Compressible Kelvin-Helmholtz instability at the terrestrial magnetopause

Compressible Kelvin-Helmholtz Instability at the Terrestrial Magnetopause

Alejandro G. González^*

Instituto de Física Arroyo Seco, Universidad Nacional del Centro de la

Provincia de Buenos Aires, Pinto 399, 7000 Tandil, Buenos Aires, Argentina

Julio Gratton^*, Fausto T. Gratton^*,

INFIP Instituto de Física del Plasma, Facultad de Ciencias Exactas y Naturales,

Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428, Buenos Aires, Argentina

and Charles J. Farrugia

Space Science Center, University of New Hampshire at Durham, NH, USA.

Received on 30 June, 2002

The compressible magnetohydrodynamic Kelvin-Helmholtz instability occurs in two varieties, one that can be called incompressible as it exists in the limit of vanishing compressibility (primary instability), while the other exists only when compressibility is included in the model (secondary instability). In previous work we developed techniques to investigate the stability of a surface of discontinuity between two different uniform ows. Our treatment includes arbitrary jumps of the velocity and magnetic fields as well as of density and temperature, with no restriction on the wave vector of the modes. Then it allows stability analyses of complex configurations not previously studied in detail. Here we apply our methods to investigate the stability of various typical situations occurring at different regions of the front side, and the near anks of the magnetopause. The physical conditions of the vector and scalar fields that characterize the equilibrium interface at the positions considered are obtained both from experimental data and from results of simulation codes of the magnetosheath available in the literature. We give particular attention to the compressible modes in configurations in which the incompressible modes are stabilized by the magnetic shear. For configurations of the front of the magnetopause, which have small relative velocities, we find that the incompressible MHD model gives reliable estimates of their stability, and compressibility effects do not introduce significant changes. However, at the anks of the magnetopause the occurrence of the secondary instability and the shift of the boundary of the primary instability play an important role. Consequently, configurations that are stable if compressibility is neglected turn out to be unstable when it is considered and the stability properties are quite sensitive on the values of the parameters. Then compressibility should be taken into account when assessing the stability properties of these configurations, since the estimates based on incompressible MHD may be misleading. A careful analysis is required in each case, since no simple rule of thumb can be given.

I Introduction

The Kelvin-Helmholtz Instability (KHI) can occur in laboratory and astrophysical plasmas when two (or more) regions are in relative motion. Other properties of the plasma (like magnetic field, density and temperature) may also change across the transition layer. These changes modify the stability and other properties of the modes, thus complicating analysis. The KHI may occur at the terrestrial magnetopause, the boundary that separates the interplanetary magnetic field (IMF) present in the magnetosheath, and the geomagnetic field inside the magnetosphere. The relative motion is generated by the solar wind flowing through the magnetosheath. The flow velocity and the magnetic field are assumed parallel to the transition layer, and the magnetic field may change its direction (magnetic shear) and its magnitude across the transition. The low frequency magnetohydrodynamic (MHD) modes shall be our main concern here since their instability may produce large-scale turbulence. The KHI is considered a major source of anomalous transport of momentum from the solar wind into the magnetosphere, and a cause of viscous drag at the magnetopause. This is so particularly during periods of northward IMF, when the reconnection of magnetic lines is less likely to occur. We shall not consider other unstable modes that can also arise close to, and inside the magnetopause, in frequency ranges higher than those treated by MHD. These modes lead to microscopic turbulence that affects the transport properties of the plasma at smaller wavelengths. We shall also neglect dissipative effects so that our treatment shall be based on ideal MHD. The KHI at the magnetopause has been examined since the 1950's, and a large literature has grown on this subject. We quote here some recent reviews [?],[?] ,[?] among several others.

The theory of the KHI for arbitrary geometry is exceedingly complicated, but fortunately in many circumstances, the wavelength of the perturbation is very small as compared to the curvature radii of the transition layer. Then the problem can be treated in a plane slab geometry, in which the unperturbed quantities (density r, pressure p, magnetic field B and mass flow velocity u) depend only on the y coordinate (perpendicular to the transition layer). A differential equation for the linear MHD perturbations of stratified plasmas [?], and its generalization to include the effect of gravity [?], can be used as a starting point to investigate these problems.

If the wavelength of the perturbation is large as compared to the thickness of the transition layer, we can ignore the structure of the transition and assume it is a mathematical surface (at y = 0) separating two infinite, uniform plasma regions. The problem can then be treated by means of standard normal mode analysis. The equations of ideal MHD and the parameters of the configuration do not involve either a frequency or a length scale. Therefore the theory only predicts the (complex) phase velocity v of the perturbation (v = /k_t), as a function of the parameters of the plasma, of the mass velocities of the uniform regions and of the wavenumber k_tº (k_x,0,k_z) of the perturbation of the interface. Then an algebraic dispersion relation is obtained, whose roots yield v.

