Journal of Atmospheric and Terrestrial Physics, 1968, Vol. 30, pp. 851-856. Pergamon Press. Printed in Northern Ireland

An effect of ohmic losses in upper atmospheric gravity waves

C. O . HINES Depar tment of Physics, Universi ty of Toronto, Toronto 5, Canada

(Received 11 November 1968)

Abstract - -The effect of ohmic losses in ionospheric gravi ty waves is analysed with respec t t o the selective dissipation of different wave modes. Preferred tilts of phase surfaces are deter- mined, as a function of azimuth of propagation, and these are related to a reported propensity toward a field-alignment of frontal surfaces in travelling ionospheric disturbances.

IT Is the purpose of this note to record an effect of ohmic losses in atmospheric gravity waves at F-region heights, and to relate that effect to an observed feature of travelling ionospheric disturbances (TID's).

The observed feature is a certain propensity for frontal surfaces in TID's to be tilted in such a fashion that they lie parallel to, rather than across, geomagnetic field lines. This propensity has been reported by a number of authors (e.g. B o w ~ , 1960; tIEmL~.~ and WHITEHEAD, 1961; MUNRO and HmST.V.R, 1956), and has given rise in some quarters to suggestions that the frontal surface is in fact determined by an electrodynamic effect mapped along the field lines.

This view contrasts with that of HIN~.s (1960), who takes the frontal surfaces to represent surfaces of constant phase in an atmospheric gravity wave, perhaps distorted in some relatively minor fashion by the effects of ion-neutral coupling that arise when the neutral-gas disturbance imposes itself on the ion distribution, and by the effects of a group envelope in the case of short-lived TID's. On the basis of this interpretation, the reported propensity for field-alignment must be explained in some other way, and three routes to an explanation come readily to mind.

The first follows the path suggested on p. 1479 of HINES (1960), where it is noted that much larger perturbation velocities might be induced in the ionization if the neutral-gas motion were along, rather than across, the geomagnetic field lines. Since gravity-wave oscillations lie very nearly in surfaces of constant phase, under certain asymptotic conditions that often obtain, waves with field-aligned phase surfaces might be expected to yield relatively enhanced disturbances of the ionization, and might therefore be particularly susceptible to detection by the usual radio techniques. This possibility can be evaluated only by means of a full analysis that takes into account factors additional to the ion velocity itself, and such analyses are now beginning to appear (e.g. HOOKE, 1968).

The second possibility entails the accidental matching of wave sources to geomagnetic tilts, the matching being executed by way of the wave tilts that a given wave period excites. This suggestion is by no means flippant, and indeed is probably relevant to the case of aurorally associated TID's, but it does not lend itself to detailed analysis in the absence of further knowledge as to source mechanisms.

851

852 C . O . Hrm~s

The third possibility is the one explored here. I t is concerned once again with t h e presence of ionization, but now with respect to the reaction tha t the ionization imposes on the neutral gas. This reaction is often te rmed 'ion drag', and it leads to what may be called hydromagnet ic or ohmic losses of energy from the atmospheric gravi ty wave. The importance of these losses varies with the parameters of the wave since the ionization introduces little resistance to neutral-gas motions along the geomagnetic field lines but much resistance (at F-region heights) to motions across those lines. As will be seen, this leads to a selective dissipation of wave energy, greater in waves whose fronts are not nearly field-aligned. The propensity for field-alignment, to the extent tha t it does indeed exist, may then result from preferentially low energy dissipation in the field-aligned waves.

The ensuing analysis treats the dissipation itself as something of a per turbat ion effect, whose magnitude can be est imated by the insertion of certain relations from the loss-free case into subsidiary formulae which then give a first approxima- t ion to the actual loss. The approach is entirely analogous to tha t adopted by I ~ E S (1960), in Section 4.4, to obtain a measure of the dissipation caused by atmospheric viscosity and thermal conduction. The necessary loss-free relations may be taken from the same paper.

