An assessment of how a combination of shears, field-aligned currents and collisions affect F-region ionospheric instabilities

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1 J. Plasma Physics (2006), vol. 72, part 4, pp c 2006 Cambridge University Press doi: /s x Printed in the United Kingdom 1 An assessment of how a combination of shears, field-aligned currents and collisions affect F-region ionospheric instabilities J.-P. S T.-M A U R I C E 1, J.-M. N O ËL2 and P. J. P E R R O N 2 1 Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E2 (jp.stmaurice@usask.ca) 2 Department of Physics, Royal Military College of Canada, Kingston, Canada K7K 7B4 (Received 31 October 2005, accepted 21 November 2005) Abstract. We present an in depth study of the fluid limit of a kinetically derived collisional, current-driven instability that includes shears in the field-aligned currents as well as collisions. We show how the theory presented here generalizes other theories, including the collisionless current-driven electrostatic ion acoustic instability and its sheared collisionless version. We offer a low-frequency generalization of the zero frequency ion shear driven instability by minimizing the relative drift magnitude as well as the shears themselves. We discuss the implication of our theoretical framework both for strongly field-aligned modes and modes where the wavevectors have arbitrary angles with respect to the ambient magnetic field. We discuss the results in terms of F -region irregularity observations of coherent echoes by ionospheric radars. 1. Introduction Coherent radars have become increasingly important for the study of the ionosphere, particularly at high latitudes. For instance, we have witnessed the deployment of a large array of high-frequency (HF) radars through the SuperDARN network. These radars rely on refraction to bend the radar waves, in order to observe field-aligned echoes in the F -region and retrieve the plasma drift in the process from the Doppler shift information by assuming that the structures are field aligned and that they follow the ambient plasma drift. Incoherent scatter radars also observe coherent echoes from irregularities. Most of the time, the irregularities are from the E-region and are field aligned (e.g. Foster and Erikson 2000). Notable exceptions to this have, however, also been recorded, particularly in association with ion-acoustic echoes observed along the geomagnetic field (e.g. review by Sedgemore- Schulthess and St.-Maurice 2001). Incoherent scatter radars have also occasionally detected powerful F -region echoes that are field aligned (Foster 1990) and even coherent E-region echoes that are not field aligned (Ruster and Schlegel 1999). The detection of unexpected coherent echoes by incoherent scatter radars raises important questions concerning the generation mechanisms that spill over into coherent radars. The latter generally operate at lower frequencies and should, if

2 2 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron anything, be more sensitive to destabilizing mechanisms: their larger wavelengths allow them to detect echoes for past the Debye sphere limitations. Thus, the clear detection by incoherent scatter radars of powerful echoes that are not field aligned suggests that unexpectedly large currents and/or shears are present at times. Recognizing the signatures from such features with coherent radars would add to their capabilities. First, however, there is the need for a systematic exploration of the mechanisms that can produce the instabilities behind the coherent echoes. Much work has indeed been done on the subject, and the gradient-drift (or universal) instability with its field-aligned structures has been the mechanism of choice (e.g. Kelley 1989). In contrast, current convective instabilities have also been invoked, but not clearly established, perhaps because they require rather intense magnetic field aligned current densities, which, while they may well be present, are highly localized. In this paper we will explore, in particular, the effects on threshold conditions of parallel currents in conjunction with shears. The interest in scrutinizing the various mechanisms lies, in part, on the interpretation of the resulting spectra. Strong velocity shears have been proposed, for instance, as a trigger mechanism for waves of the order of tens of metres at the lower hybrid frequency (Kintner and D Angelo 1977). The wavevector in that case is perpendicular to the magnetic field, which means that the waves should be Doppler shifted with a mean value equal to the E B drift of the plasma. Nevertheless, in addition, large thermal ion or electron drifts along the magnetic field line should at times be involved in such a way as to contribute modes that would not necessarily be seen to be E B drifting, to be purely field aligned and/or have high frequencies. The presence of these processes might be made manifest through unexpected Doppler shifts, unusual elevation angles or it could involve other unusual spectral signatures. For instance, the spectra might be more powerful than usual (as a result of direct excitation as opposed to a trickling of energy through cascading). Or they could also be narrower, i.e. more coherent, than usual, again indicating that the structures would not be associated with cascading or strong turbulent processes: as a result the waves could be said to be longer lived than usual (this is one interpretation of narrow Doppler widths). They could also be considered as less dependent on turbulent electrostatic electric fields for their existence (this is another way to look at spectral widths, as turbulent-induced diffusion processes). Some of the effects of large thermal electron drifts have been quite thoroughly explored before and will therefore not be looked at here. For instance, there is the generation of ion cyclotron waves by parallel currents carried by large thermal electron bulk flows (e.g. Kindel and Kennel 1971; Satyanarayana and Chaturvedi 1985). Suffice it to say that the modes so generated are perpendicular to the magnetic field, so that while their spectra may look unusual, the waves will have the usual mean Doppler shift associated with the bulk motion of the plasma, i.e. the E B drift itself. We will, however, revisit a second mechanism that can be triggered by thermal particle drifts along the magnetic field, namely, the ion shear driven instability initially proposed by D Angelo (1965). In this case, unstable waves are excited by large horizontal shears in the vertical ion drift (e.g. Basu and Coppi 1989 and references therein). This particular instability requires no density gradients as such, and the shears might be small enough to be of interest. We will also revisit a third category associated with thermal electron flows. The mechanism we have in mind here, namely, the current convective instability, has

