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On Fluctuations of Pressure and Temperature in the Atmosphere*

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On Fluctuations of Pressure and Temperature in the Atmosphere* ON FLUCTUATIONS OF PRESSURE AND TEMPERATURE IN THE ATMOSPHERE* BY R. ANANTHAKRISHNAN Received June 17, 1946 1. INTRODUOTION IN dealing with a fluid medium in motion such as the atmosphere, the properties of the medium not only differ from point to point at the same instant, but also undergo variations at the same point in course of time. If S is any physical property of the medium which we wish to study, then S is a function of the co-ordinates x, y, z and the time t. S =f(x, y, z, t) (1) ~s ~S . dy + ~_,~S bS dS =~-~ 9dx+ -b-y " dz+ ~t " dt dS bS _dx + ~S ~tŸ ~S dz ~S dt - bx " dt ~-v " + Tz " 3i + ~~ =~~-u~S +--~y.V~S +T~-~S w +--~tbS (2) where u, v, w are the x, y, z components of the velocity of flow at the point under consideration. Or (~)d) S= (~, +U~x+V~y~ ~§ W~z~) S (3) dS. d-/ ls the time rate of change of the property S over a particular element of ~S. the medium and is known as the individual variation. ~t is the time rate of change of the property S at the fixed point in the medium and is known as the local variation. The hydrodynamical equivalent of the principle of conservation of mass is the equation of continuity which states that the rate of change of density at a point in the medium is equal to the net excess of inflow of mass over outflow per unit time considered over a small volume element around * Prcsented at the Annual Session of the I.adian Acadcmy of Sciences, hr at Udaipur, ia December 1945. m 393 394 R. AnanthakHshna¡ that point, divided by the volume of the element. Mathematically this is expressed by the equation" hi + ~ (pu)+ ~ (pv) -t- ~-~ (pw) = 0, (4) where pis the density at the point under consideration. This can also be written as" ~P (~U+~v ~w) bt q- P\bx ~-y +~ =0 (4a) 2. ATMOSPHERIC PRE$SURE AND ITS VARIAT1ONS To a very high degree of accuracy, the pressure at a level h in the atmo- sphere is given by the weight per unit atea of the mass of the atmosphere above the level in question. co p=f godz (5) h The local variation of pressure at the levet h is therefore given by : ~t g-~ 9dz 3o ~p_fh bp Substituting for -~ from (4) we get: ~~- - g ~ ~ (pv) + 5-z (pw) ctz oo ---; g[~ (pu)+; (pv)] dz+g~w)h (6) h The integral on the right-hand side represents the rate of transport of mass above the level h due to horizontal motion (advection), the second term represents the rate of change of mass above the level h due to vertical motion. At the surface of the earth w = 0, so that: ~,_~___f~[~bt 7x (pu) + ~~ (pv)] dz. (7) O This equation tells us that a change of surface pressure can only be brought about by horizontal advection of mass above the place of observa- tion. On the other hand, equation (6)shows that a change of pressure at On Fluctuations o[ Pressure & Ÿ in the Atmosphere 395 a higher level in the atmosphere can occur even in the absence of hori- zontal advection of mass in the atmospheric air column under consideration. 3. FLUCTUATIONS OF TEMPERATURE IN THE ATMOSPHERE (a) General Case.--If we consideran element of air in the atmosphere, the temperature of the element can undergo changes due to two causes, viz., (1) due to direct addition of heat to or removal of heat from the element, and (2) due to external work done on the element or by the element conse- quent on the changes of atmospheric pressure. This is expressed by the well-known First Law of Thermodynamics: dQ -- d U + dW = c~ dT-- ART dp P Hence dQ = c~ dT ......ART dp -d-91 d-s p dt Of dT 1 dQ AR T dp (8) Since we can measure only local changes of pressure and temperature in the atmosphere, we transform (8) with the help of (3) and get : ~T_I dQ (ATR T ~p ~T) ~t c~ dt +u "-p ~x AR.T .