Equinox tidal heating of the upper atmosphere

EQUINOX

TIDAL HEATING OF THE UPPER ATMOSPHERE

G. V. GROVES Department of Physics and Astron~~ny. University College London, Ciower Street, London WC1 E f&T, U.K.

and JEFFREY M. FORBES Department of Physics, Boston Coflege, Chestnut Will, MA 02167, U.S.A

Abstract ---Evaluations are presented of the time-average heating at different latitudes and heights due to energy flux divergence o~theequint~x diurnal and setnidiurn~l tides calculated by Forbes (I982a. h) from 0 to 400 km. Itisfound thatdiurnal tidal lleatingma~ilnizesin thercgiorlof8Ukmandsemidiurnal ~asasharpmaximurn at 108km. ~h~rmosphericdiurnal oscillations give rise to a second region of heating that maximizes at 200 km and effectively transports energy from low to high latitudes. Giobal means are evaluated for the time-averaged vertical energy fluxes and heating rates: below 130 km. the results for the diurnal tide agree with those for the (I, I) mode alone, and for the semidiurnal tide, heating rates below 130 km arc the same as those that would he obtained without the thermospheric semidiurnal excitation. ~om~aris(~nsare madefrom~#to 170 km between tb~combincdditlrnal ~llds~midiurnal heatiIl~ratesand previously reported rates due to C.U.C.radiation, S, currents and gravity waves. Classical tidal theory treats the atmosphere as a nondissipative medium and is not therefore adequate for investigating the deposition of wave energy and the resultant heating effect. Viscosity. eddy conductivity and radiative damping were included by Lindzen and Blake (t971) in tidal calculations which showed (I, 1) amplitudes at 105 km to be about half the u~daln~ed values :and hence to imply that about 25Y;ofthe energy flux would be dissipatedabove IO5 km. The diurnal tide from the surface to 400 km has now been computed by Forbes (1987-a) using revised l-I,0 and 0, heating and including thermospheric heating and the following non-classical en‘ects : background winds, temperature, composition and their latitudinal variation, hydromagnetic coupling:, ~ewton~a~l cooling and eddy and molecular diffusion. For the (1. I) mode account is taken (uis-d-t,& the model of Lindzen and Forbes, 1983) of the generation of mesospheri~~lower thermospheric turbutonce due to cascading to smaller unstable scales of motian by for~iulating an eddy diffusion coefficient dependent on the height gradient of the tidal temperature perturbation. To obtain agreement between observed and calculated tidal arn~ljtudes~ values af the eddy di~i~sio~ coefficient were scaled upwards by a factor of 2 to have a maximum of 5 x lo2 m2 s- ’ (at 102 km). In Forbes (f982a), the (1,l) oscillation is calculated by an “equivalent” gravity

Attention was drawn by Lindzen (1967) to the upward flux of wave energy in the upper mesosphere generated by sokw diurnal atlnospheric oscillations and its possible signi~can~e to thermospheric heating. The calculated flux had a mean global value of 1.3 mW m - * almost all of which derives from the leading diurnal mode. ‘This mode propagates westwards in phase with the Sun, i.e. is a migrating mode, and is designated (1, 1, 1) or ahhreviately (I, 1). On the same ass~lm~tjolls of classical tidal theory and excitation by Ii,0 and 0, heating, Groves (1983a) showed that the (1, 1) oscillation in the upper mesosphere arises mainly from the H,Q heating and that the net effect of the O3 heating is to reduce amplitudes by roughly loam<,.Particular attention has therefore been given to the evaluation of the F-ii,0 heating, including the effect of scattering by clouds (Groves, l982a), but the accuracy of such calculations was found to be limited by that of available data for ~~*Oabsorpt~on. Va~l~esofl.34artd 2.68mW rn--‘were obtained for the upward fiux in the upper mesosphere corresponding to two different sets of published H,O absorption coefficients. 447

448

G. V. GROVES and JEFFREYM. FORBES

mode (EGM) formalism for computational economy and summed with other thermally excited diurnal oscillations to obtain the total diurnal tide. In Part I of this paper the results for the (1,l) mode alone are analyzed to obtain the corresponding deposition of energy.

and the global average of E, is 2n x is0

cos a(t, - t,).

2

IF( 1. 1 0

* @*sin

Q dO = 1.

s0 Hence by (1)

01 ;

= E,, 15.45.

