts per square meter of re-radiating area, figure. That is then reduced slightly for albedo (not perfectly black), and minor additional terms added for other sources (radiation, core cooling etc).

300 watts is the right order of magnitude for the average power therefore. The error bar is large, however, because there are a fair number of additional smaller effects in both directions.

[c3k]

What about changing albedo?

[JasonC]

If you painted the entire earth pitch black you could pick up more watts of the solar, certainly. The maximum possible added power there is something like 35-50 watts more if completely black.

Similarly, if you cover vast swaths of it with white - as ice ages managed to do - then you can reduce the effective incoming power. When you don't change the whole globe but only marginal slice of it - say, make a band 400 miles wide around the radius of the 45 parallel, which would come to something like 4% of the surface - and change that part from say a 0.85 albedo to something small, say 0.25, then you'd reduce the effect power by .04 (.85-.25) (35, 50) or 0.84 to 1.2 watts, roughly, or general scale 1 watt. That isn't corrected for a bigger impact for whiteness near the equator and less of an impact for it near the poles, though.

Clouds are also thought to be net coolers, best estimates their total effect is 20 watts or so, net, but with an error bar maybe a quarter that size. That is part of albedo effect. Clouds trap a little outgoing IR but have a bigger impact intercepting incoming visible and reflecting some rather than absorbing it, because they are white. If you had permanently cloudless skies you could pick up significant power. If instead you change average cloud cover 5% or so, you might pick up or lose 1 watt.

These are the sort of places people look for possible feedbacks and amplifiers. There are other variables too. Solar variation over its sunspot cycle may be as high as 1% of its total power, which means about 3 watts variation at the surface, but cycling on about a 14 year scale. Other greenhouse gases besides CO2 and water vapor can change a watt or so. Aerosol particles scattering incoming light are thought to be modest net coolers by about a watt.

Plausible change terms for modest variations (within observed extremes) are generally single digit watt scale (some smaller still e.g. observed cloud cover changes over decades only measure about 1%) and have scattered signs plus or minus.

[ April 17, 2006, 07:00 AM: Message edited by: JasonC ]

[John D Salt]

Originally posted by JasonC:

[snips]

a column of water 2 km long covered by a black body is in thermal equilibrium with its exterior environment at 291K.

Where does the 2km figure come from for the depth of water-column?

All the best,

John.

[JasonC]

Back of the envelope estimate of ocean depth, with the abyssmal stuff deeper but only 70% ocean coverage (not all of it that deep, etc). Consider it as having a large error bar, plus or minus half. For a physics problem rather than a the real world system, it is good to have an exact round number.

The main point of it is just to get the approximate scale of the transition time from a lower to a higher temperature. The oceans are the climate system's "heat sink", and account for the lags between new power applied and eventual equilibrium response.

[John D Salt]

Originally posted by JasonC:

Back of the envelope estimate of ocean depth, with the abyssmal stuff deeper but only 70% ocean coverage (not all of it that deep, etc). Consider it as having a large error bar, plus or minus half. For a physics problem rather than a the real world system, it is good to have an exact round number.

The main point of it is just to get the approximate scale of the transition time from a lower to a higher temperature. The oceans are the climate system's "heat sink", and account for the lags between new power applied and eventual equilibrium response.

Right -- does the ocean really count as "thermally well-mixed", as the other bits of quiz assume?

Is there, does anyone know, any evidence of any world-wide temperature increase in the abyssal depths? Or is it something we just don't have the research on yet?

All the best,

John.

[c3k]

Yes, looking for critical errors in the model: the assumption that the energy influx into the column of water gets averaged instantly throughout the entire volume is a large assumption. How would the model change using thermodynamics to model the heat dissipation?

The first meter of the 2 km column of water would have a significant temperature difference than the final meter. How does that variation effect the model?

Regards,

Ken

[JasonC]

Looking it up, I find the mass of the "hydrosphere" as it is known, is 1.4x10^21 kg. Earth surface area is 5.1x10^14 m^2, so the water mass per unit of surface area is 2.75x10^6 kg per m^2, or the same as a water column 2.75 km deep. About 37% more than my figure (meaning proporationally slower average temperature response) but the right order of magnitude.

