Internal Wave Transmission through a Thermohaline Staircase

Transcription

1 Internal Wave ransmission through a hermohaline Staircase Bruce R. Sutherland Departments of Physics and of Earth & Atmospheric Sciences University of Alberta, Edmonton, AB, Canada 6G E (Dated: January, 6) With the aim to predict energy transport by internal waves incident upon observed density staircases in the ocean, the theory for internal wave tunnelling through a single finite-sized mixed region is extended to predict the transmission of internal waves through a piecewise-constant density profile with an arbitrary number of steps. An analytic solution is found if all steps have equal vertical extent. From this solution, bounds are established for the relative incident horizontal wavenumber and frequency of waves that have negligible transmission, a result that is independent of the number of steps. If the step sizes vary randomly about a mean vertical extent, the transmission can be computed numerically for several random realizations. Provided the horizontal wavelength is much longer than the total vertical extent of the staircase, the standard deviation of the computed transmission coefficients is small and the mean agrees well with the analytic prediction for equally spaced steps. he results are applied to thermohaline staircases observed in the ocean in order to put lower bounds on the wavelengths of long inertia gravity waves that significantly transmit to great depth. PACS numbers: Hd Keywords: internal gravity waves; density staircase; tunnelling; transmission I. INRODUCION hrough double diffusion, the density stratification of the ocean, particularly at the thermocline, can evolve to have a staircase profile characterized by regions of near-uniform density separated by sharp density jumps []. his has been observed in warm regions such as the yrrhenian Sea in the Mediterranean [, ] and in the western Atlantic near the Caribbean Sea [4 6]. Recent observations have also shown that density staircases persist over large regions of the Arctic Ocean [7 ]. It is this case that is of particular interest because the reduction in sea-ice in recent years has resulted in enhanced generation of internal gravity waves (or internal waves, for short) [, ]. Because internal waves move under theinfluenceofbuoyancy[], itisuncleartowhatextent they can penetrate through well-mixed, uniform-density regions. An early examination of internal waves interacting with a thermohaline staircase was performed by Ref. [4], who examined the splitting of modes as was done for an effective two-layer system by Ref. [5]. his eigenvalue approach is analogous to the derivation for the band structure of electrons in a one-dimensional lattice. A mathematical approach more relevant to the problem of downward propagating internal waves incident upon a thermohaline staircase is that followed first by Ref. [6] who predicted the transmission and reflection of internal waves analogous to the problem of electron tunnelling across a potential barrier. hey showed that internal waves can partially transmit across a uniform-density region (where the waves are evanescent) provided it is suffi- bruce.sutherland@ualberta.ca; bsuther ciently thin compared to the horizontal wavelength. Furthermore, if the uniform-density region is flanked by density jumps (as would result from localized mixing of the background stratification), then enhanced transmission can occur for incident waves whose horizontal wavenumber and frequency are nearly resonant with those of the interfacial waves on either flank of the mixed patch. his analytic study was extended to examine the influence of shear [7]. Further numerical studies examined the transmission of waves through more complex stratification [8 ], and large-amplitude effects were also considered []. But in all these studies, the background stratification considered involved far less layers than occurs in a typical thermohaline staircase. For example, Ref. [] considered the transmission of waves across two well-mixed regions bounded by finite-depth stratified layers. Based upon their theoretical, numerical and experimental analyses they proposed that thermohaline staircases in the Arctic Ocean may effectively shield the abyss from internal waves generated at the surface by storms. he theoretical-numerical study presented here predicts the transmission of internal waves across an arbitrary number of mixed regions comprising a density staircase profile. An analytic formula is found in the case for which each step has the same depth, L. he sensitivity of this result to non-uniform step-sizes is also examined numerically. It is found that the extension from one or two mixed regions to several steps has non-trivial consequences: the wave-resonance with bounding interfacial waves in the one-step case disappears, but a new sequence of resonances becomes manifest. From the analytic solutions, bounds are found that set the size and frequency of incident internal waves that can pass through the staircase. After describing the mathematical formulation of the problem in Sec. II, analytic solutions are found in Sec. III

