Whoooooo math problems!
Ok, so before I get down to business here I just want to say this post is kind of a continuation of an impromptu "scientific" Global Warming debate
TheRedneck and I have been having on another thread.
So if you want the prequel -
it starts with this post on page 5 and moves
right on through to page 9 of that thread. It gets pretty wordy and even a bit testy at times, but overall quite constructive and (I hope) informative
- so maybe have a look if you've got nothing better to do
Anyway, at the end of that whole bit - I promised Redneck I would tackle his Numbers thread...so...here we are.
Basically there are two main problems I see here. The first is just kind of an appetizer though and the second is the main course.
Problem 1
Let me put it this way:
Redneck has used a value of 0.0001 to represent a (man-made)
increase of 100 parts per million of atmospheric CO2 to the overall existing
Greenhouse Effect. But does that mean the rest of the CO2 that was already there is
not warming anything?
This problem is more a technicality than anything because it won't really affect our results here in the end. But I think it's important to use it
to draw everyone's attention to how this overall mathematical model doesn't really explain the more complicated way Global Warming actually
works.
Although Redneck has come up with a very clever and creative way to try and quantify the problem - you can't exactly pin it down like this because
how do you determine where and when the warming even stops? I mean by this logic - yeah it's no big deal over the next 100 years, but also since our
only limit here is time - in 100,000 years the Earth would be 10K warmer. In 1 million years all the oceans would turn to steam!
Anyway, like I said - the problem's more technical than anything so don't get too hung up on it. Redneck's intent was to simply devise a generous
"upper limit" on how much we can affect things over the next 100 years, and under that pretense I think it's fine.
The lesson to take home though is that the
real way Global Warming works isn't merely by stacking numbers on top of each other. It's all
about balance - and how adding or subtracting anything ultimately disrupts that balance. This is why it's pretty inaccurate and unfair to trivialize
CO2 as simply affecting 0.0001% of our atmosphere though. Because it really doesn't show how much this supposedly minor change actually affects the
existing delicate symmetry.
But anyway, that was all just the appetizer - so let's throw it in a doggie bag for now and head to the main meal:
Problem 2
In a nutshell - for anyone who lives near a large body of water, what happens over the course of a typical summer day? The temperature of the air
might go up 15°C/30°F - but does the water do the same thing?
No, so there is an absolutely
fatal flaw here -
But to be accurate, the troposphere is not the only thing warming up. It has been often claimed (correctly) that the oceans are a major heat
sink. So let us now calculate the amount of energy required to raise the ocean temperature by a single degree Kelvin.
Sorry Redneck, but here you were definitely
not being accurate. Remember, although the oceans
are a major
heat sink, from a
thermodynamic/mathematical sense -
heat is NOT the same
thing as temperature.
This is exactly why we have "heat capacity" equations - to relate the two.
And as you can see from Redneck's calculations, not only does water have a much higher heat capacity than air - but the volume of the oceans is way
WAY bigger than the troposphere (~150 times).
So to assume they would
both just warm to the same equilibrium temperature, even after a hundred years, was a huge miscalculation, especially
since we're only talking relatively "minor" changes in the first place. A 1K rise in ocean temp is absolutely MASSIVE compared to a 1K rise in the
air.
To resolve this issue we can break the process down to heat flow instead of simple temperature rise.
To do that you need the equation for heat flow:
Q = m·C·ΔT
Where Q is heat, m is mass, C is heat capacity, and ΔT is your change in temperature.
Now to figure out what heat flows where let me first equate the two heat flow equations for the ocean and the troposphere to give you an idea of how
to express the temperature change of one in terms of the other. So with (o) for oceans and (t) for troposphere - we can write the following:
m(t)·C(t)·ΔT(t) = m(o)·C(o)·ΔT(o)
∴ ΔT(t) = (m(o)·C(o)·ΔT(o))/(m(t)·C(t))
To get my mass I'll borrow some of Redneck's numbers and also use a "conservative" value of 1,000,000,000 kg/km³ for the density of water (salty
ocean water is actually denser).
So with that we now have:
m(o) = (1,347,000,000 km³)·(1,000,000,000 kg/km³)
= 1,347,000,000,000,000,000 kg
m(t) = (8,694,154 km³)·(1,200,000 kg/km³)
= 10,432,984,800,000 kg
So plugging it all into our initial equation:
ΔT(t) = {(1.347x10^18 kg)·(4.1813 J/(g·K))/((10.4x10^12 kg)·(1.0035 J/(g·K))}·ΔT(o)
ΔT(t) = 537,963.7 ΔT(o)
So there you have it:
Any temperature rise in the ocean would be the equivalent of about half a million times that temperature rise applied to the troposphere.
Now if you look at Redneck's initial calculations, notice the way he organized the heat flow to match the temperatures, virtually
all the
energy goes into the oceans (there's actually a small error where he combined the two - but regardless, the contribution from the troposphere is only
significant to the 5th decimal place anyway - so it didn't matter).
So virtually ALL of that 0.01K warming is going into the oceans. If we put all of that warming into the troposphere - it would be the equivalent of a
5400K temperature rise!
Now to be fair, the answer will lie somewhere in between, but if we distribute the heat flow based on the relative volumes (150:1) - we still get a
tropospheric ΔT of 5400/150 =
36K