### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Analytical Solution
- 3. Kelvin-Helmholtz Stability of Tangential Discontinuities
- 4. Observations and Analysis
- 5. Conclusions
- Acknowledgments
- References

[1] Three solar wind reconnection exhaust events observed by the Wind satellite are compared to the analytical solution of the Riemannian decay of a current sheet due to reconnection of skewed magnetic fields. This process leads to high speed flows inside an exhaust region which is bounded not only by Alfvénic and slow mode waves but also, to a large extent, by tangential discontinuities (TDs). The TD portions of the exhaust boundary expand with distance from the X-line and therefore a long exhaust does not necessarily imply a long X-line. In some cases, portions of the exhaust show oscillations which might be connected to a Kelvin-Helmholtz instability in agreement with analytical estimates.

### 2. Analytical Solution

- Top of page
- Abstract
- 1. Introduction
- 2. Analytical Solution
- 3. Kelvin-Helmholtz Stability of Tangential Discontinuities
- 4. Observations and Analysis
- 5. Conclusions
- Acknowledgments
- References

[3] A genuine nonlinear model within MHD theory was first proposed by *Petschek* [1964]. Several extensions of this model to more general boundary conditions have since been published [*Levy et al.*, 1964; *Heyn et al.*, 1988; *Heyn and Semenov*, 1996; *Semenov et al.*, 2004]. An essential ingredient of this model is the initial current layer in form of a tangential discontinuity, i.e. there is neither mass nor magnetic flux across the layer and the total pressure (gas plus magnetic) is the same on both sides of the layer. Reconnection naturally starts when for some reason the plasma resistivity is suddenly increased inside a certain region of the current layer, the diffusion region. Inside the diffusion region, there appears a normal component of the magnetic field and the tangential discontinuity immediately becomes unstable. The discontinuity decays into a system of large amplitude waves in form of shocks (*S*) or rarefaction waves (*R*). MHD theory predicts four types of waves: fast waves *S*^{+}(*R*^{+}), slow waves *S*^{−}(*R*^{−}), Alfvén waves *A*, and an entropy wave *C.* The general scheme for the decay of an arbitrary discontinuous plasma surface is *S*^{+}(*R*^{+}), *A*, *S*^{−}(*R*^{−}), *C*, *S*^{−}(*R*^{−}), *A*, *S*^{+}(*R*^{+}) [*Akhiezer et al.*, 1975; *Heyn and Semenov*, 1996].

[4] Thus, a set of nonlinear waves will propagate according to their respective velocities in both directions away from the site of generation. The leading wave will be a fast shock *S*^{+} or fast rarefaction *R*^{+}, followed by a nonlinear Alfvén wave (rotational discontinuity) *A*, and a trailing slow shock wave *S*^{−} or slow rarefaction wave *R*^{−}. A contact discontinuity *C* appears in the central region. Rarefaction waves instead of shocks appear in case of strong asymmetry of the plasma parameters, in particular the plasma density.

[5] If the unstable surface respects total pressure balance, i.e. it is a tangential discontinuity, and if the diffusively produced normal magnetic field component *B*_{n} is much smaller than the tangential magnetic field component *B*_{t}, i.e., *B*_{n}/*B*_{t} = *ε* ≪ 1, the complex problem can be solved in a two step procedure: first, a 1-D nonlinear self-similar decay problem and second, a linear problem where the solutions in the different regions between the discontinuities and shocks are connected in analogy to the general boundary layer procedure.

[6] Because the total pressure *P* = *B*_{t}^{2}/2*μ*_{0} + *p* with *p* the thermal plasma pressure is constant across a tangential discontinuity, *P*_{1} = *P*_{2}, fast shocks need not be considered in the decay process. Fast waves are essentially driven by pressure gradients, whereas for a tangential discontinuity forces related to shear in magnetic field and velocity are important. Therefore, the scheme of the decay of the current sheet is reduced to the simpler sequence *A*, *S*^{−}(*R*^{−}), *C*, *S*^{−}(*R*^{−}), *A.*

[7] If we can assume that reconnection happens such that *B*_{n}/*B*_{t} ≪ 1, it follows that there must also appear a normal flow *V*_{n}/*V*_{A} ∼ *ε* where *V*_{A} = *B*_{t}/ is the Alfvén speed and *ρ* the plasma density. The finite inflow velocity is needed to replace the tangentially ejected plasma as a consequence of balancing the tangential magnetic stresses by plasma inertia.