3 Non E B drifting F-region plasma waves 3 been studied in a reasonable amount of detail. In this case it is again assumed that the destabilized modes are very nearly perpendicular to the magnetic field lines, so that their Doppler shift is once again very similar to the plasma E B drift. However, we will also show here that a second mode can be excited under the same circumstances as the current convective instability, and for which, unlike all other modes mentioned so far, a substantial difference can exist between the E B drift and the predicted Doppler shift of the waves. This mode is really an extension of the pure ion-acoustic mode that can be triggered along the magnetic field direction, but we wish to stress that it can be detected over a range of angles covering from nearly perpendicular to the geomagnetic field to parallel to it, with a Doppler shift that will vary according to the direction of the observations. This mode is not really new, but little attention has been paid to it. Its derivation also requires a kinetic derivation because of the importance of Landau damping effects, if we are to assess threshold conditions in such situations. We will, nevertheless, show that this particular eigenfrequency can be derived from fluid theory, as long as we do not wish to obtain drift threshold conditions or growth rates. To focus the mind, we will therefore use a simple generalization of the current convective derivation using fluid theory before moving on to a more appropriate kinetic-based derivation. Our paper will be organized as follows: following a quick survey of fluid modes, we will explore the fluid limits of a generalized kinetic derivation of modes excited by parallel ion and electron drifts in a collisional and sheared region. The kinetic derivation will be based on a suggestion by Gravrishchaka et al. (1998) that parallel currents with horizontal shears could destabilize oblique modes in the plasma. However, since the authors of this particular theory proposed that it had ionospheric implications, but did not include collisions in their work, we will repeat their derivation by including collisions. In the process of this generalization attempt we will show that we also run into the ion shear driven mechanism that was studied extensively by Basu and Coppi (1989). The development of one formulation that regroups a large number of theories into one single, generalized kinetic-based framework appears worthwhile to us. Finally, while in many ways the present work is similar to the approach taken by Gravrishchaka et al. (1998), we note that our derivation will proceed from a somewhat different starting point, which, to us at least appears to be more straightforward. 2. A quick look at fluid modes in the presence of parallel drifts, with and without ion inertia Ossakow and Chaturvedi (1979) first studied the current convective instability without diffusion. Soon after, they added diffusion (Chaturvedi and Ossakow 1979) to obtain full growth rate conditions. They showed that the growth rate, γ, ofthis very low-frequency (in the E B frame of reference) instability is given by γ = (E/BL)(ν eν i /Ω e Ω i )+(k V d /k L)(ν e /Ω e ) k 2 /k2 + ν ν e k eν i /Ω e Ω i Ω e Ω C 2 s 2 i k2 C2 s ν i [ 1+ ν 2 i /Ω2 i ν e ν i /Ω e Ω i + k 2 /k2 ] (2.1)

4 4 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron where E and B are the ambient electric and magnetic field, L is the density gradient scale and can have either sign, ν j is the collision frequency of species j, Ω j is its cyclotron frequency and subscripts e and i refer to electrons and ions respectively. In addition, k is the component of the wavevector along the magnetic field, and k its component perpendicular to the magnetic field, while Cs 2 =(T e + T i )/m i is the isothermal ion-acoustic speed and V d is the electron drift along the magnetic field as seen in the ion frame of reference. Equation (2.1) first has the merit of showing that in the absence of substantial parallel drifts, the maximum growth rate is for k 0, in which case the growth rate is given by γ E 1 B L ν eν i k 2 C2 s. (2.2) Ω e Ω i ν i For gradients produced on the edge of so-called ionospheric blobs, this equation shows that tens of metre sized waves can, in fact, be excited directly by the gradient-drift mechanism in the F -region if the plasma drift is of the order of several 100 m s 1 and the gradient scales are of the order of 10 km or less. Such situations are certainly possible, but our main point in this work is that while the waves could be directly excited, their mean Doppler shift would still be very close to the plasma E B drift (see (2.6) below). For the standard current convective situation (V d large and k non-zero) one could compute the angle for which the growth rate is largest (e.g. Kelley 1989, chapter 8). Angles of radians, that is, of the order of (ν i /Ω i )(E/BV d ) are possible. This emphasizes again the near perpendicularity of the instability and the consequently small Doppler shifts that will be observed in the E B frame of reference. In fact, even when V d is as large as 50 km s 1, with k /k of the order of 10 2 or greater the zero frequency waves are still strongly damped. This can be seen with the help of (2.1), which in this case gives the result γ V d k ν e k2 C2 s. (2.3) L k Ω e ν i However, this is not the full story. In the standard current convective derivation ion inertia is neglected and possible frequencies that would be comparable to, or greater than, the ion collision frequency are ruled out. If the inertial term is inserted in the momentum equation, however, we have another problem, that is kinetic effects associated with Landau damping effects should also be taken into account. The normal way to deal with these effects is to use a kinetic theory. Before going to a kinetic theory, however, we wish to highlight what happens when ion inertia is inserted into the current convective derivation. Not only is this insertion necessary away from zero frequencies at wavelengths a few tens of metres or less, but we will also show that the eigenmodes can change noticeably with direction in that case. These new eigenfrequencies (which are only new in the context presented here) will provide us with a guide for the kinetic derivation that must be used if we are to proceed with a discussion of threshold conditions or growth and decay rates in the presence of strong thermal electron drifts. To proceed with a fluid theory for a regime that can so easily be affected by kinetic effects, we follow the textbook approaches and do not worry about growth or decay rates (the full kinetic theory below will remedy this shortcoming). In the process, we can only focus on low-frequency waves, which allows us, however, to use

