-1- THE LUNAR TIDE IN THE ... by MARVIN A. GELLER

-1- THE LUNAR TIDE IN THE ATMOSPHERE by MARVIN A. GELLER B. S. Massachusetts Institute of Technology (1964) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the VMASSACHUSETTS INSTITUTE OF TECHNOLOGY February, 1989 Signature of Author................................... Department of Meteorology, February 7, 1969 Certified by..' Thesis Supervisor Accepted by ..................................................... Chairman, on Graduate Students -2- THE LUNAR TIDE IN THE ATMOSPHERE by Marvin A. Gell-,r Submitted to the Department of Meteorology on 7 February 1969 in partial fulfillment of the requirement for the degree of Doctor of Philosophy. ABSTRACT Previous studies, both observational a.nd theoretical, of the semidiurnal lunar tide in the atmosphere are reviewed. Some of the principal observed characteristics of this tide such as the tidal variation in pressure, temperature, and wind at the surface are discussed. The tidal variation in surface pressure is observed to vary annually in such a manner that the amplitude remains relatively constant throughout the year while the phase lag varies in such a manner that it reaches its maximum values during Northern Hemisphere winter and its mirimum values in Northern Hemisphere summer. The equations of tidal theory with Newtonian Cooling are derived starting with the "primitive equations" of dynamic meteorology. It is shown that the vertical structure of the lunar tide depends or. the static stability of the atn osphere. Data on the stability of the atmosphere indicate that seasonal variations in mesospheric temperatures act in the summer months to create an increased "barrier" to wave propagatior., in the Schrodinger equation sense. Calculations without Newtonian Cooling are performed using the empirical atmospheric stability curves. These calculations give results in agreement with the observations, both of annually averaged tidal effects and seasonal variation of these effects. The addition of Newtonian Cooling acts to disturb the agreement in the seasonal variations, but this is not taken to be serious as there are many uncertainties in the Newtonian Cooling -3- parameter. Calculations of the vertical energy flux give results in good agreement with the notion of an annually changing "mesospheric barrier". Finally, simple studies, both numerical and analytic, are performed to make clearer the "barrier" effects. Thesis Supervisor: Title: Victor P. Starr Professor of Meteorology TABLE OF CONTENTS I. HISTORY OF PROBLEM AND OBSERVATIONS figures for chaptei I II. III. Formulation of Equations Formulation of Variables PROCEDURE A. B. C.. D. Gravitational Forcing Stability Function Newtonian Cooling Numerical Procedure RESULTS AND OBSERVATIONAL AGREEMENT A. B. Results with No Newtonian Cooling Results with Newtonian Cooling figures for chapter IV V. ENERGETICS A. B. Formulation Results SIMPLE STUDIES A. B. 42 46 51 58 70 87 87 94 98 124 figures for chapter V VI. 27 36 42 figures for chapter III IV. 17 24 FORMULATION A. B. 6 Numerical Analytic (a) Isothermal (b) -function barrier figures and tables for chapter VI 124 129 132 135 135 139 139 145 150 -5- VII. CONCLUSION A. Discussion of Results B.. Suggestions for Further Research 170 170 173 Acknowledgements 175 References 176 Biography 179 -6- CHAPTER I HISTORY OF THE PROBLEM AND OBSERVATIONS Atmospheric tidal theory is probably the oldest branch of dynamic meteorology. The original workers in this field included such eminent names as Newton (1687), Laplace (1799), Kelvin (1882), and Rayleigh (1890). Since tides have been studied for such a'long per-.od of time, there has natural y developed an extensive literature on the subject. There are several fine review articles that cover both the observational and theoretical aspects of atmospheric tides. Among these are a book by Wilkes (1949), and articles by Chapman (1951), Kertz (1957), and Siebert (1961). All of these references include discussions of the atmospheric lunar tide. Matsushita (1967) has written a more specialized article on "Lunar Tides in the Ionosphere" that reviews observations of lunar tidal variations in ionospheric and neutral atmospheric parameters. Iri this section the author will give a brief review of lunar tidal theory and observations. Certain observations will also be cited for later comparison with the author's theoretical results. The most familiar of all the tides are the sea tides. These are observed to be a twice daily rise and fall of the ocean level -?- with the time of high tide being associated with the moon's motion. The physical understanding of these tides goes back to Newton (1687) an his hypothesis of a universtl inverse square "gravitational" attraction between all material bodies. The moon is in an equilibrium orbit about the earth with the earth's gravitational force upon the moon supplying the required centripetal acceleration. Thus, at the earth's center, Wa ro , where G denotes the gravit- tional constant, mass of the moon, earth and moon, and GL/r L is the is distance between the centers of the o W is the orbital angular velocity of the moon. Considering the spherical shape of the earth, it is easy to see that on the hemisphere nearer the moor. the gravitational force is larger while the centripetal acceleration is smaller than their values at the earth's center. This causes an unbalanced distribution of forces directed moonwards. Or the hemisphere further from the moon this type of imbalance gi.ves rise to a net force directed away from the moon. Particles free to move, like those of the sea water, will move under the influence of this force distribution. The sea surface will then tend to become spheroidal, with its long axis along the line joining the centers of the moon and earth. -8- The sun's gravitation also causes a similar semidiurnal tide. The lunar tidal action is 2.4 times as great as the solar action so that the r,-oon chiefly governs the sea tides; however, at different epochs in the month the sun acts to add to or reduce the lunar tide. The above discussion describes the "equilibrium tide";that is to say, since there has been no mention of the earth's rotation, this "equ librium tide" is a static phenamenon. Laplace (1799) recognized the importance of incorporating the effects of the earth's rotation into any reasonable theory of the tides. He carried out an analysis of ocean tides approximating the situation by an ocean of uniform depth covering the entire earth. His results are not rteally applicable to the actual ocean tides since the complex ocea a-continent topography of the earth is ignored. Newton and Laplace both realized that the atmosphere must also react to tidal forcing. In fact, the lack of lateral boundaries makes the atmosphere a more suitable subject for Laplace's idealized tidal theory. Although his theory was formulated for incompressible sea water, Laplace demonstrated that the theory could be easily applied to compressible air so long as the atmosphere was of isothermal structure with pressure changes occurring under isothermal conditions. -9- Observations of surface pressure variations indicate that, unlike the oceans, the largest tides in the atmosphere are with solar peri)ds. This has been shown to ')e a consequence of the large thermal forcing of the atmosphere with the solar tidal periods. The lunar tide in the atmosphere, in spite of its small magnitude, has been observed sufficiently to warrant attempts at modelling the observed tide. The history and results of these ob- servationv will now be reviewed. The mercury barometer was invented by Torricelli at about the time of Newton's birth. It soon became kaown, by the use of this instrument, that barometric variations were of a quite different nature in the tropics than at temperate latitudes. The surface pressure in the tropics usually varies regularly with a range of a couple of millimeters mercury with sol.tr semidiurnal frequency In temperate latitudes, the barometric readings vary in an apparently irregular fashion with amplitudes on the order of a centimeter mercury. This midlatitude beha.vior is now known to be due to the passage of migratory cyclones and anticyclones (usually accompanied by low and high pressure centers respectively). Laplace's theory predicted a "direct" lunar tidal variation of the barometer with a range of about half a millimeter in the 1. This regular small-amplitude pressure variation in the tropics is only disturbed by the passage of hurricanes. -10- tropics. Since he had no access to suitable tropical data, he attempted to detect a lunar tidal variation in barometric readings at Paris. Fe failed to get a significant rcsult since his data was not for a sufficiently long period to detect such a small regular variation imbedded in such large irregular variations. Many other 19th century investigators also failed to find evidence of the atmospheric lunar tide at midlatitudes. Dur-ing the middle of the nine- teenth celLtury, however, this lunar tic'e had been determined at such tropical stations as St. Helena, Singapore, and Batavia. Finally in 1918 Chapman succeeded in detecting the lunar tidal variation in surface pressure at Greenwich. He succeeded in this effort by only considering data on days where the barometric range did not exceed 0. 1 inch. In this way, random errors were reduced c ufficiently to reliably determine the lunar tidal variation in surface pressure at a nontropical station. Since this time, many investigations of lunar tidal effects in the surface atmosphere have been undertaken, using the basic methods of Chapman, so that the lunar variations of such parameters as wind and temperature are now at least partially known. Observations of the vertical structure of the lunar tide in the atmosphere are very poorly known. It might be possible to get such observations by using tropical radiosonde data that has been collected over a long period. To the author's knowledge, this has -11- not been tried. There is some knowledge of the vertical structure of the lunar tide that can be gained by ionospheric rmethods. As mentioned earlier, Matsushita (1967) has written an extensive article reviewing the variation of ionospheric parameters with the lunar period. Some observations pertaining to the lunar tide in the atmosphere will now be given. semidiurral tide, In these observations the local lunar LL , has been repri.sented by the formula, L2. = .92. sint where a is the amplitude, + Y)I-1 V is the phase, and e denotes time reckoned in angle at the rate 3600 per one half lunar day. L. has a maximum at (9o0 - ) or at m x a- X (in hours). A popular mode of display is the harmonic dial representation that pictures the amplitude as radial distance and the phase as time on a clock's face. formula I- 1. Thus figure I-1 would be analogous to 1-3 -12- Figures I-2 and I-3 are taken from Chapman (1951) and picture the worldwide distribution of the lunar tidal variation of surface pressure. It is seen from these figures that this lunar tide has a reasonable asymmetry that seems to be associated with the continent-ocean structure; however, certain features of this tide seem to hold throughout. In figure I-2 most arrows point to the northeast which, referring back to figure I-I, implies a phase lag in the sur.'ace pressure variation of abcut one-half hour. In figure I-3, it is obvious that the amplitude of this surface pressure variation decreases poleward. These same features are seen more clearly in figure I-4 which is also taken from Chapman (1951). This :igure shows the amplitude and phase of the zonallyaveraged surface pressure variations rlotted as a function of latitude. Here again there is seen a decrease in amplitude away from the iequator and a phase lag at nearly all latitudes. Figure I-4 also shows the observed annual dependence of amplitude and phase. There is seen a very small annual variation in the amplitude curves with the largest amplitudes in northern hemisphere summer and smallest in northern hemisphere winter. This annual variation in amplitude is a small effect, with amplitudes at the equator being approximately .065 mm. in May-Aug. and -13- in Nov. -Feb. . 055 mm The annual variation in phase is seen to be more significant with a larger phase lag in northern hemisphere winter th,:n summer. The phase lags Zt the equator are seen to be approximately 1/3 hours in May-Aug. and 1 1/3 hours in Nov. Feb. The observed lunar semidiurnal variation in surface temperature at Batavia (70 S, 1070E) is shown in figure 1-5 (also taken from Chapman (1951) ). This observedI temperature variation has an amplitude of approximately 0. 0080 C with a phase lag of around 2/3 hours. The temperature variation is approximately what one would get from the pressure variation assuming the density variations to be adiabatic. Figure I-6 (taken from Chapman (1951) ) shows the observed lunar tid.l variation of the surface wind at Mauritius (20 0 S, 550 E). The variation of the eastward component is shown to have an am-1 plitude of about 1 cm sec with a phase lag of approximately 7 2/3 hours while the northward component's variation has an -1 amplitude around 1. 2 cm sec-1 with a phase lag of around 3 hours. There have been previous attempts at modelling the observations of the lunar semidiurnal tide. viewed. Three of these will be re- These are the investigations of Wilkes (1949), Sawada (1954, 1956), and Siebert (1961). -14- Wilkes (1949) carried out his work at a time when the solar semidiurnal tide was thought to be gravitationally forced and vastly amplified through resonance. His upper boundary condition, that of an isothermal atmosphere at 1900K, served to trap all energy as would be crucial to any resonance theory. His calculations predicted a lunar tide in surface pressure with the amplitude equal -2 to that of the equilibrium tide (F 2.8 x 10-2 mb. ) with zero phase lag . In passing it should be sal I that Wilkes correctly recognized that the temperature structure of the atmosphere would act as a "barrier" to wave propagation in the mesosphere, but because of a lack of upper atmospheric observations and the supposed need of the "resonance theory", Wilkes used an improper upper boundary condition in his the)ry of the lunar tide. Sawada (1954, 1956) in a series of two papers carried out a very nice investigation of the lunar tide in the atmosphere with the goal of determining the atmospheri: temperature structure. He reasoned that a good guess for the temperature structure could be obtained by finding the temperature profile for which the lunar tide agreed most closely with observations. Sawada used a correct upper boundary condition in assuming the vertical energy flux to be directed upwards at infinity. His temperature profiles 1. With the upper boundary condition acting to trap all energy the phase lag must be either 00 or 1800 as will be shown later. -15- were comprised of straight line segments connected to form a continuous temperature structure. He obtained best agreement with luna- tide observations (amplitude -1. 