Non-Reflecting Internal Wave Beam Propagation in the Deep Ocean

Transcription

1 Non-Reflecting Internal Wave Beam Propagation in the Deep Ocean Roger Grimshaw ), Efim Pelinovsky,), and Tatiana Talipova ) ) Department of Mathematical Sciences, Loughborough University, Loughborough, UK ) Department of Nonlinear Geophysical Processes, Institute of Applied Physics, Nizhny Novgorod, Russia July 9 8 Using linear internal wave theory for an ocean stratified by both density and current, we identify several background profiles for which internal wave beams can propagate without any internal reflections. The theory is favorably compared with oceanic data.. Introduction It is well-known that internal waves can propagate downwards in the interior of the oceanic (e.g. Gill, 98) and this process has been studied experimentally in the laboratory (e.g. Stevens and Imberger, 994). Moreover, it is well-known that a monochromatic internal wave is an exact solution of the fully nonlinear Euler equations for a stratified fluid (in the Boussinesq approximation) with a constant buoyancy (Brunt-Vaisala) frequency, and such a wave propagates without change in amplitude and direction (Miropolsky, ). In the general case of the arbitrary distribution of the buoyancy frequency with depth an internal wave will transform and generate internal reflections. However, various observations (De Witt et al, 986; Pingree and New, 989, 99) and numerical simulations (Morozov and Pisarev, ; Gerkema,, 4, Gerkema et al., 6; Johnston and Merrifield, 3; Holloway and Merrifield, 3; Vlasenko et al, 3) of a nonlinear internal wave field in the ocean show that internal waves of tidal period propagate as beams with no visible internal reflection. Fig. demonstrates such behavior of internal wave beams in the Bay of Biscay (Gerkema et al, 4). Here we analyze theoretically the penetration of internal waves in an ocean with continuous stratification in background density and a shear current. We will show that for certain specific vertical distributions of the oceanic stratification internal waves can be considered as traveling waves which propagate without any internal reflection. Further these profiles can represent real oceanic data rather well. This confirms that the internal reflection of internal waves in the deep ocean is

2 quite weak when compared with the reflection from the ocean bottom and near-surface pycnocline. In section we consider the case when the background stratification is due solely to the density field. Then in section 3 we extend this to the case when there is also a background shear current, although for technical reasons, we consider only the non-rotating situation. In section 4, we examine the effect of wave reflection from a near-surface pycnocline, which is also densitystratified. We conclude in section 5. Fig.. Internal wave beams in the Bay of Biscay (Gerkema et al, 4). Internal waves in a density stratified ocean The two-dimensional Euler equations for a density-stratified fluid in the Boussinesq approximation, when considered for a linear monochromatic wave of frequency ω (> ) can be reduced to the wave equation (see, for instance, LeBlond and Mysek, 978) W W α =, N ω α =. () z x ω f Here W(x,z) is the vertical velocity of a water particle, N(z) is the buoyancy frequency, f is the Coriolis parameter, while z is the vertical and x is the horizontal coordinates. Since () is a hyperbolic equation, we can introduce the characteristics (rays) for wave propagation dz dx = ± α = ± ω f N ω, ()

3 where α(z) >, and calculate the ray trajectories z x x = α dz. (3) z Beam propagation of internal waves has been discussed many times in the literature. These rays (beams) are very visible in all computations of the internal wave field (including nonlinear computations); this is cited in the literature above. In fact, these cited papers demonstrate the beam character of internal waves in the deep ocean, and a strong reflection from the sea surface and the sea bottom. But it is well known that a wave should exhibit internal reflection in any inhomogeneous medium (see, for instance, Brekhovskikh, 98), which seems to be rather weak for real oceanic conditions. This problem is considered in detail in the following analysis. First, the existence of non-reflecting waves in the case of a constant buoyancy frequency is wellknown and this was cited in the Introduction. The same should be expected for the case of a slowly-varying buoyancy frequency, presented as N(εz) with a small parameter ε. In this case, according to the WKB-method (see, for instance, Bender and Orszag, 978; Baines, 995; and Miropolsky, in the internal wave context), the solution of () can be presented W (z,x) = A(εz)exp ik [ x x Ψ(z) ] { }, k z ( ε z) = dψ / dz, (4) with two unknown functions: A(εz) and k z (εz) for each fixed horizontal wavenumber k x (>). The equations for them are obtained after substituting (4) in () [ k k ] A = ε A z x α, (5) k + z A kz A =, (6) where a prime means a derivative with respect to the argument. Neglecting the term of order ε on the right-hand side of (5) the solution of these equations is found explicitly k z (εz) = ±α(εz)k x, k z (εz)a (εz) = constant. (7) 3