5 Non E B drifting F-region plasma waves 5 simple Boltzmann electrons. Note that, here, we will also neglect the small effect of ion electron collisions on the ion momentum. From combining the ion continuity and momentum equations and using Boltzmann electrons to remove the perturbed parallel fields we then obtain, following the usual linearization procedures for frequencies measured in the E B frame of reference iωω i δn ik y δφ (ν in iω) n 0 BL + δφ k2 B + ik E δn + T i B n 0 eb k2 δn + k 2 Ω i δn n C2 s 0 (ν in iω) 2 =0. n 0 (2.4) This equation is similar to (5) in Chaturvedi and Ossakow (1979) except for the fact that ion electron collisions have been dropped from the present work and the ion collision frequency has been replaced here by (ν in iω). This is to be expected once it is decided that ion inertia ought to be kept. Once again using Boltzmann electrons, that is, the relation δn/n 0 = eδφ/t e,we get from (2.4) iω i k y L (ν in iω) Ω 2 i T e m i + k 2 C 2 s (ν in iω) Ω 2 i + i k E B (ν in iω) Ω i + k2 C2 s (ν in iω) =0. (2.5) For a current convective type of situation, we have ν in ω.however,havingdropped the electron terms here that are related to the growth or decay rates we can only write an expression for the real part of the frequency for this case, namely, ω R = ν in k E Ω i B k y ν in T e L Ω 2. (2.6) i m i We remind the reader that this frequency is expressed in the E B frame of reference. In that frame, the frequency obtained in (2.6) is indeed small. The first term is a Doppler shift that goes at the Pedersen drift of the ions. This is about two orders of magnitude smaller than the plasma E B drift. The second term, which is typically much smaller, is the diamagnetic drift associated with zeroth order gradients. As expected for this particular case the Doppler shift in the absolute frame is therefore very close to the plasma E B drift. Of interest in the present publication is a second limit for which we can have the opposite situation, namely, ω R ν in. It is easy to show from (2.5), that we now have in the limit in the absolute frame, a frequency of oscillation given by k 2 ω R = k V 0 ± C2 s 1+k 2 C2 s /Ω 2. (2.7) i Of course, to destabilize this frequency, one must have fast enough parallel thermal electron drifts, but we stress again that the fluid theory we have used here precludes its determination at this point. Nevertheless, it is important to realize that the eigenfrequency given by (2.7) is correct (for example, one can find it in Mikhailovskii (1974, chapter 8), where it is derived for a stable plasma using kinetic theory). More importantly for the HF radar applications we have in mind here, we note that it contains a mixture of parallel ion acoustic mode and of cyclotron mode, which could be detected with SuperDARN radar geometries more easily than pure cyclotron

6 6 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron modes (the latter mode requires a wavevector nearly exactly perpendicular to the magnetic field) or pure ion acoustic modes (wavevector along the magnetic field). The reason that the mode given by (2.7) would be easier to detect is that it is not aspect sensitive. As long as the parallel electron drifts are large enough, the radars could see an instability for a large range of possible directions. A second point to note about this high-frequency (in comparison with (2.6)) mode is that it has a strong cyclotron frequency flavor at 10 m. This is because in the F -region with a typical 1500 m s 1 ion-acoustic speed, the second term in the denominator of (2.7) is large enough compared to 1 that we can write 3. The generalized kinetic derivation ω R k V 0 ± k k Ω i. (2.8) In order to study the threshold conditions needed to generate the eigenfrequency uncovered in the previous section we now proceed with a kinetic theory of streaming instabilities. In view of a proposal by Gravrishchaka et al. (1998) that shears could play an important role in destabilization, we have included their source of shears as well. In that model, we have a vertical drift in the thermal electrons, and drift has a gradient in the x direction in the plane perpendicular to the magnetic field. This situation should be expected, for instance, near steep gradients in precipitation on the boundaries of arcs (e.g. St.-Maurice et al. 1996; Noël et al. 2000, 2005). However, in order to apply the theory to the ionospheric F -region, we were required to add collisions to the treatment used by Gravrishchaka et al. (1998). Finally, we have chosen to derive the results using what is, to us, a more transparent procedure than that followed by these authors. That is to say: we will obtain similar results using a local expansion of a drifting Maxwellian about a particular position in space. But since this means that our derivation is slightly different from the one that has been used previously, we had to add some detail to clarify our procedure. On the other hand, several steps are a repetition of what can be found in textbooks. Our derivation will therefore remain concise. One final point should be mentioned before we embark on this derivation, namely: we are using a simple BGK collision model (Bhatnagar et al. 1954) to describe our electron and ion collisions. The simplicity of the collision model obviously introduces some uncertainties on the threshold conditions that we will be finding. On the other hand, this collision model is widely used, for instance in the E-region literature Salient features of the derivation We start with the method of characteristics, using the BGK collision model. After a plane wave decomposition, the perturbed distribution, δf jk of a species j is given by δf jk = q j m j where 0 dτ (δφ k ) f 0j v exp(ib j (τ))+ ν j n j δn j 0 dτf Mj exp(ib j (τ)), (3.1) b j (τ) = k (r r )+(ω + iν j )τ. (3.2) There are several ways to go about choosing an appropriate function f Mj. We will use a Maxwellian with the temperature and drift of the species as was done in the original BGK model.

