Year 12 Maths
Part of my job as a teacher of meteorology is to go to NZ universities looking for future meteorologists. It breaks my heart when, sometimes I meet a person with a genuine passion for the weather who would love to work for us as a meteorologist, but just can’t cope with the required maths. Unfortunately for them, professional meteorologists need to have some university maths under their belts. And this requirement isn’t just a local thing – it also comes from the World Meteorological Organisation of which we are a member nation.
Why is maths important in meteorology? Let me give a small example to illustrate, related to the thread of my previous blog posts, wind…
On weather maps we see isobars which are the lines going around each anticyclone (area of high air pressure “H“) and depression (area of low air pressure “L“). Have you ever noticed that the isobars seem to spread out more in the anticyclones than in the depressions?
How isobar spacing varies around anticyclones and depressions
To check out this effect, take a look at the real-life example below (covering a large part of the southwest Pacific – I have undisplayed the fronts), or have a look at a few current weather maps to see for yourself.
Mean Sea Level analysis, midday 4 July 2006
This spreading out of isobars in anticyclones is a fairly general result, and the reason comes down to the solution of a quadratic equation. If you did maths at high school you might remember these. I learnt about them in the fifth and sixth forms (now called Year 11 and Year 12). These equations look something like this: ax² + bx + c = 0, where a, b and c are given to you and you then have to work out what “x” is. You can find out lots more about them on the internet, e.g. Wikipedia, or just ask your kids!
When our trainee meteorologists study weather, they learn all about wind, isobars and air pressure, as well as the effect of the Earth’s rotation. It turns out that there is a strong connection between these factors, and it is described by a quadratic equation (I won’t write the equation here, but you can ask me for it through the comments section below if you’re interested).
The solution, x, of this particular quadratic equation holds the key to the answer, because x represents the wind speed. In this case x behaves differently depending on whether you are in an anticyclone or a depression. For anticyclones there’s a limit to how close the isobars can be and therefore a limit to the wind speed, depending on how close to the centre of the anticyclone you get (there is a small latitude effect too). However, for depressions there’s no limit to the closeness of the isobars, provided the atmospheric forces are big enough to drive the air pressure into that configuration.
I think it’s an amazing and powerful result, and it all comes down to school-level maths! This really is just the tip of the iceberg as far as maths and meteorology are concerned - you could spend a lifetime studying solely the meteorological applications of mathematics.
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Postscript:
The quadratic equation has been requested as in the Feb 2011 comment below. The equation is:
which can be solved for V:
When we were at school,there was no reason why we had to learn,we were told we Had to learn them,parabolas with quadratic equations where y and x intercepted,cosine,tangents,etc learnt as toa,matrices what use at the time they were?Find the solution to to the equation.
Not they were required for a job.
Thanks for your comment; I hope you enjoyed reading my posting.
You say “For anticyclones there’s a limit to how close the isobars can be…”. Why is there a limit?
The answer is in the nature of the solution to the quadratic equation…the solution has a square root in it (you can check out the details from the Wikipedia link in my post) and to get “real” values the bit inside this square root has to be a positive number.
I mentioned that ‘x’ represents the wind speed, but ‘a’, ‘b’, and ‘c’ also represent things – like the curving of the isobars, the latitude, and the spacing of the isobars. When the curve of the isobars is as you get in an anticyclone, the requirement for real values puts a limit on the spacing of the isobars. For a typical depression, the bit inside the square root is always positive, so in that case there’s no limit on the isobar spacing.
I hope this all makes some sense – I’m happy to explain further if you want me to.
Leaving the equations aside for a moment, what is the physical basis for anticyclones having a “limit to how close the isobars can be and therefore a limit to the wind speed”
The physical reason is to do with the balance (or imbalance) of forces acting on air as it moves around the anticyclone. To explain it all properly I may need to write another post
but I’ll give it a go here…
You’re probably familiar with the notion that air blows approximately along isobars, like slots cars on a track. And the closer the isobars, the stronger the wind. There are two main forces that control this air motion – the “pressure gradient force” and “Coriolis force” – and they both behave in well-defined ways. When air goes around an anticyclone it turns out the forces don’t quite balance and the air speeds up – the more anticyclonic curvature, the more it speeds up.
Now if the pressure gradient towards the centre of the anticyclone were to get large, the isobars would get close together, and so the wind would become very strong. But eventually a limit would be reached where the wind would no longer blow along the isobars (relates to Isaac Newton’s first law), and the air would drift across the isobars out of the anticyclone. The effect of this would be to weaken the pressure gradient and spread out the isobars in the anticyclone. Because of the orientation of the forces, this effect doesn’t apply in a depression.
Would you like me to write a full blog post to better explain this?
Appreciate the explanation. I had thought it might be gravity-related but the answer is coriolis effect. No need for full blog.
You are welcome. In a way, it is gravity-related because the pressure gradient force (that approximately balances the Coriolis force) comes about because of gravity.
An able Year 13 Calculus class has requested details of the quadratic equation. I have appended it at the end of the original post.
In the equation:
V is the wind speed,
f is a coefficient related to the Earth and its rotation,
the letter that looks like a “p” in the denominator is actually the Greek letter “rho” and represents air density, and
the dp/dn is a measure of the spacing between the isobars.
This quadratic equation can be solved to give the wind speed, V.