Alternative interpretation of results from Kelvin-Helmholtz vortex identification criteria

Authors


Abstract

Observations of lower density, faster than sheath (LDFTS) plasma at the magnetopause are believed to be specific to rolled-up vortices generated by the Kelvin-Helmholtz instability. Hence, they are used to identify vortices with single-spacecraft measurements. These vortices are expected to occur at the tail-flank magnetopause, beyond the terminator. This fact contrasts with numerous observations of LDFTS plasma far sunward of the terminator. Here we present two alternative explanations for the detection of LDFTS plasma at the dayside magnetopause: (1) the presence of a plasma depletion layer (PDL) readily featuring LDFTS plasma and (2) the plasma velocity pattern of magnetopause surface waves, by which lower/higher-density magnetosheath or PDL plasmas sensed by a wave-observing spacecraft are accelerated/decelerated in magnetosheath flow direction. Even low-latitude boundary layer (LLBL) plasma may be of LDFTS, if the LLBL background flow is antisunward at near-magnetosheath velocities.

1 Introduction

The magnetopause (MP) is the interface between the magnetosheath, which contains decelerated solar wind plasma, and the magnetosphere, where the plasma is more tenuous and the geomagnetic field is dominant [e.g., Spreiter et al., 1966]. Specifically, the MP is usually defined by the Chapman-Ferraro current sheet, but may also be regarded as the transition region from the magnetosheath proper, through the (occasionally present) plasma depletion layer (PDL) and the low-latitude boundary layer (LLBL), into the magnetosphere proper. The PDL forms under northward interplanetary magnetic field (IMF) conditions, and is characterized by increased magnetic field strengths and decreased densities with respect to the magnetosheath proper [e.g., Sibeck et al., 1990; Phan et al., 1994]. In the LLBL, both magnetospheric and magnetosheath plasma populations can be found [e.g., Sckopke et al., 1981]. Interestingly, it is also more developed under northward IMF conditions [e.g., Mitchell et al., 1987], despite diminished reconnection on the equatorial MP [e.g., Rijnbeek et al., 1984], which raises the question of how the LLBL is formed. Candidate processes are: diffusion [e.g., Treumann, 1997]; impulsive plasma penetration [e.g., Lundin et al., 2003, and references therein]; poleward-of-the-cusp reconnection [e.g., Song and Russell, 1992]; and plasma mixing within rolled-up vortices that develop at the MP in the nonlinear stage of the Kelvin-Helmholtz instability (KHI) [e.g., Fujimoto and Terasawa, 1994].

Aiming to explore this latter possibility, Takagi et al. [2006] performed three-dimensional magnetohydrodynamic (MHD) simulations of the situation at the tail-flank MP, to assess the development of rolled-up vortices at a Kelvin-Helmholtz (KH) unstable MP. As an important by-product of their work, they outlined a method to identify MP rolled-up vortices with single-spacecraft measurements by plotting density versus velocity in the direction of magnetosheath flow. Within a rolled-up vortex, magnetospheric low-density plasma should be accelerated to speeds exceeding that of the magnetosheath flow to account for pressure balance [Nakamura et al., 2004]. Hence, lower density and faster than sheath (LDFTS) plasma may be used for vortex identification.

Hasegawa et al. [2006] applied the vortex identification method for the first time to nine years of Geotail measurements. As expected, they found vortices to occur mainly at the tail-flank MP, beyond the terminator, where the velocity shear across the MP is larger than at the dayside, and KH waves have had time to develop into the nonlinear stage. In one case, however, Hasegawa et al. [2006] detected LDFTS plasma sunward of the terminator. This rather surprising result is confirmed in a statistical study based on Double Star TC1 data by Taylor et al. [2012], who relatively often find LDFTS plasma to be part of dayside MP fluctuations.

We apply the vortex identification method to THEMIS (i.e., Time History of Events and Macroscale Interactions During Substorms) spacecraft observations of dayside MP surface waves from 6 October 2011. These waves have been previously analyzed by Plaschke et al. [2013]. Although they were found not to be subject to the KHI, they unexpectedly feature LDFTS plasma. In this paper, we present alternative explanations for the presence of LDFTS plasma at the MP, other than as part of rolled-up vortices.