2.2 The time-averaged horizontal fiuxes, E,, E,. Expressions for the W-E and SN fluxes E,, E,,

km

10-a lo-”

10-e 10-s

I@

10-r

I

102

w.(mi’)

IO-’

IO-’

IO

IO’

v,,(mi’)

IO-*

I

I02

IO'

FIG. 1. pO, w”, ~1,8 FOR THE(1, I)

10-Z

MODE AND

E, is derived from equations

(4)

Figure 1 shows the adopted equatorial tidal amplitudes POand wo. Although p. generally decreases with height the ratio ofp, to ambient pressure reaches a maximum of 10.6 ‘A at 84 km. The phases of these oscillations are nearly the same at most heights (Fig. 2) and therefore by (1) are effective in producing an upward flux of mechanical energy. The global mean energy flux E,,, calculated from (1) and (4) is shown in Fig. 1 and it is seen that r?, continuously decreases from about 70 km upwards. The value of &, at 70 km is 1.6 mW mm2 which is in satisfactory agreement with the earlier values referred to in Section 1.

(2)

7Tw,(wm-? 10-a

(3)

2

I?,,,= E,, /[ 2 0

For the (1,l) mode a latitudinal profile may be assumed that is independent of height as severe damping occurs at altitudes (5 105 km) below the level (2 150 km) where the time-scale for friction is long compared with the wave period and changes in horizontal shape would be expected to occur (Lindzen, 1970). This assumption is involved in the EGM analysis and the appropriate latitudinal profile is that of the (1,l) Hough function 0, corresponding to classical conditions. At colatitude 0 the time-averaged vertical energy flux is then

6+,dal(Wme'l IO-" p. (NmeP) IO-’

;

as

(1)

E, = .wOpO COSe-(r.&)[@(0)/@(;)l’

0

= $wopocos a@,- t,)

2. EneryyJuxes ofthe (1,l) mode 2.1 The time-averaged vertical energyflux, E,. We denote equatorial tidal components of pressure and vertical velocity by p,, cos g(t - tP)and w,, cos u(t - t,), where u = 7r/12 rad h- ‘, t is local time in h and r,, t, are the local times at which these oscillations maximize, then the time-averaged equatorial vertical flux of mechanical energy is E,, = fw,p,

E, d4 sin f3 d0

E, =4’,

I%,,,,FROMTHEANALYSISOFFORBES

1 and 4; and f&,,, = -dE,/dz

(equation

(1982a). 15).

Equinox

analogous as

tidal heating

to (2) may be written, along with that for E,, E, = &J,(B)

(a = u, u, w)

449

of the upper atmosphere

(9

It,- t,j may still depart from 6 h by an hour or so : at 90 km the departure is about 3 h. E,, therefore needs to be retained in the analysis and u18 which enters into (7) is shown in Fig. 1.

where L,

= &I,

E “00 = $,,P,

J%,,

cos a@, - t,)

= h,~,

Ku,,

(6) = tw’o

(7)

3. Tidui heating 3.1 Global meun hating. The rate of heating per unit may be obtained as the negative volume, Qtidal, divergence of the energy flux, which in spherical polar coordinates (r, II, 4) is

^

$ ~(r2E.,3 +& Y$(sin @E,) (10)

Qtidal = -O,, 0, are the W-E and S-N wind expansion functions are the amplitudes of the forthe(l,l)modeandu,,,u,, horizontal tidal wind components u, u at 18” latitude. (Any latitude may be taken for defining u, L’except 0” as L+~= 0: at 18” latitude u, v are close to their maximum values.) Under conditions of classical tidal theory, the pressure and S-N wind oscillations have phases in quadrature (Wilkes, 1949) and by (6) E,, = 0. For the adopted (1,l) oscillation, t,- 6 is plotted in Fig. 2 at heights above 60 km and it is seen that from 150 to I80 km t, - 6 is less than t, by about 4 h : the cosine term in (6) is then equal to cos (n/6), i.e. 0.866, and is significant. At lower heights the differences are less but

on noting by (5) that dE,/dc#~ = 0. On neglecting term 2E,jr, (10) becomes

Qtidal = (QtidaJw + (QtidaJv

the

(11)

where

(Qtidu= -(Wv,ldzWw(@ z being the height coordinate

(12)

and (13)

where F(U) = ;;;1-i, & [sin 0H,(@] The global

average

(14)

of Qrida, is by (3), (4), (9) and

(1 lH14) 200

km

150

(15) shows evaluating that appear arise from

100

that horizontal fluxes may be ignored when Qtidal, which is plotted in Fig. 1. The wiggles in the Qtida, curve in Fig. 1 are considered to the numerical processing.

