## E

FIGURE 4.6. (a) A sketch of the laboratory apparatus used to study convection. A stable stratification is set up in a 50 cm x 50 cm x 50 cm tank by slowly filling it up with water whose temperature is slowly increased with time. This is done using (1) a mixer, which mixes hot and cold water together, and (2) a diffuser, which floats on the top of the rising water and ensures that the warming water floats on the top without generating turbulence. Using the hot and cold water supply we can achieve a temperature difference of 20° C over the depth of the tank. The temperature profile is measured and recorded using thermometers attached to the side of the tank. Heating at the base is supplied by a heating pad. The motion of the fluid is made visible by sprinkling a very small amount of potassium permanganate evenly over the base of the tank (which turns the water pink) after the stable stratification has been set up and just before turning on the heating pad. (b) Schematic of evolving convective boundary layer heated from below. The initial linear temperature profile is Te. The convection layer is mixed by convection to a uniform temperature. Fluid parcels overshoot into the stable stratification above, creating the inversion evident in Fig. 4.7. Both the temperature of the convection layer and its depth slowly increase with time.

FIGURE 4.6. (a) A sketch of the laboratory apparatus used to study convection. A stable stratification is set up in a 50 cm x 50 cm x 50 cm tank by slowly filling it up with water whose temperature is slowly increased with time. This is done using (1) a mixer, which mixes hot and cold water together, and (2) a diffuser, which floats on the top of the rising water and ensures that the warming water floats on the top without generating turbulence. Using the hot and cold water supply we can achieve a temperature difference of 20° C over the depth of the tank. The temperature profile is measured and recorded using thermometers attached to the side of the tank. Heating at the base is supplied by a heating pad. The motion of the fluid is made visible by sprinkling a very small amount of potassium permanganate evenly over the base of the tank (which turns the water pink) after the stable stratification has been set up and just before turning on the heating pad. (b) Schematic of evolving convective boundary layer heated from below. The initial linear temperature profile is Te. The convection layer is mixed by convection to a uniform temperature. Fluid parcels overshoot into the stable stratification above, creating the inversion evident in Fig. 4.7. Both the temperature of the convection layer and its depth slowly increase with time.

A heating pad at the base of the tank triggers convection in a fluid that is initially stratified by temperature. Convection carries heat from the heating pad into the body of the fluid, distributing it over the convection layer much like convection carries heat away from the Earth's surface.

Thermals can be seen to rise from the heating pad, entraining fluid as they rise. Parcels overshoot the level at which they become neutrally buoyant and brush the stratified layer above, generating gravity waves on the inversion (see Fig. 4.7 and Section 4.4) before sinking back into the convecting layer beneath. Successive ther-mals rise higher as the layer deepens. The net effect of the convection is to erode the vertical stratification, returning the fluid to a state of neutral stability—in this case a state in which the temperature of the convecting

Stratified Layer

Inversion

### Convection Layer

FIGURE 4.7. A snapshot of the convecting boundary layer in the laboratory experiment. Note the undulations on the inversion caused by convection overshooting the well mixed layer below into the stratified layer above.

layer is close to uniform, as sketched in the schematic on the right side of Fig. 4.6.

FIGURE 4.8. Temperature time series measured by five thermometers spanning the depth of the fluid at equal intervals. The lowest thermometer is close to the heating pad. We see that the ambient fluid initially has a roughly constant stratification, somewhat higher near the top than in the body of the fluid. The heating pad was switched on at t = 150 sec. Note how all the readings converge onto one line as the well mixed convection layer deepens over time.

FIGURE 4.8. Temperature time series measured by five thermometers spanning the depth of the fluid at equal intervals. The lowest thermometer is close to the heating pad. We see that the ambient fluid initially has a roughly constant stratification, somewhat higher near the top than in the body of the fluid. The heating pad was switched on at t = 150 sec. Note how all the readings converge onto one line as the well mixed convection layer deepens over time.

Figure 4.8 shows time series of T measured by thermometers at various heights above the heating pad (see legend for details). Initially, there is a temperature difference of 18°C from top to bottom. After the heating pad is switched on, T increases with time, first for the lowermost thermometer, but subsequently, as the con-vecting layer deepens, for thermometers at each successive height as they begin to measure the temperature of the convecting layer. Note that by the end of the experiment T is rising simultaneously at all heights within the convection layer. We see then that the convection layer is well mixed, or essentially of uniform temperature. Closer inspection of the T(t) curves reveals fluctuations of order ± 0.1°C associated with individual convective events within the fluid. Note also that T increases at a rate that is less than linear (this is the subject of Problem 3 at the end of this chapter).

### Law of vertical heat transport

We can use the energetic considerations discussed in Section 4.2.3 to develop a sim ple ''law of vertical heat transport" for the convection in our tank, which turns out to be a useful model of heat transport in cumulus convection. To quantify the transport of heat (or of any other fluid property) we need to define its flux. Since the quantity of interest here is the vertical flux, consider fluid moving across a horizontal plane with velocity w; the volume of fluid crossing unit area of the plane during a small time interval St is just w St. The heat content of the fluid per unit volume is pcpT, where cp is the specific heat of water; accordingly, the heat flux—the amount of heat transported across unit volume per unit time—is

In a convecting fluid, this quantity will fluctuate rapidly, and so it will be appropriate to average the flux over the horizontal plane and in time over many convective events. In our experiment we can think of half of the fluid at any level moving upward with typical velocity wc and temperature T + AT, and equal amounts of cool fluid moving downward with velocity -wc and temperature T. Then the net flux, averaged horizontally, is just 2 pfpwcAT.

Now we found that the change in potential energy resulting from the interchange of the two small parcels of (incompressible) fluid is given by Eq. 4.7. Let us now assume that the potential energy released in convection (as light fluid rises and dense fluid sinks) is acquired by the kinetic energy (KE) of the convective motion:

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