Modeling the Dawn/Dusk Asymmetry of Field Line Resonances ...
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The University of St Andrews is a charity registered in Scotland, No: SC013532 Modeling the Dawn/Dusk Asymmetry of Field Line Resonances (FLRs) Andrew N. Wright 1 , Tom Elsden 1 and Kazue Takahashi 2 1 Department of Mathematics and Statistics, University of St Andrews, St Andrews, UK 2 The John Hopkins University Applied Physics Laboratory, Laurel, MD, USA References Elsden, T., and Wright, A. N. (2017), JGR: Space Physics, 122, 3247– 3261. Elsden, T., & Wright, A. N. (2019), JGR: Space Physics, 124, 178– 196. Sandhu, J. K., Yeoman, T. K., James, M. K., Rae, I. J., & Fear, R. C. (2018), JGR: Space Physics, 123, 567–586. Singer, H. J., Southwood, D. J., Walker, R. J., and Kivelson, M. G. (1981), JGR, 86( A6), 4589– 4596 Takahashi, K., Lee, D.‐H., Merkin, V. G., Lyon, J. G., and Hartinger, M. D. (2016), JGR: Space Physics, 121, 9632– 9650. Wharton, S. J., Wright, D. M., Yeoman, T. K., James, M. K., & Sandhu, J. K. (2018), JGR: Space Physics, 123, 6231–6250. Andrew N. Wright and Thomas Elsden, (2016) ApJ 833 230. Wright, A. N., Elsden, T., & Takahashi, K. (2018). JGR: Space Physics, 123, 6443– 6456. The Problem ➢ Field line resonances (FLRs) are observed to occur preferentially and have larger amplitudes at dawn compared to dusk (e.g. Takahashi et al, 2016, Wharton et al., 2018, Sandhu et al., 2018) – see Figure 1, showing the statistically observed median fundamental toroidal Alfvén frequencies observed by THEMIS, adapted from Takahashi et al., 2016. ➢ This asymmetry is significant. From Figure 1, at dusk (red line) a 3-mHz FLR would be found at L~8.5, whereas at dawn (black line) it would be located at L~11. ➢ We present simulations which can reproduce this behaviour in a magnetospheric waveguide, and further, can provide an explanation of how it occurs. ➢ Punchline: an asymmetric equilibrium leads to asymmetries in the compressional fast modes and hence the FLRs they excite, even when driven symmetrically. FLR Driver ➢ Need to understand the fast waveguide mode which is doing the driving. The frequency of the driver can be found by looking at the time series and frequency of (field-aligned magnetic field perturbation). ➢ Figure 3 shows time series and associated FFTs at various azimuthal locations in the waveguide. ➢ There is a common waveguide frequency despite the asymmetry – thus the fast wave is global. ➢ The frequency peak corresponds with the Alfvén frequency found from integrating the Alfvén wave equation (Singer et al., 1981; Wright and Elsden, 2016) with the given polarization at the location of the black dot in the middle panel of Figure 2. ➢ Thus the largest amplitude Alfvén waves are indeed resonantly driven. ➢ Panel 4 of Figure 2 shows the absolute value of integrated in time over 5.4 < t < 9.6. The asymmetry is clear, but it is in fact the gradient of aligned with the resonance which drives resonances. ➢ Hence, panel 5 of Figure 2 plots the FFT of this azimuthal magnetic pressure gradient. This shows that at dawn, stronger gradients in the fast mode exist at the correct frequency to excite Alfvén resonances. ➢ Further, the Resonant Zone (Wright and Elsden, 2016), where permissible solutions to the Alfvén wave equation exist, overlaps the region of strong gradients at dawn, but not at dusk. Figure 1: Variation of the fundamental toroidal Alfvén wave frequency at dawn, noon, and dusk. The black dots are median f T1 frequencies at noon found from analyzing THEMIS data as described in Takahashi et al. 2016. THEMIS = Time History of Events and Macroscale Interactions during Substorms. Figure 2: From left to right: 1. Alfvén speed profile; 2. Flux tube energy density (FTED); 3. Square root of kinetic energy density; 4. Integral of the magnetic pressure for 5.4 < t < 9.6; 5. Contours of fast Fourier transform (FFT) of the azimuthal magnetic pressure gradient(∇b γ ) β . Overplotted as dot-dashed lines are the toroidal (right) and poloidal (left) boundaries of the Resonant Zone. The white line indicates the Alfvén wave polarization. Our Approach – Model ➢ 2-D dipole background magnetic field with a 3-D Alfvén speed variation. ➢ Use a field aligned coordinate system (α,β,γ), with α corresponding to L-shell, β an azimuthal coordinate and γ a field-aligned coordinate. ➢ Solve numerically the low- β, ideal MHD equations, introducing some dissipation to remove resonant singularities. ➢ Consider an Alfvén speed profile which is tailored to provide the observed fundamental toroidal Alfvén frequencies as shown in Figure 1. The Alfvén speed profile required to do this in