Identification of standing fronts in steady state fluid flows: exact and approximate solutions for propagating MHD modes

Abstract

The spatial structure of a steady state plasma flow is shaped by the standing modes with local phase velocity exactly opposite to the flow velocity. The general procedure of finding the wave vectors of all possible standing MHD modes in any given point of a stationary flow requires numerically solving an algebraic equation. We present the graphical procedure (already mentioned by some authors in the 1960’s) along with the exact solution for the Alfvén mode and approximate analytic solutions for both fast and slow modes. The technique can be used to identify MHD modes in space and laboratory plasmas as well as in numerical simulations.

Appendices

Appendix A: Changing reference frame

The calculations presented in this paper have been established in a particular reference frame (hereafter the primed reference frame) defined by the flow velocity vector $\vec{u}_{f}$ and the magnetic field vector $\vec{B}$ as illustrated in Fig. 1. This particular frame is well suited for the theoretical treatment of mode propagation related issues but is rarely the most practical one as both velocity and magnetic field orientations do generally change as a function of space. In most applications, however, there is one unique frame for the whole system (for example the simulation frame as in Fig. 14 or the frame defined by the step geometry in the Krisko and Hill 1991 experiment of Fig. 13). We denote this frame as the unprimed frame. It is therefore useful to establish the matrix $M$ which transforms the orthonormal right handed basis vectors of an arbitrary frame into the orthonormal basis vectors (also right handed) of the primed reference frame. The plasma flow being stationary in both frames there is no relative motion and the transformation matrix $M$ may be viewed as the product of two rotations implying $\det (M)=1$. $M$ is easily obtained by writing the basis vectors $\vec{e}_{x}\,'$, $\vec{e}_{y}\,'$ and $\vec{e}_{z}\,'$ of the primed reference frame in terms of the two unitary vectors $\vec{\mu }_{f}\equiv \vec{u}_{f}/u _{f}$ and $\vec{b}\equiv \vec{B}/B$ in the unprimed frame, i.e.

$$\begin{aligned} \vec{e}_{z}\,' =& -s \vec{b} \\ \vec{e}_{y}\,' =&\frac{\vec{\mu }_{f} \times \vec{e}_{z}\,'}{\sin \Delta_{f}} = -s \frac{\vec{\mu }_{f}\times \vec{b}}{\sin \Delta_{f}} \\ \vec{e}_{x}\,' =&\vec{e}_{y}\,' \times \vec{e}_{z}\,' = s \frac{( \vec{\mu }_{f}\times \vec{b})\times \vec{b}}{\sin \Delta_{f}} = \frac{- \vec{\mu }_{f} +(\vec{\mu }_{f}\cdot \vec{b}) \vec{b}}{\sin \Delta _{f}} \end{aligned}$$

(A.1)

where $s = \mathrm{sign} (\vec{\mu }_{f}\cdot \vec{b})$ and $0 < \Delta _{f}\leq \pi /2$ is the angle between the directions of $\vec{b}$ and $\vec{\mu }_{f}$. In the primed frame the magnetic field is therefore either parallel or anti-parallel with respect to $\vec{e}_{z}\,'$. If $\Delta_{f}\neq 0$, the components of the three primed basis vectors are fully specified by the constraint that $\vec{\mu }_{f}$ is in the $(\vec{e}_{x}\,',\vec{e}_{z}\,')$ plane with negative components $\mu_{fx}\,'=\vec{\mu }_{f}\cdot \vec{e}_{x}\,'<0$ and $\mu_{fz}\,'= \vec{\mu }_{f}\cdot \vec{e}_{z}\,'<0$. In the singular case $\Delta_{f}=0$, $\vec{e}_{y}\,'$ may be any unitary vector perpendicular to $\vec{b}$ and $\vec{e}_{x}\,' =\vec{e}_{y}\,'\times \vec{e}_{z}\,'$ as in the general case (A.1).

The transformation of the basis vectors $\vec{e}_{x} =(1,0,0)$, $\vec{e}_{y} =(0,1,0)$ and $\vec{e}_{z} =(0,0,1)$ to the primed frame is thus given by

$$ \vec{e}_{i}\,'= M \vec{e}_{i},\quad i=\{x,y,z\} $$

(A.2)

where the elements of the $3\times 3$ transformation matrix $M$ are merely the components of the primed basis vectors:

$$ M=\bigl(\vec{e}_{x}\,'|\vec{e}_{y} \,'|\vec{e}_{z}\,'\bigr). $$

(A.3)

Accordingly, the components of an arbitrary vector projected onto the basis vectors of the primed frame are obtained by applying the transposed matrix $M^{T}$ to the components of the vector in the unprimed frame, i.e.

$$ \vec{v} \,' = M^{T} \vec{v} $$

(A.4)

where $\vec{v}\,'=(\vec{v}\cdot \vec{e}_{x}\,',\vec{v}\cdot \vec{e} _{y}\,',\vec{v}\cdot \vec{e}_{z}\,')$.

