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336 JOURNAL OF PHYSICAL VOLUME 31

The Transparency of Barriers to Rossby Waves: The Rossby Slit Problem*

JOSEPH PEDLOSKY Department of , Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 6 August 1999, in ®nal form 29 February 2000)

ABSTRACT The transmission of barotropic Rossby wave energy through a meridional barrier pierced only by narrow gaps is studied with a quasigeostrophic model. The incident Rossby wave has the form of either a plane wave or a localized beam of wave energy. The application of Kelvin's theorem on a contour girdling the separated segments of the barrier demonstrate the necessity for wave transmission and helps determine the amplitude of the transmitted wave. When the meridional scale of the incident wave is much larger than the gaps in the barrier, the transmission becomes independent of the gap width. The problem of a barrier with two or three gaps is discussed. In the former case a single bubble of wave energy issues from the region spanned by the separated barrier segment. When a third gap or slit is allowed, the structure of the transmitted wave alters depending on the symmetry of the wave ®eld and the angle of the incidence of the ray trajectory of the incident wave.

1. Introduction is rendered problematic by several complicating factors. For large-scale oceanic motions for which the earth's The pure Rossby wave exists in a ¯at-bottomed ocean rotation is a dominant factor in the dynamics, the mes- in the absence of an advecting mean ¯uid ¯ow. For the senger of energy and is the Rossby wave. These more rapidly propagating barotropic Rossby wave the waves, which owe their existence to the variation of the neglect of the mean ¯ow is plausible, less so for the local normal component of the earth's rotation, the so- baroclinic modes. Perhaps of equal importance for the called-beta effect, have periods much longer than a day. possible modi®cation of the simple theory for the Ross- In the presence of density strati®cation, baroclinic Ross- by wave is the presence of uneven bottom topography. by waves may have timescales of months or years while If the bottom slope is mild enough, the effect on the the depth-independent, barotropic Rossby wave may wave is a quantitative distortion of the wave's propa- have timescales of the order of weeks or less. Obser- gation rather than a qualitative change in the underlying vations of Rossby waves in the are rendered physics. However, in the actual ocean the presence of dif®cult because of the large space scales and the need very steep topography such as the midocean ridge sys- for nearly synoptic observation, but recent evidence tem or extensive island arcs (such as that separating the from data has given ample evidence Caribbean from the Atlantic) present potentially signif- for their presence and their ability to propagate great icant barriers to the propagation of the Rossby waves. distances across the ocean basins (see, e.g., Jacobs et Indeed, such steep topographic or island barriers might, al. 1993; Chelton and Schlax 1996). at ®rst sight, appear to present an impenetrable barrier The theory of Rossby waves has a long history start- to the propagation of those waves whose spatial scales ing with the seminal paper by Rossby et al. (1939) and are much larger than the gaps between the segments of much of the basic nature of the wave's dynamics can the island arc or, for motion in the deep ocean, for mo- be found in current textbooks (e.g., Pedlosky 1987). The tions with scales larger than the gaps formed by trans- application of the simplest theory to the natural ocean form faults between segments of the midocean ridge system. Very little work has been done theoretically to ex- amine such problems in the context of midlatitude * Woods Hole Oceanographic Institution Contribution Number 9928. Rossby waves (for the equatorial zone, however, see Clarke 1991). In a recent paper (Pedlosky and Spall 1999) the linear Rossby normal modes of basin scale Corresponding author address: Dr. Joseph Pedlosky, Woods Hole were examined for a rectangular ocean basin that was Oceanographic Institution, Dept. of Physical Oceanography, Clark 363 MS #21, Woods Hole, MA 02543. nearly divided in two by a barrier meant to represent E-mail: [email protected] a segment of a midocean ridge. Surprisingly, normal