7 Non E B drifting F-region plasma waves 7 Since the modes under study are electrostatic we add the condition (δφ k )=ikδφ k. (3.3) From an integration over all velocities the perturbed densities are given by where H j = ω2 pj k 2 δn j = k2 δφ k 4πq j dv 0 H j 1+U j, (3.4) dτik f 0j v exp(ib j (τ)) (3.5) and U j = ν j dv dτf Mj exp(ib j (τ)). (3.6) n j 0 With the help of the Poisson equation (3.4) we then obtain the well-known dispersion relation for electrostatic modes H i + H e +1=0. (3.7) 1+U i 1+U e A plane wave decomposition is next performed along the yz-plane. In the weak shear limit we use the quasi-local approximation mentioned above, which gives f f(x)+ x f ( x f 0m + f 0m V d x m ) T (v z V d ), (3.8) where f 0m is a Maxwellian drifting along z with speed V d and V d = dv d/dx. The kinetic effects brought up by Gavrishchaka et al. (1998) are triggered by the effect of x on the H functions in (3.7) through the characteristics of the motion. Note that we are dropping subscripts from here on because the functions are treated in exactly the same way, for both ions and electrons. Next, wehavetoevaluatek f/ v, for this we need the characteristics of motion. The effect introduced by our x term is then seen to come from a v y x contribution, which contains a linear term with a slight nonlinearity that is discarded. Retaining only the linear (average) contribution we get in the end [ f k y = m ( ) 2 m v y T k y v sin(ωτ + θ)+ V T d(v z V d ) k y v 2 2Ω ] f 0m. (3.9) After the usual Bessel decomposition, it is now easy, though a bit tedious, to show that for our local Maxwellian we obtain ( ) 1/2 H = ω2 p T k 2 2πm k z (v z V d )[Γ n (b)+(k y /k )(V d dv /Ω)Γ n (b)] nωγ n (b) z ω k n z v z nω+iν exp ( m2t ) (v z V d ) 2, (3.10) where Γ n(b) = [(1 b)i n (b)+bi n(b)]e b, (3.11)

8 8 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron b = ky 2 ρ 2 L and ρ2 L = T/mΩ2 is the square of the Larmor radius. The Γ n term is the new contribution produced by shears, and the function it represents can be contrasted with the function Γ n (b) related to the usual non-shear terms and which is given by Γ n (b) =I n (b)e b. (3.12) Therefore, the only effect caused by shears in V d is the relatively simple modification of the k z (v z V d ) term in (3.10), through the Γ n term. In the absence of collisions (ν 0), and for small Larmor radius corrections (so that Γ n Γ n ), (3.10) leads to an expression similar to the expression found in Gavrishchaka et al. (1998). However, the addition ofcollisions still means that, as farash is concerned, we now have (ω + iν) replacing ω in the plasma dispersion function Z Final dispersion relation With the v z integral being performed next, we then end up, for ions, with the expression H i = 1 [ k 2 λ 2 1 k y V di + ( ( ) ω + Γ n (ky 2 ρ 2 iνi ω + iνi nω i ) )Z i di k z Ω i 2 kz v n ti 2 kz v ti n Γ n(k 2 y ρ 2 i ) k y k z V di Ω i ( ( ω + iνi nω i ω + iνi nω )Z i )], (3.13) 2 kz v ti 2 kz v ti where ω = ω k z V di. (3.14) Note that we have used the easily derivable result + Γ n =1in order to obtain this expression. For electrons the focus here on small frequencies means that ω Ω e or, equivalently, ρ 2 e =0, so that we end up with [ ( ( ) ω kz V de + iν e ω kz V 1+ de + iν )Z e 2 kz v te 2 kz v te H e = T i T e 1 k 2 λ 2 di + k y k z V de Ω e { ( ( ω kz V de + iν e ω kz V 1+ de + iν )Z e )}]. (3.15) 2 kz v te 2 kz v te The U terms are not affected by shears: they come from Maxwellians that the plasma wants to relax to, and therefore they involve no product of x with v y, unlike the H terms. The expressions for U j then take the standard values, namely, iν i ( ) ω + U i = Γ n (ky 2 ρ 2 iνi nω i )Z i (3.16) 2 kz v ti 2 kz v ti U e = n ( iν e ω + iνe + k z (V di V de ) Z ). (3.17) 2 kz v te 2 kz v te 3.3. The fluid limit Equations (3.13) (3.17) can now be put into (3.7) and the roots of the resulting dispersion relation can then be studied. The resulting expression is rather complex however, since even in the limit of zero collision frequencies it matches the already complicated expression published by Gravrishchaka et al. (1998) in their (1). The resulting roots are, consequently, a little intractable and the physics is not easy

9 Non E B drifting F-region plasma waves 9 to assess. In the present work, we have consequently chosen to limit our study to fluid-like modes (large ion and small electron arguments). A study of the full kinetic solution will be left to a future publication, once the basic ramifications of the fluid-like regime are clarified. We therefore, from now on, limit ourselves to the study of large ion and small electron arguments, which is the standard way to turn the results to a fluid-like theory, while also keeping Landau growth/damping effects. Note that, in addition to the large ion argument we have to add, for the n =0mode, a finite Larmor radius correction to H i so as to include all the leading order terms coming from the v y x integration. As a result of this procedure, we obtain a relatively simple fluid-like dispersion relation that still contains several distinct regimes. To leading order in b i we obtain ω( ω + iν i ) kz 2 Cs 2 + k2 y T e m i Ω 2 ( ω + iν i ) 2 + k y k z Cs 2 V ( ω + iν i ) di [1 2 ] 2b i 2b s i Ω i kz 2 Cs 2 + i ( ω + iνe k z (V de V di ) π )( ω + iν i )( ω + iν i b i )=0, (3.18) 2 kz v te where b s = ky 2 Cs 2 /Ω 2 i and where, in contrast to Sec. 2, we now have Cs 2 = T e /m i instead of (T i +T e )/m i. This is consistent with having used the limit T e T i in order to simultaneously satisfy the conditions for large ion and small electron arguments (the presence of collisions forces us to stay away from the limit T e T i,whichis not a large Z argument for ions). Finally, we stress that in (3.18) the i π term obviously comes from wave particle interactions involving electrons and that the exponential function that multiplies the electron argument has been set equal to 1, which is again consistent with our assumptions. From (3.18) we can study three forms of limiting cases that we discuss in more detail below. If we neglect collisions, and do not consider the ky 2 T e /m i Ω 2 i term (no finite ion Larmor radius corrections) we recover the Gavrishchaka et al. (1998) results. If we take ν i 0 and neglect shears we find that we need V de V di C s to destabilize the plasma. This is the classical current driven electrostatic ion acoustic (CDEIA) result for T e T i. This means that our derivation includes, quite obviously, a collisional generalization of the CDEIA instability. If we go to the limit of zero frequency we find that we can neglect wave particle interactions. This procedure is valid at right angles to the magnetic field. Our expression then has the capability of becoming similar to that obtained by Basu and Coppi (1989) for the ion shear driven instability, which they derived using fluid theory directly. In the Appendix we provide general expressions for the shear and drift conditions associated with the threshold conditions described by (3.18). This proves useful for a study of the relative role played by shears and drifts at different wavelengths, radar look angles and frequencies Fluid-like threshold conditions For threshold conditions, ω is strictly real. Under this condition, for non-zero k z values, the last term in the balance of the real terms in (3.18) is m e /m i smaller than the rest and can be discarded for a quick look at possible modes. It is easy to show in that case that the balance of the real terms in (3.18) yields, for non-zero