[8] In lowest order, the Rankine-Hugoniot system of equations reduces to a system for the tangential components**V**_{t}, **B**_{t}, the plasma density *ρ*, and plasma pressure *p* which depend only on the initial parameters of the tangential discontinuity and do not depend at all on first order parameters such as the reconnection rate, *B*_{n}/*B*_{t}. Moreover, they are not related to the geometry of the reconnection line and the scenario of reconnection, i.e. if reconnection is locally switched on, switched off, steady state or impulsive in character. Therefore, the profiles of **V**_{t}, **B**_{t}, *p* and *ρ* can be considered to be very general and independent of the details of the reconnection process. In the present study such characteristic profiles are looked for in solar wind data from Wind and are then compared with the predictions of the nonlinear theory outlined above.

### 3. Kelvin-Helmholtz Stability of Tangential Discontinuities

- Top of page
- Abstract
- 1. Introduction
- 2. Analytical Solution
- 3. Kelvin-Helmholtz Stability of Tangential Discontinuities
- 4. Observations and Analysis
- 5. Conclusions
- Acknowledgments
- References

[9] The solution of the Riemannian decay problem determines tangential magnetic fields and flows as well as plasma densities and pressures in lowest order and in the different regions of the boundary layer (bl) like structure. On the other hand, the detailed geometry of the bl is mainly dictated by the length of the reconnection line (X-line) and the electric field along the X-line as a consequence of the diffusive process. As sketched inFigure 1, outflow regions with high speed plasma are formed and these regions expand primarily along the guiding magnetic fields outside the bl. This picture is the result of a quantitative time-dependent model for transient reconnection of skewed magnetic fields [*Semenov et al.*, 2004] and is quite similar to the qualitative picture of magnetic flux tube reconnection first sketched by *Russell and Elphic* [1978]. An important feature of the analytical model is the fact that one expects after some time most of the exhaust being bounded by a TD. This TD has the same characteristic changes of **V**_{t}, **B**_{t}, number density *n*, and temperature *T*as a merged Alfvén-slow shock (*AS*^{−}) transition would have since in the present model the magnetic field first rotates at *A*, then the plasma is heated and compressed at *S*^{−}, and thereafter it moves inertially inside the exhaust without further changes. One can distinguish between TDs and *AS*^{−}s by the presence or absence of normal components of **V** and **B**, but normal components are notoriously difficult to determine in practice. Another possibility to distinguish between TD and *AS*^{−} would be a finite separation of *A* and *S*^{−} due to their different propagation velocities as shown in Figure 1 variants A and C. The detection of a separated pair *A* and *S* in the solar wind has also been reported by *Farrugia et al.* [2001]. Within the context of the present model we define that if one does not observe both an Alfvénic-type rotation in**B**_{t} and **V**_{t} and a *separate*slow shock-like discontinuity at the exhaust boundary, the boundary is assumed to be a TD. A further possibility is to study the stability of those surfaces. Without the normal component the surface TD might be Kelvin-Helmholtz (KH) unstable whereas for the*AS*^{−} structure we do not expect such an instability. Observational evidence for these signatures are presented below.

[10] For an incompressible plasma the explicit stability criterion is given by *Landau and Lifshitz* [1985],

where Δ**v** = **v**_{2} − **v**_{1}, *ρ* = 2*ρ*_{1}*ρ*_{2}/(*ρ*_{1} + *ρ*_{2}) and indices refer to the individual sides of the TD. To put it simply, this means that for shear velocities bigger than the Alfvén velocity and shear velocity directions essentially perpendicular to the magnetic field the TD becomes KH unstable. For a compressible plasma the dispersion equation is of 10th order [*Pu and Kivelson*, 1983; *Heyn and Semenov*, 1996] and the stability boundary is found numerically.