2 for the case of of a staircase having an arbitrary number of evenly-spaced steps. he case of an uneven staircase is examined numerically in Sec. IV, transmission is found for several stochastic representations of the density staircase. In Sec. V the results are summarized and applied to a typical oceanographic circumstance. z L ρ/ d ρ dz = ρ L ρ ρ z ρ/ ǫσ L d ρ dz = ρ L ρ ρ II. PROBLEM SE-UP he transmission of internal waves across a density staircase is computed for small-amplitude, two dimensional waves in inviscid, Boussinesq fluid. he background is taken to be stationary with density profile, ρ(z), characterized by J uniform-density steps each of mean depth L. Conceptually, it is imaged that a staircase of total vertical extent J L has developed, for example due to double diffusion, within a uniformly stratified region characterized by buoyancy frequency N. Explicitly, ( ) ρ ρ z ρ L, z > ρ (+ ( ) ) j L ρ ρ, ρ(z) = L(j +ǫσ j ) < z < L(j +ǫσ j ), j ( =,,...,J ρ ρ z ρ L ), z < J L () ρ = ρ( + ) is the characteristic density. For simplicity, whether the vertical extent of each step is fixed or variable, the density jump between steps is fixed to be ρ within the staircase and the density jump between the bounding uniformly stratified regions (for z > and z < JL) and the first and last step is ρ/. Outside the staircase the density gradient has magnitude ρ/l. Hence the squared buoyancy frequency is N = g /L in which g = g ρ/ρ is the reduced gravity. he value of ǫ ( ǫ <.5) is a measure of the fluctuation of the step extent about the mean value L. For j < J, σ j are random numbers between and but σ and σ J are both zero. he case with ǫ = corresponds to every step having equal depth L, as shown in the left-hand plot of figure. Explicit analytic predictions of the transmission coefficient can be found for this symmetric circumstance. In cases with positive ǫ, the depths of each step fluctuate about a mean value L, as illustrated in the right-hand plot of figure. In this case the transmission coefficient can be computed numerically for a given set of random variables, {σ j j =...J }. Repeating this calculation for several realizations of {σ j } provides a means to test the range of extension of the analytic predictions to more realistic oceanographic circumstances. Generally, the structure of disturbances in nonuniformly stratified fluid is cast in terms of the streamfunction by ψ(x,z,t) = ˆψ(z) exp[ı(kx ωt)], k and ω are the horizontal wavenumber and frequency, L J L ρ ρ/ ǫσ L ρ ρ/ FIG.. Schematics showing the density profiles (thick black lines) of a staircase with J equal steps of depth L (left) and a staircase with J irregular steps having mean depth L (right). In both plots the diagonal light-dashed line indicates the mean stratification in the absence of a staircase. he extent of the thick-gray line segments on the right indicate the change in the depth of the density jump from the case of equal-depth steps. respectively, and it is understood that the actual streamfunction is the real part of the right-hand expression. he (possibly complex) vertical structure function, ˆψ, satisfies [] d ˆψ dz +k ( N ω ) ˆψ ω f =, () f is the (constant) Coriolis parameter. Above and below the staircase (where N = N ) and within each step (where N = ), () can be solved explicitly. Arbitrarily assuming that internal waves are incident upon the staircase from above, the solution for prescribed k, ω, and incident streamfunction amplitude A is A e ımz +B e ımz, for z > A j e γ[z+l(j /)] +B j e γ[z+l(j /)], ˆψ(z) = L(j +ǫσ j ) < z < L(j +ǫσ j ), j =,,...,J A J+ e ım[z+lj], for z < J L () m = k (N ω )/(ω f ) (4) is the vertical wavenumber of the incident wave and γ = k ω /(ω f ) (5) is the inverse e-folding depth of the evanescent disturbances within the individual uniform-density steps of the

3 staircase. he addition of L(j /) in the exponents of () is unnecessary because they could be rolled into the A j and B j coefficients. hey are included, however, in order to take advantage of symmetry in finding analytic and numerical solutions. Provided f ω N, the wave is indeed propagating in the uniformly stratified regions surrounding the staircase. he definition of m, being positive, is consistent with the incident wave having downward group velocity. he amplitude of the reflected wave is B. By causality, only a downward-propagating wave, with amplitude A J+, can exist in the stratified region below the staircase. he J + coefficients in () are related through the requirement that the pressure and vertical velocity are continuous. his amounts to the requirement that ˆψ and dˆψ/dz g ρ k ˆψ ρ ω are continuous []. Applied to each of the J + interfaces, this gives J + equations in J + unknowns. In principle these can be solved to relate each coefficient to the incident