Take Home Final

Due April 10. For this final you may confer with any resources other than each other.

  1. Consider a cloudy air parcel rising pseudo adiabatically with constant velocity w

    1. Sketch the supersaturation S(z) as a function of height z above cloud base (S(0) = 0). Without resorting to calculations, show how S(z ) depends on w. Explain briefly.
    2. Assume that over some height interval (z1,z2) the change in S due to adiabatic cooling just balances that due to condensation. Let a1(z) and a2(z) be the smallest and the largest drops in the spectrum respectively. Again, without calculation, explain whether the width of the droplet population a2 - a1 depends on (a) z and/or (b) w in the layer.
  2. Consider a parcel of air in the mixed phase part of a cloud; ie. where supercooled droplets and vapor grown ice crystals coexist. Assume that the medium remains saturated with respect to liquid water

    1. Sketch the supersaturation with respect to ice as a function of temperature for -30 ∘C ≤T ≤0∘C
    2. Sketch the rate of vapor deposition on an ice particle of fixed size and shape over the same temperature interval
    3. Sketch the rate of nucleation of new ice particles (#/m3/s) over this temperature interval, if the parcel is rising at constant speed w, assuming a fixed temperature lapse rate dT∕dz. You may assume that the Fletcher parameterization for heterogeneous nucleation is valid, i.e.
      Ni = α exp β (T0 - T )

  3. Here we try to reproduce mathematically wave clouds using the Boussinesq equations. Consider two dimensional flow in a dry, stable, shallow atmospheric layer moving with mean velocity u¯ in the x-direction. The mean potential temperature increases with height:
    ¯ N 2 ≡ g-dθ-= constant θ¯dz

    The mean vapor mixing ratio, q¯v also changes linearly with height:

    dqv- dz = γ = constant

    The air flows over a line of hills parallel to the y-axis. The hills produce a small time-independent circulation ′ ′ ′ ′ ′ (u ,w ,p ,θ ,qv)

    1. Write the appropriate linearized Boussinesq equations for the perturbation variables. Neglect the contribution of vapor to the buoyancy.
    2. Reduce the equations to a single equation for w′
    3. Show that solutions to the equation are of the form w′(x, z) = w0 exp [i(kx + mz )] and find the relationship between k, m, ¯u, and N.
    4. Find u′(x,z) and sketch the streamlines for k > N∕¯u.
    5. Find the mean relative humidity with height RH(z) ≡ qv(z) ∕qsat( ) ¯θ (z) such that the motion w′(x,z) induces saturation: i.e. that qv(x,z) = qsat[θ(z)] somewhere. Note that the vertical velocity perturbations should induce adiabatic warming/cooling of a particular parcel.