Tsunami and Tsunami Forces

I have touched upon some of the extreme episodes in the Nature’s Action piece on the NATURE page. Tsunami is one of these Nature’s violent wraths that unleash immense trail of casualty and destruction on its path. Our memory is still fresh with the viciousness of havoc of the 2004 and 2011 tsunamis in Indonesia and Japan. Many of us have seen the live coverage of the 2011 Japan tsunami, and I have included a snapshot image (credit: anon) of it. Perhaps an image of this kind has given rise to the myth of Noah’s Ark in the ancient past. It is impossible for one not to realize the absolute shock and horror unless one is present on a tsunami scene. Let us try to talk about this interesting topic – tsunami characteristics and the loads they exert on structures standing on its path.
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1. Tsunami Intro
Tsunami is one of the rarest natural phenomena that occur with little definitive advanced notices. Tsunamis are caused by earthquake, landslide, volcanic eruption, and by rapid and high drop in atmospheric pressure. Perhaps talking about the first two will suffice for this piece. The first type triggered by underwater earthquakes – often called tsunamigenic earthquakes – causes sudden substantial rupture of the Earth’s crust displacing huge mass of water. The process gives birth to a series of impulsive waves known as tsunami that radiates out directionally from the source. Aspects of tsunamis generated by underwater volcanic eruption – in particular in light of the most recent violent eruption of Tonga on 15 January 2022 – remind us how devastating their effects could be. Tonga eruption is estimated to have measured 6 on the 1 to 8 VEI scale (Volcanic Eruption Index, C Newhall and S Self 1982). The eruption prompted Pacific wide tsunami warning – and really impacted Chile and Peru at more that 10,000 km away. On top of that, record books illustrate the largest and most disastrous tsunami generated by the Indonesian Krakatoa eruption on 27 August 1883 – it is said the tsunami was as high as 40 m – and together with the eruption killed some 36,000 people.
A first order simplistic estimate of tsunami height (trough to crest) and the period (time interval between the two successive crests or troughs) relates these two tsunami parameters logarithmically to the magnitude of earthquake in the Richter scale. For example, a submarine earthquake with a magnitude of 7.5 could generate a 3.6 meter high, 24 minute tsunami at the source. Note that the 2004 tsunami off the Indonesia coast and the 2011 tsunami off the Japanese coast were caused by 9.3 and 9.0 magnitude earthquakes, respectively.          
The second important cause is the tsunamigenic submarine or terrestrial rapid landslide. Such landslides representing a rigid body motion along a slope-failure surface, are often triggered by earthquakes. In this case, a first order simplistic estimate relates tsunami height directly to the slide volume and sliding horizontal angle, and reciprocally to the water depth of incidence. One example of such a tsunami was the 1975 tsunami that occurred at the head of the Kitimat Arm of the Douglas Channel fjord system in British Columbia. A recent example is the Palu Indonesia Tsunami.       
A tsunami with a period in the order of 10s of minutes is classified as a long wave or a long-legged wave, and as pointed out in the Ocean Waves blog on the NATURE page, such waves occur when wave length is longer than 20 times the local water depth. Both the widths and lengths of crests and troughs of such long-period waves are measurable in scales of kilometers. Like all other waves, they are subjected to the transformation processes as soon as they are born, traveling very fast in deep water. The Transformation of Waves piece on this page has highlights of some of the wave transformation characteristics. Let me briefly describe some processes specific to long wave as they enter into the shallow water.
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2. Tsunami Propagation Hydrodynamics
At least 3 processes of tsunami transformation are important – these are the processes of shoaling, funneling and resonance. The phenomena of shoaling and funneling can best be understood by applying the energy conservation principle, often known as the Green’s Law. This simple principle assuming no losses of energy by friction, etc., shows that for a gradually shoaling continental shelf, the ratio of height increase is proportional to the reciprocal of the ratio of depth decrease raised to the 1/4th power. For a channel gradually decreasing in width, the funneling effect is given by the ratio of height increase that is proportional to the reciprocal of the ratio of width decrease raised to the 1/2nd power. As a simple example of shoaling, a 3.6 meter high wave, will amplify to 6.4 meter as it transforms from 100 to 10 meter water depth.
The phenomenon of resonance is quite interesting and intriguing because of its analogy to the force-response dynamics of an oscillating system. Resonance should not be confused with funneling – funneling occurs in the process of balancing the energy, while resonance is the frequency response of the system and occurs due to the reflection of and interaction with the incident wave. Let me try to explain it more based on my paper – Modeling Tsunami and Resonance Response of Alberni Inlet, British Columbia {30th International Conference on Coastal Engineering, World Scientific, 2006. This paper is one of the many papers listed by Prof. Robert L. Wiegel at the University of California, Berkeley as Tsunami Information Sources and published in 2009 in the Science of Tsunami Hazards – the International Journal of Tsunami Society.
