TsunamiTsunamiModel and Simulation of 1-D Tsunami Model and Simulation of 1-D Tsunami

Utilizing the Non-Linear Shallow Utilizing the Non-Linear Shallow Water Wave EquationsWater Wave Equations

Joan MartinezJoan MartinezRosa Jimenez Rosa Jimenez

Ron CaplanRon Caplan

December 19, 2005December 19, 2005

What is a Tsunami?What is a Tsunami?

• Water wave caused by large displacement of Water wave caused by large displacement of water.water.

• Earthquakes most common cause, which Earthquakes most common cause, which can cause a several meter height can cause a several meter height displacement over tens of thousands of displacement over tens of thousands of square kilometers giving huge potential square kilometers giving huge potential energy to water on surface.energy to water on surface.

• Waves have wavelengths in the range of Waves have wavelengths in the range of many kilometers, with height to length many kilometers, with height to length ratios up to 1:100,000ratios up to 1:100,000

• Wave can increase height at run-up to a Wave can increase height at run-up to a steep slope up to 3 or 4 times.steep slope up to 3 or 4 times.

Motivation for Motivation for ModelingModeling

• Danger to human life Danger to human life requires a warning requires a warning system. system. “Since 1850 alone, tsunamis “Since 1850 alone, tsunamis have been responsible for the have been responsible for the loss of over 420,000 lives and loss of over 420,000 lives and billions of dollars of damage to billions of dollars of damage to coastal structures and habitats”coastal structures and habitats” (NOAA).(NOAA).

• Only possible if Only possible if dynamics are known. dynamics are known. “forecasting tsunami arrival “forecasting tsunami arrival and impact is possible through and impact is possible through modeling …”modeling …” (NOAA) (NOAA)

• Interesting Interesting phenomenon to phenomenon to model.model.

Modeling: What Equations to Modeling: What Equations to Use?Use?• Most tsunami have soliton style Most tsunami have soliton style

dynamics (with some having a “double” dynamics (with some having a “double” wave profile). So why not use KdV?wave profile). So why not use KdV?

• Simple reason: From observation, we Simple reason: From observation, we know we need an equation that includes know we need an equation that includes the depth of the water.the depth of the water.

• Solution? Start with the most Solution? Start with the most comprehensive fluid dynamics equation: comprehensive fluid dynamics equation: The Navier-Stokes Equations.The Navier-Stokes Equations.

Navier-Stokes Navier-Stokes EquationEquation

The equations require some assumptions.

They are also usually combined with the continuity equation (conservation of mass):

Can we make it simpler?Can we make it simpler?• Since Tsunami wave lengths are so long, and Since Tsunami wave lengths are so long, and

heights so small (comparable to the depth of the heights so small (comparable to the depth of the ocean), they are likened to waves in “shallow” ocean), they are likened to waves in “shallow” water.water.

• For this to hold, must have:For this to hold, must have:

A/h << 1 and kh << 1A/h << 1 and kh << 1

• If we take NS and assume zero viscosity, and If we take NS and assume zero viscosity, and reduce to 2-D, and then noting that for a thin reduce to 2-D, and then noting that for a thin layer of a liquid: layer of a liquid:

p=g ρ hp=g ρ h

We can obtain an equation for waves traveling on We can obtain an equation for waves traveling on a thin layer of fluid, which are called the Shallow a thin layer of fluid, which are called the Shallow Water Wave Equations:Water Wave Equations:

The 2-D Non-Linear Shallow The 2-D Non-Linear Shallow Water Wave EquationsWater Wave Equations

UUtt + UU + UUxx + VU + VUyy + gH + gHxx = 0 = 0

VVtt + UV + UVxx + VV + VVyy + gH + gHyy = 0 = 0

HHtt + [U(H-b)] + [U(H-b)]xx + [V(H-b)] + [V(H-b)]yy = 0 = 0

U(x,t) = Horizontal Velocity of HU(x,t) = Horizontal Velocity of H2200V(x,t) =V(x,t) = Vertical Velocity of HVertical Velocity of H2200H(x,t) = Height of the wave H(x,t) = Height of the wave b(x) = Subterrain b(x) = Subterrain g = acceleration due to gravity = 9.8 g = acceleration due to gravity = 9.8 m/sm/s22