Within the incompressible MHD approximation (IMHD) it is possible to derive a simple formula for the phase velocity. Because of this, the IMHD result is often employed in space physics to interpret the observed data and to examine the stability of the configuration. Sometimes the tacit assumption is made that the compressibility effects thus neglected should (if any) improve stability. This assumption is based on well known theorems (see for example [?]) derived from a variational principle, according to which compressibility leads to a positive contribution to the energy of the plasma and so tends to stabilize it. However, these results cannot be applied here, since due to the presence of mass flow the plasma can sustain perturbations with a negative energy density that can lead to instability. This is a consequence of the transformation properties of the energy density W of a perturbation of wavenumber k_t and frequency under Galilean transformations. Let the primes denote the quantities calculated in the reference frame of the plasma that moves with a velocity u º (u_x,0,u_z) with respect to the observer, and the quantities without primes refer to the observer frame. Then we have W¢/¢ = W/. Notice that the Doppler-shifted frequency in the plasma frame ¢ = -u·k_t can be negative if u_k = u·k_t/k_t is sufficiently large (the phase velocity in the plasma frame is then v¢ = v-u_k). Since W¢ is always positive, the energy density of the perturbation as measured in the laboratory frame is then negative. A negative energy density perturbation may lead to instability if it couples to a positive energy density perturbation, since both can grow without an external source of energy.

In IMHD, the instability arises from perturbations of the interface that are exponentially damped in both plasma regions (like surface gravity waves in water). The penetration depth of these perturbations is equal to their wavelength 2p/k_t. The instability occurs when u_k exceeds a critical value u_i (see below), such that their energy density (in the laboratory frame) is positive in one region and negative in the other. These perturbations grow because energy is transferred across the interface from the negative energy density region to the other, and remains localized as it can not be transported away across the magnetic field. This instability will be called primary KHI. Notice that the Alfvén wave only transports energy along the field lines, and does not participate in the KHI if the transition region has a vanishing thickness.

The essential difference between IMHD and compressible MHD (CMHD) is that in the latter there are perturbations (the fast and slow magnetosonic waves) that propagate in the bulk of the plasma and transport energy across the magnetic field (see for example [?]) Consequently, the effect of compressibility on the Kelvin-Helmholtz instability is complex, leading to the stabilization of certain perturbations, the destabilization of others, and the occurrence of modes that have no analogue in IMHD and that may lead to new instabilities. Let us briefly discuss the basic physics involved.

Consider first the primary KHI. Due to compressibility, if u_k is sufficiently large the perturbation may be able to propagate in both regions as fast magnetosonic waves. When this happens, the perturbation is stable, and it is seen by the observer as a pair of fast magnetosonic waves radiated away from the interface. One of these waves has a negative, and the other a positive energy density [?], [?]. In this way, perturbations that are unstable according to IMHD are stabilized by compressibility. However, in addition to this large-u_k stabilization, there is another effect of compressibility on the primary KHI, namely that the critical value u_c for the onset of the instability is lowered (u_c £ u_i). This is a destabilizing effect, since perturbations that are stable according to IMHD become unstable when compressibility is taken into account. As we shall show later, this effect is important in the magnetopause. The penetration depth of the unstable modes is also affected by compressibility, and is no longer given by 2p/k_t.

Yet, there are further effects of compressibility. Due to the existence of the slow magnetosonic waves new kinds of perturbations of the interface are possible, that have no counterpart in IMHD. Some of these perturbations are stable evanescent oscillations, and other are slow magnetosonic waves in both regions (radiation of a pair of slow magnetosonic waves). However, some of these new perturbations may be unstable. These new instabilities (called secondary KHI) are found in intervals of u_k that correspond to stable perturbations according to IMHD. They lie totally, or partially, below the critical value u_c for the onset of the primary KHI. The occurrence of the secondary KHI is an additional destabilizing effect of compressibility. The growth rate of the secondary modes is usually small. However, we shall see that these modes cannot be ignored, since in some configurations of the magnetopause they are the only unstable modes present.

A substantial contribution to the understanding of the compressibility effects on the Kelvin-Helmholtz instability was given by Miura and Pritchett [?]. However, these effects have not yet been fully explored for general configurations with density and temperature discontinuities as well as magnetic field shear, such as those occurring in the magnetopause. For an extensive revision of the literature and of the main results on this subject, see refs. [?],[?] . Recently, we have developed techniques [?], [?], [?] to analyze the stability of an interface separating two uniform flows, with arbitrary jumps in the velocity, magnetic field, density and temperature, and with no restrictions on the wave vector of the perturbation.

