2382 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 Rossby Wave±Coastal Kelvin Wave Interaction in the Extratropics. Part I: Low-Frequency Adjustment in a Closed Basin ZHENGYU LIU,LIXIN WU, AND ERIC BAYLER Department of Atmospheric and Oceanic Sciences, University of WisconsinÐMadison, Madison, Wisconsin (Manuscript received 12 May 1998, in ®nal form 12 November 1998) ABSTRACT The interaction of open and coastal oceans in a midlatitude ocean basin is investigated in light of Rossby and coastal Kelvin waves. The basinwide pressure adjustment to an initial Rossby wave packet is studied both analytically and numerically, with the focus on the low-frequency modulation of the resulting coastal Kelvin wave. It is shown that the incoming mass is redistributed by coastal Kelvin waves as well as eastern boundary planetary waves, while the incoming energy is lost mostly to short Rossby waves at the western boundary. The amplitude of the Kelvin wave is determined by two mass redistribution processes: a fast process due to the coastal Kelvin wave along the ocean boundary and a slow process due to the eastern boundary planetary wave in the interior ocean. The amplitude of the Kelvin wave is smaller than that of the incident planetary wave because the initial mass of the Rossby wave is spread to the entire basin. In a midlatitude ocean basin, the slow eastern boundary planetary wave is the dominant mass sink. The resulting coastal Kelvin wave peaks when the peak of the incident planetary wave arrives at the western boundary. The theory is also extended to an extratropical±tropical basin to shed light on the modulation effect of extratropical oceanic variability on the equatorial thermocline. In contrast to a midlatitude basin, the fast mass redistribution becomes the dominant process, which is now accomplished mainly by equatorial Rossby and Kelvin waves, rather than the coastal Kelvin wave. The coastal Kelvin wave and the modulation of the equatorial thermocline peak close to the time when the wave trail of the incident Rossby wave arrives at the western boundary. Finally, the theory is also applied to the wave interaction around an extratropical island. 1. Introduction es (Pedlosky 1987). Coastal Kelvin waves will also be excited. However, the mechanism responsible for the In a midlatitude ocean basin, the two most important coastal Kelvin wave is not fully understood. This will low-frequency waves are the Rossby wave and coastal be the major question to be explored here. Kelvin wave. Most previous works have focused on the Rossby wave. Long Rossby waves (or planetary waves) The interaction of midlatitude Rossby and coastal are important for the adjustment in the interior ocean, Kelvin waves has at least three applications. First, it is while short Rossby waves are responsible for the in- an important process that contributes to the exchange tensive current variability along the western boundary of open and coastal ocean variability. Second, the coast- (e.g., Pedlosky 1965, 1987; Anderson and Gill 1975; al Kelvin wave generated by decadal planetary waves Cane and Sarachik 1977, 1979; McCreary and Kundu in the midlatitude may provide a mechanism for the 1988). However, relatively less attention has been paid modulation of the equatorial thermocline and therefore to the role of the coastal Kelvin wave and its interaction contributes to decadal climate variability in the sub- with Rossby waves in the low-frequency variability of tropical±tropical climate system (Lysne et al. 1997). the midlatitude ocean. In particular, there have been few Third, this interaction, as will be seen later, is critical studies on the interaction of Rossby waves and coastal for the low-frequency variability of island circulation. Kelvin waves on the western boundary. When a Rossby Previous works on the interaction of Rossby and wave incident is on the midlatitude western boundary, Kelvin waves can be divided into three groups. The ®rst it is known that short Rossby waves are generated to group includes numerous studies that investigated the balance the incoming cross-shore mass and energy ¯ux- interaction of equatorial Kelvin wave and equatorial Rossby waves (e.g., McCreary 1976; Cane and Sarachik 1977, 1979; Kawase 1987). It is found that wave re- ¯ection is determined mainly by the balance of zonal Corresponding author address: Z. Liu, Department of Atmospheric mass transport. The alongshore mass distribution is ac- and Oceanic Sciences, University of WisconsinÐMadison, 1225 W. Dayton St., Madison, WI 53706-1695. complished by short Rossby waves, which contributes E-mail: [email protected] little to the zonal mass transport and therefore plays q 1999 American Meteorological Society SEPTEMBER 1999 LIU ET AL. 2383 little role in determining the re¯ected Kelvin and long address the following questions: What is the magnitude Rossby waves. These studies have laid the foundation of the slowly evolving coastal