᭧ 2001 American Meteorological Society

Unauthenticated | Downloaded 09/25/21 05:20 PM UTC FEBRUARY 2001 PEDLOSKY 337 modes were found whose frequencies were very close to the normal modes of the full basin in the absence of the barrier and the modal structure wrapped around the barrier to ®ll the basin allowing a full basin re- sponse in spite of the very narrow passages of com- munication between the subbasins. The ability of the normal modes to involve the entire basin was shown to be related to the constraint on the motion provided by Kelvin's theorem applied on a contour girdling the ridge or island segment. Key to this process is the required presence of at least two gaps in the barrier, so rendering a portion of the ridge a detached ``island.'' While a single narrow gap allows only a small, slow leakage of energy from one subbasin to the adjacent basin (see, e.g., Lamb 1932, para 300), the island char- acter of the segment involves the whole length of the segment as an ef®cient virtual wave maker acting on the ¯uid in the neighboring basin. A related Rossby wave problem in which Kelvin's FIG. 1. Schematic of a Rossby wave, represented by the thick arrow, theorem also applies has recently been treated by Firing as it impinges on a barrier pierced (in this case) by two gaps of equal, et al. (1999) in which the interaction of a Rossby wave small width d. The frequency of the carrier wave of the beam or and an isolated island is considered in the long-wave, packet is ␻ and the direction is determined by the Rossby wave's baroclinic limit. In that case the extension of the God- . frey Island rule (Godfrey 1989, see also Pedlosky et al. 1997), rather than the direct application of Kelvin's the- orem is pertinent case and the transparency of the multiply pierced barrier In an effort to understand more deeply the apparent to impinging Rossby waves is clearly illustrated. The transparency of such barriers to the propagation of Ross- introduction of additional gaps, as evidenced by the ex- by wave energy the problem of the transmission of an ample where a third gap is added, allows the wave a incident Rossby wave, either as a packet or plane wave, greater variety of passageways through the barrier and is considered here when the wave impinges on a barrier introduces more complex structure to the transmitted consisting of a solid meridional wall pierced by two or signal. The nature of the transmitted signal turns out to three gaps whose widths are small compared to the me- be a simple function of the properties of the oncoming ridional scale of the advancing wave. To ®x ideas we wave. For a wave of a given frequency, the kinematics can consider a wave with frequency ␻ arriving from the of the impinging wave is determined by the direction east, as in Fig. 1, with a group velocity as denoted by of the group velocity of the impinging beam and the the large arrow, heading toward a barrier with, in this resulting transmission is completely determined by the case, two small gaps of width d. We will consider the consequent kinematic structure. Except for special an- case where the ratio gles of incidence to be described, the barrier exhibits a d/L K 1, (1.1) remarkable transparency to the Rossby wave transmis- sion. where L is the characteristic meridional scale of the In section 2 we describe the basic dynamical model approaching wave. and the important role of Kelvin's theorem for the trans- The analysis uses the quasigeostrophic potential vor- mission dynamics. In section 3 the problem of the wave ticity equation as the basic dynamical equation (Ped- impinging on a two-gap barrier is described, while in losky 1987) and the explicit analysis will be limited for section 4 the analysis is extended to describe the nature simplicity to the case of a homogeneous ¯uid so that of the transmission when a third gap is added. Section the motion is barotropic. The baroclinic case, if the gaps 5 discusses the overall character of the results and spec- in the barrier extend over the full depth of the ¯uid, ulates on the implications of the theory for ocean dy- introduces no qualitatively new dynamical element since namics. each baroclinic mode satis®es a wave equation mathe- matically similar to the barotropic model equation. In the case of a barrier with gaps that span only partially 2. The model the depth of a strati®ed ¯uid, vertical mode mixing will For purposes of simplicity we utilize the quasigeo- occur as the wave traverses the barrier and this more strophic equation for a homogeneous complex situation is the subject of future study. ¯uid. Since the bottom of the basin will be idealized as The question of the wave transmission and the struc- ¯at except for the barrier representing the steep topog- ture of the transmitted wave is easily discussed in this raphy of the island arc or the midocean ridge, the gov-

Unauthenticated | Downloaded 09/25/21 05:20 PM UTC 338 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 erning equation in its linearized, nondimensional form is (Pedlosky 1987): ␺ץץ (2␺ ϩϭ0. (2.1ٌ xץ tץ In (2.1) time has been scaled with the characteristic Rossby timescale, (␤L)Ϫ1, where L is the characteristic scale for the impinging wave. The eastward and north- ward velocity components are determined from ␺ by the relations: ␺ץ ␺ץ (u, ␷) ϭϪ , . (2.2) xץ yץ΂΃ Plane wave solutions of (2.1) have the form: ␺ ϭ Aei(kxooϩlyϪ␻t) (2.3a)

Ϫko ␻ ϭ , (2.3b) FIG.2.Thex and y (k and l ) of a wave packet at k22ϩ l o o oo frequency ␻ ϭϪ0.08 as a function of the ratio R of the group velocity which yields a group velocity for the wave, components; Ko is the total . k22Ϫ l 2kl c ϭ i ooϩ j oo, (2.4a) g 44 KKoo K 222ϭ k ϩ l , (2.4b) 1 ooo k ϭϪ , (2.7b) where i, j are unit vectors in the eastward and northward 2␻ directions respectively. The relations (2.3) and (2.4) al- a(l) ϭ (k2Ϫ l 2) 1/2 . (2.7c) low us to determine ko and lo if the frequency and the direction of the group velocity in the x, y plane is spec- The minus sign in front of a(l) in (2.7a) is chosen to i®ed, thus satisfy the condition that the impinging wave have a

2 group velocity directed to the west. Ϫ1 R Similarly, the re¯ected and transmitted waves can be ko ϵ (2.5a) ΂΃2␻ [1 ϩ R2ϩ {1 ϩ R 2}] 1/2 represented respectively as Ϫ1 R 1 ϱ lo ϵ , (2.5b) ␺R ϭ dlAR(l) ΂΃2␻ [1 ϩ R21/2] ͙2␲ ͵ Ϫϱ where R is the ratio of the meridional to zonal group ϫ expi[(k ϩ a)x ϩ ly Ϫ ␻t], (2.8b) velocity components; that is, ϱ c 1 gy ␺T ϭ dlAT(l) R ϭ . (2.6) 2␲ ͵ ͙ Ϫϱ cgx

