THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aerospace Engineering High Altitude Engineering Laboratory Scientific Report EFFEC'T OF' MOTIO.N ON THE ALTITUDE DISTRIBUTION OF ATMOSPHERIC DENSITY'. S.:...; Qh.; "'' ". "'. ~', ORA Project 05627 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CON-TRACT NO. NASr-54(05) WASHINGTON, D. C administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR March 1967

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ABSTRACT The effect of motion on the attitude distribution of atmospheric density has been determined It has been shown, in particular, that for vertical waves moving with increasing velocity along its direction of propagation and form V < 1 t form v < i the density is given by kT = 0o exp- f| + 1 at T + 1 v3 } dz kT T Tz kT \kT T z/ where v is the velocity of the waves, Other terms have the usual significances. The three terms on the right-hand side correspond to density variation due to gravity, temperature, and the motion in the atmosphere respectively. The relative magnitudes of the three terms at 100 km altitude has'been obtained and it has been found that for v = 22 m/sec the variation of density due to the above type of motion is one-tenth of that due to temperature. The vertical components of the wind (Edwards, Ho Do, et al, JGR 68, 3021 (1963), and the dominant gravity waves (Midgley, J. Eo and Ho B. LiemohnJGR 71, 3729 (1966)), are 6 m/sec and 1 m/sec, respectively, and are small to affect the altitude distribution of atmospheric density at 100 kmo iii

The distortions of trails of long-enduring meteors (Liller and Whipple, 1954; Greenhow and Neufeld, 1959), and chemiluminescent vapor released from rockets (Rosenberg and Edwards, 1964; Kochanski, 1964), and barometric oscillations of atmospheric pressure are certain illustrations of motions in the atmosphere. Winds blow through the atmosphere with speeds which may approach 100 m sec-1 or more and having wind shears of about 0.04 sec1. Also, the atmosphere is subjected to tides having 24-hour and 12-hour periods of which the semidiurnal component is stronger. In this note the effect of motion on the altitude distribution of atmospheric density is considered. For a motion through the atmosphere, the momentum equation and the equation of continuity should hold. Neglecting the effects of viscosity and the rotation of the earth, the former can be written as Dv + -P Pg + VP Dt or ( (^v -~ = + P t + pvv. pg + VP where p = atmospheric density p = atmospheric pressure v = velocity In the steady state -p(v-)v P= g +VP -P.x x + vy + v z (ivx+jvy+kvz) pkg + (i + J + k -) (1) \ x oy oz Therefore for a constant motion the density distribution, as expected, is not affected. The equation of continuity, given by 1

- +.p v = 0 at becomes in the steady state pv = 0, or (pvx) + - (pvy) + (pv = O. (2) ^x (yy y z Broadly speaking, the effect of motion on the atmospheric density profile can be obtained by considering four specific cases. Case 1. Horizontal waves along x or y direction with increasing velocity along their direction of motion, for example vx = v O vy = vz = 0; vx - O, other derivatives are zero. ax Applying the continuity and momentum equations we have (pvx) = or p v +v- 0 aaxx ax and ~ 6v - tp X c+ p + ap ~-p l -a pkg t + + 1 J + o-iPV = x y 6z Therefore -pva- = _r or - a (3) ax ax ax ax p = 0 (4) by ap -pg (5) az Therefore in addition to the hydrostatic equation (5), Eqs. (3) and (4) are obtained. Equation (4) shows that the pressure or density along the y-direction remains constant. To obtain the density profile let us consider the equation of state 2

p = kpT (6) m where T = atmospheric temperature m = mean molecular mass k = Boltzmann's constant Equation (3) then becomes x m x (~4~p k T-~- P - kT)a = k p aT m ax m ax k aT dap m 6ax P v2 kT P = exp - I T dx _V2 + _ m 1 _T p exp - J T 6x dx Fk- T ~~~~~~~kT For m v2 << 1 P = p exp - T I + 2) x (7) 0 T 6x kT which gives the density profile along the direction of x. Combining Eqs. (5) and (6), we obtain the altitude distribution of density, namely p pexp - jT - ) dz 3+

Case 2. Horizontal waves moving along x or y direction with increasing velocity along z direction, that is v = vx Vy = vz = 0; VX f O, other derivatives are zero. An example of this case is z the wind in the mesosphere. Applying the continuity equation (2), we have a (pv) = 0 ax p av ^ = 0 - +v -- = o ax ax Since v/O, ap/xx = 0 that is, p remains unaltered along the direction of motion. Again, Eq. (1) becomes 0 pkg + i + J Y \ +x 6y kz Hence 0- O... = o (8) ax by and pg + = 0 (9) Therefore, in addition to the hydrostatic equation, these equations show that the pressure or density along the x or y directions remains constant. Applying the equation of state (6), we get in this case the usual equation of altitude distribution of density, namely P = exp - f g + 1 - T dz (10) O kT T 6k Case V. Vertical waves along z direction with velocity increasing along x direction that is v = vy = 0 and v = vz 0, 6vz/6x / 0 o: pkg+ -.-j +( k ) ax by az