In the present case, the energy dissipation per unit volume is given by J . E', where J is the current density and E' comprises an electric field E and a dynamo field U x B 0, U being the velocity of the neutral gas induced by the gravi ty wave and Bo the geomagnetic induction. The current densi ty has a component JIt ---- °0EIt directed parallel to B0, where o o is the 'longitudinal' conductivi ty and Eli is the component of E parallel to B0. I t also has a component J ± ---- olE±' transverse to B0 in the direction of the transverse component of E', where o 1 is the 'Pedersen' conductivity. (It also has a 'Hall' component, perpendicular to both E' and B0, but tha t component is relatively small in the F-region and in any event contributes nothing to J . E'.) The current is necessarily nearly divergence-free, whence [JJLii[ "" [Jz/.L±[, where LII and L± are representative scales of spatial variat ion of JLL along B0 and of J . across B0, and from this it follows tha t [Elll _~ [E. ' [ × (olLda0L±). I t follows in tu rn tha t JitEli ~ J.E.'(01/oo)(Ltl~/L±"), and, since ol is less than oo by four orders of magnitude or more, it follows finally tha t the energy dissipation derives primarily from the J . E . ' contribution to J . E' unless L± ~ Lit. Again, it may be shown tha t E~ ' results almost entirely from the U x B0 contribution to it, in the case of normal atmospheric gravi ty waves on the relevant scales (HrNEs, 1955, equation A14), and so the dissipation rate per unit volume is given approximately by 01U±2Bo 2 where U± is the component of U perpendicular to B0. These s ta tements must be amended when [U±] ~ mUll[, when too the condition L± ~ Lli may obtain, but the over-all conclusions of this note are not altered in consequence.

So long as dissipation is small, the loss rate may be evaluated by employing formulae for U . tha t derive from loss-free relations. This may be done even though U± itself will be altered by the loss process, since any such alteration results only in a higher-order correction to the loss rate. (This situation should be contrasted with tha t which impairs the work of G~.RSHMA~ and GRIGORY'EV (1965), who sought to deal with hydromagnetic effects in atmospheric gravi ty

An effect of ohmic losses in upper atmospheric gravity w a v e s 8 5 3

waves. They taci t ly assumed, at equat ion 1.14, tha t the ratio of horizontal to vertical components of U is not subject to any first-order change under the influence of the hydromagnet ic effects they discussed, and their subsequent analysis lacks generali ty for this reason. The only generally applicable dispersion equat ion tha t contains first-order corrections is I Umn] o + I umn]l ---- 0 in the notat ion of HINES (1955), equation AI3, while the modifications obtained by GERSHMAN and GRIGORH'EV represent a special case, applicable when and if their taci t assumption happens to be true.)

We may now construct the ohmic loss per wave period r, per unit mass of

gas: ralU±~Bo2/po, where U± ~ is the cycle-average of U± ~, with the elliptical polarization of U± taken into account, and P0 is the mass densi ty of the atmosphere. In order to est imate the importance of this loss, we determine it as a certain fraction, f, of the energy per unit mass carried b y the wave in the loss-free case. The lat ter quan t i ty may be taken from equation (45) of HINES (1960).

The fractional loss, f , takes on a reasonably simple form when the 'asymptot ic approximations ' of HINES (1960) are valid, as they often are in practice. I t then approximates to

j:' =- b'J~'n,][r~(s~, ~ + c~81 ~) + 2rc~8~c~ + c~][1 + r~]-J (1)

where % is the isothermal Brunt-Vais~l~ period, given b y equation (1) of Hn~Es (1960) for example; rni represents po/alBo ~, and in the F-region measures the mean t ime between successive collisions with ions by a given neutral particle; s and c denote sine and cosine of the subscribed angles; ~ is the azimuth of propagat ion measured from magnetic North; I is the magnetic dip angle, positive in the northern hemisphere; and r is the tangent of the angle b y which the wave normal is depressed below the horizontal plane. A representat ive value of r a for the F-region would be 15 rain, while a representat ive value of r,~ for the same region b y day would be 45 min, whence ~'dr,,i ~, ½.

Observationally, one detects the inclination of planes of constant phase rather than the depression of the wave normal, and it is therefore more appropriate to interpret r as the tangent of the angle by which these planes are t i l ted forward at the top, from the vertical into the direction of horizontal progression. I t should be recalled tha t this angle must lie in the range 0 ° to -$-90 °, if the energyflux of the wave has a vertically upward component, and tha t r ~_ %]r in the asymptot ic approxi- mation (HINES, 1960).

I t is evident from equation (1) tha t the fractional dissipation of energy is dependent on the tilt of the phase surfaces through r, and on the azimuth of propagat ion through s~ and c~. One may also discover tha t the middle bracketed factor in equat ion (1) vanishes when ~ ----0 and r---- - -cot I , and again when

= 180 ° and r ---- cot I . I f r is constrained to be positive, b y virtue of a restriction on energy flux, these conditions can be met only b y waves whose horizontal progression is directed along a meridian towards the equator, and whose planes of constant phase are t i l ted so as to lie along the geomagnetic field lines. Physically, the vanishing o f f ' in these circumstances results from the fact tha t the per turba- t ion velocity of the neutral gas, U, is then directed nearly along Bo; the collisional interaction of tha t gas with ions then results in no significant loss of energy, for

854 C . O . HIN~S

t h e ions are free to m o v e wi th t he gas r a t h e r t h a n resis t i ts mo t ion . D e p a r t u r e s o f f ' f r o m zero re su l t f r o m a d e p a r t u r e f r o m these f a v o u r a b l e c i r cums tances .