4 The two signs in (6) correspond to a wave propagating in the downwards or upwards direction respectively, where we have assumed that the z-axis is vertically down. For definiteness, we consider an internal wave propagating downwards, so that its amplitude and vertical wavelength increase (decrease) as N(z) decreases (increases) with depth. This asymptotic solution is valid for a slowly-varying buoyancy frequency and its derivation is well-known. The expressions (7) are the leading-order terms in an asymptotic expansion, and in principle the expansion can be continued to an arbitrarily high order, with error (ε n ), while, for an analytically smooth profile N(εz), the (internally) reflected wave is exponentially small (exp[-c/ε ]), where the constant C is found from the singularities of N(εz) in the complex plane (see, for instance, Bender and Orszag, 978; Boyd, 998). However, although the reflected wave is vanishing small as ε goes to zero, it can have a significant amplitude in practice for finite non-zero values of ε. Our goal here is to determine a large class of buoyancy frequency profiles for which an internal wave can propagate in the deep ocean with no internal reflection. That is, in essence, we seek to find those profiles for which the WKB solution can be made into an exact solution. Hereafter, the ordering parameter ε is omitted, as we are concerned with situations when it may not be small. It is convenient to use the construction of the WKB solution as a change of variables in () W (z,x) = V(Z,x) α / (z), Z = α dz. (8) After substitution of (8), equation () transforms to V Z V x = Q( Z) V d d ( α ) =. (9) α dz α α dz, Q = 3 / / In general, equation (9) is a variable-coefficient Klein-Gordon equation. The existence of waves without any internal reflection can now be achieved in the framework of a constant-coefficient equation (9), which is possible if Q = constant. Other choices for Q(Z) are possible which also provide explicit closed form expressions for V, but in general they will lead to internal reflections. For instance the choice Q ~ Z leads to a solution in terms of Airy functions, but these contain internally reflected waves with non-zero amplitudes (Bender and Orszag, 978). 4

5 There are three different cases when Q(Z) can be constant and each of them describes a buoyancy frequency profile allowing an internal wave beam without any internal reflections. The first case is Q = which leads to p α =, () ( + z / L) or N( z) = ω f ( + z / L) ω + p, () 4 where p and L are arbitrary constants; we note that L can be positive or negative in general. Mostly we will analyze the case z > (in the oceanic bulk) and assume L >, but formally the obtained solution is also valid for negative z in the range ( L <z < ). The same solution can be used for a subsurface layer (above the pycnocline) considering z < and L <. The general solution of (9) consists of two independent waves propagating in opposite directions V Z, x) = F ( x Z) + F ( x + ). () ( Z Formally, the buoyancy frequency profile () was obtained already in the literature (Kraus, 96; Vlasenko, 987), but its interpretation as a non-reflecting beam was absent. Here we will use this the non-reflecting wave beam to explain the deep penetration of internal waves in the ocean. Moreover, due to (8) the wave amplitude increases at large depths where the buoyancy frequency decreases. The rays in the ocean with this non-reflecting buoyancy frequency () can be calculated explicitly pz x x =, (3) + z / L and they have a vertical asymptote on large depths. In the limiting case of low frequency internal waves in a non-rotating ocean the non-reflecting profile is the same for waves of different frequencies, 5

