Mull, S

Rao, Y P

English

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60 THEORY OF CUMULUS CONVECTION. Bv S. MULL AND Y. P. RAO. (Rccdwd for i'ublicaliol" Sept, 15, IlN8) ABSTRACT. A new theory of convectiul haFeo (Ill lateral flow of nir int,n the clouel is discussed, a~ an alt('rnat;ve to the 'parcel nnn slice' tbe()rie~ TNTRODTlCT~ON : The fundamental assumption made in the derivation of equations re-presenting "parcel and slice" method of conv~tion is that the pressure gradient along the vertical is uetermined by the surrottlding air, As this assumption would only be valid in the l1Iost initial stage of the growth of the cOllvection clouds, i.e" in the stage \"hen the parcel is isolated and small in dimension and not at the stage when convection is represented by a rising column of cumulus or cumulonimbus, It was thought advisahle to examine the problem of cUlllulus convection tu sec if a more realistic picture of the growth of the convection clouds could 1)e given, at least qualitatively, PRHSl\NT TDEAS: The plesent ideas ahout the growth of convection clouds (cumulus and cUl1lulo-nimbus) al e represented by the parcel and the slice theories of con-vection. Whereas, ill the first process the parcel of air is assumed to ascend adiabatically with 110 lateral llIixing through an environment at rest, in the second one the environment is allowed to subside in order to preserve lIlass continuity. The two processes, however, fundamcn tally depend' upon the principle that a body floating or immcr!>ed in a fluid is subjected to an upward directed force equal to the weight of the amount of fluid that the botly dis-places. The body will rise or sink or remain at the same level depending on whether the force is greater than, less than, or equal to tespectively the downward force on the body due to acceleration of gravity. Realising that we are dealing with a parcel of air instead of a fixed body, we may write, following Petterssen (1945), for a parcel of air cmbedded in an environment in hydrostatic equilibrium . op' w'= -a'-,·- -g oz 'Or o=-CI,- -g oz where letters with indices refer to the parcel and letters without in~ices rder to the environ ment and where J-1674P- 1a 532 S. Mull and Y. P. Rao ~" == vertical acceltra t iOIl a = specific volume f> = pressure g = acceleration of gravity ;:; = di::.tance along a vertical axis. If the pressure gradient along the vertical is determined by the ~urrounding air, then we have and t11elefore or " _ (a'-a) w - g -;----. (T' T) ,,/ = g -----T where T is the absolute temperature, wllich shows that the parcel will be accelerated upward if it is wanner and doVl nwards if it is colder than the environment. As it is dear, the ahove derivation is based on the fundamental assump-tion that the pressure gradient along the vel tical is determined by tIle sur-rounding air. This assumption would cel tainly be valid in the most initial stage of cloud formation, SitlcC at that stage the parcel is lsoJated and small in dimension. It is, how('ver, difficult to c01lcciV(' how the sal11e will hold good during the suhsequent stages of cloud growth, \\"here instead of a parcel we have to deal with a li!-.ing column, since under such circulllstances the pressure difference between allY two horizontal levels ill the rising column and the surrounding environment callnot be the same. The slice method as developed by Bjerkl1e~ (I938) and Petterssen (1939) computes the excess of heating of the rising column over the surrounding descending air after taking into consideration the modification in temperature due to vertical motion. No attempt is made to explain how the temperature difference results in acceleration or deceieration of the rising air. Presumably it is taken that the accekratioll can be derived trom the temperature differ-ence as in the parcel method. It is also assumed in the slice method that there is no 11et inflow or outflow along the horizontal into the slice under consideration. Tempera-ture ascents inside convective clouds show that in a high percentage of cases, the lapse rate inside the cloud is in excess of the saturated adiabatic rate. This can only be explained on the assumption that there is appreciable lateral flow into the cloud. In situations favourable for convection, the enviroIlmental air has a lapse rate in eXcess of the saturated adiabatic lapse Theory of Cumulus Convection 533 rate. If