To get into the core concept of it, one needs to understand the behavior of an oscillating system. Such a system is characterized by a natural frequency at which it resonates to the exciting force – which means that the incident and reflected waves are virtually interlocked with each other, with very high anti-nodal amplification of the incident wave amplitude. In reality however, most natural systems do not respond to such an extent because of frictional damping etc. In addition, experiments with resonant behaviors show that a system also responds by amplification of the exciting wave both at sub- and super-resonant frequencies. This behavior is very important, and let me try to clarify how it explains the tsunami response of the Alberni Inlet.
The 1964 Alaska tsunami registered about 1 meter high with a period of 90 minute at the entrance of Alberni Inlet at Bamfield, amplified to 3-times to cause huge damages at the head of the inlet at Port Alberni. The Alberni Inlet is a 65 km long deep fjord that virtually shows no phase lag or amplification in tidal motion. Such a system is very vulnerable because its natural frequency lies within the close range of usual tsunami frequencies. Contrary to the conclusions of previous investigators, my hydrodynamic modeling investigation with Mike21 (courtesy Danish Hydraulic Institute), showed that the 3-times amplification occurred at a sub-resonant frequency – and had there been an incident tsunami close to the resonant frequency, the amplification would have been some 5 times.
Like all waves, a small tsunami in deep water shoals to monstrous waves as it propagates into the shallow water. After breaking Tsunami Run-ups flood coastal lands with enormous inbound and outbound speeds causing havoc and destruction. The arrival of Tsunami crest is preceded by the huge draw down or Sea Level Suck Out associated with the Tsunami trough. This phenomenon sucks out things from the shore out into the sea – exposing shoreline features – leaving many aquatic lives stranded in air. It catches offshore boats off-guard – and tragedies happen when people rush out to catch the stranded fishes. See more in the Frontal Wave Force Field in Force Fields in a Coastal System.
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3. Tsunami Run-up
Well, so far so good. Now let us focus our attention on the most important aspect of tsunami effect – the runup (runup is the vertical height of tsunami propagation above mean water level) and loads on structures. Part of this discussion will be based on my short article published in the 2008 ASCE Journal of Waterway, Port, Coastal and Ocean Engineering {Discussion of ‘Maximum Fluid Forces in the Tsunami Runup Zone’}.
Most tsunamis at their births belong to nonlinear wave category – and the degree of nonlinearity increases as they become taller and travel on to the shallow water. At a certain time, the tsunami wave becomes unsustainable as the water particle velocity exceeds the celerity (square root of the product of acceleration due to gravity and depth), and taking the shape of a solitary wave (a wave that does not have a trough) it breaks generating a very forceful bore that runs up the land. This process turns an oscillatory wave into a translatory one like that of a dam break flood wave. However, the difference between the flood wave and the tsunami translatory wave is that – the tsunami is not a single wave, but rather a series of waves that comes one after another with complicated hydraulic interactions of run ups and run downs.
What is the maximum velocity of the tsunami bore? Investigations show that the maximum bore velocity is about 1.4 times the celerity (up to 2 times at the leading edge; celerity becomes defined in terms of tsunami height as there is no trough), or even higher when the propagating bores become constricted by topography and structures. As an example, a 1 meter high tsunami will likely to generate a 6.3 meter per second velocity at the leading edge, followed by a propagating velocity of 4.4 meter per second. In Froude (William Froude, 1810 – 1879) terms, the wave propagation speed exceeding the celerity is called the supercritical flow. Such flows could travel upstream, cross channels, roads and embankments unleashing the huge energy it posses. A water velocity of this kind has enormous power to destroy structural members and uproot them by scouring.
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4. Tsunami Forces on Structures
Let us now turn our attention on the forces that tsunamis exert on structures standing on its way. When one thinks about it after witnessing the 2011 Japanese tsunami destructions, it is impossible for one not to wonder about the limitations of human capability in planning and designing measures to withstand the enormous wrath of a tsunami. This feeling arises because Japan is a reportedly tsunami savvy country – perhaps sophisticated in its engineering design standards and codes.
Among the limited amount works in this field, a document prepared by US FEMA (Federal Emergency Management Agency) and US NOAA (National Oceanic & Atmospheric Administration) stands out in its comprehensiveness of discussing the problem. It cites the standards of flood resistant designs developed by ASCE/SEI (American Society of Civil Engineers/Structural Engineering Institute).
According to the type and nature of forces, tsunami loads are identified as 8 types:

1. hydrostatic (horizontal water pressure if one side is empty, e.g. of a wall);
2. buoyancy (upward force that lets one feel lighter when in water);
3. hydrodynamic (primarily due to horizontal drag of speeding water);
4. impulsive or slamming (horizontal, for example by the leading edge of a tsunami slamming on a wall);
5. floating debris impact (horizontal, tsunamis carry huge amounts floating debris to cause impact);
6. damming load due to accumulated debris (elevated horizontal hydrostatic pressure);
7. uplift forces (upper floors subjected upward force while a tsunami slides and speeds underneath a floor); and
8. retained weight of water on upper floors (weight causing downward force especially when the down-floor is drained).
   

All of these forces do not occur simultaneously, and depending on circumstances some are more dominant than others, therefore a smart screening is required to choose the most possible load combinations for a particular structure or a structural element.
In addition to the enormous forces on structures, tsunamis also erode and scour shallow foundations undermining their stability. The opposite also happens in areas where sedimentation and debris dumps occur.  
Horizontal wave loads generally arise due to velocity (drag) and acceleration (inertia). Why only drag force is important for tsunamis? I would like to answer the question based on one of my papers {Wave Load on Piles – Spectral versus Monochromatic Approach, Proceedings, 18th International Offshore and Polar Engineering Conference, Vancouver, ISOPE, 2008}. It turns out that in low frequency (or high period) oscillations including flood waves, the inertial forces diminish in magnitude leaving the horizontal hydrodynamic loading processes on to the drag effect. I hope to talk more about it at some other time.
​We have talked about tsunami loads on structures located in the runup zone. How about structures – nearshore marine terminals and offshore oil platforms standing in water where a tsunami has not broken? In such cases, one needs to resort to nonlinear wave phenomena to determine the tsunami kinematics and loads.
How about the effect of sea level rise (SLR) on tsunami? Well, there are no direct effects. However, with the raised mean water level, tsunamis will be able to travel more inland, and run up higher. The argument becomes clear, if one imagines the 2004 and 2011 tsunamis to occur a century later when the SLR stand is likely to be higher.        
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Here is an anecdote to ponder:
The disciple said, “Sir, I am feeling very happy today. Someone greeted me with a smile as I was walking by.”
The master looked at his disciple and said, “Well, I am very glad to hear that. You seem to be in the right mood to receive the greeting. Reciprocally, your reaction must have made the greeting person happy.”

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- by Dr. Dilip K. Barua, 6 October 2016

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