42 x 10-2mb, phase lag - 1/6 hours at the equator) with the temperature profile that is pictured in figure 1-7. This temperature profile does compare quite well with the features of the mean atmosphere. Sawa- da's work did suffer from one fault that will become clearer later. The tidal equations depend on the atmospheric stability in a way that the shape of the stability curve with height is crucial. Discontinuities in the slope of the temperature (as in figure I-7) might introduce spurious effects into the calculations. Finally Siebert (1961) performed a theoretical calculation of the lunar tide. Like Wilkes, his upper boundary condition caused tha tidal energy to be trapped. With this erroneous upper boundary condition, he found good agreement in amplitude between his theor" and observations but contrary to observations found a zero phase lag. Siebert attributed this error in the phase to be due to the neglect of surface friction. Itis the author's view that the observed phase lag in the lunar variations of surface pressure is due to the upward energy flux and not to surface friction as suggested by Siebert. It is hoped that the observed features of the lunar tide in the upper atmosphere will become better known in the immediate fu- -16- ture through such radio methods as meteor winds. Radio methods were first used in the study of the lunar tide in 1939 when Appleton and Week s discovered an oscillation ir the height of the E-layer over Cambridge, England with a lunar semidiurnal period. This oscillation had an amplitude of . 93 km. and a phase lag of 11. 25 hours. This result is questioned in Matsushita's review on the basis that the observations of the E-layer also included E s - layers (sporadic E). When E is excluded fro-n such a study, the observations of lunar oscillations of the E-layer show an uncertain result. In Matsushita's review, lunar variations of absorption parameters and "winds" are shown for the D-region ( E-region (,100 60 km.), the km. ), and the F-region (2 300 km. ). These results are difficult to relate to neutral motions in the atmosphere though. Perhaps the situation will be clearer in the future. -17- 900 180* 1 3)00 3300 2100 2700 SE Figure I- 1 ANG L Harmonic Dial Representation t0j: t or I t CD TROPIC OF CANCER 20 -- PHA ANNUATOR 81EA 160 WEST 120 - ANTROPIC OFCIPRICRN AMPLITUDESCALE I I 40 so SCALE - MEAN 80 20 C AMPLITUDE -I-I S40 TIDE LUNAR ATMOSPHERIC AMPLITUDE S PHASE: ANNUAL 0 - 0 40 80 4040 - I ANTARCTIC CIRCLE 120 JEAST 160 Geographical distribution of the annual mean lunar semidiurnal air tide L2 in barometric pressure, indicated by dial vectors, each referring to the place at the mid-point of the arrow, whose length gives the amplitude 12 on the scale shown, and whose direction gives the time of high air tide, as shown on a local mean lunar clock. 160 120 80 0 40 40 80 It0 I0O a so o S ,10"aTO TRO. CANCER I-- .oCROBARS t do 42 -/ 2 20 MICROBARS G2 -.- 30 222 6 \5 8 20 SC.10 - - ANTARCTIC - - - r MICROBARS MCRARS -40- 30 ICROB CR O73 I S 850,M CROBARS __ 20 ---- - So CIRCLE 160 WEST 120 80 40 0 4 60 120 EAST 160 Tentative lines (broken where only weakly (letermined) of equal amplitude 12 (in microbars) of the annual mean lunar semidiurnal air tide L 2 in barometric pressure. The numbers (other than those, 10, 20,..., 80, followed by the word "microbar," which-lshow the value of 12for each line) show local values of 12. -20- S0.060 E ........ *..... 0.045 W 2 K 4s- (,5) 100(4) MEAN Z 0.0 30 (14) (9/ S(4) 0.015- 4) .. N. 50 40 (N) N0'INDICATES MEAN FROM N OBSERVATORIES 30 20 10 0 10 (3) S. 30 20 LATITUDE IN DEGREES (0) 20 2 40 0N W W W h LATE . 60 60 (14) . (9•9 h LATE * (9) (5) . O 0 (4) .Z 100 (4) -) *(2) 120 I N. 50 40 0 10 30 20 10 LATITUDE IN DEGREES 1 I 20 30 S. (b) Curves showing the average dependence on latitude of the amplitude 12 (Fig. 9a, above) and phase X2 (Fig. 9b, below) of the lunar semidiurnal air tide in barometric pressure, for the mean of the year and for groups of months J (May to August), D (November to February), and E (March, April, September, October). The curves are drawn through (or near to) points (X) each giving the mean 12 or X2 for a group of stations (the number of stations is indicated beside each point) within a moderate range of latitude. Figure I-4 -21- 900 0 0.0050C O.010 0 CO Harmonic dial specif3 ing (with probable-error circle) the lunar semidiurnal tidal v,riation of air temperature at Batavia. The point C represents the variation calculated from the lunar semidiurnal variation of barometric pressure at Batavia, on the assumption that the density variations are adiabatic. Figure I-5 -22900 I20 150" 0h Oh 60' a h 30s MAURITIUS 'WIND (I6 YEARS DATA) LUNAR TIDE e180 gh a mh 0 StN Icm sec 1 E 330 2104h an 7h 2400 O3 6h (a) 270' 1120 ' E Oh 606 2h N MAURITIU$% WIND SOLAR TIDE 10 20 cm sec (b) (a) Harmonic dial (with probable-error circles) for the annual mean lunarsemidiurnal variations in the northward and eastward components of wind velocity at Mauritius, from about 16 years' bihourly data. (b) Harmonic dial for the corresponding annual mean solar semidiurnal variations. Note the tenfold scale difference between the two diagrams. Figure 1-6 -23- t HEIGHT (KM) 0O 200 220 240 260 280 300 TEMPERATURE (oK) -Figure I-7 Sawada's Best Temperature Profile 320 -24- CHAPTER II FORMULATION List of Symbol = = () s: spherical average of ( ) tidal perturbation of ( temperature pressure density time 9 acceleration due to gravity ( = 980. 6 cm sec - 2 ) gas constant for air atmospheric scale height 3= (' RT/9) surface pressure ( = 1000 mb) solar heating term three- dimensional velocity divergence C : Cy- specific heat of air at constant pressure specific heat of air at constant volume R/cr CP/C. U- eastward wind velocity northward wind velocity latitude -25- = longitude IL = rotational frequency of the earth = radius of the earth i FAF = geopotential height = dissipative forces in eastward and northward directions : In( po/p) = altitude dt d = total derivative Q = diabatic term N = Brunt Vaisala frequency = Newtonian Cooling coefficient = V S F in~ = frequency of tide = wavenumber of tide = Hough operator = eigenvalue of Hough function m •- = Hough function = stability function = coefficient of = vertical velocity = tidal potential in expansion of () -26- 4, a = . 1gh + geopotential ( coefficient of Ii, expansion of (1r) -27- Section A: Formulation of Equations Traditionally, tidal theory has been quite inaccessible to the general meteorological community. This has been due to the use of specialized notation and the resulting difficulty in deriving the main equations of atmospheric tidal theory. These troubles are actually quite simple to surmount if one starts witl. the proper meteorological v.riables instead of the specialized tidal variables. coordinate height). Z H(s) The tidal theorist uses as his vertical ( ()=)R2T(')/ ih -II(pipo It can be easily seen that theorist formulates his basic equations in terms of as the dependent variable ( gence of the tidal wind field and is the scale . The tidal X" is the three-dimensional diver- U i the heating term in the thermodynamic equation). It will now be shown that the dependent variable of tidal theory is proportional to the vertical ve ocity in log-pressure coordinates. From the 1 st Law of Thermodynamics it follows that II A- 1 CpT Pd -28-. The continuity equation can be written as IIA- 2 P dt Hence defining = -"QI , it follows that IIA- 3 911~ Thus -notivated, the "primitive equations" of dynamic meteor1 ology will be used to derive the tidal equations . These equations are as follows: V dv dv F, V + (aflrtle)V £4bant 06 dt +1 IIA- 4 ~ IIA- 5 I I MMOMMIMONIM e -V (vcos) + -W r cocP dT - :swT _ RT "r Q = 0 IIA- 6 IIA- 7 IIA- 8 9 1. The derivation given here is essentially the same as that given in Dickinson and Geller (1968). -29- The preceding equations apply to all hydrostatic motions in the atmosphere. The following assumptions will now be made in order to deal with the specific case of +idal motions. (1) The undisturbed atmosphere is at rest relative to the earth. (2) The perturbations due to the atmospheric tides are so small that terms of second-order in the tidal variables can be neglected leading to a linear theory of tides. (3) The temperature of the undistrrbed atmosphere is uniform in the horizontal directions but varies with height. (4) The effects of viscosity and turbulence will be ignored. (5) The vertical component of the gravitational tidal forcing can be neglected. (6) Electromagnetic effects will be ignored. Other assumptions have implicitly been made in writing equations (4)-(8). These are as follows: (1) The atmosphere is in hydrostatic equilibrium. (2) A spherical earth has been assumed. (3) The atmosphere is assumed to be of uniform composition. (4) The variations of gravity with height can be neglected. Before proceeding further, it would be well to note the height limitation on the tidal calculations that is implied by the preceding assumptions. -30- The assumption of hydrostatic equilibrium will be true so long as the tidal frequency is much smaller than the Brunt-Vaisala frequency in Ihe atmosphere ( A. N) . This is true so long as the undisturbed temperature is less than about 106 OK. This assumption will certainly be justified below the exosphere (S 1000 km), where even the assumption of continuum mechanics breaks down. The a3sumption of uniform compos, tion is good up to about 120 km where the mean molecular weight falls to 93. 3% of its value at sea level I The acceleration due to gravity at 120 km is 96. 2% of its value at the ground indicating that this assumption is also good through 120 km. Kato and Matsushita (1968) have made estimates on the electromagnetic effects on tides in the upper atmosphere. They found that the electromagnetic interaction tim a-'scale becomes equal to the tidal time-scale at about 130 km. Newell 2 has performed rough calculations of the kinematic viscosity of the atmosphere at various levels. 1. CIRA 1965 2. unpublished calculations With his values it -31- can be shown that viscosity is negligible for vertical wavelengths of about 100 km so long as the motion being considered takes place below about 250 km 1 Knowing the height limitations on the equations to be used (below about 130 kin), it is possible to proceed with the derivation of the tidal equations. The set of equations (IIA-4) - (IIA-3), with the stated assumptions, bec 3mes Lkrn __ at ) V+ 00 06 OsC IIA.- 9 - IIA- 10 10 PTcos CP IIA - 12 IIA- 13 Q 1. is the total diabatic term in the thermodynamic equation The lunar tide has wavelength of this order. -32- , the solar heating, and and will be divided into two parts - the cooling-to-space term (Newtonian Cooling approxima- a tion) 1 It is now assumed that all perturbation variables and the T heating X+ lVC~?)IA are proportional to (- is changed to * latitudinal variable and that the 1V in ) . The following equations are then obtained: %k O YV~Y IIA- 14 ' L W Avy + an+ CI2Y, -- a + "moo a4QA A- W asPi %(I- -r (l.v -4 a*) T' IIA- 15 20 OI (T- + ; ) = RT' IIA- 16 4M CI IIA- 17 IIA- 18 Equations IIA.-14 and IIA.-15 may be used in solving for and . V The results are as follows: 1. Dickinson and Geller (1968) carry through this derivation with a more general operator on the perturbation temperature. -33- (2 L^) 0~r (I-) 0+ U: v-(2 ,)L S IIA- 19 and ' . ___.. F.. ...... I IIA-20 (L L ('t a, A/AL -,,) These relations for V and V can be put into equation (16), the continuity equation, to give the following equation, v FE '] IIA-21 -] ccana~[ where the operator F is defined by F.( I) F d d S-- I -~ '1 dd +, IIA - 22 Equation IIA-21 can be treated by a classical method of solution that is applicable to certain types of partial differential equations. The method to be used is that of separation of variables. The equation for Hough functions is F[H2 o +Y. H'n 0 -34- where geophysically reasonable boundary conditions are assumed at AA that is, at the polks . H (/) is assumed to remain finite The eigenvalue of the problem is taken to be n and of course is a constant. T may now be eliminated from the equations by combining equations IIA-17 and IIA-18 to yield (vam where o) 5 = 3 + (aAL' a d[3- IIA-24 is the nondimensional spherically-averaged static stability. ., It is now appropriate to expand f, sums of Hough functions. and The coefficients of 0 determined from equation IIA-24 (when (1) and W as will be is re placed by it s Hough fun tion coefficient); IIA-24' and by the separability condition 'VC jW% - wOr1 =O IIA.- 21' 1. For the gravitational modes, .A()represents wavenumber Vr motion with fl nodes. These gravitational modes are also symmetric in p (see Flattery (1967) ). -35- Eliminating AV Taking 1 16 a from these two equations gives IIA.- 2 5 da w ie leads to the vertical structure YA. equation in its simplest form 10-A JI IIA- 26 T MEMO&e~ *0" aft, M J vYAV Equation IIA-26 is a second-order ordinary differential equation; therefore, two boundary conditions are required to complete a proper formulation of the tidal problen. The lower boundary condition, at 0 , is that there ; or to express it mathema- should be no vertical motion at Iz O tically w= - IIA.