_ = O0, - 0 0^Z E = 0 6x ay pg + 6p = P _g az az Again from the continuity equation a (pv) - 0 az p aV Q P= 0 az az Since av/6z = 0 and v / O, we have ap/az = 0. From the equation of state (6) ap kT + k p T az m az m az Since aP/6z = O. p = k pT k T g.az m z m aT mmg az k 28.3x1.67x10-24x949.2 1.4x10-16 33~K/km at 100 km Therefore, in order that the pressure may satisfy the hydrostatic equation while keeping the altitude distribution of density constant, a very high temperature gradient is requiredo As such a high gradient is not present in the atmosphere (33~K/km at 100 km) this case is of little importance. Case 4- Vertical waves along z-direction which increase as they move, that is vx = vy = 0, vz = v, avz/z / 0, other derivatives are zero. In this case the equations of continuity and momentum become a (p) = or pv+ vo 0 (II) az az az

and -pkv = pkg + i + + (12) az ax ay az Hence =O- 0, p = 0 (13) ax ay and -pv = P (14) Equation (14) reduces to the hydrostatic equation if v or its derivative in the vertical direction is zero,, Combining Eqs (11) and (14-), we obtain V2 =2 C _g + p p az az g / k aT) kT _ ~m z/ m m z ~ - (v2 + k+T) +-p (g+k ), m/^Z\ m zz/ k ST o k P -v2 + k T m kT T. dz m - v2 kT For _- v2 << 1 p. kT T az kT,kT T /z Integrating

p = p exp - + + m +m__ 4 +T m a- v2f dz (15) o kT T 6z kT T Tz, If v = 0 p = poexp - f + 1 aT dz P o' kT T / Again if av/6z = 0, Eq. (14) becomes 0 = pg + = p (g + k T k 6z m az m az or p = pexp - f + dz 0 \ kT T 3z/ Therefore for both conditions av/6z = 0 and v = 0, the altitude variation of p remains unaffected. In Eq. (15), the first term under the sign of integration corresponds to the density variation due to gravity, the second term to that of temperature variation and last term due to motion in the atmosphere. To consider the relative magnitudes of these terms let us consider their values of 100 km where g = 949.2 cm/sec2 T = 208.1~K m = 28.3x1.6xl0-24gm aT = 2.95x10-5k/cm az we then have m = 1.6xlo-6cm1 kT 1 aT = 1.4x-7cm-1 T az 7

m mg lT ~ + )- -xv = 2.8x1015 v2 cm-1 kkT kT Tz Therefore, for v = 22 m/sec the variation of density due to the above motion will be one-tenth of that due to temperature. Information of the vertical motion in the atmosphere is meagre. From movements of vapor trails, Edward, et al., obtained that at about 100 km the vertical component of the wind velocity is 6 m/sec. Again the vertical wavelength of the dominant gravity waves is about 12 km up to about 100 km and then steadily increases with altitude. The period of these waves is about 200 minutes (Midgley, J. E. and H. B. Liemohn, JGR, 71, 3729 (1966). Therefore, the vertical velocity of the dominant gravity waves is about 1 m/sec. These velocities are too small to affect the altitude distribution of atmospheric density. To obtain the change of velocity of atmospheric particles due to heat input and conductivity, consider the energy equation given by pk ( + -vVT) -= Q + (AT) - pVv (16) (7-l)m at where Q = heat production in the atmosphere n(O)K(A)Eo(M) where n(0) is the concentration of O1K(\) absorption coefficient and Eo(X) ultraviolet energy flux \c = thermal conductivity of the atmosphere (Nicolet, 1960) = 1.8xl02Tl/2 where T is the abs temp (Nicolet, 1960) 7 = ratio of specific heat Expanding the above equation, we have pk 6aT aT aT aT ~+ v x Y+ Vy~ +V m(y-l) \t x y Vz/ + Q \c ( T+) __-2T p (^ + + 6vz) 6X2 ay2 aZ2J \x ay az Assuming that the velocity is directed in the vertical direction, we obtain at thermal equalibrium aT/at = 0, the gradient of vertical velocity given by 8

pkv' aT 62T av' _ ~ = Q + Tc P (y-l)m az az2 az or av' = 1 okv' aT 6 2Tc aT] (17) -z p (7y-1)m 6z C J To obtain the magnitude of av?/az, consider 100 km altitude where p = 3.1 dynes/cm2 T = 208.I~K p = 5.lx10-lg/cm3 m = 285 3xl67xl0-24gm n(O) = 5xl01lcm-3 k(A) - 1017cm2 Eo(A) = 1 erg cm-2sec-l Q = 5x106 g/cm3sec c = 1.8x102Tl/2 = 1.8x102x(2081)1/2 = 2.6x103 g/cm sec deg (Nicolet) 7 = 1.4 aT/kz = 2 95xlO105K/cm and a2T/z 2 2.3x10i10~K/cm2 For v' = 1 m/sec and 10 m/sec, - av'/z is 1 7xlO-6 sec1 and 3 3x10-5 sec-1 respectively and are small enough to be neglected. 9

REFERENCES Edwards, H. D., M. M. Cooksey, C. G. Justus, R. N. Fuller, D. L. Albritton and N. W. Rosenberg, JGR 68, 10, p. 3021 (1963). Greenhow, J. S. and E, L. Neufeld, Jo Atmoso Terr. Phys., 16, p. 384 (1959). Kochanski, A.., JGR 69, 17, po 3651 (1964) Liller, Wo, and F. Lo Whipple, Atmos, Terr. Phys., 1, p. 112 (1954). Midgley, J. Eo and H. B. Liemohn, JGR 71, p. 3729 (1966). Nicolet, M., Physics of the Upper Atmosphere, Academic Press, New York and London (1960)o Rosenberg, No WO, and H. D, Edwards, JGR 69, 17, p. 3651 (1964). 10