~p_ -bv~( ~ by ~~)~f AR.T ?p +w-~; F ~z + A R T_ . bfl (9) c~ p b t Now, if 0 is the potential temperature of the element of air, we have : AR Hence dO dT ARdp 0 T cp p Or dO = T c~ p 396 R. Ananthakrishnan Substituting this re]ation in (9) we get: ~T_ 1 dQ T( ~0 ~0 ~0) AR T ~p bt c~dt -o.Ub--x+V~+W~z_ + c: P" ~~ (10) The first term on the right-hand side of (9) or (10) represents the change of temperature resulting from direct addition of heat due to non-adiabatic processes such as radiation, condensation, etc. The second and third terms involving u and v represent the changes resulting from horizontal advection in a non-uniform pressure and temperature fiel& The fourth term involving w represents the adiabatic change of temperature due to vertical move- ments. Since ~ob~'z-~--gp and ~Ag_ P(dry adiabatic lapse-rate), this term can also be written as: w(fl- F), where fl- aT~., (prevailing lapse-rate in the atmosphere). The last term on the right-hand side of (9) and (10) represents the adiabatic heating or cooling effect due to local changes of pressure. (b) Particular Case.--If the processes are strictly adiabatic, then dQ -~ 0; also, ff the wind is geostrophic we have : bp_ u ~--Px+ v ~--0. (111 In this case, equation (9) reduces to: bT ARTbp ( bTa_vbT'~ ~t - c~ p ~t z, Tx ' ~j,j- w (F- 8). (12) Under geostrophic conditions, the relation between horizontal temperature gradient and the variation of wind with height is given by the weU-known equations : ~(~) g I~T g 1 bT (13) From this we have: u ~x+V b~= g -- u ~ --v~-~ =T. 2~~191 u --v (14) If ~ is the angle between the direction of the horizontal wind rector and the x-axis then: tan ~ = ~ l~ On Fluctuations olc Pressure & Temperatura in the Atmosphere 397 Hence 3z (tan ~k) = sec" ~b = I + h=" ,~z __ -- /g ~~ ?z z ~~- "g~J Or ~v ~u ?~b u ~- v 37=(.2+ v 2) ~-~ = Vs" 9?~, (15) ~Z where Vk is the scalar value of the horizontal wind vector. Substituting in (14) we get: ~T ~T = T 9 2oJ sin 0 ,'~b u 5Y + v ~ --~-- " Vh~ " ~z (16) Substituting (16) in (12) and transposing the terms we have: w(r_flj=A_R._T. ~p ~T 2cosin4.T.Vh, " ~._~~ (17) c~ p ~t ~t g ~z Equation (17) enables us to calculate the vertical wind component at any level in the atmosphere if we know the rate of change of pressure and temperature at the point in question, the vertical variation of the horizontal wind rector as well as the prevailing lapse-rate. The assumptions involved are that: (i) all changes are adiabatic, and (ii) the wind is geo- strophic. 4. INTER-RELATION BETWEEN PRESSURE AND TEblPERATURE IN THE ATMOSPHERE The relation between pressure and temperature in the atmosphere is given by the well-known fundamental law of atmospheric statics: dp= -- g p dz = -- g -~~F dz. On integration this leads to the familiar barometric formula: Iog = -- R = -- K " T,~ (18) o where T,,, is the mean temperature of the air colunm from the surface where the pressure is Po to the level z where the pressure is p. If we di¡ (18) keeping z constant and treating p, Po and T~ as variables, we get the following relation connecting the variation of pressure at the surface aed 398 R. Ananthakrishnan at the level z with the change in the mean temperature of the column between those two levels : .po_ gz Ap0=Ap P RT,,z'P0" AT,,,. (19) From (19) it follows that if A T,,, = 0, then AP0 = A p 9Po. That is, P a small change of pressure at a higher level in the atmosphere would give rise to much larger changes in the surface pressure when the mean tempe- rature of the air column remains unchanged. For instance, a change of pressure of I mb. at a level of 16 gms. where p~ 100 mbs. would bring about a change of nearly 10 mbs. in the surface pressure. In the case of an incompressible medium, a change of pressure at any level would produce the same change at all levels below ir ; however, in the case of the atmosphere when extra mass is added abo,,e the level z to produce ah increase of pressure at that level, the column below z gets compressed and part of the extra rnass added sinks below that level. Hence, in order to produce an increase of pressure of 1 mb. at 16 kms., additional mass eqtdvalent to 10 mbs. has to be added above that level. In (19) ifp and z relate to the level of the tropopause, then it will be seen that the change in the surface pressure can be interpreted as resulting from : (i) Change of pressure Ap at the level of the tropopause due to addi- tion of mass in the stratosphere; (ii) Change of mean temperature ATm of the tropospheric air column. In meteorological literature (i) is sometimes designated as the primary pressure wave, and (ii) as the secondary pressure wave. One of the striking results furnished by aerological ascents is that the day-to-day variations of pressure at all levels in the troposphere are of the same order of magnitude as that observed at the surface.
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