- 27 0 Linearizing IIA-27 leads to + R.,~o9 ) O IIA -28 -36- which may be rewritten in terms of the coefficients of the Hough functions as +W l fv . By substituting, as before, W " I- 29 =O eIy L),the final form of the bottom boundary condition is seen to be + ) c (2fn )'+ 9' ( 0 at . IIA- is the so-called "equivalent depth".) To complete the tidal formulation the upper boundary condition must be considered. This upper condition is taken to be the requirement that the solutions to equation IIA-26 be either evanescent or propagate outward as waves for large . Section B: Formulation of Variables The previous formulation of the tidal equations had the dependent variable where as -37- When a solution of these tidal equations is calculated, it will still be necessary to calculate the usual meteorological variables from this solution. Once these meteorological variables are computed the tidal solution is complete and can be compared directly with observations. In this study the final variables to be computed will be as follows: (1) I -- the pressure perturbation in geopotential height -- the eastward tidal wind velocity units (2) the northward tidal wind velocity (3) V (4) -- the upward tidal wind velocity (5) -- the tidal temperature perturbation. This sect .on will be devoted to finding equations for these meteorological variables in terms of %d' The starting point for this variable is eqn. (21') in the previous section -- a M~4 IIB-2 1. In this section the i -dependent Hough function coefficients will be subscripted and superscripted. The full variable (with , t ) dependence will not be scripted even though a (A , V , single tidal mode is meant. -38- Substituting for W m ,n with equation II B-1 and solving for 45v gives L (arL*b) n Since n A Y: d Yu II B-3 eqn. II B-3 can be solved for n giving (2Ay~)'I e j M Wa ' 41 -~ I~Y~rt ~n4 %.IIr II B- 4 : V Equation IIA-19 implies V t L)' "40 1~b r'-cV I exBPC-5I4!Vt3. I 011A jII B- 5 Equation II B- 3 along with equation II B- 5 yields the following formula for V IIB-6 (a A,*)- '~:I S(L / V" I I Q\~ - (ft nA 1.W() 6p() = = real part of C) imaginary part of () rd -39- V Equation II A.- 20 implies Equation II B-3 along with equation II B-7 yields the following for: mula for B-8 V The vertical velocity\J is defined by . dintt d~t Linearizing equation II B- 9 and remembering that ~g IIB-9 4 1) implies I = 3 + 40 IIB-10 Pt or alternatively A SiV 9C 4b IIB- 1 83 -40- Hence 4a=V t aYfa ,V e + I d I 111ra Afi aI ? IIB-12 Equations IIA-18 and IIA-24 may be combined to give (.a)' (iv+a 0)RT'" +' w From equatior.. II B- 13 may be seen to be given by , T Ra IIB-13 V) QG j j exP Ar +4v II B-14 Th , final formulae are found from equations IIB-4, IIB-6, IIB-8, ] B-12, and IIB-14 to be - llI Cr 4~rU Nje)7 I*. ~I Uj . ,,, ve-a)O5gtVA s- (Mvi12r f Ha~~3LL~~ da (. v u,,A .db. at f'_ T dt"4~~ +(-IYI*-j, II B- 15 sivA*-r.+YKA3 II B- 16 .} LIZ gD. oY ~yYl"') d% (*7~~, 6)"lyow4 dr e+ 4:1 coSvt+ SV t+ VX -41- Aa cos t 41.4 ;Yrd)d f V:re41L e v" VZ YIf+7 11V"~sdd +r Ff Y t I~ L~~ II B-17 rA e Of1% 4%cmr .,ey~ ~a.)3 ** y"- Y Y are real functions of Y""4 (& +A vaitl r" a t II B-18 ID LV'CllD S~~~~mi ! J + svt 4MA 60' R i' R L(vno) where V YA, g j CoS Cyt+mhVK) )* - co VT +M +J Ao II B- 1 I itA .€y yr , and ,;. /I Equations (II B-15) - (II B-19)are the formulae to be used in calculating the meteorological variables. i. . ( .a4/a) is the Coriolis parameter. -42- CHAPTER III PROCEDURE Section A: Gravitational Forcing The lunar atmospheric tide is quite unlike the solar diurnal and semidiurnal atmospheric tides. The main source of forcing for the solar tides is thought to be solar heating (see Siebert (1961), Eutler and Small (1963), Lind2en (1968), etc). This inso- lational heating displays an annual variation both due to the variation of the incoming solar radiation and due to variation in absorber amounts. The first impulse would be, for this reason, to ex- plain the annual variation of a solar tide by utilizing the variation of the solar heating. In fact Lindzen (1967) did just this for the solar diurnal t.de. The lunar tide is quite different from the solar tide since there is negligible heating of the atmo,;phere with the lunar period. The lunar tide is forced purely by the moon's tidal forces acting on the earth-ocean-atmosphere system. These lunar tidal forces may be obtained from the lunar tidal potential which was derived by Lamb (1932). this tidal potential is His expression for -43- 4-+Sih A -Via e cos(c(+A) i + AIA-1 Os 2 'e. osa- where the symbols have the following meaning: -2 S= [ E acceleration of gravity ( = 980.6 cm sec-2) = mass of the moon ( = 7. 349 x 1025 gm) = mass of the earth ( = 5. 977 x 10)27 gn) O = mean radius of the earth ( D = mean distance between the center of the earth and of = 6. 371 x 108 cm) the moon (= 3. 844 x 1010 cm) . = height above the surface of the earth At e = north-polar distance, and the hour-angle of the moon I= . colatitude and longitude. Since the lunar semidiurnal tide is known to far exceed the lunar diurnal tide the term of interest i3 the potential for the semidiurnal lunar forces, I "eJ;vrj" (' IIIA-2 A ,$ ) COS 2(.-) -44- As is usual in tidal theory the vertical tidal forces are neglected 1 and 6 is taken to be . 2 (The moon is located over the earth's equator. ) The hour-angle of the moon can also be expressed in time units so that the expression for the tidal potential becomes '4 21T -q x 10 Sec T.107 where is the lunar day which equals 24. 8412 mean solar hours. ) ( It is seen that the lunar tidal potential is proportional to btt 5' i : 2 where is the normalized (2, 2) spherical harmonic. It only remains to expand P (1) in Hough functions. expansion has been done by Flattery (1967). P:I' This His expansion of in normalized Hough functions is 1. This is true since the tidal potential varies little over the depth of the atmosphere ! 4 2. 1AS -ZCOS( -45- P cy = o.96 o70 H H'im) - O.105-93 0. onz i7 H'. () I) '0.o06~725 - 0.o31 93 + 0. Oq223 NI 31-Iti)+ 0, o.a+ O.-OtIi - H IIIA-, ( H, + .... Putting numbers into expression IIIA-3 and using the first three terms in expansion IIIA-4 yields the required expansion of 4 Cr in Hough functions: ('r)fY xd It is thus seen that the forcing for the (2, 2) mode is, roughly speaking, four times larger than the (2, 4) mode. that = n , I , 7.07009 km. while V , and ~ ID It is also true 3. 24857 km and depend linearly on Dh With this in mind, only the (2, 2) mode will be considered as representing the lunar tide in the remainder of this study. 1. These differ slightly from the calculations of Sawada (1954). -46- Section B: Stability Function The vertical structure equation was derived in a previous section (equation IIB-26) to be d ! 4~ A-] =9~ 1-4 d Y~rb~a AV ~'L III B- 1 (aRob)' ) 14' 5.LVY This equation for the lunar tide is - + Y 3Ly" dY.1+4 drL O III B- 2 since, as stated before, the atmosphere experiences negligible heating with the lunar period. For the time being, the effects of Newtonian Cooling will be ignored leaving the equation 4i~~~2I4 4 SY d': Iz% JY III B- 3 Equation III B-3 is of similar struc:ure to the time-independent one-dimensional Schrodinger equation S[-V ) + E09J 10 0 III B-4 where ,= wave-function of particle with mass M -47- = (Plank's constant) T2 VO= potential energy function E()= total energy function. The comparison between equations III B-3 and IIIB-4 is complete if V()= 2 and E _ . With this comparison, it should be possible to carry over at least part of the physical reasoning so common in Schrodinger equation usage, that of the pc tential barrier, to the tidal pl-oblem. A mean profile of g(;) is shown in figure IIIB-1. This profile above 30 km was constructed by using the 1965 CIRA mean atmosphere. The lower part of this profile was calculated by averaging temperatures collected in a five year study by the Planetary Circulations Project under Prof. V. F. Starr at the Massachusetts Institute of Technology. These temperatures were first time-averaged for the five years and then averaged with respect to area over the Northern Hemisphere 1. 3. The lower and upper parts of (1) Equations IIIB-3 and Ill B-4 are both examples of a more general class of equations -- the Helmholtz equation. 2. The hemispheric average is computed by the following formula: )l /' cN 3cosCde 3. Rigorously the calculation of A() should be carried out as follows: (1) Calculate ,9 N HOte H k, tt) i (2) Average with respect to time and area. At what is considered a small loss in accuracy, H was first averaged and was computed with this. -48- were then connected smoothly. The vertical line in the middle of the figure is where where 2V() where ( :t Y :1. =Ig - ) "CI . In the parts of~S ) , the (2, 2) tidal mode is evanescent; however, 7? 7,S) a the (2, 2) mode has the characteristics of a travelling wave. Preserving the terminology of the Schrodinger equation, the regions where ()< c I will be called "barriers". The author has named the various regions of the stability curve in figure IIIB-1 according to whether the solutions to equation IIIB-3 are evanescent or wavelike. Figures (IIIB-2) - (IIIB-13) display the author's calculations of the monthly-averaged values of and ) . The data used in these monthly calculations were as follows: (1) 1000 mb to 50 mb (approximately the ground to 20 km) The temperature data of Starr, et al. f1] 1. For example, the January (4] is used to calculate is the average of January 1957, January 1958, January 1959, January 1960, and January 1961. (2) This data is then hemispherically averaged. 30 km to 80 km Zonal averages of temperature are obtained from the CIRA 0 -49- atmospheres. W() is then calculated by hemispherically averaging these CIRA values. (3) 100 km 4- 00 The CIRA mean atmosphere is used for this region. These three regions were then connected as simply as possible with a smooth curve. This smooth curve is shown as the dashed line in figures (IIIB-2) - (IIIB-13). This curve for = 14. 8. Above calculations of a). (%) is then used to calculate t( up to L = 14. 8 a straight line was fitted to the All months are thus fitted to a standard thermosphere. The values for () , computed as explained above, are shown as the solid curves in figures The annual variation in S(') (IIIB-2) - (IIIB-13). , which is depicted in these figures, will cause a corresponding variation in the lunar tide. For this reason, the character of the seasonal changes in F:) will be discussed. The temperatures at the ground, tropopause, stratopause, and mesopause levels are marked to emphasize the regions where there is most seasonal variation. The temperature at the ground in January is 281 0 K. rises to a maximum of 294 0 K in July and A.ugust. It The range of -50- seasonal variation in T at the ground is then 130K. The temperature at the tropopause in mid-winter is 2080K and reaches its maximum value of 211°K in mid-summer. range of seasonal variation in T The at the tropopause is 30K. The minimum stratopause temperature is 253 K and occurs in February; while the maximum stratopause temperature is 278°K and occurs in June. Tat The range of seasonal variation in the stratopause is 150K. At the mesopause the minimum temperature, 183 0 K, is seen in July while the maximum temperature, in December. 2250K, is seen The range of seasonal variation in T at the mesopause is 42 0 K. From the preceding, it is clear that the seasonal variation in f() is most pronounced at the nesopause level. This seasonal variation in the region of the mesopause is also evident in the changing size of the "meso,!pheric barrier" (the shaded regions) in figures (IIIB-2) - (11IB-13). It will be noted that the "mesospheric barrier" is much larger in the summer months than during the winter. This means that the lunar tide should experience more trapping in summer than in winter. It would follow that the -51- tidal wave should more closely resemble a standing wave in summer; while in winter the tide should show more characteristics of a propagating wave. This change in wave structure could be a very important effect in explaining the observed annual changes in the lunar tide. Before proceeding further it should be noted that 41) is a Northern Hemisphere average but is assumed to be true over the entire spl ere. It is not possible to get a globally averaged stability function due to the paucity of Southern Hemisphere data at high levels. This limitation of the calculations which follow should be kept in mind. Section C: Newtonian Cooling In section II A, it was assumed tha: the diabatic term in the thermodynamic equation is expressable as the sum of a solar heating te.m and a Newtonian Cooling term, Q = being a temperature perturbation). X--MOST (ST In this section the Newtonian Cooling term will be examined. There are two distinct contributions to the Newtonian Cooling term with one part of this effect being due to the temperature dependent effect of CO 2 infra-red cooling and the other part of the Newtonian Cooling term being due to the temperature depen- -52- dence of 03 photochemistry. The contribution due to CO 2 1 will be treated first . A formula is for the net monochromatic radiant flux at a given level 2 given by III C- 1 0 . 8,(eo))TV(, z) + e,((co)T *,) where y )= net radiative flux at frequency V G(.) = temperature at level y(0) = TV({ ,Ij V and temperature = transmissivity function from level 0 V t - Plank function at frequency at frequency and level to ~' 0 evaluated . The terms in equation III C-1 represent, proceeding from left to right; the flux arriving at e flux arriving at l arriving at - from atmospheric layers below, the from atmospheric layers above, the flux from the ground, and finally the flux at g orig- inating at infinity. 3 The flux convergence, or "heating", at ferentiating equation IIIC- 1 with respect to i is given by difgiving 1. This discussion is the author's understanding of an unpublished derivation, due to C. D. Rodgers. -53- T o o dT(ioo. + ((oo)d- (0(0))d +. IIIC-2 As noted in Rogers and Walshaw (1966) the complete radiative heating equation III C-2 includes the fol'owing effects: (a) a cooling-to-space effect (b) a flux exchange with other atmospheric layers (c) a boundary flux from the ground (d) a boundary flux from infinity. Ignoring all these effects except (a) is the cooling-to-space approximation. This cooling-to-space term, integrated over all frequen- cies, is seen to be <: cooling-to- space (in 0 C/day) ...... .) 0 CFp IIIC-34 Since the main contribution to CO 2 infrared transfer is in the strong band centered at A0. I A4 , the integral in equations III C-3 may be approximated by CS where 1 V( ) w-a 8, ) dV ' ) do III C-4 (the average over the absorption p YL-I-----~i-Y ..1.