Project 3 Convection

Transcription

1 Project 3 Convection Loovica Illari an John Marshall March Backgroun The Earth is bathe in raiation from the Sun whose intensity peaks in the visible. In orer to maintain energy balance the Earth must raiate energy away. Because the Earth is so much coler than the Sun (255K compare to 6000K) the outgoing terrestrial raiation occurs at much longer wavelengths, in the infrare. Terrestrial raiation emanates primarily from the upper troposphere, rather than the groun. Solar raiation warms the groun, but because of the enveloping water vapor layer, terrestrial raiation starting out from the surface is absorbe by the water vapor layer in a mechanism that has become known as the Greenhouse e ect. Equilibrium is not establishe solely by raiative processes. Instea, warming at the surface triggers convection which transports heat vertically upwar to the emission level where, because the atmosphere above this level is transparent in the infrare, energy can be beame out to space. In this project we enquire in to the nature of the convective process. We will simulate convection in the laboratory using a tank of water with a heating pa at its base an stuy convection in the atmosphere using thermoynamic iagrams. Note that because the atmosphere is compressible it is the potential temperature (#) of the air which is equivalent to the temperature (T ) of the water in the tank. Backgroun notes are attache. Further information can be foun in Chapter 4 of notes: 1

2 1.1 Convection When a ui is heate from below (or, in fact, coole from above, as in the ocean), it often evelops overturning motions. It may seem obvious that this must occur, as the tenency of the heating (or cooling) is to make the ui top-heavy. Consier a horizontally in nite ui as epicte in Fig. 1. Let Figure 1: Schematic of convective overturning of a ui the heating be applie uniformly at the base; then we may expect the ui to have a horizontally uniform temperature, so T = T (z) only. This will be top-heavy (warmer, an therefore lighter, ui below col, ense ui above). We will see in our laboratory experiment that motion evelops from an instability of the ui in the presence of the heating. As sketche in the gure the horizontal length scale of the convective motion that evelops is comparable to the epth of the unstable layer. 1.2 Convection in an almost-incompressible liqui Let us consier the stability of a parcel of ui in an incompressible liqui. By incompressible we mean that ensity is inepenent of pressure. However, ensity will epen on temperature; a goo approximation for water in typical circumstances is = ref (1 T ) ; (1) where ref is a constant reference value of the ensity an is the coe - cient of thermal expansion. Now, consier a horizontally uniform state with temperature T (z), an (z) then e ne by (1). We focus attention on a single ui parcel P, initially locate at z 0. It has temperature T 0 = T (z 0 ) an ensity 0 = (z 0 ), the same as its environment; it is therefore neutrally 2

3 buoyant, an thus in equilibrium. Now let us isplace this ui parcel a small vertical istance to z 1 = z 0 + z, as shown in Fig. 2. The question we are Figure 2: Parcel stability going to investigate is the buoyancy of the parcel when it arrives at B. Now, if the isplacement is one su ciently quickly so that the parcel oes not lose or gain heat on the way, it will occur aiabatically. The internal energy of a liqui epens only on its temperature, so T will be conserve uring the isplacement. Therefore the temperature of the perturbe parcel at z 1 will still be T 0, an so its ensity will still be 0. The environment, however, has ensity (z 1 ) ' 0 + z ; e where (=) e is the environmental ensity graient. The buoyancy of the parcel just epens on the i erence between its ensity an that of its environment; therefore it will be 9 positively = neutrally buoyant if ; negatively e 8 < : > 0 = 0 < 0 : (2) If the parcel is positively buoyant, it will keep on rising at an accelerate rate. Therefore an incompressible liqui is unstable if ensity increases with height. It is this instability that leas to the convective motions we iscusse above. The conition for instability is just the top-heavy conition. In the laboratory experiment we now escribe, a stable strati cation is set up an estabilize by warming from below. 3

4 2 Laboratory Experiment We can stuy a turbulent convective layer in a laboratory setting using the apparatus sketche in Fig. 3 comprising a tank of water which is heate from below. Heating at the base is supplie by a heating pa, whose power can be controlle with a transformer. The motion of the ui is mae visible by sprinkling a VERY SMALL amount of potassium permanganate evenly over the base of the tank after the stable strati cation has been set up. shining a light from a projector through the evolving convection layer on to a screen (mae from tracing paper) attache to the sie of the tank. Quantitative information can be obtaine by recoring temperature timeseries from thermometers arrange vertically at the sie wall of the tank. After switching on the heating at the base, thermals will be seen to rise from the base. Successive thermals rise higher as the layer eepens. We will carry out three experiments. In one we stuy convection in to an initially unstrati e ui of constant epth; in the others we stuy convection in to (i) an initially stably strati e ui an (ii) a two-layer ui. A stable strati cation can be set up by slowly lling the tank with water whose temperature is slowly increase with time. This is one using (i) a mixer which mixes hot an col water together an (ii) a i user which oats on the top of the rising water an ensures that the warming water oats on the top without generating turbulence - see Fig.3. Using the hot an col water supply in the laboratory we can achieve a temperature i erence of 20 C or more over the epth of the tank. The temperature pro le can be measure an recore using the thermocouples provie. 2.1 Experimental proceure an observations Convection in to a linearly-strati e layer 1. Establish a linear stable strati cation within the tank an initiate convection by turning on the heating pa. Monitor the evelopment of the 4

