Basic Air Parcel Theory Modeling

[J Trueblood]

First of all I just want to say hello!
.......Hello!
I'm new here and look forward to discussing all types of weather with everyone! Forgive the lack of expertise when posting. I've been reading these threads for some time now, so I expect I won't be too terrible at it...
Anyways, I'm pursuing a degree in Physics at a small college in Northwestern Iowa with hopes of continuing through grad school and shifting my focus to meteorology/atmospheric research. (My school doesn't offer a meteorology program since it's a small christian college.)

Right now I'm taking a classical mechanics course, and am working on a semester long project of modeling the path of an air parcel as it rises through certain atmospheric conditions...
So far I've done quite a bit of research on the topic, but as you all know, there are COUNTLESS ways in which one can model the weather...

I've tried to narrow the theory down to Adiabatic moist/dry lapse rates.
I need to find some way of modeling this air parcel motion using differentials newtons laws etc etc. (I will probably use something simple like IDL or Python to model this)

Anyways, I would love to elaborate on this subject, as long as someone is willing to add some input. So before I go all out and type my brains out, I'll stop there and see if anyone has any suggestions/questions/ideas and references for me to look at? Any input is GREATLY appreciated.

Thanks everyone, and once again it's nice to meet you all!

-Jon

P.s. if there is anyone in NW Iowa area willing to talk Iowa weather let me know! Sometimes I feel all alone up here in no-mans land..

 

[J Trueblood]

Well I'm glad I could spark everyone's interest so efficiently!!

This is just an update on my project progress, maybe added info will create some opportunities for input from you guys? If not, then this will work as a nice outline for my history of project paper...

So as I stated earlier, I'm trying to model the ascent of an adiabatic air parcel in any given atmosphere.


In order to calculate the ascent of a parcel, we must first find out what the temperature profile for the surrounding air is.

Assume that the lower level air is dry, so we will use dry adiabatic lapse rate. We can call this a constant V.

Therefore, the temperature profile, wrt height, for the atmosphere is...

Ta(z)= T. - Vz

where T. is initial temperature, V is our constant lapse rate for dry air, and z is height.

Now we must calculate the lapse rate of a dry adiabatic air parcel. This is the way in which the air parcel will cool before it reaches the LFC (or cloud height).

First, we recognize that the change of pressure with altitude in this environment is given by the differential equation:

dp/dz= -M(of air) g p/RTa(z)

Now, consider the air parcel at z=0, with initial temperature T. and lapse rate of W.
Combining the pressure variation with the first law of thermodynamics and then integrating wrt initial position and temperature to final position we get the temperature profile for the air parcel.

T(z)= T. (Ta(z)/T.)^[W/V]

Where T(z) is temperature variaiton of air parcel with height
T. is initial temp
W is lapse rate for air parcel
V is lapse rate for environment

Next, we know that Acelleration is a function of the density pressure gradient.

a= g (pressure of air parcel- pressure of surrounding air)/ pressure of surrounding air

We can use Ideal gas law to see that

a= g(Ta-T)/T

Subbing in found values we finally have the acceleration of an air parcel in a given atmosphere...

a= g[(Ta(z)/T.)^[(W/V)-1]-1]

So again..I'm trying to model the position of an air parcel when placed in this type of this environment...

I believe that this makes sense...but my trouble is with units and velocity and the general modeling of the system...If I am using a system like Python for programming, do I need to integrate the acceleration equation with respect to different heights in order to find the position of the particle at a given time? The math here is somewhat confusing for me..so any help would be very much appreciated.

Also note, that at the LFC, I will change the temperature profile to accomdate for the WET adiabatic lapse rate, which BTW varies with pressure and altitude.

Thanks for taking the time to read this!
Jon

 

[Darren Addy]

I'm not sure if I'm following you. The challenge in what you are doing is in determining the "given atmosphere". Weather stations take a snapshot of the "given atmosphere" by releasing a real "packet of air" in the form of a weather balloon with http://www.erh.noaa.gov/gyx/weather_balloons.htm]radiosonde[/url] equipment attached. One way that results are used is to plot a Skew-T chart, which takes into account the variables you are discussing. Lots of good info on the Skew-T and how to interpret it at theweatherprediction.com.

Without being given a set of those variables (temp/pressure/humidity/wind by altitude) I don't see how you have anything to work with in your algorithm. But perhaps I misunderstand what you are trying to accomplish.

 

[J Trueblood]

Hm..let me see if I can clarify.
I'm using a simple modeling program, such as VPython and, using simple differential equations, want to model the path of an air parcel as it rises through a "given" atmosphere.
The variables, such as temp pressure humidity and wind by altitude, are all paramaters than when the appropriate mathematical equations are used, can simply be entered in order to create the 'given atmosphere.'

So in essence I am creating the atmosphere, and then using equations and newtons second law to path the motion of the air parcel.
My project is basically to mdoel what the NWS does, as in the releasing of a real packet of air in a balloon.
I want to model a 'real life situation' in which I release a parcel of air, and using the mathematical equations, can watch it's ascent.

Of course, this is a simple simple simple model...I'm ignoring vertical shear, entrainment, and all that good stuff...I'm simply going to forces due to negative or positive buoyancy, which is why I am focusing on lapse rates...

Does that make more sense?