Thank you for the introduction. It's my great pleasure to be invited to this nice conference with the interdisciplinary atmosphere. I enjoy all the talk.
Today, I'm going to talk about this topic. Before starting my talk, I would like to say I am a kind of a stranger because I have never been trying to investigate the tsunami phenomena so far. As a beginner, I would like to introduce some ideas and concepts that can help us understand the tsunami dynamics in offshore regions.
In the first five minutes, I will talk about the current situation. Mathematics is not a perfect tool to solve all problems, but it sometimes gives us a very strong tool because of the power of mathematics. I can say there are three kinds of powers that mathematics can give us.
So the mathematical research with mathematics is very abstract. It provides us with an abstract concept that can be applied to many problems. And the second one is universal.
The mathematics is good at extracting fundamental elements that lie behind many different problems. So the concept given by the mathematics is very universal that can be applied to many problems. And the third is rigorous. It is rigorous.
So what has been proven mathematically holds true forever. No one can change the result. This is the power of mathematics.
But and so because thanks to these powers so the research with mathematics certainly reveals many important properties in problems we consider. However generally speaking many mathematicians tends to pay attention to purely mathematical problems that are shared in a small community of mathematicians and as a consequence of this they are less interested in dealing with the real world problem through interdisciplinary collaborations with researchers from the other disciplines. On the other hand It's pretty much difficult for researchers of the other disciplines to apply many results, rigorous results, of modern mathematics to their problems, since it is very abstract to understand and to apply it to the problems they consider. So this situation suggests to us that interdisciplinary collaborations with mathematicians and other people, other researchers of other disciplines, are necessary to solve the real-world problem. Regarding the natural disasters and the mathematics, of course, natural disasters such as typhoons, earthquakes, tsunamis, tornadoes, etc. are our great national concern in Japan.
Hence, mathematical research of natural disasters is highly expected now. But on the other hand, Japanese mathematical scientists have not been strongly positive in this kind of interdisciplinary research on natural disasters in spite of its importance so far. After confronted with the destructive damages due to the tsunami hit the Tohoku area of Japan on March 2011, many mathematicians, including me, have a growing interest in the study of natural disasters, including tsunamis.
So, indeed, after the tsunami disasters, I have organized some mailing lists that consist of many mathematicians that have never been tried to consider. the natural disaster problems with mathematics. So this is why I'm here. So this is a part of my answers to this problem. Okay, so but I'm not a typical mathematician in Japanese.
And so let me introduce my standpoint in this talk. So I am not a pure mathematician. I'm a pride mathematician whose specialty is mathematical fluid dynamics.
And my research subject is vortex dynamics. I'm going to explain this talk, but I have never investigated tsunami phenomena so far. On the other hand, but I'm very interested in the application of my mathematical theory, which is a potential flow theory into dimensional, multiply connected domains to tsunami phenomena, especially offshore tsunami dynamics. So in the present talk, I would like to introduce a potential theory and discuss with participants about the future interdisciplinary collaborations based on the potential theory here is the outline of my talk In the first part, actually, so before, I didn't imagine so many mathematicians can take part in this conference. So, I need, the first part is a very review, a brief review, not a complete review, but some review of a survey of mathematical studies of the conventional waterway problems, and in which I would like to clarify some problems to be solved in regard to tsunami dynamics.
and In the second part, I would like to introduce an approach to near-shore flow dynamics, which is a mathematical theory of the potential flow in multiply connected domains and its applications. And with this approach, I would like to propose an idea of the tsunami energy dissipation with the concept of mathematics of vortex boundary interactions. Let's start the first part. Okay. This survey is based on a survey paper by Craig in 2006. In the survey paper by him, he suggested that he divided the tsunami problems into three categories.
In the first category, it is tsunami generation, in which we can see how the shape of the water surface deforms when the feed bed moves instantly due to occurrence of earthquake. This kind of study enables us to estimate the amplitude and wavelength of the tsunami caused by the earthquake. The second category is the tsunami propagation.
