Possible hydrodynamic waves in the topside ionospheres of Mars and Venus

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 17, NO. A4, 139, 1.129/21JA9142, 22 Possible hydrodynamic waves in the topside ionospheres of Mars and Venus J.-S. Wang 1 and E. Nielsen Max-Planck-Institut für Aeronomie, Katlenburg-Lindau, Germany Received 25 April 21; revised 27 August 21; accepted 5 September 21; published 1 April 22. [1] The dispersion relation for hydrodynamic waves in an ionosphere with at most a weak magnetic field shows that gravity waves as well as hydrodynamic hybrid waves may be excited in the topside ionosphere of Mars and Venus owing to fluctuations in the solar wind pressure. The gravity wave, which propagates horizontally with a frequency equal to the buoyancy frequency, belongs to the classic branches of acoustic-gravity wave (AGW) mode. The hybrid waves result from coupling between two different hydrodynamic wave modes. One of these modes is the AGW mode, and the other is excited independent of gravity but dependent on the presence of horizontal gradients in the background plasma pressure and density. The latter mode propagates horizontally but can also propagate vertically if there is a vertical gradient in the horizontal velocity. This new mode is therefore called background gradient wave (BGW). The hybrid waves will cause fluctuations with a wavelength of tens of kilometers in the vertical plasma altitude profiles when they propagate vertically. The period of the waves will be of the order around s. Further properties of possible AGW-BGW waves in Mars and Venus ionospheres are given. Radio occultation observations at Mars and Venus show electron density fluctuations in the high-altitude ionosphere. The fluctuations are mainly noise, but they may in part be caused by hydrodynamic wave activity. To verify wave activity, more detailed measurements are required and may be obtained with the low-frequency radar planned for the Mars Express mission. INDEX TERMS: 2459 Ionosphere: Planetary ionospheres (5435, 5729, 626, 627, 62); 334 Meteorology and Atmospheric Dynamics: Waves and tides; 6225 Planetology: Solar System Objects: Mars; 6295 Planetology: Solar System Objects: Venus; KEYWORDS: Mars, Venus, ionosphere, hydrodynamic wave 1. Introduction [2] Mars has localized surface regions of intrinsic magnetic field [Acuña et al., 199]. There are therefore large regions of the planet s ionosphere with negligible intrinsic magnetic field. In these regions the solar wind interacts directly with the Martian ionosphere. It is recognized that the dynamics of the interface between the solar wind and the ionosphere may be strongly influenced by this direct interaction [cf. Cloutier et al., 1999; Luhmann and Bauer, 1992; Luhmann et al., 1992, and references therein]. During times of increasing solar wind pressure the ionosphere will erode as the solar wind penetrates toward higher thermal pressures in the deeper ionosphere. The ionosphere is eroded as the plasma is transported away parallel to the piled up solar wind magnetic fields at the ionopause. The erosion may also take the form of surface waves on the ionopause, which propagate down stream toward dawn and dusk. These waves may further lead to plasma islands and plasma streamers. [3] On the basis of radio occultation observations a comprehensive review of altitude profiles of plasma density in the Martian ionosphere was presented by Kliore [1992]. It was stated that On the top, these profiles appear in the most cases to fade into the noise of the measurements (about 1 3 cm 3 ), as evident from the increasing levels of fluctuation at higher 1 Also at Department of Geophysics, Peking University, Beijing, China. Copyright 22 by the American Geophysical Union /2/21JA9142 altitude (p. 267). It does appear though that superposed on these (noisy) fluctuations, there often are spatial fluctuations present. It is too early to attribute these spatial fluctuations to waves, but they do suggest plasma density variations along the vertical direction (Figure 1a). In 23, the European Space Agency (ESA) will launch the Mars Express mission. It includes a low-frequency radar, which will sound the electron density of the topside ionosphere with a temporal and spatial resolution exceeding