On accelerated magnetosheath flows under northward IMF



[1] We study the acceleration of magnetosheath plasma using a semi-analytical magnetic string approach for a range of solar wind Alfvén Mach numbers, MA, between 2 and 20. We work with an IMF vector perpendicular to the solar wind velocity, Vsw, and pointing north. We do not invoke magnetic reconnection. Our results indicate that magnetosheath speeds can exceed the solar wind speed, and the ratio V/Vsw increases with decreasing MA. Analyzing the dependence of this ratio on MA, we find that for MA = 2, maximum V/Vsw ≈ 1.6, and for MA = 10–20, maximum V/Vsw varies from 1.21 to 1.13. Maximum speeds occur a few Earth radii (RE) tailward of the dawn-dusk terminator. The thickness of the accelerated flow layer varies as MA−2. Taking the magnetopause subsolar distance as 10 RE, we find typical values for the thickness of ∼4 RE for MA = 3 and 0.35 RE for MA = 10. The physical mechanism is that of draping of the magnetic field lines around the magnetosphere, and the associated magnetic tension and total pressure gradient forces acting on the flow. For lower MA the plasma depletion is stronger, and thus the acceleration produced by the pressure gradient is larger. An additional acceleration is produced by the magnetic tension, which is stronger for smaller MA. At the dayside the pressure gradient and magnetic tension forces both act in the same direction. But tailward of the terminator the magnetic tension starts to act in the opposite direction to the pressure gradient. When the resulting force vanishes, the highest speed is attained.

1. Introduction

[2] There has been renewed interest in accelerated magnetosheath flows, where speeds up to 1.6 times the solar wind speed (Vsw) have been reported [Lavraud et al., 2007; Rosenqvist et al., 2007]. This recent effort follows earlier work which addressed the same phenomenon in observational and theoretical work [Howe and Binsack, 1972; Erkaev, 1988; Chen et al., 1993; Phan et al., 1994; Petrinec and Russell, 1997a; Farrugia et al., 1995, 1998a; Erkaev et al., 2003]. All these works, which were case event studies, referred to an IMF pointing strongly north. In many instances, the enhanced speeds were reached relatively close to the dawn-dusk terminator [Lavraud et al., 2007, Rosenqvist et al., 2007]. The acceleration of the magnetosheath plasma was not due to magnetic reconnection, which is a known agent for accelerating plasma through conversion of magnetic to plasma kinetic energy.

[3] Chen et al. [1993] were the first to offer an explanation in terms of the magnetic tension force acting on the flow. As the magnetic field lines hang up at the magnetopause a magnetic tension force develops which can dominate the other terms in the equation of motion. Petrinec and Russell [1997b] studied several events of high magnetosheath speeds, observed by Geotail, for a variety of solar wind conditions, monitored by Wind, as a function of the angle between the magnetosheath magnetic field and velocity. To explain the enhanced speeds, Petrinec et al. advanced a phenomenological model of plasma flow around the magnetosphere. To explain speeds in excess of the solar wind speed, they invoked leakage of energetic particles from the low latitude boundary layer into a thin layer in the magnetosheath next to the magnetopause, although they did not exclude other mechanisms. They also concluded that magnetic field tension acting alone does not lead to speeds higher than the solar wind.

[4] Studying the plasma depletion layer, a low density-strong field region which forms next to the magnetopause for northward IMF [e.g., Zwan and Wolf, 1976; Erkaev, 1988], and in particular the stagnation line flow therein [Sonnerup, 1974], some authors noted a local enhancement of flow occurring just outside the magnetopause [Phan et al., 1994; Farrugia et al., 1998b]. Consistent with the stagnation line flow interpretation, this enhancement was perpendicular to the magnetic field.