amplitude A. In particular, to find the transmission coefficient it is necessary to find A J+ as a function of A. he transmission coefficient itself is defined to be the ratio of transmitted to incident energy flux [7]. Because the background is stationary and the stratification is the same above and below the staircase, the transmission is given simply by = A J+ A. (6) he reflection coefficient is given by R = B /A. However, in the absence of energy sources such as background shear, it is expected from conservation of energy that R =. III. ANALYIC SOLUION A. Finding coefficients been defined: exp(γl) N M m/γ = ω, Γ g k γ ω = N ω f ω. he results may be combined to get a formula for A in terms of A J+ alone: (8) A = [ VL C J ] V R AJ+, (9) 4M the superscript denotes the transpose, the left and right vectors respectively are V L V R ( ) / [M ı( Γ)] /, [M +ı(+γ)] ( ) / [ Γ+ıM] /, [+Γ ım] () and the matrix C is ( ( Γ C = ) ) Γ Γ (+ Γ ). () B. ransmission across one step In the special case of a single mixed layer (J = ), the expression reduces to [ A = 4M M( Γ)+ M(+Γ) +ı ( (M ( Γ) ) (M (+Γ) ) )] A. Hence, in the determination of the transmission coefficient,, one finds = A = + 6M (δ + δ ), () A In the special case of steps having equal depth, it is possible to find explicit analytic solutions by taking advantage of symmetry. After some manipulation the interface conditions give the following sequence of equations: δ + [ ( Γ) +M ] δ [ (+Γ) +M] () A = /[ ı M ( Γ)] A + /[ + ı M (+Γ)] B A j = ( Γ ) A j+ Γ B j+ B j = Γ A j+ + (+ Γ ) B j+ A J = / [ Γ+ıM] A J+ B J = / [+Γ ım] B J+, (7) the middle pair of equations is evaluated for j =,,...J and, for convenience, the following have Using the expressions in (8), the transmission coefficient can be written explicitly in terms of the relative horizontal wavenumber and by frequency, which is given implicitly in terms of the quantity Θ = tan (m/k) = tan N ω ω f. (4) Physically, Θ represents the angle between the vertical and the constant-phase lines of the incident and transmitted waves. In the circumstance f =, the formula

4 4 a) f = b) f =.N ω/n FIG.. (Color online) ransmission through a single mixed region in cases with a) no background rotation and b) f =.N. he colour scale for both plots is indicated in the upper-left of a). for transmission across a single well-mixed region is recovered [6]: = { + [ sinh() sinθ )] } (+ () cos Θ coth. his is plotted in Fig. a. As expected, the transmission is close to unity if the horizontal wavelength is much greater than the mixed-layer depth ( ). For nonhydrostatic waves, which have frequency close to N, transmission can be large even with of order unity. A transmission spike occurs for values and ω/n where incident vertically propagating internal waves are resonant with interfacial waves on either flank of the mixed region [6]. In particular, this results in perfect transmission for incident waves with ω N and.4. In the case with non-zero Coriolis parameter, the explicit formula for in terms of and Θ can be found from (), but the expression is much more complicated. he result in the case f =.N is plotted in Fig. b. (Although characteristically in the ocean at mid-latitudes f.n, throughout this paper results are shown with f =.N so that the change in the transmission for ω f is clearer.) No transmission occurs if ω < f because the incident waves would themselves be evanescent in this case. For ω > f, the transmission plot differs fromthatwithf = onlyforinertiagravitywaves, which have frequency very close to f. We find there is negligible effect of f upon the predicted transmission if ω >.5f. C. ransmission across J steps In the case of a staircase with more than one step (J > ), an explicit formula for the transmission can be found by diagonalizing the matrix C in (). his matrix has determinant equal to one, meaning that the passage of waves through the staircase changes the phase relationship between coefficients A j and B j, but does not increase or decrease the magnitude of the disturbances. he (possibly complex) eigenvalues of C are and b + λ ± = b + ±b, (5) [ ( Γ ( )+ + Γ )] b (6) b +. (7) hus we can write C = PΛP, ( ) λ+ Λ =, (8) λ and P = P = Γ 4b b ( Γ (b +b ) Γ (b b ) ( Γ (b b ) Γ (b +b ) ) [ ( Γ ( ) + Γ )] (9) ), () hus an explicit expression can be found for C J : () C J = P Λ J P ( = b Λ +b Λ + Γ Λ ) () b Γ Λ b Λ +b Λ + Λ ± = [ λ J + ±λ J ]. () his result can be put into (9). hus, after significant algebra, the transmission coefficient is found to be [ ( ) ] δ+ Γ + + δ Γ + δ Γ Λ = +, (4) 4M b δ ± is given by (), δ = Γ + M, and Γ ± = b Λ ± b Λ +. Although complicated, the explicit form of in (4) makes it amenable to asymptotic analyses. For now, it is noted that the form of guarantees that the transmission lies between one and zero as the squared expression in (4) lies between zero and infinity, respectively. Also, it is straightforward to check that if J = in (4), the result reduces to the transmission given by ().