2 MP Surface Wave Event

On 5 and 6 October 2011, THA, THD, and THE [three of the five THEMIS spacecraft (Angelopoulos, 2008)] observed fluctuations in plasma and field quantities over several hours. The three were flying in a close configuration, near their respective apogees at the MP, forming a triangle almost tangential to the boundary. Plaschke et al. [2013] focus their analysis on an interval of interest between 00:40 and 01:50 UT on 6 October. During that interval, the spacecraft separations ranged between 0.27 and 1.34RE (Earth radii), and the spacecraft's mean position was (6.1,9.7,2.4)RE (in geocentric solar magnetospheric (GSM) coordinates). Hence, they were in the afternoon local time sector, far sunward of the terminator. The IMF (as measured by THC) was steadily pointing northward and sunward. The clock angle θIMF= arctan(By/Bz) (in GSM) ranged between −16.3° and 3.9°.

Figure 1 shows density, velocity, and spectral energy flux measurements by THE (Figures 1b–1d) of the interval of interest. The corresponding THA and THD measurements are very similar. The fluctuations are caused by the oscillatory motion of the MP, by which the spacecraft changed between the LLBL (densities of ∼2/cm3) and the PDL (densities of ∼10/cm3). The oscillation periods were on the order of 1 to 2 min. The magnetosphere/magnetosheath proper regions exhibit lower/higher densities of <1/cm3and >40/cm3. Measurements of the latter region can be seen at around 01:50 UT in Figure 1 [see also Sibeck et al., 1990]. The configuration of THA, THD, and THE allowed for the determination of boundary tangential wave vectors math formulaof the MP fluctuations by cross-correlation analysis, in particular for a subinterval between 01:00 and 01:10 UT. Wave vectors are key when comparing observations to theory. Their determination enabled Plaschke et al. [2013] to relate the THEMIS observations to the ideal, compressible single-fluid MHD theory of surface waves. They found solutions of the surface wave dispersion relation to be in remarkably good agreement with observations. Most importantly, these solutions were found for vanishing imaginary frequencies, i.e., they correspond to stable-amplitude, propagating surface waves that are not subject to the KHI. This result is confirmed when using the classical KHI criterion, which is based on MHD theory for incompressible plasmas. Consequently, it is rather unlikely that the waves observed by THEMIS on 6 October featured KH-generated rolled-up vortices.

Figure 1.

(a) Depicts the three intervals I1, I2, and I3 as defined in the text, (b) THE measurements of the ion density (red, green, and blue dots mark LDFTS plasma within these intervals), (c) the ion velocity in LMN coordinates, and (d) the spectral energy flux on 6 October 2011 between 00:35 and 01:55 UT. LMN: boundary normal direction N given by the MP model of Shue et al. [1998], Earth's dipole axis in the L-N plane, and M pointing westward.

3 Application of the Vortex ID Method

The vortex identification method introduced by Takagi et al. [2006] has been systematized by Taylor et al. [2012], who specify the following four testable criteria:

  1. Clear fluctuations should be visible in measured fields and moments in the 1 to 5 min period range.

  2. Intervals should contain at least 5 wave periods.

  3. The IMF should be northward throughout considered intervals: −70°<θIMF<70°. This criterion is intended to prevent mistaking reconnection exhaust jets, which exhibit LDFTS characteristics, for rolled-up vortices.

  4. Sufficient LDFTS plasma should be observed. More specifically, a reference direction is obtained by the interval average of the bulk velocity: math formula. The reference density for the magnetosheath is set to 80%of the maximum density in the interval: nmsh=0.8nmax. The reference velocity vmsh in the direction of math formulais given by the average over those times for which densities >nmsh were observed; the corresponding standard deviation of the velocity is denoted with σ. LDFTS measurements are characterized by densities <70%nmsh and velocities >vmsh+σin the direction of math formula. Relating the number of LDFTS samples to the total number of samples (of the considered interval) that show velocities >vmshσin math formuladirection yields a percentage that we denote with %RO (index: rolled-over), following Taylor et al. [2012].

We apply these criteria to three subintervals of the interval of interest discussed in the previous section, namely: interval 1 (I1) between 00:35 and 01:55 UT, interval 2 (I2) between 00:35 and 01:48 UT, and interval 3 (I3) between 01:00 and 01:10 UT. The intervals are depicted in Figure 1a. They fulfill the first three criteria. To check criterion 4, we show results of nmsh, vmsh, σ, and %RO based on THE observations in Table 1; THA and THD measurements yield very similar results. Apparently, THE senses LDFTS plasma (%RO>0%) in all three intervals. Hence, we may conclude that rolled-up vortices were unexpectedly present at the dayside MP. However, this is not the only possible explanation for %RO>0%, as we show in the following section.

Table 1. Quantities Pertaining to Criterion 4 of the Vortex Identification Methoda
 nmshvmshσ 
 [cm−3][km/s][km/s]%RO
  1. a

    Based on THE measurements of intervals I1, I2, and I3 as defined in the text.