[5] Concentrating on the low MA regime typically realized in magnetic clouds [Burlaga et al., 1981] and ICMEs, Lavraud et al. [2007] and Lavraud and Borovsky [2008] carried out global MHD simulations and obtained high speeds (up to 1.3 Vsw). The observations presented by Lavraud et al. of an event from Cluster showed a maximum speed of 1.6 Vsw for an average MA ∼ 3.0. Similar observations were reported by Rosenqvist et al. [2007]. In their example, the magnetosheath speed peaked at 1.6 Vsw during Earth passage of the northward phase of an ICME. Lavraud et al. [2007] and Lavraud and Borovsky [2008] ascribed these accelerated flows to a combination of magnetic tension and magnetic pressure gradient forces. They thus questioned the idea that these accelerated flows could be due solely to a magnetic tension effect as the flux tubes bend around the magnetosphere. The authors also inferred that the accelerated flow region was not just a thin layer adjoining the magnetopause but was suggested to be a few RE wide.

[6] In this paper we address the topic of accelerated magnetosheath flows using our magnetic string approach. We first motivate the use of this method. We then calculate the flow speed as a function of Alfvén Mach number and investigate where the maximum speeds are attained and how thick the accelerated flow layer is. We end with a comparison of our theory with the observations and a summary.

2. Results

[7] Our approach is based on the ideal MHD equations. Using the frozen-in condition, we can represent the magnetic field lines as elastic magnetic strings (see Erkaev [1988] and Farrugia et al. [1995] for a detailed quantitative description). This approach was successfully applied to analyze flow around planetary magnetospheres [Biernat et al., 1999; Erkaev et al., 1996; Farrugia et al., 1995].

[8] Briefly, the method is as follows. It employs a coordinate system (α, ξ, τ) which is related to the vectors B, V, E, i.e., to the magnetic, velocity and electric fields. Coordinate α varies along the magnetic field lines, and τ varies along the flow streamlines. The situation is visualized in Figure 1, where the solid lines are the magnetic field lines and the dashed lines are the flow streamlines. Each field line is characterized by a certain τ which changes from field line to field line. Thus coordinate τ parameterizes the various magnetic field lines. Similarly, α parameterizes the flow streamlines. As the field lines approach the obstacle, τ may be considered as a time variable. The ξ axis is perpendicular to the plane of this plot and therefore ξ varies along the electric field (E = − v × B). Note that, since flow streamlines and field lines are generally not perpendicular, (α, ξ, τ) is not an orthogonal system. A space-fixed coordinate system (X, Y, Z) can now be expressed in terms of (α, ξ, τ).

Figure 1.

A sketch of magnetic field lines (solid traces) draping around a planetary obstacle. The dashed lines are the flow streamlines. The field lines are parameterized by τ and the streamlines by α.

[9] To describe the shape of the magnetic field lines around the magnetosphere, we have to express (X, Y, Z) in terms of (α, ξ, τ). For this we need to solve the following second order, hyperbolic partial differential equation (the so-called magnetic string equation; [see, e.g., Farrugia et al., 1995]

equation image

Here r is a position vector (X, Y, Z) normalized to the subsolar curvature radius, ρ is the plasma density normalized to its solar wind value (ρsw), MA is the solar wind Alfvén Mach number, and Π is the total pressure, i.e., the sum of the magnetic and plasma pressures normalized to the solar wind dynamic pressure (ρswVsw2). The density ρ is related to plasma pressure by the adiabatic equation. The variation of the total pressure along the magnetopause follows the Newtonian approximation [Petrinec and Russell, 1997a].

[10] The string equation (1) includes contributions to the acceleration from both the magnetic tension (first term on the right) as well as from the gradient of total pressure (second term). The reason why we work in this formulation is that it is a natural system which stresses the physics.

[11] The flow and the magnetic field are obtained by

equation image

Here the magnetic field B and velocity V are normalized to their solar wind values.

[12] The string equations were integrated by the Lax-Wendroff numerical method (as explained by Biernat et al. [1999]), and the configuration of the draped magnetic field line and plasma velocity on the streamlined surface were obtained.

[13] We work with a solar wind directed along -X and an IMF vector which is perpendicular to it (i.e. lies in the YZ plane) with Bz >0. Figures 2a and 2b show the magnetosphere with magnetic field lines draping around it as they convect anti-sunward for MA = 4 (a) and MA = 6 (b). In the projection shown, the vertical axis (Z') is parallel to Bsw and X is along the Sun-Earth line, positive sunward. Linear dimensions are given in units of R0, the radius of curvature of the subsolar magnetopause. Note that R0 is ∼1.4 Dmp, the subsolar stand-off distance of the magnetopause whose shape is approximated by the analytical formula [Shue et al., 1998]

equation image

where θ is solar zenith angle. For this surface the curvature radius exceeds the stand-off distance by a factor R0/Dmp = 2/(2 − α). In our calculations we used α = 0.6, and thus R0/Dmp ∼ 1.4.