3.2 Latitudinal projile ofheafing. We see from (I 1) to (14) that thelatitudinal profile of tidal heating due to the vertical energy flux depends on H,(O) and that due to the S-N flux on F(0). Figure 3 shows plots of H,(N) and F(O), both of which become decreasingly small at latitudes greater than about 36”. F(0) has been evaluated by expressing it in terms of O,, 0, and 0 using the relations

50

0

4

8

12

16

20

HOURS

FIG. 2. PHASESt,,,t,,.,t,,- 6 (h) FOKTHE(1,l) MODEFROMFOKHES

Key: ~---t,;---

(1982a). 1,; ~. ~.

[U-P6 (above 60 km).

450

G.

IO 0.5

R_

4

H,(e)

0

XJ

60

V. GKOVESand F(S)

0 -4 SV

~ 0

30

900-e

60

w

900-e

(equation 9) AND F(U) (equation FOR THE (I, 1) MODE.

FK;. 3. FUNCTIONSH,(O)

and Laplace’s tidal equation, ‘LJ=coso

where

,f=a/2w

(ZJf,

~~beingtheEarth’srotatio~rateand,forthe(l, s = I. We obtain after some reduction

where h is the mode equivalent

19)

depth and

h, = 4u2w2/g

(18) l)mode,

JEFFKEY

M.

FORBES

(Qtidal)J~ to give the continuous curve Qrida,/~. Above 110 km Qtida, is less than (QIidaJw corresponding to the region above 110 km in Fig. 2 where t, - 6 is less than t, At these heights the vertical energy flux decreases with height partly due to dissipation and partly due to a horizontal outflow away from the Equator. The same situation is found in Figs. 2 and 4 close to 90 km where t,>- 6 is again less than I,, and Qfida, is much less than (Qtid.&,. The opposite correspondences are found in Figs. 2 and 4 close to 80 km and again close to 105 km. (In plotting Fig. 4 various wiggles which are considered to be ofnumerica~ origin, like those apparent in Qtid., in Fig. 1, have been smoothed out to obtain a clearer presentation of the main differences between (Qtidu,),,,/~ and (QtidaJlP). At heights above 110 km and close to 90 km where there is a meridional outflow of energy from the Equator, the latitudinal profile of heating can be expected to be broadened, whereas at 80 km where there is an inflow the profile can be expected to peak at the Equator. These features are found in the plots of Qtidalin Fig. 5 and it is seen that at all heights the heating becomes insignificant at more than 30” latitude from the Equator.

u being the Earth’s radius and g mean acceleration due to gravity. For the (1,i) mode we have s = I and take 4. Disclrssion h, = 88.1 km, h = 0.6909 km to obtain the curve in From the diurnal tidal fields calculated by Forbes Fig. 3. Of interest are the relative contributions oF(Qtidaljw (1982a), the results for the (I, 1) mode have been used to evaluate the time-averaged vertical and S-V fluxes of and (Qtid& to Qtidal at the Equator where both N,(B) mechanical energy. The value obtained for the global and -F(O) maximize. Figure 4 shows that (Qid,,)Jp at mean vertical flux in the upper mesosphere, below the Equator, which is represented by the dotted curve, levels of appreciable dissipation, is 1.6 mW m ’ (at may be appreciably modified by the addition of 70 km) and lies well within the range of estimates referred to in Section 1. Particular attention was given by Forbes (1982a) to the (1,l) mode as a cause of turbulence in the upper nlesospher~~lower thermosphere(Lindzenand Forbes, 1983), and eddy diffusion coefficients were adopted which gave theoretical tidal amplitudes in line with tidal observations and reaching 5 x 10’ m2 s-’ at 102 km (Fig. 1). Such values differ markedly from that of 13.6 m2 s-r taken at all heights by Lindzen and Blake (1971) whose calculations imply a reduction in vertical energy flux by a factor of 4 at 105 km. The present calculations show a reduction by a factor of SOat the same height (Fig. 1). Whereas previously a significant fraction (25%) of the (I, 1) modal energy could be I I I f regarded as being deposited in the thermosphere, the 0 50 150 -50 100 mW/kg present calculations put the heating predominantly in the upper mesosphere with a lower boundary at about FIG. 4. TIDAL HEATING;KATE PEK UNIT MASSOF ATMOSPHFREAT 70 km (Fig. 4). The largest volume heating rates are in THE ECJUATORFOR THE (I. 1) MOIIE. the region of 75-82 km which is lower (by roughly I5 Key: ___Qii,Jp, being the sum of that dne to the vertical km) than the heights implied by the calculations of energy flux. (Q~,~~~)~/~, and that due to the S-N energy flux; Lindzen and Blake (1971). ----(Qti+a,L/‘~.