that of earlier probes [Picardi et al., 199]. These measurements will be well suited to probe for wave activity in the Martian ionosphere. [4] In this theoretical work we examine wave activity in those parts of the ionosphere that are not located in regions of crustal magnetic fields. The results are thus applicable to the Venusian ionosphere and to most of the Martian ionosphere. [5] Wave activity is one of the most fundamental processes in ionospheric and magnetospheric physics. In an ionosphere totally shielded by a strong intrinsic magnetic field, such as on Earth, perturbations of the ionosphere seldom come from the solar wind directly except in the polar regions. Instead, the waves excited in the topside ionosphere are caused by violent events such as magnetospheric substorms. In the magnetosphere the plasma is mainly controlled by the magnetic fields while gravitation of the planet is negligible. There are mechanisms including the interaction between the ionosphere and solar wind that have been suggested to produce wavelike structures in the magnetic field in the Martian ionosphere (see the review by Luhmann and Cravens [1991, and references therein]). The waves, here predicted to occur in the Martian topside ionosphere, however, are unique in that they are excited by solar wind fluctuations, are not dependent on a magnetic field, and SIA 2-1

2 SIA 2-2 WANG AND NIELSEN: HYDRODYNAMIC WAVES Figure 1. Vertical electron density profiles in the (a) Martian ionosphere observed by (top) Viking 1 in 1976 at orbits and (bottom) Mariner 9 in 1971 at orbits 21 4; and (b) Venusian ionosphere observed by Pioneer Venus at high solar activity ( , top) and low solar activity (196, bottom). Solar zenith angles are shown in both Figures 1a and 1b and values of the solar wind dynamic pressure are shown in Figure 1b next to each profile. (From Kliore [1992]). are influenced by the planet s gravitation. The weak magnetic field in the Martian ionosphere is encouraging the idea to examine if there are some plasma waves not primarily influenced by the magnetic field, e.g., are there any pure hydrodynamic waves? [6] In this work we consider what may happen to the (weakly magnetized, current free) ionosphere when an essentially steady state solar wind interacts with the ionosphere. The solar wind magnetic field may pile up against the ionosphere and diffuses into the ionosphere, where the magnetic pressure adds to the thermal pressure to deflect the solar wind. The solar wind pressure is transmitted to the ionosphere by the magnetic fields. Since the solar wind is never constant, the pressure it exerts on the ionosphere will be fluctuating. We consider the plasma waves that may be excited in the ionosphere owing to small fluctuations in the solar wind pressure. It is demonstrated that fluctuations in solar wind are a source of perturbation to the ionosphere and most likely result in waves being generated on the dayside high-altitude ionosphere. The wave activity is the ionospheric response to a fluctuation in solar wind pressure and serves to adjust the ionosphere to the change in boundary conditions. The aim of this present paper is to probe if hydrodynamic waves can exist in the Martian ionosphere, and if so, what their main properties are. [7] The plasma in the topside ionosphere is described by the continuity, momentum, and energy equations. The effects of solar wind fluctuations are simulated by considering the perturbation equations. These equations are analyzed, and the dispersion relation for possible wave modes is obtained. It is found that two wave modes are possible. One is a classic acoustic-gravity wave (AGW), and the other mode (so-called BGW) is a wave excited by horizontal gradients in background plasma pressure and density. These two modes are hybridized and result in an AGW-BGW wave. The main properties of the hybrid wave possibly existing in the topside ionosphere of Mars and Venus are outlined. 2. Dispersion Relation [] Considering the momentum equations for the ions and electrons (without electric current) r e dv e r i dv i ¼ rp e þ r e g r e v en ðv e v n ¼ rp i þ r i g r i v in ðv i v n Þ r ee ½E þ ðv e BÞŠ; ð1þ m e Þþ r ie ½E þ ðv i BÞŠ: ð2þ [9] In the equations all ions are treated as one species, i.e., with the same average velocity, temperature, and mass. The definitions used in the equations are as follows: r e, r i density of electrons and ions, respectively; v e, v i velocity of electrons and ions, respectively; p e, p i thermal pressure of electrons and ions, respectively; m e, m i mass of electron and ions, respectively; v en effective momentum transfer collision frequency of electrons with neutral species v in total effective momentum transfer collision frequency of ions with neutral species; v n velocity of neutral species; g gravitational acceleration; e magnitude of the electric charge of electron; E electric field. m i