In this problem, we are going to investigate how a tsunami spreads for a given waterway profile and the shape of seabed. This study certainly gives us the propagation speed of tsunami, with which one can predict when the tsunami reaches the coastal areas, and many other results are obtained. in terms of this problem.
And the third one, which I am concerned with in this talk, is the near-shore dynamics. Tsunamis generally show a very unpredictable behavior when they approach coastal areas, since the horizontal topography of the flow domain affects the dynamics of tsunamis greatly. This kind of study of tsunami dynamics in offshore domains presents a significant role, I think, in estimating the damages by oncoming tsunamis.
So, these are the points. The potential I'm proposing here is for this kind of problem. Okay, before starting my theory, I'm going to introduce, give you a brief survey about the waterway problems.
The waterway problem has been formulated as a free boundary problem of the equations for the incompressible and invifid fluids with a rotational condition in the presence of the gravitational force. This is a schematic picture of the waterway problems. the shape of the sea bed, which is represented by B of X. And this H represents the mean depth of the water surface, and eta, which depends on the spatial X and time T, is a deformation from the mean depth. And G is the gravitational force acting on this flow domain.
And so we can solve this problem, or flow problem, in this flow domain. The equation is quite simple. Since we assume that the flow is incompressible and inviscid and irrotational. So the velocity field is represented by the potential of the velocity field. This recovers the velocity field by this formula.
And the equation of the potential function can be written in a very simple linear Laplace equation. The equation itself is very simple, but the difficulty arises when we consider the boundary condition for the free surface of y equal to a to the x of t. the equations of the boundary condition at the free surface, and if we have another boundary condition at the seabed, the flow is parallel.
Go along the seabed. So because of non-linearity arising in these boundary conditions, the problem is very difficult to solve. The first step for the mathematical analysis is a very simple linear wave approximation.
A trivial exact solution, which is a flat still water surface where the whole equation has obtained by... This is phi and eta is 0 and... So for the flat bottom, there are no shapes, the bottom is flat. The linearized problem around this flat surface has a following wavy solution, in which k is a wave number of the waves, and omega is a growth rate, a temporal growth rate, which are connected by this famous dispersion relation, from which... We can derive the propagation speed of the water wave as a force.
This graph shows the propagation speed of the wave versus the wave numbers from 0 to 5. And as you can see, for short wave lengths, which is a large wave number, the speed is almost like, almost behaves like square root of k over z, which depends on the wave's numbers. I'm going to go ahead and On the other hand, when we consider the large wavelengths, which correspond to the short wave numbers, the speed is almost like square root of g h, which is independent of the wave numbers. And when we compare these two values like here, the most fastest speed derives when the wave number is very small.
So this indicates that short waves travel slowly, where the long waves propagate much faster. first. By using this theoretical result to the boxing day tournament, which is in 2004 in the Indian and Southeast Asian territory, 2004, the speed of the wave is like 720 kilometers per hour.
This quantity is a very nice agreement with observation. The second step is that we can take part, we include the non-linearity of the effect. So here, there are many, many studies about the non-linear effect of tsunami phenomena.
I'm going to show you one of them. So the Euler equation estimates this kind of solitary wave equations. Oh, this is a paper, this is from a paper by Craig and his colleagues in 2006. And the Euler equation estimates this kind of sort of wave solution.
and they consider the head-on collision of two solitary waves. This is a numerical computation. At the initial moment, there are two peaks, solitary waves, and they approach each other and then collide at this moment.
As you can see here, the peak height grows very rapidly, and then it disperses like this. So when we cut this solution in this direction, before Corrigion then Then the two solar waves approach each other and then merge in one point, and then after that it goes away. But the profile of the wavy solution is different because there is some strong residual wave appears after the collision.
So in this paper they conclude that feed-thrift and energy loss occur via head-on collisions. This is one of the mechanisms to loss energy of the tsunami. Okay, but it is usually very difficult to get information about the tsunami phenomena.
