*2.3.2. Green's Law approximation*

When refractive index (depth) varies slowly, Green's Law [11, 12] applies. In this limit, tsunami height and orbital velocity follow simple relationships in terms of depth, as described in reference [1]. It is convenient to normalize the height to a value of the tsunami height in deeper water; we take this depth to be 4000 m, typical for an ocean basin; other depths can be used. The approximate tsunami wave height and scalar tsunami orbital speed in water of depth *d* are then given by:

$$
\eta\_{approx} = \eta\_{4000} (4000/d)^{1/4} \tag{6}
$$

$$\mathbf{v}\_{approx} = \eta\_{a000} \sqrt{\mathbf{g} \wedge d} (4000 \wedge d)^{3/4} \tag{7}$$

where *η*4000 is the tsunami height in water of depth 4000 m.

**2.3. Ray optics and Green's Law approximations for tsunami waves**

The answer is "No: 10 km resolution is always adequate."

alternatives that will work when depth varies abruptly.

*2.3.1. Ray optics approximation*

can never predict a parallel component.

*2.3.2. Green's Law approximation*

perpendicularly.

78 Tsunami

Traveling waves, such as tsunami or electromagnetic waves, sometimes follow simple ray optics approximations where they can refract, changing their direction continuously with the refractive index. The refractive index for waves of any nature propagating through media with different or changing properties is defined as the ratio of the reference phase velocity to the phase velocity at the specific point in the medium. For light or electromagnetic waves, the reference velocity is taken as the speed of light in vacuum. For acoustic waves or water waves in the shallow‐depth limit, one normally selects a convenient reference velocity [10]. So, for example, if one selected the 4000‐m depth which is typical of a deep ocean basin, the refractive index becomes 4000/, which is sometimes referred to as the HF asymptotic limit. This approximation applies only when refractive index varies slowly and smoothly with distance. This means that the refractive index cannot have a discontinuous jump, for example, if the bottom had a significant change in depth over scales shorter than a wavelength, say 10 km. Bottom depth fluctuations over smaller scales are not important to tsunami propagation. The tsunami wave, with its massive inertia, is like a low‐pass spatial filter that effectively averages across these fine‐scale features. A wave typically does not respond appreciably to perturba‐ tions with a scale much smaller than its wavelength; this is sometimes known as the Rayleigh criterion. As tsunami wavelengths exceed tens of kilometers, this implies that perturbations with smaller spatial scales (e.g.10 km) are unimportant. There is an often‐asked question: "Do I need to use a bathymetry database for tsunami near‐field modeling with 1–2 km resolution?"

We now examine simplifications possible when depth varies slowly, and discuss exact

When depth and refractive index vary slowly, ray tracing allows a version of Fresnel's law such that the advancing wave continuously refracts, so that its direction of propagation follows the gradient of refractive index perpendicular to the isobath depth contours. This approximation has the consequence that there is always only one set of ray paths, which end up perpendicular to the coastline. The coastline boundary will reflect; outgoing rays also cross the contours

When depth and refractive index vary abruptly, it is valid to use Eqs (1–4) as an alternative. Models based on these equations, for example as described in [7–8], will predict the direction correctly and will generate components parallel to the contours and coastline, as has been observed by radars [2, 5]. Incorrect use of ray tracing for these situations will show evolution with some error of the forward ray toward the region of shallower water near the coast, but

When refractive index (depth) varies slowly, Green's Law [11, 12] applies. In this limit, tsunami height and orbital velocity follow simple relationships in terms of depth, as described in

### **2.4. Comparison of Green's Law amplitudes with exact calculations**

How accurate are the approximations in Eqs (6) and (7) for tsunami height and velocity? To get an idea of this, we compare the height approximation with solutions of the exact Eqs (1– 4) for specific cases.

**Case 1:** A typical continental shelf with a steeply sloping edge that goes from an outer depth of1000 m to an inner depth of 50 m over a horizontal distance of 20 km. Defining the trans‐ mission coefficient as the ratio R*T* of the tsunami height after passing over this shelf to that before striking the shelf's lower edge, Green's Law Eq. (6) gives *RT* = 2.11. The solution of the PDEs (2) and (3), discussed further in Section 5, gives *RT* = 1.68. These values for *RT* agree to within about 25%.

**Case 2:** A vertical shelf edge parallel to the coast falling perpendicularly from depth 50 to 1000 m. Clearly, this fails the "slow depth change vs. tsunami wavelength" criterion required for both ray optics and Green's Law and thus is beyond the scope of these approximate models. For normal incidence to a vertical escarpment, NOAA modelers [8] express Eqs (1–4) as a boundary‐value problem requiring continuity of height and transport across the escarpment and obtain the following exact expression for the transmission coefficient of the tsunami wave height:

$$R\_r = \Im / (\mathbf{l} + \sqrt{D\_z / D\_i}) \tag{8}$$

where *D*1 and *D*2 are the depths on the shelf's outer and inner edges, respectively. This is considered an exact solution for this defined geometry. Substituting *D*1 = 1000 m and *D*2 = 50 m yields a value for *RT* equal to 1.63, which is close to the prior estimate from the PDE calculation.

To conclude, the exact solutions for transmission coefficient *RT* for a depth change from 1000 to 50 m are (a) 1.68 over a steep slope and (b) 1.63 over a vertical step. These values differ by only about 3%, indicating that shelf slope does not have a significant effect on the transmission coefficient. Comparing these values with the Green's Law approximation of 2.11 suggests an error estimate of about 25% when using Green's Law for estimating tsunami height change due to sharp depth changes.

#### **2.5. Tsunami arrival time**

Solving Eqs (2–3) provides both the tsunami height and orbital velocity profiles as the wave approaches the coast, as impacted by the decreasing depth on the journey toward shore. This yields the most accurate estimate for the arrival time from any offshore point along its path and is useful in near field of the coastal radar because the computational effort is manageable.

However, for an approach from much farther out, or in order to get a quick, rough estimate of the tsunami arrival time, the phase velocity *vph*(*d*) at depth *d*, time *t*, and distance *s* along a tsunami great‐circle ray path can be expressed as follows:

$$\mathbf{v}\_{ph}(d) = \frac{\partial \mathbf{s}}{\partial t} \tag{9}$$

Eqs (5) and (9) lead to the following approximate solution for the elapsed time Δ*T* for the tsunami arrival at distance *S* from the start:

$$
\Delta T = \int\_{s}^{0} \frac{\text{\"\"\S}}{\sqrt{\text{gd}(s)}} \tag{10}
$$

Because of inherent smoothing of the integration process, the depth profile need not be defined to great resolution, allowing quick use of Eq. (10) over large ocean distances to give rough estimates of the tsunami arrival time.