Figure 2.

Calculated magnetic field lines being draped around the magnetosphere while convecting tailward for (a) MA = 4 and (b) MA = 6. The perspective is a projection on the (X, Z') plane. The solar wind flow is along -X and the IMF points along positive Z'. R0 is the radius of curvature of the subsolar magnetosphere.

[14] We integrate the magnetic string equations numerically. The field lines shown in Figure 2 are those computed at approximately the same time intervals. One can see that for MA = 4 the magnetic field lines are transferred to longer distances tailward compared to those for MA = 6 at the same times. It is also worth noting that, due to the equatorial fast flow, the magnetic field lines change their shape as they approach the terminator. The resulting draping pattern shows a magnetic field compression on the dayside and, qualitatively, also the speeding up of the plasma around the dawn-dusk terminator.

[15] Representative profiles of the ratio V/Vsw along the Sun-Earth line are shown in Figure 3 for various solar wind Alfvén Mach numbers (from top to bottom: 3, 4, 6, 10). The speed is 0 at the stagnation point (not shown) and remains below the solar wind speed up to X ∼0.75 R0. For all four cases the speed maximizes at about 0.4 R0 tailward of the terminator. The highest speed is reached for the lowest MA, and is ∼ 50% higher than the solar wind speed for MA = 3.

Figure 3.

The maximum flow speed, in units of solar wind speed, as a function of X for four Alfvén Mach numbers, as shown.

[16] A qualitative explanation of the acceleration effect is as follows. For lower MA the plasma depletion is more pronounced (i.e. plasma density at the magnetopause is smaller [see, e.g., Farrugia et al., 1995, Figures 9a and 9c) and thus the acceleration produced by the pressure gradient (∇Π/ρ) is larger. An additional acceleration is produced by the magnetic tension, which is larger for smaller MA. At the dayside magnetopause, both pressure gradient and magnetic tension forces point in the same direction (antisunward). However the accelerated flow pushes forward the equatorial part of the frozen-in magnetic field line, which overtakes the high latitude parts of the magnetic field line. This eventually leads to a reversal of the magnetic tension force. It happens slightly tailward of the terminator. After that, the magnetic tension starts to oppose the pressure gradient force, and at some point the resulting force vanishes. This is the point where the plasma velocity reaches its maximum.

[17] The explicit dependence of the maximum speed relative to the solar wind on Alfvén Mach number is displayed in Figure 4. Speeds of 1.6 Vsw are reached for MA ∼ 2. For typical values in the solar wind at 1 AU (i.e. those not ICME-related), with MA ∼10 – 15, the velocity maximum decreases from 21% to 16% Vsw.

Figure 4.

The maximum magnetosheath speed relative to the solar wind speed plotted as a function of the Alfvén Mach number.

[18] In previous work the thickness of the PDL (or “magnetic barrier”) was derived as Δ = K/MA2, in units of R0 [Erkaev, 1988]. Constant K is ∼2.5. Thus, in dimensional quantities, taking for simplicity R0 = 14 RE (corresponding to a stand-off distance of about 10 RE), we have Δ = 4 RE (MA = 3) and 0.35 RE (MA= 10).

3. Summary and Discussion

[19] We now compare our theoretical results with some magnetosheath observations, considering first the ratio of the peak magnetosheath to the solar wind speed.

[20] Howe and Binsack [1972] reported Explorer 33 and 35 magnetosheath observations over extended periods in 1967 in the X-range (−60, −20) RE. Their data (see their Figures 5–6) show speeds up to 20% higher than the solar wind speed. If we assume that the Alfvén Mach number was that typical of the solar wind over extended periods at 1 AU, i.e. of order 10, then this result is quite in accord with our theory.