5 a) J = b) J = 5 c) J = ω/n FIG.. (Color online) As in Fig. but showing the transmission through a staircase composed of a) J =, b) 5 and c) evenly spaced steps. In all cases f =.N. he shortdashed and long-dashed lines show the predicted upper and lower bounds, respectively given by (5) and (6), on the region containing the sequence of transmission spikes. D. Bounds on ransmission Figure shows the transmission coefficient for waves passing through, 5 and steps. All three plots were constructed by taking f =.N. However, as in Figure, the transmission is found to be approximately the same if f = except for ω.5f, case the transmission drops off to zero at the low-frequency cutoff ω = f. As the number of steps increases, the transmission spike apparent in the single-step case weakens (for J = ) and disappears for J 4. Instead a new sequence of transmission spikes appear for which for finite values of at frequencies between f and N. If the staircase has J steps, there are J distinct bands of transmission spikes, including the spike at =. o illustrate this more clearly, Figure 4 shows values of the transmission coefficient as a function of with ω/n fixed and computed for J = 5, and steps. Focusing on any one plot, it is obvious that doubling the number of steps doubles the number of transmission spikes. Where the peaks occur for J = 5 they also occur for J = and. But the widths of the peaks become narrower. Whatever the value of J, the transmission is negligibly small if exceeds a cut-off, which is larger as ω/f increases. Finally the difference between a transmission peak and an adjacent transmission low is greater if ω is closer to f or if is closer the cut-off relative wavenumber. Some of these observations can be quantified by examination of the analytic solution. In particular, the bounds on the region in -ω/n space for which the transmission spikes occur are given by the condition that the discriminant of the eigenvalues given by (5) is zero: transmission spikes occur where λ ± is complex-valued; transmission is negligible if is large and λ ± are realvalued. he condition b = ( b + = ) in (7) becomes the following when written in terms of the relative horizontal wavenumber and frequency: ω = N γl { coth(γl) tanh(γl), γ is given by (5). If ω f, then γ k, in which case the upper transmission cut-off is given by ω/n = tanh(). (5) he lower bound on the region of transmission spikes (except that for ) is given by ω/n = coth(). (6) Both these curves are drawn in Fig.. It is clear that even for ω f, the approximation given by (5) wellrepresents the region between transmission spikes and negligible transmission. he energy-containing internal waves in the ocean are dominantly inertia gravity waves, which have long horizontal scale. hus we are particularly interested in the transmission occurring in the limit of small. If ω is sufficiently large that the effect of f can be neglected, the transmission decreases with increasing as () (J ) (kd) 6(ω/N ) [ (ω/n ) ] 4sin Θ, (7) Θ is given by (4) with f = and D = (J )L is approximately the total depth of the staircase if there are many steps (J ). he expression is accurate provided kd and ω f. More relevant for inertia gravity waves is the case with ω close to f. Assuming kd and (ω/f), the transmission decreases from unity as N 4ω f (kd) (kd) 4cos Θ, (8)

