Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies

Abstract

In this paper we introduce a new method for exact decomposition of propagating, nonlinear magnetohydrodynamic (MHD) disturbances into their component eigenenergies associated with the familiar slow, Alfvén, and fast wave eigenmodes, and the entropy and field-divergence pseudo-eigenmodes. First the mathematical formalism is introduced, where it is illustrated how the ideal-MHD eigensystem can be used to construct a decomposition of the time variation of the total energy density into contributions from the eigenmodes. The decomposition method is then demonstrated by applying it to the output of three separate nonlinear MHD simulations. The analysis of the simulations confirms that the component wave modes of a composite wavefield are uniquely identified by the method. The slow, Alfvén, and fast energy densities are shown to evolve in exactly the way expected from comparison with known linear solutions and nonlinear properties, including processes such as mode conversion. Along the way, some potential pitfalls for the numerical implementation of the decomposition method are identified and discussed. We conclude that the exact, nonlinear decomposition method introduced is a powerful and promising tool for understanding the nature of the decomposition of MHD waves as well as analysing and interpreting the output of dynamic MHD simulations.

Figures (9)

• Figure 1: Left of the vertical separator line: (a-l) $x$-$z$ plots of the eigenenergy time derivatives summed over both characteristic directions ($\pm$ terms) given by \ref{['eq:EDT']} computed for simulation Ia ($\beta = 9.03$) at $t = 17.95$ and $y = -0.024$. The top nine panels (a-i) in a row-wise order from top to bottom are the $x$, $y$, and $z$ components of the (row 1) slow, (row 2) Alfvén, and (row 3) fast eigenenergy time derivatives, given by \ref{['eq:ESlowdt']}, \ref{['eq:EAlfdt']}, and \ref{['eq:EFastdt']}, respectively. Panels (j) and (k) depict the field-divergence and entropy pseudo-mode eigenenergy time derivatives provided by \ref{['eq:EDivdt']} and \ref{['eq:EEntdt']}, respectively, after summing over $q$. Panel (l) is the net rate of change of the total wave-energy recovered by summing over all the previous plots from (a) through to (k). Right of the vertical separator line: all quantities in panels m to p are computed directly from numerical differentiation of the primitive code output variables. (m, n) are the rates of change of the magnetic and internal (acoustic) energies, respectively (the fourth and third terms in \ref{['eq:ddtentot']}). Panel (o) depicts the direct numerical results of $\hbox{\boldmath$\nabla$}{\mathbf{\cdot}}{\boldsymbol{B}}$. Finally, panel (p) represents the original time derivative of the total wave-energy (\ref{['eq:EnTot']}). The four coloured dots appearing in the plots are set up to coincide with the initial wavefront and move upwards at the three characteristic speeds, namely, the slow (red), Alfvén (green), fast (blue), plus the (noncharacteristic) sound (black) speed, provided for tracking the speed of the individual simulated wave branches post-decomposition. Given the different ranges of magnitude of the variables and in order to maximise the contrast between the waves and the background color for better visibility, as well as better categorization, we use three different color maps. Panels (d), (e), (f), (j), and (k) on the left of the vertical separator, as well as panels (m), (n), and (o) on the right of the separator have all been scaled according to their individual maxima and minima, using two different color maps. The rest of the plots are scaled with reference to panel (p).
• Figure 2: Plots of the energy contents corresponding to simulation Ia ($\beta = 9.03$) taken at $y = -0.024$ and $t = 17.95$, given by the individual terms in \ref{['eq:totendecomposed']} after summing over the characteristic directions ($\pm$), and presented using the same layout as that of \ref{['fig:highBetaDDTEnergy']}. Notice the visible nonlinearity in the magnetic energy plot (m) manifested by the bends in the field lines at the location of the (magnetic) slow/Alfvén wavefront. The negative energy values in the internal energy (n) imply that acoustic energy is being extracted from the background state, meaning that the magnetically dominated (magneto-acoustic) slow wavefront causes rarefaction in the gas (downtick in the thermal pressure) as it travels. This, however, is countered by larger positive values of magnetic energy (m), implying that magnetic energy is being deposited by the wave (uptick in the magnetic pressure), leading to an overall positive energy state for the slow branch. Finally, note the extremely small (numerically zero) amounts of energy in the $x$ and $y$ components of the plots, as expected by the $x$- and $y$-independent nature of both the driver and the background magnetic field.
• Figure 3: 1D plots of wave-energy propagating along the $z$-axis belonging to simulation Ia ($\beta = 9.03$), taken at $x = y = -0.024$ and $t = 17.95$. Top: Post-decomposition wave-energy components associated with the pseudo and physical modes given by \ref{['eq:totendecomposed']}, individually summed over the $\pm$ terms and $q$. Bottom: The original and recovered energies of the wave.
• Figure 4: Time-distance ($t$-$z$) plots of the post-decomposition wave-energy components of simulation 1a ($\beta = 9.03$) at $x = y = -0.024$. The layout of the panels is carried over directly from \ref{['fig:highBetaDDTEnergy']}.
• Figure 5: 2D plots of the post-decomposition wave-energy components of simulation Ib ($\beta = 0.54$) at $y = -0.024$ and $t = 0.99$, presented in the same format as that of \ref{['fig:highBetaDDTEnergy']}.