^II..LI(-IUI_..I-__II* ..I1LII~ -U-(-~ -54- band). The emissivity is defined to be equal to I-T V so that, and adjusting to proper after putting in numerical values for Cp time units, the CO 2 cooling is approximated by Coolihi9g g.)Ixo • Bg5)1 ( de dp OC III C- 5 I- Rodgers (1967) has fitted analytical expressions to approximate the CO 2 emissivity. His expressions are as follows: (vU) % 160 v above 10 mb III C- 6 EM" 20 Inu below 10 mb III C- 7 These emissivity formulae are e valid for given by Cp) pP V( pipl 6 PC with ratio. (the optical path) -2 where p being 10 6 dynes cm-2 while is in atmospheres (P) is the CO 2 mixing As is customary, when treating CO 2 , C(p) is taken to be constant giving Ve Uop Z III C- 8 Using these approximations, the cooling due to CO 2 may be calculated by noting 1. dp dp dE dv dvdp Note the change to pressure coordinates. . For the log region -55- d?. 20 dV9 U thus, do d4 while PO dt - P III C- 9 below 10 mb. P dF = The band-averaged Plank function is approximated by the value of the Plank function at the center of the band, kv i.e. : A I) a k 3 =wow le III C- 10 (This approximation is good to about 5%1/ for the 15 A band at reasonable temperatures.) Using formulae III C- 5, III C- 9, and III C- 10 the cooling for the log region (below 10 mb) is given by q60 Cooling 6 ' with P P III C- 11 atmospheres and T in K. := 1S pSmW. CM. For the square-root region atm. ) if the constant CO 2 mixing ratio is taken to be 250 atm. cm. dp P X I. Hence in this region, d"zSRXI which implies 3 III C- 12 Therefore, using formulae III C- 5, III C- 9, and III C- 12, Cooling I ,0 e 7 above 10 mb . IIIC-13 -56- In summary, the approximate formulae for CO 2 cooling are as follows: 960 below 10 mb CO 2 cooling III C- 14 = 60o e above 10 mb The corresponding Newtonian Cooling fTrmulae are obtained by linearizing expressions IIIC-14 with 1= T. + T . These formulae are as follows: I 960 3.5 xl0 Le below 10 mb SM IIIC-15 above 10 mb An average profile of OCOz is calculated from the hemi- spherical ty-averaged temperature profile. file is shown in figure III C-1. a O This temperature pro- The Newtonian Cooling function, (figure IIIC-2) is arbitrarily set to zero at Z = 14. 8. This should not affect any calculations since all energy reaching S= 14. 8 leaves the system because of the upper radiation condition. It is important to have a constant value of 2 CO. above = 14. 8 for consistency of this radiation condition. -57- The Newtonian Cooling effects due to ozone will now be discussed. Classical ozone photochemistry predicts that ozone concentrations should be strongly temperature dependent between about 40 and 70 km. (see Leovy (1964) and Lindzen and Goody (1965) ). This is due to the strong temperature dependence of the reaction o + 03 -- . 0 IIIC-16 Because of the ozone photochemistry, the ozone heating rate should be strongly temperature dependent. This heating rate is given by the following equation which has been taken from Leovy (1964): L where 4 a Q dQ - Hcost. IIIC-17 is the rate of absorption of solar photons by an ozone molecule. A Q ( a g is the activation energy for reaction III C- 16 6 Kcal. mole- ), and RI is the universal gas constant. is dependent on the ozone concentration at all levels above that being considered and therefore is dependent on the atmospheric temperature structure. of T3 This temperature dependence is,however, taken to be secondary to the temperature -58- dependence of the ozone concentration this reason.. and is disregarded for Differentiating equation III C-17 with respect to gives the following: Ch Upper and lower limits for Ot 9T O IIIC-18 are shown in figure III C-3. These profiles were computed from curves of "observed" ozone heating rates prepared by Leovy (1967), and the temperature structure in figure III C- 1 Section D: Numerical Procedure In section (IIA) the lunar tidal equations were formulated and in section.) (IIIA), (IIIB), and (III C) the coefficients in the tidal equations and boundary conditions were found. It only remains for the means of solution to be discussed. The vertical, equation along with boundary conditions can be written as S+11 + - ) . Y" a 0 III D- 1 with 1. The author is very grateful to Capt. Thomas Dopplick and Mr. Herbert Jacobowitz for their help in understanding the radiative transfer considerations contained in this section. -59- VT" dom 9D DIII and Radiation Condition D-2 . as 0 OO III D- 3 Although some solutions to this lunar tidal problem can be obtained analytically, any realistic problem must be treated numerically, The plan of attack is illust rated in figure III D-1. Equation III D- 1 along with conditions III D- 2 and III D- 3 constitute a boundary-value problem. Such problems are amenable to two types of solution (1) an iterative procedure and (2) a matrix approach. The iteratve procedures are quite tedious and the rate of convergence of such procedures is sometimes a problem. For this reason, the nLethod used in this study is a matrix approach. Equation IIID-1 should more precisely be written as the two equations d9 and emus lap Y . r 6~ q 6 a6 YIIID-4 -60- a#~L~h ~/ vi.5 ( v,., Xz. Q" P .d=__ ~~YII~) O III D- 5 In order to simplify the notation, the following variables are defined: -' ;-,C -_I~,- cr wCt) - - WL 'a) L' LM "IS ca) Y Y (;E) V 1+t ~A? Equations III D-4 and III D- 5 become 4qIL -39cE)w = O 9(Z) 0= III D- 6 III D- 7 Equations III D- 6 and III D-7 will now be put into finite-difference form. The following finite-difference notation will be needed: N = 5%() 9 ) C ) K, evaluated at ZY . z ( ) K -t- ( )1<-1 -61- The starting point for the finite-difference equations will be the following formula which is taken out of Hildebrand (p. 241). III D- 8 ) cL %i 1V,S +1 V n where the truncation error is given by h6 R40-- J) (z.,(f T < 4,) III D- 9 Equation III D-6 can be put into finite-difference form, by , as using equation III D- 8 and ignoring 74I -V+-I..', = h4(1+ .,+ III D- 10 , 7v)7 III D- 11 -12. )(- f. while equation III D- 7 is written = v,, -l*,w,.., By expanding S S '9(-"w,- , in equations III D-10 and IIID-11, the following equations are derived: 12- 0 3sl n am b- (I+ h11 + Oft% '., .. awIX % SUL. 4- 6 017'n we't - III D- 12 III D- 13 -62- Equations III D-12 and III D-13 give a fifth-order system of finite-difference equations as an approximation to equations III D-6 and III D- 7. The boundary conditions must now be put into finite-difference form in order to complete the formulation of the numerical problem. It will, for this purpose, be necessary to expand d4 numerically at both the upper and lower boundaries. Using finite-difference operators (;ee Hildebrand,p. 138) the following relations are found to exist ! - ---- III D- 14 - III D- 15 I! where A( - )+ -( >~- a( ). -dOl Using equation III D-14 at IIID-15 at TO= P ( VI N s (Vr O ) and equation ), the following expressions are obtained: vl 1h 0 V, + v' +( III D-16 -63- I .-=h a+ + () III D- 17 Hence to second-orderI III D- 18 /,i Kn - aw and C~-vg*-' 'T'I PfN ,kfNV'M VV +3jNwN] + fNI)41 %NU-l L~fl~ III D- 19 g The lower boundary condition III D-2 can then be written in finite- difference form as 1 , [vw 0 - 1~% - o V~ . +~' J~9 I- [-fE- 3,V. ~ ro= 0 A - III D-20 SIII D- 21 The upper boundary condition is handled in a different way. The upper boundary condition is taken to be the requirement that 1. In approximating to second-order, the writer has compromised the accuracy in order to keep the boundary conditions near the boundary. -64- there be no downward energy flux at infinity. In order to treat this condition numerically the stability function above FTOp assumed to be O= ~ is (assumption of a thermosphere). In the last section it was stated that the Newtonian Cooling coefficient was assumed to vanish above i-fp ; thus, the vertical structure equation, above ZTO9 , becomes Y 4 .7 III D-22 n The solution to equation III D-22 is Y_ e- z(VW),--j) III D-23 where 1 and & are the two Airy functions (A and B are complex constants)I Since the radiation condition applieo3 at infinity, it is of inter- G est to look at the asymptotic behavior of as X-- oo -x . (- ") and f&(-X) These asymptotic formulae are - TT III D- 24 and p 1. (-x) Pk TT ' os( x/ Airy functions satisfy the equation If ) If V IIID-25 D e VU -65- From expressions IIID-24 and IIID-25 it follows that The combination fl 8JX x X)^T)Trr +i ('-K) III D- 26 is thus seen to lead to O-X) downward phase propagation or, since the lunar tidal wave is a gravity wave, upward energy propagation. 40 above Z = 0'f W (b ~~l a - Therefore, III D- 27 ) jl ) rToP For the numerical solution it will be assumed that and are continuous across "TP . Y 7) Mathematically speaking,these conditions are as follows = ntp) A -y o ( I III-28 hIDD)29 Equation III D- 28 can be separated into real and imaginary parts giving two equations. QL (A) and Sm two equations: When these two equations are solved for (A) , equation IIID-29 becomes the following -66- - o -cv~" a~ op + bro? r Cv V ; avt %,V%) dYA t- III D- 30 -A ZTD? :vvv ,TIYC, + STOP' III D- 31 where L,TO III D- 32 ( -[ (-p and b T0 I b r -? ' ( Cvi a l (Y '? 1k) ) (with TOP : - p- (aaft')l z,,, - - 1l ai III ) ( -- III D- 33 t Cf~op (b -IY (r4 6vym sl Equations III D- 30 and III D- 31 can be put into finite difference form by the use of relations III D-19 giving LYN -,[I-* 4NY[ N N m tr0JTOP W7V III D- 34 and ILW-WN-- h-fNW -9N-1 =ToPKYN apA/WN III D- 35 The total finite-difference system can be written in matrix form as -67- I ' III D- 36 - where 1 ' O and A isa (ar42)x(N+) matrix given on the following page. The set of simultaneous linear equations given by the matrix equation III D- 36 is then solved numerically by GELB, an IBM subroutine suitable for solving such matrix equations as III D-36 with a band-structured matrix such as A The numerical method derived in this section is shown to be accurate with about 1-2% error. In testing the accuracy, the analytical results of a later section were used to compare the nu- r- + Nr- [d@L9~~t 4~ 0 o-~~~ *Nt 000 * .0 o 0 o 0 C)0 SIO 0 0 * * 00 00 S 10 1 0 0 0 * IZ 0 *0 ** 0 * 0 0 0 5 0 0 1 LR 0 .iqq* IN I DtS%-i+~ 1,Nr- NC I-Nr £-Nr, C-INr -69- merical solution to the analytic solution in the special case of an isothermal atmosphere 1. In both this test and actual calculations the following para= 14. 8, and N = 148. = 0. 1, 7F meters are used: -70- t h (altitude) km 100 90 80 70 60 50 40 30 20 10 0 .0 I .02 Figure III B- 1 .03 -71- I (scole height) 8.0km 7.Okm 5.Okm JANUARY - 22 s =1/4 km 100 S ------ H 90 80 224 K I 70 'I S) 2640 K 50 40 20 208* K L 2810 9-,' Figure III B-2 .03 -72- 5.0km 6.0km H (scale height ) 7.0km I I 8.0km I I I RUARY 2 r.1/4 h (altitude) -0 km - 100 $ / / -90 / -80 80 224K - 70 -60 263 OK -50 -40 - 30 /- 20 2820K : ,02 .01 02 Figure III B- 3 03 0 -73- fH 5.0km 6.0km (scale height) 7.0km 8.0km h(altitude) km Figure III B-4 -74- IH (scale height ) -- 5.0km 6.0km 7.0km 8.0km h (altitude) km 100 90 80 70 60 50 40 30 20 10 1igure III B- 5 -75- H (scale height ) -- h( altitude) Figure III B- 6 -76- H (scale 60 km 5.0km height ) 7.0 km I I 8.0 km I I JUNE 1/4 Sh(oaltitude) Skm -- 15- -~ 14 7 k S - 100 - - 13- / 90 1840K .. ... 12 -70 I - 60 .- 8,278 .01 .02 g Figure IIIB-7 ... .03 K 5O -77- H (scoale km height) --7.0km JULY S 1/4 h(altitude) so km 100 000, 50 40 30 2110K -120 / 294*K .03 Figure III B- 8 -78- f h (altitude) km 100 90 80 70 60 50 40 30 20 10 0 .01 .02 Figure III B- 9 .03 -79- I h (altitude) km 100 90 80 70 60 50 40 30 20 10 0 Figure III B- 10 -80- R (scole height) 7.0km 6.0km 5.0 km I p 8.0km I I OCTOBER I h(altitude) s $ =1/4 km Sl 15- 15 H 100 / 90 14 13 12- 197*K -80 7 II S70 1060 9 8- 6 40 - 30 4- 2090K k 11 2 Figure III B-11 -81- !5.0km . 6.0km H (scole height) 7.0 km 8.0km t h(al titude) NOVEMBER . 2; r21/4 km S - 100 14 - 90 13 12 - 215 K I I -80 II - -70 10- - 60 8- 2700 k 7 6- / 50 -40 5-30 432 20 / - 208 K Figure IIIB-12 -82- F (scale height) 5.0km 6.0km 7.0km 8.0km I h (altitude) km 100 0.1 .02 Figure III B- 13 .03 -83- h (altitude) km -100 -90 80 70 60 50 40 -30 20 -10 0 2900 T ( K) O- Figure IIIC- 1 -84- t h altitude km 100 90 -80 -70 -60 -50 - 40 -30 -20 - 0to O ff 0.00 I I I I I I 0.05 Oco0(doy -) Figure III C- 2 I 010 10 0.14 -85- NEWTONIAN COEFFICIENT 03 I COOLING h (altitude) FOR km 100 1490 13 80 12- 70 60 upper limit curve .......... 81 50 Slimit lower 6 40 curve 5 30 20 2110 I 0.0 I I I I 003 (day "1) Figure III C- 3 0.5 70 0 .7 -86- RADIATION CONDITION REGION ( ANALYTIC SOLUTION) II n= N n=N-1 n=N-2 n=N-3 n=N-4 n=N-5 INTERNAL RE( (SOLUTION I nz0 *= n= n=2 * I n 5 nl 3 bT h Lower Figure III D- 1 BY EQUAT -87- CHAPTER IV RESULTS AND OBSERVATIONAL AGREEMENT Section A: Results With No Newtonian Cooling In the previous sections the methods of calculating the lunar tidal response were presented. The inputs for such calculations were shown along with discussions of them. Having done all of this, the results of these computations are given in this chapter. The first results will be calculations of the lunar tide with no Newtonian Cooling for each of the twelve months. These calculations were carried out for the twelve monthly stability profiles which were presented previously in figures (IIIB-2) - (IIIB-13). These stability curves were all analyzed before any calculations had been performed, with a view towards elimir ating any bias in the analysis. Figu es (IVA-1) - (IVA.-12) show these results. The final variables are presented as being proportional to COSE'Lt.a*s Here 4?) )] . (Oa is the phase of the variation; that is, the variation and reaches maximum at fore lunar transit. A. negative 300 ( Swill + I lunar hours be- imply a phase lag. Each of the variables is represented b:y a curve in which the phase plotted as a function of - . is I'he amplitudes of the variables at -88- the equator are also put on the curve at integral values of kI, Each of these variables V , V , Wh , and T . will be discussed sUparately. (geopotential height) The pressure variation is expressed in geopotential height units for two reasons. First, where equations have been formulated in pressure coordinates it is custo nary in meteorology to use the geopotential height variable. Second, geopotential height is a more convenient variable for upper atmosphere work. Geopoten- tial height variations may be related to pressure variations as follows: J- P For instance at the ground for hydrostatic motions. / '.3 I. XIO -3 gm Cm , so a 1 cm change in geopotential height implies a change of approximately -3 1. 