The content of this paper is as follows. In Section II we outline the basic theory. In Section III we explain the techniques for analyzing the dispersion relation and the characteristics of the unstable modes. Section IV discusses the secondary KHI. A useful graphical technique that simplifies the complexity of the stability analysis is given in Section V. The physical parameters used in the applications to the magnetopause stability are introduced in Section VI. The results for two examples, (i) a model of the front side magnetopause, and (ii) an event observed at the near equatorial flank of the magnetopause are given in Section VII. We determine the stability of these configurations, and study the unstable modes that may occur in them. The final remarks are in Section VIII. We conclude that the effects of compressibility may introduce important changes with respect to the predictions of IMHD.

II Theory

We consider two semiinfinite, uniform plasma regions in relative motion. The geometry of the problem is shown in Fig. 1. Region 1 (y > 0), moves with a constant velocity , and region 2 (y < 0) moves with a constant velocity . The velocities and and the magnetic fields B₁ and B₂ are parallel to the interface (suffixes i = 1,2 denote quantities pertaining to region 1 and 2, respectively). At the interface, p, r, and B can have arbitrary discontinuities, subject to the equilibrium condition

We shall assume ideal CMHD and consider the linear adiabatic perturbations of this configuration. Since the growth rate of the instability does not depend on and separately, but only on their difference 2u = -, we use a reference frame moving with the average velocity u¢ = (+)/2. In this frame u₁ = u = ue_x, u₂ = -u, and a is the angle between u and the wavenumber k_tº (k_x,0,k_z) of a perturbation whose frequency is ; the phase velocity is then v = /k_t. Once the phase velocity of the perturbation is found, we can obtain its value in the observer frame by means of the Galilean transformation v¢ = v-.

The (Doppler-shifted) frequencies in the plasma frames are

₁ = -u_kk_t in region 1 and ₂ = +u_kk_t in region 2, and the corresponding phase velocities are

We shall denote by y_i the angles between B_i and k_t, and by j_i the angles between B_i and u. The angle between k_t and u is a, so that u_k = ucosa. We employ the following notation:

In each one of the uniform regions, the general form of the perturbation is

In this formula, z_i denotes the y-component of the displacement, and C_i and D_i are constants. The penetration depth of the perturbation in region i is proportional to 1/Re(G_i).

When Re(G_i) = 0, it can be recognized that eq. (6) is the dispersion relation for the fast and slow magnetosonic waves in a semiinfinite uniform moving plasma; their wave vector is (k_x,k_t Im (G_i),k_z). When Re (G_i) ¹ 0, eq. (6) is the dispersion relation for evanescent perturbations, and of course in this case the amplitudes of the exponentially growing perturbations (D₁ and C₂) in eq. ( 7) must vanish since the plasma extends to (+ or -) infinity.

The boundary conditions at y = 0 (continuity of z and of the normal stress) couple the perturbations of regions 1 and 2. The coupling leads to a matrix equation that relates the amplitudes C_i, D_i:

Here

The matrix equation (8) may be used to investigate many problems involving localized modes as well as the reflection and transmission of magnetosonic waves. Since unstable modes and stable surface oscillations have Re(G_1,2) ¹ 0, we must have C₁ = D₂ = 0 and then the corresponding dispersion relation is

The propagating modes have Re(G_1,2) = 0, so that there is no restriction on the C_i, D_i. In this case, the conditions T = 0 and S = 0 determine the poles and the zeros of the reflection coefficient. Notice that the sign of the y-component of the group velocity of magnetosonic waves may be different from the sign of Re(G_1,2). Then one must examine the group velocities in each region, to ascertain which of these conditions yields the dispersion relation for radiation of a pair of waves, or for the total transmission of waves (and their time-reversed processes).

III Classification of the modes

To investigate the roots of the dispersion relation (10) it is useful to introduce a polynomial P associated to it. It is obtained by taking the numerator of the product TS, thus eliminating the square roots:

Clearly, the roots of P = 0 encompass the solutions of T = 0 ( 'true' roots) as well as those of S = 0 ('spurious' roots), and we must discard the latter when we look for localized modes. Accordingly, we call 'true branch' (T) the set of solutions of T = 0, and 'spurious branch' (S) the solutions of S = 0. The T branch and the S branch consist of several pieces (that we call 'segments'), each of which is a continuous manifold in the parameter space of the problem. They are arranged in such a way that any given T segment is bounded by one or more S segments (and vice-versa). Let us consider a root v (real or complex) of P that belongs to a certain segment T_j. If the parameters change, v moves along T_j until eventually it arrives to the boundary between T_j and a segment S_l of S. As it crosses this boundary, v ceases to belong to the true branch and becomes spurious. It is important to notice that the T-S boundaries occur only for v real, at the points where G_1,2 = 0,¥. This is fortunate, since it simplifies considerably the analysis of the topology of the T and S branches.

