Calculus and Meteorology

[lookingnorth]

I am a high school student who is taking AP Calculus BC. I have already taken my AP test and now we are having to work on end of year projects. I chose to do mine on the applications of calculus in meteorology. I have found a few basic websites, but nothing really in-depth. The main problem I'm running into is I haven't taken any courses on atmospheric science or the like, though I do have some basic knowledge from reading books. One thing I'm thinking about doing is using vector math, but most of that seems to require multivariable calculus. I'm trying to teach myself a bit of it, but I think it will be hard to learn before the project is due. Do you have any good ideas or resources?

[Jim Marusak]

one of the best sources in the use of calc in meteorology is in dynamics. and any meteorologist in here would know this last name when it comes to dynamics: James Holton.

 

http://www.amazon.com/Introduction-Dynamic-Meteorology-Fifth/dp/0123848660/ref=sr_1_1?s=books&ie=UTF8&qid=1400445988&sr=1-1&keywords=holton%2C+meteorology%2C+dynamics

 

the textbook is now in the 5th edition. when I went to school, my version was the third edition (yes that's how long ago I got my degree).

 

and for what you'd use calc-wise in dynamics, there is some vector math, but a lot of multi-differential and multi-integral calc. and no matter how much anyone may think the class was hell to go through (and how much the class can scare people off from the major), they also know how crucial it is for not only how to explain how things work in the atmosphere, but also how we relate that into numerical weather prediction down the line.

 

and one of the big ideas that come out of those atmospheris dynamics: the quasi-geostrophic vorticity equation.

 

http://en.wikipedia.org/wiki/Quasi-geostrophic_equations

 

with that equation, you get to see the slow and microscopic spin that is one of the little whirls that helps make up the storms that the models predict.

 

but the holton text i previously mentioned would be a good first reference.

[ohleary]

A major (maybe the largest?) application of calculus in Meteorology is in numerical weather modeling.  There is a good presentation on this and it might give you some background and good ideas for a project.

 

http://weather.ou.edu/~scavallo/classes/metr_5004/f2013/lectures/NWP_LecturesFall2013.pdf

[LocoAko]

Since almost everything in meteorology is calculus based, there could be plenty of things for you to investigate that involve calculus but not vector calculus. I'm not sure what the project entails but could you do something related to the growth of a raindrop (or anything else with cloud microphysics, really). Most of those are just rate equations for the growth of a droplet radius, etc.

[lookingnorth]

one of the best sources in the use of calc in meteorology is in dynamics. and any meteorologist in here would know this last name when it comes to dynamics: James Holton.

 

http://www.amazon.com/Introduction-Dynamic-Meteorology-Fifth/dp/0123848660/ref=sr_1_1?s=books&ie=UTF8&qid=1400445988&sr=1-1&keywords=holton%2C+meteorology%2C+dynamics

 

the textbook is now in the 5th edition. when I went to school, my version was the third edition (yes that's how long ago I got my degree).

That's a cool idea, but I don't really have $40 to spend on my project.

 

A major (maybe the largest?) application of calculus in Meteorology is in numerical weather modeling.  There is a good presentation on this and it might give you some background and good ideas for a project.

 

http://weather.ou.edu/~scavallo/classes/metr_5004/f2013/lectures/NWP_LecturesFall2013.pdf

Ok. I'll look through it tonight.

[Jim Marusak]

That's a cool idea, but I don't really have $40 to spend on my project.

 

Ok. I'll look through it tonight.

who said you necessarily had to buy it right now. how about seeing about borrowing a copy from the met department over at NC State? from what I am seeing on the maps, if your hometown is correct, that should be 30-40 mins away driving-wise. and maybe they might even have a copy that can be done as a lending between NC State and UNC libraries, if the both universities have some type of sharing/lending agreement?

[hm8]

If you wanna go really simple you can do something on deriving the hydrostatic equation. It's pretty easy to understand conceptually and has some really basic integrating and stuff... But then again it probably would be much to make a whole project out of. Might be something you can work with though... Go ahead and Google it there's a plethora of pages on it.

[lookingnorth]

If you wanna go really simple you can do something on deriving the hydrostatic equation. It's pretty easy to understand conceptually and has some really basic integrating and stuff... But then again it probably would be much to make a whole project out of. Might be something you can work with though... Go ahead and Google it there's a plethora of pages on it.

I think I will incorporate that. Assuming you mean this:

And not this: ,

[Spiraliform]

I think I will incorporate that. Assuming you mean this:

And not this: ,

 

Deriving the hydrostatic equation that, at least I am familiar with, is done considering a motion balance in the vertical sense between buoyancy and the pressure gradient force. In other words, a fluid is in hydrostatic balance when the vertical pressure gradient force (always upward, as pressure decreases with height) and the gravitational force (downward) are equal in magnitude. The key is that the hydrostatic assumption neglects vertical accelerations, dw/dt . It also assumes that friction and rotation-based terms are negligible.

 

A good starting point is to consider a slab of air with two horizontal faces at different vertical levels, and try to find expressions for the gravity force per unit area and pressure gradient force upon the slab. For example, the force of gravity per unit area on the slab would look something like g*m*dz/(dxdydz) or g*rho*dz . The vertical pressure gradient, which would also have to be in the same units for the equation to make sense, must be in units force per unit area, which is just a unit of pressure by itself. Note that you'd be trying to find the gradient of pressure through the slab and not just the pressure at some point on or in the slab.

 

Then, equate those two forces and the rest is just rearranging the terms and taking limits. When you arrive at the hydrostatic equation from this, you can use integration with limits from the surface to the top of the atmosphere (or "infinity") to derive an equation for pressure at any height assuming hydrostatic conditions. There are a few websites online that have conceptual images to help explain the balance that can be found pretty easily through Google.