Note that for ko Ͼ 0, ␻ Ͻ 0. For example, for Ϫ␻ ϫ expi[(k Ϫ a)x ϩ ly Ϫ ␻t]. (2.8c) ϭ 0.08, Fig. 2 shows the associated wavenumbers for a range of R Ͻ 0 representing a wave packet propagating Note that for those values of l for which a is imag- northwestward. Here R ϭϪ1 corresponds to a beam inary there will be wave energy trapped to the vicinity traveling at 45Њ to the northwest while R ϭ 0 corre- of the barrier. Hence some of the incident energy will sponds to a purely westward approach to the barrier remain trapped rather than re¯ected or transmitted. shown in Fig. 1. We assume that the input wave amplitude, AI(l)is The incident wave can be represented by a Fourier known as well as both the (single) frequency of the wave integral over all y wavenumbers. If the wave is ap- and the direction of the incident wave and the goal is proaching from the east, it is easy to establish that the to determine the re¯ected, and more interestingly, the appropriate solution to (2.1) is transmitted wave amplitudes. ϱ Consider the islandlike ridge segment shown in Fig. 1 1 and consider a contour that encircles the segment and ␺I ϭ dlAI(l) 2␲ ͵ is coincident with the perimeter of the segment. For a ͙ Ϫϱ ¯uid that is inviscid and homogenous, Kelvin's theorem ϫ expi[(k Ϫ a(l))x ϩ ly Ϫ ␻t], (2.7a) applied to that contour, which is always a streamline, where yields

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We arrange our coordinate system so that the isolated ץ u ´ ds ϭ 0, (2.9) ridge segment occupies the region Ϫyo Յ y Յ yo and, t Ͷץ C for the sake of simplicity, we assume that the two gaps where C is the contour girdling the segment. It is im- have equal width d. As noted above, along the segment portant to note that (2.9) is valid for both linear and of the barrier for Ϫyo Յ y Յ yo, the streamfunction must nonlinear motions. equal the constant ⌿I. On the other hand, on the re- For motion that is periodic with frequency ␻; (2.9) maining length of the barrier the streamfunction is zero. requires that the circulation itself vanish, that is, that, For very narrow gaps satisfying (1.1) we will assume that the streamfunction smoothly and linearly interpo-

lates between 0 and ⌿I. In the paper of Pedlosky and Ͷ u ´ ds ϭ 0. (2.10) Spall (1999) this assumption was carefully checked by C direct numerical integration of the governing equations and found to yield excellent agreement between this Consider an incident wave that has a component of analytical ansatz and the numerical results for the large- velocity parallel to the island segment with a scale lon- scale wave ®eld in the normal modes. We employ this ger than the segment itself. In that case the contribution simplifying assumption here. Thus, on x ϭ 0, the po- of the incident wave to (2.10) on the western side of sition of the pierced barrier, we insist that ␺ satisfy, the segment will be nonzero. Unless the re¯ected wave  happens fortuitously to cancel both the normal velocity 0, y Յ Ϫyo Ϫ d to the segment as well as the average tangential velocity (y Ϫ (y ϩ d))/d, Ϫy Ϫ d Յ y Յ Ϫy along the segment, (2.10) will be violated. Neglecting  ooo  such special cases for now, it is intuitively clear that ␺ ϭ⌿Ioo1, Ϫy Յ y Յ y the satisfaction of (2.10) generally requires the existence (y ϩ d Ϫ y)/d, y Յ y Յ y ϩ d of a compensating oscillating motion parallel to the seg-  ooo ment on the western side of the barrier. That is, Kelvin's 0, yo ϩ d Յ y. theorem implies that generally the Rossby wave must (3.1) slip through the narrow slits and yield a balancing wave ®eld of large scale on the side of the barrier opposite This condition is an approximation to the full boundary to the impinging wave. Naturally, direct calculation is conditions of continuity of streamfunction and ®rst de- required to verify this and such calculations are given rivative in the region of the gaps. The nearly zonal nature below. It is important to note, however, that the general of the ¯ow there, imposed by the narrowness of the gaps, requirement for this type of Rossby wave tunneling is renders the second condition essentially redundant. a direct consequence of the robust requirement of (2.10), Therefore, on x ϭ 0, the streamfunction must satisfy which is valid even for nonlinear wave problems as long ⌿ ϱ as the circulation around the ridge is initially zero, for ␺ ϭ I g(l)edl,ily (3.2) 2␲ ͵ example, before the impinging wave arrives. ͙ Ϫϱ On that part of the barrier solidly attached to the outer where boundary of the basin, that is, the long northern and southern segments of the barrier in Fig. 1 the stream- 211/2 function must take on the value of ␺ on the basin perim- g(l) ϭ {cos(lyoo) Ϫ cos(l[y ϩ d])} (3.3) ␲ ld2 eter, which we may take to be zero. On the island segment ΂΃ itself, the streamfunction must also be spatially constant is the Fourier transform of the function of y given by (although in the linear problem it will be oscillating pe- (3.1). Ϫi␻t riodically in time) with a value ␺ ϭ⌿Ie . The value Note that g(l) is not singular at l ϭ 0 and that in the of ⌿I must be determined as part of the solution to the limit, as d → 0, problem and its value determines the ¯uid ¯ux through each gap. In the case of two gaps as in Fig. 1 the ¯ux 21/2 sinly g(l) → o (3.4) is equal and opposite in each gap at each moment. ΂΃␲ l so that in the small d limit the only scale against which 3. The two slit problem l is measured is the length of the island segment ya rather We begin by examining the wave transmission prob- than the gap width. lem for the case where the barrier contains two narrow On x ϭ 0 one matching condition for the wave ®eld gaps, or slits, as shown schematically in Fig. 1. The is, using (2.7), (2.8), and (3.3), barrier then consists of an islandlike segment between A ϩ A ϭ A ϭ⌿g(l). (3.5) the two slits while north and south of the slits the barrier I R T I continues until, as imagined in this simple example, it The second condition that must be applied in order encounters the distant boundaries of the . to relate the re¯ected and transmitted wave amplitudes