A convenient way to visualize the real roots of the dispersion relation (10) and so unravel the topology of the T and S branches is to use a graphical method introduced by Chandrasekhar [?]. This method takes advantage of the fact that the problem of finding the roots v of P = 0 is equivalent to solve for v₁ and v₂ the coupled equations

From any given solution (v₁,v₂) of the system (12), one can obtain a root v = (v₁+v₂)/2 of P = 0. The real solutions of this system can be easily found graphically in a (v₁,v₂) diagram, as the intersections of the line L given by (12b) with the curves that are obtained solving the bicubic (in v₁ and v₂) equation (12a) for v₂.

In Fig. 2 we show a typical Chandrasekhar diagram, corresponding to the plasma configuration (d) investigated in Section VII (notice that the scale of v₂ is different from that of v₁ for convenience in drawing). It suffices to consider a single quadrant, since the curves v₂ = v₂(v₁) are symmetrical under reflections on both axes. In Fig. 2, the lines v₂ = v₁ and v₂ = -v₁ represent the v- and u_k- axes (except from a scale factor ).

The perturbation in the region i (i = 1,2) is a slow magnetosonic wave if b_i £ | v_i| £ m_i, and a fast magnetosonic wave if q_i £ | v_i| . It is evanescent if 0 £ |v_i| £ b_i or m_i £ | v_i| £ q_i (we shall call the latter the 'slow' and 'fast' evanescent perturbations, not to be confused with the slow and fast magnetosonic waves). A real root belongs to the T branch if Sign(–) ¹ Sign (–) (i. e., if | v₁| < a₁ and | v₂| > a₂, or | v₁| > a₁ and |v₂| < a₂); otherwise, it belongs to the S branch.

Both the T and S branches must be considered for the y-propagating modes (radiated or totally transmitted magnetosonic waves). It can be shown that all the roots of this type in the second and fourth quadrants of the (v₁,v₂) diagram correspond to radiation and those in the first and third quadrant to total transmission. According to these criteria we can classify the branch segments shown in Fig. 2. The true evanescent segments are represented by solid curves (labeled T₁, , T₂), the spurious evanescent segments by gray dotted curves (labeled S₁, S₂, S₃, ), and the radiation segments by gray solid lines (labeled R₁, , R₂, R₃, ).

For any given u_k there are five pairs of roots of P(v) = 0. Two pairs are always real, and are uninteresting for us since they are always spurious, or correspond to totally transmitted modes (they do not appear in Fig. 2 since they lie in the first and third quadrants). One root of a third pair lies on the curve formed by the segments labeled with the suffix 3 in Fig. 2 (S₃, R₃, , ), or on its continuation in the first quadrant. The other root of this pair is found on a symmetrical curve lying in the third and fourth quadrants. This pair of roots is also uninteresting, since they are always either spurious or propagating modes. The remaining two pairs of roots, labeled A, A¢ and B, B¢, are important to us since they can be unstable. For the value of u_k represented in Fig. 2 all these roots are real and represent stable surface oscillations, since the points lie on the segments and T₂, respectively.

If we consider a larger value of u_k, the line L must be displaced upwards, parallel to itself. Then the points B, B¢ move on the segment T₂ until they coalesce for u_k = u_d2 when L is tangent to T₂. For u_k > u_d2 the roots v_B, v_B¢ are complex conjugate, but they still belong to the T branch (since roots can pass to the other branch only for v real). Then u_k = u_d2 corresponds to marginal stability, and one of the roots v_B, v_B¢ is unstable for u_k > u_d2 (the other is damped). This instability persists until u_k = u_d6, when L is tangent to the segment R₂. Then u_d6 corresponds also to marginal stability, and for u_k > u_d6, v_B, v_B¢ are stable radiation modes. On the other hand, if one considers values of u_k larger than that represented in Fig. 2, the points A, A¢ move along and eventually pass into the segments R₁, , and then v_A, v_A¢ correspond to radiation modes.

Let us now consider smaller values of u_k. As u_k is reduced, L moves downwards. The points B, B¢ move on T₂ until they pass into S₂. It can be shown that for u_k > 0, no new instability arises from the roots v_B, v_B¢. On the other hand, the points A, A¢ move along until they coalesce for u_k = u_d3. When this happens, the double root v_A = v_A¢ is marginally stable. For u_d1 < u_k < u_d3 we have instability, and v_A, v_A¢, are complex conjugate. Marginal stability is again achieved at u_k = u_d1. For u_k < u_d1 the points A, A¢ move along T₁ until they pass into S₁. No other instability related to v_A, v_A¢ occurs for u_k > 0. Using diagrams like Fig. 2 it is possible to investigate the stability of any configuration of interest. However, this method is unpractical, since we need a new diagram for each orientation of k_t, and in addition, we cannot determine the growth rate of the unstable modes.