So in order to obtain more information on the water wave dynamics, we need to consider the order equation. itself. However, the nonlinear character of the Euler equation for the propagation of ocean waves makes its mathematical analysis very difficult. So, conventionally, we consider the two-scale limit to derive weakly nonlinear models for the water wave for the for the sake of further mathematical analysis. So here I'm showing you some examples, typical examples, but we introduce a two scaling parameter, alpha and beta.
Alpha is h over a, h is a mean depth, a is an amplitude of waves. And beta is a l over h, and l is a length of waves. So then we consider two scale limits.
The first one is alpha grows like a beta square, and both of them are very small, which is called a dispersive nonlinear. regime. The other one is alpha is like beta and both of them are very small.
Then it is called the weakly nonlinear shallow water regime. For example, when we compute these kind of parameters for the boxing day tsunami in 2004, alpha is like 10 to the minus fourth and beta is 10 to the minus second, minus two, which means that this kind of wave is like in the dispersive wave regime. So in this phenomena, the dispersion nonlinear regime is effective for the mathematical analysis. So I'm going to show you the two limit water equations. And when we take a shallow water scale limit like this, then we have obtained the shallow water equations.
This is in which the wavelength of the wave is much longer than the depth. These are the boundary conditions. And so what we have to say, the mathematicians do some works on this equation by justifying the 2D shallow water equations. These are the authors justifying this equation.
Mathematical justification means the solution of the 2D Euler equations is converged to the solution for the Hello-Water equations as alpha tends to 0. This kind of justification is a very mathematical... point of view and a mathematician concerned the mathematicians and justification for the 3d problem has been studied very recently by these researchers and This shall all the equation can be applied to the tsunami generation by considering by by replacing the second equation and by the temporal deformation rate of a seed bed and This equation are used as a computational model for the tsunami generation due to earthquake Of course, EDC is one of the examples. application of shallow water equations.
And you can see many shallow water equations in this conference. So there are many wide variety of applications about these shallow water equations. The other one is very long wave approximation, where we take the scale limit of where is the square is goes to zero. Then we obtain the linear wave equations, which admit this kind of wave equations.
This kind of linear wave equations are used as a computational model over tonight. propagation in the open sea. And when we plug into this and that into the equation, we have obtained the evolution equation for the wave shapes, which is known as the Kotov-Beck-Dufferis equation for the weather wave equation, weather wave shapes. And so this mathematical justification of this kind of approximation has been considered by many mathematicians.
And here is a weak note about this equation. If the parameter mu equals one-third, the equations are reduced to the historical inviscid Burgess equation. This permits a shockwave equation. This is a kind of explanation why the tsunami has a very sharp profile when time proceeds. Okay, the last problem.
is the offshore dynamics. The problem is very complicated, very difficult to deal with the waterway problem because the complex topography of coastal flow domains makes it difficult to deal with the waterway equation in offshore domains. mathematically so I'm gonna explain I'm gonna show you some research by these guys saying the importance of vortex structures which is which is a critical macro macro-vortices is defined as a topographically induced vortices in coastal areas.
So this is excerpt from their paper. So this is a water wave here, and the wave coming from this direction to this way. And as you can see, in the size of the water break, as time goes A, B, C, D, you can see a very circular flow streamline structure and detached from the tip of the border break and this vortex structure can interact and interact and very and show a very complex behavior so they conclude that it is very important to deal with this kind of interaction vortex structure interactions in order to understand the offshore tsunami dynamics so this study says interaction between macrobots is important to understand offshore tsunami with dynamics, but this is difficult.
It's naively difficult to manage by conventional waterway problems. So a new mathematical approach is required to consider this kind of problem. That's why I'm introducing a new approach to deal with near-shore dynamics here in the second part.
Okay. So in the second part, we are considering a multiple connected domain. So everyone does not know about this, may not know about the name of this domain, so I will explain it very briefly.