Resu l t s f r o m one c o m p u t a t i o n o f f ' a re d i sp l ayed in l~ig. 1, for ] I I = 57 °. T h e hor i zon ta l axis r ep re sen t s t he a z i m u t h of ho r i zon ta l p rogress ion of t h e w a v e ,

I L l

u_ < I N T E R N A L G R A V I T Y W A V E S i . !i:: ::

r," . . . . . S E V E R E L Y D A M P E D B Y

6o o_ ~ ~ O H M I C L O S S E S "

< ~ . A B O V E ~ 1 5 0 KM i.- z o i " i L 3 0 ° " ~ . . . . . . ~ ' ' ~ ' " ~ ' - o , , , • I

LI_ o

I- ._1 F o ' - - - : : _-

° : ~ -50"- L J - T ~ " "

TOWARO 3 0 ° 6 0 0 TOWAR 0 120 ° 150 ° TOWARD MAGNETIC MAGNETIC MAGNETIC EQUATOR EAST OR WEST POLE

AZ MUTH OF F R O N T A L A D V A N C E Fig. 1. Ohmic losses in F-region gravity waves, and TID observations. The shaded regions represent waves with azimuth-tilt parameters that lead to severe ohmic losses in the F-region by day, with about a third of the locally available wave energy being lost per cycle. The continuous, lightly drawn curves represent waves for which this fraction is reduced to a ninth. The dotted curve represents optimum tilt for minimum ohmic loss, as a function of azimuth, and the broken curve represents field-aligned tilts. The data bars and points are derived from BowM~r (1960). The theoretical curves are drawn for a dip angle Z = ±57 °,

equal to that which applies in B o w e l ' s observations.

m e a s u r e d f r o m the d i rec t ion t h a t po in t s t o w a r d s t he m a g n e t i c e q u a t o r ; th is a z i m u t h is ident ica l to ~ w h e n I = - -57 ° ( sou the rn hemisphe re ) b u t equa l t o

+ 180 ° w h e n I = + 5 7 ° (no r the rn hemisphere ) . The ve r t i ca l ax is m e a s u r e s a rc t a n r; t he t e r m ' f r on t a l su r face ' is e m p l o y e d in i t s label in t h e be l ie f t h a t t he t i l t ed f ron t s t h a t a re o f ten r e p o r t e d f r o m T I D o b s e r v a t i o n s a re in f a c t closely r e p r e s e n t a t i v e of phase sur faces as p r e v i o u s l y no ted .

The shaded po r t i ons of Fig. 1 r e p r e s e n t t he c o m b i n a t i o n s of a z i m u t h a n d t i l t t h a t l ead to f ' > %/wn~, or f ~ ½ in t h e d a y t i m e _/~-region. These c o m b i n a t i o n s c lear ly lead to s t rong d i s s ipa t ion of t he ava i l ab l e w a v e energy , to t h e e x t e n t t h a t

An effect of ohmic losses in upper atmospheric gravity waves 855

the use of loss-free formulae in the actual est imation o f f is probably no longer valid. The computat ion is not wi thout significance, however, for it does reveal the region in which the loss is intense, even if it cannot provide a reliable quant i ta t ive est imate of tha t loss.

The unshaded regions are of more interest. They represent waves which are not so severely dissipated, and they therefore represent waves which are more likely to be found at a given level of intensity in the F-region (unless variations of source strength or refractive effects introduce overriding considerations). Within the unshaded regions lie two lightly drawn continuous curved lines. These represent the combinations of azimuth and tilt for which f ' ---- rd3rin, or f ~ ~ in the dayt ime F-region. Between them lies an area for which hydromagnet ic damp- ing is extremely small indeed. At two points in the diagram, one at a tilt of 33 ° for propagat ion toward the magnetic equator and the other at a tilt of --33 ° for propagat ion toward the magnetic pole, f ' ---- 0 and ohmic losses are negligible. The J . E loss is no longer a negligible contr ibutor to the total 7 . E ' loss in these special circumstances, and the asymptot ic approximation tha t led to (1) breaks down at tilts very close to 0°; bu t detailed computat ions have shown tha t the dissipation is not increased seriously b y the additional terms tha t a more complete analysis would introduce.