In the case shown in Fig. 2 there are two unstable intervals (for u_k > 0). The first (in order of increasing u_k) involves v_A, v_A¢ and occurs for u_d1 < u_k < u_d3. The second occurs for u_d2 < u_k < u_d6, and involves v_B, v_B¢. This result is due to the special choice of parameters of Fig. 2. In general, there can be up to six positive values of u_k corresponding to significant (i.e., belonging to the T branch) double roots of P(v) = 0, that we label u_d1,...,u_d6. Correspondingly, there can be up to three unstable intervals, namely

As the parameters are varied, it may happen that the pair (u_d3, u_d5), or the pair (u_d4, u_d5), coalesces and becomes complex. In the first case, the secondary interval A merges with the primary interval. In the second case, the secondary interval B merges with the primary interval (as in Fig. 2). In addition the secondary intervals (13a, b) may also overlap partially or totally. The secondary intervals can be extremely narrow for some values of the parameters, in particular they vanish when the perturbations are flutes in either region 1 or 2. For negative u_k, a completely symmetrical result is obtained, in which the signs of u_d1,...,u_d6 are changed, the inequalities (13a-c) are reversed, and the roles of the root pairs (A, A¢) and (B, B¢) are exchanged. There are no simple formulae for u_d1,...,u_d6.

To find the double roots v_d1,...,v_d6 of P(v) and the corresponding u_k values u_d1,...,u_d6 we must solve numerically for (u,v) the system

and discard the roots belonging to the S branch. The unstable intervals can then be mapped in the parameter space of the configuration (see Section V).

IV The secondary Kelvin-Helmholtz instability

With the help of diagrams like Fig. 2 it is easy to understand the role of the compressibility on the KHI. As S_i increases, b_i and m_i approach a_i, and q_i increases without bound, so that in the incompressible limit (S₁,S₂® ¥) the secondary unstable intervals disappear, and we are left only with the primary unstable interval, that now extends to infinity. The secondary KHI is then a consequence of compressibility, as well as the large u_k stabilization of the primary KHI at u_d6. It is enlightening to see in more detail the origin of the secondary KHI. Notice that P(v₁,v₂) can be written in the form

In (15),

and P₁, P₂, P₃, are polynomials whose coefficients are functions of a₁, a₂, A₁, A₂, r₁ and r₂. Then, in the limit S₁,S₂® ¥, if we assume that v_1,2 remain finite, eq. (12a) reduces to P_inc = 0, whose solutions are:

These solutions yield, respectively, the roots

The roots (17a) and (17b) are stable, and correspond to Alfvén waves propagating in regions 1 and 2, respectively. The roots (17c) have been discussed by Axford [?],[?], Chandrasekhar [?] and Southwood [?]. They yield the incompressible (primary) Kelvin-Helmholtz instability for

Finally, the roots (17d) are spurious.

Some of the roots (17a-d) can coincide for special values of u_k. For example, when

the roots v_a1–, v_a2+, v_K± (– or + according to the sign of r₁-r₂) and v_S- coincide and their value is

This degeneracy corresponds to the point D₁ (v₁ = -a₁, v₂ = -a₂), where the lines (16a – d) cross. It can be verified that u_i ³ u_D, and that the equality holds only for r₁– r₂ = 0. Let us now assume that S₁,S₂ are large, but finite. Then the term P₁(v₁,v₂)+P₂(v₁,v₂) in (15) can be treated as a small perturbation that couples the degenerate roots v_a1- , v_a2+, v_K+ (or v_K-), v_S-. It can be shown that part of the degeneracy is removed, and some roots acquire an imaginary part. The unstable modes so arising (secondary KHI) have no analogue among the purely incompressible modes. It is interesting to observe that this instability appears around u_k» u_D, that is, in a range of u_k that is stable according to IMHD, since it lies below u_i.

In addition to D₁, there are three other points where degeneracy of the roots (17a-d) occurs, namely D₂ (u = -u_D, v = -v_D), D₃ (u = v_D, v = u_D) and D₄ (u = -v_D, v = -u_D). The removal of the degeneracy at D₂ due to compressibility leads to overstable modes like at D₁. On the other hand, the removal of the degeneracies at D₃ and D₄ does not lead to instabilities.