The connectivity is a very, very important geometric concept that characterizes domain, which is coming from one of the mathematical subjects, field topology. So we can see the four kinds of... domain, we need to identify each of the, what we call, this kind of four domains. When we compare the domain A and B3D, the difference is very clear because in the domain A, the two domains are separated.
And B3D, they are only one domain. So in this sense, we call these three domains are connected to domain and this is disconnected domain and the second one When we compare B, phi, D, there is a difference. In B, there is no inner boundaries inside the domain, which we call the simply connected domain. In C and D, there are some holes in the domain, which we call the multiply connected domains. Depending on the numbers of the holes inside of the circle, we can say that C has a doubly connected region, and D has a quadruply connected region.
So when we consider the coastal flow problem, for example, this is Osaka Bay. So you can see a very big island and a very big airport here. And here is some islands along the seashore. So when we consider the water problem in this kind of domain, this kind of domain is usually represented as multiply-connected domains.
This is why we need to consider multiply-connected domains mathematically. Okay. So let me introduce the basic equation. So in here we also assume that the flow is two-dimensional but the difference is the waterway problem we consider the vertical direction but in here we consider the horizontal direction but 2D and the flow is inviscid and incompressible which are the same condition for the waterway problems but the difference only one difference is we do not assume the irrotational condition. In the waterway problem, we assume that there is no vortex structure, but in here we consider the vorticity, which is introduced from the velocity by this formula, omega, and here we introduce another function, which is called string function phi, that recovers the velocity field by this relation.
Then we have obtained the evolution equation of the vorticity here, and these two functions are connected through the Poisson equation. According to the mathematical analysis of the two-dimensional Euler equations, we have obtained the very famous Lagrange's theorem, in which, without external force and boundary in the domain, vorticity, omega, is conserved in 2D Euler equations. This is a mathematical result, and this theorem indicates that It's sufficient to consider vortex structures that exist at the initial moment. In other words, vortex structures never emerge nor dissipate in the 2D oil flows without external force and boundary.
So, in order to deal with the problem for this mathematical setting, the evolution of 2D flows can be expressed in terms of vortex dynamics, whose equation I would like to introduce here. Here is, this is a strategy on how to derive the equation of the vortex dynamics in multiply connected domains. So, since we are considering a two-dimensional space, we identify the two-dimensional space as a complex plane, which is Z-plane. For a given domain in the Z-plane, which is, which has, which is a multiply connected domain here with three irons, but this is generally very difficult, and the boundary, shape of the boundary is...
is generally very difficult to describe. But the strategy is, when we construct some mapping from this domain to another very high-symmetric circular domain in the other complex zeta plane, but the multiplicity is the same. There are three islands. But the shape of the boundary is circular, which is very high-symmetric, which acquires higher symmetry than this given domain. And suppose that There are n-vortex structures in this domain, and these n-vortex structures are mapped into the other n-vortex structures in this circular domain.
So, here's the strategy is, we are going to construct the evolution equation of the vortex structures in this canonical circular domain, and then modify the equation for this domain so that it describes the evolution vortex structures in the target domain, with using this conformal mapping. This is a strategy. So I'm now focused on the derivation of the equation for this circular domain.
And this is a very mathematical part. So here I'm going to introduce the Schottky-Klein-Plein function. And so before studying the definition of this function, we have to give you an exact definition of the circular domain.
circular domain is a domain in the complex plane inside the unit circle that contains m circular holes. This is an example of the circular domain with three holes. The boundaries of the circular obstacles are denoted by cm, in this case c1, c2, c3. And the configuration of these three islands are identified and characterized with the centers of these circles and radii of these circles. And using these kind of parameters, we can define a transcendental function, which is called a Schottky-Klein-Plein function.
associated with the circuit domain. This is the definition. This is the representation of this function. The definition is quite very difficult and very technical, so I hope you don't care about the exact definition.
