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tokens: ψ at 89°, as a function of the resonance order N for both inside (a) and outside (b) the plasmapause. Here the results are presented for the broadband MS wave model at the wave lower ( f lc ), peak ( f m ), and upper ( f uc ) frequencies, respectively. Clearly, to evaluate the resonant scattering effect for the ring current protons with energies ∼1 keV–1 MeV, the high order cyclotron resonances must be considered. As shown in Figures 1 (a) and (b), due to the variation of the k ∥ which is modified by the background plasma density, the MS waves with the given frequency distributions can cyclotron resonate with ring current protons at orders N = 3–11 inside the plasmapause, while at orders N = 4–9 outside the plasmapause. The determined resonance orders can be applied to QLT method for calculating the diffusion coefficients. Figure 1. Resonance energy of ring current protons for interactions with MS waves with different resonance orders at L = 4.5: (a) inside the plasmapause; (b) outside the plasmapause. Here the blue, green, and red lines denote the upper cutoff frequency, the peak frequency, and the lower cutoff frequency, respectively. 3.2. Scattering of Ring Current Protons In this part, we examine the MS wave-induced scattering effect of ring current protons in the form of the bounce-averaged diffusion coefficients, and compare the results obtained from both TP simulations and QLT calculations (e.g., Ni et al. 2008 ) with the given MS wave model. Figure 2 shows the 2D plots of bounce-averaged diffusion coefficients , , and as a function of equatorial pitch angle α eq and proton kinetic energy E k at L = 4.5, both inside ((a)–(c)) and outside ((d)–(f)) the plasmapause. The diffusion coefficients obtained from TP simulations, QLT calculations, and the corresponding normalized differences are presented in each column (from left to right, respectively). Here the cyclotron resonance orders discussed above are adopted for the QLT calculations. First, MS waves ∼218 pT can remarkably affect the ring current protons with the equatorial pitch angles ∼50°–90°. Specifically, the energy diffusion () dominates the wave-driven interactions with the rates at the order of ∼10 −4 s −1 for both inside and outside the plasmapause (Figures 2 (b1) and (b2), (e1), and (e2)). The pitch angle scattering () and the mixed diffusion ( ) effects are weaker than the energy diffusion effect, with the rates ∼10 −6 s −1 and 10 −5 s −1 , respectively. The results imply that the MS waves can effectively accelerate and heat the ring current protons, rather than pitch angle scattering. Clearly, the accelerating becomes more important for ring current protons at high pitch angles. Second, the maximum diffusion rates are located at several keV for ring current protons inside the plasmapause, while at several tens of keV outside the plasmapause. These differences in the diffusion rates are mainly caused by the change in the MS wave phase velocities, which are modified by the