Physics of Tsunamis. This is an article from my home page: Ole Witt-Hansen 2011 (2016)

Transcription

1 Phsics of Tsunamis This is an article from m home page: Ole Witt-Hansen 11 (16)

2 Contents 1. Constructing a differential equation for the Tsunami wave...1. The velocit of propagation in deep and medium deep water Estimating the destructive force of a Tsunami...8

3 Phsics of Tsunamis 1 1. Constructing a differential equation for the Tsunami wave In all those ears, where I have been teaching the phsics of waves, I have emphasized, that the propagation of a wave is not a transportation of matter, but rather a periodic propagation of energ and momentum. And almost each time the class has confronted me with the fact that sea waves break and subsequentl rush on the shore. In these cases I normall choose the classic evasion technique, saing: Yes that is correct, but unfortunatel it is far to complicated to comprehend at this level of education. True enough, if we are talking hdrodnamics, but if an audacious student asks: But do ou understand it ourself? Then m answer has been evasive, in the sense that I merel explain that a rising water crest is created, because the dept of the water decreases. But this is just an phenomenological ascertainment, it is not an explanation of phsics. When I have been seeking for an analtic treatment founded on theoretical phsics I have onl found some less conspicuous articles, so I decided to find out for mself. At the Universit of Copenhagen we were in 1967 taught hdrodnamic from the book b Arnold Sommerfeld: Mechanics of Deformable Bodies, from 1964, but without a thorough knowledge of vector analsis, that is, the operator s of vector analsis grad, div and rot including the theorems of Stoke and Green, Sommerfeld s book is not reall applicable. In 11 the subject of the ear at the Danish gmnasium (senior high) was catastrophes, and man students chose to write about the Tsunami in 4, and I was their phsic teacher. I certainl searched for extern assistance, but the onl thing, that was reall tangible, was a remark in Sommerfelds book, accompaning two formulas for the velocit of propagation in deep water and in medium deep water. (1.1) Deep water: v g g k Medium deep water: v gh h is the depth of water, and g is the gravitational acceleration and λ is the wavelength. In deep water the velocit of propagation is independent of the depth, but in medium water it is not, and that is the ke to understand the phsics of the tsunamis. The derivation of the formulas (1.1) is however far from simple. The derivations are postponed until section. One ma obtain a phenomenological understanding of the creation of a tsunami, propagating towards lower water from the velocit formula. Namel the back end of a wave packet will steadil run faster than the front end, because the velocit decreases with the water depth. In the long run the back of the wave packet will crawl up on the front of the wave, and create a higher wave crest. This happens continuall and the result ma become a wave crest which is rising and finall breaks before reaching the shore, flooding the surroundings of the coast line, that is, a tsunami.

4 Tsunami Phsics To do a more formal analsis, it will be necessar with some simplifing assumptions. First we assume that the wave has a rectangular shape, where the bottom floats on the surface of the water. Further we assume that the water depth decreases linearl with distance from a reference point. At x =, the seabed is at h and at x it is h h x tan The velocities of the wave-packet (the box) are at the rear end and at the front end, v 1 and v respectivel, where v 1 > v. Since the wave-packet (the box) holds the same amount of water when moving, but changing its shape, it follows that the area A of the two rectangles shown in the figure must be the same. A = Δx = (Δx+dx)(+d), which gives: Δxd + dx = Solved with respect to d to give: (1.1) d dx x For medium deep water, we have the formula: (1.) v gh g( h x tan ). Because of the different velocities of the wave-packet at the rear and at the front, the wave-packet (the box) will be compressed an amount dx during the time dt. When inserted in (1.1) gives: dx v v ) dt vdt ( 1 (1.3) Furthermore d dx vdt x x v dv d dt dt x dx d dv dt dx d dt d dx dx dt v d dx This leads to the wanted differential equation.