6 N N e ( + z / L) =, (4) and the wave parameters vary as N( z) N A e ( z ) = A ( + z / L ), k z = =, (5) c c( + z / L) where c = ω/k x is the wave speed in the horizontal direction. The second class of profiles can be obtained from the condition Q >. In this case a suitable solution for α(z) is [ ], z( Z) = α(z) = α exp( mz) dz p, α(z) = α ( Z ) + z /L, (6) and then N( z) = ω f ( + z / L) ω + p. (7) The rays in this model of a non-reflecting buoyancy frequency are described by x x = plln( + z / ), (8) L and now rays have no vertical asymptote for large depths, but slowly turn to the vertical. In the limiting case of low-frequency waves in a non-rotating ocean, the non-reflecting profile of the buoyancy frequency is N N e =. (9) + z / L It differs from (4) in that it provides a slower change of the buoyancy frequency with depth. 6

7 The elementary downward traveling wave solution of the Klein-Gordon equation (9) with positive Q has a sinusoidal shape V( x, Z) = V sin[ kxx KzZ], Kz = kx, () 4 p L if k x > /pl. In the opposite case, the wave exponentially attenuates with depth. The general solution for a traveling wave is obtained by Fourier superposition of the elementary solutions (), { i[ k x K ( k Z ]} V ( x, Z) ), () = V ( kx )exp x z x dkx where K z and k x satisfy the dispersion relation (). As a result, the wave field disperses with depth, and the beam width increases with depth. The third class of profiles can be obtained from the condition Q <. In this case the solution for α(z) is also found explicitly α(z) = C cos (mz + ϕ), z(z) = mc tan(mz +ϕ), α(z) = p + (z /L). () The non-reflecting buoyancy frequency profile in this case is described by N = ω f [ + ( z / L) ω + p. (3) ] For z > the solution () and (3) are monotonically decreasing functions of depth. At large depths the asymptotic α(z) ~ z - is valid, as in the case (). It is important to mention that this obtained solution can describe the buoyancy frequency profile in the vicinity of a pycnocline (if its centre is located at z = ), and this is the first example of non-monotonic non-reflecting profile. The wave field again has the shape () and () but the dispersion relation now is K =, (4) 4p L z kx + 7

8 with no formal limitations on the value of k x. In this case the beam width increases with depth due to the dispersion of the wave field. The rays in this model of a non-reflecting buoyancy frequency are x x = platan( z / ), (5) L and the rays again have a vertical asymptote for large depths. In the limiting case of lowfrequency waves in a non-rotating ocean, the non-reflecting profile of the buoyancy frequency in the third case is N N e + ( z / L) =. (6) It has the same asymptotic behaviour as (4). Thus, we have found four cases of a non-reflecting internal wave beam propagation in the deep ocean. The classical example N = constant can be obtained from () as L (as formally from all other profiles) and here it will be not analyzed separately. In the case of () (which will be called case I below) the wave shape does not vary with depth (but the wave amplitude and wavelength do change with depth) and the beam width remains constant. In the cases (7) and (3) (called below case II and case III respectively) the wave field is dispersive and the beam width increases with depth. All the non-trivial non-reflecting profiles in the low-frequency limit (in variables N/N and z/l) are presented in Fig.. In case I the profile (4) corresponds to a faster decrease of the buoyancy frequency with depth, in case II the profile (9) varies slowly with depth, and in case III the profile (6) tends to (4) for large depths, but is only slowly varying for small depths. The rays for the same non-reflecting buoyancy frequency profiles are displayed in Fig. 3. All curves are qualitatively the same, but their asymptotic behaviours for large depths are different. A comparison of observed profiles of the buoyancy frequency with the non-reflecting profiles (), (7), and (3) is given in Fig. 4 for the Pacific, Atlantic, Indian and Arctic Oceans. Fig. 4a shows results of measurements on the north-west shelf of Australia in the Indian Ocean at latitude (Holloway et al, 997; Holloway, ); the Malin Shelf Edge stratification in the Northern Atlantic (latitude 56.5 ) is illustrated in Fig. 4b (Pelinovsky et al., 999; Grimshaw et al, 4); the third example (Fig. 4c) is the buoyancy profile on the Hawaiian Ridge in the Pacific Ocean, latitude (Johnston and Merrifield, 3); the last example (Fig. 4d) is for the Laptev Sea in the Arctic Ocean, latitude 73 (Polukhin et al, 3ab; Grimshaw et al, 4). In all the 8

