Lund, Fernando

Umeki, Makoto

Kyoto University Research Information Repository

Title Scattering of dislocation in shallow waterAuthor(s)Umeki, Makoto; Lund, FernandoCitation数理解析研究所講究録 (1995), 908: 40-48Issue Date1995-05URL http://hdl.handle.net/2433/59504RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityScattering of dislocation in shallow waterMakoto Umeki $a$ and Fernando $\mathrm{L}\mathrm{u}\mathrm{n}\mathrm{d}^{b}$$a\mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}$ of Physics, University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo, 113 Japan$b\mathrm{D}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}_{0}$ de Fisica, Facultad de Ciencias Fisicas $\mathrm{y}$ Matem\’aticas,Universidad de Chile, Casilla 487-3, Santiago, ChileAbstractInteraction of surface waves in both shallow water and deep water with a verticalvortex is studied analytically. A dislocated wave may exist on the vortex and itsstrength (degree of dislocation) is characterized by the circulation of the vortex andthe frequency and speed of the wave. Using an analogy between Aharonov-Bohm effectin the quantum mechanics and this hydrodynamic system, a scattering problem withthe incident dislocated wave is solved and scattering amplitudes are derived.1 Shallow waterWe consider a scattering problem of $\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{U}\mathrm{o}\mathrm{W}$ water waves of inviscid incompressiblefluid by a vertical vortex. The coordinate system is expressed by $(x,y)=x$ in thehorizontal direction and by $z$ in the vertical. The velocity and the surface displacementare denoted by $v(t, x, z)=(v_{\perp}, w)$ and $\eta(t,x)$ , respectively.The equation of motion is$\partial_{t}v+(v\cdot\nabla)v=-\rho^{-1}\nabla p-g$, (1)where $\rho$ is a density of the fluid, $p$ a pressure, $g$ an acceleration due to gravity, $\partial_{t}=$$\partial/\partial t$ and $\nabla$ a three-dimensional gradient. The kinematic boundary condition at thesurface is given by$w=\partial_{t}\eta+v\perp\cdot\nabla\perp^{\eta}$ (2)where $\nabla_{1}$ is a horizontal gradient.In the shallow water case we may write the velocity as$v= \sum_{n}v_{n}(X,t)(\frac{z}{h})^{n}$ . (3)The equation of continuity $\nabla\cdot v=0$ leads to$h\nabla_{\perp}\cdot v_{1n}+(n+1)w_{n+1}=0$, $n=0,1,$ $\cdots$ . (4)At the lowest order, the expressions of the velocity and the pressure become$v=(v_{\perp}(x,t),$ $\frac{z}{h}w(x,t))$ , (5)908 1995 40-48 40$p=p \mathrm{o}(X,t)+\frac{z}{h}p1(x,t)$. (6)From $z$-component of (1) and the pressure condition at the surface, we have$p=\rho g(\eta-z)+pa$ ’ (7)where $p_{a}$ is the atmospheric pressure. The $x$-component of (1) is written as$\partial_{t}v_{1}+(v_{\perp}\cdot\nabla_{\perp})v\perp=-g\nabla_{1}\eta$ . (8)The kinematic boundary condition leads to$\partial_{t}\eta+h\nabla\perp\cdot v_{1}+\nabla\perp(\eta v\perp)=0$ . (9)The existence of vortices with a non-zero total circulation produce a steady solenoidalbackground flow $U(x)$ and a surface deformation $\eta \mathrm{o}(x)$ . Substituting $v_{\perp}=U(x)+$$u(x,t)$ and $\eta=\eta_{0}(x)+\eta_{1}(x,t)$ into (8) and (9) leads to$-g \nabla_{\perp\eta_{0}}=(U\cdot\nabla_{\perp})U=\frac{1}{2}\nabla_{\perp}U^{2}-U\mathrm{X}$ rotU, (10)$\partial_{t}u+(U\cdot\nabla_{\perp})u+(u\cdot\nabla\perp)U+g\nabla_{1\eta_{1}}=-(u\cdot\nabla_{\perp})u$ (11)and$U\cdot\nabla_{\perp}\eta 0=0$ , (12)$\partial_{t\eta_{1}\dagger}h\nabla_{\perp}\cdot u+\eta_{01}\nabla\cdot \mathrm{u}+(u\cdot\nabla\perp)\eta 0+(U\cdot\nabla\perp)\eta 1=-\eta 1\nabla\perp\cdot u-u\cdot\nabla_{\perp^{\eta_{1}}}$ . (13)(10) and (12) are equations for the background field. Using (10), we can express $\eta_{0}$ by$U$ . Linearizing (11) and (13) with respect to $u$ leads to$\partial_{t}u+(U\cdot\nabla\perp)u=-(u\cdot \mathrm{v}_{\perp})U-g\nabla_{\perp}\eta 1$ , (14)$\partial_{t}\eta_{1}+(U\cdot\nabla\perp)\eta_{1}+h\nabla_{\perp}\cdot u=-[\eta 0\nabla\perp\cdot u+(u\cdot\nabla\perp)\eta 0]$ . (15)Taking the divergence of (14), we obtain$\partial_{t}\nabla_{\perp}\cdot \mathrm{u}+g\triangle_{1}\eta 1=-\nabla\perp\cdot[(U\cdot\nabla_{\perp})u+(u\cdot\nabla_{\perp})U]$ , (16)where $\triangle_{1}=\nabla_{1}^{2}$ . Using an alternative expression of the right hand side of (16),$\nabla_{\perp}\cdot[(U\cdot\nabla_{1})u+(u\cdot \mathrm{v}_{\perp})U]=2(\partial_{i}U_{j})(\partial jui)+(U\cdot\nabla 1)(\nabla_{\perp}\cdot u)$,we have$D_{t}\nabla_{\perp}\cdot u+g\triangle\perp\eta 1=-2(\partial iU_{j})(\partial jui)$ , (17)41where $D_{t}=\partial_{t}+U\cdot\nabla_{\perp}$ Taking $D_{t}(15)-h\cross(17)$ , we have$D_{t}^{2}\eta_{1}-\mathrm{c}^{2}\triangle\perp\eta 1=-D_{t}[\eta 0\nabla_{\perp}\cdot u+(u\cdot\nabla_{\perp})\eta 0]+2h(\partial_{i}U_{j})(\partial jui)$, (18)where $c=\sqrt{gh}$ is a phase velocity of shallow water waves.We consider the case that the Mach number $M=U/c$ is much smaller than 1. Thesquare of $M$ is also called Froude $\mathrm{n}\dot{\mathrm{u}}$mber. We denote a typical length scale of vortexby $a$ and a wavelength and a frequency of shallow water waves $\lambda$ and $f$ where $c=\lambda f$ .We also assume that $a/\lambda\equiv\beta\gg 1$ . Then the right hand side of (18) will be order of$M$ or $\beta^{-1}$ compared with the left hand side. Neglecting them, we have a final equation$D_{t}^{2}\eta_{1^{-}}c\triangle_{1}2\eta_{1}=0$ . (19)The localized vortex with the circulation $\Gamma$ produces the background flow $U\approx\Gamma/(2\pi r)\hat{\theta}$ .The assumption $\beta\ll 1$ is complementary to the condition for the Born approximation(e.g. Kambe (1982)) to hold.2 Deep waterWe may treat the scattering of dislocated waves in deep water as follows. First, wewrite the velocity, the surface displacement and the pressure as$v=(U+u, w)$ , (20)$\eta=\eta_{0}+\eta_{1}$ , (21)$p=P+p_{1}$ , (22)where $U,$ $\eta 0$ and $P$ denote a steady field due to a vertical vortex and depend onlyon $x$ , and surface-wave components $u,$ $w,$ $\eta_{1}$ and $p_{1}$ are functions of $x,$ $z$ , and $t$ . Thesteady flow field satisfies the same equations as (7), (10) and (12).The linearized equations of motion are given by$\partial_{t}u+(U\cdot\nabla_{1})u+(u\cdot\nabla_{1})U=-\rho^{-1}\nabla_{\perp p_{1}}$ , (23)$\partial_{t}w+U\cdot\nabla_{1}w=-\rho^{-1}\partial_{Z}p_{1}$ , (24)and the equation of continuity is$\nabla_{\perp^{u+}}\partial_{z}w=0$ . (25)The boundary condition at the surface is$w=\partial_{t}\eta_{1}+U\cdot\nabla_{\perp\eta_{1}}+u\cdot\nabla_{\perp^{\eta 0}}$ (26)42and$p(z=\eta)=p_{a}$ . (27)Here we may assume that the wave components are given in a form of separationof variables as$u=u_{0}(x,t)+\hat{u}(x,t)\cosh k(z+h)$ , (28)$w=\hat{w}(X,t)\sinh k(z+h)$ , (29)$p_{1}=p\mathrm{o}(_{X,t)(x}+\hat{p},t)\cosh k(z+h)$ . (30)Substituting these into the equations of motion leads to$D_{t}u_{0}+(u0^{\cdot}\nabla_{\perp})U=-\rho^{-}\nabla_{\perp}1p0$ , (31)$D_{t}\hat{u}+(\hat{u}\cdot\nabla_{\perp})U=-\rho-1\nabla_{\perp}\hat{p}_{1}$ , (32)$D_{t}\hat{w}=-\rho^{-1}k\hat{p}1$ . (33)The continuity equation reduces to$w=-k^{-}1\nabla_{\perp}\cdot$ \^u, (34)$\nabla_{\perp}\cdot u_{0}=0$ . (35)Thus the flow $u0$ is incompressible. The kinematic boundary condition is replaced by$\hat{w}(x)\sinh kh=D_{t}\eta_{1}+(u_{0}+\hat{u}\cosh kh)\cdot\nabla\perp\eta 0$, (36)where we assumed $k\eta\ll 1$ . The pressure condition becomes$-\rho g\eta_{1}+p0+\hat{p}1\cosh kh=0$ . (37)Here we assume $\beta\gg 1$ and $M\ll 1$ . Then we can neglect the second term in the lefthand side of (31), (32) and in the right hand side of (36). Under these assumption, itcan be shown that $p_{1}$ satisfies the Laplace equation. Then we have $\triangle_{1p_{0}}=0$ . Takingthe laplacian of (37) leads to$\triangle_{\perp\hat{p}_{1}}=(\rho g/\cosh kh)\triangle\perp\eta_{1}$ . (38)Taking the divergence of (32), neglecting $\mathrm{O}(\beta^{-1})$ term, and eliminating $\nabla_{\perp}\cdot u$ by using(34) and (36), we have the same equation as (19) with $c=\sqrt{(g/k)\tanh kh}$, which is aphase velocity of deep water waves.433 Dislocated wave and scatteringAs is already pointed out by Berry et al. (1980) and Cerda and Lund (1993), the equa-tion (19) possesses a close analogy to the $\mathrm{w}\mathrm{e}\mathrm{U}$-known quantum mechanics of Aharonov-Bohm effect, in which a potential gives physical effects without accessible electromag-netic fields. It is a scattering problem of a beam of particles with charge $q$ and mass$m$ incident normally on a long thin cylinder containing a magnetic field $B(x)$ parallelto its axis. The Schr\"odinger equation in the presence of the magnetic vector potential$A$ due to the magnetic field is given by$\frac{1}{2m}(-i\hslash\nabla-qA(X))2\psi(x)=\frac{\hslash^{2}k^{2}}{2m}\psi(x)$ , (39)where $h$ is a Plank constant, $A(x)=(\Phi/2\pi r)\hat{\theta},$ $\Phi$ is a magnetic flux and $\hat{\theta}$ is anazimuthal unit vector.Equations (19) and (39) have a solution of a dislocated wave of the form $\exp[-\dot{i}(k\cdot$$x+\alpha\theta+\nu t)]$ , where $\alpha=\nu\Gamma/(2\pi c^{2})$ in the fluid mechanics and $\alpha=-q\Phi/h,$ $(h=2\pi h)$in the quantum mechanics. It is noted that, while this dislocated wave is an exactsolution in quantum mechanics, it is an approximate one in the water wave problemvalid if $M\ll 1$ . We are now interested in the case that the effect of dislocation issignificant, i.e. $\alpha=$ 0(1). Using $\Gamma=2\pi Ua$ , we have a