2 x 10-3 mb in pressure. The observed lunar variation in surface pressure at the equator averages about 0.06 mm (.079 mb) which implies a variation in geopotential height of about 66 cm. The vertical structure of the geopotential variation is given by the solid lines in figures (IVA-1) - (IVA.-12). The h -curve, as all of the other curves, shows a positive slope as is required in order that there should be an upward energy flux. This lunar variation in geopotential height varies from tens of centimeters at -89- the surface to fractions of a kilometer at about 95 km. The lunar semidiurnal variation in pressure changes appreciably with latitude. This latitudinal dependence is given by the principal lunar semidiurnal Hough function (figure IVA-13), n. ; thus, by latitude 300 the lunar variation in geopo- (() tential height has fallen to less than half of its amplitude at the 1 equator It is important to note that there is a phase lag in the h -curve at the surface in every month. U and V (eastward and northward wind) The eastward wind, L/ , and the northward wind, V , due to the lunar semidiurnal tide are shown in figures (IVA-1)-(IVB-12) by the dotted line and the dot-dash-dot line respectively. are no amplitudes next to the V -line, ;ince the northward wind velocity is identically zero at the equator. the V Note that the phase of -curve is always 900 ahead of tie V -curve. be seen from the formulae for V There This can also and Y' in section IIB. The eastward wind varies in amplitude from a fraction of a cm sec-1 at the surface to some m sec-1 at 95 km. The latitudinal variations of the lunar tidal wind velocities are 1. is shown normalized so that C -90- shown in figures IV A- 14 and IVA-15. V or V , the amplitude of LI To obtain the amplitude of at the equator is multiplied by the value that is read from these curves. W (vertical velocity) The vertical tidal velocity is shown in figures (IV A-1) - (IVA-12) by the dash-dot-dash line. The "W -curve does not ex- tend to the surface, since the bottom boundary condition forces T~ to be identically zero at the surface thus making the phase indeterminate. The 17T -curve has been arbitrarily cut off at 7 = 1. 0. The vertical wind velocity varies in amplitude from thousanths of a cm sec-1 at the surface to some cm sec-1 at 95 km. The latitudinal variation of this tidal vertical velocity is given by T H'IL() which is pictured in figure IVA-13. (temperature) The lunar tidal temperature variation is shown in figures (IV A- I)-'IVA.- 12) by the dashed line. This temperature variation varies from thousanths of a degree Centigrade at the surface to 1-7 degrees Centigrade at around 95 km. The latitudinal dependence of the lunar tidal temperature variation is given by which is pictured in figure IVA.- 13. After looking at the results of these calculations of the lunar semidiurnal tide, neglecting the effects of Newtonian Cooling, -91- these results will now be compared with observations that were cited in Chapter I. Recalling figure 1-4, it was seen that there was a marked annual variation in the phase lag of the surface barometric lunar tide but not in its amplitude. The amplitude of the pressure variation was seen to vary between about . 055 mm - . 065 mm (or in geopotential height units 61 cm - 72 cm). The phase lag at the equator wvas seen to vary from about -100 (20 min) in northern hemisphere summer to about -400 (-1 1/3 hrs) in northern hemisphere winter. The annual variations in the amplitudes and phase lags in the calculated lunar tidal variations of geopotential height at := O as read off from figures (IVA-1) - (IVA-12) are shown in figures IVA-16 and IVA-17. In figure IVA-16, there is no ap- parent or ganized annual variation of the amplitude even though there is a great scatter of the points from a minimum amplitude of 9. 2 cm in May to a maximum amplitude of 49 cm in January. average amplitude seems to be about 28 cm . The The phases in figure IVA-17 do, on the other hand, show an organized seasonal behavior with greater phase lags in winter than in summer, as is observed. The maximum phase lag of -590 ( ruary with a minimum phase lag of -40 (e 2 hrs) occurs in Feb8 min) occurring in 1. The smallness in the calculated amplitudes relative to the observed amplitudes will be explained later. -92- July. The observed annual variation in the phase lags seems to be explained by the present theory while the calculated amplitude is too small by a factor of about two. An observation of the lunar tidal variation in temperature at Batavia (70 S, 1070E) was shown in figure 1-5. It showed a temperature variation of . 008 0 C with a phase lag of about - 2 3 o (. 75 hrs). The amplitude of the calculated temperature variation varies from a minimun of . 000360C in June to a mL.ximum of . 0053oC in Feb- ruary with the average amplitude being around . 0025 0 C. The calculated phase lags lie between -590 (& 2 hrs) in February and -40(% 8 min) in July. hr). The average phase lag is around -300 ( 1 The calculated phase lag is in good agreement with observations while the calculated amplitude is too small by approximately a factor of three. An observation of the lunar tidal wind variations was shown in figure I-6 at Mauritius (20 S, 55 E). The variation of the eastward wind component showed an amplitude of 1 cm sec-1 and phase lag of 2300 (7 2/3 hrs) while the variation of the northward wind compo-10 nent had an amplitude 1. 2 cm sec and phase lag of 900 (3 hrs). The average calculated wind variations, when adjusted to the latitude of Mauritius show amplitudes and phases as follows: -93- amplitude . 5 cm/sec phase lag -2400 (8 hrs) amplitude . 5 cm/sec phase lag -1500 (5 hrs) There is also a considerable variation in the calculated amplitudes and phase lags from month to month. The average calculated value for the phz se lags of the wind observations compare reasonably well with the observations, but once again the amplitudes are too small by a facto.: of two. In this section calculations have been presented for the atmospheric response to the lunar gravitational forcing. only lunar tidal forcing of the atmosphere, though. This is not the The oceans move up a:id down with the lunar semidiurnal period thereby causing an additional forcing. Sawada (1965) investigated the interaction betwe mn the lunar tides in the ocean and in the atmosphere. His result ; indicate that for an ocean covering the entire earth's surface to a depth of 4-5 km, the ocean forcing would cause an atmospheric tide of about the same amplitude as the tide caused directly by the moon's gravitation, while the phase lag would remain unchanged. Since the earth's surface is 71% ocean which, if covering the entire globe, would have a mean depth of about 4 km, -94- the lunar tide should have about twice the amplitude that was calculated but still have the same phase lag. The comparison between the theoretical results presented in this section and observations is seen to be quite good when considering the simple model used. Section B: Results With Newtonian Cooling In the previous section, it was seen that the calculations which were performed with classical tidal theory gave results that, in many cases, compared quite favorably with observations. The tidal equations were derived, in this study, to include Newtonian Cooling. In this section calculations will be presented for various types of Newtonian Cooling. Such calculations were performed for all of the monthly temperature profiles, but it would seem redundant to present all of them. Instead the January and July vertical tical structure is presented along with the monthly values of the geopotential height variation at the surface. In the previous section it was seen that the calculations with no Newtonian Cooling gave a surface geopotential height variation with amplitudes ranging from 9. 2 cm in May to 49 cm in January and phase lags ranging between -590 in February to -40 in July. -95- The surface geopotential height tidal variations with CO 2 cooling, with CO 2 and small 0 3 cooling, and with CO 2 and large 0 3 cooling are pictured in figures (IV B- 1) - (IVB-3) respectively. The calculations of the tidal variation of geopotential height at the surface, including CO 2 cooling (figure IV B-1), show amplitudes ranging from 14 cm in May to 53 cm in February and phase lags ranging between -590 in May to -40 in July. performed with CO 2 and small 0 3 The same calculation cooling (figure IVB-2) gives amplitudes between 24 cm in May and June and 56 cm in February with phase lags between -610 in May and -150 in July. The calculations with CO 2 and large 03 cooling (figure IVB-3) gives amplitudes from 31 cm in June to 65 cm in February with phase lags from -580 in May to -210 in July. The effects of increasing the Newtcnian Cooling is to decrease the seasonal differences in the amplitude and phase of the surface pressure tidal variation (both in absolu.e difference and ratio) while both the amplitudes and phase lags are generally increased. The seasonal variation in phase lag in figure IV A-17 (no Newtonian Cooling) becomes more obscure with increased Newtonian Cooling while a seasonal variation in amplitude becomes evident with increased Newtonian Cooling (counter to observations). The vertical structure of the January lunar tide with CO 2 -96- cooling, CO 2 and small 0 3 cooling, and CO2 and large 0 3 cooling are shown in figures (IVB-4) - (IV B-6) respectively while the corresponding calculations for July are presented in figures (IVB-7) (IV B- 9). Figures (IVB-4) - (IVB-9) generally show increased amplitudes with increased Newtonian Cooling at lower altitudes while the amplitudes decrease at higher altitudes. The slopes of the curves also tend to increase with added Newto iian Cooling. The vertical structures, however, do seem qualitatively similar to the results with no Newtonian Cooling. These Newtonian Cooling calculations are subject to question on at least two grounds. The first is that monthly calculations are being performed with annual Newtonian Cooling profiles. This is certainly not a desirable situation, especially when interested in the seasonal variations of such calculations. This situation is very difficult i: remedy at present since ozone observations are poorly known at higher levels (above 30 km), much less the seasonal variation in ozone concentrations at these levels. The seasonal variation in the CO 2 cooling coefficient could have been included but, since there was no hope of treating the variations of ozone, it was not. The second aspect to be questioned is whether the classical -97- ozone photochemistry is correct. Hunt (1966a, 1966b) has brought up this question and suggested that the photochemistry of ozone in the upper atmosphere is controlled by a different set of reactions than heretofore had been supposed. Hunt's reactions are not strongly dependent on temperature considerations so that the Newtonian Cooling due to ozone might be significantly smaller than that used in the present investigation. * / km -90 80 70 60 50 ' co 40 30 20 10 I 32 320 50000 h(cm) T (*C) FEBRUARY 14 2.9 [/O1100 6.3 1 7'34000 .-34ooo/ 23000 . u (cm sec /km 90 e. * - 750 ,* 4.9 .3.3 500oo 8 S.89 / .64 I- 1-80 ' 15000 / 9700 SI 32.2 320 ;210 1.4 6600 *.76 46/ * -70 * 140 S-60 .34 4800 * .56 * 100 .3o0 3200 @66 S21 I" o -- I I 1600 ! -076 280 I 032 -8o* .21 15 012 S.0052 .023 cm\ 053 02.85 -40* 30 ." / .* .97 10 1.0 i o -40 o.9 \/ .053 CD 45.6 . /40 39 120 - 45 ,O34 I .094 .0141 62 0 I 41 690 5 -. 12 O -52 45 .59 85 400 800 10 1200 1600 2000 240 2800 3200 MARCH .MARCH S 14 32000 h(cm) h 300 - T (0C) 4.2 1.6 / m 0- 710 / 90 *490 *330 0 I Sz, / tO - 7500 .35/ I .- 230 03.0 / 160 1.0 I 0 *.86 1 i *60 / C o 32 .095 5 50 40 ;14 /I0 30 *5.7 057 028 .' 2.0 2001).086 1 290 10I2 53 .013 0O -80 60 I .16 0044 80 70 120 rT km 36 . 55 10 /0026 .'37 I -40* mm 20 m - .- 00 400 800 120 .0O2 ° I/ i I 1600 2001 iOi 240 I 28U 0 320U 3AArw o 26000 h(cm) T(*C) 26000 h(cm) -T(C) 1.4 . /" .68 / 2.2 APRIL APRIL 20000 -- 7' 10 1.0 I * 170 70 1 .23/ -2 * 59001!.53 o130 I / .98 / ?. * 64 .25 I. .25, 80 *220 I 77 I0 S300 I1.5 10000 km 90 S410 / 14000 .36, * - 23 S/O0 -580 . .12 .064 300 I10 .0320 110 40 .014a cm)23 .0050 ScE -s -800 20 -400 30 .. * 1.8 20 . S...29 *.013 I *.0021 00 40 / *6.0 .041 ./00q94 0 S16 I I 50 0 -34 7401.21 51 I -' 1I 16001.37 .18 60 -me"'0 I l4Q0 40* .• 00 I 800 2I 0 / 1200 I 1600 I I 2001 240u 10 .9 20 20O0 3 km 15000 ' .4 7 12000 22; .22 Z 1 .2 10 - 0 330 *.93.93 * / / .11 1.5 .68 - 90 260 1•8 * 210 ** - 9500 I -70 - 70 * 170 7700 1.57 I .19 -80 6100 1.50 I 130 S130 .23 4700 45 !100 .25 3300 .53 72 -60 -50 Ln 1800 .20 .130' 45 -40 790 .24 320 0721 w 036 cm 9.00 S100 016* 17 0019: *.046 27 *.0I3 . I.6 1600 , 01 .*. ..* 00571wm7 0 c5019 • I sec 9.2 -4RO0 .0"0043 _0n* 00 ' '**" - 30 I . *6.4 .11 * 400 000 80 " -. 19 320 o .1 ..51 .57 I 400 1 ." :0* 1200 n 160* I 2000 I 2400 t 2800 0 3200 .%f 13000 .47 9800 .19 .097. ~ '7600 r 10 93 / O I --.4 . / .89 • 280 0200 80 62 * 170 7 44 130 I 6100 • 110 .- 60 4100 .32 89 S•50 20 3000 S.18 .12 067 43 "!-40 7_90 .23 e17 320 .11 *6.4 -o 110 ~.045.1.8 015 - 4 0052 - 18 cm sec --80 / p00 -4 0 0 ° o 0, - 30 " 1. .il 15 Aoop3 0 64 1700 .39 / .0341 w - 70 15000 .33 .16 a km - 90 i 34 10 .40 0 400 800 1200 1600 2000 2400 2800 320* T (*C) 13000 h(cm) JULY JULY 14 / 08 80 . 5200 4300 1.32 ;93 .24 .47 12' 1300 .28 .28 .011. w( cm} WImm)~ 014 280 .085 * 5.7 40 34 * 0 / . 00IR . 20 .,51 10 .. 18 .0 019 00 30 - 2.0 I I -400 -40 .17 - .00351 50 650 11O !.037 .0241 60 S62 2200 .28 048 70 * 76 .13 5ie 90 * 120 2900 .21 0811 S km 150 .45 3500 -A *210 ".69 .069,/ .066 / / 10 I / 7000 .15 sec) s cm - 280 .1 990 .29 (s5 u 1 / 1.6 .67 0 ,* 4Q 04O0 400 I O 8OI20~I6O~ZOO 1200 80 I, . -- 1600 200- I 2400 2800 320 3200 0 h(cm) AUGUST .074 .076 * t T(0C) l" " 6 0 00 .94 / 12000 *-.35 9000 .16 7500. * .56 / sec) u (m c " 350 1.8 . 260 90 IS I* *210 .74 80 : 160 SI 6000 *.45 * 130 4800 .38 *110 3800 .38 .82 2600 .40 * 57 120 i .18 - 70 tO 60 I .19 .15 1500, 34 _ .101 5 .058 .029 * 15 30 - 300 1.096 * 6.0 - II O .. 