there is no immediate and thorough mixing between the air inflow-iog from the sides at any level and the air coming fr011l below, it is very likely that the air which has newly come from the sides being unsaturated will first cool at dry adiabatic lapse rate. For the above two reasons the lapse rate inside the cloud will definitely be in excess of the saturated adiaba-tic. The simple picture of a convective cloud fed fro111 below the base with no lateral inflow beyond tUI bulent mixing with the environment is insuffici-ent to account for the observed lapse rates. SUGGESTED PICTURE OF CO~VECTrON IN CUMULUS AND CUMULONIMBUS (a) Initial Stage: Let us consider a barotropic atmosph~re with horizontal isobaric sur-faces. A pel turLation in thl' fOlm of a vertical displaccment is applied to the system, the 11Iagnitude of the displacemeS1t being maximum at C (Fig. I) and zero at the ground as well as at a higher level R. The displacement al~o decreases in the horizontal with dIstance fr0111 the vel tical ABCDE. The pressure surfaces, represented by dottcd hnes, \vill consequently be perturbed as shown in Fig. I, so that at any horizontal level the pressure is higher along the vertical ,lBCDE than ill the surrounding. At the levels A and E, the pressure surface ih still hOrIzontal. The above 1<; merely a descrip-tion of a vertical perturbation. , , , , , E -- ----c ...... , , ~ ' .... A , .... --. , " ' .... ---534 S. Mull and Y. P. Rao Owing to tht.! outward pressure gradient at all levels between A and E, air flows out radially and tends to decrease the pressure, Its fall at any level being cumulative of the divergence at all levels aloft. This will be a maximum at the ground and as initially the pressure surface was horizontal at the ground, a low pressure will develop at the ground and progressively build upwards. This stage is rl.!presented in Fig. 2. Air will then flow ------- .-..-------,'------ ... - - ... +- ~----l!'IG. 2 ............. --... .... -.... ~ radially inwards into the low pressure zone and outwards ill the high pressure zone aloft. The inflowing air in the lower levels will then set up continuity with the outflowing air aloft in the form of a vertical current or in other words, the air in lower levels will tend to rise. TIlC risilll?, air will cool either according to dry adiabatic or saturated adiabatic rate depending upon 111e humidity condition:'>. If the rising air is denser than the air it replaces (~tahl!;! stratification), it will counteract the development of low by outBow of air aloft. The perturbation will be damped out in such a case. If on the other lland, the rising air is less dense than the air it replaces (unstable stratification), it will intensify the development of the low pressure below. The radial inflow into the low pressure will therefore increase and feed the rising column. (b) Large Cumulus Stage: Fig. 3 shows the outline of the cloud, preSsure surfaces (broken lines) and air flow (arrow heads) at the stage of a large cumulus. The top of the growing cloud will push upwards the isobaric surfaces and consequently a high pressure will occur immediately above the top of the cloud. Owing to the lower density of the cloud air, the high pressure will change to a low one Theory 0/ Cumulus Conoection some distance below the top of the cloud. The low pressure will intensify froln there up to the ground. Air will flow into the low pressure zone of ---+--.... - -- - ---,~~, ~ , ., , ,. , ...... , .... .. fill' ~ -J """', --.... ... ....... , .... -+t-.... ........ --... t " ~ ... .... ........... ..... -' ,-Cloud air less dense than environment. the cloud from the l>idcs while there will he ~0l11c lateral outflow in the shallow high pressure zone in the top part of the cloud. The total convergence of the air into the low pressure zone of the cloud is appreciably greater than the divergence in the top portion of the cloud so that the cloud is still grow-ing. It should be notl:.d that though the top of the cloud forms a high pres-sure zone, its density is shlllcss than that of the environment and as the cloud grows, the high pressure zone of the cloud is being changed into low pressure. Hence there is no persistent outflow from the top of the cloud and it does not spread out. (c) Beginning oj Cumulo·nimbus Stage: The next stage in the growth of the cloud will be when with fUrtber ascent the top portion of the cloud commences to be more dense than the environment. This is shown in Fig. 4. As in the plevious stage there is a high pressure zone just at the top of the cloud but due to the higher density of the cloud air (over the envir,,'" 'l1ent) the high pressure intensifies below 536 S. Mull and Y. P. Rao the top up to the level where the density in the cloud is the same as that of the environment. As a result of this high pressure aloft, the high pressure ----.