One-Dimensional CaseOne-Dimensional CaseIf we assume that the velocity in the y-direction is If we assume that the velocity in the y-direction is

negligible, then the non-linear SWWE reduces to:negligible, then the non-linear SWWE reduces to:

UUtt + UU + UUxx + gH + gHxx = 0 = 0HHtt + [U(H-b)] + [U(H-b)]xx = 0 = 0

U(x,t) = Horizontal Velocity of HU(x,t) = Horizontal Velocity of H2200H(x,t) = Height of the wave H(x,t) = Height of the wave b(x) = Subterrain b(x) = Subterrain g = acceleration due to gravity = 9.8 g = acceleration due to gravity = 9.8 m/sm/s22

The SWWE – Is it perfect?The SWWE – Is it perfect?There are some limitations to the model:There are some limitations to the model:

• Vertical velocity is considered negligible, Vertical velocity is considered negligible, therefore cannot model some situations such therefore cannot model some situations such as asteroid impacts, underwater landslides, as asteroid impacts, underwater landslides, wave traveling over submerged barrier, or wave traveling over submerged barrier, or model short-wave-length tsunami.model short-wave-length tsunami.

• Obviously this also means that the equations Obviously this also means that the equations are inadequate to model the wave crashing are inadequate to model the wave crashing over on the beach front, however since over on the beach front, however since tsunami tend to not break at all at run-up, tsunami tend to not break at all at run-up, and if they do, do so somewhat inland, this and if they do, do so somewhat inland, this limitation is not such a concern.limitation is not such a concern.

Lets AnalyzeLets Analyze• Since a tsunami is a wave which travels across distances, Since a tsunami is a wave which travels across distances,

we might expect to find a traveling wave solution in the we might expect to find a traveling wave solution in the form: form:

H(x,t) = h(z) z = x-ct H(x,t) = h(z) z = x-ct U(x,t) = u(z) z = x-ct U(x,t) = u(z) z = x-ct

• However, when one tries (However, when one tries (and triesand tries, , and triesand tries, , and triesand tries and triesand tries ……), you end up not being able to find such a solution, or ……), you end up not being able to find such a solution, or even a phase-plane analysis form.even a phase-plane analysis form.

• The reason this is so becomes clear on second inspection The reason this is so becomes clear on second inspection of the SWWE. Since the wave velocities (thus shape), of the SWWE. Since the wave velocities (thus shape), depends on the depth below it (which is variable), we depends on the depth below it (which is variable), we cannot expect to find a solution valid for all t in which the cannot expect to find a solution valid for all t in which the overall velocity of the wave is constant. For example, if overall velocity of the wave is constant. For example, if half of the wave is over one depth, and half is over another half of the wave is over one depth, and half is over another depth, the total wave’s overall velocity will change with depth, the total wave’s overall velocity will change with time, and thus cannot be a constant.time, and thus cannot be a constant.

If Tsunami is far from shore, then the ocean depth does not If Tsunami is far from shore, then the ocean depth does not change much and therefore can take (H-b) to be a constant, change much and therefore can take (H-b) to be a constant, d.d.

Also, as can be seen in numerical simulation, the horizontal Also, as can be seen in numerical simulation, the horizontal velocities do not change much from the initial background velocities do not change much from the initial background velocities, so can replace U with a constant, u.velocities, so can replace U with a constant, u.

