Applicability of Bernoulli's theorem to upstream and downstream of the bow shock

In this research, the validity of Bernoulli's theorem for the solar wind crossing the Earth's bow shock has been investigated based on space explorations. Using solar wind plasma and interplanetary magnetic field data from the OMNI database, along with magnetosheath plasma and magnetic field data measured by the magnetospheric multiscale satellite in 2016 and 2018, 95 events were selected to study the Bernoulli invariant from the upstream to the downstream of Earth's bow shock (from the unshocked solar wind region to the shocked magnetosheath region). Statistical investigations show that the differences in the characteristic energy of plasmas between the solar wind region and the magnetosheath region are less than 5%, regardless of the upstream solar wind speed and the satellite's location in the magnetosheath. These results quantitatively support the generalization of Bernoulli's equation to the solar wind flowing through the bow shock, including the contribution of magnetic field energy.