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

᭧ 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 ϭ (gЈD) principle mechanisms for the interaction of Rossby and is independent of forcing frequency, where gЈ is the coastal Kelvin waves are contained in a QG system. reduced gravity and D is the mean depth. Thus, along However, in none of these works are the physical mech- a straight coast, the alongshore wavenumber diminishes, → → anisms, especially those responsible for the quantitative k ϭ ␻/co 0, as the forcing frequency vanishes (␻ aspects of the wave interaction, fully studied. 0). In other words, a low-frequency coastal Kelvin wave We will investigate the basinwide adjustment process has a long alongshore wavelength and therefore resem- to an initial planetary wave packet incident on a mid- bles a coastal current. In a closed basin, this mathe- latitude western boundary, with the focus on the quan- matical argument, although not strictly true, is still help- titative aspects of the interaction. We will attempt to ful. The adjustment time for the completion of a CKC 2384 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 around the basin is the traveling time of the Kelvin wave around the basin. For the ®rst two baroclinic modes, co is no less than 1 m sϪ1. Thus, the boundary adjustment time ranges from days to months and is less than half a year even for a basin of the size of the North Paci®c Ocean. As a ®rst step, a linearized 1.5-layer QG model will be used to study the interaction between the Rossby wave and coastal Kelvin wave in a midlatitude basin. Mean advection and nonlinearity are not included at this stage. The most serious impact of the mean boundary current advection is likely to be on slow short Rossby waves, although coastal Kelvin waves may also be af- fected. Since short Rossby waves contribute little to the FIG. 1. Schematic ®gure of an extratropical basin: various solution mass redistributionÐthe key process that determines the regimes for the adjustment process are plotted. The initial planetary modulation of the coastal Kelvin wave, the absence of wave is plotted in solid lines against the western boundary. The dashed lines separate the basin solution into western (hW), southern mean boundary current advection may not affect our (h ), eastern (h ), and northern (h ) regions. conclusion on the coastal Kelvin wave signi®cantly. The S E N equation can be written in terms of sea surface height (SSH) h as contribution to local potential vorticityÐa feature that can be veri®ed directly for a shallow-water Kelvin wave (h ϭϪrٌ 2h, (1a ץ␤ 2h Ϫ h/d 2) ϩ ٌ) ץ t x along a straight coast on an f-plane. The implication is where d ϭ co/ f 0 is the deformation radius, co ϭ that the interaction between the Kelvin and planetary (gD⌬␳/␳)1/2 is the gravity wave speed, and D is the mean wave is not caused by potential vorticity ¯ux. thickness of the upper moving layer. A linear momentum The structure of the CKC can be derived using the damping is also used.1 The boundary condition is no standard boundary layer method (Pedlosky 1987). In normal ¯ow across the boundary the following, we will only consider a rectangular basin h| ϭ K(t), (1b) of 0 Յ x Յ X and YS Յ y Յ YS ϩ Y (see Fig. 1). The ⍀ boundary layer solution of the CKC is then derived fromץ where K(t) will be determined by mass conservationÐ (5) as a key point to be returned to later. Ϫx/d Ϫ(XϪx)/d Ϫ(yϪY )/d Ϫ(Y ϩYϪy)/d Following MM, the total SSH (h) can be decomposed h* ϭ e ϩϩeeSS ϩ e, (6) into two parts: the Rossby wave part (hR) and the Kelvin where the ®rst through fourth terms on the rhs represent wave part (hK): the CKC along the western, eastern, southern, and north- h ϭ h ϩ h . (2) ern boundaries, respectively. This is a coastal current R K of width d, or the CKC. Here a small correction due to The Rossby wave satis®es the QG equation corners (at the order of d/L K 1) is neglected as in the 2 2 2 Fofonoff basin mode (Fofonoff 1954). This neglection (xh ϭϪrٌ h (3aץ␤ t(ٌ hR Ϫ hR/d ) ϩץ seems to be justi®ed based on the numerical simulation within the basin and has no pressure anomaly around of MM, which shows an insigni®cant effect of the corner ⍀: on the modulation of the boundary pressure. The smallץ the basin boundary

⍀ ϭ 0. (3b) effect of the corner has also been discussed in the caseץ|hR of an inertial western boundary current by Liu (1990). The Kelvin wave has an amplitude K(t) and a ®xed Now we derive a basin solution for the adjustment spatial structure h*(x, y): in response to an initial planetary wave packet

hK ϭ K(t)h*(x, y), (4) h ϭ hi(x ϩ Ct, y), (7) where where C ϭ ␤d 2 is the speed of the planetary wave. The 2 2 ⍀ ϭ 1. (5) planetary wave satis®esץ|*h* Ϫ h*/d ϭ 0, h ٌ

2 (xh ϭ 0 (8ץ␤ t(ϪhR/d ) ϩץ Equation (5) implies that the Kelvin wave makes no in the interior ocean. In addition, the Kelvin wave van- ishes in the interior ocean according to (6), leading to 1 All major conclusions can be derived in a generalized QG equa- h ϭ hR. For the long wave in (8), we have assumed a tion and are independent of the form of dissipation. More speci®cally, weak dissipation such that the dissipation timescale (1/r) the solution along the eastern boundary in a generalized QG equation 2 has been derived in McCalpin (1995). The solution along the western is much longer than that of the planetary wave (xp/␤d ). boundary in a generalized QG equation is derived in Liu et al. (1999), For an initial planetary wave packet of zonal scale xp 2 2 which is also similar to Godfrey (1975). k d, this requires 1 K (1/r)/(xp/␤d ) ϭ d /␦xp, where SEPTEMBER 1999 LIU ET AL. 2385

␦ ϭ r/␤ is the width of the Stommel boundary layer. Here we assumed that the pressure anomaly of the in- Therefore, the dissipation effect on long waves is neg- cident planetary wave vanishes in both the southern and ligible if ␦/d K d/xp K 1. northern boundary layers. Otherwise, an additional dif- The planetary wave ®rst reaches the western bound- fusive boundary layer need to be added (Pedlosky et al. ary, where Eq. (1a) can be reduced to the western bound- 1997). ary layer equation: Along the eastern boundary there is no viscous h (9) boundary layer; therefore, the solution consists of a ץh ϭϪr ץ␤ x xx planetary wave that radiates away from the eastern on the western boundary layer. boundary. This wave should satisfy the planetary wave Here we have assumed that re¯ected short Rossby equation (8) and, consequently, should have the form waves are damped within the boundary layer in a time- of hE(x ϩ Ct). The boundary condition (1b) requires scale (1/r) much shorter than that of the short Rossby hE(X ϩ Ct) ϵ K(t) along the eastern boundary. There- wave adjustment (Tbdry). The temporal variability term fore, the solution in the eastern part is in (1) can be shown negligible with ␦/d K 1. The re- sulting boundary layer pressure ®eld has a quasi-steady- hE ϭ K[t ϩ (x Ϫ X)/C], X Ϫ Ct Ͻ x Ͻ X. (14) state response (9). The western boundary layer solution has to satisfy the no-normal ¯ow boundary condition Notice that the two diffusive boundary layers on the (1b). In addition, it has to match the incident wave in northern and southern boundaries have been neglected the interior as in (14). The hE is dominated by an eastern boundary planetary wave (EBP). It can be considered as the sum h → h (x ϩ Ct, y)| ϭ h (Ct, y) i xϭ0 i of the forced planetary wave K[t ϩ (x Ϫ X)/C] Ϫ as x/␦ → ϱ. (10) K(t)eϪ(XϪx)/d and the CKC along the eastern boundary Ϫ(XϪx)/d The western boundary layer solution can therefore be K(t)e in (6). Solution (14) can be shown to be the derived as low-frequency limit in a generalized QG equation (McCalpin 1995; Liu et al. 1999). Thus, except for the Ϫx/␦ Ϫx/␦ h ϭ hi(Ct, y)(1 Ϫ e ) ϩ K(t)e ;0Ͻ x/␦ Ͻϱ, unknown amplitude K(t) of the CKC, Eqs. (11)±(14) where we have used Eqs. (2), (3b), (5), and (10). Thus, give the complete basin solution, valid until the EBP the solution in the entire western part of the basin (see reaches the western boundary, or t Ͻ X/C. Fig. 1) can be written as

hW ϭ hWL ϩ hWS ϩ hWK,0Ͻ x Ͻ xp Ϫ Ct, (11a) b. Mass conservation and Kelvin wave amplitude where The CKC amplitude is determined by total mass con-

