*5.1.2. Two-dimensional tsunamis*

**5. Calculation of simulated tsunami velocities and heights**

**5.1. Simulation based on solution of the fundamental equations of motion**

simulated band velocities.

*5.1.1. One-dimensional tsunami*

*d*−1/4, while velocity varies as *d*−3/4.

region.

98 Tsunami

by the video:

Tsunami simulation provides an understanding of many of the factors affecting the capability of coastal HF radars to provide tsunami observation and warning. Ultimately, this can lead to performance assessment for a radar at a given site based on local bathymetry. Orbital velocities are tracked vs. time and related to the tsunami wave intensity. Comparisons with the back‐ ground current field allow the assessment of possible warning time and wave amplitude as the tsunami approaches the coast near the radar. In this section, we describe two methods for simulating tsunami velocities: the first based on solving the fundamental equations to give total velocity/height maps and the second based on application of Green's Law to give

To simulate tsunami height and velocity, Eqs (2) and (3) are solved numerically within the radar coverage area, typically out to ∼50 km from the coast. The offshore bathymetry is included as the depth variable, *d*(*x*,*y*), and the coastline becomes a boundary for the domain. First, the scalar Eq. (3) is solved for the tsunami wave height. Then, velocity is obtained by integrating the left side of Eq. (2) over time, after linearization as described in Section 2.1. This establishes the relations between the orbital velocity measured by the radar and the tsunami wave height, as well as provides the time of arrival at the coast from any point in the near‐field

We examine here how a simple one‐dimensional wave approaching normally to the coast behaves when it encounters a steep continental slope starting from depth 1000 m at a distance of 70 km from shore and sloping upwards to depth 100 m, at a distance of 50 km from shore. How do both height and orbital velocity change as they traverse this shelf? How much is transmitted across the shelf and how much gets reflected? On the return of the ray reflected by the coast, is there a second reflection going back toward shore? Answers to these questions are provided by Video 2 that can be viewed at http://bit.ly/29nKuLh Elapsed time in minutes is given in each frame of the movie. Colors represent the wave height (blue) and the orbital velocities (red). The coast was taken to be a Neumann reflecting boundary, that is, velocity stops perpendicular to the coast, where its magnitude is zero. The bottom profile including the shelf is shown as the heavy black curve at the top in the video. We note several points indicated

**•** The normalized height wave and orbital velocity wave come in from the right, toward the coast at the left. For this exact solution, the orbital velocity grows much faster than height as the wave advances onto the shallower shelf. This also follows from the Green's Law approximation given by Eqs (6) and (7), which indicates that height depends on depth *d* as We now examine two more realistic scenarios: in the first, a plane wave tsunami approaches the Portuguese West Coast and in the second, a tsunami is generated by a point source in the Alboran Sea. Videos are provided, which show tsunami height and velocity normalized by their initial values, as the tsunami is refracted by the bathymetry and reflects from the coast. In these videos, the background color represents the tsunami wave height normalized by its initial value, with the magnitude indicated by the color bar. Velocity vectors overlain on top of the height background represent the orbital velocities normalized by the initial value, with the magnitudes indicated by the vector length. Elapsed time shown on each frame indicates the time taken for the tsunami to reach various points along the coast. These simulated values can be tested against radar observations of real tsunamis. In both cases, reflection of the wave from the coast is clearly visible and the tsunami velocity increases more rapidly than the height as depth decreases, indicating that observed velocities provide a sensitive alert flag for a tsunami approaching the coast. Future work on this simulation will study results related to actual heights and velocities, rather than normalized values; output values can then be tested against radar observations of real tsunamis.