[21] Rosenqvist et al. [2007] examined Geotail data for one example on 22 November, 1997 during Earth passage of an ICME, monitored by Wind. The IMF was oriented mainly northward and duskward (By > 15 nT; Bz > 25 nT). Quantity MA was of order 1.5. Geotail recorded plasma flow speeds above those in the solar wind by ∼ 60% when located at the equatorial dawnside magnetotail flank (X = −25 RE). The strongly northward field makes reconnection at low latitudes unlikely. Similarly, the observation of these flows near the equator rules out a source in reconnection poleward of the cusp. The authors proposed the magnetic tension force associated with draping as the source of these fast flows.

[22] Lavraud et al. [2007] presented a case study, supported by global simulations, of an accelerated flow event seen by Cluster when located just tailward of the dusk terminator (X ∼ −5 RE) and at small southerly latitudes (Z ∼ −5 RE). The interplanetary medium was an ICME during the northward-pointing phase. The event occurred close to the magnetopause when a pressure pulse pushed the boundary over the spacecraft. They argue compellingly that the event was not an accelerated flow occurring earthward of the magnetopause, such as would be caused by magnetic reconnection. The average MA during the event ≈ 3 (average before and after the dynamic pressure pulse). They used multi-spacecraft discontinuity analysis [Dunlop et al., 2002] to obtain the normal speed of the magnetopause and from this and the event duration the authors could derive an estimate for the width of the “burst” of high-speed flow of order 10 RE.

[23] These observations are well in line with our theory, also based on a IMF pointing north. The agreement is good with respect to the flow speed ratio at the reported MA. In the one case where the thickness of the accelerated layer is inferred, it is in general agreement with our theoretical estimate of a few RE. The underlying “magnetic field draping cause”, with both gradient of total pressure and magnetic tension, is seen to be capable of causing these accelerations.

[24] We now compare our results with those obtained from global MHD simulations. Lavraud and Borovsky [2008, Figure 7] showed that the acceleration effect may also be expected for southward IMF, a conclusion also reached by Chen et al. [1993]. We have, however, in this paper not investigated this and concentrated on a northward-pointing IMF. The global simulation emphasized the importance of both magnetic tension as well as magnetic pressure gradient in causing this acceleration. The thickness of the layer of fast flow was addressed by Lavraud and Borovsky [2008], whose conclusions provide further support for our analytical results. The dependence of the maximum speeds on MA we inferred from our analytical work may be compared with those obtained from the global MHD simulations of Lavraud and Borovsky [2008, Figure 4]. Qualitatively similar profiles are obtained.

[25] In the circumstances we discuss, where the IMF was pointing predominantly northward, the flow velocity perpendicular to the magnetic field is a favorable situation for Kelvin-Helmholtz instability. Indeed, the observations of Chen et al. [1993] of accelerated flows were also made during an interval of magnetopause surface waves ascribed to the Kelvin-Helmholtz instability. The possibility of reconnection poleward of cusp could also provide accelerated magnetosheath plasma over and above those discussed here, particularly at high latitudes. Since our theory does not include this, the most suitable observations to compare the theory against are those near the equatorial plane.

[26] In summary, we carried out a semi-analytical treatment of magnetosheath flow past the magnetosphere for a northward IMF. We parameterized our study by solar wind MA. Our treatment shows that acceleration occurs for all MA, but is most pronounced for low MA. For MA = 2, the speed exceeds the solar wind speed by about 60 %. We also addressed the location where the highest speeds are reached and estimated the thickness of the fast flow layer. Our approach is through the magnetic string equations, a natural system for studying flow around planetary obstacles in the ideal MHD limit. Agreement with observations of reported cases of accelerated flows is very good.


[27] This work was done while N.V.E. was on a research visit to the Space Science Center of UNH. This work is supported by RFBR grant N 09-05-91000-ANF _a and also by the Austrian “Fonds zur Förderung der wissenschaftlichen Forschung” under project I 193-N16 and the “Verwaltungsstelle für Auslandsbeziehungen” of the Austrian Academy of Sciences. Work by C.J.F. was supported by NASA grants NNX10AQ29G and NNX08AD11G.