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Pressure gradients

Air and fluids · level 8, lesson 3 of 6

A question to start

Wind blows from high pressure towards low pressure. Mars's air pressure is tiny, so its pressure differences are tiny too. Does that mean Mars's winds are weak?

The idea

Take a slab of air of area \(A\) and thickness \(\Delta x\), with pressure \(\Delta p\) higher on one side. The net force is \(A\,\Delta p\), and the mass is \(\rho A\,\Delta x\). So the air speeds up at

\[a = \frac{1}{\rho}\,\frac{\Delta p}{\Delta x},\]

towards the low pressure. The small \(\rho\) on Mars makes up for the small \(\Delta p\). In fact, writing \(\Delta p\) as a fraction of the pressure \(p\):

\[a = \frac{p}{\rho} \times \frac{\Delta p/p}{\Delta x}, \qquad \frac{p}{\rho} = \frac{RT}{M}.\]

So what matters is the fractional pressure difference, and \(RT/M\), which is similar on the two planets.

A 1% pressure difference over 1,000 km Mars Earth
\(p/\rho\) 39,800 J/kg 83,300 J/kg
\(\Delta p\) 6.4 Pa 1,014 Pa
Acceleration of the air \(4.0\times10^{-4}\) m/s² \(8.3\times10^{-4}\) m/s²
Wind after an hour, if nothing else acts 1.4 m/s 3.0 m/s
high plow pslab of airnet push: towards low p

Mars also has big fractional pressure differences: sunlight heats its thin air strongly, and its whole surface pressure swings from about 4.0 to 8.7 mbar through the seasons.

Worked examples

1. Mars. \(\Delta p = 0.01 \times 636 = 6.4\) Pa; \(a = 6.4/(0.016 \times 10^6) = 4.0\times10^{-4}\) m/s². After an hour: \(4.0\times10^{-4} \times 3600 = 1.4\) m/s.

2. Earth. \(\Delta p = 1014\) Pa; \(a = 1014/(1.217 \times 10^6) = 8.3\times10^{-4}\) m/s².

3. \(RT/M\). Mars: \(8.314 \times 214/0.04349 = 40{,}900\) J/kg, close to \(636/0.016 = 39{,}800\).

Try it

  1. Why does the air speed up towards low pressure?
  2. A 2% pressure difference over 500 km on Mars: find the acceleration.
  3. Why do Mars's tiny pressure differences still make real winds?
  4. In practice, a wind does not keep speeding up for hours. Give one thing that stops it.
Answers
  1. The pressure pushes harder on the high-pressure side, so the net force points towards the low.
  2. \(\Delta p = 12.7\) Pa over \(5\times10^5\) m: \(a = 12.7/(0.016 \times 5\times10^5) = 1.6\times10^{-3}\) m/s².
  3. Mars air is so light that a small force gives it a large acceleration.
  4. Friction with the ground, or the Coriolis effect turning the wind sideways.

Common mistake

"Low pressure means weak winds." Winds are set by the pressure difference compared with the air's density. On Mars both are small, and their ratio is similar to Earth's.

Where it leads