The upstream-propagating Alfvénic fluctuations with power law spectra in the upstream region of the Earth's bow shock

Based on theories, the beam instability induced by shock-accelerated ions can generate upstream-propagating Alfvén waves (UPAWs) with a power spectral bump near 0.03 Hz, while the nonlinear wave-wave interaction favors an inverse cascade to create a power law spectrum. Here we present the first observational evidence for the upstream-propagating Alfvénic fluctuations (UPAFs) with power law spectra. We utilize a new criterion to identify the upstream-propagating Alfvénic intervals: the propagation direction is opposite to that of solar wind strahl electron outflow. Besides 35 UPAWs, we find 47 UPAFs with power law spectra, and ∼47% of these UPAFs are associated with energetic ion events (>30 keV). These UPAWs and UPAFs are mostly observed in the slow solar wind. However, their occurrence rate and power behave differently in dependence on the radial distance from the Earth. These results provide new clues on understanding the dynamic equilibrium between the nonlinear inverse cascade and the linear ion beam instability.