;t ##⍀ hdxdyϭ 0 (Godfrey 1975ץtM ϵץ servation Ϫx/␦ hWL ϭ hi(x ϩ Ct, y), hWS ϭϪhi(Ct, y)e , McWilliams 1977; Flierl 1977; Dorofeyev and Larichev h ϭ K(t)eϪx/␦. (11b) 1992; MM). In the following, the incident planetary WK wave is assumed to arrive at the western boundary at t Here hWL is the incident planetary wave, hWS is the ϭ 0 and is of the form (Fig. 1) damped re¯ected short Rossby wave, and hWK is the sum Ϫx/d of the CKC along the western boundary K(t)e [see hi(x, y) ϭ h0 sin(␲x/xpcp) cos[␲(y Ϫ Y )/y ], (6)] and the damped short Rossby wave that is forced by the western boundary Kelvin wave K(t)(eϪx/␦ Ϫ for 0 Յ x Յ xp, Ϫx/d e ) [see Liu et al. (1999) for further discussions]. Note Ϫ0.5y Յ y Ϫ Y Յ 0.5y (15a) that the spatial structure of the total response to the pcp

Kelvin wave forcing along the western boundary (hWK) hi(x, y) ϭ 0 elsewhere, (15b) is determined by the viscous boundary layer scale ␦, which is independent of the structure of the Kelvin wave where the initial amplitude is h 0, and the center latitude having a deformation radius d. The hWK solution is sim- of the initial pulse is Yc ϭ YS ϩ 0.5Y. The initial mass ilar to the Kelvin±Munk wave of Godfrey (1975) where is therefore he used a no-slip boundary condition. Along the southern and northern boundaries (Fig. 1), m ϭ hdxdyϭ (2x /␲)(2y /␲)h , (16) there are no re¯ected Rossby waves. Thus, the solutions o ͵͵ ipp0 in the southern and northern boundaries consist only of ⍀ the CKCs [note (6)]: where the factor 2/␲ arises from the integral of the half- Ϫ(yϪYs)/d hSsϭ K(t)e ,0Ͻ (y Ϫ Y )/d Ͻϱ (12) sine pro®le. The mass in the western region is contributed by the Ϫ(YsϩYϪy)/d hNsϭ K(t)e , Ϫϱ Ͻ (y Ϫ Y Ϫ Y )/d Ͻ 0. incident long wave (MWL), the re¯ected damped short

(13) wave (MWS), and Kelvin wave (MWK) [per (11)]: 2386 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

YSpϩYxϪCt Ct Yx T , b p , Mw ϭ dy hWWLWSWK dx ϭ M ϩ M ϩ M . ϭ ϭ ͵͵ xp ␲(2Xd ϩ Y␦) YS 0 (17a) m a ϭ 0 . (23) 2(2Xd ϩ Y␦) Using the hWL, hWS, and hWK in (11b), one can show that The time MWLϭ 0.5m o[1 ϩ cos(Ct␲/x p)]H(x p Ϫ Ct) (17b)

TEI ϭ 1 (24a) MWSϭ 0.5m o[Ϫq sin(Ct␲/x p)]H(x p Ϫ Ct) (17c) is the time when the incident long wave trail just reaches MWK ϭ K(t)Y␦, (17d) the western boundary (end time hereafter), and the time respectively, where q ϭ ␦␲/xp and H is the Heaviside TPI ϭ 0.5 (24b) function: H(z) ϭ 1 for z Ն 0 and H(z) ϭ 0 for z Ͻ 0. Clearly, the mass of the short Rossby wave M is neg- is the time when the peak of the incident wave reaches WS the western boundary (peak time hereafter). The non- ligible compared with the incident planetary wave MWL as long as x k ␦. Along the southern and northern dimensional parameter b (except for a factor ␲)isthe p area ratio of the ®nal EBP area Yx and the CKCs area boundaries, the mass is due to CKCs (12) and (13) as p along the southern, northern, and western boundaries YSϩYX 2Xd ϩ Y␦. This is an important parameter, because it M ϭ dy h dx ϭ K(t)Xd S ͵͵S measures the relative importance of the slow (EBP) and YS 0 fast (CKC) processes. Finally, a is proportional to the and ratio of the initial perturbation area to the coastal wave YSϩYX area. The amplitude K(t) can be derived from (22) as M ϭ dy h dx ϭ K(t)Xd. (18) N ͵͵N a YS 0 K(T) ϭ {eϪb␲T Ϫ cos(␲T) ϩ b sin(␲T) 1 ϩ b2 Finally, the mass in the eastern region is dominated by the EBP and can be derived from (14) as ϩ q[b(cos(␲T) Ϫ eϪb␲T ) ϩ sin(␲T)]}

YSϩYX t for T Յ 1 (25a) M ϭ dy h dx ϭ CY K(tЈ) dtЈ. (20) E E 2adX(1 ϩ eϪb␲) ͵͵ ͵ Ϫb␲T Ϫb␲T YXS ϪCt 0 K(T) ϭ K(1)e ϵ e 1 ϩ b2(2Xd ϩ Y␦) Therefore, the amplitude K(t) is determined by the total mass conservation for T Ͼ 1. (25b) The complete basin adjustment solution is therefore giv- M M M M m . (21) W ϩ S ϩ E ϩ N ϭ 0 en by regional solutions in (11), (12), (13), (14), and the amplitude (25). The mass source MW is redistributed by two processes: MN and MS represent the fast mass redistribution process due to CKC along the northern and southern coasts (MWK c. Mass exchange and modulation of CKC is also a fast process), while ME represents a slow mass redistribution process toward the interior ocean by the Figures 2a±d plot four snapshots during an adjustment EBP. The fast process redistributes mass over the entire process in a basin of X ϭ 20Њ and Y ϭ 20Њ, where 1Њ coastal region, which has a width d and, therefore, a ϭ 111 km. Other parameters are ⌬␳/␳ ϭ 0.002, f o ϭ ®xed area. The slow process, however, redistributes 7.3 ϫ 10Ϫ5 sϪ1, ␤ ϭ 2 ϫ 10Ϫ11 mϪ1 sϪ1, and D ϭ 500 mass over an area that slowly expands westward. It will m. These parameters give a Stommel boundary layer be seen that the evolution of the CKC amplitude K(t) width of ␦ ϭ 19 km and a deformation radius of d ϭ is determined predominantly by these two mass redis- 44 km. Zonal and meridional sections of the four snap- tribution processes. shots are also plotted along the central latitude (Fig. 3a) The mass conservation (21) can be differentiated to and longitude (Fig. 3b), respectively. A planetary wave yield an ordinary differential equation patch (of size xp ϭ 10Њ, yp ϭ 10Њ) is initiated near the western boundary at T ϭ 0 (Fig. 2a and circle-connected dK lines in Figs. 3a,b). At the peak time, T ϭ ½ (Fig. 2b ϩ bK(T) ϭ a[sin(␲T) ϩ q cos(␲T)]H(1 Ϫ T) PI dT and solid lines in Figs. 3a,b), half of the patch has moved (22a) out of the western boundary, a sharp viscous boundary layer is formed along the coast, a signi®cant CKC is with the initial condition of established around the basin, and an EBP radiates into K(0) ϭ 0. (22b) the eastern ocean. At the end time TEI ϭ 1 (Fig. 2c and dash±dot lines in Figs. 3a,b), the initial perturbation Here we have used the normalized quantities planetary wave has just passed the western boundary SEPTEMBER 1999 LIU ET AL. 2387

FIG. 2. Four snapshots of the analytical solution in Eqs. (11)±(14). (a) Initial condition, (b) T ϭ 0.5, the peak time of the initial long wave, (c) T ϭ 1, the end time of the initial long wave, and (d) T ϭ 1.5. The contour interval is 0.05 for the solid lines and 0.2 for the dotted lines.