-. -FIG. 4 - -- - --" " .... Cloud air denser than environment. ('loud air les5 dense than environ. ment zone extends down Ul) to some level into that portion of the cloud which is less dense than the environment. Below that lev~l, the low pressure zone builds up. The convergence in the low lJre~surc Zoone is greater than the divergence in the high pressure portion of the cloud and hence the cloud is still growing. The cloud also spreads out in the high pressure zone unlike in the previous stage, as the high pressure is steadily maintained. (d) Fully Grown Cumulo-nimbus: At this stage represented in Fig. 5, the pressure right at the top of the cloud is the same as in the environment, as the cloud is no longer growing, and hence no more perturbing the upper pressure surfaces. A high pressure zone builds up below the top and is most intense at the level, S, where the cloud air comes to have the same density as the environment. Below that the cloud air is less dense than the environment and the high pressure zone first diminishes in intensity and finally changes into low pressure zone which attains the maXllllUtn intensity at the ground. The convergence in the low pressure zone below and the divergence in the high pressure zone aloft are Theory of Cumulus Conoection 537 in this stage equal so that there is no further growth of the cloud. lateral inflow or outflow and the vertical currents do 110t cease. But the Thus the .... .--- -, ... - " - ... ... FIG. 5 ..... ' ............... - - -- --Cloud air denser than environment . Cloud air less dense than environment. fully grown cumulonimbus represents a state of dynamic equilibrium. On this basis it is possible to understand how a cumulonimbus cloud can main-tain itself for considerable time without further growth or decay. The low pressure below the cloud is a necessity for can sing inflow of air into the cloud. It is also to be noted that the high pressure zone extends from the top of the cloud iuto a portion of the dond which is warmer than the environ-ment and hence the ol1tflow compensating the inflow lower down occurs also in a portion of the cloud which is rarer than the environment. Hence the vertical portion which is· less dense than the environment, is generally grea-ter than the portion which is denser. Thus, according to the above discussion, it is possible to conceive purely 011 theoretical grounds, a stage in the development of the cumulonimbus cloud when the cloud, with its fully grown anvil, can maintain itself without any further growth or decay for hours together. As the cloud represents a system in dynamic equilibrium and as the up currents are stable, 'it may be expected that the cloud will not break i11to thunderstorm with the usual 538 S. Mull and Y. P. Rao downdraft, squall and precipitation unless an additional mechanism comes into operation. Such an equilibrium stage in the life cycle of cumulonimbus is frequently obsel ved in the tropics, wherc the cloud persists for a consider-able time without any apparent change. The discussion also clearly reveals how a low pressure occurs below con-vection clouds at all stages of its growth. This explains the antic10ckwise circulation reported in cumulus and cUlllulonimbus clouds and the low pres-sure recorded with thl' passage of such clouds, though the fall of pressure is generally very small. This paper is restrictcd to the stages prior to the breakdown of the cloud into a thunderstorm. The mechanism which brings out this change and the resulting circulation are heing discl1ssed in a separate paper. RI!GIO>;AL MltTI!OROLOGlCAL CI!NTRIt. NAGPUR. REFERENCES Bjerkne~,J., (1938), Qualt. Jo",. Roy. Met. Soc., 61, 325, -Petter~sen, ~., (1939), Geof· Pub. (Oslo), 12, 9. Petterssen, S., (1945), Geol. Pub. (Oslo), 16, to. 

Theory of Cumulus Convection

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Theory of Cumulus Convection

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