life time of the helix turns observed in “blob” B, which is on the order of 20 sec. Higher resolution observation is nevertheless necessary to identify whether the phenomena is indeed RT-CD coupled instability, or just purely kink. This is because RT-CD coupled instability is also paramagnetic and could be confused with current-driven instability. However, if high resolution observation is indeed able to distinguish the two instabilities, it can be used to estimate magnetic field configuration of the system using the RT-CD theory.
Numerical simulation of buoyant magnetic flux tube in solar convection zone has found mushroom-shaped RT-type disturbance at bottom of the flux tube [26, 27], and this RT- type disturbance is significantly stabilized by adding more toroidal magnetic field. This is consistent with the RT-CD theory because larger toroidal magnetic field gives smaller Φ2 and hence the RT effect is weaker. As the flux tube emerges from the convection zone to the solar corona, a filamentary structure can form at the top of the emerging flux tube as a result of MRT instability [52, 53] under solar gravity. The filamentary structure is found to be parallel to the surface magnetic field, consistent with the interchange mode of MRT instability. We suggest that the geometry of the cylindrical emerging flux may also be important and could modulate the behavior of the RT instability. However, we admit that the RT-CD theory presented in this chapter should not be directly used in situations like buoyant and emerging flux tube, because in these cases the flux tube is surrounded by dense plasma that can be denser than the flux tube. Nevertheless the theory suggests that in cylindrical geometry the RT could be fundamentally different from 2D planar case.
This work is supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences under Award Numbers DE-FG02-04ER54755 and DE-SC0010471, by the National Science Foundation under Award Number 1059519, and by the Air Force Office of Scientific Research under Award Number FA9550-11-1-0184.
Here we prove thatJ1 is zero in side the plasma.
Consider the momentum equation Eq. 3.18 inside plasma, where ρ1 = 0, B0 = bzzˆso B0· ∇=bz∂z=ikbz. Hence
γρpU1 =−∇
P1+B0·B1
µ0
+ikbz
µ0 B1. (3.86)
The dimensioned version of Eq.3.38 gives U1 = γ
ikαbθB1= γ
ikbzB1. (3.87)
Use this equation to eliminateU1 in Eq. 3.86 and get γ2ρp
ikbz
B1=−∇
P1+B0·B1 µ0
+ ikbz
µ0
B1. (3.88)
Take the curl of this equation and the first term on the right hand side vanishes. There is γ2ρp
ikbz −ikbz
µ0
∇ ×B1= 0. (3.89)
In general,γ2ρ0/ikbz 6=ikbz/µ0. Thereforeµ0J1 =∇ ×B1 = 0 inside plasma. HenceJ1 is also confined in the interface.
3.7.2 Asymptotic behaviors of modified Bessel functions
Asymptotic behaviors of modified Bessel functions are very useful for semi-quantitative analysis in different limits. Here we relist some results from Ref. [1].
In large argument approximation, xm2/2 and Im(x) = ex
√ 2πx
1−4m2−1
8x +O(x−2)
Km(x) = r π
2xe−x
1 +4m2−1
8x +O(x−2)
for |argx|< π/2. Therefore Im(x) and Km(x) are weakly dependent in the limit of x m2/2 and hence
Im0
Im ≈1− 1
2x ≈1 Km0
Km ≈ −1− 1
2x ≈ −1.
In small xapproximation, 0< x√ m+ 1, Im(x) ≈ 1
|m|!
x 2
|m|
Km(x) ≈ (|m| −1)!
2
2 x
|m|
m6= 0,
so that
Im0
Im ≈ |m|
x and Km0
Km ≈ −|m|
x m6= 0.
3.7.3 Eigenvalues of matrix G
In this section we show that the eigenvalues ofiGare purely real and are in positive-negative pairs.
In the following derivation we will always use a finite symmetrically truncated matrix Gp (see Eq. 3.61) instead of the original infinitely large matrixGdefined in Eq. 3.54.
MatrixGis a tridiagonal matrix with zero diagonal, negative super-diagonaland positive sub-diagonal entries. A finite truncatedGp has the form of
Gm,m+1 =ωm <0, Gm,m = 0, Gm+1,m=βm >0.
Define a real diagonal matrix D with
D1,1= 1, Dm,m= s
βm−1
−ωm−1· · · β1
−ω1
and real matrix H
H=D−1GpD.
His also a tridiagonal matrix with zero diagonal elements and the off-diagonal elements are
Hm,m+1 = D−1m,mGm,m+1Dm+1,m+1=−p ωmβm
Hm+1,m = D−1m+1,m+1Gm+1,mDm,m=p ωmβm.