Equinox

FIG.5. KEY:-Q

tidal heating

of the upper atmosphere

451

,,,,,rNuNrrsorW/m3vs LATIT~~~~~~~T~~(~,~)~~~~ATA~~L~~TI~~\~~F~IEIGHTS;----(&,.,),,,,THE CONTKIBUTION TO f&,.,FKO~'TWEVEKTlCALeNEKCiY FLUX. The horizontal lines on the right of each diagram show the values of Qtida,.

The latitudinal profile of heating is examined in Section 3.2 and is found to be significant only within about 30” of the Equator in accord with the lowlatitude confinement of this mode. The contribution of S--N energy fluxes to the heating is comparable with that of vertical fluxes (Fig. 5), their effect at a few heights being to augment theequatoria~ peak due to the vertical flux but at most heights they resutt in adisplacement of the peak from the Equator to give peaks at about 14”N and S latitudes. The above calculations have con~rmed the magnitude of the vertical energy fh.tx of the (1,l) mode in the upper mesosphere and defined in more quantitative detail than previously the height and latitude distributions of the heating that would be produced by the dissipation of this oscillation acting alone. Il.

HEATING

D1SIPATION

OF THE UPPER

OF THE DIURNAL

ATMOSPHERE TIDAL

RY

OSCILLATION

In Fart I we evaluated the time-averagedmechani~a) energy fluxes of the (1,l) mode in the vertical and S--N directions and derived the resultant heating rates at different heights and latitudes.. Global mean values were also evaluated. a value of I.6 mW me2 being obtained for the global mean vertical energy flux at 70 km above which height the Aux rapidly decreased. Other diurnal oscillations considered by Forbes (1982a) are those excited by the (1, - 2) modes of H,Q and 0, heating and the diurnal Fourier component of thermospheri~ heating. Together with the {I, 1) mode these oscillations are taken to be responsible for the major part of the diurnal oscillation. The purpose of

Part II is toextend the evaluations of Part I to the total diurnal oscillation and to make comparisons with the results that have been obtained for the (1.1) mode above.

Diurnal tidal fields were evaluated by Forbes( 1982a) from 0 to 400 km altitude and at latitudes q6)78~ for both equinox and solstice conditions. From the tidal components of pressure p cos cr(t-- I,), vertical velocity ~1cos ri.(t-t,*) and S-N wind u cos off-t,) at colatitude 0, the vertical and S--N energy fluxes E,(fl), E,.(0) are obtained as E,(O) = fzp cos rL?(t,- t,)

(3: = t’,w).

(20)

Qtia,, then follows from (10). Equation (I 5) again holds for the global mean heating QtidJ, as may be shown by appropriately modifying the analysis of Section 3.1. The global mean vertical energy flux is obtained by evaluating x E,,. = 4 E,,(O) sin U do. (21) s0 7. t Lutitudirrulpr~~leso~~~I~nting. Qlids,,evaluated by (lo)> is shown against latitude in Fig. 6 for a selection of heights which are the same as in Fig. 5 plus 250 and 400 km. The dotted curves in Fig. 6 are Qfidalfor the (1, I) mode alone, being the same as the continuous curves in Fig. 5. Figure hshows that the latitudinal profile of Qlida,for the total diurnal tidechanges radically between 125 and 150 km from one with negligible high-latitude heating to one with significant hi~il-latitude heating. Also

452

G. V. GROVES and

JEFFREY

150 km

60km

0

‘lo

20

FI~;.~.KEY:--~Q,~~~,IN

UNITSOF

60

W/m” vs

OF HEIGHTS;.."

M. FORBES

80'

0

20

40

80'

60

LAT~T~I~EF~RTHESOLARDIUR~AL~S~~LLAT~~NATASELECTION

Qlid.,FOR THE(

1.1) MOUEALONE (Fig. 5).

