ATMOS 301: The Hydrostatic Equation

Thursday, October 31, 2013
3:20 a.m.

I got a lot of sleep today, OK? So I’m allowed to do this. Plus, I don’t have class until 1:30 tomorrow. Well, that’s technically not true… I do indeed a class, but truth be told, I don’t really learn much in it. It’s not that I skip it on a regular basis… I’ll just refrain from going to it tomorrow.

Seriously though, I’m going to go to bed soon. I’ve just always found that I’ve been more effective at getting a blog done once I’ve already started it.

You ever feel your ears pop in an airplane? That’s because you are going into the upper levels of the atmosphere where the pressure is lower. Remember, pressure is equal to force/area, and force is equal to mass*acceleration. When we increase in elevation, all that mass of the atmosphere that was previously weighing down on us is now below us, and the pressure on us is therefore lower. When we undergo this change in elevation and babies of all shapes and sizes begin to cry, we are directly experiencing the effects of the pressure gradient force. And, of course, as I’m sure all of you know, airplanes, while not counteracting the effects of gravity, keep us afloat at 35,000 feet so we can travel from point A to point B. These two forces – the pressure gradient force and the gravitational force – play a huge role in what is called hydrostatic equilibrium.

The gravitational force pulls objects towards the center of the Earth, so theoretically, it should pull all the atmosphere down to the ground. However, the pressure gradient force causes air to flow from high pressure to low pressure, so according to it, the atmosphere should have a homogeneous pressure distribution. So why don’t we see either of those outcomes? Well, that’s where the hydrostatic equilibrium comes into play.

Here are the individual forces that contribute to hydrostatic equilibrium.

g = gravity (9.8 m/s^2)
ρ = density
A = area
p = pressure
Δ = change in (insert variable here)

Gravitational force downward: g*ρ*A*Δz
Pressure force downward: (p + Δp)*A
Pressure force upward = p*A

Ok, it’s now 4:49 a.m. I should sleep. Goodnight.

Alright, 11:45 a.m. and back to work. I have class at 1:30. I’d like to take a shower and eat before then as well. I hope you realize how epic this will be if I get all this done.

Anyway, we were talking about the hydrostatic equilibrium and why the atmosphere is structured the way it is. Since we have these forces that work in opposite directions, they get to a point where they equalize and the general structure of the atmosphere doesn’t change. The equation for this is given by:

p*A =  (p + Δp)*A + g*ρ*A*Δz

This equation shows the pressure force upward having an equal magnitude to the sum of the pressure force above the specified level of air downward and the gravitational force downward. We can also restate this as:

Δp/Δz = -ρ*g

And by taking limits and partial derivatives, we end up with our final hydrostatic equation that we all use… an equation so important that it deserves its own picture.

Geopotential Height:

I know what you’re thinking. These terms aren’t getting any easier to understand. I’m right there with you. Although, I have to admit, I’ve always seen this term thrown around on the UW mm5/WRF model charts, and I’ve always wondered what it was but for some reason or another never actually tried to figure it out. Well, now that I have a midterm in 25 hours and 20 minutes, I’m suddenly a lot more interested in it. Imagine that!

The geopotential Φ at any point in the Earth’s atmosphere is defined as the work that must be performed to raise 1kg of something to that point. The units are J/kg or m^2/s^2, suggesting that Φ is the gravitational potential per unit mass. The force (in newtons) acting on 1 kg at a given height z above sea level is the same as g, as g is, for our purposes, constant with altitude. The work is given by the integral of gdz from 0 to z, where z is equivalent to the height. The resulting work ends up just being equal to gravitational acceleration (9.81 meters/second^2) multiplied by the height the parcel is lifted. The equation below is just basic calculus.

The Hypsometric Equation:

Would you like fries with that?

Everybody loves a combo. And that’s exactly what the hypsometric equation is. It’s a conglomerate of our equation of state and our hydrostatic equation. Remember, the equation of state is P=ρR_dT_v and the hydrostatic equation is ∂p/∂z = -ρ*g. Now watch what we do here when we combine the equations. To make things easier to read, I’m just going to take a snapshot of Professor Houze’s slide instead of trying to make a bunch of ugly subscripts and fractions.

 Once you have dZ = (R_d/g_o)*T_v*d*ln(p), you take the integral of both sides and you get the hypsometric equation below. For those of you taking the test, I would highly recommend memorizing the equation and what each of the variables mean. It showed up on one of the quizzes and I couldn’t remember it, so I tried to try and BS it by using the hydrostatic equation, which is much simpler. It didn’t really work, but I managed to scrape a few points. 🙂

The Scale Height:

In atmospheric science, the scale height is the distance over which a substance decreases by a factor of e (2.71828… the base of natural logarithms). Because the atmosphere is well-mixed below the turbopause (about 105 km), the pressures and densities of individual gases decrease at about the same rate as a function of altitude with a scale height directly proportional to their gas constant R. Since R* is just a universal constant and the only thing that results in R being different for different gases is the apparent molecular weight of the mixture in the denominator, the scale height is inversely proportional to the apparent molecular weight. The average temperature of the troposphere and stratosphere is around -18 degrees Celsius (255 degrees Kelvin), and this gives us a scale height of around 8 km.

To be honest, we took some notes on this in class but I barely understood any of it, and our book only barely touches on it and we don’t have any lecture notes on it. So hopefully it won’t be on the midterm. If you are interested in researching it by yourself, by all means, go ahead, especially if you can get it done by the morning of November 1, 2013.

Thicknesses and Heights:

Many things in atmospheric science intersect. Fronts intersect. Winds intersect. In our neck of the woods, the Puget Sound Convergence Zone is a classic example of winds intersect. But pressure ALWAYS decreases with height, and because of that, pressure surfaces (imaginary surfaces on which pressure is constant) never intersect.

If we take a look back at our hypsometric equation, check out the term on the left. This difference in height represents the thickness of the atmosphere (usually in decameters) from one pressure level to another. The thickness of the atmosphere is determined by the temperature. Remember, a cold atmosphere is more dense, so the thickness between the different pressure levels will be more compact than it would if the atmosphere was warmer. I feel like the diagram below shows this well.

Here are some common manifestations of this effect that we see in storm systems. Mid-latitude storms (cold-core cyclones) have low heights associated with them because, as the name suggests, the air at their core is cold and therefore more dense. The opposite is true with tropical cyclones, which have very high heights.

All these diagrams, with the exception of Thumper, the Super Cool Ski Instructor, were taken from Professor Houze’s lecture on thermodynamics here. Now, onto the first law of thermodynamics!

Charlie

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