Overview
This lecture discusses the role of mathematics, especially potential theory and vortex dynamics, in understanding and modeling tsunami phenomena, with a focus on offshore and near-shore dynamics, and proposes new ways to enhance tsunami energy dissipation through vortex-boundary interactions.
Mathematics and Tsunami Research
- Mathematics offers abstraction, universality, and rigor to study natural disasters like tsunamis.
- Interdisciplinary collaboration is necessary, as mathematical results are often too abstract for direct application by other fields.
- Japanese mathematicians are increasingly interested in natural disaster research after the 2011 Tohoku tsunami.
Categories of Tsunami Studies
- Tsunami phenomena are divided into three areas: generation, propagation, and near-shore (offshore) dynamics.
- Generation: Study of surface deformation after bed movements (e.g., earthquakes).
- Propagation: Analysis of tsunami travel given seabed and water profile, used to predict arrival times.
- Near-shore dynamics: Focuses on unpredictable behaviors, significantly affected by coastal topography.
Mathematical Modeling of Water Waves
- Waterway problems modeled as free boundary problems for incompressible, inviscid, irrotational fluids under gravity.
- Linear wave theory uses Laplace’s equation and shows short waves travel slowly, long waves faster.
- Nonlinear effects, such as solitary wave collisions, lead to amplitude growth and energy loss.
- Weakly nonlinear models use scaling parameters (alpha, beta) to create shallow water or dispersive wave approximations.
- Shallow water equations and linear wave equations serve as computational models for tsunami generation and propagation.
Offshore Dynamics and Vortex Theory
- Offshore tsunami dynamics are complex due to coastal topography and the formation of macro-vortices.
- Traditional models struggle with these complexities, motivating new mathematical approaches.
Potential Theory in Multiply Connected Domains
- Multiply connected domains (regions with holes or islands) are common in coastal areas and require specialized mathematical treatment.
- Vorticity (omega) and stream function (ψ) describe the velocity and evolution of 2D flows, with Lagrange’s theorem stating vorticity is conserved absent boundaries and external forces.
- Conformal mapping transforms complex domains to symmetric circular domains for simplification.
- The Schottky-Klein prime function enables analytic solutions for uniform flows, source/sink pairs, and point vortices in these domains.
- The evolution and interaction of vortices are strongly influenced by domain topology and boundary effects.
Vortex-Boundary Interaction and Energy Dissipation
- Boundaries generate vorticity, forming macro-vortices behind obstacles, which can dissipate energy.
- Practical examples include airplane wings (lift/stall), automotive torque converters (energy loss), and biological cases (efficient insect flight, slow-falling seeds).
- Trapping vortices behind obstacles in coastal waters could enhance tsunami energy dissipation via viscous friction.
Proposed Applications and Future Directions
- The theory suggests engineering obstacles to trap macro-vortices offshore can dissipate tsunami energy.
- Potential theory can be applied to design new coastal protections based on vortex dynamics.
Key Terms & Definitions
- Potential Theory — A mathematical approach using potential functions to describe fluid flow.
- Multiply Connected Domain — A region with more than one boundary, such as with internal holes or islands.
- Vorticity (omega) — A measure of local rotation in a fluid flow.
- Stream Function (ψ) — A function whose contours represent streamlines of the flow.
- Conformal Mapping — A mathematical transformation preserving angles, useful for simplifying complex domains.
- Schottky-Klein Prime Function — A transcendental function used to represent solutions in multiply connected domains.
- Macro-vortices — Large-scale vortices formed due to topographic effects in coastal waters.
Action Items / Next Steps
- Review the mathematical models of tsunami generation (shallow water equations).
- Explore potential theory and conformal mapping techniques in fluid dynamics.
- Investigate further literature on vortex-boundary interactions and applications to tsunami mitigation.