5 Phsics of Tsunamis 3 (1.4) d dx dv v dx Substituting: v g( h x tan ) and dv dx g tan h x tan We finall arrive at a differential equation, which determine how the height of the wave crest depends on the distance to the shore. (1.5) d dx 1 tan h x tan d 1 tandx h x tan The last differential equation can then be integrated. (1.6) 1 1 x ln ln( h x tan ) d x tandx h x tan When the equation is solved for it ields: (1.7) h h x tan Even if we have used a highl simplified model, the solution reflects the behaviour of a huge wave approaching the shore. The wave crest becomes infinite, when the depth approaches zero, but this is a mathematical fact, not a phsical one. One could easil plot, the height of the wave crest, as a function of the distance to the shore, but it is more illustrative to plot a wave-packet rolling in towards the shore. tkx) A harmonic wave can be written (as the real part of) ( x, t) e Integrating over the wave number k / one obtains a wave-packet. ikx it ikx it e it cos( kx) isin( kx) (1.8) e e dk e e, ix ix sin(kx t) Which after rewriting using trigonometric formulas: Re x sin kx At t = the shape of the wave-packet is:, and the shape in x, ma be found multipling x the height of the wave at x. sin x Since 1 for x the maximum value of the wave packet is (x). x We the multipl (1.8) b the height of the wave crest, we will get a model that illustrates what happens, when a Tsunami wave approaches the shore.

6 Tsunami Phsics 4 (1.9) sin k( x x ( x) ) x x The figure shows a wave approaching the shore. The rising wave is shown at various instants from 1.5 km from the shore until it reaches the shore, where it goes to infinit, in accordance with (1.9). Although hdrodnamics is highl complex, the simple model developed above gives at least a qualitative answer on the dnamics of a tsunami.. The velocit of propagation in deep and medium deep water The derivation of the two formulas for the velocit of propagation of a wave, require vector analsis, and we shall recall the notation: (,, ) x z Gradient of a scalar field φ. v v x vz v x z Divergence of a vector field v. The Laplace operator x z v v v z vx v z vx v (,, ) The operator rot. z z x x From the vector analsis we know that for a rotational free vector field i.e. v, it is possible to define a potential from which the vector field is its gradient. Appling this to the velocit vector field v in a two dimensional field of flowing liquid without turbulence, we have (.1) v v v (, ) x If the velocit field is also divergence free: ( v ), which means that the fluid is incompressible, then the potential will satisf the Laplace equation, since: v (.) x

7 Phsics of Tsunamis 5 For an analtic function: f ( z) f ( x i) ( x, ) x, ), both the real and the imaginar part are solutions to the Laplace equation. This follows, because the satisf the Cauch-Riemann differential equations: (.3) and x x Using (.3) (.4) x x x x x x If v, then Φ(x,)= Φ, will represent curves with the same velocit, and through the Cauch-Riemann equation, it follows that: x, is orthogonal to x,, So the curves Ψ(x,) = Ψ and Φ(x,)= Φ, will be orthogonal curve sstems, and for that reason Ψ(x,) will represent the streamlines. The most general velocit potential, that represents the propagation of a wave, can be written: kxt ) k k (.5) e ( Ae Be ) Where we fix = at the surface. The boundar condition at the bottom ( = -h), is that the vertical velocit is zero. ( v = or using the potential: h ). Inserted in (.5) it leads to the equation: Introducing the constant C b kh kh Ae Be. 1 kh kh 1 kh kh C Ae Be so that A Ce og B 1 Ce, the potential takes the form: kxt ) 1 k ( h ) k ( h ) kxt ) (.6) e C( e e ) e C cosh( k( h)) The derivation of the expression for the velocit field, takes its starting point in the Navier-Stokes equation.

8 Tsunami Phsics 6 (.7) d v p F dt If the fluid is rotational free and divergence free i.e. v and v, Then (using vector analsis) (.7) can be reduced to the following form: (.8) t v 1 v p F If F is a conservative force (the gravitational force) then F can be expressed as the gradient of a potential U. (.9) F U Finall inventing the velocit potential v, and moving the gradient outside a parenthesis we get Bernoulli s law. (.1) ( 1 ( ) p U ) t For moderate velocities, we ma discard the term ( ) and if we preliminar are interested onl in the surface profile of the wave, we ma put the pressure p =. Hereafter the equation becomes substantiall simplified. (.11) U const t The constant ma in principle depend on time, but for a non forced periodic movement, one can show that the onl possibilit is that it is zero. At the same time U = ρg, where ρ is the densit of water and g is the gravitational acceleration. Inserting this in (.11) gives: (.1) g t If we consider a progressive harmonic wave, the surface profile is of the form: (.13) kxt) u( x, t) ae, Where a in general is a complex number possibl containing a phase. Using (.1) g t kxt ) with (.6) e C cosh( k( h)) and (.13) we find: kxt) kxt) (.14) ie C cosh( k( h)) age ic cosh( k( h)) ag