FIG. 3. Sea surface height pro®les of the snapshots in Fig. 2 along (a) the zonal section of the middle basin latitude, y ϭ 30ЊN, and (b) the meridional section of the middle basin longitude, x ϭ 10Њ. 2388 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 4. (a) Evolution of the amplitudes for the Kelvin wave K(T) with (solid line) and without (b ϭ 0, dashed line) the EBP and the magnitude of the incident long wave at x ϭ 0Њ, y ϭ 30ЊN (dotted line). (b) The evolution of each mass component of the solution with the EBP [solid line in (a)]. (c) The evolution of each mass component of the

solution without the EBP [dashed line in (a)]. The de®nitions for each mass component are [Eqs. (17)±(21)] MWL,

incident long wave; ME, eastern boundary planetary wave; MN, northern boundary Kelvin wave; MS, southern boundary

Kelvin wave; MWK, western boundary Kelvin wave and the resulting short Rossby wave; and MWS, damped western boundary short Rossby wave. Both the amplitude and mass are normalized by the initial perturbation amplitude. and only a weak short Rossby wave (forced by CKC) this case, K(T) is determined predominantly by the slow exists. Around the coast, the CKC weakens, yet is still transfer from the incident planetary wave to the ex- visible. On the east, the EBP has left the coast. Finally, panding EBP, as shown in Fig. 4b. at T ϭ 1.5 (Fig. 2d and dashed lines in Figs. 3a,b), the CKC has virtually disappeared along all boundaries and d. Slow and fast mass redistribution the EBP has propagated to the middle basin. All of the major features of our theoretical solution closely resem- The mass balance for slow redistribution occurs be- ble the QG model experiment of MM. One difference tween the incident planetary wave and the EBP: is that the initial wave packet in the QG model also generates dispersive waves that trail the leading wave MWL ϩ ME ϭ m0, slow mass redistribution. (26) packet. In addition, in the shallow-water model, the This mass balance, together with (17) and (20), leads planetary wave front propagates faster at lower latitudes to the amplitude due to decreases in the Coriolis parameter. dM a The evolution of the CKC amplitude K(T) is plotted K(T) ϭϭWL sin(␲T)H(1 Ϫ T), in Fig. 4a (solid line). For comparison, the SSH variation dT b of the incident planetary wave at the western boundary X [hWL(0Њ,30Њ)] is also plotted (dotted line). It is seen that for 0 Ͻ T Ͻ . (27) the K(T) maximum is about 0.3 times the initial anomaly xp and is achieved at about T ϭ 0.58, slightly after TPI ϭ The maximum amplitude is therefore 0.5. The mass exchange among various wave components am0 2 yp KMooϭϭ ϭ h Յ h , (28a) is shown in Fig. 4b. The mass of the incident long wave b 2xYp /␲␲Y (solid line) decreases monotonically, losing all the mass or at the end time TEI ϭ 1. The mass loss is little affected by the damped short Rossby wave (dotted line). Instead, m0 most mass is initially lost to the CKC along the southern, ϭ 1. (28b) 2KxY/␲ northern, and western boundaries (dash±dot). Later, Mp most mass leaks into the EBP (dashed line), which even- Equation (28b) states that the maximum amplitude is tually completely absorbs the initial mass. Therefore, in such that the total initial mass is completely converted SEPTEMBER 1999 LIU ET AL. 2389

to that of the EBP that has an amplitude KM and an area xpY. In x direction, both the initial and ®nal planetary waves have the same width xp and the same shape (a half sine patch). Therefore, Km is determined only by the ratio of the widths of the two waves in y direction as shown in (28a). The 2/␲ factor arises from the half sine shape in y for the initial wave, which differs from the eastern boundary wave that always has a uniform sea level in y as shown in (14). The amplitude in (27) ®rst increases to its maximum at the peak time TM ϭ TPI ϭ 0.5, then decreases to zero at the end time TEI ϭ 1 and remains unchanged there- after. This behavior is similar to the example in Fig. 4a

(solid line), which has KM ഠ 0.31, slightly smaller than the 1/␲ ഠ 0.32 estimated from (28), and which has TM 0.58, slightly later than the peak time T 0.5. ഠ PI ϭ FIG. 5. The time for maximum Kelvin wave amplitude (TM)asa The fast mass redistribution may also be important function of the area ratio parameter b [de®ned in (23)]. Friction is in determining the amplitude evolution K(t), as will be neglected in this case (q ϭ 0). seen later in section 4 for the case of a combined ex- tratropical±tropical ocean basin. The evolution of K(t) that is driven by the fast process differs signi®cantly of the EBP. This offset precludes an increase of ampli- from that by the slow process. Indeed, if the mass is tude. In the extreme case, when all the mass is redis- completely redistributed by the fast process along the tributed by the slow process, as in (26), K(T) reaches northern and southern boundaries (neglecting friction so its maximum at TPI ϭ 0.5 when the rate of incident q ϭ 0), mass reduction is the greatest [see (27)]. The relative importance of the slow EBP versus the MW ϩ MS ϩ MN ϭ m 0, fast mass redistribution. fast CKC processes is measured by the area ratio pa- (29) rameter b in (23). A large (small) b favors the slow The amplitude can be derived as (fast) process. The solutions for the complete slow and fast redistributions [(27) and (30), respectively] are the K(T) ϭ a[1 Ϫ cos(␲T)] for T Յ 1 (30a) asymptotic solutions of the full solution (25) in the lim- → → K(T) ϭ 2a for 1 Ͻ T Յ X/x . (30b) its of b ϱ and b 0, respectively (neglecting p friction). Figure 5 plots the time of maximum amplitude

Now, K(T) continues to increase until the end time and TM as a function of b. In general, TM is later than TPI remains unchanged thereafter, so TM ϭ 1. The dashed ϭ 0.5, but earlier than TEI ϭ 1 due to the competition line (b ϭ 0 curve) in Fig. 4a illustrates this response. between the fast and slow processes. If the slow process

This evolution is in sharp contrast to the slow process dominates (large b), TM is closer to ½, and vice versa. discussed above, where TM ϭ 0.5. The different peaking Even for a modestly large b (b ϭ 2), TM is already fairly time for fast and slow mass redistribution can be un- close to ½. The example of Fig. 4b has b ഠ 5.7; there- derstood as follows. In the absence of an EBP, the CKC fore, its evolution (solid line in Fig. 4a) resembles the can only redistribute mass to a ®xed coastal area (of slow process. In an extratropical basin, b k 1 is usually width d on the southern, and northern, and ␦ on the true because the EBP has an area comparable to the western boundaries).2 Therefore, as the incident wave basin area, while the CKC has an area of subbasin scale injects more mass into the Kelvin wave, the amplitude due to the narrow offshore width d. However, if the of the CKC increases. The maximum amplitude is there- equatorial region is included, the dominant role of EBP fore achieved at the end time, when all the incident mass will no longer be true, as will be discussed in section 4. is accumulated to the CKC. As shown in Fig. 4c, the decreasing mass of the incident planetary wave (solid line) is balanced completely by the increasing mass in e. Energetics the CKC (dash±dot line). In contrast, when the slow The slow EBP process and the fast CKC process are process becomes important (Fig. 4b), as time progresses, the two major mass sinks and, therefore, determine the the continued addition of mass to the CKC is offset by modulation of the CKC. However, neither of them is an increasing area due to the slow westward expansion the dominant energy sink of the initial long wave dis- turbance. Instead, most of the initial energy is converted in the western boundary to the kinetic energy of the 2 The effect of the Kelvin wave on the eastern boundary is always short Rossby wave (Pedlosky 1965), which, because of much smaller than the EBP and therefore will not be discussed sep- its dispersive nature and the small area occupied, plays arately. little role in the mass balance. The consistency of the 2390 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 mass and energy balances can be understood as follows. First, we notice that the energy of the QG model, after neglecting a constant factor, is

2 2 2 , (yh) ϩ (h/dץ) xh) ϩץ) E ϭ where the ®rst two terms are the kinetic energy and the last term is the available potential energy (APE). The initial mass and energy (mainly APE) of the incident planetary wave are at the order of

2 M0 ϳ A 0h 0, E 0 ϳ A 0(h 0/d) , (31) where A 0 ϭ xpyp is the initial area and h 0 is the initial amplitude. Eventually, the EBP has the mass and energy (again mainly APE) of

