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Scattering of internal waves from small sea bottom inhomogeneities 7 0 0 2 A. D. Zakharenko n a J Il’ichev Pacific oceanological institute, Baltiyskay St. 43, 9 Vladivostok, 41, 690041, Russia 1 ] h p Abstract - o Theproblem of scattering of linear internal waves from small com- a . pact sea bottom inhomogeneities is considered from the point of view s c of mode-to-mode scattering. A simple formula for modal conversion i s coefficients C , quantifying the amount of energy that is scattered nm y into the n-th mode from the incident field m-th mode, is derived. In h p thisformulatherepresentation ofinhomogeneities bytheir expansions [ into the Fourier and Fourier-Bessel series with respect to angular and 1 radial coordinates respectively are used. Results of calculations, per- v formed in a simple model case, are presented. The obtained formula 1 2 can be used for a formulation of the inverse problem, as it was done 2 in the acoustic case [2, 3]. 1 Keywords: internal wave, scattering 0 7 0 / 1 Introduction s c i s The concept of mode-to-mode scattering was considered in the context of y h the acoustic scattering fromsmall compact irregularities of theocean floorby p Wetton,Fawcett[1]. Intheirworksomesimpleformulasformodalconversion : v coefficients, quantifying the amount of energy that is scattered from one i X normal mode of the sound field to another, were derived. Recently new r formulas for these coefficients were obtained by Zakharenko [2] and applied a to the inverse scattering problem in the subsequent work [3]. This paper contains the detailed derivation of such formulas in the case of scattering of linear internal waves from small compact sea bottom inhomogeneities. Some numerical examples are presented. 1 2 Formulation and derivation of the main re- sult We shalluse thelinearized equations forinviscid, incompressible stably strat- ified fluid, written for the harmonic dependence on the time with the factor e−iωt in the form iωρ u+βP = 0, 0 x − iωρ v+βP = 0, 0 y − iωρ w +βP +βρ = 0, (1) 0 z 1 − iωρ +wρ = 0, 1 0z − u +v +w = 0, x y z where x, y, and z are the Cartesian co-ordinates with the z-axis directed upward, ρ = ρ (z) is he undisturbed density, ρ = ρ (x,y,z) is the pertur- 0 0 1 1 bation of density due to motion, P is the pressure, and u, v w are the x, y and z components of velocity respectively. The variables are nondimen- ¯ sional, based on a length scale h(a typical vertical dimension), a time scale N¯−1 (where N¯ is a typical value of the Brunt-Väisäla frequency), and a density scale ρ¯ (a typical value of the density). The parameter β is g/(h¯N¯2), where g is the gravity acceleration. The boundary conditions for these equations are w = 0 at z = 0, (2) w = uH vH at z = H, x y − − − where H = H(x,y) is the bottom topography. We introduce a small parameter ǫ, and postulate that the components of velocity and the pressure are represented in the form u = u +ǫu +..., v = v +ǫv +..., 0 1 0 1 w = w +ǫw +..., P = P +ǫP +... 0 1 0 1 We suppose also that the bottom topography is represented in the form H = h + ǫh , where h is constant and h = h (x,y) is a function of x, 0 1 0 1 1 y vanishing outside the bounded domain Ω, which in the sequel is called a domain of inhomogeneity. Excluding from the system (1) ρ and substituting the introduced expan- 1 sions, we obtain 2 iωρ (u +ǫu +...)+β(P +ǫP +...) = 0, 0 0 1 0x 1x − iωρ (v +ǫv +...)+β(P +ǫP +...) = 0, 0 0 1 0y 1y − (3) (ω2ρ +βρ )(w +ǫw +...)+iωβ(P +ǫP +...) = 0, 0 0z 0 1 0z z1 (u +ǫu +...)+(v +ǫv +...)+w +ǫw +... = 0, 0x 1x 0y 1y 0z 1z with the boundary conditions w +ǫw +... = 0 at z = 0, 0 1 w +ǫw +... = (u +ǫu +...)(h +ǫh ) (4) 0 1 0 1 x0 1x − ǫ(v +ǫv +...)