3 WANG AND NIELSEN: HYDRODYNAMIC WAVES SIA 2-3 [1] The friction term is ignored based on the assumption that the velocity differences among the electrons and ions in the plasma (no current) and the neutral gases are small, i.e., v e v i v n. This is valid if the interactions with temporal scale longer than several hours between ionosphere and atmosphere are not focused, as in this present paper. Adding the two equations together, the momentum equation for the plasma is obtained as r dv ¼ rpþrg ; ð3þ where r e r i r (plasma density), p i + p e = p (plasma pressure), and v is the velocity of the plasma. It is notable that the magnetic and electric fields disappear naturally in the equations since the electric current is not considered. [11] As Barth et al. [1992] summarized that transport processes take over from photochemical ones somewhere between 17 and 2 km in the dayside ionosphere of Mars, the photochemical production and loss terms in the mass equation can be neglected over 2 km. [12] Hence the governing equations of plasma including the continuity, momentum, and energy equations are r dv þr ð rv Þ ¼ ¼ rpþrg; dp ¼ g p r where g = 5/3 for a perfectly conducting plasma. [13] Now consider a two-dimensional coordinate system with x (horizontal; parallel to the planet s surface and pointing in the antisolar direction) and z (vertical; pointing toward zenith). Let a certain disturbed parameter q be expressed as q = q + q, where q is the background value, q the perturbation value, and q q. Substituting this into (4), we obtain a set of equations of zeroth (background parameters), first (products of background and perturbation parameters) and second (products of perturbation and perturbation parameters) order terms. The zeroth-order terms on the left side of the equations equals exactly the zeroth-order term on the right side; the reason is that the background parameters satisfy (4), and therefore they can be erased. Furthermore, we reasonably omit the second-order terms because q q. In this way the governing equations for the perturbation, q (first-order terms), are obtained. (For the methodology, see Boyd and Sanderson [1969]). To simplify the expression, we replace the symbols of perturbation value q with q and obtain the following equations: dr @z ¼ ; >< du þ ¼ ; dw þ þ rg r ¼ ; dp þ p >: g r þ Þ¼; where u and w is the x and z component of velocity, /@x /@z are reasonably assumed to equal zero for simplification [Theis et al., 194; Krymskii et al., 1995; Vaisberg et al., 1976; Zakharov, 1992], i.e., the unperturbed vertical velocity has no gradient in the vertical and horizontal direction and the unperturbed horizontal velocity has no horizontal gradient. [14] If the equations have single wave solutions with a form of exp[i(k x x + k z z st)], in which k x, k z the horizontal and vertical wave number, respectively, and s the angular frequency of the wave. Let w = s k x u k z w (the angular frequency dr ð4þ ð5þ measured in the coordinate moving with the plasma), then (5) can be rewritten þ ik 1 zr iw ik x r B W ik iw z g C@ r r A p g iw iwg p þ xr p g C A ¼ ; ð6þ where U, W, P, and R is the amplitude of u, w, p, and r, respectively. [15] If the equations have nonzero solutions, the coefficient determinant must be zero. This requirement leads to the dispersion relation w 4 w 2 b w2 p k z w Jk x k z r þi Jk x w 2 w 2 b þ kz gw 2 k x k z g w ¼ ; where is the buoyancy frequency, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w b ¼ x ¼ ; J ¼ g ¼ ð1 gþr T r ð7þ ðþ ð9þ ð1þ with p = r R /@x, and where R = R/m, R is the universal gas constant, and m is the mean molecular weight of the ions. [16] Let the