Homework 3
Earth’s Atmosphere and Climate
ENVST-UA0360 - MATH-UA0228
Due: Monday, February 24th
The homework focuses on hydrostatic balance and convection - Chapters 3 and 4 in Marshall and Plumb. Show how you obtained your answer. You only need to get numerical answers with 2-digit accuracy. Please submit it through Brightspace.
1. (10pt) Starting from the hydrostatic balance
show that the air mass per unit area is given by ps/g, where ps is the surface pressure. Use your answer to estimate the mass of air per unit area and the total mass of the atmosphere.
2. (20pt) Consider an Earth-like atmosphere at a uniform. temperature of 255K in hydrostatic balance and surface pressure of 1000hP a.
(a) Consider the level that separates the atmosphere into two equal parts in mass (i.e. half the mass is above and half is below that level). Find the pressure, altitude and air density at that level.
(b) Repeat the calculation above, but for a planet where the gravitational acceleration is 3ms−2 and the atmosphere is 100% carbon dioxyde, with a molar mass of 44 grams per mole.
3. (20pt) We consider a column of water where the density is given by
We take two liquid parcels of the same volume ∆V , one initially at level z1 and density (z1) and the second at a lower level z2 < z1 and density (z2).
(a) Compute the combined geopotential energy of the two parcels (E = mgZ).
(b) Compute the combined geopotential energy of the two parcels after they’ve switched positions (i.e., the first parcel moves to z2 and the second to z1). Determine under which values of λ the geopotential energy would decrease after the switch.
(c) Compare this to the stability conditions (eq. 4-5) and discuss how these are related to each other.
4. (10 pt) In the absence of heating and moisture, the tempertaure of an air parcel evolves as
where g is the gravitational acceleration, Cp is the specific heat at constant pressure (Cp = 1007Jkg−1K−1
for dry air) and w is the vertical velocity.
(a) Show that the quantity E = CpT + gZ is conserved during the ascent of an air parcel (i.e. we have = 0).
(b) Consider now an atmospheric column where the temperature profile is given by
where Γ is an arbitrary parameter. An air parcel from level z1 and temperature T(z1) is displaced upward by 1 km. Find its new temperature and determine its buoyancy.
(c) For which value of Γ is the atmospheric column convectively stable or unstable. Explain your answer based on question 4b.