2 ME ϳ AEK, EE ϳ AE(K/d) , (32) where AE ϭ xPY is the area of the EBP. Mass conser- vation of the initial and ®nal ¯ow ®eld requires M 0 ϳ ME, or, with (31) and (32), K ϳ (A 0/AE)h 0. Therefore, 2 the ®nal energy of the EBP is EE ϳ AE[(A 0/AE)h 0/d] ϳ (A 0/AE)E 0. Since the area of the initial wave cannot be larger than the basin, A 0/AE ϭ yp/Y Յ 1, we have EE Յ E 0. Thus, as long as the initial pulse occupies an area smaller than the basin scale, the energy of the EBP is always smaller than the initial energy and, therefore, cannot be the dominant energy sink. Physically, al- though the initial mass is transmitted to the EBP, the larger area of the EBP reduces the response amplitude and, in turn, the center of gravity. Consequently, part of the APE is lost. This energy loss can be seen clearly in Fig. 6, the energy evolution diagram for the example in Fig. 2. The energy associated with the incident plan- FIG. 6. Energy evolution for different basin regions (Fig. 1): the etary wave (EW) decreases monotonically, vanishing at western region (EW, circle marked), the eastern region (EE, plus about the end time 1. The energy of the EBP (EE) in- marked), and the northern and southern regions (EN ϩ ES, star creases monotonically, but reaches a ®nal energy level marked). The total energy is also plotted (ETotal, solid). Kinetic energy that is much smaller than the initial distrubance energy for each region is indicated by symbols without line connection. level. The CKC energy (EK) is even less then the EBP in the later stage of the evolution. The remaining energy the area occupied by the short Rossby wave has the is lost mainly to the short Rossby wave along the west- meridional extent yp and the zonal extent of CgTb, where 2 ern boundary. Since the short Rossby wave is associated Cg ϳ ␤/k is the group velocity of the short Rossby principally with the kinetic energy of the western bound- wave, Tb ϳ xp/C is the time for the incident planetary ary current, its energy should dominate the kinetic en- wave to pass through the western boundary, and k ϳ ergy of the western region (EW). However, in our viscous 1/␦ is the dominant short zonal wavenumber. The energy case (1), the damped short Rossby wave (17c) has the of the short Rossby wave is then on the order of EWS 2 2 area integrated energy within the viscous boundary layer ϳ (h 0k) CgTbyp ´ ϳ (h 0/d) xpyp ´ ϳ E 0. Thus, the initial (mainly kinetic energy from the alongshore current) of energy can be converted completely to the kinetic en- 2 EW ϳ (h 0/␦) yp␦ ϳ (d/xp)(d/␦)E 0. For an incident plan- ergy of the short Rossby wave. This discussion also etary wave that is much wider than the deformation helps to understand the QG model in which, if one only radius (d K xp), the damped short wave energy is much considers Rossby waves, the energy is conserved, but smaller than the initial energy EW K E 0. This difference not the mass. can be seen in the kinetic energy (circles) for the western In short, our study above suggests that the incident region, diagrammed in Fig. 6. In the presence of dis- planetary wave ®rst loses its mass to CKC. Along the sipation, the total energy (ETotal) is reduced as shown in eastern boundary, the Kelvin wave sheds mass into a Fig. 6. planetary wave, generating an EBP. Available energy is In an inviscid case, however, the lost energy of the reduced compared to the initial energy because of the incident long wave can be absorbed by the re¯ected EBP's smaller amplitude, which is caused by the spread- short Rossby waves. To estimate the area integrated ki- ing of the EBP. Instead, most of the incident energy is netic energy of the short Rossby wave, we notice that lost to the short Rossby wave along the western bound- SEPTEMBER 1999 LIU ET AL. 2391 ary. In spite of its importance in energy balance, the case (Fig. 9a), slightly less than the theoretical value short Rossby wave contributes little to mass redistri- KM ഠ 0.3 (Fig. 4a). The maximum occurs at about TM bution and, therefore, has little impact on the modulation ϭ 190 days in the shallow water case (Fig. 9a), cor- of the CKC. The EBP balances most of the mass and responding to a nondimensional TM ϭ 0.54, close to the determines the modulation of the CKC. theoretical value TM ϭ 0.57 (Fig. 4a). The mass exchange process and energetics are also similar between the shallow-water model (Figs. 9b,c) 3. Shallow-water model experiments and the QG theory (Figs. 4b and 6). Figure 9b plots the

Here, we present numerical experiments to further mass evolution of the western region [MW as in (11a)] substantiate our theory. The model is a 1.5-1ayer shal- that is dominated by the incident long wave, the EBP low-water model on a beta plane (Wallcraft 1991). The (ME), and the CKC along the southern and northern 1 horizontal resolution is ⁄8Њ. The control run has the mod- boundaries (MN ϩ MS). As in the theory (Fig. 4b), the el con®guration and parameters the same as those of the mass of the incident wave decreases monotonically until analytical solution in Fig. 2. The coef®cient of Laplacian the end time about day 300. Initially, the mass loss to diffusion is 140 m2 sϪ1, which gives a Munk boundary the CKC and EBP is comparable, but the mass loss to layer width 19 km, the same as the Stommel boundary the EBP dominates in the later stage. A comparison of layer width in the case of Fig. 2. The initial SSH is also the theoretical solution, Fig. 4b, with the shallow-water chosen to be the same form as (15), and the initial cur- solution, Fig. 9b, reveals a somewhat stronger CKC in rents are in geostrophic balance. The initial amplitude the shallow-water case, especially in the later stage after is small so that the solution is essentially linear. day 400. This difference is due to a more realistic EBP Figure 7 shows four snapshots of SSH, and Fig. 8 in the shallow-water model. First, the EBP front is ac- shows the SSH pro®les along the zonal and meridional companied by a meridional current that intersects the midbasin sections (all SSH are normalized by the am- southern and northern boundaries, forcing CKCs there. plitude of the initial perturbation wave). Overall, the Second, in the later stage, the faster EBP front in the shallow-water solutions in Figs. 7 and 8 resemble close- southern basin reaches the western boundary (Fig. 7d), ly the corresponding QG theoretical solutions in Figs. generating an additional CKC. 2 and 3, respectively. There are also two differences. The energy evolution of the shallow-water solution First, the initial shallow water wave has a weak dis- is plotted in Fig. 9c for the same three components as persion during its westward propagationÐa feature that the mass. For each component, both the total energy also exists in the QG model simulation of MM. Second, (solid line with marks) and the kinetic energy (uncon- the planetary wave propagates faster at lower latitudes. nected marks) are plotted. It is seen that the initial en-

This feature can be simulated in a generalized QG model ergy in the western region (EW) is lost rapidly, initially if the wave speed is allowed to vary with latitude to the CKC (EN ϩ ES) and then to the EBP (EE). There (McCalpin 1995) and therefore can be readily included is a small amount of kinetic energy in the western re- in our theory. gion, mostly due to the damped short Rossby wave: the The evolution of the CKC in Fig. 9a closely resembles CKC kinetic energy is about half the total energy, while that in the theoretical QG solution in Fig. 4a. For ref- the EBP kinetic energy is negligible, as they should be. erence, the time series of midlatitude SSH outside the All of the features agree with the theory (Fig. 6) re- western boundary layer at (0.6Њ,30ЊN) is used to cal- markably well. culate the proxy for the SSH amplitude evolution of the Finally, sensitivity experiments are performed for incident planetary wave on the western boundary (dot various sizes and locations of the initial anomaly. Figure in Fig. 9a) corresponding to hWL(0Њ,30ЊN) in the QG 10 presents the normalized maximum amplitude (KM) theory (dotted line in Fig. 4a). The theoretical peak and (Figs. 10a,c) and time of maximum (TM) (Figs. 10b,d) end times of the incident wave can be calculated as CKC amplitude for four sets of sensitivity experiments.

