We consider the resonant coupling of fast and Alfvén magnetohydrodynamic (MHD) waves in a 3D equilibrium. Numerical solutions to normal modes (∝ exp(−iωt)) are presented along with a theoretical framework to interpret them. The solutions we find are fundamentally different to those in 1D and 2D. In 3D there exists an infinite number of possible resonant solutions within a "Resonant Zone," and we show how boundary conditions and locally 2D regions can favour particular solutions. A unique feature of the resonance in 3D is switching between different permissible solutions when the boundary of the Resonant Zone is encountered. The theoretical foundation we develop relies upon recognising that in 3D the orientation of the resonant surface will not align in a simple fashion with an equilibrium coordinate. We present a method for generating the Alfvén wave natural frequencies for an arbitrarily oriented Alfvén wave, which requires a careful treatment of scale factors describing the background magnetic field geometry.

A novel simulation grid is devised that is optimized for studying magnetohydrodynamic (MHD) wave coupling and phase mixing in a dipole‐like magnetic field. The model also includes flaring on the dawn and dusk flanks. The location of the magnetopause is quite general. In particular, it does not have to coincide with a coordinate surface. Simulations indicate the central role of global fast waveguide modes. These switch from being azimuthally standing in nature at noon, to propagating antisunward on the flanks. The field line resonances (FLRs) seen in the simulation results are three dimensional and not strictly azimuthally polarized. When a plume is present, the FLRs cross a range of 2 in L shell, and have a polarization that is midway between toroidal and poloidal.

We present results from a 3‐D numerical simulation which investigates the coupling of fast and Alfvén magnetohydrodynamic (MHD) waves in a nonuniform dipole equilibrium. This represents the time‐dependent extension of the normal mode ( ∝exp(−iωt)) analysis of Wright and Elsden (2016) and provides a theoretical basis for understanding 3‐D Alfvén resonances. Wright and Elsden (2016) show that these are fundamentally different to resonances in 1‐D and 2‐D. We demonstrate the temporal behavior of the Alfvén resonance, which is formed within the “Resonant Zone”; a channel of the domain where a family of solutions exists such that the natural Alfvén frequency matches the fast‐mode frequency. At early times, phase mixing leads to the production of prominent ridges in the energy density, whose shape is determined by the Alfvén speed profile and the chosen background magnetic field geometry. These off resonant ridges decay in time, leaving only a main 3‐D resonant sheet in the steady state. We show that the width of the 3‐D resonance in time and in space can be accurately estimated by adapting previous analytical estimates from 1‐D theory. We further provide an analytical estimate for the resonance amplitude in 3‐D, based upon extending 2‐D theory.

This paper considers the resonant coupling of fast and Alfvén magnetohydrodynamic (MHD) waves. We perform numerical simulations of the time‐dependent excitation of Alfvén resonances in a dipole magnetic field, with nonuniform density providing a 3‐D equilibrium. Wright and Elsden (2016) showed that in such a system where the poloidal and toroidal Alfvén eigenfrequencies are different, the resonance can have an intermediate polarization, between poloidal and toroidal. We extend this work by driving the system with a broadband rather than monochromatic source. Further, we investigate the effect of azimuthal inhomogeneity on the resonance path. It is found that when exposed to a broadband driver, the dominant frequencies are the fast waveguide eigenfrequencies, which act as the drivers of Alfvén resonances. We demonstrate how resonances can still form efficiently with significant amplitudes, even when forced by the medium to have a far from toroidal polarization. Indeed, larger‐amplitude resonances can be generated with an intermediate polarization, rather than purely toroidal, as a result of larger gradients in the magnetic pressure formed by the azimuthal inhomogeneity. Importantly, the resonance structure is shown to be independent of the different forms of driving, meaning their locations and orientations may be used to infer properties of the equilibrium. However, the amplitude of the FLRs are sensitive to the spatial structure and frequency spectrum of the magnetopause driving. These results have implications for the structure of field line resonances (FLRs) in Earth's magnetosphere, although the focus of this paper is on the underlying physics involved.

Field line resonances (FLRs) are observed to occur preferentially and have larger amplitudes at dawn compared to dusk. We present simulations of FLR excitation in a magnetospheric waveguide that can reproduce this behavior. Crucially, our equilibrium is asymmetric about noon. Even when this system is driven in a symmetric fashion about noon, the fast waves that are established in the magnetosphere develop asymmetries—as do the FLRs they excite. Fast mode ray trajectories are employed to show that the asymmetry evolves due to refraction. Preferential FLR excitation at dawn is further reinforced by calculating the Resonance Map. This shows that the Resonant Zone at dawn coincides with a large‐amplitude coherent fast mode driver, which is not the case at dusk. These factors result in FLRs having a larger amplitude at dawn compared to dusk.