9 figures the vertical coordinate z is measured from the centre of the pycnocline. The theoretical non-reflecting profiles of the buoyancy frequency (), (7) and (3), computed for the semidiurnal tide (.4 hr), are also presented in Fig. 4. The fitting parameters are given in Table for all three cases (for the Laptev Sea L is given above/below pycnocline). N/N Depth, z/l Case I Case II Case III 3 Fig.. Non-reflected profiles of the buoyancy frequency x/pl Case I Case II Case III z/l 3 Fig. 3. Rays for various non-reflected profiles of the buoyancy frequency 9

10 N(z), s N(z), s z, m 8 z, m 4 6 a) b) N(z), s N(z), s z, m z, m 3 4 c) d) Fig. 4. Buoyancy frequency profile and its approximation by a non-reflecting profile: a) North-West Shelf of Australia; b) Malin Shelf Edge; c) Hawaiian Ridge; d) Laptev Sea. Red dashed lines - fit (), green dash-dotted lines - fit (7) and purple dotted lines - fit (3); black lines with points - the measured data

11 Table. Fitting parameters of non-reflecting buoyancy frequency profiles Case I Case II Case III N, Depth, L, m p L, m P L, m p s - m Australian NWS Malin Shelf Edge Hawaii Ridge Laptev Sea /4 7.5/ From Fig. 4a for the NWS Australian Shelf the second fit (7) (green dash-dotted line) seems to be the best but the first fit () (red dashed line) may also be used for a description of this buoyancy frequency profile. Again, from Fig. 4b for the Malin Shelf Edge stratification, the best fit seems to be the second approximation (7) (green dash-dotted line), but both other approximations () (red dashed line) and (3) (purple dotted line) also may be used. For the Hawaiian Ridge (Fig. 4c) we see that both fits, the first () (red dashed line) and the second (7) (green dash-dotted line) describe the buoyancy frequency profile quite well, but the third fit (3) (purple dotted line) is not so good. The third fit (3) (purple dotted line) describes the buoyancy frequency profile in the Laptev Sea (both above and below the pycnocline). Also, the density stratification for the Laptev Sea is fitted by the profiles of cases I and II with different values of L above and below pycnocline, see Table where x/x gives values of L above/below pycnocline. So, all theoretical non-reflected profiles can be used to fit the measured data. It is important to mention that all the theoretical curves (), (7) and (3) formally include a dependence on the wave frequency, and therefore, for a fixed stratification, only one wave with the appropriate frequency will be have no internal reflections. However, this dependence is negligible for the Malin Shelf Edge (Fig. 5) where the buoyancy frequency profile is fitted by formulae (7) for three different wave periods T: semidiurnal tide (T =.4 hours), T = 6. hours and T = 4. hours. All curves almost coincide and therefore, various spectral components of the low frequency internal wave field propagate at depth with no internal reflection. Thus our theoretical analysis shows the existence of several density stratifications in the ocean which can provide non-reflecting propagation of internal waves in the deep ocean. Further we have shown that the observed density stratification in various areas of the world s oceans can be fitted by these profiles. This means that the vertical propagation of internal waves with minimal

12 internal reflection is typical for real oceans, thus explaining the results of various numerical simulations cited in the Introduction. N(z), s z, m 4 6 Fig. 5. Buoyancy frequency profile and its approximation by the non-reflecting profile (7) for Malin Shelf Edge: T =.4 hours (red line); T = 6. hours (green line); T = 4. hours (purple line); the black line are the experimental data. 3. Traveling waves in an ocean stratified in density and current The same approach can be applied for the propagation of internal waves in an ocean with both density and current stratification. For simplicity we consider here monochromatic low-frequency waves in a non-rotating ocean, when the basic equation for the vertical velocity (or streamfunction) reduces to the Taylor Goldstein equation (Baines, 995; Miropolsky, ). d w + κ w =, (7) dz where