10 10 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron k z (Perron 2004) [ ( )] ( )( ) 1 ζ i 1 2bs Ti /T e νi 2/k2 z Cs 2 + ν 2 i /Ω 2 i k 2 y /kz 2 ω R = ±k z C s, (3.19) 1+bs (1 2ζ i ) where ζ j = k y V dj / k z Ω j should be less than 1. Also recall that we have assumed 0 <b i = b s T i /T e < 1. We note that the frequency does not depend on the relative ion electron drift, and that shears can still play an important role. This is explored in more detail below. A balance between the various imaginary terms yields expression (A.4) in the Appendix for the relative drift V d =(V de V di ). However, it can be argued that ν er is normally ɛ less than the second term in (A.4), i.e. m e /m i times smaller than that term when k z is large enough. In that case, to a very good degree of approximation, we find V d = (V de V di ) = ω [ ] R ω R ν i 1+2bs (1 2ζ i ) + C s C s k z C s ( ω 2 R νi 2b, (3.20) i) ɛ where ɛ = πm e /2m i 1. We note that for sufficiently frequent collisions, the electrons now play a potentially important role, through ɛ, in the determination of the relative drift strength needed in order to destabilize the plasma The weakly collisional shear-free CDEIA limit. We can recover the fluid results of Sec. 2 with ω R >ν in if we suppose that no significant shears are present and if the wave vector is sufficiently far away from perpendicular that we can also drop the νi 2/Ω2 i term in (3.19). The result, in the absolute frame is then essentially the same as (2.7), namely, ωr A k z C s = k V 0 + k V di ± 1+ ( ), (3.21) ky 2 Cs 2 /Ω 2 i where we have used the symbol ωr A to stress the fact that we refer to frequencies in the absolute frame of reference that is E B drifting, as well as moving along the magnetic field line with the ion drift V di. In (3.21) there is a direct association between k z and k whereas k in (2.7) has been replaced by k y,wherethey direction is perpendicular to B and to the direction of the shears. However, since our shears are infinitesimal here, there is, in effect, no special direction to be associated with y as long as it is perpendicular to the magnetic field. The difference from Sec. 2 is that we now have an expression for the growth rate and can establish threshold conditions as a result. These can be obtained from (3.20) in the limit of small, though non-negligible, collision frequencies, still in the absence of shears. That is to say, for ionospheric F -region situations, when we drop the shears, take ω R ν i (consistent with Sec. 2), and keep all remaining leading order terms, the relative ion electron drift condition from (3.20) becomes V d = (V de V di ) = ω R + ν i 1+2b s. (3.22) C s C s k z C s ω R ɛ This shows that, even when small, the collision frequency can play an important role in determining the drift threshold conditions, because ɛ is also very small. In Fig. 1 we display how ω/kc s changes with V d /C s in an O + ionosphere, when ν i =0.01Ω i for three values of η =0.4, 1.0and2,whereη =Ω i /kc s.inorder

11 Non E B drifting F-region plasma waves 11 1 η = 0.4 η = 1 η = 2 ω/kc s V/C s Figure 1. Threshold conditions for the weakly collisional shear-free CDEIA mechanism. The frequency is expressed in units of kc s and the relative speed between ions and electrons along the B field is also in units of the ion-acoustic speed C s. The three curves correspond to Ω i /kc s = η = 0.4, 1, and 2, as we proceed from left to right. to be self-consistent with the finite Larmor radius correction, the results are only shown for η =0.4, whencos θ<η. Clearly, from Fig. 1 we can conclude that as η increases (or as the wavelength increases), the minimum drift requirement for destabilization also goes up. In addition, the parallel drift requirement is smaller for wavevectors that are more closely aligned with the magnetic field direction. This certainly means that the weakly collisional shear-free CDEIA (WCSFCDEIA) mode could have implications for HF radars in places for which parallel thermal electron drifts of the order of 5 10 km s 1 would exist. The lower threshold at smaller wavelengths also deserves to be studied for higher frequency radars, although in that case a fully kinetic study of that regime will probably be required The collision-free sheared CDEIA limit. Gavrishchaka et al. (1998) explored the regime ω R kc s in the presence of a non-zero ζ i. They neglected both collisions and finite Larmor radius corrections, which means that in the fluid limit discussed here they simply got from (3.19) (a valid procedure if we assume small growth rates), k z ω R kc s 1 ζi. (3.23) k Likewise, having neglected collisions, their threshold conditions in our fluid-like limit is equivalent to having (using (3.20)) V d = ω R. (3.24) C s k z C s As a result of (3.23) and (3.24), it was suggested by Gavrishchaka et al. (1998) that even in the presence of very small shears, S i = V di /Ω i = k z ζ i /k y, the plasma could be destabilized with rather small relative ion electron drifts, at least in the collisionfree situation. While there is truth to this, we should, however, first introduce finite Larmor corrections to the eigenfrequency expression, because of a singularity introduced by this correction in terms of the shears in (3.19). Specifically, for the

