Are differential equations some sort of elitist trick?

Well I know that climate models are based on differential equations. So one would expect that you could certainly use a climate model to fix CO2 increases and then project temperatures out for 86 years.

There would certainly be a lot of assumptions due to a lack of complete understanding of the physics (e.g. cloud processes) but you would certainly be *calculating* future temperatures (ranges) using differential equations.

A linear approximation can be justified in cases where a variable is known to be slowly varying in the region of interest. In the calculation referenced, the strongly non-linear part is neatly encapsulated in the GWP numbers. The approach implicitly assumes that the cross terms are negligible (no spectral overlap). One should check and know the error introduced when making approximations. Perhaps a statement at the end indicating that a rigorous calculation gives the same result within x% would be helpful. In this forum there is a trade-off between mathematical rigor (which would use DE) and simpler explanation the captures the essence of the science for presentation to a non-technical audience.

I have to admit that I wondered how he did it too--I was impressed that he could knock that answer relatively quickly to a problem that did not sound that easy. Differential equations are not the ONLY way to solve problems, though. Sometimes an integral approach is more useful. Stephen Wolfram seems to think that much of science could be done using cellular automata, and he is definitely smarter than I am. You'd be surprised how ingenious people are that are not mathematically oriented. Geologists have invented all kinds of charts and nomograms to avoid doing mathematics. Some equations that can be solved as differential equations are also open to solution by iteration.

EDIT: You see the world in terms of differential equations--I understand that. In physics, certainly, that is the way we typically see the world. However, that does not mean that EVERYONE sees it that way. And not everyone arrives at an iterative solutions by starting with a continuous differential equation and discretizing it. The equation may be discrete to begin with--population dynamics is like that, think of the Fibonacci sequence or the logistic map. You may see them as the discretized version of a differential equation, but that's because of your own biases--other people think about them differently. And of course there is an equivalence between differential and integral equations, but they ARE different approaches to problems. Maxwell's Equations, for example, are more correctly expressed in integral rather than differential form.

2 x 4 = 8

4 x 2 = 8

2 + 2 + 2 + 2 = 8

4 + 4 = 8

Same thing, same result, different way of doing it. Some ways are better than others but the end result is the same.

If you want to do the calculation that I did using differential equations then please do so, I would be interested in seeing your result.

Instead I did the obvious thing and programmed the figures into an Excel spreadsheet and let it take care of the numbers.

Why would I have wasted so much time doing something the long winded way when I have a computer to do it all for me?

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RE: YOUR ADDED DETAILS

I have mentioned differential equations once, you’ve even quoted what I said in this question. I stated “I do know what a differential equation is. I don’t particularly like them and tend to avoid using them” by some weird and wonderful method you’ve managed to translate that into me denying that differential equations govern dynamical systems. So I have to ask… what on Earth are you going on about?

As I’ve already said and as Baccheus has said in response to this question – if you don’t like the way I did the calculations then do them yourself. If you want to use differential equations then by all means do so. If the end result is the same then it makes no difference what method is used.

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FURTHER DETAILS

The maths involved is simple, it’s long-winded and there’s a lot of it but there’s nothing complicated.

• We know what the atmosphere consists of today, we know what it consisted of X number of years ago.

• We know what the current temperature is today and what it was X years ago.

• We can assign that change in temp to the change in atmospheric composition.

• We know how long each greenhouse gas molecule resides in the atmosphere and we know how effective each of the greenhouse gases are at causing warming.

• We know when the gases were put into the atmosphere therefore we know when they will become ineffective.

In it’s very simplest form, let’s assume that CO2 resides in the atmosphere for 10 years. It has an initial GWP of 10 which declines by 1 each year. It dissipates fully out of the atmosphere after 20 years, declining proportionately with time. Again, to keep it simple, the only greenhouse gases we’ve released were 5 tons of CO2 in 2010 and 10 tons in 2011. Since 2010 the temp has increased by 5°C.

In 2010 we released 5 tons of CO2, 85% of it is still there in the atmosphere (ARP = 20, 3 years passed) but because it’s been for 3 years the GWP has dropped to 7. 5 x 7 x .85 =29.75. In 2011 we released 10 tons, after 2 years the GWP is 8, 90% remains. 10 x 8 x 0.9 = 72. 29.75 + 72 = 101.75. So the current atmosphere contains 101.75 CO2 ‘warming units’ and this creates 5°C of warming. 1°C temp rise = 20.35 units.

In 5 years time there will be 5 tons with GWP of 2 and 10 tons with GWP of 3. (5 x 2 x 0.75) + (10 x 3 x 0.8) = 31.5. 31.5 ÷ 20.35 = 1.55, so in 5 years the level of warming will have fallen to 1.55°C.

It’s not quite as simple because GWP’s aren’t linear so values have to be assigned for each year of ARP, this means you need 115 different values for CO2, 12 for CH4, 116 for N2O etc.

Differential equations are nice for closed-form solutions. Trevor outlined one spreadsheet approach for solving difference equations, which are much more commonly used in this age of computers. Multivariate equations normally turn out to be partial differential equations, which usually can't be solved without using approximations that turn them into difference equations anyway. A severe prejudice in favor of using differential equation is usually an indication that 1) you're old enough (like me) to have had more experience in pencil and paper solutions than computer ones, and 2) most of the problems you work on can be reduced to bivariate ones.

My suggestion, since you don't want to hear it from a climate scientist, and you don't want to go through the work of the previously published studies, is just do the calculations yourself.

there are too many variables to accurately predict the weather 86 years from now

they can't even accurately predict the temperature for tomorrow

De Financial Geek Asians be much much ill eats. Avoid 'em like the play egg!

And dona get me started on fracking fractions like parts vermillion.

http://www.youtube.com/watch?v=9QBv2CFTSWU