Assignment 9. Due April 24

  1. Read and discuss the paper by Randall (1980)
  2. The virtual potential temperature $\theta_{v}$ of a parcel of temperature $T$, vapor mixing ratio $q_{v}$ and liquid water mixing ratio $q_{l}$ is

    \begin{displaymath} \theta_{v}=\theta\left(1+0.61q_{v}-q_{l}\right)\end{displaymath}

    The total water mixing ratio $Q$ in the parcel is

    \begin{displaymath} Q=q_{v}+q_{l}\end{displaymath}

    and the moist static energy $h_{m}$ is

    \begin{displaymath} h_{m}=c_{p}T+L_{v}q_{v}+gz\end{displaymath}

    1. Consider small isobaric fluctuations in temperature and water properties around a mean state. These give rise to fluctuations $\theta_{v}'$, $Q'$ and $h_{m}'$. Show that

      \begin{displaymath} \theta_{v}'\simeq\alpha\left(p,T\right)Q'+\beta\left(p,T\right)h_{m}'\end{displaymath}

      We solved results for an unsaturated layer in class. Find $\alpha$and $\beta$ (i)  for saturated air in terms of the constants given and $\frac{dq_{s}}{dT}]_{p}\left(T,p\right)$, the slope of the saturated vapor mixing ratio with temperature at constant pressure. (Hint: see Randall (1980), JAS 37, 125-130)
  3. Draw a graph with $h_{m}$ as the abscissa and $Q$ as the ordinate.

    1. Sketch the curve $Q=q_{sat}\left(h_{m}\right)$ on this graph, assuming a constant pressure level.
    2. Consider a sounding made from sea surface to a level 100 mb above a solid stratocumulus deck capping a well-mixed boundary layer. Sketch this sounding on your graph.
  4. Consider a well-mixed marine cloud-topped boundary layer at equilbrium at night. The downward $LW$ flux a cloudtop is 280 $W\, m^{-2}$. The sea surface temperature and pressure are $T_{0}=287.5$ K, $p_{0}=1010$ mb, and the inversion is a $p_{TOP}=915$ mb. The air just above the inversion has vapor content $Q_{+}=$3 g/kg and potential temperature $\theta=300$ K. The horizontal wind speed at 10 m above the sea surface is $V_{H}=8$ m/s, the mean divergence in the layer is $D=3\times10^{-6}$s$^{-1}$ and the transfer coefficient for heat and moisture is $C_{T}=0.001$. There is no precipitation. The pressure, height and temperature at cloud based are $p_{CB},$$z_{CB}$ and $T_{CB},$ respectively.

    In this problem you are to find the properties in the equilibrium layer. You may assume a hydrostatic pressure distribution, with:

    \begin{displaymath} \rho_{air}=1\, kg/m^{3}\end{displaymath}


    \begin{displaymath} \frac{dq_{sat}}{dp}\left(subcloud\right)=4\times10^{-5}\left(\frac{kg\, H_{2}O}{kg\, air\, mb}\right)\end{displaymath}


    \begin{displaymath} \frac{dq_{sat}}{dp}\left(in\, cloud\right)=2\times10^{-5}\left(\frac{kg\, H_{2}O}{kg\, air\, mb}\right)\end{displaymath}


    \begin{displaymath} c_{p}\simeq1000\,\left(J/\left(kg-K\right)\right)\end{displaymath}


    \begin{displaymath} L_{v}=2.5\times10^{6}\, J/kg\end{displaymath}

    Finally, you may assume that $T_{air}$$(z=0$) = $T_{0}$ and $Q\left(z=0\right)=q_{sat}\left(T_{0},p_{0}\right)$

    1. Show that $p_{CB}-p_{0}=A\left(Q_{+}-Q_{0}\right)$. Identify $A$ in terms of the parameters given.
    2. Find $Q$, $T_{CB},$ $z_{CB}$, and $T\left(p_{top}\right)$. (You may use pseudo-adiabatic chart or estimates based on facts given.)
    3. Find $\bar{h_{m}}$, the mean moist static energy in the layer.
    4. Estimate the magnitude of the net radiative flux out of the layer.
    5. Are the conditions for CTEI met in this case?
  5. For the above problem, consider the system if it is not in equilibrium.

    1. Find an expression for $\bar{Q}\left(t\right)$, the mean total water content in the layer at later time $t$. Assume constant boundary conditions.
    2. Estimate the magnitude of the time scale for mixing water throughout the layer (Assume a layer depth typical for the marine boundary layer off the coast of California).