The horizontal lines on the right of each diagram show the values of e,,,,,.

between 125 and 150 km any similarity with the (1,l) heating profile disappears, heating rates being greatly inexcessofthoseofthe(1, 1)modeatandabove 150km. Agreement with (1,l) heating rates is very close at 80 and 90 km, but dissimilarities appear in the 100 and 125 km plots which show significant heating over a wider range of latitudes than with the (I, 1) mode alone. 1.2 Global mean values. Figure I shows I?,*,evaluated by(21)andQtidaI( = -dE,,,/dz).Thedashedcurvesin Fig.

km

400 ‘\ x)0-

‘\

‘\

‘\

‘t

7 are for the (1, I) mode alone (Fig. 1). At heights from 0 to 130 km there is no plottable difference between values for the total diurnal oscillation and those for the (1,1) mode alone, but from about 130 km, they rapidly diverge. Above this height J?, increases to a maximum at about 180 km and Qtidal correspondingly shows a second region of heating above 180 km with cooling from 130 to 180 km. In the second region of heating slowly with height in Qtidal and also /?, decrease comparison with therate ofdecreasein thelower region of X0 to 110 km. .4bove 230 km. the scale heights of QtidaI

‘\

&+,dol(WCT-‘) Id3

16”

lO-8

IO-’

FIG. 7. I?,, &d,,.

Key

: -----derived

for the equinox diurnal tidal fields of Forbes (1982a); alone (Fig. 1) where different from the continuous

~~- --derived for the (I, 1)mode line.

Equinox

tidal heating

453

of the upper atmosphere

and E,,, are nearly 100 km (and therefore exceed density scale heights by a factor of about 2).

The diurnal tidal fields adopted in Part II were derived by Forbes (1982a) from the (I, 1) and (1, -- 2) modes of 0, and H,O heating and the Fourier component oftll~rmospheric heating. The{ 1, I) mode is capable, under conditions of classical tidal theory. of propagating energy away from a height region of excitation, whereas a (1, -2) oscillation decays exponential-like away from a region of excitation. It is therefore of interest to examine the extent to which the (1,l) mode alone represents the energy flux and mechanical heating of the total diurnal tide. Figure 6 shows that the diurnal tide has latitudinal profiles of Qlida, similar to those of the (I, 1) mode alone at 80 and 90 km. This result would be expected as the diurnal tidal fields of Forbes (1982a) are closely approximated by those of the (I, 1) mode at these heights. But above 90 km, the thermospheric diurnal ( excitation generates an in situ oscillation which with increasing height becomes the major component of the diurnal tidal field. This change is seen in Fig. 6 where the two profiles still show general similarities ofmagnitude at 100 and 125 km, whereas at and above 150 km the (1, I) profile is negligible compared with that of the diurnal tide. It may be noted from the 150-400 km plots in Fig. 6 that the in situ oscillation at these heights gives rise to a second region of tidal heating and that the latitudinal profile of the heating is such that energy is transferred from lower to higher latitudes. Very large S-N velocities are generated in the high-latitude thermosphere (Forbes, 1982a, Fig. 12) amounting to 150 m s- ’ at 60” latitude (and over 200 m s - ’ at 70” latitude) and the large high-latitude heating rates in Fig. 6 are a consequence of these velocities. Comparisons of the global means E,,,, &da, of the diurnal oscillation with those of the (I, 1) mode alone show very close agreement up to 130 km and then rapid divergence above this height. These results imply that the effect of the in situ thermospheric oscillation on the (1,l)mode is to redistribute its energy flux and hence its heating in latitude without affecting the global mean up to 130 km, and that above this height the in situ oscillation itself carries out mechanical work on the atmosphere, which soon becomes greatly in excess of that of the much reduced (1,l) oscillation at these heights. The comparisons in Fig. 7 also show that E,, Qfidal are not affected by the oscillation generated by the (1, -2) modes of H,O and 0, heating. This result is to be expected under conditions ofclassical tidal theory(or of

extended classical theory with separable modes having Hough-function latitudinal dependences). The contributions from products ofdifferent modes then becomes zero on global averaging due to the orthogonality property of Hough functions. In particular the global mean vertical energy flux for a summation of modes equals the summation of the global mean vertical energy fluxes ofthe individual modes (Groves, 1982b). For a summation of( 1,l) and (I, - 2) modes, the global mean vertical energy flux is that of the (I, 1)mode, as the (1, -2) is a “trapped” mode with no propagation of energy : and Fig. 7 confirms this theoretical result. 111.HEATlNG

