**2.1. Fundamental equations describing a tsunami in the near-field region**

Two equations form the basis of tsunami wave theory and propagation modeling. They are the essence of NOAA's Method of Splitting Tsunami (MOST) over near-field distances but away from the coastal run-up zone where flooding is experienced [7–8]. The first is essentially Newton's second law, that is, force = mass times acceleration, which for fluids gives the Navier-Stokes vector equation to the lowest order. For horizontal coordinates *x* and *y* and time *t*, this is given by:

$$\nabla \eta(\mathbf{x}, \mathbf{y}, t) = -\frac{1}{\mathbf{g}} \frac{\partial \tilde{\mathbf{v}}(\mathbf{x}, \mathbf{y}, t)}{\partial t} \tag{1}$$

where *η*(*x, y, t*) is the tsunami wave height, *g* is the acceleration due to gravity, and (, , ) is the orbital velocity of the wave. We assume, as in [7–8], that the orbital velocity at a particular location is independent of depth. The second equation is the continuity equation that expresses the incompressibility of water:

$$\nabla \cdot \left[ (d(\mathbf{x}, \mathbf{y}) + \eta(\mathbf{x}, \mathbf{y}, t)\tilde{\mathbf{v}}(\mathbf{x}, \mathbf{y}, t)) \right] = -\frac{\partial \eta(\mathbf{x}, \mathbf{y}, t)}{\partial t} \tag{2}$$

where *d*(*x*,*y*) is the depth below a mean datum reference, from which the tsunami wave height is measured. The left side of this equation is the net horizontal water volume transport per unit time into a vertical column. Eq. (2) simply expresses the fact that the net flow per unit time into the column of incompressible water must be matched by the rate of rise in the water elevation.

As it stands, Eq. (2) is nonlinear because the two dependent variables, height and velocity, on the left side are multiplied together. Because the tsunami elevation is small compared with the water depth (typically less than 0.5 m in deep water), the height in the left-hand side of the equation can usually be neglected, upon which the equation becomes linear.

## **2.2. Reduction to partial differential equations (PDEs)**

STV2: Fort Stevens, Oregon, USA TRIN: Trinidad, California, USA

76 Tsunami

YHS2: Yaquina Head South, Oregon, USA

"near field," typically of ranges up to 50 km.

is given by:

the incompressibility of water:

**2. Tsunami theory and modeling applicable to HF radar observations**

**2.1. Fundamental equations describing a tsunami in the near-field region**

h

h

equation can usually be neglected, upon which the equation becomes linear.

(, ,) ¶ Ñ =-

*vxyt* % *xyt*

where *η*(*x, y, t*) is the tsunami wave height, *g* is the acceleration due to gravity, and (, , ) is the orbital velocity of the wave. We assume, as in [7–8], that the orbital velocity at a particular location is independent of depth. The second equation is the continuity equation that expresses

(, ,) [( ( , ) ( , , ) ( , , ))]

where *d*(*x*,*y*) is the depth below a mean datum reference, from which the tsunami wave height is measured. The left side of this equation is the net horizontal water volume transport per unit time into a vertical column. Eq. (2) simply expresses the fact that the net flow per unit time into the column of incompressible water must be matched by the rate of rise in the water elevation. As it stands, Eq. (2) is nonlinear because the two dependent variables, height and velocity, on the left side are multiplied together. Because the tsunami elevation is small compared with the water depth (typically less than 0.5 m in deep water), the height in the left-hand side of the

¶ Ñ× + = - ¶ % *xyt d xy xytvxyt*

In this section, we provide a brief synopsis of tsunami theory and numerical modeling, as required to explain and develop the HF radar coastal observation/warning capability. We refer to a region of interest near the coast, or within the coverage area of a coastal HF radar, as the

Two equations form the basis of tsunami wave theory and propagation modeling. They are the essence of NOAA's Method of Splitting Tsunami (MOST) over near-field distances but away from the coastal run-up zone where flooding is experienced [7–8]. The first is essentially Newton's second law, that is, force = mass times acceleration, which for fluids gives the Navier-Stokes vector equation to the lowest order. For horizontal coordinates *x* and *y* and time *t*, this

1 (, ,)

¶

h

*g t* (1)

*<sup>t</sup>* (2)

Eqs (1) and (2) represent coupled equations in the unknown tsunami wave height and orbital velocity. By differentiating with respect to time and/or space, as relevant, we eliminate one variable, arriving at the following two hyperbolic PDEs for tsunami height and velocity:

$$\nabla \cdot (d\nabla \eta) - \frac{1}{\mathbf{g}} \frac{\partial^2 \eta}{\partial t} \tag{3}$$

$$(\nabla \nabla \cdot (d\vec{v}) - \frac{1}{\text{g}} \frac{\partial^2 \tilde{\nu}}{\partial t^2} = 0) \tag{4}$$

We first solve the scalar Eq. (3) for height. Then orbital velocity is obtained by integrating the left side of the linearized Eq. (2) as a function of time. Radars measure orbital velocity, rather than the tsunami height measured by other tsunami sensors.

These are well‐known equations for waves in shallow water. They are justified when water depth is much less than the horizontal scale of the water wave, for example, its wavelength. Horizontal scales of a tsunami wave exceed tens of kilometers, so that even in water several thousand meters deep *the tsunami is always a shallow-water wave everywhere on the planet*. The time scales (periods) for tsunami waves that represent hazards are large, varying from 20 to 50 min.

Another useful quantity applied to shallow‐water waves is their phase velocity *vph* that is given in terms of the water depth by:

$$\text{vol}\_{ph}(d) = \sqrt{\text{gd}} \tag{5}$$

The phase velocity for a tsunami wave train traveling across the ocean with depths from 100 to 1000 m typically exceeds 100 km/h, while the orbital velocities encountered are tens of centimeter/s or less. The water particles themselves move at the orbital velocity, while the surface velocity that the eye would follow as the wave rushes across the ocean is the phase velocity.

We define the "near field" over which the linear model applies as ranging from about 2 km from shore (beyond first radar range cell) out to as far as the radar can see, ∼50 km from the coast. The PDEs (3) and (4) are exact under the accepted linearity assumptions and can handle sharp bottom depth changes and reflections or partial absorptions at coastlines. Typically, they are solved with standard finite‐element methods [9]. Over near‐field distances, they can be solved on personal computers.
