Were all familiar with Isaac Newton. You know, that "apple" guy. Or was it a peach? Either way, I'm sure most of you are also familiar with his second law, which says that F=ma, or that Force = mass*acceleration. There are three types of horizontal forces in the atmosphere that sum up to make the forces that we observe with the weather everyday, so let's start out by talking about these individual components.
First, let us define our coordinate system so I don't have to keep running through it again. If you know your calculus, you should understand this. If you don't know calculus, just pretend that the u axis is the x axis and the v axis is the y axis.
Here is our equation for the sum of the total horizontal forces in the atmosphere:
When do some math and derivations that I don't think we'll have to do on the exam, we get the following equation as the end-all-be-all vector equation of horizontal motion.
f = Coriolis Parameter = 2*omega*sin(phi)
phi = latitude
omega = angular velocity of earth's rotation
The air can only get started moving by means of the pressure gradient force, and it moves from high pressure to low pressure. Once it is moving, the Coriolis and frictional forces can act on it.
We can actually write this equation in two different ways. The way just mentioned is used for application to a constant geopotential height in the atmosphere with the pressure changing, while the second equation below with the -g_0 term is used to apply a constant pressure surface to the atmosphere when the height is changing.
Geostrophic balance refers to when the red (either one) and fk x v terms are equal, meaning there is no friction in the atmosphere. In this case, the velocity of the wind is directly parallel to the isobars.
Wind in the Boundary Layer:
The lowest part of the atmosphere (the boundary layer) has lots of friction, so the Equation of Motion goes changes to suit it. The vector r must balance the PGF.
Below is a series of diagrams to explain how this process works. They are taken from Professor Houze's presentation on dynamics and should be able to explain how it works better than I could ever dream of doing so with. First, we start out with some wind going to the right, and I'll let the diagrams explain the rest from there.
The Thermal Wind:
This is were things got interesting and difficult for me, so I'm going to spend a bit of time on it. First off... let's recall the first law of thermodynamics, which says that the change in internal energy is equal to the heat added to the system minus the work done by the system.
We can let a variable H=dq/dt, however, and we can get the following relationship.
Dry Static Stability:
The dry static stability equation gives us the stability of the atmosphere. The atmosphere is usually either stable or conditionally unstable, so
and omega is greater than 0, which means there is local warming. If omega is less than 0, than there is local cooling and adiabatic expansion.
There are two types of horizontal advection: cold advection and warm advection. In cold advection, wind blows from lower to higher temperature, and in warm advection, the opposite happens. Warm advection takes place ahead of a cold front whereas cold advection takes place behind it.
If we let delta T = change in temperature and delta s = change in distance, warm advection occurs when delta T/delta S is less than 0 (wind is blowing from an area of higher temperature to an area of colder temperature, and the temperature is therefore getting colder with distance) and cold advection occurs when it is greater than 0.
A good deal more confusing than thunderstorms, and not nearly as fun to write about. Well, let's move on to our next topic: numerical weather prediction!