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FIG. 3. (a) Plot of ⌿I as a function of R for the case ␻ ϭϪ0.08 for the incident wave amplitude given by the Gaussian distribution of (3.8); the parameter b measures the width of the Gaussian. (b) For

b ϭ 1, lo ϭ 5 the dotted curve shows the Gaussian wave amplitude distribution while the solid curve shows the modulus of the kernel

of the integral in the nominator of (3.8) for ⌿I. (c) As in (a) but for an incident plane wave.

to the incident wave comes from the application of excite a Rossby wave to the west of the barrier with an Kelvin's theorem (2.10) amplitude of the order of the incident wave amplitude Since the barrier is idealized as a very thin meridional and with a frequency and wavenumber that depend only segment, this condition reduces to on the frequency of the incident wave. The magnitude yy y of the response is relatively independent of the gap ␺ץ␺ oץ ␺ץoo IRTdy ϩ dy ϭ dy, width d when d/L is very small, as can be seen above. xץ xץ xץ͵͵͵ ϪyooϪy Ϫy o The structure of the transmitted wave will depend, rath-

er, on the scale yo. x ϭ 0. (3.6) Figure 3a shows the response diagram for the island

Using (2.7), (2.8), and (3.5) this yields the constant constant ⌿I as a function of the slope of the incident

⌿I as wave energy trajectory, that is, as a function of R [see (2.5) and (2.6)], which for a given ␻ determines both ϱϱcomponents of the wave vector of the incident Rossby ⌿ϭ dlA (l)a(l) sin(ly )/ldlg(l)a(l) sin(ly )/l, I ͵ Io͵ o wave. In Fig. 3a the incident Rossby wave is imagined Ϫϱ΋ Ϫϱ to have a wave amplitude distribution, (3.7)

1 22 Ϫ(lϪlo)/b where a(l) is given by (2.7c). AI ϭ e (3.8) Unless the numerator in (3.7) happens to vanish, the b Ϫlax oscillating streamfunction on the island, ⌿Ie , will so that, while the carrier wave has wavenumber lo in

Unauthenticated | Downloaded 09/25/21 05:20 PM UTC FEBRUARY 2001 PEDLOSKY 341 the y direction, the energy is contained in a beam of 2 1/2 ⌿ r1/2|x| ␺ I ei(k(rϩx)Ϫ␲/4) width 2/b. The magnitude of ⌿I depends on the integral T ഠ 3/2 2 of the product of this wave amplitude with the kernel ΂΃␲ dky a(l) sin(ly)/l, which is shown in Fig. 3b. Note that ϫ {cos(kyyoo/r) Ϫ cos(ky(y ϩ d)/r)}, (3.11) depending on the value of l the center of the Gaussian o 2 2 1/2 in (3.8) will contribute a greater or lesser amount to where r ϭ (x ϩ y ) . The stationary phase approxi- mation to the solution is shown in Fig. 4c. The agree- the integral in (3.7). As lo increases (as R increases in magnitude), the integral will tend to diminish as shown ment between (3.11) and the transmitted wave obtained in Fig. 3a. Similarly, for small magnitudes of R, that by the numerical evaluation of the Fourier integral so- is, for the packet moving nearly zonally westward, the lutions, which led to Fig. 4b, is very good. Note that smaller the value of b, and thus the wider the beam of in the limit as d tends to zero, (3.11) tends to a limit independent of d. In Figs. 4d and 4e the calculation is incoming energy, the greater will be the value of ⌿ . I repeated for a smaller value of b for the amplitude On the other hand for larger values of R, that is, for Gaussian. Here b ϭ 2 instead of 5 as in Fig. 4a. This more glancing angles of incidence, the carrier merid- wider beam at the lower angle of incidence is more ional wavenumber l goes up and the product of A o I successful at transmitting wave energy. with the kernel diminishes. In this case increasing the The value of meridional wavenumber l correspond- packet width in wavenumber space, by increasing b s ing to the stationary wave point is easily shown to be allows a greater overlap of the incident wave amplitude with larger values of the kernel and the response, that ls ϭ k(y/r), (3.12) is, ⌿I , increases. In Fig. 3a, this occurs at a value of R ഠ Ϫ0.32. and so depends only on the frequency of the incident A considerable simpli®cation of the algebra occurs wave. The amplitude of the transmitted wave, as seen when the incident wave is a plane wave. This occurs from (3.11), depends on the length of the island segment y and goes to zero as this length diminishes to zero. when we choose the incident amplitude to be o When the angle of incidence is greater, as Fig. 3 pre- dicts, the transmitted wave amplitude is much smaller. AI ϭ ͙␲␦(l Ϫ lo), (3.9) Figure 5 shows the wave ®eld for the case where ␻, where the constant before the delta function is chosen again, ϭϪ0.08 but where the angle of approach of the to yield the same overall integrated amplitude in (3.9) wave is about 30Њ to a circle (R ϭϪ0.5). There as in (3.8). Figure 3c shows the response curve in this is no indication of a transmitted wave. Of course, as case and it is clearly similar to that of 3a, although the Fig. 3 implies, there will be a small wave ®eld trans- response is a bit higher at small angles of incidence. mitted but its magnitude is too small to be seen in the Figure 4 shows the wave solution for the case R ϭ contour plot of Fig. 5. The wave is very nearly perfectly Ϫ0.1, that is, very nearly a head-on approach of the re¯ected at these large angles. The reason is easy to wave energy to the barrier. In this case a plane wave comprehend. As R increases, the meridional wave- number goes up as shown in Fig. 2. As l gets larger, with frequency ␻ ϭϪ0.08 is shown. These ®gures are there is greater cancellation in the integral condition obtained by numerically evaluating the Fourier integrals imposed by Kelvin's theorem so that the incident wave at each ®eld point. Figure 4a shows the real part of the makes a very small contribution to the circulation on wave ®eld at t 0 and t /4. The transmitted ␻ ϭ ␻ ϭ ␲ the eastern side of the island segment and, as we have wave ®eld is more zonal than the ®eld of the incident seen, this is the essential ingredient in determining the and re¯ected wave, as it is shaped by the position of tunneling of the Rossby wave through the narrow gaps. the gaps and the length of the island segment. The mod- The wavenumbers of the incident wave depend on the ulus of the wave ®eld is shown in Fig. 4b with several frequency as well as on R. At lower frequency the wav- contours numerically labeled. The transmitted wave, enumbers increase as shown by (2.5). This has important which has an amplitude of the same order as the incident consequences for the transmission properties. Figure 6 and re¯ected wave ®elds, is seen to balloon out from shows the response curve for ⌿I as a function of R when the two gaps in the barrier as a single bubble achieving ␻ ϭϪ0.02 for an incident plane wave. Note that ⌿I a scale greater than the island segment as well as the goes through zero at R ഠ Ϫ0.27. This follows from the gap widths. numerator of (3.7) when (3.9) applies, for then ⌿I will A straightforward application of the method of sta- be proportional to sin(loyo), which vanishes whenever loyo tionary phase, when applied to the transmitted wave (the ϭ n␲. The ®rst such root occurs for the case yo ϭ 0.5, harmonic dependence on time has been suppressed), as in these calculations, when lo is 2␲, which in turn occurs at the value of R ഠ Ϫ0.27. At this exceptional ⌿ ϱ I i(lyϩ(kϪa(l))x) value the Kelvin integral for the sum of the incident and ␺T ϭ g(l)edl,(3.10) 2␲ ͵ re¯ected wave is exactly zero so that no transmission is ͙ Ϫϱ necessary to satisfy Kelvin's theorem. For this ®nite set yields for large negative x and large y, of approach angles the transmission will be zero. For all