I1392151161%
I212192754%
I3111432411%

4 Discussion

The results of Table 1 are illustrated in Figure 2. It shows measurements of THE with respect to density and velocity in math formuladirection. I3 samples are depicted by blue crosses, I2 samples in green, and I1 samples in red. As I1⊃I2⊃I3, blue crosses cover a number of green crosses, which themselves cover most of the red crosses. Solid horizontal lines show the vmshσlevels; LDFTS samples are in the lower left areas detached by dashed lines.

Figure 2.

THE measurements of I3 (blue), I2 (green and blue), and I1 (all colors) with respect to density and velocity in math formula direction. Solid lines depict vmshσ for I1 (red), I2 (green), and I3 (blue). Samples are identified as LDFTS if in the lower left areas detached by dashed lines.

As shown in Table 1, %RO of I1 is significantly higher than that of I2. This results from the inclusion of the high-density interval around/after 01:50 UT in I1 (excluded from I2), during which THE moved from the PDL into the magnetosheath proper. Measurements of the latter region determine nmsh=39/cm−3, which is much higher than nmsh of I2. The corresponding velocity measurements of around 200km/s (see Figure 1c) are, however, quite low in comparison to the PDL (just before 01:50 UT), as PDL plasma is accelerated in tailward direction due to the release of magnetic tension and magnetic pressure gradient forces [Lavraud et al., 2007]. Hence, simple PDL plasma is identified as LDFTS, and a large number of crosses are found within the red dashed rectangle in Figure 2. These samples are highlighted by a red background in Figure 1b, confirming that they mainly belong to PDL measurements around 00:40 and before 01:50 UT.

The difference between PDL and magnetosheath proper characteristics explains the large value of %RO for interval I1, but not %RO>0%for I2 and I3. The reason for this may be the velocity disturbance pattern associated with a MP surface wave. We explore this possibility using the same, ideal, single-fluid MHD model that THEMIS observations were compared to by Plaschke et al. [2013].

The basis for this model are the MHD equations of momentum balance, mass continuity, the induction equation, and generalized Ohm's law. They are linearized using a plane wave ansatz

display math(1)

for the pressure p, the mass density ρ, the velocity math formula, the magnetic field math formula, and the electric field math formula[Plaschke and Glassmeier, 2011, equations (5)–(8)]. By combining the linearized momentum balance equation with the induction equation, we obtain [Plaschke and Glassmeier, 2011, equation (14)]

display math(2)

Here math formula denotes the disturbance of the total pressure, ω is the wave's frequency in the respective plasma rest frame, math formula is the vector of the Alfvén velocity, and math formula.

We are interested in the math formulaand boundary normal (pointing toward the magnetosheath, index: N) components of equation (2). During the interval of interest, math formulawas approximately perpendicular to both directions [see Plaschke et al., 2013, Table 2].

Hence, we can neglect the second term on the right side and obtain

display math(3)

Equations (1), (3), and math formula, where math formulais the boundary displacement, yield the basic velocity disturbance pattern math formula(see equation (1)) associated with the observed waves in the plane spanned by math formula and the boundary normal direction. The pattern is given in Table 2 and is illustrated in Figure 3a by red arrows. Additional relations and assumptions on which the pattern computations are based are given in section A1.

Table 2. Wave Phase Dependent Disturbance Patternsa
WaveMagnetosheathMagnetosphere
PhaseeiωtδξNδvNδvsheareiωtδξNδvNδvshear
  1. a

    Wave phases an observer moving at background plasma flow speeds will experience with increasing time (math formula assumed in equation (1)). Complex-plane phases of the solutions of equations (1) and (3), and math formula, assuming math formula and real, as well as the relations given in the lower rows of the table (see also section A1). Red arrows in Figure 3a depict δvshear in wave phases 1 and 3, and δvN in wave phases 2 and 4.

11−1i−11−1i−1
2ii1iii1i
3−11i1−11i1
4ii−1iii−1i
 math formula, math formulamath formula, math formula
 kshear>0, realkshear>0, real
 ω<0, realω>0, real
Figure 3.

(a) Shows the velocity disturbance pattern (math formula, red arrows) associated with the surface wave; numbers mark different wave phases (see Table 2).(b) Shows the effect of the disturbed flow on the distribution of measurements with respect to density and velocity in math formula direction. (c) I3 samples are depicted as a function of these quantities. This shows the I3 part of Figure 2, enhanced by solid red lines that connect subsequent samples creating a hodogram-style plot. Corresponding arrows in Figures 3a and 3b are connected by yellow shaded areas. Figures 3b and 3c: threshold levels vmshσ are indicated by solid blue lines; (lower left) areas of LDFTS samples are detached by blue dashed lines (same as in Figure 2).