12 12 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron ω R /kc s = Ion Shear, S i ,000 Relative Drift, V d /C s Figure 2. Threshold conditions for the collisionless shear CDEIA instability using (3.26) and (3.27) for ω/kc s from 0.02 (dot-line) to 0.1 (dashed line) to 0.5 (dotted line) to 0.9 (dash-dotted line) to 1.01 (solid line). case at hand (no collisions and τ 0) (3.19) becomes 1 ζi ω R = ±k z C s 1+bs (1 2ζ i ). (3.25) Note that in the absence of shears, this mode is identical to the mode presented in the previous section for (3.21). However, also note that positive values of ζ i are capable of lowering the threshold speed, though the effect is only seen in the presence of collisions and will be discussed in the next subsection rather than now. In Fig. 2 we have plotted various threshold conditions associated with our collision-free sheared CDEIA limit. The vertical axis shows how the threshold conditions change with the ion shear, while the horizontal axis describes the corresponding values of V d /C s. Note that the line S i =0corresponds to a collision-free version of the shear-free CDEIA limit. The figure describes the results for a range of values of ω R /kc s between 0 and 1. Each curve is traced by sweeping over the angle θ given by tan θ = k /k y. For smaller values of V d /C s we find that the angle is larger (i.e. k /k y is largest). Note that we have refrained from showing results when S i > 1 so that the starting angle is smaller as we move from the curves on the left to those on the right, which correspond to larger values of V d /C s. The threshold loci are obtained using (A.1) and (A.4) in the Appendix and sweeping over the angle θ, in the absence of any collisions. This means that the zero growth rate condition is simply V d = C s ( ωr kc s Likewise, in the absence of any collisions, (A.1) becomesw ) 1 sin θ. (3.26) S i = tan θ 1 ((ω R/kC s ) 2 / sin 2 θ)(1 + cos 2 θ/η 2 ) 1 2((ω R /kc s ) 2 /η 2 tan 2, (3.27) θ) where η =Ω i /kc s.wechoseη =2in Fig. 2 to ensure that b s =cos 2 θ/η 2 would be less than 1 under all circumstances in order to remain consistent with the finite Larmor radius series expansion. Note, once again, how the S i =0solutions are

13 Non E B drifting F-region plasma waves 13 1, V d /C s, relative drift 10 1 ω R /kc s = θ, angle of wavevector Figure 3. Relative drift as a function of wavevector angle for the case described in Fig. 2. The angle is 0 when the wavevector is perpendicular to B and 90 when parallel to it. The line scheme is the same as in Fig. 2. described by (3.21). However, the threshold conditions are very different even when S i crosses 0 in Fig. 2 (near V d = C s ) because we included collisions in Fig. 1, but not in Fig. 2. Figure 2 confirms that there is a range of conditions over which, in the collisionfree situation, the ion shear does indeed seriously reduce the requirement for relative electron ion drifts. However, this is only true of the smaller values of ω R /kc s and for a large enough value of θ or k /k y. For instance, at ω R = 0.02kC s ashear S i 0.1 is sufficient to decrease the requirement on V d /C s down to 0.1, this when the wavevector is just 5 away from being perpendicular to B. Figure 3 shows more precisely how the threshold requirement goes down as the wavevector becomes increasingly aligned with the magnetic field. By comparison to Gavrishchaka et al. s (1998) analysis, we note that, while in the end, our leading order expression (3.25) differs from theirs (3.23), we have similar threshold conditions (3.24). Furthermore, our frequency expression only affects the solution in the regime V d /C s > 1. Since we are interested in finding the smallest possible drifts to excite an instability, our correction therefore takes a secondary character in the collision-free context The collisional sheared CDEIA limit. Unless shear contributions are large and positive, adding collisions to the fluid sheared CDEIA modes tends to put a larger requirement on the threshold conditions in terms of the relative ion electron drift. There is a very large difference between the collisionless and collisional cases because of the amplification produced by 1/ɛ in (3.20) and/or (3.22) (and, more generally, (A.4)). The fact that the threshold conditions are pushing the drift up when ζ i 0.1 can be seen from (3.20), or even from (3.22), which is the small ζ i limit of (3.20): with b i 0, because of the small T i /T e ratio used in the fluid limit, we are left with (3.22) as long as ζ i is of order 0.1 or less. However, things become more difficult to predict when ζ i = S i / tan θ approaches +0.5 and goes beyond that value. In that case (3.20) shows that the sign of the correction to the collision-free case can

14 14 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron Ion Shear, S i ω R /kc s = ,000 Relative Drift, V d /C s Figure 4. Same as Fig. 2, but for the collisional case ν i /Ω i =0.01 and ν e /ν i =10.0. Also T i =0.5T e. actually be negative. As a result, the threshold conditions might actually decrease. However, the negative correction can, by the same token, become so important that V d can even take a large negative value. No matter what, a study of the threshold conditions becomes a lot more complicated because the shear is involved in both the eigenfrequency and the determination of the drift threshold condition. Figure 4 illustrates how complex the analysis becomes when collisions are added in the sheared CDEIA study. For Fig. 4 we used (A.1) (A.5). However, we did not include the V d contribution in (A.2) as we are not interested in large values of this parameter anyway. A numerical check indicated that for ω R < 0.1kC s it would be best to solve the full equation, namely (3.18) instead. However, these are still solutions for which the threshold drift is rather large. At any rate, for the cases showninfig.4wehavealsoheldν i /Ω i fixed at 0.01, assumed ν e =10ν i, T i =0.5T e, andusedo + ions so as to consider typical F -region situations. We have also held η =Ω i /kc s fixed at 2, as we did for Fig. 2. We considered the same values of ω R /kc s as we did in Fig. 2, namely, 0.02, 0.1, 0.5, 0.9 and One point to note is that the ratios T i /T e (up to 0.5) and ν e /ν i (up to 10) do not affect the answer significantly for our choice of other parameters. In other words, the answers would basically be the same if we had used 0 for both T i and ν e. It should be noted that all of the branches that exhibit an increase in S i with V d are associated with negative values of V d. This makes little difference except for the ω R = 0.1kC s case, where the negative drifts are associated with lower thresholds. In all other cases, the positive and negative values are associated with particular shears where the drift itself has little importance (see the next section). One easy conclusion can be drawn from a comparison of Fig. 4 with Fig. 2, namely: for S i < 0.2 the threshold conditions are more stringent in terms of V d, as expected. The feedback of the shears on the requirement for a larger V d only pushes the curves further to the right once the shear term takes a large negative value (we only consider tan θ>0bydealing with the absolute value of k z,asa result ζ i and S i always have the same sign). If we are interested in lowering the drift requirements for possible instabilities, this result is of little physical interest.