real and imaginary parts of (7) both equal to zero, and we obtain >< k x ¼ >: k z ¼ ð 2 w 2 b =x J Þ 2J 2 gxr w b; 2 w 2 b =xþjþ 2 Jg 2 þg 2 xr 2 T 2 =@zþ 2 ½ Š gxr T where = w/w b w b ð Þ 2 1 ; (11a) ð11bþ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 w 2 2 b J 41 ð gþ2 g 2 2 w 2 b x g 2 ð@u =@zþ 2 : ð12þ [17] Note that for a system in which there are no horizontal gradients in the background plasma, i.e., J = x =, (7) reduces to the standard dispersion relation for the AGW mode. On the other hand, if there are gradients in the background plasma, then the AGW mode may still be excited, but now in addition, a new wave mode can also be excited and together with the AGW mode form a hybrid wave. As a matter of fact, it will be shown below that even in the absence of gravity, when no AGW can be excited, the new wave mode may be active. This new wave mode is called the background gradient wave (BGW), because it

4 SIA 2-4 WANG AND NIELSEN: HYDRODYNAMIC WAVES depends critically on the presence of background gradients in pressure and density and a vertical gradient in the horizontal velocity. 3. Estimates of Wavelength and Frequency [1] First, some considerations are made in order to simplify (11a) and (11b), and then the frequencies and wavelengths of the standard gravity wave and of the hybrid wave are estimated. [19] Without calculating x directly, its magnitude can be estimated from (9) by the following deduction. It is known that r 1 (@r /@z) = H 1, where H is the scale height of electron density; for Mars H is 3 km [Hanson et al., 1977]. For a stationary /@t =, and for a nearly incompressible gas, rv, it follows from the continuity equation in (4) that v rr =, and that x ¼ ¼ w u ¼ w u H : ð13þ [2] We write the vertical variation of the horizontal velocity as u = U exp((z z )/H u ) where U the magnitude of u and normally H u a constant with average value of 1 km for Mars [Shinagawa and Cravens, 199; Krymskii et al., 1995]. The order of magnitude of w is taken from theoretical studies [e.g., Shinagawa and Cravens, 199; Krymskii et al., 1995], and normally it is a few m s 1 for Mars. Substituting typical values observed for the concerned parameters 3 km in the ionosphere, we get w b.1 s 1 and x 1 7. By checking the magnitude of every term in (11a) and (11b) we find some terms are negligible compared with others, and the equation can be simplified as k x ¼ gu3 Hw3 b >< 3 ð1 gþ 2 w 2 GgH ; u gu 2 k z ¼ HH uw 3 b ð 2 1Þ >: ð1 gþw Hu 2g2 þg 2 GHu 2 w ; where G = R T /H 5 m s 2. (14a) (14b) 3.1. Gravity Wave [21] It is notable that if = 1 (i.e., w = w b ), neither (11a) and (11b) nor (14a) and (14b) is valid. However, from (7) it can be seen directly that k z must be zero, while k x may be chosen freely in order for a solution to exist. Since w is the frequency observed in the frame moving with the plasma with velocity v ¼ u ^x þ w ^z, this wave propagates horizontally in the coordinate system moving with the plasma flow. It is a standard gravity wave only controlled by gravity and buoyancy forces [Yeh and Liu, 1972] Hybrid Wave [22] Equations (14a) and (14b) can be rewritten to express the horizontal and vertical wavelengths of the hybrid wave: L x ¼ 2p 2p ¼ ð 1 g Þ2 w 2 GgH u >< k x gu 3 ; Hw3 b 3 (15a) L z ¼ 2p ¼ 2p ½ ð1 gþw Hu 2g2 þg 2 GHu 2 w Š >: k z gu 2 HH uw 3 : b 2 ð 1Þ (15b) [23] The vertical wavelength of waves in the ionosphere must be limited in magnitude: it can neither be too long (e.g., more than 5 km on Mars) nor too short (e.g., less than 1 km). In the former case, it cannot propagate in the relatively narrow space (the depth of the ionosphere above peak altitude is only 15 km on Mars). In the latter case, the wave energy will easily be damped by factors such as collision, and in any event it cannot be detected by available or planned measurements. Although the horizontal wavelength can be much longer up to some thousand kilometers, it also should not be too short. Since (15a) and (15b) show L x 1 4 / 3 km and L z 1 1 /( 2 1) km, it can be argued that is around or w is around s 1 on Mars (see Figure 2). 