0.5xp/C ϭ 180 days and xp/C ϭ 360 days, respectively. The ®rst set has seven experiments (pluses in Figs. These correspond roughly to the peak and end times of 10a,b) in which the initial patches are all centered in the western boundary SSH [hWL(0Њ,30ЊN) in Fig. 9a]. the middle of the basin (x ϭ 10Њ, y ϭ 30ЊN) but have The evolution of the CKC can be seen in the coastal different zonal widths; the second set also has seven SSHs on the southern (solid), eastern (star), northern experiments (pluses in Figs. 10c,d), which are the same (plus), and western (circle) boundaries (Fig. 9a). The as the ®rst set, except that meridional sizes are varied; coastal SSHs evolve with virtually the same amplitude the third set has four experiments (stars in Figs. 10a,b), and phase, although the amplitude is slightly larger on which are the same as the ®rst set, but the initial patch the western coast. This supports the speculation that a has its western edge on the western boundary; the fourth low-frequency coastal Kelvin wave has a uniform SSH set has four experiments (stars in Figs. 10c,d), which along the coast, as assumed in (1b). The coastal SSHs are the same as the third set except with different me- in Fig. 9a, therefore, correspond to K(t)Ðthe amplitude ridional widths. The experiments with varying zonal of CKC in the theory (solid line in Fig. 4a). The max- size (Figs. 10a,b) all have yp ϭ 6Њ, while the experiments imum amplitude is KM ഠ 0.26 for the shallow-water with varying meridional size (Figs. 10c,d) all have 2392 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 7. Four snapshots of sea surface height of the shallow-water experiment (normalized by

the initial perturbation amplitude) at (a) initial time, (b) t ϭ 0.5TEI, (c) t ϭ TEI, and (d) t ϭ 1.5TEI,

where TEI is the end time of the initial planetary wave (360 days). Contour interval is 0.05. Contours larger than 0.4 are not plotted.

xp ϭ 6Њ. The corresponding theoretical QG solutions are mainly by the slow mass spreading of the EBP. As a also plotted in Fig. 10 (circles). It is seen that the model result, the amplitude of CKC is weaker than the initial and theory have a good overall agreement, with the error pulse, and the maximum is achieved slightly after the usually within about 20%. The agreement is particularly peak time of the incident wave. Most of the initial en- good for the two sets of experiments with the initial ergy is lost to short Rossby waves, which are then dis- patch against the western boundary (stars) because of sipated on the western boundary. Finally, it is important the small wave dispersion during the initial wave prop- to point out that our theory, although derived in an agation toward the western boundary. The major fea- extratropical basin, can be applied directly to the ad- tures of the theoretical solution are clearly captured. For justment process around an island. This will be dis- example, the amplitude does not change much with the cussed in the appendix. Furthermore, it can be shown zonal size (Fig. 10a), but increases linearly with the that the formation of an island circulation (Godfrey meridional size (Fig. 10c). This result is easy to un- 1989) can be clearly understood in terms of the inter- derstand, because, for the extratropical basin case, the action of coastal Kelvin waves and Rossby waves (Liu dominant mass redistribution is the slow process due to et al. 1999). the EBP. Therefore, the amplitude is determined, to ®rst order, by the ratio of the meridional size y /Y, but in- p 4. Application to extratropical±tropical interaction dependent of xp, as shown in (28a). Overall, the theory tends to agree better with experiments of larger xp. This The QG theory discussed above, after some modi®- is expected because of the role of the long wave ap- cations, can also be applied to a basin of combined proximation in the theory. extratropics±tropics with the inclusion of the equator. In short, our theory and model experiments suggest For the application to decadal climate variability (Lysne that in an extratropical basin the CKC is modulated et al. 1997), it is important to keep in mind that we are SEPTEMBER 1999 LIU ET AL. 2393

FIG. 8. Normalized sea surface height pro®les for the shallow-water solution snapshots in Fig.7, along (a) the middle basin latitude y ϭ 30ЊN, and (b) the middle basin longitude x ϭ 10Њ.

FIG. 9. (a) Evolution of coastal sea surface heights on the eastern (hEB, star), southern (hSB, solid), northern (hNB, plus), and western (hWB, circle) boundaries (in the middle of each boundary). These sea surface heights can be taken as the proxies for the Kelvin wave amplitude

K(t) in the theoretical QG solution. The proxy time series of the incident long wave at the west coast [hWL (x ϭ 0Њ, y ϭ 30ЊN), dot] is calculated using the sea surface height outside the western boundary layer (x ϭ 0.6Њ, y ϭ 30ЊN) by adding a time offset of 0.6Њ/C. (b) The mass evolution in the western region MW, eastern region ME, and the combined northern and southern boundary region MN ϩ MS (see Fig. 1). The boundary of the western and eastern regions change with time as in Fig. 1 to track the planetary wave packets. For comparison with the theoretical results in Fig. 4, the nondimensional time T ϭ 1 there corresponds to the dimensional time of about 360 days. 2394 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 10. Shallow-water sensitivity experiments on the size (xP and yP) and location of the initial

planetary wave packet. The normalized maximum amplitude KM (a), (c) and time of maximum

amplitude TM (b), (d) are plotted against the different zonal (a), (b) and meridional (c), (d)

dimensions. (a) and (b) Two sets of experiments with different zonal widths (xP): one set of seven experiments (pluses) have the initial perturbation patch centered in the middle of the basin, while the other set of four experiments (stars) have the west edge of the initial pulse on the western

boundary. All 11 cases have the same yP ϭ 6Њ. The maximum amplitude in (a) is normalized by the maximum derived from the time series of SSH outside the western boundary layer h(0.8Њ,

30ЊN), while the time for maximum in (b) are normalized by the theoretical end time TEI ϭ xP/ C. The QG theoretical solutions are also plotted (circles). (c) and (d) Same as (a) and (b), except

that the experiments have different meridional dimensions while using the same zonal width xP ϭ 6Њ. studying the slow decadal modulation of the equatorial al. 1994; McCalpin 1995). These fast processes in an thermocline, rather than the fast (intraannual) equatorial combined extratropical±tropical basin will be referred waves. First of all, we should realize that, in a combined to as the ``Equatorial±Coastal Kelvin Wave Current'' extratropical±tropical basin, the fast mass redistribution (ECKC), although one should keep in mind that equa- involves not only coastal Kelvin waves but also equa- torial Rossby waves are also included. torial Kelvin and Rossby waves, all of which propagate The conceptual model that we obtained in a midlat- with a speed comparable to that of the coastal Kelvin itude basin can be generalized to a combined extra- wave c0. The slow process, however, is still due to the tropical±tropical basin. The major modi®cation is to re- extratropical EBP, now at decadal timescale. Along the place the coastal Kelvin waveguide (equator side) width western boundary, an equatorward coastal Kelvin wave of d for the equatorial region of fast mass redistribution will ®rst be converted to an eastward equatorial Kelvin due to equatorial waves. Realizing that the so-called fast wave, which is then re¯ected on the eastern boundary or slow process is relative to the adjustment of the initial by shedding most its mass and energy into a westward long wave, the latitude width y of the fast equatorial equatorial Rossby wave (Cane and Sarachik 1977, E 1979). The Rossby wave then re¯ects as an equatorial region (Fig. 11) can be estimated as follows. First, the Kelvin wave and so forth. These re¯ections of equatorial equatorial region has to adjust faster than the incident waves rapidly build up an equatorial quasi-basin mode extratropical planetary wave. Second, the slowest pro- (Cane and Moore 1981). A small amount of mass leaks cess in the equatorial region is the eastern boundary out poleward along the eastern boundary in each hemi- planetary wave radiating at yE. Therefore, for this wave sphere as a coastal Kelvin wave. The Kelvin wave then to be a part of the fast redistribution process, this wave 2 generates the EBP in the extratropics (Miles 1972; En- must cross the basin width X at the speed ␤gЈD/f (yE) ®eld and Allen 1980; White and Saur 1981; Jacobs et by the time the initial midlatitude planetary wave (cen- SEPTEMBER 1999 LIU ET AL. 2395

western boundary region due to the short Rossby wave, a widespread equatorial region due to equatorial waves, a widespread region of EBP, and a poleward boundary region of coastal Kelvin waves. The evolution of the monthly mean SSH associated

with the ECKC is plotted on the equator (hEQ), the east- ern (hEB), and northern (hNB) boundaries (Fig. 13). The three SSH time series show virtually the same amplitude and phase, as in the case of the extratropical basin, in Fig. 9a. However, different from Fig. 9a, the coastal SSH along the western boundary (not shown) is much larger than the others. One possible explanation is that FIG. 11. The schematic ®gure of different solution regimes in a the mass ¯ux of the western boundary coastal Kelvin combined extratropical±tropical ocean basin. Compared with an ex- wave, which is trapped in a western boundary layer in tratropical basin, the equatorial region replaces the equator-side coast- al Kelvin wave region. the extratropics (tens of kilometers), is spread onto the equatorial Kelvin wave, which has a much wider equa-