5 Figure 3: Convection experiment. Figure 4: Evolving convective bounary layer above a heating pa. 5

6 bounary layer visually an through the temperature recore by the sensors. Plot graphs of the epth an the temperature of the bounary layer as functions of time an interpret in terms of the following theory. The thermoynamic equation (horizontally average over the tank) can be written: T c p t = H (3) h where h is the epth of the convection layer - see g.4 - H is the heat ux coming in at the bottom from the heating pa, is the ensity an c p is the speci c heat. We observe that the temperature in the convection layer is almost homogeneous an joins on to the linear strati cation in to which the convection is burrowing. Thus T = T z h (see g.4) an so the above can be arrange thus: c p T z 2 t h2 = H The solution of the above is: 2Ht h = c p T z Thus h an T of the convective bounary layer shoul evolve like p t. 2. Plot the theoretical preiction along with your observations an iscuss. 3. Investigate the iameter an spees of the thermals as a function of the epth of the mixe layer. The thermals have a size spectrum for a given epth, so several observations will be require to enable averages to be taken. Use eqs(5) an (6) erive in the theory section to estimate typical w s an T 0 s of the bounary layer in terms of the applie heating rate. Are they in accor with the observations Convection of a homogeneous ui of constant epth Moify your experiment an the theory evelope in the previous section to iscuss convection in to a homogeneous ui of constant epth (h =constant). How oes the T evolve in time in this case? 6 1 2

7 2.1.3 Convection of a two-layer ui Set a up a two-layer ui with fresh water overlaying salty water. Measure the ensity jump across the interface an compute how long the heating pa must warm the salty layer before convection breaks through the salt inversion. At this point the plumes will exten through the fresh water layer all the way up to the surface. Check your estimates against experiment. 3 Theory 3.1 Energetics of convection Available Potential energy Show that the change in potential energy resulting from the interchange of the two small elements (of incompressible) ui shown in g.2 (of equal volume but i ering ensities) is given by: P = P initial P final = gz 2 (4) where P initial is the potential energy of the two particles before they are swappe an P final is there potential energy after they are swappe. Interpret the stability conition (2) in terms of eq(4) Law of vertical heat transport By equating the release potential energy to the acquire kinetic energy of the convective motion, show that: w 2 gzt (5) where T is the i erence in potential temperature between a parcel an its environment after it has been isplace a height z, an w is a typical vertical velocity. Hence show that a law of vertical heat transfer appropriate to the convection in our tank is: 1 P is calle available potential energy: if e e < 0, then potential energy is available for conversion in to kinetic energy of the growing convection cells: see chapter 7 of notes. 7

8 H = c p wt time = c p (gz) 1 2 T 3 2 (6) 4 Convection an Atmospheric Thermoynamics As an introuction to this assignment rea notes on ry an moist convection from web site Convection in a compressible atmosphere ( When a ui is heate from below (or coole from above), it often evelops overturning motions. We are all familiar with summer time convective clous an thunerstorms. They evelop because solar heating uring the ay warms the surface making the air in contact with it buoyant. Here we will use observe temperature pro les from raiosone sounings to stuy the onset of convection in the atmosphere. 4.1 Dry Convection in a compressible ui Let s consier the stability of a parcel of air in a compressible ui such the atmosphere. As the air parcel rises, it moves into an environment of lower pressure. The parcel will ajust to this pressure; in oing so it will expan an thus cool aiabatically. The rate at which the temperature ecreases with height uner aiabatic isplacement is calle: the ry aiabatic lapse rate = g=c p 10 o K=kilometer The stability of the pro le epens on how T= of the environment varies relative to. We n: 9 8 T UNSTABLE = < NEUTRAL ; if < E T : = E : STABLE The non-conservation of T uner aiabatic isplacement makes it a less than ieal measure of atmospheric thermoynamics. However we can e ne 8 T E >

9 a temperature calle potential temperature which is conserve in aiabatic isplacement. The potential temperature of an air parcel, enote by, is the temperature it woul have if it were compresse aiabatically from its existing p an T to a stanar pressure. The e nition of potential temperature is: p0 = T (7) p with k = R=c p = 9=7 an conventionally p o = 1000mb. The stability criterion, expresse in terms of potential temperature becomes: UNSTABLE NEUTRAL STABLE 9 = ; if 8 < : Stability of observe raiosone pro les < 0 E = 0 E E : (8) Examine the latest surface maps an satellite image. Where woul you expect ry convection to occur? Upper air sounings from weather balloons (also calle raiosones) are collecte twice aily (00z an 12z) at all major airports over the US an worlwie. Plot temperature pro les at selecte locations an look for examples of ry convection. Stuy the pro les an ientify: 1. tropopause height 2. regimes of static stability (stable, unstable, neutral) 3. note any temperature inversions an its iurnal variation A Case Stuy: Yuma, AZ June 18, 2007 Yuma is in the Arizona esert an a great location for ry convection uring the hot summer months. Here you will stuy the evolution of the bounary layer temperature uring July 18, 2007, using observe raiosone pro les every two hours. See instructions at: Comment on how the temperature pro les evolve uner the e ect of ry convection. This is a nice analogue to the tank experiment. 9