13 N (z) κ (z) = [ c U(z) ] + c U(z) d U dz. (8) Here U(z) is the background shear flow (assumed stable) and c is the wave speed in the horizontal direction. Using again the WKB construction V = κ / w, Z = κ dz, (9) equation (7) reduces to d V dz d + V = Q( Z) V, Q( Z ) = [ κ / ( Z )]. (3) / κ ( Z ) dz Non-reflecting wave solutions are obtained from (3) if Q does not depend on Z and < Q <. (3) Again three classes of background profiles are possible. The first case is Q = leading to N + 4 [ c U ] c U dz ( + z / L) d U = κ, (3) which determined a relationship between N(z) and U(z). So, the main difference with the previous cases is that now there are an infinite number of background density and shear flow stratifications providing non-reflecting propagation of internal waves in the deep ocean. For each density profile N(z) from (3) we can find the corresponding shear flow U(z) in principle. But equation (3) with respect to U(z) is second-order linear ordinary differential equation, with variable coefficients, and to find its analytical solutions is very difficult. It is more simple is to find N(z) from (3) for a given U(z). If the shear flow has the Gaussian shape (it is shown in Fig. 6) ( z z U = u exp ), (33) l 3

14 the computed buoyancy profile satisfied to (3) is displayed in Fig. 6 for l/l =., z /L =.5, c /l N =.3. It is clearly seen the non-reflecting buoyancy profile is deformed in the zone of the shear flow forming fine structure elements. Such elements are visible on many observed profiles (see Fig. 4) and they together with the shear flow (which is very often not measured together with the density profile) can provide non-reflecting downward propagation of an internal wave in the ocean interior. Another example is the flow with a constant shear in the sub-surface layer u( z / l) < z < l U =. (34) l < z < L The non-reflecting buoyancy profile is modified in the subsurface layer also (Fig. 7 for l/l =. and u/c =.5). In fact, the point z = l is singular (U zz is infinite) and the stratification should be smoothed in the vicinity of this point. N(z)/N.5 z/l U(z)/c Fig. 6. Non-reflecting buoyancy profile in a basin with a shear flow. Numbers values of U/c. 4

15 N(z)/N.5 z/l U(z)/c Fig. 7. Non-reflecting buoyancy profile in a flow with a constant shear. Numbers values of U/c. If the background is weak, all non-reflecting density profiles are described approximately by N( z) N( z) N U c d U c N dz, (35) obtained from (3) at the first order of a perturbation theory. Here N (z) is the non-reflecting profile (4) with no background shear flow. In zones of weak density stratification even a smooth shear flow leads to strong variability of the buoyancy frequency profile. Especially we would like to note the solution of (3) for L (that is κ is a constant). If there is no shear flow, this case corresponds to the classical case of constant buoyancy frequency, providing propagation of a wave with constant amplitude. If the shear is constant (du/dz = constant) the same result holds when N(z)/(c-U(z)) is a constant, namely an internal wave propagates with no reflection. The second case is < Q < when κ(z) is described by an expression analogous to (6) and the relation between N(z) and U(z) is 5

16 N + [ c U ] c U dz ( + z / L) d U = κ. (36) Formally κ L >/ is needed to ensure that Q <. This limitation occurred also for the case with density stratification only, see (). It is important to note that in the case of a weak current, the expression (35) is again valid again, but now N (z) satisfies (9). In the third case (Q < ) equation (36) is replaced on N + [ c U ] c U dz + ( z / L) d U = κ. (37) Again for a weak current the expression (35) is valid, with N (z) now satisfying (6). Fig. 8 demonstrates the influence of a current with the Gaussian shape (33) on the nonreflecting buoyancy profile for all three cases described by (3), (36) and (37) for l/l =., z /L =.5, c /l N =.3, u/c =.. It can be seen that the perturbations of the buoyancy profile are different, and such perturbations are sensitive to the shape of unperturbed buoyancy profile..5 N/N z/l.5.5 Case I Case II Case III Fig. 8. Non-reflecting buoyancy profiles with Gaussian-shape current. There is also an alternative approach to that described above, in which we assume at the outset that the current shear is weak, and so we replace (8) with 6