OF THE UPPER 4TMOSPHERE

DISSIP~~TION OF THE S~~~IDIURN~L

BY

TIDAL

OSCILLATION

9. Introduction Calculations by Lindzen (1970) showed that the main semidiurnal mode, i.e. the (2,2) mode, is dissipated above 130 km and should therefore be relatively effective in heating the thermosphere. Further calculations (Lindzen and Blake, 1971) gave a timeaveraged upward vertical flux ofabout 0.3 mW rn-’ at the Equator for the oscillation excited by the (2,2) modes of H,O and O3 heating and a region of dissipation extending from 120 to 160 km. The mean heating of the thermosphere by dissipation of this mode was investigated at that time by considering upward vertical fluxes at the Equator of 0.3-0.4 mW rn-’ (Lindlen and Blake, 1970). Evaluations of the (2,2) modes of H,O and 0, heating have more recently been undertaken (Forbes and Garrett, 1978; Walterscheid et al., 1980; Groves, 1982a,c) which significantly differ from those upon which the above calculations were based. For a nondissipative atmosphere satisfying the conditions of classical tidal theory, Groves (1983b) found that the upward energy flux of the (2,2) mode became nearly constant in height above 105 km as the atmosphere becomes rapidly non-reflective to this wave. An equatorial upward energy flux of 0.12 mW me2 was obtained at these heights which is significantly lower than the earlier estimates. The corresponding global mean was 0.051 mW m-‘. Classical tidal theory is inadequate for evaluating the deposition of tidal heating as firstly dissipation is not taken into account and secondly the (2,2) and other semidiurnal modes are coupled through the latitudi~lal structure of the background atmosphere. These limitations do not apply to the semidiurnal tidal fields computed from 0 to 400 km by Forbes (1982b) for equinox and solstice conditions utilizing the same numerical procedure that yielded the diurnal tides

G. V. G~ovss and JEFFREY M. FORBE~

454

IOOkm

. ..‘. x10-’

.

.:.

.’

x,.

..

0-

0 FIG. 8. KEY:

-Qlidd

20

..._

, . .. . . . .. .

40

IN UNITSOF W SELEcTIONOFHEIGHTS;"'-

175km

‘t

:.

,:ym ]~‘o-~:~~ 60

60'

0

20

40

60

60'

II-3 VS LATITUDE FOR THESOLAR SEMIDIURNAL OSClLLATION AT A Qlid., FOR THESOLAR DIURNALOSCILLATION (Fig.6).

A horizontal line on the right of a diagram

shows the value of Qida, for the semidiurnal

analyzed in Part II. The sources of tidal excitation were taken to be the(2,2),(2,3),(2,4),(2,5)and(2, @modes of H,O and 0, heating (the (2,3) and (2,5) modes being zero at the equinox) and the semidiurnal Fourier component of thermospheric heating. The purpose of Part III is to evaluate the heating arising from the equinox semidiurnal tide in terms of latitude and height. The eddy diffusion coefficients upon which the adopted tidal fields (Forbes, 1982b) are based are those ofthe “moderate” profile ofForbes (1982a, Fig. 6) which has values of the order 50 m2 s ’ m the mesosphere and lower thermosphere. 9. Results for equinox conditions 9.1 Latitudinal projles ofheating. Qtida,, evaluated by (10) and (20), is shown against latitude in Fig. 8 at a selection ofheights from 90 to 250 km. At 90 km, Qtidalis negligible compared with that of the diurnal tide which is plotted as a dotted curve, but by 100 km the two sources of tidal heating become comparable and at 125 km the semidiurnal is the greater. No radical change is found in the semidiurnal heating profile at 150 km and above, as with the diurnal profile, and by 250 km the semidiurnal is again generally small compared with the diurnal tidal heating. 9.2. Global mean value. The continuous lines in Fig. 9 show E, evaluated by (21) and Qfidal (= - d&,/dz). E, decreases at all heights above 103 km where its value is 0.043 mW m ‘, which is in reasonably good agreement with 0.051 mW mm2 obtained (Groves, 1983b) for a non-dissipative atmosphere. Qtida, decreases at all heights above 108 km where it peaks sharply with a maximum of 2.8 x lo-’ W rn- 3 which compares with 1.5 x lo-’ W me3 for the diurnal tidal heating. The

oscillation.