our model is shown as a colour contour plot in the first panel of Figure 2. This clearly bears out the asymmetry seen in the frequencies. ➢ The simulation is driven with a localised `push’, centred on noon, achieved by imposing a perturbation to the linear magnetic pressure / 0 . This acts as a broadband stimulus. Field Line Resonances ➢ FTED – Flux tube energy density in panel 2, is the energy density integrated along an elemental flux tube, and can be used to study the location of FLRs (Elsden & Wright, 2017). ➢ Clearly a stronger response at dawn than dusk. ➢ Kinetic energy (panel 3) shows structure of ridges, whose phase motion is from high to low Alfvén frequency. ➢ Overplotted black line in panel 3 gives the polarization of the Alfvén wave, which is aligned with the direction of the ridges. The angle is θ=0.09 radians, and so these FLRs are not toroidal (θ=0), but in fact cross L-shells. Figure 3: Variation of b γ (t) for several points in the equatorial plane (left) along with the associated frequency spectra (right). Figure 4: Schematic: Ray trajectories and their orientation (ϕ) in the equatorial plane. Color panels: snapshots of |b γ (α,β,t)| in the equatorial plane. The dot represents the current location of the maximum, and the line is its locus/path at earlier times. Waveguide Refraction ➢ The fast mode has the structure shown in Figure 2 due to refraction in the non-uniform equilibrium. ➢ By driving the waveguide with a very quick push, the ray traced by an incident fast mode can be revealed, as shown in Figure 4. ➢ A straight wavefront enters in the first colored panel, but soon refracts duskward, since the left side of the wavefront experiences a higher Alfvén speed (recall Alfvén speed profile panel 1 Figure 2). ➢ It is this refraction which allows the stronger gradients to form at dawn to drive larger FLRs there. Conclusions ➢ Simulations of an asymmetric waveguide using an Alfvén speed profile consistent with an observed toroidal Alfvén frequency variation reproduces a similar to observed dawn/dusk FLR asymmetry. ➢ Asymmetry in the fast mode due to refraction generates gradients in the magnetic pressure which are larger and more coherent at dawn than dusk, hence favouring FLR excitation at dawn. ➢ Interesting propagation, refraction and coupling process between fast/Alfvén waves is required to explain the asymmetry. ➢ This poster has summarised the findings of Wright et al., 2018.
Transcript of Modeling the Dawn/Dusk Asymmetry of Field Line Resonances ...
The University of St Andrews is a charity registered in Scotland, No: SC013532
Modeling the Dawn/Dusk Asymmetry of Field Line Resonances (FLRs)Andrew N. Wright
1, Tom Elsden
1and Kazue Takahashi
2
1 Department of Mathematics and Statistics, University of St Andrews, St Andrews, UK2 The John Hopkins University Applied Physics Laboratory, Laurel, MD, USA
ReferencesElsden, T., and Wright, A. N. (2017), JGR: Space Physics, 122, 3247– 3261.Elsden, T., & Wright, A. N. (2019), JGR: Space Physics, 124, 178– 196.Sandhu, J. K., Yeoman, T. K., James, M. K., Rae, I. J., & Fear, R. C. (2018), JGR: Space Physics, 123, 567–586.Singer, H. J., Southwood, D. J., Walker, R. J., and Kivelson, M. G. (1981), JGR, 86( A6), 4589– 4596Takahashi, K., Lee, D.‐H., Merkin, V. G., Lyon, J. G., and Hartinger, M. D. (2016), JGR: Space Physics, 121, 9632– 9650.Wharton, S. J., Wright, D. M., Yeoman, T. K., James, M. K., & Sandhu, J. K. (2018), JGR: Space Physics, 123, 6231–6250.Andrew N. Wright and Thomas Elsden, (2016) ApJ 833 230.Wright, A. N., Elsden, T., & Takahashi, K. (2018). JGR: Space Physics, 123, 6443– 6456.
The Problem➢ Field line resonances (FLRs) are observed to occur preferentially and have larger amplitudes at dawn
compared to dusk (e.g. Takahashi et al, 2016, Wharton et al., 2018, Sandhu et al., 2018) – see Figure 1, showing the statistically observed median fundamental toroidal Alfvén frequencies observed by THEMIS, adapted from Takahashi et al., 2016.
➢ This asymmetry is significant. From Figure 1, at dusk (red line) a 3-mHz FLR would be found at L~8.5, whereas at dawn (black line) it would be located at L~11.
➢ We present simulations which can reproduce this behaviour in a magnetospheric waveguide, and further, can provide an explanation of how it occurs.
➢ Punchline: an asymmetric equilibrium leads to asymmetries in the compressional fast modes and hence the FLRs they excite, even when driven symmetrically.
FLR Driver➢ Need to understand the fast waveguide mode which is doing the driving. The frequency of the
driver can be found by looking at the time series and frequency of 𝑏𝛾 (field-aligned magnetic field
perturbation).
