HOMEWORK
Macrophysical cloud properties
1. Radiative cooling at top of stratocumulus
A cloud-topped boundary layer has a depth h = 0.5 km and a mean potential temperature
θ = 293 K. Across the top of the boundary layer the jump in total water mixing ratio is given
by qT = -5 g kg-1 and the jump in equivalent potential temperature ∆θe = 3.5 K. These
differences are calculated as the value above the boundary layer minus the value in the
boundary layer. The evaporation rate at the surface is 2.5 x 10-5 kg m-2 s-1 and the sensible heat
flux is zero. Assume that h, θe and qT are constant in time (and independent of height in the
boundary layer) and that precipitation is negligible. Use the budgets of mass (h), water, and θe to
calculate the rate of change of θe by radiative flux convergence in the boundary layer. Express
this rate in degrees per day. Calculate the vertical air motion in cm s-1 at the top of the boundary
layer.
2. Cloud Topped Boundary Layer Entrainment
A cloud-topped boundary layer initially has a depth h = 0.5 km. The entrainment velocity
is 5 mm s-1. Across the top of the boundary layer the jump in water mixing ratio is
qT = -7 g kg-1 (value above minus value below). If these conditions were to prevail for
a day, what would be the change in height of the mixed layer in meters, and what would
be the change in mean total water mixing ratio in the boundary layer. Show your work.
3. Vorticity in a convective cloud:
Consider the inviscid Boussinesq equation for the time rate of change of the vertical
component of vorticity in a frame of reference moving horizontally with a convective
cloud. Show that if this equation is linearized around an environment basic state:
̅
, 0,0 , the result is:
where the equation is applied at the level where the cloud's horizontal motion vector
equals ̅ . Draw a sketch showing that this relationship implies that an updraft in an
environment of unidirectional shear results in a pair of oppositely rotating vortices
located on either side of the updraft.
4. Gust Front
Assume an environment with air temperature 300 K, pressure 1000 hPa, relative humidity
70%. Assume the top of gust front propagating through this environment is 0.5 km above
the ground. Assume the air behind the gust front is 15 K cooler than the environment and
has a relative humidity of 50%. Calculate the gust front speed relative to the environment.
For simplicity assume that there is no pressure difference across the gust front.