But anyway, this is a complex valued function which can be used to describe fluid evolution. in the multiply connected circular domains. I'm going to show you the example.
The Schottky-Klein-Plein function is used to describe analytic formula of the complex potential of many fluids, fundamental many elements, in multiply connected circular domains. I'm going to show you the three examples. The first one is the uniform flow. When we use the Schottky-Klein function, omega, like here, then we can represent this. Complex potential represents a uniform problem.
I'm going to show you some examples later. And the second one is a pair of source and sink, in which an M is the strength of the source and sink pair, and alpha and beta are the locations of the source and sink, respectively. The last one is a representation of the vortex structure, in which a kappa is the strength, and alpha is the location of the point vortex.
When we use this analytic representation, we can very easily... reconstruct the flow fundamental flow field in the multiplicant domains. So for example when we use the first analytic formula we can construct the uniform flow coming from this direction in the exterior domain with five holes.
This is an example of the streamline of the pair of source and sinks. As you can see here there is a source here and a sink here. the streamline going out this direction and then go back to the another sink. The third example is a point vortex.
As you can see, there are four point vortex structures in the multiply connected circular domains. All of these streamlines can be constructed by using the analytic formula. This analytic formula can be applied to many problems. For example, when we apply the... formula to the river flows which is a multiply connected channel flows we can construct the uniform flow and a pair of source and sink and the point vortex can be constructed by using a short key client prime function so even using even though the the flow is not circular and the flow domain is not circular domain by using the conform mapping technique we can construct the flows are the many areas in two-dimensional space.
So this is an example. So this is an aerial picture of Toyohira River in the city of Sapporo. When we extract the boundaries of these domains and applying the numerical conformal mapping technique and the Schottky-Klein prime function, we can construct the uniform flow and the point vortex flow and the source and sink pair.
So in this sense, the potential theory can be used to represent real flow problems. So let's now move to the derivation of the vortex dynamic. Equation of vortices, which is in some sense a model of interacting macro vortices proposed by the preceding papers in multi-connected domains, is obtained with the Schottky-Klein prime function. This theorem, mathematical theorem, shows the equation of the point vortices in canonical circular domains.
I'm going to read this. Let zeta lambda for n-point vortices are located in the domain d-veta, and for the strength kappa lambda, then the interacting z-point vortices is given by this equation. In here, we use the Schottky-Klein-Planck function here and there. Based on this, following the strategy I'm showing in the first slide of the second part, I can derive the equation of motion for the target domain, for arbitrary domains, by constructing the conformal mapping, which are given by this theorem too.
The rate Z lambda for lambda equal 1 through n denotes the rate of motion. Locations of point vortices with strength kappa lambda in the multiply connected arbitrary domain VV. Assume that we can construct conformal mapping from this target domain to the circular domain, and the point vortices are mapped. to Z lambda in the canonical circular domain. Then the motion of the point vortices in the target domain is given by this.
The evolution of the point and vortex structures consists of two parts. The first part is the evolution equation for the endpoint both C's in the canonical circuit domain with some scaling factor and there are additional term coming from conform mapping reconstruct so this equation can be used to describe the vortex dynamics in the many multiply connected domains I'm going to show you some example the most simplest case is we consider a single vortex structure. So Johnson and MacDonald, the two guys, have been considering the evolution of a single vortex. So in this equation, this picture shows the three islands, which one is a circular, but one is a gap in a slit-like region.
These each curves represent the orbit of the one vortex point. As you can see here, when a... vortex point starting from here goes through around this line and then goes away.
And on the other hand, when the point vortex starting from here then goes around this direction and go this direction. But it is important to see that the vortex can intrude, can get into this direction, but through the gaps, because of the existence of this kind of separatrix orbit. Because of the existence of this kind of structure, the vortex point can get into this region.
Here's another example by a vortex trajectory near two islands. So there are two pictures. You don't see any difference between these two figures, but there is a very big difference. For example, When we pay attention to this trajectory, the one-point vortex can go around the two islands.