torial deformation radius (dEQ ഠ 300 km). Fortunately, as in an extratropical basin, the western-boundary tered at latitude Yc) passes the western boundary after 2 Kelvin wave contributes little to the total mass redis- traveling xp at the speed of ␤gЈD/f (Yc); that is, tribution and therefore has little impact on the modu- Xx ϭ p . lation of the ECKC. Ϫ2 Ϫ2 f (yEc) f (Y ) The amplitude of the ECKC (multiplied by 10 in Fig. 13a) is about an order smaller than in the extratropical The latitude width of the fast equatorial region is there- basin in Fig. 9a. This is due to the mass spreading over fore determined by a much larger area, which now includes a large equa- 1/2 f(yE) ϭ f(Yc)(xp/X) . (33) torial region and the Southern Hemisphere. As for Fig. 9, the end time for the initial pulse is about 360 days, For a large-scale initial pulse in the midlatitude, this both from the theory and the proxy time series of the width is usually much larger than the deformation radius incident wave (Fig. 13a, dot). However, the time of 1/2 in the extratropics (d) or in the equator (dEQ ϭ [(gЈD) / maximum ECKC is achieved at about day 270, in Fig. 1/2 3 ␤] ). This implies that, if an extratropical basin is 13a, almost 100 days later than in Fig. 9a. This is con- extended to include the equator, the amplitude of the sistent with the analysis above. The fast ECKC, which ECKC will be reduced due to the spreading of the initial is dominated by the fast equatorial region, has a much mass into a larger area. Furthermore, the time of max- larger area than the CKC in the extratropical basin. imum ECKC should be delayed closer to the end time Therefore, the mass balance is more like the fast balance TEI. This is because the increased area is contributed (29) than the slow balance in (26). Indeed, the mass predominantly by the equatorial region, which has a fast evolution diagram in Fig. 13b clearly shows that much mass redistribution. Therefore, the area that the fast pro- of the western-boundary incident mass (MW) is rapidly cess occupies is signi®cantly increased relative to the injected into the ECKC (M ϩ M ϩ M ) due to the slow process. The mass balance should resemble more N S EQ fast redistribution. The EBP (ME) never becomes com- the fast process (29) than the slow process (26) of an parable to the ECKC except at the very latest stage. extratropical basin. These speculations are con®rmed by This is in sharp contrast to the extratropical basin case the shallow-water experiments below. (Fig. 9b), where the EBP is the dominant mass sink. The major features of the adjustment process in a The relatively greater importance of the ECKC can also combined extratropical±tropical basin are demonstrated be seen in the energy evolution diagram in Fig. 13c. In in the shallow-water experiment illustrated by Figs. 12 contrast to the extratropical basin case (Fig. 9c), the and 13, which should be compared with the correspond- energy in the ECKC (E ϩ E ϩ E ) is much stronger ing ®gures for the extratropical basin case, Figs. 7 and N S EQ than the EBP (EE) until after the end time. 9, respectively. The initial anomaly (Fig. 12a) is the Most of the ECKC energy is now APE (Fig. 13c). same as in Fig. 7a. The snapshots at 180, 360, and 540 This is different from Fig. 9c. The reason is that the days are plotted in Figs. 12b, 12c, and 12d, respectively. fast process in the equatorial region is dominated by In spite of strong inertial±gravity waves, the large-scale low-latitude Rossby waves, rather than the equatorial pattern in each hemisphere is characterized by a narrow Kelvin wave. Indeed, (33) gives the width of the fast 1/2 equatorial region as yE ϭ f(30Њ) ϫ (10Њ/20Њ) ഠ 20Њ. This is much wider than the equatorial deformation ra-

3 dius, dEQ ഠ 3Њ, which is also the width of the equatorial The equatorial deformation radius dEQ is the latitude where the speeds of the equatorial Kelvin wave and extratropical Rossby wave Kelvin wave. To further examine the relative contri- 1/2 2 equal each other: (gЈD) ϭ ␤gЈD/f (dEQ). bution of each subregion within the equatorial region, 2396 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 12. As in Fig. 7 but in a combined extratropical±tropical basin. Now, the contour interval is 0.005 (label scaled by 1000), and contours above 0.1 are not plotted. the equatorial region is subdivided into four bands: |y| of both the EBP and ECKC should be doubled to take Յ 5Њ,5ЊϽ|y| Յ 10Њ,10ЊϽ|y| Յ 15Њ, and 15ЊϽ|y| into account the Southern Hemisphere. The amplitude Յ 20Њ. The evolution of mass and energy in each band of the ECKC can be derived from the same equation as is plotted in Figs. 14a and 14b, respectively. It is seen the CKC in (25), except that the coef®cients a and b in that each band contributes about the same amount of (23) need to be replaced by mass and energy. The mass and energy of the two bands within 10Њ of latitude peak earlier at about day 220, b ϭ 2YxpE/{␲[2X(d ϩ y ) ϩ Y␦]}, while the other two bands peak later at day 300 and day a ϭ m0/{2[2X(d ϩ yE) ϩ Y␦]}. (34) 360. All these peaks occur before the end time TEI (ϳ360 days) and therefore represent the fast process. For the shallow-water example in Figs. 12±14, the mod- For a quantitative comparison between our QG theory i®ed QG theoretical solution produces ECKC amplitude and the shallow-water experiments, two modi®cations in Fig. 15a (solid line). The maximum amplitude is KM are needed for our QG solution (11)±(14). First, the ϭ 0.039, which is achieved at the time TM ϭ 0.9. This coastal Kelvin wave solution on the equatorial boundary is in good agreement with the shallow-water solution

(12) should be replaced with the width of yE in (33), (Fig. 13a), which has the normalized KM ഠ 0.036 and representing the fast equatorial region. Second, the areas TM ഠ 0.80. The theoretical mass redistribution (Fig. SEPTEMBER 1999 LIU ET AL. 2397

FIG. 13. Similar to Fig.9 but for the combined extratropical±tropical basin case in Fig. 12. (a) Evolution of boundary

sea surface heights multipled by a factor 10: along the equator (hEQ, solid), northern boundary (hNB, dash), and eastern

boundary (hEB, dash±dot). The western coastal sea surface height is not plotted. The dot is the proxy for the incident long wave amplitude calculated from the sea surface height outside the western boundary at (0Њ,30ЊN) (similar to Fig.9). (b) Mass evolution and (c) energy evolution. The fast process region now consists of the northern boundary,

southern boundary, and the equatorial region. The equatorial region has the latitude span yE ϭ 20Њ, as calculated from Eq. (33).

FIG. 14. The mass (a) and energy (b) evolution in each subequatorial region (5Њ latitude in both hemispheres) for the case of Fig. 12. The mass (energy) from both hemispheres, within the speci®ed bond, is combined in the plotted totals. 2398 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 15. Evolution of the QG theoretical solution: same as in Fig. 4 but uses the modi®ed theoretical QG solution [Eqs. (25) and (34)] to simulate the case in a combined extratropical±tropical basin (Fig. 12). The b ϭ 0 case is without the EBP.

15b) also resembles the shallow-water solution (Fig. creases from 0.039 to 0.043 and 0.048, while TM de- 13b) remarkably well. To examine the role of the fast creases slightly from 0.9TEI to 0.88TEI and 0.86TEI. The mass balance (29) in determining the modulation of agreement is good, especially on the amplitude. Finally, ECKC, the area ratio parameter can be calculated from the mass evolution also agrees well between the model

(34) as b ഠ 0.3. Therefore, the fast process is dominant and theory (Figs. 16b and 17b). As Yc moves toward in modulating the ECKC. This can be seen in Fig. 15a the equator, the mass controlled by the fast process de- from the strong resemblance between the K(T) with (sol- creases, while the mass in the slow process increases. id) and without (dashed) the EBP. The mass exchange Correspondingly, the time of the maximum ECKC also diagram (Fig. 15b) also resembles that without EBP in becomes earlier. All these features can be understood Fig. 15c except in the later stage. This resemblance from a stronger in¯uence of the fast process of ECKC becomes much clearer if one recalls the larger differ- in the case with a Yc closer to the equator. Thus, the ences between Figs. 4b and 4c for the extratropical ba- modi®ed QG theory is able to capture the major features sin. of ECKC modulation. In contrast to an extratropical If the initial pulse is located closer to the equator basin, the most important mass redistribution mecha-

[smaller Yc in (33)], the width of the fast equatorial nism in a combined extratropical±tropical basin, be- region shrinks [smaller yE according to (33)], while the comes the fast equatorial waves. Consequently, the time latitudinal width of the EBP increases (larger Y). There- of maximum ECKC is delayed substantially, approach- fore, the area increases for the EBP, but decreases for ing the end time of the incident wave. The maximum the ECKC. The maximum amplitude of ECKC should, amplitude is usually small. therefore, decrease and the time of that maximum should occur earlier. This is con®rmed by three shallow-water 5. Summary and discussions sensitivity experiments in Fig. 16a, which plots the SSH averaged along the equator and outside the western The adjustment of a midlatitude ocean basin to a plan- boundary layer. One experiment is the same as in Figs. etary wave packet incident onto the western boundary