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FIG. 4. (a) The wave ®eld at t ϭ 0 (upper panel) and ␻t ϭ ␲/4 lower panel for R ϭϪ0.1 and ␻ ϭϪ0.08. The transmitted wave amplitude is shown in x Ͻ 0. (b) The modulus of the wave amplitude. (c) The stationary phase solution for the wave amplitude modulus. Note the agreement as indicated by the labeled contours in (b) and (c). (d) As in (a) but for a wider beam of incident wave energy (b ϭ 2). (e) As in (b) except for b ϭ 2.

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FIG.4.(Continued) other angles there will be some transmission while Figs. 4. The three slit problem 3 and 6 show that the maximum transmission occurs when the incident wave is propagating zonally, directly When a third opening is added to the barrier, as shown at the meridionally oriented barrier. schematically in Fig. 7, new routes for wave transmis-

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FIG.4.(Continued)

FIG. 5. As in the upper panel of Fig. 4a except that now the angle of incidence is steeper so that R ϭϪ0.5. Very little wave energy tunnels through the gaps in the barrier. At the same level of contouring as Fig. 4 it appears as though the wave is completely re¯ected by the barrier although

⌿I is not precisely zero.

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FIG. 7. Schematic of the barrier with three slits. On the northern separated segment the streamfunction, aside from a periodic function FIG. 6. (a) Plot of ⌿ as a function of R for ␻ ϭϪ0.02. Note that I of time, equals ⌿ , while on the more southern segment the stream- ⌿ passes through zero for R ϭϪ0.28. (b) The wavenumbers of the 2 I function equals ⌿ . The northern gap of width d , the middle gap of incident wave as a function of R for ␻ ϭϪ0.02. I n width dm, and the southern gap of width ds will all be taken to be of equal width, d, in the calculations reported in the text. sion become available to the oncoming wave. To keep the analysis as simple as possible, the widths of the relevant to the wave transmission. In the second case, three gaps will each be taken to be of the same width when ⌿1 ϭϪ⌿2, the ¯ow through the northern and and equal to d. Only quantitative differences occur if southern gaps are in the same direction and are com- they are different as long as they each are small in the pensated by a return ¯ow of twice these individual trans- sense of (1.1). There are now two island constants to ports through the middle gap. This solution is antisym- determine, ⌿1 on the southern segment and ⌿2 on the metric about the midpoint of the barrier and so involves northern segment. In the case where the three slits have wave scales that are somewhat shorter in the meridional the same width, the symmetry of the geometry implies direction. Such antisymmetric motions would automat- that the solution for the transmitted wave can be thought ically satisfy Kelvin's theorem for the combined broken of as the sum of a solution in which the two island segments in the absence of a middle gap. Now, since constants are the same and one in which they are equal Kelvin's theorem must apply to each segment separately, in magnitude and opposite in sign. In the ®rst case, when such antisymmetric motions will force an antisymmetric