6 6 a) ω/f =.5 b) ω/f =..6 c) ω/f = FIG. 4. ransmission as a function of for a) ω/n =.5 (ω/f =.5), b) ω/n =. (ω/f =.), c) ω/n =. (ω/f =.). Plots show transmission through J = 5 steps (thin solid line), steps (thin red-dashed line) and steps (thin blue-dotted line). he thick short-dashed lines in each plot gives the approximation (8) to for and ω f for the case J = 5 and the thick long-dashed line in c) gives the approximation (7) to for and ω f for the case J = 5. Note that the vertical scale in c) is larger than in a) and b). In theory all peaks do reach unity, but the finite vertical resolution of the plots means that the peak is not always computed. the last expression supposes f N, and Θ is close to 9 for ω f. hese approximations are superimposed on the plots of versus in Fig. 4 for the case of J = 5 steps. he approximation given by (8) agrees well with the exact solution in the cases ω/f =.5 and. with better agreement to lower values of if ω is closer to f. In the case ω = f, (8) predicts a faster drop-off in than actually occurs. he exact result is closer to the approximate formula (7). hese approximate formulae have the potential to make predictions for inertia gravity waves transmitting through density staircases in the ocean. But, before pursuing this, it is necessary to consider how well the analytic results extend to predict transmission through realistic, uneven density staircases. IV. NUMERICAL SOLUION FOR UNEVEN SEPS Although density staircases observed in the ocean typically have a characteristic length scale, L, determining the depth of each step, there is nonetheless some variation in the extent of each step over the total depth of the staircase. his is represented in the density profile given by () in the case ǫ >. It is not feasible to find analytic solutions for the transmission of internal waves through an uneven staircase. But it is straightforward to compute the transmission numerically. he formula relating A to A J+ is given by (9) but with C J given by the product of the J matrices: Π J j= ( ( Γ ) ) Γ ǫσj Γ ǫσj (+ Γ ). (9) Of course this reduces to C J in the case ǫ = and, like C, the product of matrices has determinant equal to unity. For a given set of random values {σ j j =...J } with σ j, the product of matrices is computed iteratively to give a two-by-two matrix with real-valued elements. he result inserted in (9), the real and imaginary components separated out, and the transmission coefficient is then computed from the inverse of the sum of the squares of the real and imaginary parts of A /A J+. As a test of the efficacy of this method, it is confirmed that the procedure reproduces the analytic solution (4) for ǫ =. For comparison with Fig. c, Fig. 5a plots the transmission coefficient for a staircase with J = unevenly spaced steps in the case f =.N. Qualitatively, the plots appear similar with transmission spikes occurring

7 7 4 a) ransmission with unevenly-spaced steps b) ω/f =.5 c) ω/f =..6 d) ω/f = FIG. 5. a) As in Fig. c, but showing transmission through J = unevenly-spaced steps computed from one random realization of {σ j} with ǫ =.. he average transmission (solid lines) computed in random realizations is plotted as function of for b) ω/n =.5 (ω/f =.5), c) ω/n =. (ω/f =.) and d) ω/n =. (ω/f =.). he thin-dashed lines in the bottom three plots show the mean plus and minus the standard deviation. he thick-dashed lines show the analytic prediction (8). between the bounds predicted by (5) and (6). he obvious discrepancy between this transmission plot and the analytic solution is that the spacing between the peak transmission curves is more irregular. Just for what values of the transmission peaks occur more closely together or further apart depends upon the set of random values, {σ j }, that determines the depths of each step relative to the mean depth L. he plots of transmission versus for three different frequencies, shown in Fig. 5b-d, result from computing the transmission for different realizations of the set of random variables, {σ j } and then averaging the results. he solid line in each plot is the mean transmission and the dashed lines plot the standard deviation above and below the mean. Although the variation of the transmission becomes larger as approaches the upper critical value given by (5), it is obvious that the transmission is insensitive to the variation in step sizes if is very small. In particular, the value of the transmission between.8 and is