sum of the diurnal and semidiurnal heating rates is shown by the dotted line in Fig. 9 to fall into two regions : the lower one has peaks at 80 km due to the diurnal tide and at 108 km due to the semidiurnal tide, whereas the upper region has a broad peak at 200 km that arises mainly from the diurnal tide at high latitudes (Fig. 6). The dashed curves in Fig. 9 show the results obtained without the thermospheric semidiurnal heating: although the effect of the thermospheric source is to increase E, and Qtidal in the mid- and upper thermosphere by the order of SO%, the magnitude ofthe effect is insignificant compared with that arising from the thermospheric diurnal heating (Fig. 7). 10. Discussion The equinox semidiurnal tidal fields adopted for the calculations in Part III were derived by Forbes (1982b) from the(2,2),(2,4)modesofH,O and 0, heating(plus a fairly insignificant (2,6) mode) and the Fourier component of thermospheric heating. Calculations have also been carried out in Part III with the Fourier component of thcrmospheric heating omitted and the comparisons in Fig. 9 show that the thermospheric excitation only augments the vertical flux significantly above 120 km and the heating rate above 130 km. In particular, we have that at 103 km, which is the height above which the energy flux is affected by appreciable dissipation, the oscillation would be generated predominantly by the H,O and 0, heating. The value of 0.043 mW m - 2 for E,, at this height is in reasonably good agreement with that of 0.051 mW me2 obtained by classical tidal theory for the (2,2) mode excited by H,O and 0, heating in a non-dissipative atmosphere. An important effect that is not taken into account by classical tidal theory is “mode coupling”: below 60 km

Equinox

tidal heating

455

of the upper atmosphere

400 km

200

KEY: ___ DERIVED FOR THE EQUINOX SEMIDIURNAL TIDAL FIELDS OF FORBES (1982b); '..'DIURNAL +~EMIDIURNALTIDALHEATING,BEINGTHE~~M~FTHEC~NTINUOUS LINESIN Figs.7 AND 9;---ASFORTHE CONTINUOUS

the

atmospheric

semidiurnal

attributed to the (2,2)

LINES BUT WITHOUT

response

can

THERMOSPHERIC

be

mode, but between 50 and 70 km its exponential growth is interrupted, in part due to a tendency towards evanescent behavior connected with the background thermal structure, and in part due to “coupling” into higher order modes (Forbes, 1982b). Under equinox conditions it is mainly the (2,4) mode to which energy is transferred from the (2,2) mode and although this may not affect the total vertical energy flux as the above comparison with the classical analysis indicates. the heights of energy dissipation are lower

0

SEMIDIURNAL

TIDAL

HEATING.

due to the shorter vertical wavelengths of higher order modes. Thus, Fig. 9 shows that most of the flux is dissipated between 103 and 120 km in contrast to the earlier result that dissipation of the semidiurnal tide is over the region extending from I20 to 160 km (Lindzen and Blake, 1971). The eddy diffusivity of 13.6 m2 s- ’ taken by Lindzen and Blake (1971) is much lower than the value of 50 m2 s- ’ taken by Forbes (1982b) but this alone does not account for the lower heights of appreciable dissipation as mesospheric eddy diffusivities in excess of 200 m2 s-’ are required for

d W/m'

IO-

IO+

FIG. 10. KEY :~TIUES;---TIME-AVERAGED JOULE Qma,DUETODISSIPATIONDFDIURNAL+SEMIDILJRNAL HEATINGTAKEN AS7[ -’ TIMES NOON VALUES(FoKREs, 1975);...'TlME-AVERAGED C".V.HEATING TAKEN AS 7[-1 TIMES NOON VALUES(FORBES, 1975;Ron~~ AND DICKENSON, 1973);~.~.~~ HEATING DUETO MEDIUM SCALE GRAVITYWAVESWI-1HVERTICAL.ENERGYFLUXOF 0.1 mb'm-'AT 120km (SHOWNONLYONC‘ENTERDIAGRAM) (VIDAL-MADJAR, 1979).

456

G. V. GROVES and

amplitudes above (Forbes, 1982b).