⌿1 ϭ⌿2, there will be no ¯ow through the middle gap response on the western side of the barrier and en- but equal and opposite ¯ows through the northern and courage wave transmission hitherto prohibited. southern gaps. This is essentially identical to the two The formulation of the problem is nearly identical to slit problem, the middle gap is inactive and largely ir- that of section 2 for the two-gap problem. On x ϭ 0,

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where the barrier is located, the streamfunction is taken AI(l) ϩ AR(l) ϭ AT(l) as ϭ⌿g (l) ϩ [⌿ Ϫ⌿ ]g (l) ϩ⌿ g (l),  1 1 2 1 12 2 2 y Ϫ (y1s Ϫ d) (4.2) ⌿11, y s Ϫ d Յ y Յ y1s Ά·d where 1 ⌿11, y s Յ y Յ y1n ilyo ild g1 ϭ e (1 Ϫ e )/d (4.3a)  ͙2␲l2 y Ϫ y  1n 1 ␺ ϭ⌿ϩ121(⌿Ϫ⌿),y 1n Յ y Յ y2s ild/2 Ϫild/2 d g12 ϭϪ (e Ϫ e )/d (4.3b) ͙2␲l2

⌿22, y s Յ y Յ y2n 1 Ϫilyo Ϫild g2 ϭ e (1 Ϫ e )/d. (4.3c) ͙2␲l2  y ϩ d Ϫ y 2n ⌿22, y n Յ y Յ y2n ϩ d, Kelvin's theorem (3.6) must now be applied separately  Ά·d to both island segments. After this is done, two equations (4.1) in the constants ⌿1 and ⌿2 are obtained. It is convenient where y1s, y1n, y 2s, y 2n are the southern and northern to consider the sum and difference of those equations to termini of the segments 1 and 2, respectively. For the obtain, separately, equations for the sum and difference sake of symmetry and simplicity we will consider the of the two island constants since, as noted above, they case where y 2n ϵ yo ϭϪy1s so that with equal gap widths refer to two opposite symmetries of the solution for the the geometry of the barrier and gaps is completely given transmitted wave and are the natural building blocks for by the two parameters yo and d. the total solution. With a little algebra, the application of Thus on x ϭ 0, (3.5) is replaced by the condition, (2.7), (2.8), (3.6), (4.2), and (4.3) yields,

ϱ Aa{sinly Ϫ sinld/2} dl/l ͵ Io Ϫϱ ⌿ϩ⌿ϭ and (4.4) 12 1 ϱ a dl [sinlyoooϪ sinld/2][cosly Ϫ cosl(y ϩ d)] d 2␲ ͵ l3 ͙ Ϫϱ

ϱ Aa{cosld/2 Ϫ cosly } dl/l ͵ Io Ϫϱ ⌿Ϫ⌿ϭ . (4.5) 12 iaϱ dl [cosld/2 Ϫ coslyoo][sinly Ϫ sinl(y oϩ d) ϩ 2 sinld/2] d 2␲ ͵ l3 ͙ Ϫϱ

Thus, if the incident wave were symmetric in y (and at this angle the difference of ⌿1 and ⌿2 is zero. Thus, thus in y wavenumber), the numerator of (4.5) would at this angle the three slit barrier really still acts as a two vanish and both island constants would be identical. On slit barrier. As the angle of incidence increases, that is, the other hand, if the incident wave were antisymmetric as R becomes numerically larger, the sum of the island about y ϭ 0, which is the midpoint of the barrier, then constants decreases while their difference increases. The the sum of the island constants would vanish. For a gen- difference peaks at a value of R slightly smaller, alge- eral form of the incident wave the structure of the trans- braically, than Ϫ0.5. At this angle the transmitted wave mitted wave is a sum of these two extreme symmetries. will have a dipolar character as strong ¯ow through the Again for simplicity we consider the case where the center gap is compensated for by ¯ow in the opposite incident wave is a plane wave so that AI is given by (3.9). direction through the two gaps at the farther termini of It is then particularly easy to calculate the sum and dif- the segments. It is important to note that, since the trans- ference of the island constants ⌿1 and ⌿2. Figure 8a mission of the difference of ⌿1 and ⌿2 occurs at larger shows the island constants for the case where ␻ of the values of R (numerically) where the sum has already oncoming wave is Ϫ0.08. In that panel the sum of ⌿1 substantially diminished, the range of values of R where and ⌿2 is indicated with a ϩ sign while the difference transmission occurs is considerably increased. More gaps, of ⌿1 and ⌿2 is marked with a ⅜. Note, that as in the naturally, implies more transmission, but the nature of two slit case the sum of ⌿1 and ⌿2 attains its maximum the transmitted wave changes as shown below. at a directly zonal approach to the barrier (R ϭ 0) and At lower frequencies for the oncoming wave a rough-

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FIG. 8. (a) The response curve at ␻ ϭϪ0.08 for the sum difference

of ⌿1 and ⌿2 as a function of R. The sum is shown with a plus sign while the difference is indicated with an open circle. (b) As in (a) but for ␻ ϭϪ0.04. (c) As in (a) but for ␻ ϭϪ0.02

ly similar situation occurs. However, at lower frequen- coslyo Ϫ cosld/2 ϭ 0. (4.6b) cies the same phenomenon occurs as in the two slit Since these points do not coincide, there is no simple problem. Namely, for some special angles of approach rule for which of the two dominate at any particular one or the other of the two numerators in (4.4) and (4.5) frequency. For example, Fig. 8b shows the response will vanish. For the plane wave this will occur for the curve for the sum and difference of ⌿1 and ⌿ 2 when sum of ⌿1 and ⌿ 2, where ␻ ϭϪ0.04, while Fig. 8c shows the same response sinly Ϫ sinld/2 ϭ 0, (4.6a) curve for the case when ␻ ϭϪ0.02. o The form of the transmitted wave is clearly shown while for the difference of ⌿1 and ⌿ 2 this occurs when by the stationary phase approximation. Since