This gives us:This gives us:UUtt + uU + uUxx + gH + gHxx = 0 = 0HHtt + dU + dUxx = 0 = 0

If we take the background velocities, u, to be 0, then by If we take the background velocities, u, to be 0, then by differentiation, the linear SWWE reduce to:differentiation, the linear SWWE reduce to:

HHtttt = (gd)H = (gd)Hxxxx

Which is a linear wave equation with a traveling wave solution Which is a linear wave equation with a traveling wave solution with velocity dependant on the depth of the ocean: v = with velocity dependant on the depth of the ocean: v = √√(gh)(gh)

If ocean is 1km deep, this velocity is ~100m/s ~ 400mph!!If ocean is 1km deep, this velocity is ~100m/s ~ 400mph!!

Tsunami in Open WatersTsunami in Open Waters

Linear Equation Break DownLinear Equation Break Down• The linear SWWE does simulate a tsunami in The linear SWWE does simulate a tsunami in

open waters but since the waves are so long, open waters but since the waves are so long, as they approach shore, the height of the as they approach shore, the height of the waves increases too much, and break too waves increases too much, and break too early.early.

• As noted before, Tsunami typically do NOT As noted before, Tsunami typically do NOT break at all during run-up, or if they do, much break at all during run-up, or if they do, much later than the linear theory predicts. later than the linear theory predicts.

• Thus, linear theory breaks down close to shore.Thus, linear theory breaks down close to shore.

• Therefore, we use the non-linear SWWE for our Therefore, we use the non-linear SWWE for our simulation.simulation.

Numerical AnalysisNumerical Analysis

• Non-linear SWWE requires numerical Non-linear SWWE requires numerical solutions.solutions.

• Much research has been done on how to Much research has been done on how to implement efficient numerical schemes, implement efficient numerical schemes, which are quite complicated.which are quite complicated.

• Simple explicit finite-difference methods Simple explicit finite-difference methods cannot be used due to instability.cannot be used due to instability.

• To ensure stability, we use a “leap-frog” To ensure stability, we use a “leap-frog” method. method.

• Taking a gaussian hump of water 40km Taking a gaussian hump of water 40km long, and 3 meters high, centered 28km off-long, and 3 meters high, centered 28km off-shore, we obtained the following simulation:shore, we obtained the following simulation:

Simulation of 1-D Simulation of 1-D TsunamiTsunami

Another run, this time with a 10km initial Another run, this time with a 10km initial widthwidth

Is this Is this accurate?accurate?

1. We see a wave propagating over large distance.We see a wave propagating over large distance.

3. The velocity in open water is about 70m/sThe velocity in open water is about 70m/s

2. The wave piles up on itself, as its velocity decreases The wave piles up on itself, as its velocity decreases close to shore.close to shore.

4. The wave’s amplitude increases by about a factor of 2 The wave’s amplitude increases by about a factor of 2 towards the shoreline.towards the shoreline.

5. The wave causes coastal flooding, but then reflects back The wave causes coastal flooding, but then reflects back towards the ocean.towards the ocean.

6. The wave does NOT try to break over itself, but rather The wave does NOT try to break over itself, but rather flows onto shore.flows onto shore.

“… and can travel great trans-oceanic distances … ” (wikipedia.org)

“ …reach shore, they slow down… The waves scrunch together like the ribs of an accordion and heave upward. ” (PBS)

“… across the ocean at speeds from 500 to 1000 km/h” (wikipedia.org)

“..run-up heights, ..which are greater than the height of the tsunami … by a factor of two or more…” (Bryant, 48)

“After runup, part of the tsunami energy is reflected back to the open ocean. ” (USGS)

“Tsunami in most cases do not break, but surge onto shore…” (Bryant, 35) “(i.e., a rapid, local rise in sea level) “ (USGS)

What do we see?What do we see?What do they say?What do they say?

ConclusionConclusionss

• Despite limitations mentioned previously, the Despite limitations mentioned previously, the SWWE seems to simulate a non-breaking SWWE seems to simulate a non-breaking tsunami very well, both in open waters as well tsunami very well, both in open waters as well as run-up and coastal flooding.as run-up and coastal flooding.

Any Any Questions?Questions?