This blog-post dives into the measurements of the 2011 11’th March Tsunami from the ‘Great Sendai Earthquake’. The Great Sendai Earthquake, caused a loss of lives in the order of 20.000 and insurmountable amounts of economic damages.

- Tsunami measurements from NDBC buoys
- Buoy data and placements
- Tsunami wave celerity
- Tsunami wave trains
- Why are Tsunamis wave heights increasing at shorelines?

### Tsunami measurements from NDBC buoys

Lets focus on measuring the Tsunami itself. We do this to try and understand how the tsunami spread across the globe. To investigate this we will have a look into some wave buoy measurements from the National Data Buoy Center in short, NDBC.

The NDBC wave buoy measurement equipment, are located at different distances from the earthquake epicenter across the Pacific Ocean. Figure 1, showcases all buoy measurement locations of the NDBC Tsunami program within the pacific ocean. These buoys are specifically designed for Tsunami detection purposes.

The water depth and distances vary with thousands of kilometers between the buoy observation stations. However we will see that measuring a tsunami is still possible even across the vastness of the pacific ocean.

**Related read:** *What caused this catastrophic event? How do we mitigate future catastrophes?*

**Measurements from the (**N**ational **D**ata **B**uoy **C**enter) **NDBC** buoys – (**N**orth **O**ceanic and **A**tmospheric **A**dministration) – **NOAA

Measurements are extremely valuable for understanding the natural world in general and in particular the oceans dynamics.

Although the vastness and shear scale of the oceans makes such a task almost implausible we still try and measure wave heights, wind speeds and current speeds by placing buoys out at sea.

### Buoy data and placements

The NDBC-buoys are located offshore and for this exercise we will specifically look into buoys 43451, 21418 and 21413 which was in place during the March 11’th 2011 Tsunami. The driving mechanism behind the Tsunami was the “Great Sendai Earthquake”. We can see the Earthquake epicenter and aftershock locations in Figure 2.

**Related read**: *Top ten misconceptions about earthquakes*, *Top ten misconceptions about tsunamis*

In order to showcase the differences in measured surface elevations between the NDBC-buoys. I have created Figures 3-8 where measurements are shown and GIS (Geographical Information System)-figures are created of the buoy locations in question. Here we can see that the measurements of the Tsunami are occurring at different times across the pacific ocean.

### Tsunami wave celerity

The Figures above show that there are differences in the Tsunami surface elevations. The Tsunami wave disturbance develops and travel across the globe changing underway.

We can calculate the average wave celerity using the time difference of the measured wave disturbance between individual buoys. The arrival time of the wave at the eastern NDBC buoy 43412 is approximately 30 hours after the first measured signal. See Figures 7 and 8 for locations.

The approximate distance between NDBC buoy 21418 and 43412 is 15.000 km’s. Using this approximate distance together with the time difference of 30 hours and we get that the celerity of the Tsunami wave is approximately 500 km/hr!

In conclusion, the Tsunami moves with approximately 500 km/hr across the deep ocean of the Pacific. That is faster than a racing formula 1 car!

lets now try and examine the wave-shape of the Tsunami itself.

## Tsunamis – Single or multiple waves?

Lets investigate Figures 3-6 more in-depth. In all figures, we can see that the initial wave disturbance comprises of multiple individual waves with different frequencies. Each of these waves travel independently and exchange energy with one-another. For instance, Figure 6 show individual wave buoy signals plotted together. In order to process the wave buoy data to reach the above signals we do the following steps:

- Firstly, we download the individual raw wave data.
- Secondly, we preprocess the data series removing unwanted data.
- Thirdly, we detrend the data removing mean water depths from the equation.
- Finally, we plot the results against each other with date-stamps.
- In conclusion we can now investigate the Tsunami wave amplitude individually.

We can see that the large amplitude in buoys 21418 and 21413 are smoothed out in buoy 43412. This confirms the hypothesis that the wave dispersion relation transforms energy from bound high frequency harmonics towards carrier low frequency harmonics resulting in the smoothing out the of wave signal across distance.

### Tsunami wave trains

In reality, Tsunamis consist of a plethora of waves with different amplitudes, frequencies and directions, all interacting resulting in what we call a Tsunami. When we think of an individual Tsunami wave, in reality we think of a wave-train of multiple waves with different frequencies, amplitudes and directions whose sum and difference make up the Tsunami.

In linear wave theory, we assume that these individual waves are all travelling independent of one-another, resulting in a wave-train of motion where the net wave train moves at speeds different from the individual frequencies.

This agrees with the above measured wave celerity as the individual tsunami wave-train moves with 500 km/hr while the incoming individual Tsunami wave frequency only travels of few tens of kilometers an hour.

## Tsunami amplitude

Lastly lets look into the tsunami wave amplitude. We have now concluded that the Tsunami wave consists of a train of individual waves whose frequencies and directions all interact. We have also concluded that the wave-train travels across the ocean with approximately 500 km/hr whenever the water depth is large O(4000 m’s). The last interesting piece of the puzzle is the most feared one, the Tsunami amplitude.

We can see that the amplitudes out in the deep sea are O(1/10 m’s) above figures and small compared to the amplitudes experienced near the shore O(10 m’s). Now one may wonder why are there so large differences in the observed wave amplitude between the deep and shallow ocean?

### Why are Tsunamis wave heights increasing at shorelines?

The answer lies in the dynamical nature of wave shoaling. As waves near the shore, the wave-energy transforms from the sub-harmonic space into the super-harmonics. Shoaling causes the wave to slow down and transform increasing the wave amplitudes of the incoming wave until breaking occurs. Check out this blogpost explaining shoaling in detail.

The wave period of the Tsunami wave endures throughout the breaking causing the water to keep rushing in over shores, low breakwaters damaging infrastructure and potentially people.

## Conclusion

In this blogpost we have examined the measurements of the 11’th March 2011 Tsunami caused by the Great Sendai Earthquake.

We have looked into the placements of NDBC buoys along with measured amplitudes of the Tsunami.

We have estimated the wave celerity at deep water O(4000 m’s) to be 500 km/hr, faster than a racing formula 1 car.

We have explained in short details why wave heights at nearshore is substantially different from the deep water measurements.

## References

https://www.britannica.com/event/Japan-earthquake-and-tsunami-of-2011

https://www.sciencelearn.org.nz/resources/596-tsunami-shoaling