We can see how this is consistent with the development of small scales via phase mixing by introducing local wavenumbers for the variation with α and β, (3)\begin{eqnarray} \xi \propto \exp {\rm i}\left[ \int \kappa_\alpha {\rm d}\alpha + \int \kappa_\beta {\rm d}\beta \right]\!. \label{number3} \end{eqnarray}ξ∝expi∫καdα+∫κβdβ.Here κα and κβ are the wavenumbers in α and β and have units that are the inverse of the units of their respective coordinates. These wavenumbers should be distinguished from the perpendicular components of the usual wave vector k, which has units of 1/length. The different wavenumbers may be related through the scale factors (h) that relate elemental coordinate increments to physical distances: dr = eαhαdα + eβhβdβ + eγhγdγ, where eα is a unit vector in the α direction, etc. In this notation ∇⊥ = (eα/hα)∂/∂α + (eβ/hβ)∂/∂β, and noting that ∇⊥ξ ≈ ik⊥ξ, Eq. (3) yields \begin{eqnarray} \bdelp\xi &amp;=&amp; {\rm i} \vec{k}_\perp \xi = {\rm i} \left( \frac{\vec{e}_\alpha}{h_\alpha}\frac{ \partial}{\partial\alpha} +\frac{\vec{e}_\beta}{h_ \beta}\frac{ \partial}{\partial \beta}\right)\xi \\ &amp;\equiv&amp; {\rm i}\left( \frac{\vec{e}_\alpha}{h_\alpha}\kappa_\alpha +\frac{\vec{e}_\beta}{h_ \beta}\kappa_\beta\right)\xi. \label{number4} \end{eqnarray}∇⊥ξ=ik⊥ξ=ieαhα∂∂α+eβhβ∂∂βξ≡Equating components of the second and fourth expressions in Eq. (5) gives the expected relations between the various wavenumbers, (6)\begin{eqnarray} k_\alpha = \kappa_\alpha /h_\alpha, \qquad k_\beta = \kappa_ \beta /h_ \beta. \label{number5} \end{eqnarray}kα=κα/hα, kβ=κβ/hβ.Equations (2) and (5) give a direct and elegant expression for the perpendicular wave vector as (7)\begin{eqnarray} \vec{k}_\perp \approx -(\bdel \omega_{\rm c}) t, \label{number6} \end{eqnarray}k⊥≈−(∇ωc)t,which is a generalisation to three dimensions of the results of Mann et al. (1995), (Wright et al. 1999) and Kaneko & Yokoyama (2015) for lower dimensional systems, which developed phase mixing in only one perpendicular coordinate. The above expression allows phase mixing in both perpendicular directions, giving physical phase mixing lengths (or wavelengths) in the α and β directions of (8)\begin{eqnarray} L_{{\rm ph}\alpha} = \frac{2\pi}{|k_\alpha |} \equiv \frac{2\pi h_\alpha}{|\partial \omega_{\rm c}/\partial\alpha |t}, \qquad L_{{\rm ph}\beta} = \frac{2\pi}{|k_ \beta |} \equiv \frac{2\pi h_\beta}{|\partial \omega_{\rm c}/\partial \beta |t}\cdot \label{number7} \end{eqnarray}Lphα=2π|kα|≡2πhα|∂ωc/∂α|t, Lphβ=2π|kβ|≡2πhβ|∂ωc/∂β|t·If the phase mixing lengths are expressed in the same units as α and β, rather than physical length as in Eq. (8), slightly simpler expressions are found, i.e. (9)\begin{eqnarray} \ell_{{\rm ph}\alpha} = \frac{2\pi}{|\kappa_\alpha |} \equiv \frac{2\pi}{|\partial \omega_{\rm c}/\partial\alpha |t}, \qquad \ell_{{\rm ph}\beta} = \frac{2\pi}{|\kappa_ \beta |} \equiv \frac{2\pi}{|\partial \omega_{\rm c}/\partial \beta |t}\cdot \label{number8} \end{eqnarray}ℓphα=2π|κα|≡2π|∂ωc/∂α|t, ℓphβ=2π|κβ|≡2π|∂ωc/∂β|t·The development of the phase mixing length can be pictured simply as the tendency for each field line to oscillate with its own natural frequency. Even if all the field lines start to oscillate with the same phase, they soon drift out of phase with one another as time passes. Not only does the phase mixing process generate perpendicular scales, but points of constant phase can be seen to move across field lines. This phase motion has been seen in magnetospheric data of Alfvén waves (see the review by Wright & Mann 2006) and the simulations of coronal oscillations by Kaneko & Yokoyama (2015). These studies note that the direction of motion is related to the spatial variation of ωc. The results of these papers for the perpendicular phase velocity in physical space generalise to Vph = ωc/k⊥, giving the components (10)\begin{eqnarray} V_{{\rm ph}\alpha} = \frac{-\omega_{\rm c} h_\alpha}{(\partial \omega_{\rm c}/\partial\alpha )t}, \qquad V_{{\rm ph}\beta} = \frac{-\omega_{\rm c} h_ \beta}{(\partial \omega_{\rm c}/\partial \beta )t}, \qquad \label{number9} \end{eqnarray}Vphα=−ωchα(∂ωc/∂α)t, Vphβ=−ωchβ(∂ωc/∂β)t, If the excitation occurred at a time ti, the subsequent properties are found by replacing t with t−ti in the above formulae.
Citation: Kaneko & Yokoyama (2015)
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Citation: Kaneko & Yokoyama (2015)
Citation function: Uses; Uses; Uses