041 * 1.8 .013 42 .00451 27 " 40 720, .20 I sec 0=80° 50 I -40* I 20 .0 12 0* 'a- 10 .0020 . .15 22 /1.9013,~.3, 00 400 /K I * I 80" I I . 120I" IbU I * * 'u- A 240 I I I%0 * E'% 480 2 3L n 0 " 0 S EPTEMBER EPTEMBER 14 14000 h(cm) T(C) .66 10000 * 7 .1 / 4100 .0 241 .141 I I i * 230 * 180 ;.4 m km 90 a I */ 80 * 140 . -6500 I 5200 .41 * A .300 .97 .oooo. .31 81001 .14 - 10- 07 6 vV(~C)sc ) 14000 h(cm) T (C) i .110 70 * 89 .34 3200 .30 *69 23O0 '.33 *50 1300 .30 .28 60 i o 50 0 I 40 13 630 .17 *5.6 280 .082 30 I 0S 110 T.036 .0036 .01 55 -- .0036l 0 0 -- _0 °0 *2.0 .011 .* -400 20 I 0.18 34 r.0019 0° * 10 40 /P019 I / -0 400 800 12_0 . 078 8 1600 I 2000 I I 2400 1 2800 10 3200 OCTOBER 23000 -rrh (cm) T (0C) 1.2 /2.8 1600 O/ .55 [ I0.91 .32 * 13 / 12000 / 8600 .23.' .19d 10o 18* D / 390 SII .059 - .013 .87 *190 .59 0140 42 * 110 .38 I .0044*sec 800 8O0 C 70 60 0 C S32 300 .. 095 I 110 I .041 .6.0 .* o.012 50 40 30 1.9 *.*.31 80 I *60 S15 km 90 I * 85 700 .19 44 C * 260 1500 .37 .16, I S360 2700 .44 217- - I 5000 I7 .20 - 6500 * v ( c) sec) 7 500 20 .0 10 29 e.0023 24 .00 Ir ' . 1 -400 0 ~2i .26 400 I 800 80 ., 1200 1200 1600 1600 2000 2000 240 2400 2800 S0 3200 NOV/EMBER 14 30000 --,(cm) ,.6---**/ 210 0 0 4 L, oo* - 56 15000 / 10000 .43P S.32/ •/ 10 /I 3650 2,6 /I.8 I 3i 3900 0 I i 70 S160 .53 * 110 S 0 SI .35 ;85 i I I .22 2700 .47 * 59 .17 1500 .34 *32 II 5 300 .059, 029 110 - 46 .013, 60044 w 0O 50 1* 0 0 700 .20 * 15 .094 S6.0 '041 .*' cm __o 30O 631 , 40 30 i 20 .42 . . 10 0021 )_ 6.35 "oo ,,o -40 / S.9 /.Of 60 I0 I cD 80 I *220 0.35 km 90 * 320 1 5100 / *460 . 1.3 I ,85 7 100 6. .2 I / v(~) ( ) T(C) 3.7 /"T(G) 0o @33 .35 , , ,, 400 800 1200 1600 II 2000 2400 1 2800 j0 3200 DECEMBER 38000 h(cm) T(*C) 2.1 / '4.9 14 .89 18000 o -6 - .34i 5500 pSI" 2 5 -* S * 1 06 I 110 0131 : cm 37 sec32 / . / - 400 89 50 50 - 40 1 15 30 2.0 014. ... o10 / 0 70 " .0018 00181o -80 S * 5.9 .039 54 - 120 32 I .030 * 61 .092 290 800 67 1 -* 0 ;180 1500 *.3 I 690 .i9 * /. .260 4100 e.34 2800 49 4 .004 k 8390 1l.7 I km km 90 580 390 8100 0/ . 46 .460/ IO * *2.5 11000 66* .: 5 o° 35 26000/ / / v'(cm . 820 / / 1.4 u( .54 I 400 , 9 800 1200 1600 2000 I 2400 2800 0 3200 0 H2 1.00 0.75 0.50 0.25 t Au 1.5 1.0 0.5 0.0 Latitude Av 1.5 1.0 0.5 0.0 50" Latitude Amplit de Annual variation of h' (surface amplitude) cm 50, 0- 30 - 20t- IOF OL !I F M ! t ]. L .L I .i. • I I I I A M J J A S 0 I z| .[• N D I months -- Annual variation of h'-phase (surface phase) Phase lag - 90 -600 - 300 0I JF J I I I I I I 1 I I I I F M A M J J A S 0 N D months ampl itude Annual variation of h' phase lag cm (CO 2 cooling) 60 a -90 - - o -- phase lag 50 amplitude -40 -60 30 U 20 -30* 10 U 001 J I t f I I I I I F M A M J J A S I 0 .N f D 0 t ampl itude I Annual variation of h' pha lag 0 (CO 2 and 23 small 3 cm cooling) 60 * --- phase lag amplitude 50 - 9 0 °* o 40 -600 30 U a 0 0 a 20 - 300 10 (00 J I v F -M A I I I I M J J A S I I O N I I D 0 t amp litude O (CO Annual variation of h' pho se 2 and large O0 cm 60 cooling) log -900 0 w s phase 0 amplitude log -50 0 40 -60O 30 - 20 a -300 U I J = I I I , I I I J J A S I F M A MI F M A M rO I I I 0 N D JAN (Oco 2 ) 31000 h(cm) T(oGC) ~~1AN~' (Oo)300hc)T0 2.2 / /-4.0 3.0 14 1 t .98 1400 0 '" w S670 . 460 /2.1 90 / S300 / 0 91000 .630/ 46 / 10 km /0- 1.4 -- .32 200 I 1/84 I 5800 4100 2900 1 i1 2100 - t 1.45 o86 .28 S62 ,3 2 I I I ;45 0 1200 1.27 I 80 70 0 I .191 - 0 * 130 I .23 I * 25 60 50 I 40 I 0 570 1 14 5.0 .05 270 5 12 .071 I .033 130 .02 0 0 * '5.1 0 /.0oI 74 7- 0 003 -o 59 sec -o -E 0 ° a 10 /0028 152 /.00 29 -40 ° 0 20 , .96 I 71 30 / 0la I .01 ; * 40 S ° 80 80 ° 120 1200 S.52,1600 160 / I 200 2000 , I, - 2400 2800 S10 3200 JAN. (Oco2? + O(small + 0 (small (Oco2 1JAN. 03 30000 h(cm) )) 03)) 2.2. v 2.9 14000/ 11.3 5700 .451 *- I * 190 I / 1.83 km 90 / * 290 I 8700 v 660 * 450 2.0 / .68 / vt ( cm u (scM) 1/w 3.9 -1.4, %1000 o .96 T(*GC) I 0 0 80 - ; 120 70 I S .31 3900 /.48 I I .28 I 23. _. /I 2800 I 1100 i 0* I 0 60 - 50 *44 .26 - I*1 ;61 2000 1.31 .13 I ;24 0 i 40 S .0 881 5 .0 491 560 2 70 140 251 I 78 00 431 ;84 62 w (cm sec i 54 O0 -40 " -80* .14 S12 .. 069 25.3 1.032 /! o 2.3 I o .011 - 30 20 ' 1.0 -/0026 /.00 30 O II 1I0 , .7 5 I. .61 40* 80" 1200 1600 2000 S0 240* 2800 3200 u( c 14 I0 I0 5, 01 0L -800 sec JULY(Oco 2 h(cm) ) 13000 9300 .095 .C (~)v(_ 1 JULY(0co) v 1.0 . v (sc m ) hcm) 280 km * 200 / 036 .17 .6 8 I o150 5.45 4000 .23 .13 ' 2800 28 4300 .L.32 e ;93 I I 0 o 80 70 .75 0 .62 60 4 1 I It 0 (D td 90 0 * 120 .080 / /o .09 5/" io - T (*C) ___._ I .16P 2200 .28 o47 .14i 1300 .27 e28 .17 * [4 .084 * 5.7 650 '0 I .0571 280 I S 120/.037 I I .0131 sec -80 40 ) 5 -40 / *2.0 20 '.53 I .00511wI 50 40 .0971 030 SI I 10 .0019 ,* 21 .,0018 00 0 13 40 ° 800 1200 I - 1600 2000 2 2400 r ° Z80 320 3200 JULY (Oco2 +O(sma l l 03)) h(cm) T(C) -T 13000 / 1.6 .76 9300 /1./ // .35 / 6900n .68 .17 .094 * 1.45 -5300 1 .080 ./ 4200 ~.31 IA I0-- v (cm) u(e) * 280 F I IAiJ / 0200 I i/ * 150 S * 120 km 90 80 0 .92 /- 4000 .23 I S 70 ;75 -0 .14 S.7 2800 .21 S61 .15 2100 46 .13, 1300 ,27 .28 0 0 0 I 60 S 50 I 0 098 640 5 120 .17 57 42 .00 49 ---Osc 80 ° S35 -400 / *2.0 1.010 I S *0 20 ,.58 o0019 10 * .30 .Opi9 00 30 e5.6 I .013 4 40 14 1.084 I o .036 280 056 ( .0 31 0II I 0 o 28 40" 80" ~ ,22 1200 1600 1*~I 2000 1 0o I 2400 2800 320" JULY (Oco JUL (h + a(l0ore O)) 2 12000 9100 /.99 0 _ .34 .16 O T(*C) ( sc *270 1.5 I / .' 1.43 km 90 1 e 150 I ,80 i!i 70 I 60 SI110 "5200 I 4100 t.30 .078/' I * 200 J.65 6700 .092 / 10 h(cm) .90 0 D94/ 3800 23 * 73 .13 2700 .20 e59 * 0 0 13 2100 .27 t 1 I 45 50 I 1300 0 27 09 I2 5 T 630 1.16 ,030 .013 ! 4 .0048 w (~c 0 30 "--E 13 I 280 . 082 120 O .036 .055 K - 40 30 0" "2.0 20 / 00122 0 10 o.44 *.0020 1 ° / .. 010 56 35 40 iI .5.6 I 42 1 S27 I 400 800 00 ",.35 1 120 ° 0 1600 2000 2400 2800 i0 3200 -124- CHAPTER V ENERGETICS Section A.: Energetics Formulation It is of interest to look at the energetics of the lunar tidal problem both to investigate the effectiveness of the "mesospheric barrier" and to look at the distribution and amount of energy depletion due to Newtonian Cooling. A convenient starting point for looking at energetics is the set of linearized primitive equations from which the tidal equations were derived. ao + These are as follows: citsin)v 21o _ _ + 0 C?) ocaos- I A a c sc w(T)-) T RT j VA.-I VA.- 2 VA-3 VA-4 VA.- 5 -125- These equations hold for any linearized hydrostatic motions, neglecting viscous effects, with the Newtonian Cooling approximation. Multiplying equation VA - 1 by U 7.nd equation VA-2 by V gives the following equation: os'Y _+v')] (ac: _m (v_ +V VA-6 O0 atwhich ma be written as which ma,- be written as E[(vw-,]. +(cos (v4) C(aCs O' A V A-7 dam 0 LCcSYI Da -r &coY'FVA (vcoso?) ] The continuity equation VA-3 is now used to rewrite equation VA-7 as [j (U%_#V'%)] a.C. 0'Saa(Vus0 16 +(oACos'4 VA-8 -v'Vo) Now 0 ~SL4d -01- , so equation VA.-8 becomes VA-9 (OcOSZ SI (,%.tV-)] aPC40) - t&a ;4P ) Co'Sv, (Vl (aI4CoS -126- Solving equation VA-4 for aI+ 0 (t T ~/ '-! I P W implies VA-10 C3 T",i~5 T' - With equation VA-10 and the hydrostatic equation, "P P the following equation is finally obtained from equation VA-9: V' fY9 C?-t-Ak j 7i OI al (4) Wf (&COSY) V~ VA3 P _ a .2 TV 1z: -+ SON . le l RL (f4 /or IV Equation VA-11 may be expressed in words in the following manner. The time derivative of the kinetic plus "available potential energy" is balanced by: 1. 2. the convergence of the energy flux, the creation of energy by heating, aind 3. the loss of energy due to cooling-to-space. If equation VA-11 is integrated over a sphere and over a tidal period for the lunar tide (J = 0), it becomes S ) = average value of (( ) = ) over a period); thus, the divergenceA12 average value of () over a period); thus, the divergence -127- of the vertical energy flux must balance the loss of energy due to infrared cooling. In order to obtain results with this tidal energy formulation,it is convenient to express (d Recalling the definition of in the tidal variable y" Y 'se ), it is seen that or A -po a H,, ( N Cos Cs(WtMA) Cem ° 1 l VA.- 13 AIV S'iM cvrz* i It was also seen that 'p I12 V'P ,-taeL so Anr4l .EL eV" aI. 14 dy ) Ycm," I A (equation IIB- 3) dVA.-14 1-Y' 8' ) d-..Y a% 51 V% [Vt -0VV% Al3.- Multiplying equation V A-13 by equation V A.-14 gives -128Io V o"H 'V Y s i l c SyC' CSEvWC4IVA .1 > .,Z'( ( It d~Lyl~ Zlr $Z +yt Now ds - - W0 'ty tVUX7 VEv Yrr d(T'zl S ;VI [ 'r C0 [V -C -r fw 'dI jr+VC VA.-15 ,,,%,X cscvti7 , so performing the indicated averaging on equation VA-15 and using the well-known relations 7T 77 Scos rrT dx _Tr J~sinx~r1i- -71 implies that 7"r - 9D~ v y(ft%) Y 1b) % -" A.- 16 jy Cai 8e ,I As is usual in these wave propagation problems, the energy flux proves to be proportional to the Wronskian of n" -129- Section B: Energetics Results Having formulated the energetics of the lunar tide in the last section, some energy calculations will be presented in this section. More specifically calculations of the averaged vertical energy flux, , will be given. W At this point, it should be recalled that the lunar tidal problem is being t:'eated in a pressure coordinate system; thus, the energy flux is in units appropriate to such a coordinate system. also be noted that the following results are shown in CGS ~( A It should units -5 &gm cm sec). It has been mentioned throughout that the enhanced "mesospheric barrier" in summer is of great importance in dealing with annual changes in the lunar semidiurna' tide in the atmosphere. The monthly values of the vertical energy flux, with no Newtonian Cooling, is shown in figure VB-1. It i, evident that there is much less vertical energy flux in summer than winter. 1 This result fits nicely into the idea of there being an increased barrier to wave propagation in the summer. Since there is no dissipation in this calculation, the vertical energy flux must be constant with height which, along with assuming a steady state tidal oscillation, would 1. Irregularities such as that in January and February are probably due to arbitrary differences in the analysis of the stability function. -130- Be- imply such an annual variation in the vertical energy flux. cause this energy flux is smaller in summer, the lunar tidal wave more closely resembles a standing wave in summer than in winter, thus leading to the smaller summer phase lag in the surface pressure oscillation both in the observed and calculated tide. The effects of Newtonian Cooling on the energy flux are shown in figures VB-2 and V B-3. These figures show the vertical energy flu:: calculations for January and July with 1. no Newtonian Cooling, 2. CO 2 Newtonian Cooling, 3. CO 2 and small 0 tonian Cooling, and 4. CO 2 and large 0 3 3 NewAs Newtonian Cooling. expected, it is seen that more energy is lost when there is a greater amount of cooling present, but there are also other things to be seen. The addition of Newtonian Cooling has the effect of increasing the flux at lower levels and decreasing the flux at higher levels relative to the results with no cooling. It is also seen that approximately the same amount of energy is dissipated in July as in January giving a much greater fraction of energy lost in July. It thus becomes clear why the seasonal variation in phase lags were lost with the addition of large Newtonian Cooling. The proposed explanation for the annual variation in the phase lag was the variation in trapping due to the changing "mesospheric barrier". This explanation becomes largely irrelevant if most of -131- the tidal energy is dissipated before it even reaches this "barrier". This is the case if the energy is dissipated in the ozonosphere as shown in fEgures VB-2 and V B- 3. This disappearance of the seasonal behavior does not negate the validity of the explanation of the phase lag variations in terms of the "mesospheric barrier". The annual behavior was only seen to vanish with- the addition of the 0 1 CO 2 . 3 Newtonian Cooling and not with The cooling due to ozone is suspect not only because of uncertainties in 0 3 amounts but also to even greater uncertainties in 03 photochemistry. It is even possible to view the present work as contributing some small evidence in favor of the modified 0 photochemistry as the energy losses with 0 3 3 present seem a bit excessive. 1. The use of an annual average for OkCO. mask such a seasonal variation. might also tend to Monthly values of vertical energy flux in CGS units 1009080706050403020e I J F M A M i I I I J J A S 0 N C 50 w 0' - 0. altitudeI I i km -90 JULY - i -80 o10 -70 60 50 - 00 -- - 0 C0 2 40 5- ~\ \ \ OC0 2 + 0 (smoll c3) CO2 + O(lorg e 03) 30 I 20 I I. iI ,10 I I O 10 20 03 30 I 40 40 50 50 0 60 70 70 0 80 9 90 IC0 ) K0 -135- CHAPTER VI SIMPLE STUDIES Section A: Numerical Studies In the two previous chapters, calculations of the semidiurnal lunar tide performed with the monthly stability profiles gave a seasonal variation in the tide that was correlated with changes in these stability profiles. The writer att ibutes these changes in the calculated tide to the seasonal variation in the strength of the "mesospheric barrier". This interpretation would be clearer if this "mesospheric barrier" we-s the only portion of the stability curve that changed with season. This is not the case. Although the most marked changes in the stabilit,- curve do occur in the mesosphere, there are also changes in the "Iropospheric barrier" and "stratospheric cavity". There is also a seasondi variation in the surface temperature which affects the lower boundary condition. The purpose of this section is to isolate these varying "parameters" and see what happens to the tidal calculations as these "parameters" are changed, keeping everything else constant. The complete tidal calculations will not be presented in this section, since the author believes that the number of pages required to give the complete results would not only be excessive but -136- also be confusing to the reader. For this reason, only the calculated amplitude and phase of the surface variation in geopotential height along with the vertical energy fl'x are shown. The stability curves that were used in making the mesospheric test are shown in figure VIA-1. It was the author's intention to progress smoothly from a mid-winter mesosphere (# 1) to a midsummer mesosphere (#6). The results of the mesospheric test are average s;ability curve. shown in table VIA-11. The solid curve (# 7) is the annual These results show both the phase lag and energy flux decreasing markedly with an increasing "mesospheric barrier" while the amplitude of the surface pressure oscillation is seen to increase slightly. The curves for the tropospheric test are shown in figure VIA-2 wiih the corresponding results il table VIA-2. curve is again the annual average. The solid The amplitude increases while the phase lag is seen to decrease with increasing "tropospheric barrier" strength. The energy flux first decreases slightly and then increases slightly with increasing barrier strength. The stratospheric test was carried out using the curves shown in figure VIA-3 with the results in table VIA-3. These calcula- 1. Throughout this section parantheses, ( ), will denote results for the annual average curve and brackets, [ ], will denote results for the annual curve with a winter mesosphere. LII_ _.._ ~I _~_~ I1 .