(h +...) at z = H. 0 1 1y − − Separating terms in various orders of ǫ, we obtain a sequence of boundary problems. At order O(ǫ0) we have iωρ u +βP = 0, 0 0 0x − iωρ v +βP = 0, 0 0 0y − (5) (ω2ρ +βρ )w +iωβP = 0, 0 0z 0 0z u +v +w = 0, 0x 0y 0z with the boundary conditions w = 0 at z = 0 0 w = 0 at z = h . 0 0 − Differentiating the third equation in (5) twice with respect to x and twice with respect to y, summing obtainedequations andreplacing β(P +P ) 0zxx 0zyy by iω(ρ w ) , we obtain 0 0z z − (ω2ρ +βρ )(w +w )+ω2(ρ w ) = 0. (6) 0 0z 0xx 0yy 0 0z z We seek a solution to this equation in the form of the sum of normal modes w = ei(kx+ly)φ(z), where φ is the eigenfunction of the spectral boundary 0 problem (ω2ρ +βρ )(k2 +l2)φ+ω2(ρ φ ) = 0, 0 0z 0 z z − (7) φ(0) = φ( h ) = 0, 0 − with the eigenvalue λ = k2 + l2. It is well known that the problem (7) has countably many eigenvalues λ , which are all positive. The corresponding n real eigenfunctions φ we normalize by the condition n 0 ω2 0 ω2ρ +βρ φ2dz = ρ (φ )2dz = 1. (8) −Z−h0(cid:0) 0 0z(cid:1) k2 +l2 Z−h0 0 z 3 The eigenfunctions φ and φ with n = m are also orthogonal n m 6 (φ ,φ ) = 0 (9) n m with respect to the inner product 0 (φ,ψ) = ω2ρ +βρ φψdz (10) 0 0z −Z−h0(cid:0) (cid:1) In our scattering problem w is the incident field, and we shall calculate the 0 main term of scattering field w , so we act in the framework of the Born 1 approximation. At the first order of ǫ we obtain the following system of equations: iωρ u +βP = 0, 0 1 1x − iωρ v +βP = 0, 0 1 1y − (11) (ω2ρ +βρ )w +iωβP = 0, 0 0z 1 1z u +v +w = 0, 1x 1y 1z with the boundary conditions w = 0 at z = 0, 1 (12) w = u h v h at z = h ǫh . 1 0 1x 0 1y 0 1 − − − − Sofarasweareinteresting intheconnection ofmodalcontents ofincident and scattering fields, we suppose that the incident field consists of one mode w = ei(knx+lny)φ (z). Reducing the second boundary condition (12) to the 0 n boundaryz = h withtakingintoaccounttheexplicit formofw , weobtain 0 0 − the new boundary condition for w at the boundary z = h : 1 0 − ik il w = h n h n h ei(knx+lny)φ . (13) 1 1 1x 1y nz (cid:18) − k2 +l2 − k2 +l2 (cid:19) n n n n Reducing the system (11) in the same manner as it was done for the system (5), we obtain the equation for w : 1 (ω2ρ +βρ )(w +w )+ω2(ρ w ) = 0. (14) 0 0z 1xx 1yy 0 1z z We seek the scattering field in the form w = N C (x,y)φ , the func- 1 m=1 nm m tions C (x,y) are called the modal conversioPn coefficients. To obtain the nm equationforC wesubstitutethepostulatedformofw tothe(13),multipli- nm 1 cate it by the function φ and integrate from h to 0. Using the conditions m 0 − of orthogonality and normalization (8), (9) and the boundary condition (13), we finally obtain 4 ∂2 ∂2 C + C +(k2 +l2 )C = F, (15) ∂x2 nm ∂y2 nm m m nm where ik il F = ω2ρ h n h n h ei(knx+lny)φ ( h )φ ( h ). 0 1 1x 1y nz 0 mz 0 (cid:18) − k2 +l2 − k2 +l2 (cid:19) − − n n n n Writing the solution to the equation (15) as the convolution of the fundamental solution (Green function) of the Helmholtz operator G = ( i/4)H(1)( k2 +l2 R) with the right-hand side F, we have − 0 m m p i C (x ,y ) = FH(1)( k2 +l2 R)dydx, (16) nm r r −4 Z Z 0 m m p x y where R = (x x )2 +(y y )2 andby theindex r we designate thepoint r r − − of registratipon of the field. Integrating by parts the terms containing h ,h and passing to the 1x 1y cylindrical coordinatesystem with the origin in our domain of inhomogeneity and such that k = κ , l = 0, x = rcosα, y = rsinα, we obtain n n n ∞ 2π 1κ C = mG h eiκnrcosαcos(ψ α )H(1)(κ R)rdαdr, (17) nm −4 κ Z Z 1 − r 1 m n 0 0 G = ω2ρ φ ( h )φ ( h ), R = r2 +r2 2rr cos(α α ), (r ,α ) are 0 nz − 0 mz − 0 r − r − r r r the polar coordinates of the registraption point, tan(ψ) = rsin(α