9 Phsics of Tsunamis 7 To determine the velocit of propagation of the wave at the surface, we need et another condition, which we choose that the velocit V n of a point at the surface, normal to the surface must be the same as the corresponding velocit of the fluid element v n, at the same point. Expressed b the velocit potential: v n. n We can obtain the velocit on the surface in the same point V n as the velocit in the wave profiles u up and down movements: V n, where u(x,t) = ae kx- ωt ). t If the wavelength, is substantiall larger than the amplitude, we ma replace v n b n v n Appling the last expression for the surface profile: u(x,t)=ce kx- ωt ), and the velocit potential kxt ) kxt) e C cosh( k( h)), dropping the factor e it gives: u (.15) t kc sinh( k kh) ia If we put = (close to the surface) the two equations (.14) (.15) kc sinh( k kh) ia become: i C cosh( k( h)) ag and (.16) i C cosh( kh) ag and kc sinh( kh) ia : Thus C g i g v tanh( kh) a i cosh( kh) k sinh( kh) k k And indeed. (.17) g v tanh(kh) k k We shall then look into two limiting cases: Medium deep water: kh 1 tanh( kh) kh Deep water: kh 1 tanh( kh) 1 This gives the velocities: (.18) Deep water: As stated in the beginning. g v and medium deep water: v gh k

10 Tsunami Phsics 8 3. Estimating the destructive force of a Tsunami This section does not pretend to be a credible estimate of the destructive effect of a Tsunami, corresponding to a certain earthquake in a certain depth at a certain distance from shore, (which is probabl impossible), it is onl an attempt to give a qualitative understanding of the phenomena. If an earthquake triggers a power P measured in Watt along the axis of a clinder, with radius r and height h, then the radial intensit I(r) measured in W/m, at the distance r from the axis, will be the power divided b the area of the clinder. (3.1) P I ( r) rh We shall the tr to estimate the intensit of a tsunami, which is triggered at a distance of km from the earthquake. But first, we shall estimate the power that is triggered b a lift of the seabed, as a consequence of an earthquake. We shall do some prerequisites more or less at random, and the formulas derived ma be used with other initial conditions. Consequentl we suppose that the seabed is lifted h = 1 m, at a circular area with r = 1 km. We put the depth at that place to h = 3 km. This will result in an increase in energ (3.) E mgh V gh The volume of water is water water r h. If the numbers above are inserted, the energ released is V water (3.3) E J If the duration of the earthquake is t =5 min = 3 s, it will correspond to a released power: (3.4) E P = W t We presume that the tsunami propagates from a depth of 1 m to the surface, and we subsequentl calculate the intensit at a distance km from the epicentre. (3.5) P I ( r) rh I 3, ( km) kw / m 4.6 kw / m 6 In deep water the velocit is dependent on the wavelength, but we don t know the wavelength. When the tsunami approaches the shore however, we ma use the formula for the velocit in medium deep water: v gh.

11 Phsics of Tsunamis 9 At a depth at 3. km it gives 17 m/s, and at a depth 1 m it is about 1 m/s. The Power of a force, acting on a bod with velocit v is: P F v It then follows that the Intensit = Power/area is equal to the pressure times the velocit: I p v And if we assume that the tsunami moves with velocit 1 m/s when it reaches the shore, this will trigger a pressure = force/m amounting to: (3.6) p I v 4.6 kw / m 1 m / s.5kn / m Which corresponds to a weight of 5 kg per m, being absolutel terrifing. As mentioned above, the data are chosen much at random, but the calculation ma be carried out with other data giving other results. However, when ou look at a movie, observing the devastating force of a tsunami, the values obtained above do not seem unreasonable. And now we understand better wh!