➢ Figure 3 shows time series and associated FFTs at various azimuthal locations in the waveguide.
➢ There is a common waveguide frequency despite the asymmetry – thus the fast wave is global.
➢ The frequency peak corresponds with the Alfvén frequency found from integrating the Alfvén wave equation (Singer et al., 1981; Wright and Elsden, 2016) with the given polarization at the location of the black dot in the middle panel of Figure 2.
➢ Thus the largest amplitude Alfvén waves are indeed resonantly driven.
➢ Panel 4 of Figure 2 shows the absolute value of 𝑏𝛾 integrated in time over 5.4 < t < 9.6. The
asymmetry is clear, but it is in fact the gradient of 𝑏𝛾 aligned with the resonance which drives
resonances.
➢ Hence, panel 5 of Figure 2 plots the FFT of this azimuthal magnetic pressure gradient. This shows that at dawn, stronger gradients in the fast mode exist at the correct frequency to excite Alfvén resonances.
➢ Further, the Resonant Zone (Wright and Elsden, 2016), where permissible solutions to the Alfvén wave equation exist, overlaps the region of strong gradients at dawn, but not at dusk.
Figure 1: Variation of the fundamental toroidal Alfvén wave frequency at dawn, noon, and dusk. The black dots are median fT1 frequencies at noon found from analyzing THEMIS data as described in Takahashi et al. 2016. THEMIS = Time History of Events and Macroscale Interactions during Substorms.
Figure 2: From left to right: 1. Alfvén speed profile; 2. Flux tube energy density (FTED); 3. Square root of kinetic energy density; 4. Integral of the magnetic pressure for 5.4 < t < 9.6; 5. Contours of fast Fourier transform (FFT) of the azimuthal magnetic pressure gradient(∇bγ)β. Overplotted as dot-dashed lines are the toroidal (right) and poloidal (left) boundaries of the Resonant Zone. The white line indicates the Alfvén wave polarization.
Our Approach – Model➢ 2-D dipole background magnetic field with a 3-D Alfvén speed variation.
➢ Use a field aligned coordinate system (α,β,γ), with α corresponding to L-shell, β an azimuthal coordinate and γ a field-aligned coordinate.
➢ Solve numerically the low- β, ideal MHD equations, introducing some dissipation to remove resonant singularities.
➢ Consider an Alfvén speed profile which is tailored to provide the observed fundamental toroidal Alfvén frequencies as shown in Figure 1. The Alfvén speed profile required to do this in our model is shown as a colour contour plot in the first panel of Figure 2. This clearly bears out the asymmetry seen in the frequencies.
➢ The simulation is driven with a localised `push’, centred on noon, achieved by imposing a perturbation to the linear magnetic pressure 𝐵𝑏𝛾/𝜇0. This acts as a broadband stimulus.
Field Line Resonances➢ FTED – Flux tube energy density in panel 2, is the energy density integrated along an elemental flux
tube, and can be used to study the location of FLRs (Elsden & Wright, 2017).
➢ Clearly a stronger response at dawn than dusk.
➢ Kinetic energy (panel 3) shows structure of ridges, whose phase motion is from high to low Alfvénfrequency.
➢ Overplotted black line in panel 3 gives the polarization of the Alfvén wave, which is aligned with the direction of the ridges. The angle is θ=0.09 radians, and so these FLRs are not toroidal (θ=0), but in fact cross L-shells.
Figure 3: Variation of bγ(t) for several points in the equatorial plane (left) along with the associated frequency spectra (right).
Figure 4: Schematic: Ray trajectories and their orientation (ϕ) in the equatorial plane. Color panels: snapshots of |bγ(α,β,t)| in the equatorial plane. The dot represents the current location of the maximum, and the line is its locus/path at earlier times.
Waveguide Refraction➢ The fast mode has the structure shown in Figure 2 due to refraction in the non-uniform equilibrium.
➢ By driving the waveguide with a very quick push, the ray traced by an incident fast mode can be revealed, as shown in Figure 4.
➢ A straight wavefront enters in the first colored panel, but soon refracts duskward, since the left sideof the wavefront experiences a higher Alfvén speed (recall Alfvén speed profile panel 1 Figure 2).
➢ It is this refraction which allows the stronger gradients to form at dawn to drive larger FLRs there.
Conclusions➢ Simulations of an asymmetric waveguide using an Alfvén speed profile consistent with an
observed toroidal Alfvén frequency variation reproduces a similar to observed dawn/dusk FLR asymmetry.
➢ Asymmetry in the fast mode due to refraction generates gradients in the magnetic pressure which are larger and more coherent at dawn than dusk, hence favouring FLR excitation at dawn.
➢ Interesting propagation, refraction and coupling process between fast/Alfvén waves is required to explain the asymmetry.
➢ This poster has summarised the findings of Wright et al., 2018.