~.*.~-~I-IIILLI---1Xr-- -137- tions show the amplitude decreasing with increasing "stratospheric cavity" through curve (5) with the amplitude increasing by an order of magnitv-de for curves (6) and (7). The phase lag shows some- what irregular but small variations through curve (5) with large variations in curves (6) and (7). The energy flux increases with increasing "stratospheric cavity" with very large increases for curves (6) and (7). It should be kept in mind when looking at these results that stratospheres (6) and (7) appear to be quite unrealistic when compared with observations. Taking the annual stability curve, calculations were performed for various values of the surface temperature. this experiment are shown in table VIA-4. The results of Both the amplitude and energy flux are seen to decrease slightly with increasing surface temperature while the phase lag remains constant. Referring back to figure VI A-1 one can see that the annual stability curve is closer to the summer situation than to the winter, in the sense of there being a fairly strong "mesospheric barrier". It is of interest to ask whether with a lesser "mesospheric barrier" similar changes in the non-mesospheric portion of the stability curve might give different variations in the results. For this reason the same tropospheric curves, stratospheric curves, and changes in surface temperature as before were stu- -138- died with a winter mesosphere. These tropospheric and stratospheric tests with winter mesospheres a-e shown in figures VIA-4 an VIA-5. The corresponding results are shown in tables VI A.-5 and VI A-6 with the results of the surface temperature test being shown in table VIA-7. The results with winter mesospheres are qualitatively the same as the corresponding results with the annual mesospheres. As would be sugges'ed by the mesospheric test, the amplitudes of the surface oscillation in geopotential height are less with the winter mesosphere while the phase lags and energy fluxes are greater. In all these tests the variations of amplitude, phase lag, and energy flux, while being qualitatively the same for the winter and annual mesospheres, are roughly in proportion to tl eir magnitudes. In summary, these simple numerical experiments have shown that the seasonal changes in strength of the "mesospheric barrier" dominate the seasonal variations of the lunar tide. This is con- cluded since the observed changes in the mesospheric portion of the stability curve cause greater variations in the results than would the observed variations in the troposphere, stratosphere, or in surface temperature. Finally it will be remembered that a standard thermosphere has been assumed for all months. This standard thermosphere -139- has been used in formulating the "radiation condition". In the following numerical experiment the slope of the thermosphere is varied and results are presented for these thermospheres with an annual mesosphere and a winter mesosphere. The curves and results for this thermospheric test are shown in figures VIA-6 and VIA.-7 and tables VIA.-8 and VIA-9. It is seen that in the case of the annual mesosphere the amplitude o0: the surface oscillation in geopotential height is rather constant Nhile the phase lag and energy flux decrease for an increasing slope of the thermosphere. In the case of the winter mesosphere the amplitude is seen to decrease substantially while the phase lag and energy flux vary a bit irregularly by comparison. While the variation in the slope of the thermosphere causes some variation in the results, the assumption of a standard thermosphere for all seasons should not affect the qualitative results in this study of the atmospheric lunar tide. Section B: Analytic Studies subsection a: Isothermal calculation Analytical solutions to the lunar tidal problem may be obtained for certain model atmospheres. The simplest of these is the isothermal atmosphere, since it is specified by only one parameter, the temperature. It is quite worthwhile to see this -140- isothermal solution for the lunar tide because, in this case, the details of calculation do not pose such a barrier to understanding as in the general case. There are two distinct solutions to be considered for the lunar tide in an isothermal atmosphere. The first of these is the situation in which energy is propagating upwards, and the second is where energy is being prevented from leaving the system (trapping). Due to its greater plysical significance, the first situation will be treated first. Since the atmosphere is isothermal, the stability function is given by a constant x ga Dn-o The veitical tidal equation is = Yn (., + ..... (O 0 VI Ba- 1 ) O VIBa-2 d " If " + K ( K' O for the propagating case), the solution to equation VI Ba-2 may be written Y where A - Ae' and 8 VIBa-3 are complex constants. The "radiation con- 1. In the remainder of this and the next section the super and will be dropped. subscripts () -141- dition" requires that the phase velocity of the wave be directed downwards, since in a gravity wave the group velocity (velocity of energy propagation) is directed opposite to the phase velocity; hence, Y AeK The constant A VIBa- 4 may be obtained by using the lower boundary condition Y-- D d YD D VI Ba- 5 giving M + . K + 2(. VIBa-6 D thus, for the propagating case P Kr- 90 (e For '.he trapped case ( VI Ba- 7 < O ) the solution to equation VI Ba-2 is Y= A e + . VI Ba-8 Instead of the "radiation condition", it is required in this case that the vertical energy flux remain finite at OO so that the so- -142- lution becomes Y VI Ba- 9 . K Putting equation VI Ba-9 into the lower boundary condition VIBa-5 gives the solution in the trapped case as VIBa-10 According to the form of equatioi VI Ba-7, it is easily seen that resonances are not possible in the propagating case since the denominator is always positive. In the trapped case, equation VIBa-10, the denominator becomes zero when K ; or, solving for 14, when H-7 (assuming - )= -7 in the case of a predominantly diatomic gas such as air). as This is as it 3hould be, since resonances in forced oscillations should only occur when the energy being put into the system, through the forcing, is trapped in the system. In previous sections, it was seen that a phase lag of the surface pressure oscillation was a feature of both the observations and the calculated results. It is of interest to compare the surface pressure oscillation in the propagating isothermal case with that in the trapped case. cillation is given by The geopotential height os- -143- C4. ( 7e ~H~t1 Y, -r -iv~ co cas r- .9 C t% 3 da °tAyr VIBa- 11 but may be rewritten as VI Ba- 12 awhere where M (MOO dY4 3 D~ I n coSct) VIBa- 13 and d: dc-N VI Ba- 14 = d ON.0A IM M I'tY inls Since If I + VIBa- 15 it is easily seen that the propagating and trapped regimes are given by the following inequalities: ul 47 7 /implies propagating VI Ba- 16 trapped It is easily verified from equations VI Ba-6 and VIBa-10 that the decomposition of 1. Y) into real and imaginary parts for the In the remainder of this section it is assumed that 7 -144- two regimes are given by Trapped Propagating Yrl.) _ Yr 1 _ -Co I_ " YAP K+() VI Ba- 17 6ito CS '-O K+t ~ 0 Ie~-f~ K' K9p 2- Dy -0 d:Z IZ -oD d I Putting VIBa-17 into equations VIBa-13 and VIBa-14 gives the following: Propagatin g YK :O PO ~t> 1"rapped N) 10 IIOB K%( (ID;P) i~ XA~wa qCo$M eDrl)I/IBI 1<2+ R 2. VIBa- 18 Lfl B D12 - d ,0 cos4j( 05d- 4-C <o (;v 70 (iC -i7,E 'Ir) ci7) Thus, it is shown that there is always a phase lag (with -<U4CO in the propagating case with an isothermal atmosphere. In the trapped case the surface pressure oscillation is always exactly in phase or out of phase with the forcing (00 or 1800 phase lag) ) -145- depending on whether the scale height (temperature) is greater or less than the resonance value. This isothermal calculation does not prove, for an arbitrary atmospheric temperature structure, that a phase lag is necessary if the wave energy is being propagated to infinity. It is quite sug- gestive of this result, nevertheless. S subsection b: -FCN barrier It was seen in the last section that a calculation of the lunar tide using an isothermal atmosphere (propagating case) reproduced some important observed features -- most notably, the phase lag of the surface pressure oscillation. In this section, a simple model atmosphere will be utilized to investigate how changes in atmospheric "larriers" affect the lunar semidiurnal tide. This model atmosphere is shown in figure VIBb-1. It is seen that this model consists of an isothermal atmosphere with a -function "barrier". With this model, it is possible to look into the effects of a strength, a0 , S- function "barrier" varying in both and location, 0 In the case of this special atmosphere, the vertical tidal equation becomes -146- d7 VI Bb- 1 The boundary conditions are as follows: (1) at z =O (2) at : 00 dZ A , no energy is propagating downwards. 2 The :;olution to equation VIBb-1 so)long as 210 Y- Ae"KI' + B fKwhere ' ( A and 8 is K,.I:-', are arbitrary complex constants). With boundary condition (2), the situation is pictured in figure VIBb-2. ITI is the magnitude of the wave transmitted through t'he "barrier" while IRI is the magnitude of the reflec- ted wave. The three constants A , R , and T may be obtained by the following three conditions: (1) boundary condition (1) (2) Y (3) the "jump condition" on d iscontinuous across VO across obtained by integrating equation VI Bb-1 across These conditions are respectively: , which is o -147- R -((L H)(A+ R) aZ iKA- VI Bb- 2 t + Wo AeZAe R T el 'o O VI Bb- 3 and .KO TZ K,e - .AelK"o- .,,Ri ' -4a o -,SeaI VI Bb-4 Equa.tions VIBb-2, VIBb-3, and VIBb-4, when used to solve for A , R , and T I give 4e -Z -L J 1T1'T .K, ,R r= ' 7Al~ 1<reU' Bb-5 A: -.-JiJ(g [I - (a ,ar o .I It is seen from VIBb-5 that when ~ while if ISo = .o +[I =, O , T= O ,ITI=IAl and I/ and IIAI This is as it should be, since there should be perfect transmission with no "barrier" and no transmission with an infinite "barrier". There exists a resonance when 4.4 (a ?4 - fSTo I =(&i 4 2+gto LCI Klc- Yfd46 VI Bb- 6 -148- or alternati-vely when - . Bb- 7 .VI Taking the absolute value of equation VI Bb- 7 implies that which can only be true if 4o' SSI ; otherwis a, I . As expected then, resonances can only I I occur with an infinite barrier. Even with - o , it is seen, going back to equation VI Bb-7, that resonances can only take place when In o:-der to learn about partial resonances, ,'o the solution to the vertical tidal equation for various values of o and Zo were found. S -function More specifically a "barrier" was moved from 2 =0 to : = 14 and solutions for the amplitudes and phases of the surface pressure oscillation were found. Figure VIBb-3 shows the amplification, A , of the surface pressure tide over that of an isothermal atmosphere with I = 7. 07034 ( = .025 ) as a function of Za for -149- .010, = .005, O 0 .040, .0801. Figure VIBb-4 shows , of the surface pressure oscillation as a func- the phase lag, tionof .020, = .000, for .050, .010, .020, .040, .080. In both these figures there are seen partial resonances; that is to say, regions where there is both a rapid rise in amplitude and a rapid change in phase lag. These partial resonances become much larger for larger values of resonanc regions (from about amplitude with ;0 . Between these 7o = 2 to Z4 = 9) the change of is rather irregular and small, while the phase lag seems to decrease monotonically with . This behavior is quite consistent with both the observed and calculated surface pressure oscillation. 1. For reference, the area of the "mesospheric barrier" varies from about 0. 005 in mid-winter to 0. 0275 in mid-summer. -150- 15 10 5 O .01 .02 S Figure VIA-1 p .03 -151- TABLE VI A.- . MESOSPHERIC TEST 4(CG) Profile C S) [ 1] [44] [-26 ° ] [53] 2 47 -120 26 3 51 - 50 13 4 54 - 30 7.8 5 56 - 20 5. 3 6 58 - 1o 3.6 (7) (52) (- 20) ( 5.7) -152- 15 10 5 Figure VIA-2 -153- TABLE VIA-? TROPOSPHERIC TEST (ANNUAL MESOSPHERE) Profile -w S(c i) (CGS) - 5c 5. 5 5. 3 t3 - 6 (7) 5. 2 - 20 5. 2 20 5. 3 - 52 30 5.2 (- 2° ) (5.7) -154- Stratospheric test 2 1/4 = 0 .01 .0 2 Figure VIA.- 3 S -> - - w .03 -ft ft -155- TABLE VIA-3 STRATOSPHERIC TEST (ANNUAL MESOSPHERE) Profile (CGS) ( 1 82 - 30 1.1 2 76 - 50 1.7 3 68 - 10 2.7 (4) (52) (- 20) ( 5.7) 5 29 - 80 6 120 -1630 100 7 200 - 630 490 11 -156- TABLE VIA-4 SURFACE TEMPERATURE TEST (ANNUAL MESOSPHERE) T0 (oK) h (c w) o (CGS) 2770K 54 - 20 6.4 280 0 K 53 -20 6.1 2840K 53 -2o 5.9 (287°K) (52) (-20) (5. 7) 2900K 52 -20 5.4 2940K 52 -20 5.2 2970K 51 -20 5.0 1_L__ ___I1Ll_^__l______i___~li- I y-~l_ _Y~I^I~I~_~_ -157- Tropospheric test (winter mesosphere) 22 S = I/4 .01 .02 Figure VIA-4 -- 0 .03 -158- TABLE VIA--5 TROPOSPHERIC TEST (WINTER MESOSPHERE) Profile (c 1 ) o (CGS) 23 -580 54 2 25 -490 52 3 27 -420 50 4 30 -360 50 5 35 -300 49 6 39 -.270 50 [7] [44] --260 53 -159- Stratospheric test (winter mesosphere) y 2 1-/4 15 10 5 00' pop 00 .01 .02 Figure VI A- 5 W,*1 .10-0-1 .03 o -160- TABLE VI A-6 STRATOSPHERIC TEST (WINTER MESOSPHERE) Profile [4] (CGS) 79 - 20 72 - 40 62 - 70 [44] [-20] [53 40 -800 110 280 -620 680 220 -17 0 1880 7. 