⌿ ϱ cosly Ϫ cosl(y ϩ d) sum i(lyϩ(kϪa)x) oo ␺T ϭ dle 2␲ ͵ ld2 ͙ Ϫϱ [] i⌿ ϱ sinly Ϫ sinl(y ϩ d) ϩ 2 sinld/2 ϩ diff dlei(lyϩ(kϪa)x) oo , (4.7) 2␲ ͵ ld2 ͙ Ϫϱ []

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FIG. 9. The stationary wave approximation for the transmitted wave in the three-gap model. (a) For ␻ ϭϪ0.08 and R ϭϪ0.01, the modulus of the transmitted wave. (b) The wave ®eld at (top) ␻t ϭ 0, and (bottom) ␻t ϭ ␲/2. (c) and (d) As in (a) and (b) except that R ϭϪ1.0. Note the change in the structure of the transmitted wave. In the case of R ϭϪ1.0 the middle cap has roughly twice the ¯ux as the two gaps at the extreme termini of the islands.

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FIG.9.(Continued)

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FIG. 10. (a) Four snapshots of the Rossby wave normal mode excited by localized forcing in the northeast quadrant of the basin at four phases of the wave oscillation. The wave passes through the barrier by a process similar to the one described by the theory of section 2. (b) The square of the wave amplitude in a basin, open on its eastern end and forced by a localized periodic forcing nearly coincident with the natural frequency of oscillation of the western subbasin. Again the wave energy passes through the narrow gaps of the barrier to excite strong motion west of the barrier.

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where ⌿ϭ⌿Ϫ⌿diff 1 2, (4.8b) ⌿ϭ⌿ϩ⌿and (4.8a) sum 1 2 it follows by standard asymptotic analysis that in the stationary phase approximation,

eri(k[rϩx]Ϫ␲/4) 1/2|x| ␺ ϭ͗⌿{cosky y/r Ϫ cos[k(y ϩ d)y/r]} T 2␲ kyd3/2 2 sum oo

ϩ i⌿diff{sinkyoo y/r Ϫ sin[k(y ϩ d)y/r ϩ sin[k(y/r)d/2]}͘. (4.9)

Figure 9 shows the stationary wave solution for the The cause of this transparency is connected fundamen- transmitted wave in the region x Ͻ 0. For R ϭϪ0.01 tally to Kelvin's theorem, which, in the present case, and ␻ ϭϪ0.08 the transmitted wave is dominated by will not allow motion with a large meridional scale on the symmetric part of the solution, as anticipated from only one side of the barrier. This forces a compensating the results shown in Fig. 8a. Figure 9a shows the ab- tangential ¯ow on the far side of the barrier, so the solute magnitude of the transmitted wave. It is clearly transmission follows directly. In order for Kelvin's the- monopolar in form with the wave amplitude issuing as orem to be applicable there must be a circuit in the ¯uid a single bubble from the two narrow slits at the northern that can completely encompass a segment of the barrier. and southern extremities of the two island segments. That is, the region of the ocean basin, barrier, and gaps Figure 9b shows the wave ®eld itself at two distinct must yield a multiply connected domain for the ¯uid. phases, that is, at ␻t ϭ 0 and ␻t ϭ ␲/2. On the other Thus, a single gap, which renders the ¯uid region still hand, when the angle of incidence of the approaching singly connected, will not allow substantial transmission wave is at 45Њ to a latitude circle so that R ϭϪ1, the unless the gap width becomes broad so that the wave symmetric response is very small and the dominant can proceed unhindered by the barrier on its own scale. wave transmission occurs as a dipolar pattern in a double Similarly, if viscous processes become important bubble as shown in Figs. 9c and 9d. In this case where enough in the gaps to enter signi®cantly into the inte-

⌿diff is so much larger than ⌿sum the middle gap carries grated balance, which is the statement of twice the ¯ux as each of the extreme termini of the Kelvin's theorem, the degree of transmission can be islands and, of course, in the opposite directions. This reduced but, as shown in Pedlosky (2000), this requires mode of transmission of the antisymmetric mode is al- a substantial amount of friction to choke off the trans- lowed only because of the new center gap. As more mission. gaps are opened, higher multipoles in the transmitted The theory considered here has been linearized and wave ®eld will be generated but, since the response, as nonlinear effects can be expected to alter the structure (4.9) shows, will go to zero as the segment length di- of the wave ®elds by self-interaction and by interaction minishes in size, the higher multipoles will generally with mean ¯ows and other waves. However, the con- have smaller amplitude. This clearly arises because the straint of Kelvin's theorem in the form (2.9) will still radiation from each segment can destructively interfere. hold, and it therefore seems unlikely that transmission Naturally, at the special values of R where one or the can be choked off by purely nonlinear effects. Indeed, other of the components of the transmitted wave vanish, and on the contrary, it will be of interest to examine the symmetry of the transmitted wave will correspond the possibility that large-scale nonlinear waves of the to the alternative symmetry. Generally, though, glancing solitary wave type, or isolated but propagating eddies angles of approach will generate the antisymmetric form in their interaction with such pierced barriers will ®nd of wave transmission while the more frontal approach similar ways of squeezing through the gaps to emerge of the incident wave will yield a single balloon of trans- beyond the barrier by virtue of the Kelvin constraint. mission. In any case, the transparency of the barrier to In the problem described here, I have examined the transmission of Rossby wave energy is increased as new, problem of a wave packet or plane wave approaching small gaps are added to the barrier. the barrier to render the problem as simple as possible and to also isolate the transmission problem from the complexities of the full basin response. The conse- 5. Conclusions and discussion quences of the barrier's transparency shows up just as The transparency of narrowly pierced barriers to clearly in the normal mode problem described by Ped- Rossby wave transmission is a striking theoretical pre- losky and Spall (1999), which illustrates the same diction. The wave scale of the transmitted wave is much points. The analysis of the basin-scale problem is similar larger than the narrow gaps that allow the transmission. to the one we have described here, so the details of the