Kelvin wave, and when to our understanding of seasonal- to interannual climate does it peak? What are the major factors that determine variability, such as the ENSO in the Paci®c Ocean (e.g., the strength and timing of the Kelvin wave amplitude? McCreary 1983; Cane and Zebiak 1985; Battisti 1989, Furthermore, in a basin that includes the equator, what 1991; Kessler 1991) and the Somali Current system in is the impact of the wave interaction in the midlatitude the Indian Ocean (e.g., McCreary and Kundu 1988). on the modulation of the equatorial thermocline? A sim- However, it will be seen later that interaction of the ple analytical theory is ®rst developed to determine the equatorial Rossby and Kelvin waves differs signi®cantly amplitude evolution of the coastal Kelvin wave. Exten- from that between a midlatitude Rossby wave and a sive numerical experiments are performed to substan- coastal Kelvin wave (see more discussions in section tiate the theory. The modulation of the Kelvin wave is 5). found to be determined by two nonlocal mass redistri- The second group focused on the extratropical ocean, bution processes: one fast and the other slow. The for- but only along the eastern boundary. The primary issue mer is due to the coastal Kelvin wave current, while the is the generation of planetary waves by coastal Kelvin latter is attributed to the eastern boundary planetary waves along the eastern boundary (e.g., En®eld and wave. The amplitude evolution of the coastal Kelvin Allen 1980; White and Saur 1983; Jacobs et al. 1994; wave depends critically on the mass redistribution pro- McCalpin 1995). At low frequencies, coastal Kelvin cess. Different from the re¯ection of equatorial Rossby waves change the eastern boundary pressure, which in and Kelvin waves, which is determined by the balance turn generates planetary waves radiating westward into of cross-shelf mass ¯ux along the equator, the coastal the interior ocean. These studies are important to un- Kelvin wave is generated to balance the alongshore mass derstand the impact of annual to decadal variability of ¯ux of the incident long Rossby wave in the midlatitude. coastal upwelling on the interior ocean. The paper is arranged as follows. A QG theory is The third group investigated the interplay of midlat- developed in section 2 to illustrate the major processes itude Rossby waves and coastal Kelvin waves, mostly that determine the interaction between planetary waves along the western boundary. Most of these works orig- and coastal Kelvin waves in a midlatitude basin. Sup- inated from the issue of mass conservation (or the con- porting numerical experiments are carried out in a shal- sistency condition) in a quasigeostrophic (QG) model low-water model in section 3. The theory is then ex- in a midlatitude basin (McWilliams 1977; Flierl 1977; tended to a combined extratropical±tropical basin in sec- Dorofeyev and Larichev 1992; Millif and McWilliams tion 4 to study the modulation of midlatitude planetary 1994, hereafter MM). All these works strongly sug- waves on the equatorial thermocline. A summary and gested that the coastal Kelvin wave plays a key role in a brief comparison with the interaction of equatorial mass conservation. Our work is a further extension of Rossby and Kelvin waves are given in section 5. Finally, these works. Godfrey (1975) and McCreary and Kundu a direct application of our theory to the wave interaction (1988) have discussed some initial re¯ection processes around an island is given in the appendix. and some local dynamics with the focus on low lati- tudes. Dorofeyev and Larichev limited their discussion to a half-plane midlatitude ocean, and therefore it does 2. The theory not directly apply to a closed basin. Kelvin wave prop- a. Model and solution agation around a basin has also been discussed in nu- merical models (Cane and Sarachik 1979; McCreary The modeling study of MM assures convincingly that 1983; MM). As will be seen later, the modulation of a the initial adjustment of Rossby and coastal Kelvin coastal Kelvin wave is a nonlocal process due to the waves in a midlatitude ocean basin can be simulated in rapid propagation of the coastal Kelvin wave. Conse- a QG model. Indeed, a low-frequency Kelvin wave can quently, the response differs dramatically for different be approximated as an alongshore current because of ocean basins. The modeling study of MM shows a re- its rapid along boundary propagation. This so-called markable resemblance between the shallow-water model coastal Kelvin wave current (CKC) is nearly geostrophic and the quasigeostrophic model, after the consistency and can therefore be represented in a QG model. Math- 1/2 condition is imposed in the latter. This assures that the ematically, the speed of the Kelvin wave co 5 (g9D) principle mechanisms for the interaction of Rossby and is independent of forcing frequency, where g9 is the coastal Kelvin waves are contained in a QG system.