336 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 The Transparency of Ocean Barriers to Rossby Waves: The Rossby Slit Problem* JOSEPH PEDLOSKY Department of Physical Oceanography, 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 vorticity 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 oceans 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 satellite altimeter 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 q 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 v and the direction is determined by the Rossby wave's baroclinic limit. In that case the extension of the God- group velocity. 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 v 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 potential vorticity 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): ]]c ¹2c 150.