But on the other hand, when the distance of these two islands are separated away a little bit, this kind of structure has collapsed, so there is no vortex trajectory that goes around the two regions. indicate that the topography of the flow field can affect the vortex dynamics. This is the most simplest case.
The next one is the topography and the vortex dynamics for a vortex pair. Vortex pair is a vortex, two-point vortexes, which has the opposite sides with the same strength. Without boundaries, These two vortex structures can go like this.
But with boundaries, the point vortices show a complex behavior. This is an example of the vortex trajectory. When we consider a symmetric domain like here, so the two irons...
the symmetry with respect to this real axis so i can this picture shows that the vortex trajectories of two point vortices for example when we take the point borders here and the other one here the two vortex is goes like along this line with the pairing and in order to see the structure of this flow flow trajectories it is we must mention sometimes simplified the the representation of the flow, which we extract the capital. characteristic structure in this streamline field, then we represent this kind of one is like represented by this one. This corresponds to this island, and this point corresponds to this point. And by showing this kind of topological representation, we can consider the classification of trajectory patterns and transition between these patterns when we change the distance of two islands and the size of two islands.
And here is a mathematical a description. So the one trajectory can change through this bifurcation and change this kind of global transition, and then here. So we can show that the trajectory of two-point vortexes can be represented among these five patterns. This is another example for the application of the vortex dynamics to vortex pairs.
And on the other hand, if we consider the asymmetric topography, like here, the trajectories of a pair of two-point vortices in the multiplicant domains looks very complicated. As you can see. See here the trajectory looks very complicated and you can it is very hard to predict the future future positions of these two-point vortices. So generally speaking the evolution of many vortices became chaotic, which is unpredictable behavior.
But even though the solution is very chaotic, but we have still have a very nice tool called the dynamical system theory that can be applicable to understanding of these complex phenomena. In the second part, I'm going to show you some mathematical theories to deal with flow problems in multiply connected domains. Based on this potential theory, I would like to show and propose an idea about and discuss about an idea to dissipate the tsunami energy. Okay, the present talk is based on the current project, Crest project.
funded by Japan Science and Technology, that I am reading. And the title of this project is The World of Problems Shift, Created by Mathematics of Vortex Boundary Interactions, starting from 2010 through 2016. And in the project, we are carrying out mathematical and numerical analysis of vortex boundary interactions in multiply-connected domains. Based, we have developed the potential of vortex dynamic theory in multiple domains that I have explained in the second part, and combined with a large-scale numerical computation and some new topological method to represent the flow field. And with these three, applying these three topics, we developed some new approach to the sports science and environmental flows and a new design of vortex wind turbines.
So in this project we are proposing use of the vortex structure can make can give us a new idea of to solve the problem. So based on this concept I would like to discuss how to dissipate the energy of tsunami. Okay here is a quick review of a mechanism of both creation which is known as boundary radius even though we are considering The two-dimensional inviscid irrotational model, the situation has changed when there is an object in the boundary, I mean the object inside the flow domain. So this is a schematic picture, how to generate the vorticity.
So in the object, the flow is coming from this direction, but because of the existence of this object, the flow field has to be slow. And because of the difference of these two velocity fields, this kind of... rotational flow can be created from this boundary.
This is a very simple explanation why the boundary layer forms. So this is an experimental result. There is a boundary here and you can see around the red region inside the red region you can see a very complex turbulent boundary layer.
But in here we can you can see this is very laminar which is very can be an approximate inviscid and potential problem. Here is a numerical example, shows that here are two boundaries here and there, and as you can see, a red region and a blue region represents a vorticity area. This numerical computation indicates that vorticity are created from the boundary, and then it gets into the flow domain like this. So, even if we assume that the flow is irrotationally inviscid in two-dimensional space, Because of the existence of the boundary boundaries we need to get into take into consideration of the vortex creation Okay Of course the mechanism of The creation of the macro-vortexes from the water breaks has is partly due to the existence of boundary radius And this generation of the vortex structures has a positive or negative effect to the real fluid problems.
