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A Modi®ed Sverdrup Model of the Atlantic and Caribbean Circulation
ROXANA C. WAJSOWICZ* Department of Meteorology, University of Maryland at College Park, College Park, Maryland
(Manuscript received 9 October 2000, in ®nal form 6 August 2001)
ABSTRACT An analytical model of the mean wind-driven circulation of the North Atlantic and Caribbean Sea is constructed based on linear dynamics and assumed existence of a level of no motion above all topography. The circulation around each island is calculated using the island rule, which is extended to describe an arbitrary length chain of overlapping islands. Frictional effects in the intervening straits are included by assuming a linear dependence on strait transport. Asymptotic expansions in the limit of strong and weak friction show that the transport streamfunction on an island boundary is dependent on wind stress over latitudes spanning the whole length of the island chain and spanning just immediately adjacent islands, respectively. The powerfulness of the method in enabling the wind stress bands, which determine a particular strait transport, to be readily identi®ed, is demonstrated by a brief explanation of transport similarities and differences in earlier numerical models forced by various climatological wind stress products. In the absence of frictional effects outside western boundary layers, some weaker strait transports are in the wrong direction (e.g., Santaren Channel) and others are too large (e.g., Old Bahama Channel). Also, there is no western boundary current to the east of Abaco Island. Including frictional effects in the straits enables many of these discrepancies to be resolved. Sensitivity in strait transport to friction parameter is explored for the Caribbean island chain. Transport reversal in the minor passages around the Bahama Banks and Windward Passage as the friction parameter increased is noted. The separation latitude of the western boundary currents on Cuba's east coast moves southward as the friction parameter increases from zero, so making the Great Inagua Passage transport a better proxy for the Windward Passage transport. Major discrepancies with observations, namely, eastward instead of westward ¯ow in Grenada Passage, a southward instead of northward Guyana Current, and hence a Caribbean circulation and Florida Current fed wholly by water masses of North Atlantic origin, cannot be resolved. However, they are simply overcome by extending the model to three layers with the wind-driven and upper limb of the thermohaline circulation con®ned to the top layer, and the lower limb of the thermohaline circulation to the bottom layer. If it is assumed that over the latitudes of the Caribbean there is no signi®cant upwelling/downwelling between the layers, then the thermohaline-driven circulation is effectively a western boundary current, and all of the results for the analytical wind-driven-only model carry over, but with the value of the upper-layer transport streamfunction on the boundary of the American continent set to the magnitude of the thermohaline circulation rather than that on Africa. Exploration of strait transport sensitivity to friction parameter gives that realistic transports through the passages of the Windward Islands are only obtained if the friction coef®cient in these passages is an order of magnitude larger than that in the western passages. Windward Passage transport reverses from south to north for a smaller value of the friction parameter than in the absence of the thermohaline circulation; Anegada and Mona Passages are robust in¯ow passages for the Caribbean Sea. South Atlantic water masses enter the Caribbean Sea through the passages from Grenada Passage to Martinique Passage. As the friction coef®cient in the Windward Islands passages increases from zero, South Atlantic water mass is partially de¯ected northward along the outer arc of the islands and enters the Caribbean Sea through the passages up to Anegada Passage. The model suggests that for realistic friction parameters, South Atlantic water masses are unlikely to be found in the more western passages, or in the western boundary current skirting the edge of the Bahama Banks.
