SDO/AIA OBSERVATION OF KELVIN–HELMHOLTZ INSTABILITY IN THE SOLAR CORONA

The Kelvin–Helmholtz instability (KHI), produced by two fluids undergoing differential shearing motion across an interface, was described by Kelvin (1871) and Helmholtz (1868) almost one and a half centuries ago. The KHI is observed over a wide range of gaseous, fluid, and plasma regimes on the Earth and in space. Chains of vortex-shaped features have been observed in clouds on the Earth, and in the atmospheric cloud belts of Jupiter and Saturn that exhibit shearing wind motions. These vortices were interpreted as signatures of the KHI. KH vortices have been observed in the Earth's aurora (Farrugia et al. 1994) and the magnetosphere, where there is evidence that the instability plays an important role in the energy transport of solar wind into the magnetosphere (Hasegawa et al. 2004). Observations indicating the presence of KHI have also been made in the magnetospheres of Mercury (Slavin et al. 2010), Saturn (Masters et al. 2010), and Ganymede (Kivelson et al. 1998). Recently, evidence of KH development in solar prominence material was found in observations from the Hinode Solar Optical Telescope (Ryutova et al. 2010; Berger et al. 2010).

The growth rate of the instability was calculated analytically for sheared flow in an idealized fluid (Chandrasekhar 1961). This instability can also occur in magnetized plasma of the solar corona; in this case, the magnetic field component along the direction of the shear can have a stabilizing effect (Chandrasekhar 1961) and can also affect the growth rate and structure of the vortices (Lysak & Song 1996). Non-uniform (sheared) magnetic field across the velocity shear layer can have stabilizing or destabilizing effect on the KH mode, depending on the properties of the magnetic shear (e.g., Ofman et al. 1991). The KHI has been studied theoretically in the context of coronal plasmas, where it is believed to play an important role in the transition to turbulence and plasma heating (Heyvaerts & Priest 1983; Ofman et al. 1994; Karpen et al. 1994). KHI was also proposed as a mechanism that produces the observed quasi-periodic pulsation in flare high-energy emission (Ofman & Sui 2006). In this study, we report the first observation of KHI in the solar corona in EUV coronal lines by Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA). The interpretation of observations is supported by linear analysis and the magnetohydrodynamic (MHD) model of KHI. This Letter is organized as follows: Section 2 is devoted to the description of SDO/AIA observations, Section 3 described briefly the theory of KHI, Section 4 is devoted to numerical result of the MHD model, and the conclusions are in Section 5.

The AIA on SDO is an array of four telescopes that captures images of the Sun's atmosphere out to 1.3 R_☉ in 10 separate wave bands. Seven of the 10 channels are centered on EUV wavelengths: 94 Å (Fe xviii), 131 Å (Fe viii, Fe xx, Fe xxiii), 171 Å (Fe ix), 193 Å (Fe xii, Fe xxiv), 211 Å (Fe xiv), 304 Å (He ii), and 335 Å (Fe xvi), representing a range of effective temperatures from the upper chromosphere to the corona. The images are 4096 × 4096 square with a pixel width of 06.

SDO was launched on 2010 February 11, and on 2010 April 8 the observatory was still undergoing its post-launch commissioning phase. AIA was taking a full set of images every 20 s in preparation for the transition to nominal science operations which would have a 10–12 s cadence. On 2010 April 8 beginning at 02:34 UT, AIA observed a flare and coronal mass ejection (CME) coming from an active region located 16°E and 29°N of heliographic disk center. The flare peaked at 03:25 UT at a flux of B3.7 on the GOES magnitude scale, and STEREO/SECCHI measured the CME's speed to be around 500 km s⁻¹. In addition to the flare, the primary indicator of the CME in the AIA images was the formation of large-scale dimming regions adjacent to the active region (see Figure 1). These dimming regions are the sites of plasma evacuated by the eruption.

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Along the boundary of one of the dimming regions, we identify the formation, propagation, and decay of vortex-shaped features ranging from several to 10 arcsec in size. They traveled at speeds ranging from ∼6–14 km s⁻¹ along the boundary, which we identify as the shearing interface between erupting magnetic fields and the surrounding, non-erupting corona. Based on the structure, evolution, and decay of these features and supported by theoretical results (see below), we believe that these are the first observations of the KHI in the corona in EUV.