4. Two Wave Modes [24] Acoustic-gravity wave is a mode formed by coupling between an acoustic wave and a gravity wave. The wave discussed previously, which propagates only horizontally (in the frame moving with the plasma) at the buoyancy frequency, is a special branch of the AGW, where the acoustic wave is not included, i.e., it is a pure gravity wave. It is the eigenmode in a horizontally stratified incompressible fluid in a gravitational field [Yeh and Liu, 1972]. [25] If the density and horizontal velocity of the background has no gradient, i.e., x = /@z =, equation (7) is actually the standard dispersion relation of the AGW mode. It is a mode controlled by gravity and buoyancy forces as well as by the compressibility of the plasma [Yeh and Liu, 1972]. The wave with frequency of s 1 described by (14a) and (14b) or (15a) and (15b) seems also an AGW, which is influenced by the gradients in the background parameters and is therefore more complicated than normal AGWs. However, we would like to argue that this wave is actually a hybrid wave resulting from coupling between AGW and another different hydrodynamic wave mode. It is even conceivable that the other mode may be excited even in the absence of gravity, i.e., without simultaneous excitation of the AGW mode. [26] To illustrate this point the physical processes in the ionosphere associated with this second mode are further analyzed for the case of zero gravity. It is shown to be possible for the new wave mode to be excited even in such a case. If the gravity force is not considered, (7) leads to a frequency and wave number of the second wave mode: < w ¼ : k z ¼ p ffiffiffiffiffi Jx g x R : ð16þ [27] Even though the values of J and x for the case of no gravity is not known, the implication of (16) is that the new wave mode is caused by the existence of horizontal gradients in pressure and density (nonzero values of parameters J and x). This wave propagates horizontally freely but can also propagate vertically if there is a vertical gradient of horizontal velocity /@z 6¼ ). [2] Since the background ionosphere is in the gravitational field, one cannot conclude that the second mode is independent of gravity. However, it is safe to state that the gravity force is not involved in the generation process of the second wave mode; this is clear from (16). The point is that the second mode is dependent on gravity in as far as the gravity influences the background plasma, but the excitation of the second mode is not dependent on gravity, i.e., the condition for excitation are given by (16), and that equation does not involve the gravity. This is analogous to acoustic waves in the Earth s atmosphere where the background atmosphere is controlled by gravity, while the excitation of the wave is independent of gravity. [29] What are the driving forces of this new type of wave? From the equation for the horizontal momentum in (5) it can be seen that the horizontal component of the pressure gradient force r 1 (@p/@x) acts to balance another force : the horizontal

5 WANG AND NIELSEN: HYDRODYNAMIC WAVES SIA 2-5 Figure 2. Wavelengths with frequency parameter at Mars. (Solid lines indicate normal case; dashed lines indicate the cutoff case; Thick lines show horizontal wavelengths; thin lines show vertical wavelengths.) component of term v rv, i.e., v ru (= w@u /@z). Thus, if there is a horizontal pressure gradient in one direction, then plasma will be moved in the horizontal direction in accordance with the gradient. The mass conservation law then comes into work, and v ru (= w@u /@z) will act as a restoring force to force the plasma back against the pressure gradient. In a weak viscous liquid this interplay between the pressure gradient force and the momentum force leads to a plasma oscillation. During this balancing process between r 1 (@p/@x) and v ru the horizontal acceleration changes its direction continually and results in a horizontal oscillation. This oscillation spreads away p ffiffiffiffiffi horizontally and becomes a wave with an eigenfrequency Jx. If the horizontal velocity is in a vertical shearing pattern /@z 6¼ ), the wave will also propagate vertically according to its dispersion relation. [3] Since the enhancement of the second mode depends on the existence of the gradients in background plasma pressure and density and also the vertical propagation relies on the vertical gradient of background horizontal velocity, we therefore suggest this mode could be called background