relation $\alpha=2\pi M\beta$ .As an example, we consider a scattering problem by a circular uniform vortex withvorticity $\omega$ and a radius $a$ surrounded by an irrotational flow. Using polar coordinates$(r,\theta)$ , the background flow is given by$U= \frac{1}{2}\omega r\hat{\theta}$, $r\leq a$ ; $\frac{\mathrm{I}}{2\pi r}\hat{\theta}$ , $r>a$ ; $\Gamma=\pi\omega a^{2}$ . (40)Inside the vortex we have from (19)$[(\partial_{t}+(\omega/2)\partial_{\theta})^{2}-C2(\partial_{r}^{2}+(1/r)\partial_{r}+(1/r^{2})\partial_{\theta}^{2})]\eta 1=0$ . (41)Assuming the solution of the form $\eta_{1}=\sum_{n}\tilde{\eta}_{1n}\mathrm{e}^{i(n}\theta-\nu t$), we obtain$( \partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{n^{2}}{r^{2}}+k_{n}2)\tilde{\eta}1n=0$ , $k_{n}= \frac{|\nu-n\omega/2|}{c}$ . (42)Equation (42) has solutions of Bessel and Neumann functions. The non-singularity atthe origin will exclude the latter. Thus we have$\eta_{1}=\sum_{n}a_{n}j\mathrm{I}n|(k_{n}r)^{i}\mathrm{e}(n\theta-\nu t)$ . (43)Outside the vortex, $r>a$ , the assumption $U^{2}/c^{2}\ll 1$ may reduce (19) into$[ \partial_{t}^{2}+\frac{\Gamma}{\pi r^{2}}\partial_{\theta}\partial_{t}-c(2\partial_{r}2+(1/r)\partial_{r}+(1/r2)\partial\theta 2)]\tilde{\eta}_{1n}=0$. (44)44Assuming the above form of solutions, we have$( \partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{n^{2}+2n\alpha}{r^{2}}+k2)\tilde{\eta}1n=0$ , $k= \frac{\nu}{c}$ . (45)Since $\partial_{r}\eta_{1}\gg\alpha^{2}\eta_{1}/r$ , we may replace (45) by$( \partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{(n+\alpha)^{2}}{r^{2}}+k2)\tilde{\eta}_{1}n=0$ . (46)The replacement of (45) by (46) is based on the assumption $n\gg\alpha$ . Since $(1/r)\partial_{\theta\eta}\approx$$(n/a)\eta\approx k\eta$ , the representative value of $n\approx ka=a/\lambda$ is much larger than $\alpha=0(1)$ .Then the surface elevation can be given in the form of$\eta_{1}={\rm Re}(\psi_{AB}+\psi_{R})$, (47)where ${\rm Re}$ denotes a real part and$\psi_{AB}=\sum_{n}b_{n}J_{m}(kr)\mathrm{e}^{(-})in\theta\nu t$ , $m=|n+\alpha|$ , (48)$\psi_{R}=\sum_{n}C_{n}H^{1}m(kr)\mathrm{e}^{i}\mathrm{t}n\theta-\nu t)$ . (49)In order to obtain the coefficients $a_{n},$ $b_{n}$ and $c_{n}$ , we first require the continuity of$\eta$ and $\nabla_{\perp}\eta$ at $r=a$ . It gives two relations:$a_{n}J_{|n|(a)}k_{n}=b_{n}J_{m}(ka)+c_{n}H_{m}^{1}(ka)$, (50)$a_{n}k_{n}J’|n|(k_{n}a)=k(b_{nm}J’(ka)+c_{n}H_{m}^{1’}(ka))$ . (51)The last condition comes from that the asymptotics of $\psi_{AB}$ should coincide with thedislocated wave, which leads to$b_{n}=(-i)^{m}=\mathrm{e}^{-}i\pi|n+\alpha 1/2$ . (52)The limit $rarrow\infty$ gives the wave function in the form$\psi_{AB}arrow \mathrm{e}^{i\langle t)}-kr\cos\theta-\alpha\theta-\nu-\frac{\mathrm{e}^{i\mathrm{t}^{kt})_{\sin\pi}}r-\nu\alpha}{(2\pi ikr)1/2\cos(\theta/2)}(-1)[-\alpha]\mathrm{e}\mathrm{t}i[-\alpha]+1/2)\theta$ , (53)$\psi_{R}arrow[\frac{2}{T\dot{i}kr}]1/2\mathrm{e}\langle ikr-\nu t)_{\sum_{n}C_{n}}\mathrm{e}^{\mathrm{t}n}i\theta-\pi|n+\alpha|/2$), (54)where $[x]$ is a notation of Gauss. If $\alpha$ is an integer, the second term of (53) vanishes.Figure 1 shows the dislocated wave given by (48) and (52). This asymptotic is validexcept in a narrow sector centered at $\theta=\pi$ , where we cannot separate $\psi_{AB}$ into45incident and scattered waves. The scattered wave $\psi s$ can be defined by the sum of thesecond term of (53) and (54). The general asymptotic form of $\psi_{S}$ is$\psi_{S}\sim f(\theta)r^{-}\mathrm{e}^{i}1/2\mathrm{t}kr-\nu t)$ , (55)where $f(\theta)$ is a scattering amplitude$f( \theta)=\frac{1}{\sqrt{2\pi ik}}\tilde{f}(\theta)$ ,$\tilde{f}(\theta)=-\frac{\sin\pi\alpha}{\cos(\theta/2)}(-1)[-\alpha 1i([-\alpha]+1/2)\theta 2\mathrm{e}+\sum_{n}C_{n}\mathrm{e}i(n\theta-\pi|n+\alpha|/2).$ (56)The coefficients $c_{n}$ are$c_{n}=b_{n}[-(k_{n}/k)J_{|n|(ka)(k}\prime nJma)+J_{|n|}(k_{n}a)j’m(ka)]/\triangle$ , (57)where$\Delta=(k_{n}/k)J’(|n|k_{n}a)H^{1}(mka)-J_{||(}nk_{n}a)H_{m}^{1’}(ka)$. (58)Using a notation $\gamma_{n}=|1-\delta n|$ , with $\delta=\omega/2\nu=M/(2\pi\beta)=\alpha/(4\pi^{2}\beta^{2})$ , we maysimplify the formula of $c_{n}$ as$c_{n}=-(-i)|n+ \alpha|\frac{\gamma_{n^{j_{1n}’}}|(\beta\gamma n)J_{m}(\beta)-J_{1}|(n\beta\gamma n)J^{;}(m\beta)}{\gamma_{n}J_{1n}’(|\beta\gamma_{n})H_{m}^{1}(\beta)-J||n(\beta\gamma_{n})H_{m}^{1’}(\beta)}$ . (59)The coefficients $c_{n}$ are parametrized by only two dimensionless numbers $\alpha$ and $\beta$ .Figure 2 shows absolute values of $c_{n}$ versus $n$ for $\alpha=1$ and $\beta=0.1,1,5$ and 10. Wecan evaluate the convergence of the sum (49) from this figure.According to the scattering theory, the differential cross section may be defined by$\frac{d\sigma}{d\theta}=|f(\theta)|^{2}=\frac{1}{2\pi k}|\tilde{f}(\theta)|^{2}$ . (60)Figure 3 shows polar plot of $|\tilde{f}(\theta)|^{2}$ for $\alpha=1$ and $\beta=10$ . It is extremely anisotropic;the amplitude is very large in the forward direction $(\theta\approx\pi)$ and oscillates in thebackward direction.4 ReferencesM. V. Berry, R. G. Chambers, M. D. Large, C. Upstill and J. C. Walmsley, (1980)Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,Eur. J. Phys. 1 pp. 154-162.E. Cerda and F. Lund, (1993) Interaction of surface waves with vorticity in shallowwater, Phys. Rev. Lett. 70 pp. 3896-3899.T. Kambe, (1982) Scattering of sound waves by vortex systems, (in Japanese) Nagare1 pp. 149-165.46(a) $\alpha=0$ (b) 0.5 (c) 1(e) 2 (f) 2.5Figure 1. Density plots of incident dislocated waves for $\alpha=(\mathrm{a})0,$ $(\mathrm{b})0.5,$ $(\mathrm{c})1,$ $(\mathrm{d})$$1.5,$ $(\mathrm{e})2$ and (f) 2.5 The summation in (48) is truncated at $n=\pm 20$ . The plottedregion is-10 $\leq x\leq 10$ and-10 $\leq y\leq 10$ . Lines denote the zero displacement.Figure 2. Absolute values of $c_{n}$ versus $n$ for $\alpha=1$ and $\beta=0.1$ (denotedby solidcircle), 1 (open circle), 5 (triangle) and 10 (square).47Figure 3. Polar plot of differential cross section for $\alpha=1$ and $\beta=10$ . The incidentwave comes from the right side.48

Scattering of dislocation in shallow water

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Scattering of dislocation in shallow water

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