Unauthenticated | Downloaded 09/25/21 05:20 PM UTC 352 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 calculation are not given but Fig. 10a shows the normal constraint enters only implicitly. Instead, Kelvin waves mode excited in a full basin by a localized delta function running around the island with a velocity CK are the forcing in the northeastern quadrant of the basin in the messengers that establish the equivalent of the island presence of weak bottom friction. The Rossby normal constant value by determining the ¯uxes through the mode, with its pattern of continuous westward phase gaps. This establishes a timescale L/CK for the estab- propagation is seen to easily squeeze through the gaps lishment of the situation described in this paper. This in the barrier from the eastern subbasin where it is forced time will be short compared to a Rossby wave timescale to involve the total basin by the process discussed in (␤L)Ϫ1 as long as L is subplanetary, that is, for L large detail in this paper. If instead, the region is open to the enough for geostrophy but small enough for the ␤-plane east so that the region as a whole is not closed, resonance approximation to be valid. with a full basin mode is no longer possible and the It should also be apparent that the considerations of analysis of the present paper is more apt. However, it wave transmission considered here are applicable to oth- is of interest to consider cases where the barrier is lo- er wave types as well, such as both surface and internal cated not too far from a closed western boundary, per- gravity waves. haps modeling the situation such as corresponds be- tween the open North Atlantic and the adjacent Carib- Acknowledgments. This research was supported in bean separated by an island arc barrier. In that case part by a grant from the National Science Foundation waves transmitted through the barrier at the right fre- OCE 9301845. quency might reach large amplitudes in the adjacent if the frequency of the incident wave closely corre- REFERENCES sponds to a natural frequency of the adjacent basin. Such Chelton, D. B., and M. G. Schlax, 1996: Global observations of a situation is shown in Fig. 10b where the variance of oceanic Rossby waves. Science, 272, 234±238. ␺ is shown. Again the forcing is localized in the north- Clarke, A. J., 1991: On the re¯ection and transmission of low-fre- quency energy at the irregular western Paci®c Ocean boundary. east part of the domain and the eastern boundary is open J. Geophys. Res., 96 (Suppl.), 3289±3305. so that wave energy leaks eastward, but a near-resonance Firing, E., B. Qui, and W. Miao, 1999: Time-dependent island rule occurs in the adjacent sea so that the amplitude of the and its application to the time-varying North Hawaiian Ridge variance there strongly exceeds that near the forcing in Current. J. Phys. Oceanogr., 29, 2671±2688. Godfrey, J. S., 1989: A Sverdrup model of the depth-integrated ¯ow spite of the presence of weak frictional dissipation. De- for the World Ocean allowing for island circlations. Geophys. tails of the calculation are not given here since the pur- Astrophys. Fluid Dyn., 45, 89±112. pose of the present discussion is to describe the trans- Jacobs, G. A., W. J. Emery, and G. H. Born, 1993: Rossby waves in parency of the barrier to Rossby wave transmission, but the Paci®c Ocean extracted from Geosat altimeter data. J. Phys. Oceanogr., 23, 1155±1175. these examples are presented to demonstrate the im- Lamb, H., 1932: Hydrodynamics. 6th ed. Dover, 738 pp. portant consequences such transparency can have. Pedlosky, J., 1987: Geophysical . Springer-Verlag, In quasigeostrophic theory the need to determine the 710 pp. value of the streamfunction on each separated segment , 2000: The transmission of Rossby waves through basin barriers. J. Phys. Oceanogr., 30, 495±511. of the barrier leads naturally to a consideration of , and M. A. Spall, 1999: Rossby normal modes in basins with Kelvin's theorem and, since that theorem holds at each barriers. J. Phys. Oceanogr., 29, 2332±2349. instant, the determination of the streamfunction constant , L. J. Pratt, M. A. Spall, and K. R. Helfrich, 1997: Circulation is determined at each instant by the application of the around islands and ridges. J. Mar. Res., 55, 1199±1251. Rossby, C. G., and Coauthors, 1939: Relation between variations in theorem. Models of the motion that resolve faster time- the intensity of the zonal circulation of the and the scale physics, such as (coincidentally) Kelvin waves, displacements of the semi-permanent centers of action. J. Mar. are not formulated in terms of a streamfunction and the Res., 2, 38±55.

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