For example, when we consider airplane flight, the flows around the wing of airplanes can be represented schematically as this. This kind of situation has been theoretically studied by the conventional wing theory. In this theory, we can say the upward lift is obtained from the uniform flow.
for example this is schematic pictures when the one object is in the potential flow When we have the rotation flow around the wing, then it gains upward rift. This is a mechanism how we get the upward rift from the viewpoint of the wing theory. But, on the other hand, if... The angle of attack is getting steeper like this. In the upper side of the boundary, there is a boundary layer, and then it sheds many vortex, turbulent many vortex structures.
gives us the difference between this ideal situation. And as a result of this vortex creation, the stall phenomenon occurs. So this indicates that in the airplane flight, generation of vortex...
rupture from the wing causes the loss of the lift which is a bad effect of the vortex generation here is another example in the true converter in automobile transmissions this is a cut model of the automatic transmission in automobiles when we magnify this region into this one we can see a true converter which consists of a pair of rotating fans the fluid flows is fused inside of this transmission, so this part, this device can transmit the torque generated by the engine to the shaft of the axle of the automobile. So this is designed so that the engine shaft is rotating faster than the axle of the automobile, which is we accelerate the car, then the torque transmission is smooth. This is because design is the true combinator.
body design so that the torque transmittance moves but on the other hand when the axle of the automobile automobile is rotating faster than the engine shaft which is the situation when the engine braking is in effect the performance of torque transmission is getting worse this is due to the generation of vertices in the fans between fans and these creator vortex structures are interrupt between these fans which results in the energy dissipation of the flow. So in this example indicates that the energy loss occurs because of the generation of the vortex structures inside of the fans. On the other hand, generation of vortices has a very good effect for the insect fright.
So this is a model study of butterfly fright by these Japanese guys in 2001, one in which they consider two flapping plates that generate the vortex shedding from the tip of the plate. This is a numerical computation. When the plate is going upward to downward then the because of the existence of the boundary the vortex structure are generated from the tip of the wing and this is the bone structures and then it goes upward then there's another board structure appears and then merges and then repeating these processes there are very complex vortex structures but this vortex configuration can give an additional lift to the butterfly wing so in the insect flight problem we can say that vortex structures are shed from the dropping wind and butterflies obtain additionally from the vortices. So in this sense, the generation of vortex structures is necessary for the efficient flight, for the insect flight.
Here is another example for the efficient use of vortex structures. So this is falling of the rotating plant seeds. These kind of plant seeds can fall with rotation like this.
According to the numerical computation and the particle image velocimetry measurement, for example, in this slide, this is a two-dimensional cross-section with a velocity field. You can see a very wide range of blue region above the wing. This blue region represents the vortex structure. And here is another this particle image velocimetry measurement indicates that there is a very huge vortex structure and entrapped above the wing.
Because of the existence of this entrapped vortex structures, these plant seeds can gain additional lift upward. And as a result, as a consequently, it gains initially and they fall very slowly. And it has a wider chance to spread horizontally because of the slow descent. So in this sense, seeds are falling very slowly by entrapping a big vortex structure above it. So this is the and I'm example.
The effective use of the vortex structure setting from the boundary. Okay, this kind of idea has been applied in many flight problems. For example, the most famous one is the delta wing structure.
This kind of triangular like wings goes this way. The vortex structure appear and generate from the edge of the boundary. And this vortex structure can gain additional lift to the delta wing.
Here is an example of the vortex structure. another conceptual pictures given by Rafaano Cho in 1981. They consider the Jugo-Suki wing, which is a model of the wing of an airplane. When they consider the entrapped vortices above the plate, it gives this wing the additional lift.