I f the sources of the wave energy lie below the F-region, then the energy flux in the F-region must have an upward component, and only forward tilts may be present. ~ e are therefore led to direct our a t tent ion to the portion of the diagram above the line of 0 ° tilt, and below the shading or below the lightly drawn curve, f ' ---- rd3rin. I t is evident tha t azimuths directed more-or-less toward the mag- netic equator are generally favoured over azimuths toward the east or west, and they in tu rn are favoured over azimuths more-or-less toward the magnetic pole, insofar as freedom from ohmic dissipation is concerned. This general t rend is consistent with observations of the azimuthal distribution of TID'S, al though details of tha t distribution and of its changes with t ime make it clear tha t other f ac to r s - -no tab ly the sources of the waves - -mus t be impor tant as well.

We may now enquire as to the opt imum angle of tilt in the sense tha t ohmic dissipation should be minimized, for a given azimuth of advance. In the approxi- mation given b y equat ion (1) this evident ly requires Of ']Or ---- 0, and it leads to angles of tilt as a function of azimuth given by the dot ted line in ~ig. 1 for azimuths equatorwards of magnetic east or west. At azimuths polewards of magnetic east or west, the condition af'/Or ---- 0 leads to negative values of r, which may be derived by extending the do t ted line in a fashion tha t maintains the type of sym- met ry tha t the other curves of the diagram possess. Negative values of r would be excluded, however, if the energy sources lie below the F-region, and the least absorption tha t can be obtained under this restriction arises at r ---- 0, or a ti l t of 0 °. For this reason, the dot ted line is extended into the poleward half-space along the axis of zero tilt. The dot ted line as a whole then gives the opt imum tilt as a function of azimuth, when at tent ion is confined to waves whose sources lie at lower levels.

The foregoing results may now be compared with observations. As was remarked earlier, it is often said tha t T ID fronts are inclined so as to include t h e

856 C . O . Hi~r~s

geomagnetic field lines. In gravi ty-wave theory, this would imply tha t the wave normal was perpendicular to the field lines. This condition may be wri t ten

r = cos ~ cot I , (2)

and it is i l lustrated in Fig. i by the broken curve. Data tha t appear to confirm this curve have been presented b y BowMA~ (1960, Fig. 18), and are reproduced here as vertical bars in Fig. 1. These data were collected in Australia, at a location whose magnetic dip was very nearly --57 °. Fur ther measures of tilt may be derived from another of BOWMAN'S diagrams, Fig. 20, which displays a series of somewhat irregular but rather continuous ripples propagating nearly in the direction of magnetic north. Values obtained from this diagram, by scaling the parallelograms drawn by BowmAn, are shown b y small boxes along the left-hand edge of Fig. 1.

The fit of the data points to the field-alignment condition, given by the broken curve of Fig. 1, is clearly adequate to just i fy the claim that a field-alignment does occur. But it is equally evident from Fig. 1 tha t an excellent fit is found between the data points and the curve of minimum ohmic losses. Indeed, the two theoreti- cal curves differ only slightly except for azimuths polewards of magnetic east or west, and the one observational bar tha t is available there favours the curve of minimum ohmic loss, if either.

No theory espousing field-alignment of TID fronts, as a basic characteristic, h a s been developed in detail. One might suppose, however, tha t any such theory would demand something close to field-alignment in all cases. The present approach, on the other hand, yields an approximation to field-alignment as a somewhat favoured situation, bu t permits wide variations from tha t si tuation at the penal ty of somewhat enhanced dissipation. This approach appears to be far more closely in accord with the bulk of observational evidence. M u ~ o and H~. iSL~ (1956), for example, find that the frontal tilt is generally fo rward- -o r positive, in present notat ion--regardless of the azimuth of advance, including azimuths toward east or west and polewards of those directions. This result is clearly consistent with the gravi ty-wave analysis, together with its emphasis on waves whose sources lie beneath the F-region, and is clearly in conflict with any strict field-alignment hypothesis.

We may conclude tha t the gravi ty-wave analysis continues to provide a satisfactory explanation of TID behaviour, including the reported propensi ty for frontal field-alignment.

~ E F E R E ~ C E S

Bow~a~ G.G. 1960 GERSH~h~ B. N. and GRIGORY'EV G.I. 1965 HEISLER L. 2~k. and Wm'~HEA~ J .D. 1961 Hr~rES C.O. 1955 HI~rES C. 0 . 1960 HOO~E W . H . 1968 MU'~-RO G. H. and I-IEIsLER L . H . 1956

Planet. Space Sci. 2, 133. Geomag. and Aeron. 5, 656. Aust. J. Phys. 14, 481. J. Atmosph. Terr. Phys. 7~ 14. Can. J. Phys. 38, 1441. J. Atmosph. Terr. Phys. 80~ 795. Aust. J. Phys. 9, 359.