17 κ(z) = N(z) c U(z). (38) Then the same change of variables (9) leads to d V dz + V = Q( Z) V, where now Q(Z) = κ / (Z) d dz [ κ/ (Z)] (κu ) Z Z κ(c U). (39) Essentially the former procedure placed the term involving the curvature U zz into the phase of the wave, whereas now we place in the amplitude of the wave. The simple choice Q= leads to the solution B = κ / d B (c U), where = dz. (4) The choice B=constant leads to N(c-U) = constant, which generalizes the classical case when N(z) =constant. Otherwise, the choice B = B Z leads to the solution κ(c U) = N(c U) = B z ξ, ξ = Z = dz. (4) (c U) z This generalizes the case when Q = leads to () in section. Overall, this alternative class of non-reflecting buoyancy profiles will exhibit qualitatively the same sensitivity to the presence of even a rather small shear flow as the cases we have discussed above. 4. Reflection properties of a pycnocline If the background stratification differs from any of the non-reflecting profiles given above, the internal wave will reflect in the ocean interior. This effect is important for the passage of an internal wave through a pycnocline where there is typically a fast variation of the buoyancy frequency. We consider here the simplified example of internal wave reflection from a pycnocline which is presented as superposition of two non-reflecting profiles (4) 7

18 z < (+ z / L ) N = N (4) z > ( + z / L ) with different characteristic scales. As is shown on Fig. 4 this distribution of the buoyancy frequency may be used for the approximation of many observed profiles. The fitted parameters are given in the Table and are used here for the quantitative estimation of wave reflection from the pycnocline. For simplicity we will ignore in this section any effects of a shear flow, and use the low frequency approximation. Let the internal wave be incident on the pycnocline from z <. Writing the solution as the sum of incident and reflected waves above the pycnocline and a transmitted wave below it, and using continuity conditions for the vertical velocity and its derivative at the point z =, we find the reflection coefficient in the form R =, (43) inlp + c where L p = L L /(L +L ). The complex value of R is the ratio of the amplitude of reflected wave to the amplitude of the incident wave at z = (away from this point the amplitudes grow linearly with distance). Modulus of R is R = N L + c p. (44) When the pycnocline depth is much less than the ocean depth, the modulus of the reflection coefficient tends to (strong reflection). In a shallow ocean when the pycnocline depth is comparable with the ocean depth, the modulus of the reflection coefficient may be quite weak. The square of the modulus of the reflection coefficient characterizes the energy of reflected wave. This coefficient is computed for all the buoyancy frequency profiles presented in Fig. 4. The value of c in each case is found as the maximum value of the wave speed in the horizontal direction computed from the eigenvalue problem 8

19 d w N + w = dz c (45) with zero values on the sea surface and sea bottom. The results of these calculations are presented in Table. The value of R is quite large for the North West Shelf of Australia, and is about the.5 for the Hawaiian Ridge and the Laptev Sea. It equals. for the Malin Shelf Edge. So overall, internal waves are reflected from the pycnocline (and of course, from sea surface and sea bottom). An example of strong reflection from a pycnocline computed for the Faeroe- Shetland Channel is shown in Fig. 9 (taken from Gerkema, ). Table. The coefficient of internal wave reflection on the pycnocline N, s - H, m L p, m C, m/s R North West Shelf of Australia Malin Shelf Edge Hawaiian Ridge Laptev Sea a) b) Fig. 9. Faeroe-Shetland Channel: a) buoyancy frequency profile; b) beam structure demonstrating the strong reflection of internal waves from the pycnocline (Gerkema, ); 9