10 4.2 Moist Convection in a compressible ui Except for esert or few other hot ry locations, the atmosphere is almost always stable to ry aiabatic isplacements. On the other han we know that the atmosphere is a mixture of ry air an water vapor. It is because of the presence of water vapor that the atmosphere is convectively unstable. Let s consier what happens if we isplace an air parcel vertically. If the air is unsaturate, no conensation will occur an so our previous results for ry air hol. However, if conensation occurs, the release of latent heat will make the air parcel warmer an therefore more buoyant. The atmosphere is estabilize by the presence of moisture Stability of observe raiosone pro les Examine the latest surface maps an satellite image. Where woul you expect moist convection to occur? Plot temperature pro les at selecte raiosone locations an look for examples of moist convection. Stuy the pro les an ientify: 1. tropopause height 2. regimes of moisture (ry, moist) 3. regimes of static stability (stable, unstable, neutral) 4. if clou is expecte, estimate pressure at clou base an clou top How oes the temperature pro les evolve uner the e ect of moist convection? Comment on any analogy to the tank experiment. 5 Appenix: stability of a "stanar" atmosphere In meteorology it is customary to stuy the stability of observe vertical pro les of temperature using thermoynamic iagrams. The most common form of thermoynamic iagram use is one in which pressure forms the orinate an temperature the abscissa. While the temperature scale is linear the pressure scale is logarithmic because logp is proportional to altitue in an isothermal atmosphere. Fig. 5 shows an example of a commonly 10

11 use iagram, the skewt logp iagram in which the isotherms runs iagonally across the iagram rather than vertically. This is one so that typical atmospheric sounings take up less area on a piece of paper. In aition to the aforementione coorinates, lines of constant potential temperature (aiabats) are marke. 5.1 Stability of "stanar" atmosphere to ry processes Plot the appene tabulate temperatures for stanar atmospheric conitions in mile latitue (40 o N) an the tropic (10 o N). Make two separate plots. From each plot: 1. compare the observe tropospheric lapse rate to the ry aiabatic lapse rate. You will see that the atmosphere is stable to ry aiabatic isplacements. 2. if a parcel is isplace ry aiabatically from its equilibrium conition it will return. From your ata estimate the perio of the resulting buoyancy oscillations in mi-troposphere (see Notes, Chapter 4 - Eqs an 4.21). 3. ientify the tropopause for the stanar atmosphere - is it higher in the tropics or mile latitues? 5.2 Stability of "stanar" atmosphere to moist processes From the pro les of temperature for the stanar atmosphere we can estimate the amount of water vapor that the air can hol an stuy how it epens on height an latitue (i.e. assuming that the air is saturate): The atmosphere is a ry air an water vapor mixture. The equation of state of ry air is: an that for water vapor is: p = RT e = v R v T 11

12 where e is the vapor pressure, R v is the gas constant for the vapor an v is the ensity of water vapor. The speci c humiity q is a common measure of the amount of water vapor e ne by where q = v = R R v e = " p e p mol. wt. water vapor " = = 18 mol. wt. air 29 = 0:62: Note that q is unchange until conensation takes place. Then it becomes q, the saturation speci c humiity, which is a function of p an T : q = "e s (T ) p where e s is the saturation vapor pressure (only a function of T ) given by: e s = A exp (T ) with A an constants an T is in o C. Some measure values of e s (in mb) are: e s (mb) 6:11 13:0 23: T ( o C) Evaluate an A to t the ata in the table. From your tables an the information above euce the value of q at 850mb an hence how much water vapor the atmosphere can hol at saturation in mile latitues an in the tropics. Quote your answer in g/kg of ry air. Comment on your result. Since q is only function of temperature an pressure, lines of constant saturation speci c humiity can be plotte on a thermoynamic iagram. These are the otte lines bening to the right in Fig.6. They are labele in g/kg. Given an observe temperature pro les, we can use the q lines to estimate the maximum amount of water vapor the air can hol before conensation occurs. 12

13 As the water vapor conenses, latent heat is release an the parcel is warme, thus we expect the lapse rate of the moist parcel to be less than if it were ry. As shown in the attache Notes (Moist Convection), for a saturate parcel unergoing aiabatic isplacement: T= = s < where s is the saturate aiabatic lapse rate. Compare the observe lapse rate in the tropics an in mile latitues for the given stanar atmosphere pro les. In a thermoynamic iagram lines showing the ecrease in temperature with height of a parcel of air, which is rising or sinking uner saturate aiabatic conitions are calle saturate aiabats (or pseuo-aiabats). In Fig. 7 they are the ashe lines slightly curve to the left. A saturate atmosphere is unstable if: T= < where s <. This instability is known as conitional instability, since it is conitional on the air being saturate. Typically s 0.7 7K/km. s 13