Bow Shock and Local Bubble Plasma Unveiled by the Scintillating Millisecond Pulsar J0437$-$4715

The ionized interstellar medium contains au-scale (and below) structures that scatter radio waves from pulsars, resulting in scintillation. Power spectral analysis of scintillation often shows parabolic arcs, with curvatures that encode the locations and kinematics of the pulsar, Earth, and interstellar plasma. Here we report the discovery of 25 distinct plasma structures in the direction of the brilliant millisecond pulsar, PSR J0437-4715, in observations obtained with the MeerKAT radio telescope. Four arcs reveal structures within 5000 au of the pulsar, from a series of shocks induced as the pulsar and its wind interact with the ambient interstellar medium. The measured radial distance and velocity of the main shock allows us to solve the shock geometry and space velocity of the pulsar in three dimensions, while the velocity of another structure unexpectedly indicates a back flow from the direction of the shock or pulsar-wind tail. The remaining 21 arcs represent a surprising abundance of structures sustained by turbulence within the Local Bubble -- a region of the interstellar medium thought to be depleted of gas by a series of supernova explosions about 14 Myr ago. The Local Bubble is cool enough in areas for sub-au density fluctuations to arise from turbulence.