1.1 A.1 Special case: $\mu_{fy} = b_{y} = 0$

The transformation matrix $M$ is particularly simple in the case where the $y$ components of $\vec{\mu }_{f}$ and $\vec{b}$ are zero in the unprimed frame. In this case the basis vectors (A.1) reduce to:

$$\begin{aligned} \vec{e}_{z}\,' =& -s(b_{x},0,b_{z}) \\ \vec{e}_{y}\,' =& -s(0,\sigma ,0) \\ \vec{e}_{x}\,' =& +\sigma (b_{z},0,-b_{x}) \end{aligned}$$

(A.5)

where $\sigma \equiv \mathrm{sign} (-\mu_{fx} b_{z} + \mu_{fz} b_{x})$. Accordingly, the transformation matrix $M$ for this particular case is

$$\begin{aligned} M=\left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c} \sigma b_{z}& 0 &-sb_{x} \\ 0&-s\sigma & 0 \\ -\sigma b_{x}& 0 & -sb_{z} \end{array}\displaystyle \right ) . \end{aligned}$$

(A.6)

Appendix B: Parallel compressibility and Alfvén ratio

2.1 B.1 The parallel compressibility

The parallel compressibility of a plane mode with wave vector $\vec{k}$ is defined as

$$ C_{\parallel }= \frac{\delta n}{n}\frac{B}{\delta B_{\parallel }} $$

(B.1)

where $\delta n$ and $\delta B_{\parallel }$ are the variations of density and magnetic field along a path parallel to $\vec{k}$. The subscript ∥ in $\delta B_{\parallel }$ denotes the variation of the magnetic field parallel to itself so that (for example) $\delta B_{\parallel }=0$ through a rotational discontinuity. Assuming a wave vector $\vec{k}=(k,0,0)$:

$$ \delta B_{\parallel }= dx \frac{\partial \vec{B}}{\partial x}\cdot \frac{ \vec{B}}{B} = dx \frac{\partial B}{\partial x} $$

(B.2)

and

$$ \delta n=dx \frac{\partial n}{\partial x}. $$

(B.3)

In case of an arbitrary orientation $\vec{k}$ the variations $\delta B_{\parallel }$ and $\delta n$ over an infinitesimal distance $\vec{\delta x}=\epsilon \vec{k}$ are

$$ \delta B_{\parallel }= \vec{\delta x}\cdot \nabla B =\epsilon \vec{k} \cdot \nabla B $$

(B.4)

and

$$ \delta n=\epsilon \vec{k}\cdot \nabla n $$

(B.5)

respectively. The parallel compressibility can then be written as:

$$ C_{\parallel }= \frac{\vec{k}\cdot \nabla n}{n}\; \frac{B}{\vec{k} \cdot \nabla B}. $$

(B.6)

For the incompressible Alfvén mode the parallel compressibility is zero. For both the slow and the fast mode the parallel compressibility is given by

$$ C_{\parallel }(\theta ) = \frac{c_{A}^{2}}{u_{\phi }^{2}(\theta ) -c ^{2}} $$

(B.7)

where $\theta$ is the angle between $\vec{k}$ and the magnetic field $\vec{B}$, $c_{A}$ is the Alfvén speed, $c$ the adiabatic sound speed and $u_{\phi }=\omega /k$ the phase velocity (3) of the corresponding mode. We note that for non-zero values of the Alfvén speed, the fast mode does always propagate faster than the sound speed. Thus, according to (B.7) the compressibility of the fast mode is always positive. On the contrary, slow modes do always propagate slower than the sound speed (except for the special case $c=c_{A}$ and $\theta =0$). Thus, the denominator in (B.7) is always negative implying a negative compressibility for the slow mode. A typical example of parallel compressibility profiles for both compressible MHD modes is shown in Fig. 16.

Fig. 16

Parallel compressibility and Alfvén ratio for the 3 MHD modes and particular values of $\beta$ and $\gamma$

2.2 B.2 The Alfvén ratio

The Alfvén ratio is defined as

$$ R_{A}= \frac{\delta v_{\perp ,k}^{2}}{c_{A}^{2}} \frac{B^{2}}{\delta B_{\perp ,k}^{2}} $$

(B.8)

where the perpendicular direction is now to be considered with respect to $\vec{k}$, i.e. $\delta v_{\perp ,k}=|\vec{\delta v}\times \vec{k}/k|$.

For the three MHD modes the Alfvén ratio is given by

$$ R_{A}(\theta ) = \frac{c_{A}^{2}}{u_{\phi }^{2}}\cos^{2}\theta . $$

(B.9)

The phase velocity of the Alfvén mode being $u_{\phi }^{2}=c_{A} ^{2} \cos^{2}\theta$ implies that $R_{A}=1$, independently of the propagation angle $\theta$. Sample profiles of the Alfvén ratio for the three MHD modes are shown in Fig. 16.