The vortex-shaped features are clearly visible in six of AIA's EUV wave bands and are partially visible in the 94 Å images. The corona imaged in the 94 Å is the hottest of the AIA wave bands (around 6 MK), while the plasma at the dimming boundary is closer to typical coronal temperatures of 1–2 MK. The observed vortices are small (∼7000 km) and evolve rapidly, disappearing on timescales of tens of minutes. In Figure 1 we show the solar disk observed in the EUV 171 Å wave band. The outer box shows the location of the CME eruption, while the inner box shows the boundary of the dimming region where the vortices were observed. In the right panel, we highlight the evolution of the vortices in a sampling of 211 Å images ranging from t = 3:20:53 UT to t = 3:37:53 UT (see the accompanying animations of this event). The vortices propagate from right to left at speeds ranged from ∼6–14 km s⁻¹. All in all, aspects of these vortices were visible for more than an hour.

The first sign of vortex formation is at 3:00:13 UT, and the flow of the vortices along the boundary was fully developed by 3:13:13 UT. The vortex and associated features lasted until about 4:47:13 UT. In order to study the motion of the vortices, in Figure 2 we show the interface boundary stacked in time, where the x-axis is the location across the solar coordinate in arcseconds, and the y-axis is time in seconds (lower panel). Cuts were taken from a total of 51 images in 211 Å from a period of 3:20 UT to 3:40 UT, which corresponds to the period in which the vortex motion was most visible. The average difference between consecutive images was 20–25 s. In order to obtain the speed relative to the curved interface, the stacks are shown along the coordinate of the interface s on the x-axis, and time on the y-axis. The dashed lines in the figure indicate slopes corresponding to speeds of 6 km s⁻¹ and 14 km s⁻¹, after compensating for the velocity of solar rotation.

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The motions of the magnetic loops associated with the erupting CME on 2010 April 8 are shown in Figure 3. The five frames show difference images that were created from consecutive pairs of 211 Å images taken at t = 2:51:13, 2:55:13, 3:00:33, 3:08:33, and 3:13:55 UT. The AIA image of the region at 3:13:53 UT is shown for context. The difference images enhance the structure and the evolution of the loops. We have estimated the upper limit of the speed of the shearing field motion in the initial stages of the eruption by tracking several loop features as a function of time and found that their velocity was ∼20 km s⁻¹. We assume that this velocity is close to the velocity at which the material is evacuated from the dark regions, and that the projection effects are small. Thus, we can estimate the relative velocity between the non-erupting (stationary) and erupting regions. By observing the width of the interface between the two regions before the formation of the vortices, we estimate the thickness of the interface as 1–3 pixels. Thus, on average the interface is ∼800 km thick, and the velocity must change from zero to ∼20 km s⁻¹ over this thickness. This is the initial driving shear of the KHI (see below).

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The first indication of the appearance of the dimming structure is around 2:42:53 UT, and it is fairly well defined by 2:51:13 UT. The dimming structure expands and reaches the location where the KH vortices eventually develop. The first image where we see any motion along the KH front is 3:13:53 UT. Thus, we can roughly estimate that the growth time of the KHI to the nonlinear stage is on the order of 14 minutes.

The KHI in fluids or plasmas arises when there is a velocity discontinuity or finite thickness velocity shear over an interface between two regions (Kelvin 1871; Helmholtz 1868; Chandrasekhar 1961), and it is observed in many natural phenomena as well as in laboratory fluids or plasmas. In the initial linear stage, the KHI exhibits exponential growth, followed by nonlinear saturation and the formation of the typical KH vortices on the interface between the fluids, thus broadening the interface and reducing the magnitude of the shear. The kinetic energy of the shear is converted to the kinetic energy of the vortices that further transport it to smaller scales. In magnetized plasma, the orientation of the magnetic field has an important role on the growth of the KHI (Chandrasekhar 1961). When the magnetic field has significant component parallel to the interface, the growth of the KHI can be suppressed, while perpendicular field does not affect the growth of KHI. The strength of the magnetic field affects the size and the complexity of the vortices produced by KHI in magnetized plasma. The linear growth rate of KHI in an incompressible fluid with a discontinuous velocity jump between the two regions is (Chandrasekhar 1961; Frank et al. 1996)