gradient wave mode. So the hydrodynamic wave suggested here is actually an AGW-BGW hybrid wave. 5. Discussion [31] We have considered a planetary ionosphere, possibly with a weak magnetic field, which is interacting with the solar wind. Fluctuations in the solar wind pressure cause the ionospheric plasma to continuously seek toward a new equilibrium. The processes that will bring the ionosphere to equilibrium are probably wave activity in the ionospheric plasma. It has been demonstrated that two different hydrodynamic modes may be excited in the topside Martian ionosphere. One mode is an AGW as known from the Earth s atmosphere [see Yeh and Liu, 1972] and also found in the lower Martian atmosphere [Zurek et al., 1992, and references therein]. The other mode (BGW) excited owing to horizontal velocity gradients in the background plasma flow and this mode is here predicted to couple with AGW and form an AGW-BGW hybrid wave, which propagates both horizontally and vertically in the ionosphere. [32] The AGW component generated on top of the ionosphere has no relationship to the gravity waves comprehensively discussed by Mayr et al. [1992], which are regarded as internal gravity waves enhanced in lower thermosphere. These waves are not considered here because they are not generated by the solar wind. Furthermore, on the basis of the theory of Hines [196], which points out that propagating gravity waves tend to grow exponentially with altitude, Seiff et al. [1992] concluded the lower-source gravity wave in Venus and Mars will reach its saturation below 2 km and never affect the topside ionosphere. [33] As argued by Krymskii et al. [1995], if the solar wind pressure is very strong the value of the parameter H u may decrease from a value of 1 to 2 km. Then L z increases violently, while the depth of the topside ionosphere decreases rapidly, up to so large a value that the ionosphere can not contain this wave any longer (see Figure 2). That means there will be no waves excited when the Martian ionosphere is cut off during strong solar wind pressure. Thus, during typical conditions, i.e., relatively weak solar wind pressure, there are expected to be AGW-BGW hybrid waves in addition to pure gravity waves excited in the Martian topside ionosphere. The waves are likely to be suppressed only during strong solar wind pressure. [34] Even though no observations of the Martian ionosphere can be used directly to search for wave activity, these theoretical results are at least not inconsistent with available measurements. Kliore [1992] showed that there are often vertical variations in the electron density exceeding the statistical fluctuations associated with the measurements (see Figure 1). More recent observations on board Mars Global Surveyor (MGS) also shows, on occasion, a trend toward vertical wavelike structures in the electron altitude profiles of Martian ionosphere (L. Tyler, private communication, 21). [35] Since the ionosphere of Venus is free of intrinsic magnetic fields and spacecrafts penetrating close to Venus have detected wavelike plasma irregularities in the topside ionosphere [Brace et al., 19], it is of interest to also apply these theoretical considerations to ionospheric observations at Venus [Luhmann and Cravens, 1991; Kliore, 1992; Miller et al., 19; Knudsen et al., 192; Cravens et al., 194; Kasprzak et al., 19; Whitten et al., 194; Theis et al., 194; Shinagawa and Cravens, 19; Zhang et al., 199]. We find that the trend in electron densities at Venus also tends to support the concept of hydrodynamic waves. Figure 3 is the diagram of wavelength versus frequency on Venus. Although the allowed vertical wavelength on Venus is obviously longer than that on Mars, it is nevertheless reasonable because of a thicker topside ionosphere on Venus (the depth of the Venusian ionosphere over peak altitude is 3 km, two times of Martian (see Figure 1b). At low solar activities the vertical wavelength is larger than 5 km. At high solar activities, however, it can be larger than 1 km. It is unlikely that a 1-km wavelength wave can always exist in the ionosphere, even though at that time the ionosphere is much thicker [Knudsen et al., 197]. That means at high solar activities the AGW-BGW waves in the Venusian ionosphere may disappear, but they could be present at other times. [36] Table 1 shows parameters relevant to the wave activity at both Mars and Venus. Since magnetic fields are not involved in the discussion above, the waves generated in the ionosphere can be regarded as pure hydrodynamic ones. However, to what extent a magnetic field, i.e., electric currents considered, would affect the features of the hybrid waves, a new dispersion relation will be needed then for discussions in details. Nevertheless, it is safe to state that the magnetic field will not prohibit the enhancement of