α )/(r r r − − rcos(α α )). r − Using the addition theorem for the Bessel functions we express contained (1) (1) in (17) cosψH (κ R) and sinψH (κ R) in the form: 1 m 1 m ∞ cos(ψ) H(1)(κ R) = H(1) (κ r )J (κ r) cosk(α−αr) . sin(ψ) 1 m k+1 m r k m sink(α−αr) n o kX=−∞ n o From now on we shall assume that the distance r to the registration point r (1) is big enough to replace the functions H (κ r ) by their asymptotics k+1 m r (1) H (κ r ) 2/(πκ r )exp[i(κ r (π/2)(k+1) π/4)]. k+1 m r ≈ m r m r − − p Then, expanding h (r,α) as function of α in Fourier series with the coeffi- 1 ˜ cients h (r), after integration with respect to α, we obtain 1ν 5 ∞ i√2π√κ exp(iκ r iπ/4) C = m m r − Gcosα (i)νe−iνα0 nm r 2 κ √r n r νX=−∞ (18) ∞ ∞ e−ikαr h˜ (r)J (κ r)J (κ r)rdr 1ν k m ν+k n × Z kX=−∞ 0 Changing the order of integration and summation we can achieve further simplification by using the formula ∞ J (κ r)J (κ r)e−ikαr = J (ξr)e−iνθ, k m ν+k n ν kX=−∞ κ sinα where ξ = κ2 +κ2 2κ κ cosα , θ = arctan m r . We ex- m n − m n r κ κ cosα p ˜ n − m r pand now the radial coefficients h (r) on the segment [0,L], where they do 1ν not vanish, in the Fourier-Bessel series ∞ γν h˜ (r) = fνJ pr , 1ν p ν (cid:18) L (cid:19) Xp=1 where γν are the positive roots of the function J , J (γν) = 0. Substituting p ν ν p this expansion in (18) and taking into account that L γν L2γνJ (ξL)J′(γν) J pr J (ξr)rdr = − p ν ν p , Z ν (cid:18) L (cid:19) ν γν2 ξ2L2 p − 0 we obtain the final expression for modal conversion coefficients iL2√2π√κ exp(iκ r iπ/4) m m r C = − Gcosα nm r − 2 κ √r n r (19) ∞ ∞ γνJ′(γν) (i)νJ (ξL)e−iν(α0+θ) fν p ν p . × ν p γν2 ξ2L2 νX=−∞ Xp=1 p − 6 3 Numerical examples For a model example we choose ρ = e−λz,β = λ−1 and H = 1 . Then the spectral boundary problem is written in the form ω2φ ω2λφ κ2(ω2 1)φ = 0, zz z − − − φ(0) = 0, φ( 1) = 0. − The eigenfunctions of such a problem are φ = Aeλz/2sin((l+1)πz) with the eigenvalues ω (l+1)2π2 +λ2/4 κ = . p √1 ω2 − Here A = √2/(√1 ω2) by the condition (9). For the calculations the value − of parameter λ was taken to be equal to 0.003 , which corresponds to the typical stratification in the ocean shelf zones. The domain of inhomogeneity has the form of the ellipse with the big and small radii a and b of which were taken in proportion a : b = 2 : 1, and in this region x2 y2 h (x,y) = 0.05 1 . 1 r − a2 − b2 In the figure are presented the results of calculations with ω = 0.5 and the angle of incident field α = 0, conducted for various wave sizes κa of the 0 scatterer. We note that according to the meaning of the small parameter ǫ, in these calculations ǫ = 0.05. For the presentation of results we use the scattering amplitude eiκmrr −1 F (α ) = C (α ). nm r nm r (cid:18) √r (cid:19) r References [1] Wetton, B. T. R., Fawcett, J. A. Scattering fromsmall three-dimensional irregularitiesintheoceanfloor.J.Acoust.Soc.Am., vol.85.(1989),No4, pp. 1482-1488. [2] Zakharenko, A. D. Sound scattering by small compact inhomogeneities in a sea waveguide. Acoustical Physics, vol. 46 (2000), pp. 160-163. [3] Zakharenko, A. D. Sound scattering by small compact inhomogeneities in a sea waveguide. Acoustical Physics, vol. 46 (2000), pp. 160-163. 7 90 90 0.015 0.02 120 60 120 60 0.015 0.01 150 30 150 0.01 30 0.005 0.005 180 0 180 0 210 330 210 330 240 300 240 300 270 270 (a) F11 (b) F12 | | | | 90 90 0.05 0.08 120 60 120 60 0.04 0.06 0.03 150 30 150 0.04 30 0.02 0.02 0.01 180 0 180 0 210 330 210 330 240 300 240 300 270 270 (c) F11 (d) F12 | | | | 90 90 0.8 0.15 120 60 120 60 0.6 0.1 150 0.4 30 150 30 0.05 0.2 180 0 180 0 210 330 210 330 240 300 240 300 270 270 (e) F11 (f) F12 | | | | Figure 1: Absolute value of scattering amplitude: κa = 1 (a,b), κa = 2 (c,d), κa = 8 (f,g) 8