15 Non E B drifting F-region plasma waves 15 1,000,000 10, V d /C s 1 ω R /kc s = θ, angle of wavevector Figure 5. Same as Fig. 3, but for the case described by Fig. 4. Put another way: in the collisional regime, negative shears make the plasma more stable. As we look at the positive shears, things become more complicated. The destabilizing effects of the shears are felt by the lowest frequencies, first because they are associated with larger values of ν i /ω R (see (3.20) when b i 0), which amplifies the response to the positive shears. As the traces show, however, at lower frequencies, the threshold conditions on the magnitude of V d are only lowered for a very narrow range of angles θ: this can be seen by looking at Fig. 5, which shows how the angles change with V d in the sheared case. In that sense, at the higher frequencies at least, the threshold lowering can be called a resonant condition, since the angular requirement on the direction of the wavevector is so specific. Otherwise, for ω R of order kc s, the requirements on the relative ion electron drift can become quite modest, while tan θ is of the order of 15 20, at least for the range of parameters considered here. The only catch is that S i now has to be of the order of 0.8, making the shear scales uncomfortably close to the ion gyroradius, which seems excessive. This being stated, Fig. 2 also shows that less excessive values would be required for S i and the associated shears if the relative drifts were to be of the order of 0.5 5C s and ω R was between 0.1 and 0.5kC s. The shear-free mode discussed in the previous subsection in relation to (3.19), can be identified by the points where various frequencies cross the S i =0point. We can conclude from this that the excitation of those modes in collisional F -region situations would require V d 5 15C s, provided the frequency is in the range kC s. This threshold requirement is noticeably higher than for the collision-free case, but not impossibly high, corresponding to thermal electron drifts of the order of km s 1. Finally, we have to consider the fact that our collisional fluid results were only posted here for T i T e. Under these conditions, we have checked that the fluid results agree with the kinetic calculations. While the results will be shown in detail in a future paper, it is sufficient to say that from a qualitative point of view the kinetic results are similar to the fluid until T i /T e approaches 0.4. By then (Perron

16 16 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron 2004), higher frequencies require narrow resonance conditions, while increasingly large positive shears are required in order for V d to remain of order C s. Alternately, this means that higher altitudes with lower collision frequencies are required for a CDEIA destabilization by the ion shears The small frequency ion shear driven instability. While we have seen in the previous subsection that our analytical solutions are not as accurate at ω R < 0.1kC s, (3.19) indicates in a simple way that some zero frequency modes can be excited even when k z 0. The analysis seems, at first sight, to be particularly simple in the long wavelength regime, where it would appear that b s could be neglected. In that case we obtain from (3.19) V di = k ( ) 2 z νi k y + Ω i k y Ω i k z. (3.28) This corresponds to the zero-frequency ion-shear driven instability that was studied in detail by Basu and Copi (1989). In particular, if or when the conditions for (3.28) are met, we find that a minimum shear condition is obtained when the wavevector is in a direction given by tan θ = k z = ν i. (3.29) k y Ω i In that case the minimum possible shear required to destabilize the plasma is given by ( ) V di Ω i min =2 ν i Ω i. (3.30) This condition can be found in the Basu Coppi (1989) work. Unfortunately, it proves inconsistent to drop b s from (3.19) because b s νi 2/(k2 z Cs 2 ) is comparable to (νi 2/Ω2 i )(k2 y /kz 2 ). Even if we still drop from (A.1) the contribution from T i /T e (or, equivalently, assume we have a small enough k y magnitude), we obtain the following at zero frequency, instead of (3.28), V di = k z 1+(ν i /Ω i ) 2 (k y / k z ) 2 Ω i k y 1+2(ν i /Ω i ) 2 (k y / k z ) 2. (3.31) While there is no extrema for this particular solution, at small angles the solution remains interesting since it identifies small shears at values of the order of k z /k y to 0.5 k z /k y, depending on the value of ν i /Ω i. However, another problem now arises, namely: while the shears might be small near zero frequency, the relative drift requirements might be large, given that some of the denominators in the expression for V d (A.4) are very close to zero. Finally, while we may have an expression that relates the shears to the aspect angles, we still do not have a determination of the said aspect angle. Yet, the requirement for small (zero) relative drifts should tell us what this aspect angle should be. The small frequency regime must therefore be explored more carefully before we can draw any conclusions A systematic study of the small aspect angles modes. To start with, we note that we do not require the frequency to be zero in order for V d to be zero. We recall that Fig. 5 showed an example of such a solution at ω R =0.02kC s ( V d is negative on the left of all the absolute values minima shown in Fig. 5). We take a clue from the previous subsections to assume that the small shear/small drift solutions are features associated with very small aspect angles. We therefore