8 8 well predicted for f < ω.f. on the wavelength of transmitting waves by λ l, we have λ > λ l πd [ ω f / R]. () N V. DISCUSSION AND CONCLUSIONS An analytic prediction was made for the transmission of internal waves through a staircase density profile with an arbitrary number of evenly spaced steps. Compared with the numerical solution computed for waves passing through an uneven-staircase, the analytic predictions for the high- and low- wavenumber cut-offs bounding the region of transmission spikes were found to remain accurate. Crucially, the transmission as it falls from unity for increasing from zero was found to well-predict the transmission of waves through an uneven staircase. In particular, for inertia gravity waves having ω f f, (ω/f) and kd (with L and D, respectively, the depths of each step and the whole staircase), the transmission is given approximately by (8). In practice, (8) is found to well-predict the transmission as it falls to.8 if f < ω.f. And so we can invert this approximate equation to answer the following question: what is the lower bound on the horizontal wavelength of incident inertia gravity waves such that their transmission lies between and R, with R (the reflection coefficient) small? Denoting the lower bound In particular, taking f =.N, ω =.5f and R =., the lower bound is λ l.6 D. So for inertia gravity waves incident upon a staircase of total depth m [], the waves transmit with little reflection if their horizontal wavelength is larger than 5km. his is on the horizontal scale of internal waves generated by storms. And so, consistent with the assertion of Ghaemsaidi et al [] based on their study of transmission across two weakly stratified layers, this work predicts that density staircases may act as an effective filter allowing waves with longer horizontal wavelength and larger frequencies to transmit through to the abyss. Shorter waves may reflect or transmit substantially. According to (5), strong reflection of inertia gravity waves is certain if f/n. aking f.n, the upper bound on the horizontal wavelength of waves that must reflect from a staircase with m deep steps is λ Ru πln /f m, ACKNOWLEDGMENS his work was supported by funding through the Discovery Grant program of the Natural Sciences and Engineering Research Council of Canada (NSERC). [] R. W. Schmitt, Annu. Rev. Fluid Mech. 6, 55 (994). [] O. M. Johannessen and O. S. Lee, Deep-Sea Res., 69 (974). [] R. Molcard and A. J. Williams, Mem. Soc. R. Sci. Liege 6, 9 (975). [4] P. A. Mazeika, J. Phys. Oceanogr. 4, 446 (974). [5] R. B. Lambert and W. Sturges, Deep-Sea Res. 4, (977). [6] R. W. Schmitt, H. Perkins, J. D. Boyd, and M. C. Stalcup, Deep-Sea Res. 4, 697 (987). [7] R. A. Woodgate, K. Aagaard, J. H. Swift, W. M. S. Jr., and K. K. Falkner, J. Geophys. Res., C5 (7). [8] L. Rainville and P. Winsor, Geophys. Res. Lett. 5, L866 (8). [9] M.-L. immermans, J. oole, R. Krishfield, and P. Winsor, J. Geophys. Res., CA (8). [] M.-L. immermans, S. Cole, and J. oole, J. Phys. Oceanogr. 4, 659 (). [] L. Rainville and R. A. Woodgate, Geophys. Res. Lett. 6, L64 (9). [] L. Rainville, C. M. Lee, and R. A. Woodgate, Oceanography 4, 6 (). [] B. R. Sutherland, Internal Gravity Waves (Cambridge University Press, Cambridge, UK, ) p. 78. [4] E. Salusti, Deep-Sea Res. 5, 947 (978). [5] C. Eckart, Phys. Fluids 4, 79 (96). [6] B. R. Sutherland and K. Yewchuk, J. Fluid Mech. 5, 5 (4). [7] G. L. Brown and B. R. Sutherland, Atmos. Ocean 45, 47 (7). [8] J.. Nault and B. R. Sutherland, Phys. Fluids, 66 (8), doi:.6/ [9] M. Mathur and. Peacock, J. Fluid Mech. 69, (9). [] M. Mathur and. Peacock, Phys. Rev. Lett. 4, 85 (). [] K. Gregory and B. R. Sutherland, Phys. Fluids, 66 (), doi:.6/ [] S. Ghaemsaidi, H. V. Dosser, L. Rainville, and. Peacock, J. Fluid Mech., in press (6). [] G. L. Brown, A. B. G. Bush, and B. R. Sutherland, Phys. Fluids, 66 (8), doi:.6/.9968.