$$\begin{eqnarray} &&\gamma =\frac{1}{2}|{\bf k}\cdot \Delta {\bf V}|[1-(2V_{\rm A} {\hat{\bf k}}\cdot {\hat{\bf B}})^2/({\hat{\bf k}}\cdot \Delta {\bf V})^2]^{1/2}, \end{eqnarray} \tag{ 1 }$$

where  ΔV is the velocity jump over the interface,  B is the magnetic field, k is the wavevector, $V_{\rm A}=B/\sqrt{4\pi \rho }$ is the Alfvén speed, and ρ is the density. When the interface region is of a finite width, the growth rate is reduced compared to the case with discontinuous jump in velocity.

Based on the parameters of the observed velocity shear in the range 6–20 km s⁻¹, wavelength of 7000 km (based on the size of the initial ripples), and assuming that the velocity jump is discontinuous since it is much smaller than the wavelength, we can estimate the upper limit of the linear growth rate of the KHI using the above equation: γ_KH ≈ 0.003–0.009 s⁻¹. For simplicity, we have assumed that the magnetic field has no parallel component to the interface in Equation 1. The nonlinear saturation is reached at timescale of several γ⁻¹ or in about 10 minutes is in agreement with the observed timescale of the evolution. We relax some of the simplifying assumptions in the MHD model. Note that the KHI is not strongly suppressed, implying that the magnetic field was mostly radial (i.e., perpendicular to the plane of the velocity shear) at the interface. This is in agreement with the observations of the erupting loops, which stretch the magnetic field and evolve to a nearly radial orientation as the eruption proceeds.

Here, we describe the results of the numerical 2.5-dimensional (2.5D, i.e., two spatial dimensions and three components of magnetic field and velocity) MHD model of KHI as a simplified model of the magnetic and velocity structures seen by AIA. Numerical studies have been developed in the past to model the behavior of KHI in a magnetized plasma, with initial velocity shear and weak magnetic field in the plane of the shear using a 2.5D MHD model (e.g., Miura & Pritchett 1982; Miura 1984; Frank et al. 1996, and similar studies). Here, we solve the compressible resistive MHD equations in the two-dimensional Cartesian x–y plane, without gravity (since it is perpendicular to the plane), and isothermal energy equation (see Ofman & Thompson 2002 for the description of the equations and the normalization) with the following initial state (Figure 4):

$$\begin{eqnarray} &&V_{x0}(y)=(V_0/2){\rm tanh}(y/a), \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\rho _0(y)=\frac{1}{2}(\rho _{{\rm max}}+\rho _{{\rm min}})-\frac{1}{2}(\rho _{{\rm max}}-\rho _{{\rm min}}){\rm tanh}(y/a), \end{eqnarray} \tag{ 3 }$$

where a is the width of the shear layer in V_x0 and ρ₀, and the density is varied similarly between ρ_max and ρ_min, and the width of the interface 2a = 800 km, guided by AIA observations. In order to determine the density ratio between the bright and dark (evacuated) regions across the interface, we have examined the emission ratio in EUV AIA images for several wave bands. We found that the emission ratio is on average ∼5 with a range of 3–10. Since the density is proportional to the emission square for the same column depth in the line of sight, we have used ρ_max/ρ_min = 5^1/2 in the model to represent the average observed value. We have also performed runs with density ratios of 2 and 3 and found similar results, with the growth rate of KHI decreasing by ∼15% with the density ratio in this range. The initial magnetic field is taken to be $\mbox{\bf B}_0=B_{x0}\hat{x}+B_{z0}(y)\hat{z}$, with constant B_x0, and the form of B_z0(y) determined from pressure balance over the interface, i.e., in normalized units B_z0(y)²/2 + βρ₀(y) = const. Since we assume that the magnetic field is opened by the erupting CME, we take the value of B_x0 ≈ 0.011〈B_z0〉 resulting in small Alfvén speed in the x–y plane $V_{{\rm A},xy}=B_{x0}/\sqrt{4\pi \rho }=4$ km s⁻¹ compared to the total Alfvén speed $V_{{\rm A}}=B/\sqrt{4\pi \rho }=368$ km s⁻¹, and the value of V₀ = 5V_{A, xy} = 20 km s⁻¹ consistent with the observed velocities of the overlaying loop motion (see section 2). The resulting Alfvén time τ_A = a/V_{A, xy} = 400/4 = 100 s, and the value of β ∼ 0.15. The value of the magnetic field in the observations is unknown. However, the results of the model are not sensitive to the magnitude of B_z0 and B_x0 as long as V_{A, xy} < V₀. The boundary conditions are periodic in the x-direction and open in the y-directions. The size of the modeled region in the y-direction is 10a, and in the y-direction ∼5πa corresponding to the wavelength of the fastest growing mode (e.g., Miura & Pritchett 1982). The time is in units τ_A. The calculations are initialized with a small amplitude fundamental harmonic perturbation in velocity in the x–y plane with a magnitude of ∼0.1V_{A, xy}. The MHD equations are solved using the modified Lax–Wendroff method with fourth-order artificial viscosity term on 260 × 260 grid. The resistivity does not play a significant role on the timescale of the evolution of KHI that grows in few Alfvén times, and the Lundquist number is set to 10⁴.