6 SIA 2-6 WANG AND NIELSEN: HYDRODYNAMIC WAVES those waves. It is not reasonable to think that the weak magnetic field can stop the excitement of AGW since this wave mode can survive even in Earth s topside ionosphere where the field is much stronger [Yeh and Liu, 1972]. In the BGW mode the plasma particles oscillate horizontally, and the oscillations are surely permitted by a horizontal magnetic field. Usually, the ionospheric magnetic field on the two planets can be treated as nearly horizontally except the cusp-like regions on Mars [Acuña et al., 199]. Thus it is to be expected that the BGW mode can also be excited when the magnetic field is included (i.e., currents are considered). Of course, the waves occurring in the cusp-like regions in Martian magnetic field will be different from waves in regions of horizontal field. Because the ionosphere background in cusp-like region is rather different [Ness et al., 2; Mitchell et al., 21], many assumptions about the background in this present paper could be invalid in the magnetic anomaly regions. However, it is difficult to discuss because of deficiency of observations in the cusp-like regions. [37] The perturbation equations on the dayside interaction surface between the solar wind and ionosphere of a weakly magnetized planet has been analyzed, and pure hydrodynamic waves are found. Using parameter values appropriate for Martian and Venusian ionospheres, some conclusions about these waves can be listed as follows: 1. A wave hybridized by two wave modes could be generated in the topside ionosphere. 2. One of the modes is an acoustic-gravity wave (AGW). Specially, there is a branch of pure gravity wave, with a frequency equal to the buoyancy frequency, and it propagates only horizontally. 3. The other wave mode (BGW mode) is caused by the significant horizontal gradients of plasma pressure and density and propagates horizontally. If there is a vertical gradient of horizontal velocity, the wave will also propagate vertically. Table 1. Main Wave Parameters on Mars and Venus Parameter Mars Venus H, km 3 5 H u, km 1 14 w b, Hz.1.1 G, ms (low solar activity) 3 (high solar activity) w,ms Absence under very strong solar wind pressure sometimes at high solar activity 4. Although the frequency of the AGW-BGW hybrid wave is limited by the equations to only occur for frequencies not equal to the buoyancy frequency, the frequency of long-lived BGW in the ionosphere has generally to be some hundrehs (on Mars) or some tenths (on Venus) times the buoyancy frequency. 5. Horizontal wavelength of the hybrid wave might be of global scale, while the vertical wavelength is of the order of some to some tens of kilometers, and usually is larger on Venus than on Mars. 6. In the Martian ionosphere the hybrid wave will disappear when the topside ionosphere is cut off. 7. In the Venusian ionosphere the hybrid wave is always present at low solar activities but may disappear at times of high solar activities. [3] The theoretical results of this paper suggests that there might be some unique waves in the ionosphere of weakly magnetized planets and also gives some properties of those waves in the ionospheres of Mars and Venus. A new type of topside remote sensing HF radar for exploring the Martian topside ionosphere [e.g., Picardi et al., 199; Nielsen et al., 1994] may contribute to observations of these waves. [39] Acknowledgments. The authors thank Q. G. Zong and S. Y. Fu for the fruitful discussions and S. Mathew for his help with the figures. This research was funded by Bundesministerium für Bildung und Forschung through Deutsche Zentrum für Luft- and Raumfahrt e.v. (DLR) grant 5 QM 4. J.-S. 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