12 and 13, with Yc ϭ 30ЊN. The other two experiments is studied both analytically and numerically. The focus have Yc ϭ 25ЊN and 20ЊN. Equation (33) gives the is on the low-frequency modulation of the resulting widths of the fast equatorial region as yE ഠ 20Њ, 17.5Њ, coastal Kelvin wave. This coastal Kelvin wave, at low and 14Њ, respectively. Figure 16a shows that the max- frequencies, can be approximated as a coastal Kelvin imum amplitude KM increases from about 0.036 to 0.042 wave current (CKC) along the boundaries of the basin. and 0.048. The time for maximum (TM) tends to de- It is shown that the CKC is modulated mainly by two crease only slightly from about 0.8TEI. In comparison, mass redistribution processes: a fast process due to the the theoretical solution in Fig. 17a shows that KM in- CKC, and a slow process due to the eastern boundary SEPTEMBER 1999 LIU ET AL. 2399

FIG. 16. Three shallow-water sensitivity experiments in the combined extratropical±tropical

basin with different center latitudes Yc ϭ 30ЊN (solid), 25ЊN (dash), and 20ЊN (dash±dot) for the initial planetary wave patch. (a) The evolution of the zonal mean equatorial sea surface height

(hEQ, multipled by 10) and the proxy of incident long wave sea surface height calculated from the sea surface height outside the western boundary layer at (0.6Њ,30ЊN). Time is normalized by the end time of each corresponding initial patch. (b) The evolution of mass of the western

(MW), eastern (ME), and the combined equtorial, northern and southern boundary regions (MN ϩ

MS ϩ MEQ). planetary wave (EBP). Because the mass is eventually waves is constrained by the balance of cross-shore mass redistributed to a much larger area of basin scale, the ¯ux at the coast. In the midlatitudes, however, the coast- amplitude of the CKC is usually much smaller than that al Kelvin wave is generated to balance the alongshore of the initial disturbance. In a midlatitude basin, the EBP mass ¯ux of the incident long wave, while the cross- dominates mass redistribution, which results in a Kelvin shore mass ¯ux is completely balanced by the re¯ected wave peak time close to that of the incident wave. The short Rossby wave. Second, the equatorial Kelvin wave short Rossby wave, although contributing little to mass and low mode Rossby waves have comparable wave redistribution, absorbs most energy at the western speeds, while the midlatitude Rossby wave is far much boundary. slower than the coastal Kelvin wave. This has several The QG theory is further extended to a combined consequences. First, the equatorial interaction is a local extratropical±tropical basin. Now, the fast equatorial process. In the midlatitudes, however, the interaction waves, both Kelvin and Rossby waves, form the equa- depends critically on the remote area that is affected by torial coastal Kelvin wave current (ECKC) and domi- the coastal Kelvin wave during the slow modulation nate mass redistribution. As a result, the coastal Kelvin time. Given an incident long wave, the coastal Kelvin wave peaks near the end time of the incident wave. wave evolves differently for various basins, as seen in the cases of the midlatitude and extratropical±tropical basins (compare Figs. 7 and 12). A closed basin also a. Rossby±Kelvin wave interaction in the midlatitude differs from a half-plane ocean, in the latter of which and equatorial regions Kelvin waves eventually diminish because of the ever It is interesting to compare our results of Rossby± expanding coastal area (Dorofeyev and Larichev 1992). Kelvin wave interaction in the midlatitude with that in Second, the comparable speed of equatorial waves im- the equatorial region (Cane and Sarachik 1977, 1979). plies that the incident and re¯ected waves occupy about In both cases, mass redistribution is the major constraint the same area and therefore both have a comparable for the interaction. However, signi®cant differences ex- amplitude. Therefore, the re¯ected wave can balance ist. First, the re¯ection of equatorial Kelvin and Rossby not only the mass but also the energy ¯uxes of the 2400 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. 17. As in Fig. 16 but for the modi®ed QG theoretical solution of (25) and (34): (a) K(t) (multipled by 10) and sea surface height on the west coast (dot); (b) the mass evolution. incident wave. In the midlatitudes, however, the coastal boundary coastal Kelvin wave is about 0.2ЊC (in com- Kelvin wave is important for mass balance, but not for parison, their modeled subduction anomaly can reach energy balance. The latter is accomplished by re¯ected 1ЊC in the extratropics). These anomalies, although short Rossby waves. Indeed, regardless of mass con- small, could be ampli®ed by positive ocean±atmosphere servation (or the consistency condition of McWilliams feedbacks and therefore may still be important for cli- 1977), the QG system conserves energy within a closed mate variability. basin. Third, the comparable speed and local re¯ection One alternative way of thinking may also be helpful nature of the equatorial waves also ensures that both in understanding extratropical±tropical interaction (Bat- incident and re¯ected waves always peak at the same tisti 1989, 1991; Kessler 1991). In terms of the classical time. For an incident midlatitude Rossby wave, how- theory of equatorial wave (latitudinal) modes, an extra- ever, the resulting coastal Kelvin wave could peak at tropical long Rossby wave packet could have signi®cant different times, depending on the basin. In a midlatitude projection on some higher mode equatorial Rossby basin, the coastal Kelvin wave does peak around the waves, whose turning latitudes, and in turn the maxi- same time as the incident Rossby wave did at the west- mum pressure anomalies, are located in the extratropics. ern boundary. However, when the equator is present in For example, mode 15 has the peak pressure at about the basin, the coastal Kelvin wave peaks substantially latitude 17Њ (Fig. 1 of Kessler 1991). The re¯ected later. Kelvin wave, calculated by zonal mass ¯ux balance, has a small amplitude, less than 10% of the incident wave (Fig. 3 of Kessler 1991). The afterward equatorial wave b. Extratropical±tropical wave interaction adjustment should have an amplitude comparable to this Our study suggests that an extratropical thermocline equatorial Kelvin wave. Qualitatively, this small am- anomaly can modulate the mean equatorial thermocline plitude of equatorial response is consistent with our con- through the ECKC. However, the resulted equatorial clusion in section 4. This modal approach easily produce thermocline anomaly is usually much smaller than the the ®rst re¯ection equatorial Kelvin wave, but it is not initial extratropical anomaly. Nevertheless, a large-scale convenient to describe the low-frequency modulation of anomaly at a relatively low latitude may still produce the equatorial thermocline, which is accomplished after a non-negligible anomaly in the equatorial region. This multire¯ection in the equatorial wave guide. The modal seems to agree qualitatively with the numerical exper- approach is also inconvenient to describe a temporal iment of Lysne et al. (1997). The equatorial thermocline adjustment process. For example, it is dif®cult to obtain temperature anomaly that is transmitted by the western the timing of maximum modulation of equatorial ther- SEPTEMBER 1999 LIU ET AL. 2401 mocline. Indeed, it gives the timing of the ®rst re¯ection Acknowledgments. We would like to thank Dr. P. equatorial Kelvin wave the same as the incident Rossby Chang for a discussion on extratropical±tropical inter- wave, which is earlier than the peak time of the mod- action, which provided a strong motivation for this ulated equatorial thermocline. In addition, a higher work; Drs. J. A. Young and H. Hurlburt for helpful mode of equatorial Rossby wave, although having a discussions in the early stage of the work; Dr. J. pressure peak in the extratropics, has signi®cant, alter- McCreary for his comments on an early version of the nating zonal velocity all the way to the turning latitude paper; and Dr. J. S. Godfrey for bringing our attention (Fig. 2 of Kessler 1991); these higher modes are also to his early relevant work during our revision. We also very dispersive. Therefore, strictly speaking, any single thank Dr. A. Wallcraft for his help in the set up of the high mode is dif®cult to represent a nondispersive plan- shallow-water model. Comments from two anonymous etary wave packet localized in the midlatitude. Finally, reviewers have helped to improve the paper substan- the modal approach still uses zonal mass ¯ux balance, tially. This work is supported by ONR, NSF,and NOAA. while our approach uses alongshore mass ¯ux balance. The calculation of the shallow-water model is performed In spite of these differences, the modal approach and on the supercomputer at the Stennis Space Center, which our approach could still be complementary to each other. is supported by ONR. Our study may have important implication to the un- derstanding of general oceanic circulation and its low- APPENDIX frequency variability. Past attention has focused on Rossby waves. Our study shows that the coastal Kelvin Wave Interaction around an Island wave is an inseparable part of the general circulation system especially for low-frequency variability. The as- The basin solution (25) can be applied directly to an sociated CKC could be of critical importance in deter- island of the size X and Y, shown schematically in Fig. mining the boundary pressure distribution of the general A1. The incident wave now hits the east coast of the island (x ϭ X ϩ X), producing a CKC around the circulation, which in turn affects the interior ocean. In I northern (y ϭ Y ϩ Y), western (x ϭ X ), and southern an accompany paper (Liu et al. 1999), our theory is I I (y ϭ Y ) boundaries of the island. A new planetary wave extended to study the variability of the general circu- I is radiated westward from the west coast of the island lation around an island (Godfrey 1989). The interaction (corresponding to the EBP in the basin case, and thus between the coastal Kelvin wave and Rossby waves, will still be called the EBP). Solution (25) is valid until especially the short Rossby waves, will be seen as crit- the EBP hits the western boundary of the basin (at x ical elements in the establishment of the island circu- ϭ 0) or t Ͻ X /C. All the discussions on the modulation lation. This effect of short Rossby wave is reminiscent I of the amplitude of CKC, the mass exchange, and en- of the wave interaction on an equatorial island (Cane ergetics will be the same as in the QG theoretical basin and du Penhoat 1982). solution. Furthermore, later adjustment after the EBP Our study so far remains preliminary. Many important hits the western boundary of the basin can also be dis- processes need to be studied in order to gain a full cussed similarly, if one replaces the basin size with X understanding of the realistic ocean. One limitation of B ϫ Y and the incident wave packet size with x ϫ Y. the present work is the linear wave assumption. In the B P Figure A2 shows one example of SSH snapshots in real ocean such as the North Paci®c, the main ther- a basin including an island. The evolution is essentially mocline tends to outcrop at mid- and high latitude. As the same as that in a basin (Fig. 2). The corresponding a result, a Kelvin wave may not be able to propagate shallow-water experiment is shown in Fig. A3, whose around the basin. We speculate that this latitudinal con- mass and energy evolution are shown in Figs. A4a and straint of the Kelvin wave propagation will not change the major features of our work here. This is because, in our theory, the polar boundary plays a negligible role in both the mass and energy balance. The reduction of the latitudinal extent of the eastern boundary Rossby wave, however, may result in a larger amplitude of the Kelvin wave because the initial mass is redistributed onto an area smaller than the entire basin. This spec- ulation remains to be investigated. A study of the effect of the mean ¯ow advection is also in progress. The adjustment in a realistic Paci®c Ocean is also under current study (Wu and Liu 1999, manuscript submitted to J. Phys. Oceanogr.). Our preliminary results suggest that the simple theory here can be applied successfully to understanding the interaction of open ocean with the western coastal Paci®c Ocean and its marginal seas, as FIG. A1. Schematic ®gure of different solution regimes for an ini- well as the equatorial Paci®c Ocean. tial planetary wave incident on the east coast of an island. 2402 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