3 ] II-Nlil-_ -~L~^_I-L--i_._l ... I._I_ -161- TABLE VIA-7 SURFACE TEMPERATURE TEST (WINTER MESOSPHERE) T (°K) (c rn) a (CGS) 277 -270 58 280 -27, 56 -260 55 284 44 [287] [44] [-26 ] [53] 290 44 -250 51 294 44 -250 50 297 44 -250 48 -162- .01 .0 2 Figure VI A- 6 -163- TABLE VI A- 8 THERMOSPHERIC TEST (ANNUAL MESOSPHERE) Profile (c 4o (CGS) 1 53 -4 11 2 53 -40 11 3 52 -40 10 4 52 - 30 8.3 5 52 - 3o 8.6 ( 6) (52) (-20) ( 5.7) 7 53 -20 4.6 -164- I Thermospheric test (winter mesosphere) I 11 I j I I (I) (2) )22 .01 .02 Figure VIA-7 = 1/4 I I / ' (3) -165- TABLE VIA-9 THERMOSPHERIC TEST (WINTER MESOSPHERE) Profile cC M) O (CGS) 1 57 -200 55 2 57 -210 58 3 56 -230 60 4 51 -24o 59 5 53 -260 65 [6] [44] [-260] [53] 7 40 -230 43 -166- I I I I I II I I " I I I i'~$-a(-o I5 Figure VI Bb- I " -function barrier" model atmosphere -167- i S:S I4 I I Teiki s'-I + S- 4W zo -I I I ReAi, Ae ik Figure VIBb-2 oa8(tz-zo ) -168- I 1 I 4~1 I I I I I So = .000 So = .005 S o = .010 So 3J i I = I .020 So S .040 So = .080 Ir\ .1r I,~d~ %h. N5 / .,,-. 10 5 Figure VI Bb- 3 -169- -1500 S\ 50 = .000i . . I I -1200 -90 I I I 50o =,020 I So = .04C) I I 5 = o080 *I I I I I I I 'I I 'I j-I.. 'I -60 Ii *1 •II .. _- = "V " O I -- -- '...........I \ ,." -. . --- T - " ... I -- I---- --.-- .-. '...//..,. , . . " . rI - ~~'10000 00 . zo Figure VI Bb-4 10 14 -170- CHAPTER VII CONCLUSION Section A.: Discussion of Results In the previous sections a simple model of the lunar semidiurnal tide in the atmosphere was presented. It was thus possible to account, quite well, for the observed features of the annually averaged Lunar tide. Moreover, it gave results that resemble the observed annual variations in the surface pressure tide. feature of this model is its simplicity. The best This simplicity allows one to look more easily through the mathematics into the basic physics of the lunar tide. In pai'ticular, the simplicity of the model allows for a very straightforward explanation of the annual changes in the surface pressure oscillation. As stated previously the monthly stability profiles were analyzed at the beginning of this study, before any calculations of the lunar tide were attempted. The calculated annual variation in amplitude and phase of the surface pressure oscillation (with no Newtonian Cooling) are thus assumed to be free of any bias resulting from the author's analysis of the stability profiles. In fact, the annual variation of phase lag is shown to be a consequence of the annual variation in the "mesospheric barrier". -171- This point is demonstrated by the monthly calculations, simple numerical calculations, and simple analytical work. This 4ffect is also easily explained physically. An increase in the "mesospheric barrier" will tend to make the tide assume the character of a standing wave in the vertical (zero vertical energy flux). A standing wave will be in phase with the forcing and thus will have a zero phase lag. This annual variation from a propaga, ing tidal structure in the wint er months to a standing tidal structure in the summer months, is graphically demonstrated in the annual variation of the vertical energy flux. The addition of Newtonian Cooling effects upsets not only the annual variation in the tidal oscillation of surface pressure, but also the previous line of reasoning. Th calculation of vertical energy flux with Newtonian Cooling sho wed that a substantial amount of tidal energy was being lost before it reached the "mesospheric barrier". For this reason, the changing characteristics of the tidal structure with variations in the "mesospheric barrier" become largely irrelevant. The greater the proportion of energy lost, the more irrelevant this line of reasoning becomes. The Newtonian Cooling results should not be taken as being wholly negative, however. The treatment of Newtonian Cooling in this study is to be questioned since a Newtonian Cooling coefficient that is based on the annually averaged state of the atmosphere -172- is used in the monthly calculations. This might have the effect of smearing out month-to-month variations. The uncertainty in the value of the 0 3 Newtonian Cooling coefficient is still more significant. The value of the Newtonian Cooling coefficient depends cru- cially on the assumed 03 photochemistry, and this photochemistry is still undetermined. It was mentioned earlier that calculations in this study suggest that the classical 03 photochemistry is not correct. Perhaps the weight of evidence from similar theoretical calculations witn 0 3 Newtonian Cooling will serve to settle the question of 0 3 photochemistry to some extent. Just as the simplicity of this theory of the lunar tide leads to manyadvantages, it also is responsible for some shortcomings. To derive the tidal equations the mean state of the atmosphere was assumed t) be free of any mean wind ar d horizontal temperature gradients. The addition of these effects to the present theory is not an easy task, since the added terms in the "primitive equations" would prevent the separation of variables. By ignoring these effects, the author does not mean to imply that they are unimportant. It is his view, however, that when trying to model an observed phenomenon one must include physical effects in an orderly fashion so that the influence of each physical effect can be seen. ______IIIJ__L~LI ____LI -173- Some comment on the neglect of horizontal temperature gradients in the mean state-is in order, however. In the present in- vestigatio n only Northern Hemisphere Jata was used, since only Northern Hemisphere data was available. This gave rise to a seasonal variation in the surface pressure oscillation; whereas, an annual variation is observed. The present theory would yield an annual variation if the behavior of tie temperature in the Southern Hemisphere is other than exactly equal and opposite to the behavior in the Northern Hemisphere. This would lead to an annual variation in the globally-averaged mesospheric temperature which would then lead to an annual variation in the lunar tide, as is observed. In fact, the present theory might be taken to suggest such a hemispheric asymmetry in mesospheric temperature behavior, Section B. Suggestions For Further Research There is much to be done in atmospheric tidal theory of both an observational and theoretical nature. The author intends to investigate the solar tides in the spirit of the present research, that is to say, he will try to model the observations of the solar tides using empirical parameters as inputs into a simple theory. -174- More upper atmospheric observations in the Southern Hemisphere are required for tidal calculations. These observations are needed for all studies of the upper atmosphere, of course, but are especially needed for studies of global scale phenomena such as tides. Theoretical studies of tides including effects such as winds and horizontal temperature gradients should be attempted. These s:udies will, no doubt, have to be carried out numerically due to their great mathematical complexity. One such study has already been carried out by Hunt and lVanabe (1968). Of course, there is also room for many more observational studies of the lunar and solar tides. It would be very good to have more information on the yearly i ariations and vertical structure of these tides. Finally, a very fertile field for siudy is the interaction of tidal wind fields with the electromagn tic fields in the ionosphere. Such studies would not only -.dvance the theoretical work on tides, but also would help in interpreting data on oscillations in ionospheric parameters. -175- Acknowledgements I wish to thank Prof. Victor P. Starr for being, what I consider, the ideal thesis advisor. His advice and encouragement were invaluable to me. The basic idea for this research had its beginning in many discussions with my colleague and friend Dr. Robert E. Dickinson. I also app:-eciated having the opportunity to engage in many helpful discussions with Professors Norman Phillips and Reginald Newell. Most of the computational work was performed at the M. I. T. Computation Center with other parts being carried out by Mr. Howard Frazier al Travelers Research Center in Hartford, Connecticut. The manuscript was typed by Mrs. Cynthia Webster while the figur -s were drafted by Miss Isabelle Kole and photographed by Mr. John Cook. Financial support during my tenure as a graduate student at M. I. T. came from NASA, in the form of a NASA traineeship, and from the National Science Foundation, Grant (GA - 1310 X). -176- References Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1965, 1046pp. Appleton, E. V. and K. Weekes, 1939: "On lunar tides in the upper atmosphere," Proc. Roy. Soc. (A), 171, p. 171-187. Butler, S. J. and K. A.. Small 1963: "The excitation of atmospheric oscillations, " Proc. Roy. Soc. (A), 274, p. 91-121. Chapr nan, S., 1951: "Atmospheric tides and oscillations," Compendium of Meteorology, Waverly Press Inc., p. 510-530. Chapman, S. and K. C. Westfold, 1956: "A comparison of the annual mean solar and lunar atmospheric tides in barometric pressure, as regards their worldwide distribution of amplitude and phase'" J. Atmos. Terr. Phys., 8, p. 1-23. CIRA 1965 (COSPAR Intern. Reference Atmosphere), NorthHolland Publishing Comp., 313pp. Dickinson, R.E., 1966: "Propagators of atmospheric motions, " Sci. Rept. No. 18, Planetary Cir ulations Project, Dept. of l eteor., M.I. T., 243 pp. Dickinson, R. E. and M.A. Geller, 1963: "A generalization of 'Tidal theory with Newtonian Cooling', " J. Atmos. Sci., 25, p. 932-933. Flattery, T. W., 1967: "Hough functions," Tech. Rept. 21, Dept. of G oph. Sci., U. of Chicago, 175pp. Hildebrand, F. B., Introduction to Numerical Analysis, McGrawHill, 1956, 511pp. Hunt, B. G., 1966a: "The need for a modified photochemical theory of the ozonosphere, " J. Atmos. Sci., 23, p. 88-95. "Photochemistry of ozone in a moist atmosphere," J. G. R. , 71, p. 1385-1398. Hunt, B. G. , 1966b: -177- Hunt, B. G. and S. Manabe, 1968: "An investigation of thermal tidal oscillations in the earth's atmosphere using a general circulation model, " Mon. Wea. Rev., 96, p. 753-766. IBM System 360 Scientific Subroutine Package, (C360A-CM-03X) Version II, Programmers Manual, 329pp. Kato, S. and S. Matsushita, 1968: "Tidal theory in the ionosphere, " Trans. A. G. U., 49, p. 145. Kertz, W., 1957: "Atmospharische Gezeiten, " in Handbuch der Physik, (S. Flugge ed. ), 48, p. 928-981. Lamb, H., Hydrodynamics, Dover, 1945, 738pp. Laplace, P. S. (later Marquis de LaPlace), Mecanique Celeste, 2(iv), 1799, p. 294-298. Leovy, C., 1964: "Simple models of thermally driven mesospheric circulation," J. Atmos. Sci., 21, p. 327-341. Leovy, C., 1967: "Energetics of the middle atmosphere, " The RAND Corporation, P-3679, 30 pp. Lindzen, R. S., 1967: "Thermally driven diurnal tide in the atrrosphere," Quar. J. Roy. Met. Soc., 93, p. 18-42. Lindzen, R. S., 1968: "The application of classical atmospheric tidal theory, " Proc. Roy. Soc. (A), 303, p. 299-316. Lindzen, R. S. and R. Goody, 1965: "Radiative and photochemical processes in mesospheric dynamics: Part I, Models for radiative and photochemical processes, " J. Atmos. Sci., 22, p. 341-348. Lindzen, R. S. and D. McKenzie, 1967: "Tidal theory with Newtonian Cooling, " Pure Appl. Geophys., 66, p. 90-96. Matsushita, S., 1967: "Lunar tides in the ionosphere, " In Handbuch der Physik (S. Flugge, ed. ),49/2, p. 547-602. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, Inc., 1953, 1978pp. -178- Newton, I., Philosophiae Naturalis Principia Mathematica, London, 1687, Bk. 1, Prop. 66, Cor. 19, 20; Bk. 3, Prop. 24, 36, 37. Rayleigh, Lord (Strutt, J.W. ), 1890: "On the Vibrations of an Atmosphere, " Phil. Mag., 5th Ser., 29, p. 173-180. Rodger s, C. D., 1967: "The useof emissivity in atmospheric radiation calculations, " Quart. J. Roy. Met. Soc., 93, p. 43-54. Rodgers, C. D., and C. D. Walshaw, 1966: "The computation of Quart. infrared cooling rate in planetary atmospheres, J. Roy. Met. Soc., 92, p. 67- 9,. Sawada, R., 1954: "The atmospheric lunar Iides, " Meteor. Papers, New York Univ., 2, no. 3, 31 pp. Sawada, R. , 1956: "The atmospheric lunar tLdes and the temperature profile in the upper atmosphere, " Geophys. Mag., 27, p. 213-236. Sawada, R., 1965: "The possible effect of oceans on the atmospheric lunar tide, " J. Atmos. Sci., 22, p. 636-643. Siebert, M., 1961: "The atmospheric tides, physics, 7, p. 105-187. Advances in Geo- Thompson, P. D., Numerical Weather Analysis and Prediction, The MacMillan Company, 1961, 170 pp. (later Lord Kelvin), 1882:' "On the thermodynamic acceleration of the Earth's r2otation, " Proc. Roy. Soc. Edinb., 11, p. 396-405. Thompson, W., Wilkes, M. V., Oscillations of the Earth's Atmosphere, Cambridge, University Press, 1949, 74 pp. -179- Biography Marvin Alan Geller was born on March 19, 1943 to Mr. James Geller and Mrs. Saide Geller in Boston, Massachusetts. During his childhood years his family lived in Brockton, Massachusetts. He graduated from Brockton High School in 1960, re- ceiving the Rensselaer Medal, and subsequently enrolled at the Massachu:-etts Institute of Technology ireceiving his Bachelor of Science degree in Mathematics in June, 1964. He enrolled at M. I. T. in meteorology in September, 1964. While in graduate school at M. I. T. he married the former Lynda Louise Grafinger. A daughter, Stephanie Lynn, was born from this union, M '. Geller was supported duriitg his first and last years as a gradluate student by a research assistantship under Prof. V. P. Starr ind during the middle three years was a NASA trainee. M2. Geller was elected to full membership in the Society of Sigma Xi in 1968. His publications to date are as follows: (1) "The effect of seasonal changes in atmospheric temperature structure on the lunar tide, " Trans. A. G. U., vol. 49, no. 1, March 1968, p. 179 (abstract of talk delivered at 1968 A. G. U. meeting in Washington, D. C.). (2) "The effect of changes in atmospheric temperature -180- structure on the lunar tide, " Bull. A. M. S., vol. 49, no. 8, Aug. 1968, p. 847 (abstract of talk delivered at 1968 Conference on Composition and Dynamics of the Upper Atmosphere in El Paso, Texas). (3) "A. generalization of 'Tidal theory with Newtonian Cooling'," J. Atmos. Sci., vol. 25, no. 5, Sept. 1968, pp. 932933 (with R. E. Dickinson).

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