Therefore HT = −H, where T is the transpose operation. Hence H is a real skew- symmetric matrix. Furthermore, (iH)† = −iHT = iH, where † is complex conjugate
transpose operation. ThereforeiHis a Hermitian matrix. Since matrix iGp is similar to a Hermitian matrix, all eigenvalues of iGp are pure real.
A matrix is similar to its transpose, so H and HT =−H must have same eigenvalues.
This immediately leads to the results that the eigenvalues of Halways come in pairs ±λ, where λare imaginary. Therefore the eigenvalues of iGp always come in real positive and negative pairs. SinceiGp is a (2p+ 1) by (2p+ 1) matrix, by symmetry, 0 must also be an eigenvalue of iGp. Therefore iGhas p positive eigenvalues andp negative eigenvalues (in pairs) and one zero eigenvalue. This is also consistent with the fact that the summation of all eigenvalues of a matrix, which is zero foriG, equals to the trace of the matrix, which is also zero becauseiGhas zero diagonal entries.
Chapter 4
Circularly polarized Magnetic Field of Obliquely Propagating Whistler Wave during Fast
Magnetic Reconnection
The whistler wave is a cold electromagnetic plasma wave ubiquitous in collisionless mag- netic reconnection [29, 32, 54], Earth’s magnetosphere [24,108, 116, 117], and laboratory antenna excitation experiments [108, 109]. Whistler waves can be excited in a magnetized plasma when the frequency is much higher than the ion cyclotron frequency but much lower than the electron cyclotron frequency, i.e., ωci ω |ωce| [5, 6, 54]. The conventional derivation of whistler waves shows that both the electric component and the magnetic com- ponent of the wave are right-hand circularly polarized when the wavevector is parallel to the background magnetic field. Tsurutani et al. [117] discovered that the magnetic compo- nent of a magnetospheric whistler wave is circularly polarized even when the wavevector is oblique to the background magnetic field. Theoretical analyses [6, 119] confirm that the whistler wave always has a circularly polarized magnetic component regardless of the wave propagation direction, but the wave electric component is circularly polarized only when the wavevector is parallel to the background magnetic field.
Lab experiments [54, 80], electron magnetohydrodynamic (EMHD) simulations and particle-in-cell (PIC) simulations [28,29,32], and space observations [24,117] suggest that the whistler wave is an integral component of fast magnetic reconnection. However it is still unclear what role the whistler wave plays in the reconnection processes. Theoretical modeling shows that the equations that govern the fast reconnection process may involve
whistler waves [7].
In the Caltech plasma jet experiment, a current-carrying collimated jet is created from the merging of eight plasma-filled flux ropes. When the current-carrying jet undergoes a kink instability, a lateral Rayleigh-Taylor instability occurs on the jet surface. A linear theory of this Rayleigh-Taylor instability has been presented in Chapter 3. The Rayleigh- Taylor instability quickly evolves to a nonlinear phase, and the plasma jet is eroded by the instability to have a width smaller than the ion skin depth. At the ion skin depth scale, MHD is no longer valid because ion and electron motion are decoupled. A fast magnetic reconnection induced by the Rayleigh-Taylor instability then occurs and eventually breaks the plasma jet structure [80]. A capacitively coupled probe placed near the jet measured fast electric field fluctuations with frequencies in the whistler regime [79,80].
In this chapter, we present the measurements and analyses of the magnetic components of the whistler waves associated with the fast magnetic reconnection in the Caltech plasma jet experiment. A 3D high speed magnetic probe with excellent electrostatic rejection has been specifically designed and constructed to perform the measurement (see Chapter 5for details about the magnetic probe). High-frequency magnetic fluctuations in the whistler regime are detected by the probe at the time of Rayleigh-Taylor instability and reconnection.
The magnetic fluctuations span a wide frequency range and have a very steep power spec- trum. Circularly polarized magnetic components of obliquely propagating whistler waves are identified. The detection of whistler waves is important evidence indicating that the reconnection process is in the two-fluid, not MHD regime.
This chapter is arranged as follows. First we briefly review the theories of whistler wave and its association with fast reconnection. Then we analyze the measurements from the high-speed 3D magnetic probe and show that the high-frequency magnetic fluctuations are an ensemble of whistler waves generated during the reconnection. We use a hodogram technique to resolve the circularly polarized magnetic components of the whistler waves.
A polarization recognition algorithm was developed to automatically identify the type of polarization. Useful signal processing algorithms are listed and briefly reviewed in Ap- pendix B, such as Fourier transform, finite impulse response digital filters and principal component analysis.