V Stability diagrams

The problem of finding the localized eigenmodes is straightforward, if tedious, since we must calculate numerically the roots of the polynomial P (of degree ten), that depends on seven independent parameters. Six of them characterize the plasma configuration; they can be taken as the ratios r_b = B₂/B₁, r_s = S₂/S₁ and r_d = r₂/r₁, the magnitude u of the relative velocity, and the angles j₁ from B₁ to u, and q from B₁ to B₂ (magnetic shear angle). The remaining is the angle y₁ from B₁ to k_t, and identifies the perturbation we are considering.

Notice, however, that P (and its roots) depends on u only through the combination

Accordingly, we shall represent the results of the analysis by means of diagrams in which we plot the growth rate Im(v) of the unstable modes as functions of y₁ and u_k, keeping fixed the remaining parameters (r_b, r_s, r_d, q) that determine P. An example is the stability diagram shown in Fig. 3, in which we have drawn a contour plot of Im(v) for the parameters of a case investigated in Section VI (case (d), r_b = 1.5, r_s = 2.67, r_d = 0.117, q = -80^º). In Fig. 3 we have also drawn the curve u = u_i(y₁) that gives the marginal stability condition (18) according to the incompressible model. All the points of the stability diagram above u_i(y₁) correspond to instability according to IMHD. Notice that the contour plot does not refer to a unique plasma configuration, since we have not yet specified u and j₁. Then it can be used for all the configurations having the same values of r_b, r_s, r_d and q. This is a distinct advantage since with a single graph we can study the effects of changing the magnitude of the relative velocity and its orientation with respect to the magnetic field.

In Fig. 3, we can see the main effects of compressibility: the large-u_k stabilization and the small-u_k destabilization of the primary instability, and the presence of the secondary instability in u_k intervals that are stable according to IMHD. It should also be mentioned that for u_k appreciably larger than u_i(y₁) (i.e., not very close to it) the growth rates calculated with CMHD are always smaller than those calculated with IMHD (17c).

To investigate a particular configuration by means of the stability diagram we must draw the curve C given by eq. (21), that depends on the two remaining parameters (u, j₁) that characterize it. Following C we find the value of Im(v) for this configuration, for any perturbation characterized by the angle y₁. Since the problem is invariant under the transformation (v® –v, u_k® –u_k), it suffices to calculate the contour plot for u_k ³ 0; of course, we must then draw the curve C¢ given by u_k = -ucosa in addition to C.

In the example shown in Fig. 3 we see that the configuration described by C and C¢ (u = 1.7A₁, j₁ = -89.9^º) is stable according to IMHD, but is unstable if compressibility is taken into account. The unstable modes occur in the (approximate) range 40^º£ y₁£ 50^º and they belong to the secondary KHI. Their maximum growth rate is small (Im(v) » 0.04A₁), but significant. It can be observed that other configurations with the same values of r_b, r_s, r_d and q can be more unstable. The extreme of Im(v) in Fig. 3 is » 2.115A₁; it occurs for y₁ » 25^° and u_k » 5A₁, a value that requires a very large u.

The meaning of the large-u_k stabilization can be clarified by means of an example. Let us consider in Fig. 3 a hypothetical configuration with u = 10A₁ and j₁ = -89.9^°. The new curves C_n and are similar to C and C¢, only scaled by the factor 10/1.7. Part of C_n (for 45^°£ y₁£ 113^°) lies in the upper part of the stability diagram (u_k > u_d6) corresponding to stable perturbations: these are the large-u_k modes stabilized by compressibility. However, this configuration is unstable, since C_n and pass through the unstable part of the diagram (actually, its most unstable modes have Im(v) very close to the extreme value for the diagram). Clearly, configurations having very large u are always unstable, although large-u_k stabilization may occur for some modes.

A few comments must be made concerning stability diagrams like Fig. 3. To draw them we calculate the roots of the dispersion relation at discrete points of a grid in the (y₁,u_k) plane. Sometimes, at the same point (y₁,u_k) there are two unstable roots (this happens when the secondary intervals A and B overlap, or when any one of them overlaps parts of the primary interval). In these instances we show only the larger Im(v). The secondary intervals are very narrow in some (y₁,u_k) regions, and may not show up, due to the coarseness of our grid (for the same cause some contours appear wavy in Fig. 3). Nothwithstanding these drawbacks, stability diagrams are extremely useful tools, since they provide an economical and fast way for finding the most interesting domains in parameter space. It is then possible to perform more precise and detailed calculations in these regions only, with confidence that we shall not miss anything relevant.

