Up to this point, we have focused primarily on the height of tsunamis and how far onshore they can travel. We have paid less attention to the speed of the wave. However, the speed of the wave determines whether people have time to evacuate from the coast. The sooner the wave arrives, the less time there is to move people away! And as we talked about in lecture, some people may move more slowly: children, older people, or people with disabilities. Even fit, young people might have trouble moving quickly if they have to navigate buildings or other infrastructure that collapsed in the earthquake. This means that a crucial element of tsunami preparation is determining how fast tsunami waves move.
We have the scientific understanding of tsunami wave propagation in the ocean to estimate the wave speed and therefore the warning time. As we talked about in class, the speed at which a tsunami travels depends on the water depth. The deeper the water, the faster the tsunami travels:
c=√(gL/2π tanh(2πd/L) )
c: the speed of the tsunami wave (we use the letter “c” for “celerity”)
g: the acceleration due to gravity (9.8 m/s2)
L: the wavelength of the tsunami (typically 100s of kilometers)
π: 3.14159265…
d: the water depth (0-10 km)
tanh: the hyperbolic tangent function
This is a complicated equation! But when the tsunami wavelength L is much larger than the water depth d (specifically, when d≤L⁄10), then the equation becomes much simpler:
c=√gd
We will apply this formula to the observed tsunami in Japan to determine how much warning time they had in 2011, and then investigate the warning time before a potential tsunami in Cascadia.
What happens if you cannot evacuate in time? We know the wave is powerful, but how fast does it move onshore? Maybe you could outrun the water! The physics of this problem is more difficult to solve, and scientists are still working on understanding it. But we can measure how fast the tsunami travels using video footage from Japan and determine whether you could outrun the wave.
As usual, explain your reasoning and show your work to get full credit!
- First, we should double check that the tsunami speed equation actually works. In other words, we need to “calibrate” our tsunami speed calculations. We will do this using observations from the 2011 Japan tsunami. Below is a map of water depth along a line from the coast of Japan to a DART buoy. Remember: DART buoys measure tsunami waves in the ocean by measuring the pressure of the water above them. They record the arrival time of tsunami waves, allowing us to evaluate the speed equation.
We will work through this calibration calculation systematically (you will use the same steps in later questions, so pay attention and take notes!):
1A. What is the distance between the tsunami source region and the DART buoy? Provide the answer in meters.
1B. What is the average depth of the ocean along the path from the tsunami to the DART buoy? Provide the answer in meters.
1C. Multiply the acceleration due to gravity (g) (provided in the Introduction) and the average depth you measured in Part 1B. Double check your units!
1D. Take the square root of your answer. You can do this on your calculator or you can use the graphs of the square root function provided below. This is the tsunami speed in meters/second.
1E. You now have the distance the tsunami traveled from the earthquake to the DART buoy (from Part 1A) and the speed of the tsunami (from Part 1D). To calculate the tsunami travel time, divide the distance by the speed. This gives you the travel time in seconds.
1F. What is the tsunami travel time in minutes?
1G. Finally, we can compare our predicted travel time (from Part 1F) to the observed travel time at the DART buoy. The figure below is the record of wave height over time at the DART buoy. The time axis starts at the time of the earthquake and the units are hours. Identify the time when the tsunami arrives at the DART buoy and convert this time to minutes. How does this compare to your calculation from Part 1F? How well does the tsunami speed calculation work?
- After validating the speed equation, we can calculate how much warning time people living on the east coast of Japan had before the tsunami hit.
2A. Using the figure of water depth offshore of Japan and the same steps as in Part 1, this time calculate the time the tsunami takes to travel from the earthquake to the coast. Give your answer in minutes.
2B. In order to warn people that a tsunami is on the way, we have to have to know it has begun. One indicator that a tsunami might be coming is the strong shaking produced by the earthquake surface waves. These large amplitude seismic waves travel at ~3 kilometers/second. Calculate the amount of time between the earthquake and the arrival of these surface waves at the coast of Japan, in minutes.
2C. For people on the coast of Japan, how much warning time did they have between the strong shaking produced by the earthquake and the arrival of the tsunami wave? Explain whether you think this enough time to evacuate a town.
- Now that we have practiced calculating the speed and warning time for a tsunami that already occurred in Japan, we can apply our knowledge to another subduction zone and use this result to prepare for a potential tsunami. We will turn again to Crescent City, CA (the same town where we determined potential inundation areas in Lab #3). Below is a figure showing the water depth along a line from the Pacific Ocean to Crescent City.
Using the same approach as in Parts 1 and 2, calculate the warning time (in minutes) that people in Crescent City would have if a large earthquake occurred and generated a tsunami. How does this compare to the tsunami warning time people in Japan had in 2011?
- Finally, what happens if you cannot evacuate on time? Maybe you were able to evacuate away from the coast, but the tsunami was higher than you anticipated. Could you outrun the tsunami wave onland?
4A. Video footage of the 2011 Japan tsunami from a news helicopter allows us to calculate the tsunami speed. This video is available on Canvas. Below are two still frames from the clip, highlighting some of the features visible on the ground before they are covered by the tsunami. Write down the time of these two frames and calculate the difference (in seconds).
4B. Below is a map of this region showing the features from the video. Measure the distance the tsunami traveled between these two frames in meters.
4C. Calculate the speed of the tsunami (speed = distance/time) in meters/second. Convert this to miles/hour (1 m/s = 2.24 mph).
4D. How long does it take you to run or quickly walk 1 mile (in minutes)? Your speed in mph is:
60/(("your mile time in minutes" ) )
Can you outrun the tsunami onshore? You might consider how long can you maintain your top speed. Could you run at top speed around buildings, over fences, across ditches, etc?
Sample Solution