An observer comoving with the background flow on either side of the boundary will experience wave phases 1 to 4 (first column of Table 2 and numbers close to the red arrows in Figure 3a) in ascending order. At wave phase 1 (3), plasma on both sides of the MP is moving against (with) the magnetosheath flow, in addition to the respective background flows. At wave phases 2 and 4, plasma velocities are unaffected in math formuladirection, but plasmas move toward or away from the boundary while adapting to its shape.

As illustrated in Figure 3a, a spacecraft observing the wave will mainly sense plasma in wave phase 1 on the magnetosheath side and plasma in wave phase 3 on the magnetospheric side. Hence, it will not see the full background flow of the magnetosheath, but a flow velocity diminished by δvshear in the layers of highest density, most distant from the MP, that are reached by the spacecraft. Similarly, on the magnetospheric side, layers of lowest density sensed by the spacecraft will be accelerated in the direction of the magnetosheath flow.

The consequences for the distributions of samples in velocity-density-space (Figure 2) are illustrated in Figure 3b: a smooth, linear relation of background density and flow velocity in the transition region between LLBL and PDL (grey line) will become inverse S shaped (red line), which we can compare to observations. Figure 3c shows the distribution of THE I3 measurements; red lines connect subsequently observed samples creating a hodogram-style plot that exhibits remarkable similarity to the structure predicted by the red line in the panel above. Most importantly, the deceleration of the higher-density PDL plasma at wave phase 1 lowers the reference level vmsh and, consequently, the detection threshold vmsh+σfacilitating LDFTS plasma identification. In addition, lower density PDL plasma at wave phases 2 and 4 is not decelerated in math formuladirection, and spacecraft sensing the wave closer to the PDL side (only skimming the LLBL) might also see accelerated, lower density PDL plasma in wave phases between 2 and 3 or between 3 and 4, prone to be identified as LDFTS. If background flow velocities in the LLBL are antisunward and approach magnetosheath/PDL values, then LLBL plasma accelerated in wave phase 3 may also be identified as LDFTS.

5 Conclusions

We applied KHI-vortex identification criteria originally suggested by Takagi et al. [2006] to intervals of THEMIS observations of dayside MP surface waves that were previously found not to be subject to the KHI. Unexpectedly, lower density and faster than sheath (LDFTS) plasma is identified, which is thought to be a marker of KHI-generated rolled-up vortices. We present the following alternative explanations:

  1. Under northward IMF conditions, a PDL may form whose plasma exhibits lower densities and higher plasma velocities than the plasma in the magnetosheath proper. Hence, PDL plasma will be identified as LDFTS if threshold levels pertaining to the magnetosheath proper are used.

  2. The flow disturbances associated with surface waves can decrease the observable velocities (in math formuladirection) in layers of higher plasma density on the sheath side of the MP (e.g., in the PDL). Furthermore, the disturbances can enhance these velocities in lower density regions, also in the LLBL. There, the velocity enhancements may become important if the background flow is antisunward and comparable in velocity to the magnetosheath/PDL. Altogether, the flow disturbances will increase the likelihood of LDFTS plasma detection.

These explanations may apply at least to some of the numerous LDFTS plasma observations at the dayside MP, sunward of the terminator, where the KHI is less likely to reach the stage of vortex formation than beyond the terminator.

Appendix A:: Additional Relations

Pattern computations are also based on the following relations and assumptions: math formula is continuous over the boundary and (without loss of generality) is assumed to be real and negative. The amplitudes of surface wave disturbances (δQin equation (1)) decrease exponentially with increasing distance to the MP. Hence, math formula and real, and kN is purely imaginary and positive (negative) on the magnetosheath (magnetospheric) side. The boundary tangential wave vector math formula, instead, is real, continuous over the boundary, and, in this case, points tailward. Hence, kshear>0 and real on both sides of the MP. On the sheath side, the PDL flow overtakes the wave resulting in ω<0 due to Doppler shift. Note that frequencies are real as the wave is not subject to the KHI.

Acknowledgments

We acknowledge NASA contract NAS5-02099 and V. Angelopoulos for use of data from the THEMIS Mission. Specifically: C. W. Carlson and J. P. McFadden for use of ESA data; K. H. Glassmeier, U. Auster, and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302.

The Editor thanks David Sibeck and Johan De Keyser for their assistance in evaluating this paper.