20 In fact, the reflection properties of a pycnocline are sensitive to the shape of the pycnocline. For instance, if the pycnocline in the Laptev Sea is approximated by (6) (Case III), the reflection will be absent. In this case, use of the buoyancy profile (4) (Case I) can give an upper limit of the reflection coefficient due to big value of jump in dn/dz at the centre of pycnocline. 5. Conclusion Much observed data and many results of numerical simulations demonstrate the deep penetration of internal wave beams in the interior of the ocean with no visible internal reflection (excepting rather strong reflections from the sea surface, pycnocline and the sea bottom). This effect is related to the existence of non-reflecting background profiles in density and current. In this paper we have described as a set of such profiles, found within the framework of the linearized theory for internal waves in a stratified fluid. In the absence of a background shear flow, three possible profiles of the buoyancy frequency are found. In case I the buoyancy frequency below the pycnocline is described by (+z/l) -, and the wave shape remains constant with depth but its amplitude and wavelength vary with depth; the wave field is concentrated in beams. The wellknown example of non-reflecting wave propagation in an ocean with a constant buoyancy frequency is a particular case for L. In case II the buoyancy frequency profile is described by (+z/l) -, and internal wave field disperses with depth leading to an increase of the beam width. In case III the buoyancy frequency profile is non-monotonic [+(z/l) - ] -, and can describe the pycnocline structure. The wave field in this case is also dispersive. Observed buoyancy frequency profiles in many areas of the world s oceans can in many cases be well approximated by the obtained non-reflecting profiles. We have demonstrated this here for the North-West Shelf of Australia, the Malin Shelf Edge, the Hawaiian Ridge and the Laptev Sea. This good fitting of observed data by the theoretical profiles allows us to conclude that nonreflecting beam propagation of the internal waves in the ocean is a frequent phenomenon. If the ocean is stratified in both density and current, we have found an infinity number of nonreflecting background profiles. In fact, for each density profile one can find an associated nonpropagating shear flow, and some examples of such profiles are given in this paper. In the pycnocline, where the buoyancy frequency varies significantly, a downward propagating internal wave reflects from the pycnocline. The reflection coefficient is calculated for the North- West Shelf of Australia, Malin Shelf Edge, Hawaiian Ridge and Laptev Sea. It varies over a wide range, and is sensitive to the details of stratification in the vicinity of the pycnocline. It is also demonstrated that if the buoyancy profile in the vicinity of the pycnocline is symmetrical

21 and described by the profile (6) (Case III), the internal wave can pass through a pycnocline with no reflection. Theoretical non-reflecting stratifications have been obtained here in the framework of the linearized theory of internal waves. It is well known that in the case of N(z) = constant the nonlinear wave also propagates without reflection. The analysis of nonlinear downward propagation of internal waves will be given in the future. It is pertinent to note in this context that nonlinear theories for slowly-varying internal gravity waves have demonstrated that the main nonlinear effect is the generation of a wave-induced mean flow (e.g. Grimshaw, 974, 975; Sutherland. 6); since here we have shown that the class of non-reflecting profiles can in principle allow arbitrary background shear flows, we expect that the results we have obtained here may indeed extend to the nonlinear case. Further, as we have shown above, the results of observations and numerical simulations made for real parameters of the internal wave field demonstrate the beam character of the wave propagation, irrespective of the nonlinearity of the wave amplitude. Acknowledgement Grants from INTAS ( ) and (RFBR KO_a) for all co-authors, Leverhulme Trust for EP, and RFBR ( and HBO_a) for TT are acknowledgement. References Baines, P.G. Topographic effects in stratified flows. Cambridge University Press, 995. Bender, C.M., and Orszag, S.A. Advanced mathematical methods for scientists and engineers. McGraw-Hill, 978. Boyd, J.P. Weakly nonlocal solitary waves and beyond-all-orders asymptotics. Mathematics and its Applications, v. 44, Kluwer, 998. Brekhovskikh, L.M. Waves in Layer Media. 98. De Witt, L.M., Levine, M.D., Paralson, C.A., and Burt, W.W. Semidiurnal internal tide in JASIN: Observations and simulation. J. Geophys. Research, 986, vol. 9, Gerkema, T. Application of an internal tide generation model to baroclinic spring-neap cycles. J. Geophys. Research,, vol. 7, No. C9, 34, doi:.9/jc77. Gerkema, T. Internal-wave reflection from uniform slopes: higher harmonics and Coriolis effects. Nonlinear Processes Geophysics, 6, vol. 3,

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