### 4. Observations and Analysis

- Top of page
- Abstract
- 1. Introduction
- 2. Analytical Solution
- 3. Kelvin-Helmholtz Stability of Tangential Discontinuities
- 4. Observations and Analysis
- 5. Conclusions
- Acknowledgments
- References

[11] In this section we discuss the analysis of three events of magnetic reconnection observed by the Wind satellite in the solar wind. Since the analytical solution is very sensitive to the decomposition of the vectors **B** and **V** into normal (with respect to the initial current sheet) and tangential components, it is an important task to present the data in the associated normal coordinate system (L, M, N). In principle, the normals of the discontinuities *A* and *S*^{−} can differ from the normal of the initial current sheet because shape and inclination of the discontinuities depend on the behavior of the reconnection rate. Therefore, it is better to determine the normal vector to the initial current sheet by the tangential discontinuity method, i.e. taking the cross product of the magnetic field vectors from magnetic field data well outside the reconnection layer rather than using the MVA (minimum variance analysis) method which utilizes data of the internal discontinuity structure. The present MHD model is invariant with respect to the choice of the tangential vectors L and M. Those vectors are chosen such that the L and M directions correspond to the directions shown in Figure 1.

[12] Wind solar wind data have been looked through and three characteristic events have been chosen: i) an event with two KH stable TDs, ii) an event with clearly separated *A* and *S*^{−} and a KH unstable TD, iii) an event with one KH stable TD and one KH unstable TD.

[13] We first analyze the reconnection event on 1999-11-12 observed by the Wind spacecraft in the solar wind. From the upper panel ofFigure 2 it follows that *B*_{n}/*B*_{t} as well as *V*_{n}/*V*_{A} is smaller than 0.2 and, in addition, that the total pressure is essentially constant across the whole outflow (exhaust) region, *P* = 0.0104 ± 0.0008 nPa. Thus, this event complies with the requirements for the Riemannian decay analysis. We calculate analytically the magnetic field and plasma parameters for outflow regions (**B**_{t}, **V**_{t}, *T*, *ρ*) using for input average values of the ambient solar wind.

[14] Figure 2 shows the Wind spacecraft measurements (3 s resolution) of magnetic field and plasma parameters across the reconnection layer in normal coordinates (L,M,N) in the solar wind reference frame. The tangential vectors, **L** and **M**, and the components of the normal, **N**, in GSE coordinates are, respectively **L** = (−0.64, −0.06, 0.74), **M** = (−0.46, 0.81, −0.33), **N** = (−0.59, −0.57, −0.56). The solar wind velocity external to the exhaust (*V*_{L}, *V*_{M}, *V*_{N}) = (335, 235, 280) km/s has been subtracted from all velocities to fix the frame of reference.

[15] In this frame clear signatures of reconnection during the spacecraft crossing of the outflow exhaust can be observed: the sharp rotation of magnetic field vector at the boundaries (at 19:08:00 and 19:12:00) of the outflow layer; accelerated, compressed and heated plasma inside the exhaust while the magnetic field strength being slightly weaker there. One should note the correlated changes in *V*_{L} and *B*_{L}at the leading edge and anti-correlated changes in*V*_{L} and *B*_{L} at the trailing edge, which indicate that the waves were propagating in opposite directions along the field.

[16] These main signatures of the reconnection based on the classical Petschek model have been reported many times [*Phan et al.*, 2006; *Paschmann*, 2008]. In the present study, a detailed analysis of all MHD discontinuities generated by reconnection is made. For this purpose we calculate the background values of plasma and magnetic field (**B**_{t}, **V**_{t}, *T*, *ρ*) outside the reconnection layer from time averages for the intervals 19:05:00–19:08:00 and 19:12:00–19:15:00 UT.

[17] The mutual positions of the individual discontinuities depend on geometry and reconnection rate but the profiles of *B*_{L}, *B*_{M}, *V*_{L}, *V*_{M}, *T*, and *ρ* between the discontinuities do not. At the exhaust boundaries (19:08:00, 19:12:00) one can see a sharp rotation of the magnetic field (Figure 2, bottom) which is the signature of an Alfvén discontinuity. But at the same time, one can also see an increase of density, temperature and entropy, and a decrease of the magnetic field strength in accordance with changes expected across a slow shock. In view of the fact that both signatures happen simultaneously, we interpret this as a single TD surface rather than a separated *AS*^{−} structure. Therefore, this crossing corresponds to variant B shown in Figure 1.

