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.