Assignment 8 Due Mar 25

  1. It is often argued that mammatus clouds, often seen as bulbous protuberences below anvil cloud base, are the consequence of hydrometeor sublimation. This occurs when cloudy air is advected over unsaturated air with a higher value of the conserved variabile liquid water static energy
    hl = cpT + gz - Lvql

    Such a scenario is quite plausible in regions associated with wind-shear near cloud base. The argument is that if cloudy air is mixed downward into unsaturated air with a higher value of hl, and just enough mixing occurs to evaporate all of the liquid water in the cloudy air, the resulting mixture will be negatively buoyant with respect to the surrounding air. The thermal then mixes with the sub-cloud air causing evaporation, and the negatively buoyant parcel accelerates downward, disappearing at the point that all liquid water in the parcel fully evaporates.

    Here we explore this hypothesis in more detail. Consider an initially saturated penetrative downdraft in the form of a spherical similarity thermal of radius R, vertical velocity w, total water surplus Q = qvs + ql - qve that descends from cloud base into clear air with vapor content qve.

    The entrainment parameter is α: i.e. in the absence of sources or sinks a parameter X inside the thermal obeys the equation D(ψX)- Dt = D-(R3X) Dt = 3R2αwX e + SourceψX
    At cloudbase R = 0, w = 0, ψX = ψX0

    1. At time t the thermal has descended a distance ζ. Let w be positive downward, so in the frame of the thermal D- Dt = wd- dζ. Show that the equations for the water surplus and heat in the plume, at distance ζ below cloud base are
      d-R3 (b - M q ) = - R3N 2 dζ l

      d 3 3dqve ---R (qvs + ql - qve) = - R ---- d ζ dζ

      where

      L g M = --v-- cpTv

      hankhan

      T---Te- b = T

      HiCoudl

    2. Supposing that the sub cloud mixing ratio is uniform, show that the liquid water mixing ratio in the thermal is given by
      ( ) ψQo-- --Lv-- ql = α3ζ3 + qve - qes exp Rv ¯Tg b

    3. Find an expression for the value of ζ at which the mammatus lobe disappears, assuming the surrounding cloud is saturated and hl(ζ ) is not very different from hle(ζ).
    4. What conditions favor deep mammatus lobes? Make some reasonable calculations for ζ, as a function of a range of plausible initial disurbances in ψQ0, for a characteristic case in ambient conditions.
  2. The Ozmidov length scale represents the maximum size of eddies in the inertial subrange
    ( 3)1∕2 L0 ~ ε∕N

    where ε is the eddy dissipation rate and N is the buoyancy frequency in a vertically stratified fluid.

    1. Derive the above expression using scale analysis. Hint: The Ozmidov length scale corresponds to the height at which all turbulent kinetic energy is converted to potential energy from to the restoring force of buoyancy.
    2. Based on the Ozmidov length scale, for a typical stratiform cloud, estimate the time it takes for an entrained parcel of air to be mixed completely into the cloud.
  3. Read and summarize Entrainment, Detrainment and Mixing in Atmospheric Convection by C. Bretherton.