IV. COMPARISON

100 km to be significantly

affected

OF TIDAL HEATING WITH OTHER

LOWER THERMOSPHERIC

HEAT SOURCES

The continuous curves in Fig. 10 show the heating from 90 to 170 km due to the combined effects of the diurnal and semidiurnal tides at 0” (low), 45” (mid) and 75” (high) latitudes. Features of this heating have been identified above and are again apparent. At low latitudes the maximum heating is in the region of 80 km (which is off the diagram) and is due to the diurnal component, whereas at high latitudes it is at 108 km and is due to the semidiurnal component: at most latitudes both tidal heat sources contribute below 140 km. Above 140 km there is cooling at low latitudes and heating at high latitudes due to the diurnal tide. The dotted curves in Fig. 10 show the 24-hourly average of e.u.v. heating for average solar conditions and we see that below 110-120 km, these are exceeded by tidal heating at all latitudes; whereas at greater heights, it is only at high latitudes that the two are comparable. A comparison between tidal heating and Joule heating due to the S, current system is appropriate as the Iatter is also dependent on the tidal fields. The dashed curves in Fig. 10 are taken from Forbes (1975). The plot for 0” latitude shows heating by the equatorial electrqject to be small compared with the tidal heating, whereas the S, currents at high latitudes produce heating above 120 km that is comparable with both tidal and e.u.v. heating. At midlatitudes above 120 km both tidal and Joule heating are small compared with the e.u.v. heating. Joule heating due to currents of non-S, origin have not been considered in these comparisons. Heating by dissipation of medium-scale gravity waves (horizontal wavelength 150 km, period 20 min, horizontal phase velocity 125 m s- ‘) has been calculated by Vidal-Madjar (1979) for a vertical energy flux at 120 km of 0.1 mW m-‘. The maximum deposition of energy is obtained around 160 km and although the vertical heating profile is modified by interaction with the background wind, a mean heating rate of 5 x 10-r” W me3 between 120 and 200 km is obtained whether or not the background wind is taken into account. The dot-dash curve in Fig. 10 is for waves propagating northward without a background wind and the values are seen to be small compared with the e.u.v. heating.

JFFFRCY M. FORBES

Acknowledyements~-One of us (G.V.G.) gratefully acknowledges the award of a National Research Council Research Associateship during the tenure of which this work was carried out in the Atmospheric Structure Branch, Aeronomy Division, Air Force Geophysics Laboratory, Hanscom AFB, Massachusetts. J. M. Forbes received support for this work under Grant ATM-81 13078 from the National Science Foundation to Boston College. REFERENCES CIRA (1972) COSPAR International Reference Atmosphere 1972. Akademie, Berlin. Forbes, J. M. (1975) Atmospheric solar tides and their electrodynamic effects. Ph.D. Thesis, Harvard University, Cambridge, MA 02138, U.S.A. Forbes, J. M. (1982a) Atmospheric tides 1. Model description and results for the solar diurnal component. J. geophys. Rex 87,5222 Forbes, J. M. (1982b) Atmospheric tides 2. The solar and lunar semi-diurnal components. J. geophys. Rex 87, 5241. Forbes, J. M. and Garrett, H. B. (1978) Thermal excitation of atmospheric tides due to insolation absorption by 0, and H,O. Geophys. Rex Lett. 5, 1013. Groves, G. V. (1982a) Hough components of water vapour heating. J. atmos. terr. Phys. 44,281. Groves, G. V. (1982b) The vertical structure of atmospheric oscillations formulated by classical tidal theory. Planet. Space Sci. 30, 219. Groves, G. V. (1982~) Hough components of ozone heating. J. atmos. terr. Phys. 44, 111. Groves,G. V.(1983a) Energyfluxesofthe(1, 1,l)atmospheric oscillation. Planet. Space Sci. 31, 67. Groves, G. V. (1983b) Thermospheric energy flux of the semidiurnal tide. Planet. Space Sci. 31, 1183. Lindzen, R. S. (1967) Thermally driven diurnal tide in the atmosphere. Q.J.R. Meteorol. Sac. 93, 18. Lindzen, R. S. (1970) Internal gravity waves in atmospheres with realistic dissipation and temperature, Part I, Mathematical development and propagation of waves into the thermosphere. Geophys. Fluid Dyn. 1, 303. Lindzen, R. S. and Blake, D. (1970) Mean heating of the thermosphere by tides. J. geophys. Rex 75.6868. Lindzen, R. S. and Blake, D. (1971) Internal gravity waves in atmospheres with realistic dissipation and temperature, Part II, Thermal tides excited below the mesopause. Geophys. Fluid Dyn. 2, 3 I. Lindzen, R. S. and Forbes, J. M. (1983)Turbulence originating from convectively stable internal waves. J. geophys. Res. 88, 6549. Roble, R. G. and Dickinson, R. E. (1973) Is there enough solar extreme ultraviolet radiation to maintain the global mean thermospheric temperature? J. geophys. Res. 78, 249. Vidal-Madjar, D.(1979)Mediumscalegravity wavesand their non linear interaction with the mean fow: a numerical study. J. atmos. terr. Phys. 41, 279. Walterscheid, R. L., DeVore, J. G. and Venkateswaran, S. V. (1980) Influence of mean zonal motion and meridional temperature gradients on the solar diurnal atmospheric tide: A revised spectral study with improved heating rates. J. Atmos. Sci. 37,455. Wilkes, M. V. (1949) Oscillations of the Earth’s Atmosphere. Cambridge University Press, Cambridge