An actual figure for average ocean depth I find given as 3711 meters over an area of 361 million square kilometers. Roughly half are more than 3 km deep. The deepest bit, Marianas trench, is just under 11 km deep.

As for ocean warming measurements, they have certainly been taken, mostly because they would be a sign of radiative imbalance. A very slight upward slope to temperature so measured has been reported in the papers I've seen on it.

Clearly an upper layer can warm or cool faster, and then dump its excess heat into the oceans. Similar with land surface. All vary drastically from the mean value anyway. A sunny day in the tropics warms the sea surface beyond its average value, it falls some the next night etc.

But if the surface layer as a whole gets appreciably warmer, two things happen. Some heat is exported to the "sink". And in the meantime, you get higher re-radiation from the warmer outer layer. Both act to cool the outer layer again. If it is in thermal equilibrium with its sink, though, and has a power term as high as its radiative "exports", then it can actually stay hotter. Not just as a temporary or local deviation, but a higher sustained mean temperature.

Also notice that the sink issue only effects the time scale of the response. Once the temperature reaches the higher level the new power sustains, radiative exports prevent further heating without additional net new power, even if the heat capacity of the heated object is tiny. The equilbrium temperature is thus set purely by the power term (an energy rate per unit time), not the heat capacity term (an energy "stockpile").

[JasonC]

Consider this layer example. Suppose a region only 10m thick has somehow been pushed to a current temperature of 38C, 20 degrees above the equilibrium value for the system as a whole. Stipulate that its incoming power - whatever deviation led to the temperature bump - has reverted to the mean 300 watts that was compatible with the 18C average temperature of the rest of the system. What happens in pure radiative balance terms?

Well, the output goes as the 4th power of the absolute temperature. It was 300 watts per square meter at 18C = 291K. At 38C = 311K the absolute temperature is 6.873% higher, thus the radiative output is that to the fourth higher or 1.3046 times the original 300 watts, gives 391.4 watts. So being 20 degrees hotter locally, creates a local radiative imbalance of -91.4 watts.

The same low heat capacity that let the region move to a temperature far from the average value from some transient cause, now means that big restoring force power term is going to deplete the region of its excess energy much more rapidly than in our original.

Instead of half a degree in 133 years, we have 45.7 times the power as in the original problem pushing in the other direction, and by hypothesis only 1/200th the mass. That means a temperature response that is 200*45.7 = 9140 times faster than in the original problem - heading back to 18C. With 20 degrees to go, it has 40 times as far to get as the half a degree in the original problem, so the time it will take (roughly, same averaging endpoints method as before) is 133 years * 40/9140 = 0.58 years, or 7 months.

You can get modest seasonal lags that way, all the "weather" you want, but you can't get a permanently hotter layer aka "climate". Note also that the rate above is pure radiative cooling. If the layer is in direct contact with cooler stuff (e.g. below it), it will lose its excess energy even more rapidly as energy flows across its temperature gradients, could be faster still if there is convection and mixing with cooler water from elsewhere, etc.

[Corvidae]

I know jacksquat about physics,

Thermodynamics is as alien to me as the folks on alpha centari,

[BigDog944]

Originally posted by Ike:

...

It is my opinion that the answer ... is "No, we don't need to do anything except buy more shorts and short-sleeved shirts."

Ike

I think the real concern for most thinking people around climate change is long term local climate shifts (aka "Dust Bowl" prairies) and critical food chain species extinctions, such as the loss of certain plankton species in large regions of the world's oceans.

Though a couple of extra C average temperature could possibly help with our average tanning rates, it's likely to harm far more people than are benefited by it.

One thing I really dislike about this issue is that policy is generally set by politics, not science. But then again, we lack the science to adequately explain the phenomenon.

Wouldn't it be ironic if it were man's political nature that doomed the species, rather than his war-like nature?