Tsunami and induced magnetic anomalies generated by slender fault

We present a mathematical model for tsunami and induced magnetic anomalies originating from a time-dependent seabed deformation in an otherwise quiescent ocean over a conductive seafloor. The deformation is assumed to be a slender fault, whose lateral extension is much larger than the longitudinal scale. Using a perturbative method with multiple time scales and Green's function approach, we examine the slow evolution of the wave field and induced magnetic anomaly over transoceanic distances from the fault. The model is validated against deep-ocean observations from the 2011 T & omacr;hoku-oki tsunami. Our study reveals that lateral propagation in two horizontal dimensions decreases the period of both the surface wave and induced magnetic signal compared with one-horizontal-dimension scenarios. Over time, initially longitudinal wave propagation alters as wave fronts bend and stretch, affecting the magnetic signal accordingly. Interestingly, the magnetic anomaly gradually separates from the leading tsunami wave and travels ahead of the tsunami by a distance proportional to the fault's longitudinal scale. We show that increased lateral propagation reduces the detectability of magnetic anomalies. Finally, we derive an asymptotic formula valid for the long leading wave that travels ahead of the dispersive group over transoceanic distances. This formula holds promise for the rapid assessment of tsunami risk. These findings advance fundamental understanding and may inform the development of future tsunami early warning systems relying on magnetic field detection.