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At time t ≈ 7τ_A = 700 s the vortex size reaches the width of the interface (2a). The results of the 2.5D MHD model of KHI are shown in Figure 5 at t = 16τ_A = 1600 s in the nonlinear stage when the maximal size (in y) of the vortex is reached. The temporal evolution is on similar timescale as the observed one. The density structure and the corresponding velocity fundamental vortex generated by the KHI are evident. The structure and the temporal evolution of the density correspond to the traveling structures seen in the time sequence of the SDO/AIA EUV emission in Figure 1 and in the corresponding animations of the data and the model. The timescale of the evolution is similar in the model to the observed timescale. The model shows that later evolution leads to formation of smaller scale vortices, and the development of low-density "bubbles" in the high-density region, similar to observational features (see the animations). Although the present model is for simplified 2.5D configuration, it reproduces the main observational features, supporting the interpretation of the observations in terms of KHI.

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We report the first EUV observations of the KHI in the corona, which occurred along the interface between erupting and non-erupting material associated with the CME of 2010 April 8. The KHI features are small (7000 km) and evolve on timescales of seconds, so it would be difficult to observe this phenomenon with an imager that has a lower resolution and cadence than SDO/AIA. The detailed formation and evolution of KH vortices is observed and analyzed. It was found that features along the velocity shear first exhibit the linear growth stage, but the instability quickly reaches nonlinear saturation (within tens of minutes), exhibiting the typical formation of KH vortices.

The growth and the evolution of the instability is compared to theoretical predictions in the linear stage, and to the results of the 2.5D MHD model in the nonlinear stage, and good qualitative agreement is found. In particular, we find that the modeled vortices and associated density structures reach the saturated, nonlinear stage, and the dynamics of the features is qualitatively similar to the observed evolution in AIA images. The purpose of the idealized MHD model is to capture the main physical effect in the observed evolution, and it supports our interpretation of the evolution in terms of KHI. The idealized model contains simplifying assumption on the magnetic geometry, and on observationally unknown parameters. The model cannot capture all the details of the event and is not designed to do so. Other interpretations of this intriguing observation in principle are possible (for example, in term of traveling waves, if the low propagation speed across the field can be justified). However, the proposed interpretation of the observations seems to us most supported by known parameters, KHI linear theory, a 2.5D MHD model, and by analogy with observations of KHI elsewhere in the solar system.

The detection of the KHI in the solar corona further shows that KHI is a ubiquitous process in natural fluids and plasmas that can play an important role in energy transport in the corona on all scales. Theoretical studies find that KHI is important in dissipation of free energy in shear flows and jets, and in the transition to turbulence. KHI can also occur on small scales in solar coronal plasma in the presence of shear flows, leading to enhanced dissipation of waves and super-Alfvénic jets produced by impulsive events (such as flares) in the solar corona, and facilitates the heating of the solar coronal plasma.

We are grateful to SDO/AIA team for providing the data used in this study. L.O. was supported by NASA grants NNX08AV88G and NNX09AG10G.