Third and last batch of chapter 1 notes. This is the part with the actual derivation, and honestly the part I almost skimmed past, which would have been a mistake.

Hydrostatic equilibrium is the idea that a thin horizontal slice of air has more pressure pushing up on its bottom face than pushing down on its top face (since pressure decreases with height), and that this net upward push is exactly balanced by the weight of the slice pulling it down. Balance the two and you get delta P = rho times g times delta z. Plug in sea-level density (1.225 kg/m^3) and g = -9.8 m/s^2, and a 100 m rise in altitude works out to about a 1.2 kPa drop in pressure, which lines up with the rough “10 kPa per km near the ground” number from earlier in the chapter. There’s a whole callout box just about getting the sign of delta right (top minus bottom vs bottom minus top, and keeping numerator and denominator consistent), which felt like overkill until I actually tried the derivation myself and immediately flipped a sign.

Combine that hydrostatic balance with the ideal gas law and, after some calculus I won’t reproduce here, you get the hypsometric equation: the thickness of a layer between two pressure levels equals a constant (about 29.3 m/K) times the average virtual temperature times the natural log of the pressure ratio. This is the actual honest way to convert between pressure and altitude, better than assuming a single constant temperature for the whole atmosphere. What stuck with me is a worked example in the book: computing the thickness of the 100-to-90-kPa layer in one big step gives 864.38 m, but chopping the same layer into thirty 0.5 kPa sub-layers and summing the thicknesses gives 864.11 m, a slightly different and more accurate answer. The difference is small here, but it’s a clean demonstration of why you’d bother iterating in small steps at all: virtual temperature isn’t actually constant across a thick layer, so pretending it is over a big jump quietly costs you accuracy.

Then there’s a section just naming things: isothermal for constant temperature, isobaric for constant pressure, isotherm and isobar for the map lines connecting equal values of each. Then it just keeps going, there’s an “iso-“ word for almost anything you’d want to contour on a weather map: isohyet for equal precipitation, isotach for equal wind speed, isodrosotherm for equal dew point, isoceraunic for equal thunderstorm frequency, isopycnic for equal density (borrowed by oceanographers, where both temperature and salinity affect density). It reads like a joke but apparently every one of these is a real, used term.

Last bit was barometers, which was more mechanical/historical than I expected. Aneroid barometers use a corrugated metal can with a vacuum sealed inside, atmospheric pressure squeezes the can and levers amplify that tiny deflection into a needle position. Mercury barometers go back to Torricelli in the 1600s: a vacuum-topped glass tube with a column of mercury whose height balances atmospheric pressure exactly, which is where “inches of mercury” comes from. Huygens later figured out you could put a different fluid on the open side to amplify the height signal even further. Modern versions ditch actual mercury (it’s poisonous) for a silicone-based fluid, or skip fluid columns entirely and use strain gauges or capacitance changes in a sealed can, digitized straight into a computer.

That closes out chapter 1. Next chapter is thermodynamic diagrams.