Wednesday, March 05, 2008

The return of Maclaurin and Taylor

About a month ago, Eli made a serious error in expanding a function from which he has received a serious amount of grief. I suppose the point being that the rabett hole has its uses. The point was to show that there was little difference between using a linear function for modeling thermal emission and a function which varies as the fourth power of the temperature. There was a graph at the end which showed that this was true, but it got lost in the forth and back.

The Maclaurin series is an expansion around a point


f(a) means the function evaluated at point a, f'(a) is the first derivative of the function at point a, f"(a) the second derivative of the function at point a, etc. If we apply this to the Stefan Boltzmann formula for emission as a function of temperature
then (UPDATE: AARGH forgot the 2! (which I neglected anyhow) Thanks Nick (I think?)))What happens if we keep only the first two terms and expand the function around 300 K for temperatures in the range 250-350K
which is not bad. We could improve this by adding a small offset to the linear term

which is what we originally showed as the best linear fit.

23 comments:

Nick Barnes said...

Your 12T^2 term should be 6T^2 (you have to divide by 2!).
I still think that Maclaurin is a bizarre way to treat a harmless fourth power. The explicit expansion of (T_0 + dT)^4 is so trivial. Look ma, no calculus.

Anonymous said...

yep, what Nick said. Yet another simple error. Maybe it's a sign Eli.

Anonymous said...

Aaargh!

Why does everyone else catch those errors before me?

As Nick said, draw a Pascal triangle :-)

Martin

Anonymous said...

Give it up Rabett, try something you think you're good at which is.....?

Return to the ice growth, Rabett. Getting close to that 14m sq km number.

JohnS

Anonymous said...

Theres a new solar variability article out in physics today: Scafetta & West. I'd be really interested to hear what you think.

Steve Bloom said...

anon 7:01 pm, the S+W article is just a retrospective of their work. Interestingly it was listed in the PT table of contents as an opinion piece. Was that a message?

As for what's wrong with their ideas, searching on their names over at RC will turn up plenty of material.

Anonymous said...

\begin{nitpick}
What you have there is called a Taylor series, a Maclaurin series is where $a=0$.

Also, since you are expanding a polynomial of degree four, your Taylor series will stop with the degree four term. Notice that the Taylor series will in fact be the original polynomial.

REALLY, just foil it all out and you get $\sigma T^4$

And yes, this always happens when you use a polynomial.
\end{nitpick}
elspi

llewelly said...

Off topic but Eli, you may be interested in this . Pay close attention to the closing sentences of the paragraph labelled 'Arctic sea ice recovers a bit'


The extra sea ice extent will help to reduce the amount of melting this summer, but this effect will probably be overshadowed by the fact that natural wind patterns have forced a large amount of thick, multi-year ice out of the Arctic this winter. This has left much of the sea ice very thin, making it very vulnerable to melting. For the first time on record, the edge of thin first-year ice has pushed beyond the North Pole. IF we get another relatively warm and sunny summer in the Arctic in 2008, we will likely see Arctic sea ice loss surpassing last year's astounding collapse.


Recall that you bet 10 carrots on this outcome.

EliRabett said...

With the way the $ has been going, Eli is gonna need those carrots.

Anonymous said...

Hey Rabett, can I use your approximations of above to say the more than 13.8m sq km of Arctic as per Cryosphere Today, is 14 m sq km. That figure so horrified various Anons only a couple of months ago, as it would be UNPRECEDENTED ice return. Old Gaia, what a joker he/she is!

JohnS

Anonymous said...

John S

The peak winter number of square Km of ice means very little with regard to long-term trends.

Try end of summer.

New thin ice melts away.

Anonymous said...

Thanks Anon, and if you have been around for several months, the old story of winter maximum not meaning anything has been thrown around quite a lot. Why do I have a feeling that if the winter ice maximum was less than 13 m sq km, you among many here would have gladly used it to shout another example of AGW. Gaia gives us 14m sq km(allowing for Rabetts approximation), and winter maximum suddenly means nothing.

But okay Anon, what ice is Gaia going to leave in the Arctic at the end of September. Is it 4m sq km, 3 m sq km, less than 3 m sq km. Rabett has a bet with Connolley of less than last summer of 3 m sq km. I say, because of upcoming cool northern summer, that ice at end of September will be 4m sq km or greater. Thin ice and all.