FIG. A2. Four snapshots of the theoretical QG solution, similar to Fig. 2, but for the corresponding island case.

A4b, respectively. The agreement of the mass evolution component in the basin case in Fig. 9b, compared with Figs. A4a and the theoretical solution (the same as Fig. the theory, Fig. 4b, virtually disappeared in the island 4b) is remarkable. Indeed, the agreement is better than shallow-water experiment (Fig. A4b). Correspondingly, the basin case of the shallow-water experiment (Fig. the energetics (Fig. A4c) also show a better agreement 9b). For example, the overestimation of the CKC mass with the theory (Fig. 6) than the basin shallow-water

FIG. A3. Four snapshots of a shallow-water experiment, similar to Fig. 7, but for the corresponding island case. SEPTEMBER 1999 LIU ET AL. 2403

FIG. A4. The same as in Fig. 9 but of the shallow-water experiment for the corresponding island case. The notation of the solution regime is the same as in Fig. A1 with subcripts ``W'' for the region east of the island and ``E'' for the region west of the island. case (Fig. 9c). The reason is simple. In the island case, a consistent quasigeostrophic model in a multiply-connected do- the EBP does not interact with the northern and southern main.' Dyn. Atmos. Oceans, 1, 443±453. Fofonoff, N. P., 1954: Steady ¯ow in a frictionless homogeneous boundaries of the basin, even in the shallow-water mod- ocean. J. Mar. Res., 13, 254±262. el. This is the case of the QG theoretical basin solution Godfrey, J. S., 1975: On ocean spindown. I: A linear experiment. J. (25), which has neglected the Kelvin wave produced by Phys. Oceanogr., 5, 399±409. the EBP on the northern and southern boundaries. How- , 1989: A Sverdrup model of the depth-integrated ¯ow from the world ocean allowing for island circulations. Geophys. Astro- ever, in a basin model, either QG or shallow water, the phys. Fluid Dyn., 45, 89±112. EBP will interact with the northern and southern bound- Jacobs, G. A., H. E. Hurlburt, J. C. Kindle, E. J. Metzger, J. L. aries, generating additional Kelvin waves. Mitchell, W. J. Teague, and A. G. Wallcraft, 1994: Decade-scale trans-Paci®c propagation and warming effects of an El Nino anomaly. Nature, 370, 360±363. REFERENCES Kawase, M., 1987: Establishment of deep ocean circulation driven by deep-water production. J. Phys. Oceanogr., 17, 2294±2317. Anderson, D. L. T., and A. E. Gill, 1975: Spin-up of a strati®ed ocean Kessler, W. S., 1991: Can re¯ected extra-equatorial Rossby waves with applications to upwelling. Deep-Sea Res., 22, 583±596. drive ENSO? J. Phys. Oceanogr., 21, 444±452. Battisti, D. S., 1989: On the role of off-equatorial oceanic Rossby Liu, Z., 1990: On the in¯uence of the continental slope on the inertial waves during ENSO. J. Phys. Oceanogr., 19, 551±559. western boundary currentÐthe enhanced transport and recir- , 1991: Reply. J. Phys. Oceanogr., 21, 461±465. culation. J. Mar. Res., 48, 254±284. Cane, M. A., and E. S. Sarachik, 1977: Forced baroclinic ocean , L. Wu, and H. Hurlburt, 1999: Rossby wave±coastal Kelvin motions. II. The linear equatorial bounded case. J. Mar. Res., wave interaction. Part II: Formation of island circulation. J. Phys. 35, 395±432. Oceanogr., 29, 2405±2418. , and , 1979: Forced baroclinic ocean motions. III. The linear Lysne, J., P. Chang, and B. Giese, 1997: Impact of the extratropical equatorial basin case. J. Mar. Res., 37, 355±398. Paci®c on equatorial variability. Geophys. Res. Lett., 24, 2589± , and D. Moore, 1981: A note on low frequency equatorial basin 2592. modes. J. Phys. Oceanogr., 11, 1578±1584. McCalpin, J. D., 1995: Rossby wave generation by poleward prop- , and Y. du Penhoat, 1982: The effect of islands on low frequency agating Kelvin waves: The midlatitude quasigeostrophic ap- equatorial motions. J. Mar. Res., 40, 937±962. proximation. J. Phys. Oceanogr., 25, 1415±1425. , and S. E. Zebiak, 1985: A theory for El Nino and the Southern McCreary, J., 1976: Eastern tropical ocean response to changing wind Oscillation. Science, 228, 1084±1087. systems with application to El NinÄo. J. Phys. Oceanogr., 6, 632± Dorofeyev, V. L., and V. D. Larichev, 1992: The exchange of ¯uid 645. mass between quasi-geostrophic and ageostrophic motions dur- , 1983: A model of the tropical ocean±atmosphere interaction. ing the re¯ection of Rossby waves from a coast. I. The case of Mon. Wea. Rev., 111, 370±387. an in®nite rectilineary coast. Dyn. Atmos. Oceans, 16, 305±329. , and P. Kundu, 1988: A numerical investigation of the Somali En®eld, D., and J. Allen, 1980: On the structure and dynamics of Current during the southwest Monsoon. J. Mar. Res., 46, 25±58. monthly mean sea level anomalies along the Paci®c coast of McWilliams, J. C., 1977: A note on a consistent quasi-geostrophic North and South America. J. Phys. Oceanogr., 10, 557±578. model in a multiply-connected domain. Dyn. Atmos. Oceans, 1, Flierl, G. R., 1977: Simple applications of McWilliams' `A note on 427±441. 2404 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29

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