17 Non E B drifting F-region plasma waves 17 rewrite our general threshold condition equations (A.1) and (A.4) with θ 1 in mind. We then obtain from (A.1) S i = V di [ θ = Ω i 2(1 Y 2 1 Y 2 (1+η 2 )+ ɛ ν i η 3 ( {Y 2 1+ τ ) θ Ω i η 2 + ν ) e τ }] ν e ν i η 2, (3.32) ν i where we have introduced the symbols τ = T i /T e and Y =( ω R /kc s )/(ην i /Ω i ). Since we are searching solutions for which we do not just get a small aspect angle, but also V d =0, we now use (A.4) to solve for S i under small aspect angle conditions and also under the condition V d =0. After a bit of algebra we obtain [ S i = ɛη3 ν i Y 2 τ 4 Ω i η 2 ν e (1+ τη )] ν 2 + θ ) (1+ η2. (3.33) i 2 2 All that remains to be done is to equate our two expressions for S i. We obtain the following expression for the aspect angle as a result θ = ɛ ην i Y 4 +(1+α)Y 2 + α Ω i 1+Y 2, (3.34) where α = τ η 2 + ν e ν i (1 τη 2 ). (3.35) This result allows us to find an explicit expression for the shear, namely, [ θ S i = 2(1 Y 2 1 Y 2 (1 + η 2 )+η Y ] 4 (1 + τ/η 2 + ν 2 e /ν i )+(1+α)Y 2 τ ν e η 2 ν i ) Y 4 +(1+α)Y 2. + α (3.36) There are many situations for which Y 2 1 and yet θ is small. For instance, our standard choice of parameters reveals that θ < 1 even when Y =50.Atthat point, for our standard set, ω R = kc s.wheny 2 1 our results take the simple forms θ = ɛ ην i Y 2 = ɛ Ω ( ) 2 i ωr (3.37) Ω i ην i kc s and S i = θ 2 (1 + η2 )= ɛ 2 (1 + η2 ) Ω ( ) 2 i ωr. (3.38) ην i kc s We also note that if we were to neglect τ and ν e /ν i (making α =0in the process) we would obtain the same results. Finally we see that there is an apparent singularity in S i when Y 2 =1( ω R = 0.02kC s with our standard parameters). The singularity is only apparent, however, since (Y 2 1) can be factored out of the numerator in the general S i expression in that case. This leaves a root that turns out to be identical in form to the above limit, namely, S i = θ(1 + η 2 )/2. In that case the expression for the angle θ differs only slightly from the Y 2 1 limit, namely, we obtain θ =(1+α)ɛην i /Ω i (recall that this is now the Y 2 =1case). We should add that we have checked our expressions against the numerical results obtained with the full expressions and that the comparison worked everywhere except for small angle cases with Y 2 =1, which is subject to obvious round-off problems with the numerical approach.

18 18 J.-P. St.-Maurice, J.-M. Noël and P. J. Perron As for the importance of the present results, we have obtained an expression for which S i is of the order of the aspect angle when the latter is small. This compares with the Basu Copi regime and its generalization from the addition of finite Larmor radius corrections, as discussed in the previous subsection. However, by solving for the zero relative drift condition instead of postulating a zero frequency (this comes from considering ɛ 0 which, in turn, means adding finite Landau damping effects to our description) we have also been able to determine under what value of the aspect angle itself the small shear condition would correspond to a zero relative drift condition between electrons and ions. The frequency, it turns out, can be of order kc s and still produce small shear results, although the smaller the frequency is, the smaller the shear requirements. We could even derive a zero frequency (Y =0) result from our work, but this is probably taxing the accuracy of the power series that we used in order to get fluid results from the kinetic theory. The point is that even without resorting to that extreme Y 2 =0case, we get interesting small shear results with Y 2 > Summary In our survey of some of the small scale instabilities in the F -region, we have argued that a fluid-derivable mode given by the frequency ω R = k V 0 + k Ω i /k could be directly excited in the ionospheric F -region. This mode can be contrasted with the usual zero frequency modes and ion cyclotron modes that exist much closer to perpendicularity to the magnetic field. We have obtained the relatively unknown eigenfrequency from a kinetic theory as well, using a somewhat simpler procedure than outlined by Gravrishchaka et al. (1998) while at the same time adding collisions to their work. With our finding that relative ion electron drifts in the range 3 10C s can be involved at threshold, we feel that the instability should take place, particularly on the edge of precipitation regions, where large enough thermal electron drifts are possible, if not likely, as argued in St.-Maurice et al. (1996) and Noël et al. (2000, 2005). In general, we have found that collisions reduce the effect of shears in ionospheric F -region CDEIA instabilities. This is caused by the dampening effect of collisions. However, while focusing on situations for which the wavevector is close to being perpendicular to the magnetic field, we also found that small drift and shear threshold conditions could be obtained in spite of collisions. In those cases, we have found the instability to be restricted to very narrow angular intervals (fractions of a degree) that deviated from perpendicularity by 5 or less for the least stringent threshold requirements. We have suggested that these solutions were in fact related to the zero frequency ion shear driven instability discussed at length by Basu and Coppi (1989) once finite Larmor radius corrections and weak Landau damping effects were introduced in the equations. An ideal region for the production of the ion shear driven modes should be regions of ion outflows, which tend to be intermittent and seem to have a preference, once again, to occur on the edge of precipitation regions (e.g. Wahlund et al. 1989; Forme and Fontaine 1999; Kagan and St.-Maurice 2005). Our general conclusion has to be that the edge of precipitating structures should be a fertile ground for short-scale high-latitude instabilities. This is in addition to the likely trigger of large-scale gradient-drift (universal) instabilities at larger wavelengths in the same regions. According to our fluid results at least, the short scale structures should be triggered by relatively modest parallel thermal electron