JohnS

bi said...

JohnS screams,

"Why do I have a feeling that if the winter ice maximum was less than 13 m sq km, you among many here would have gladly used it to shout another example of AGW."

Because you denialists love making up crap. That's why it's called a "feeling", not a fact.

Anonymous said...

what ice is Gaia going to leave in the Arctic at the end of September. Is it 4m sq km, 3 m sq km, less than 3 m sq km"

Even this confuses the issue.

The trend over the long term is the only really meaningful thing.

ice extent goes up and down from one year to the next becasue there are lots of factors involved contributing "scatter".

but the fact is, the trend has been downward (both in ice extent and in thickness0 over the past few decades.

You would need more than a singe year -- or probably even more than a decade -- to be confident that that trend had reversed.

So your bet is without meaning, John.

Anonymous said...

Trends are for those who don't want to think. Lets only use satellite coverage of ice area and thus only have a trend for 30 years. Why is a thirty year trend meaningful. Bi of above has the right word to describe your trends and his own faith warmer garbage, ie "crap". The Hockey Stick is not so straight, only when you so want the Mannian trends(go look at CA for recent Mannian "crap"). And the blade has gone flat these last ten years.

You see trends, Anon, good, so you and Bi can go and pay the carbon tax. Do it while switching off your power and freezing in the dark. You don't do that, easier to be a run the lights, power and heater while writing useless comments on trends on your computer.
JohnS

Anonymous said...

John S says it all: "Trends are for those who don't want to think."

So, John.

Which means more from a long term investment standpoint:

The ups and downs in the market from one year to the next?

Or the long term trend?

Note: obviously, to the day trader , it is the ups and downs that are his bread and butter, but for the long term investor, it is the long term trend that is most meaningful -- just as with global climate.

Your focus on one year's ice extent (either in winter or summer) is simply misguided.

There are so many factors that can affect a single year that it becomes almost meaningless to use one year to indicate anything with regard to the climate over the long term.

Anonymous said...

Incidentally, John S. I would urge you to think about the above question about the stock market in the context of 9/11.

I really pity those who dumped all their stocks based on where the stock price stood in the year following 9/11.

Anonymous said...

Any old ice? Any old ice?
Any, any, any old ice?

(with apologies to Collins / Sheppard / Terry).

Cymraeg llygoden

Anonymous said...

Rabett attracts all types, and now a financial advisor. Good man Anon, for advice ,but think how well off the investor would be who sells his investment just before the fall, and then buys just as the turning point at the low point.

I'm sure we'll all look back in ten years time at the current stock market dip and have a laugh. "What fools they were to sell their stock in the December 07 to say June 08 period". A bit different here and now as one tries to keep their wealth. But stock market armchair analysts like you Anon, are always around. A millionaire in their minds.

Nothing to do with Arctic ice, however.

JohnS

Anonymous said...

John S says:

"think how well off the investor would be who sells his investment just before the fall, and then buys just as the turning point at the low point."


Ah, yes, the proverbial "Buy low and sell high" (or "sell high and buy low" in John's case)

It's always a joke when someone suggests as much ... until now, that is.

Anonymous said...

There speaks an Anon who will always remain a poor man. "The trend is his friend" will remain his refrain. Unfortunately the Government is currently bailing out a lot of trend slaves with tax payers money, or providing the final guarantees that could cost the tax payer.
Let the Anon trend slaves go to the wall, maybe then they'll be a little wiser about their money.

Again, nothing to do with Rabett's growing Arctic ice. Except Anon has the same mentality of I want the government to fix my problems, be it finance or climate

JohnS

EliRabett said...

Hmm, Eli remembers in school learning about a certain document that began, We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.

Of course, John's MMV as he obviously enjoys his autarchy. OTOH working together does help a whole lot of things.

Anonymous said...

"There speaks an Anon who will always remain a poor man."

Ah, yes. The "You too can get rich with just this one lottery ticket" claim.

"Anon has the same mentality of I want the government to fix my problems, be it finance or climate"

All I said was "The trend is the important thing" and that's what it means at a deeper level?

I had no idea.

John, you are simply amazing.

Are you psychic or just schizoid?