[18] Even a structure similar to a contact discontinuity might be identified in this event, although it is generally believed that contact discontinuities cannot persist in collisionless plasmas. As can be seen in Figure 2this is the region where entropy, temperature and density change noticeably whereas changes in magnetic field and plasma velocity are small. Checking the conditions for KH stability, it turns out that the initial current sheet as well as both subsequent TDs bounding the exhaust are KH-stable.

[19] We conclude that the reconnection exhaust observed by the Wind spacecraft in the solar wind on 1999-11-12 is bounded by KH stable TDs. The values for magnetic field, plasma flow, density, and temperature expected from the Riemannian analysis are in good agreement (shown in red inFigure 2) with the data.

[20] The next reconnection event discussed was observed by the Wind spacecraft on 1998-09-01. InFigure 3 the time interval 01:14:00–01:56:00 UT is shown. The *L*, *M*, *N* vectors for this event are **L** = (−0.31, 0.23, 0.88), **M** = (−0.93, −0.21, −0.33), **N** = (0.14, −0.95, 0.29), and the solar wind velocity for fixing the frame is (*V*_{L}, *V*_{M}, *V*_{N}) = (142, 461, −100) km/s.

[21] The exhaust boundaries marked by vertical solid lines at 01:24:00 and 01:36:00 are again TDs which is evident from the rotation of the magnetic field, the changes in velocity and the magnetic field amplitude occurring simultaneously. The KH analysis for this event shows that the initial current sheet as well as the left hand side TD (at 01:24:00) are stable whereas the right hand side TD is KH unstable. We conjecture that this instability leads to the oscillating structures seen in the interval 01:36:00–01:46:00 in Figure 3. These oscillations are largely coupled variations in velocity and magnetic field similar to Aflvénic-like oscillations but also show clear signatures of temperature, density, and entropy increase. It should be pointed out that the increase in entropy is a signature of some dissipative process starting at the edge of the oscillatory region at 01:46:00 and gains its maximal value at 01:40:00 outside the exhaust and the oscillating structure. This might be explained by energy transfer through the instability from bigger to smaller scales on which it is eventually dissipated.

[22] We conclude that the reconnection exhaust observed by the Wind spacecraft in the solar wind on 1998-09-01 is bounded again by two TDs. Due to the fact that one of the TDs is predicted to be KH unstable, the entropy change inside the exhaust is not as localized as it is in the example above (region*C*). At the same time, also no sharp transition of |**B**|, *n* and *T* can be observed at the unstable TD.

[23] The last event to be discussed was observed on 1998-03-25. InFigure 4 the time interval 16:00:00–17:00:00 UT is shown.The *L*, *M*, *N* vectors are **L** = (−0.06, 0.71, −0.7), **M** = (−0.023, −0.68, −0.68), **N** = (−0.97, 0.11, 0.21). The solar wind velocity is (*V*_{L}, *V*_{M}, *V*_{N}) = (70, 125, 395) km/s.

[24] The exhaust boundaries are identified and marked with vertical solid lines at 16:09:00 and 16:25:00. Again, oscillations can be seen in the interval 16:25:00–16:45:00. It is a remarkable feature that for this event the Alfvén discontinuity *A* at 16:09:00 (changes in rotation, *V*_{M}, *V*) and, at the same time, the slow shock at 16:11:00 (changes in entropy, density, temperature, magnetic field strength, *V*_{L}, *V*_{M}, *V*) were well separated.

[25] The KH analysis indicates a marginal stable initial current sheet. Since the exhaust can be clearly distinguished we can conclude that reconnection can probably stabilize potentially unstable surfaces. The right hand side TD at 16:25:00 is again predicted to be KH unstable which is reflected in oscillations propagating in both directions from the site of the TD. Again, the entropy decreases with increasing distance from the exhaust boundary.