Problems

Show that the buoyancy frequency, Eq. 4-22, may be written in terms of the environmental temperature profile thus:

Te \ dz where rd is the dry adiabatic lapse rate.

2. From the temperature (T) profile shown in Fig. 4.9:

(a) Estimate the tropospheric lapse rate and compare to the dry adiabatic lapse rate.

(b) Estimate the pressure scale height RT0/g, where To is the mean temperature over the 700 mbar to 300 mbar layer.

(c) Estimate the period of buoyancy oscillations in mid-troposphere.

3. Consider the laboratory convection experiment described in Section 4.2.4. The thermodynamic equation (horizontally averaged over the tank) can be written:

where h is the depth of the convection layer (see Fig. 4.28), H is the (constant) heat flux coming in at the bottom from the heating pad, p is the density, cp is the specific heat, t is time, and T is temperature.

We observe that the temperature in the convection layer is almost homogeneous and ''joins on'' to the linear stratification into which the convection is burrowing, as sketched in Fig. 4.28. Show that if this is the case, Eq. 4-31 can be written thus:

pCPTz d 2

2 dt where T is the initial temperature profile in the tank before the onset of convection, and T-stratification, assumed here to be constant.

dT is the initial

(a) Solve the above equation; show that the depth and temperature of the convecting layer increases by Vi, and sketch the form of the solution.

(b) Is your solution consistent with the plot of the observed temperature evolution from the laboratory experiment shown in Fig. 4.8?

(c) How would T have varied with time if initially the water in the tank had been of uniform temperature (i.e., was unstratified)? You may assume that the water remains well mixed at all times and so is of uniform temperature.

4. Consider an atmospheric temperature profile at dawn with a temperature discontinuity (inversion) at 1 km, and a tropopause at 11 km, such that

(where here z is expressed in km). Following sunrise at 6 a.m. until 1 p.m., the surface temperature steadily increases from its initial value of 10°C at a rate of 3°C per hour. Assuming that convection penetrates to the level at which air parcels originating at the surface and rising without mixing attain neutral buoyancy, describe the evolution during this time of convection

(a) if the surface air is completely dry.

(b) if the surface air is saturated.

You may assume the dry/wet adiabatic lapse rate is 10Kkm-1/7Kkm-1, respectively.

5. For a perfect gas undergoing changes dT in temperature and dV in specific volume, the change ds in specific entropy, s, is given by

(a) Hence, for unsaturated air, show that potential temperature 9

is a measure of specific entropy; specifically, that s = cp ln 9 + constant, where cv and cp are specific heats at constant volume and constant pressure, respectively.

(b) Show that if the environmental lapse rate is dry adiabatic (Eq. 4-14), it has constant potential temperature.

6. Investigate under what conditions we may approximate c^dq by d (Lq/cpT) in the derivation of Eq. 4-30. Is this a good approximation in typical atmospheric conditions?

7. Assume the atmosphere is in hydrostatic balance and isothermal with temperature 280 K. Determine the potential temperature at altitudes of

5 km, 10 km, and 20 km above the surface. If an air parcel was moved adiabatically from 10 km to 5 km, what would its temperature be on arrival?

8. Somewhere (in a galaxy far, far away) there is a planet whose atmosphere is just like that of the Earth in all respects but one—it contains no moisture. The planet's troposphere is maintained by convection to be neutrally stable to vertical displacements. Its stratosphere is in radiative equilibrium, at a uniform temperature of -80°C, and temperature is continuous across the tropopause. If the surface pressure is 1000 mbar, and equatorial surface temperature is 32° C, what is the pressure at the equatorial tropopause?

9. Compare the dry-adiabatic lapse rate on Jupiter with that of Earth, given that the gravitational acceleration on Jupiter is 26 m s-2 and its atmosphere is composed almost entirely of hydrogen and therefore has a different value of c

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