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Science / Forces / Undergraduate / sfd30

Wind and Air: Derive an Atmospheric Profile

Combine hydrostatic balance and an ideal-gas model with explicit assumptions.

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5 questions0m 0s

Prerequisites

Differential equations, ideal gases, dimensional analysis, and hydrostatic balance.

Learn the skill

For an isothermal ideal-gas atmosphere, dp/dz = -rho g and p = rho Rs T. With constant g, T and specific gas constant Rs, define H = Rs T/g. Then p(z) = p0 exp(-z/H), and density has the same exponential shape.

Worked example

At z = H, p/p0 = exp(-1), about 0.368. At z = 2H it is exp(-2), about 0.135. With Rs = 287 J/(kg K), T = 280 K and g = 10 m/s^2, H = 8036 m. Real temperature profiles need not be isothermal.

At constant temperature and gravity, hydrostatic ideal-gas pressure follows p/p0 = exp(-z/H). At heights zero, H and 2H the relative pressures are 1, about 0.37 and about 0.14.
At constant temperature and gravity, hydrostatic ideal-gas pressure follows p/p0 = exp(-z/H). At heights zero, H and 2H the relative pressures are 1, about 0.37 and about 0.14.
Question 1 For the stated Rs, T and g, what is H?
Question 2 What is p(H)/p0 to three decimal places?
Question 3 If temperature doubles while Rs and g stay fixed, H does what?
Question 4 If top pressure tends to zero and g is constant, total column mass per area is what?
Question 5 What most directly breaks the single constant-H profile?

Further inquiry

Derive the exponential profile by separating variables, then integrate density to obtain the column mass per area. Replace constant T with a specified positive T(z) and write the integral solution for pressure. Explain why this mathematical atmosphere is not a full weather model.

Review criteria

  • Declare the pressure boundary condition and constant-g assumption.
  • Keep absolute temperature and specific-gas-constant units consistent.
  • Distinguish the isothermal model from an observed atmospheric profile.