VI Parameters of magnetopause configurations

Before discussing the application of the theory to the magnetopause, it is convenient to review how the parameters used in our formulae can be obtained from the theoretical models or from spacecraft data. We assume that the outside region (as seen from the Earth) is region 1. In the applications, the parameters of the configuration (the flow velocities and and the angles yB₁,and yB₁,from B₁ to and , respectively) are given in a reference frame at rest with respect of the Earth. In our theory we use a reference frame moving with the average velocity u¢ = (+)/2, then u₁ = u, u₂ = -u, with u = (-)/2. In what follows, all velocities are given in km/s. We employ the following formulae:

The angle d from u to u¢ is given by

In eqs. (23) and (24) the factor Sign(sinq_u) is needed to obtain the correct determination of j₁ and d. Finally, the phase velocity of the perturbation in the Earth's frame is given by

To consider the remaining parameters we must discuss separately the theoretical and spacecraft data.

Data derived from theoretical models

The conditions for the excitation of the Kelvin-Helmholtz instability on the dayside magnetopause were recently investigated in two papers. One, by Miura [?], reports numerical MHD simulations on the development of the instability, starting from hyperbolic tangent profiles of the unperturbed field quantities. One of the main results is that the magnetopause appears to be more KH unstable for northward IMF conditions than for southward pointing magnetic fields. However, the choice of parameters for the simulations is more adequate for the analysis of near flank configurations rather than dayside proper. The second paper [?] is based on a linear incompressible theory, and specifically addresses the distribution of the KH activity over the magnetopause surface. Charts of KHI growth rates, relying on input parameters provided by a MHD code that simulates the physical conditions of the frontside magnetosheath, are given. One of the main results is that under northward IMF conditions, the KH activity is restricted to some well defined regions of the magnetopause. Even when the IMF is due north, the activity decreases as the latitude increases from the magnetopause equator. In an third paper [?] the latitude dependence of the KHI is examined with a linear incompressible theory based on hyperbolic tangent profiles, using our incompressible theory for a tangential velocity discontinuity.

The data used here are given in Tables 1 and 2. Those of Table 1 (also used in [?]) proceed from ref. [?] and were calculated with the magnetosheath model of ref. [?], which is also described in [?]. The IMF is assumed to point exactly north. The positions are at equator, middle latitude, and high latitude, along a meridian about 3 hours east of noon. The main difference among these positions is the increase of the local magnetic shear angle.

We have in this case = 0, then u = u¢ = /2, j₁ = yB₁,and v¢ = v-u¢cos(y₁-j₁). The model provides yB₁,, the shear angle q, the ratio r_b = B₂/B₁, as well as the following quantities:

In (??), u_¥ and r_¥ are the velocity and the density of the solar wind, far from the interaction region.

Because the magnetosheath model does not provide magnetospheric values for temperature and density, Table 2 complements the set of parameters needed in the theory. We assume u_¥ = 500 km/s and use typical values of the density ratio and of the temperatures in both regions, often observed in spacecraft crossings of the magnetopause (expressed as r_d = r₂/r₁, r_s = S₂/S₁ and f_s = S₁/u_¥). Clearly r_d, r_s and f_s must satisfy the equilibrium condition (1). In terms of the f 's and r's this implies 2fd(1–r_d) = g(–1). So we first assign r_d and f_s, and then use this relation to calculate r_s. Since must be positive, we need > g(–1)/2f_d. When r_b > 1 (as in case (a) below) this condtion sets a lower bound on f_s.

Using the data of Tables 1 and 2 we then find

and from these quantities we calculate the remaining parameters of our theory.

Spacecraft data

At the flanks of the magnetopause, the magnetosheath flow is supersonic and the effects of compressibility on the KHI are expected to be important. We shall examine a configuration at the near equatorial magnetopause at dusk, observed by Interball/tail during the event of January 11, 1997. This was at the tail of a coronal mass ejection passing Earth, during which the dynamic pressure of a high density solar wind strongly compressed the magnetosphere. As a consequence, Interball/tail was for some time in the magnetosheath, and reentered the magnetosphere, after the dynamic pressure returned to lower values. During the crossing of the magnetopause, from magnetosheath to low latitude boundary layer, the spacecraft measured the physical parameters necessary for a stability analysis. A detailed study of the event is given in ref. [?], together with a discussion of the Kelvin-Helmholtz instability, based on an incompressible model. We intend to revisit that configuration with our compressible theory. The interface that we shall consider is that between the magnetosheath and the low latitude boundary layer.

In the first line of Table 3 we give the parameters of the event (we use the definitions M_A,1º A₁/ , M_S,1º S₁/). The second line contains the parameters of a hypothetical configuration with a slightly larger value of M_A,1, to ascertain the effect of this change.