So in this study set, it's also possible to obtain the additional vortex induced for the lift for the wings. Based on this idea, an engineer, Casper, proposed a new design of a wing, which is known as a Casper wing, which is a US parent in 1978. And the basic idea, rising the Casper wing, is explained in this schematic picture. The Casper wing consists of the one main wing attached with the two auxiliary flaps in which he argued that we can possibly interrupt the vortex structure like here and there by adjusting a two flaps very efficient way.
But, okay, so he said the lift can be enhanced by interrupting both around the wing. But unfortunately this kind of wing design have not been realized in the commercial basis because of the lack of of the theoretical study of this kind of flow. The difficulty lies in this domain is represented by the multiple-connected domain. At this time, the potential theory in multiple-connected domains are not available.
Now we can do this problem by using the potential theory I am explaining in the second part. Here is a brief summary of the idea. So I have said, and some people said, macro-vortices are created by the topographic effect in near-shore region.
And on the other hand, a potential theory of mathematics or vortex-boundary interaction gives us entrapped configuration of these created vortex structures. And as I have shown in the AT transmission structure, The interrupted vortex structures can dissipate the flow energy due to the viscous friction, vorticity friction. So from this fact, we can imagine that the interruption of macro vortices behind the obstacles in near-shore domain can enhance the energy dissipation of tsunamis.
This is an idea proposed by these observations. Okay, in order to see how it's possible, I'm going to show you a very simple example to... to give you the stationary configuration of behind two plates. In the model, we consider two plates to be of equal lengths, which are slanted at the angle phi, and they are separated by the distance 2s.
And suppose that there exists a uniform flow, which represents a current of the flow line, and there is a speed of u and a parallel to the rear axis here. and two axial point vortices which represent, which is a model of the macro vortices behind the two plates. And the question is can we interrupt the vortex, these axial vortex structures behind this object? The solution is here.
By applying the potential theory I have proposed in the second part, we can derive the many configurations of the vortex, axial vortex structures behind the plate. This picture shows an example. As you can see there are two parallel plates here and then there is another big actual vortex structure here. This configuration is stationary which is fixed.
So this suggests that entrapment of two actual vortices behind two plates is possible. Okay so based on this observation here is an idea for the energy dissipation through the vortex generation. So in the paper by this guy, it shows energy dissipation of waves is mainly due to viscous friction of vortex structures. And so I have obtained a station vortex structures entrapped behind the obstacles can be a source of effective energy dissipation or one of the effective. ...dissipation.
So that I mean there is the two obstacles and then coming from the current then there exists a very big macro vortex structure which are fixed and because of the existence of this entrapt vortex... structures, the bollistic dissipation due to the viscous friction in large bollic structures, which may result in energy dissipation of the plographically induced macrobot in offshore coastal flow domains. This is a kind of kind of a trial, not exact mass studies for dealing with the tsunami phenomena, but what I'm now proposing here is a new application of the mathematics of boundary, both boundary interactions using the potential theory I have used. Okay, let me summarize my talk. In the first part, I'm going to give, I gave a survey of mathematical studies for tsunamis and in which the mathematical studies of the waterway problem has been...
successfully providing us with useful information on tsunami dynamics. This is a very striking result. There are many striking results on tsunami studies.
And the linear and non-linear equations of water wave problems are used as a computational model of tsunami generation and propagation. Also offshore dynamics. But regarding offshore dynamics, which is very complicated, this is a very important problem to be answered with mathematical studies in future.
One study indicated the importance of macro-vortices, which are topography in these both structures. So in the present talk, based on this observation, I will give you a potential theory for theory in multiple domains that can be applicable to the interaction of macro-vortices. I hope the theory has the potential to investigate interaction of macro-vortices in offshore coastal domains. And here I am proposing a mathematics about boundary interactions, this may be a key to realize efficient, this may give us some idea to realize efficient energy dissipation of tsunami energy through the viscous dissipation of vortex structures.
This note brings us to the end of my talk. Thank you for your attention.