We must recognize that owing to measurement uncertainties, the spacecraft data do not comply with the equilibrium condition (1) at the interface. To emend this flaw, we replaced the measured r_s (that is the less reliable datum) by the value derived using the remaining data in the equilibrium condition, expressed as 2,₁ (1 – r_d) = g_,1(-1).

The expressions of u, u¢, j₁, d and v¢ are given by eqs. (22)-(25), and the remaining quantities we need are:

We now investigate the stability of these configurations of the magnetopause.

VII Stability of some magnetopause configurations

Configurations near the front of the magnetopause

In the low latitude case (a) there is no magnetic shear, so that the flute modes are unstable, both in the incompressible limit and if compressibility is taken into account. In Fig. 4 we compare the growth rate obtained with the compressible model with that predicted by IMHD (eq. 17c). It can be noticed that there is no significant difference. An increase of the relative velocity by a factor of 3-4 does not change this conclusion.

The intermediate and high latitude cases are stable when compressibility is taken into account, a result that coincides with the predictions of IMHD. In Fig. 5 we show the stability diagram for the high latitude parameters. We can see the lowering of the critical value u_c, due to compressibility (already mentioned in Section I). This effect is of no consequence for the configuration (c) owing to the low relative velocity, but it may be relevant for configurations with a large relative velocity (say, u » 1.5A₁).

A configuration in the near flank of the magnetopause

The stability diagram corresponding to these parameters is shown in Fig. 3. It can be appreciated that this configuration is stable according to the IMHD model, but is unstable when compressibility is taken into account. Moreover, this instability corresponds to the secondary modes, which have no counterpart in IMHD. The growth rate of the instability is small, but significant, as can be seen in Fig. 6.

When the line C of the configuration passes close to the marginal stability line for the primary modes, as happened in this case, the stability properties of the configuration are very sensitive to slight changes of the parameters. This can be appreciated in Fig. 7, in which we show the growth rate of the instability, for a slightly different orientation of the relative velocity u with respect to the magnetic field (j₁ = 75.5^° instead of 89.9^°). It can be noticed that this change produces a fourfold increase of the growth rate of the secondary instability (and a shift towards larger y₁). The primary instability is also present now, although its growth rate is much smaller than that of the secondary modes. This is due to the small-u_k destabilization phenomenon, since the modified configuration is still stable according to IMHD.

If we change the relative velocity, keeping the remaining parameters fixed (as if the solar wind had been slightly faster at that event), the line C of Fig. 3 shifts upwards, and the primary instability becomes dominant for a 30% (or larger) increase of u.

In Fig. 8 we show the stability diagram for case (e), where M_A,1 differs from case (d) by 14%. The effect of this change is to increase the growth rate of the secondary instability with respect to the results of Fig. 6. This modified configuration is stable according to IMHD.

VIII Conclusions

Compressibility affects the Kelvin-Helmholtz instability by changing the unstable domains of the modes already present in the incompressible model as well as their growth rate. Modes with large u_k (the projection of the relative velocity on the wave vector of the perturbation) are stabilized. At the same time the lower boundary of the unstable u_k interval shifts downwards, thus destabilizing perturbations that are stable according to IMHD, for u_k < u_i (the critical value for marginal incompressible instability). On the other hand, the growth rate of the unstable modes above u_i is reduced by compressibility. In addition, compressibility introduces new unstable modes (the secondary modes) that do not exist in IMHD, and that occur for u_k below u_i. For these reasons, compressibility may play an important role in determining the stability of configurations with a relative velocity of the order of u_i.

For configurations of the front of the magnetopause, which have small relative velocities, the IMHD model gives reliable estimates of their stability, and compressibility effects do not introduce significant changes. However, compressibility must be taken into account when assessing the stability properties of configurations at the flanks of the magnetopause. As shown in Section VII, in these cases the occurrence of the secondary instability and the shift of the boundary of the primary instability play an important role. Then, configurations that are stable if compressibility is neglected turn out to be unstable when it is considered. In these situations, the stability properties are quite sensitive on the values of the parameters. The estimates based on IMHD may be misleading, and a careful analysis is required in each case, since no simple rule of thumb can be given. In these respects the stability diagrams introduced in Section V may prove useful, since a single graph allows to visualize the effect of changing the magnitude and orientation of the relative velocity.

Acknowledgments

We acknowledge grants from CONICET (PIP 4521/96 and PIP 4535/96), the University of Buenos Aires (Projects X151 and TX32) and NASA (grant NAG5-2834).

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^*

Researcher of the Consejo Nacional de Investigaciones Científicas y Tecnológicas

Publication Dates

• Publication in this collection
  11 Feb 2003
• Date of issue
  Dec 2002

History

• Received
  30 June 2002