1. Introduction subject of much controversy since Leetmaa et al.'s Whether models based on Sverdrup dynamics explain (1977) note that the observed magnitude of Florida the circulation of the North Atlantic Ocean has been the Strait transport is consistent with that required to bal- ance the southward Sverdrup transport in the ocean in- terior. Wunsch and Roemmich (1985) showed that the * Additional af®liation: Earth System Science Interdisciplinary Center, University of Maryland at College Park, College Park, Maryland. diagnosed northward heat transport for the North At- lantic is not consistent with a Sverdrup model in which the ¯ow through Florida Strait is the return ¯ow of the Corresponding author address: Roxana Wajsowicz, Dept. of Me- northern subtropical gyre. Also inconsistent with this teorology, University of Maryland, 3433 Computer and Space Science Bldg., College Park, MD 20742-2425. simple type of Sverdrup model is evidence of South E-mail: [email protected] Atlantic water masses ¯owing through the passages of
᭧ 2002 American Meteorological Society
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC 974 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 the Lesser Antilles (Wilson and Johns 1997), and as tilles and exit through the Florida Straits. The exact much as 11 Sv (Sv ϵ 106 m3 sϪ1) through Florida Strait pathway for the South Atlantic water mass depends on (Schmitz and Richardson 1991). However, realistic the amount of frictional resistance in the passages. The GCM simulations [see, e.g., Maltrud et al.'s (1998) Plate results are summarized and discussed in section 5. 2] show that a Sverdrup model actually describes well the mean transports in the ocean interior of the North 2. The multiple island rule Atlantic over the latitudes of interest from 10Њ to 30ЊN. Observations of South Atlantic water masses pene- The island rule and multiple-island rule were de- trating the Caribbean can be reconciled with Leetmaa scribed in general form in Wajsowicz (1993). Their der- et al.'s (1977) note if it is recognized that estimating ivation is recapped below in section 2b for the case of transports from the Sverdrup balance does not neces- frictional effects con®ned to oceanic western boundary sarily imply anything about the water mass origin of the layers and assuming that all of the passages are dynam- ¯ow. The properties of the western boundary layer in ically wide and deep. In section 2c, modi®cations to the which the Sverdrupian gyres close are crucial, as de- rules, assuming frictional effects are important, are de- scribed in Wajsowicz (1999b) in the context of the In- scribed with emphasis on the in¯uence of wind stresses donesian Through¯ow. With this revised perspective, for outside the latitude band of the island under con- the wind-driven circulation within the North Atlantic sideration. Asymptotic solutions in the limit of small and Caribbean is reexamined. Sverdrup dynamics are and large friction coef®cient are presented for an ar- shown to be consistent with observations provided a bitrary length island chain, assuming a simple north- thermohaline circulation, which is con®ned to the west- west±southeast skew. ern boundary layer over the domain of interest, is in- cluded, as noted by Townsend et al. (2000). Also, the a. Equations of motion transports through certain straits need to be limited; fric- tional effects are assumed. The streamfunction on island To keep the discussion succinct, the description is boundaries, and so strait transports, are calculated using given in terms of quasi-steady motion of an active layer a frictional form of the multiple-island rule (Wajsowicz above an inert deep layer in which all bottom topog- 1993). This is a quite powerful result, as it enables the raphy is assumed con®ned; the only external forcing is wind stress bands determining a strait transport to be due to surface wind stresses. Let describe the transport identi®ed, and so in turn the sensitivity of the transport streamfunction of the active layer so that the depth- to changes in the wind stress and strength of the ther- integrated zonal and meridional velocities for the layer mohaline overturning circulation to be better under- are given by u ϭϪy, ϭ x. Then, the depth-inte- stood. grated momentum equations are The basic multiple-island rule (Wajsowicz 1993) is P x Ϫ f ϭϪx ϩ ϩF x (2.1a) recapped in section 2, and several extensions appropri- x ate for the Caribbean derived. The wind-driven trans- oo ports through the major passages of the Caribbean are P y f ϭϪy ϩ ϩF y, (2.1b) calculated using the multiple-island rule and Hellerman y and Rosenstein (1983) wind stress climatology in sec- oo tion 3. The cases of dynamically wide and narrow chan- where f is the Coriolis parameter, P is the depth-inte- x y nels are considered. Determining strait transports by the grated pressure, ϭ ( , ) is the wind stress, o is the x y multiple-island rule enables the effect of using different density of the layer, and F ϭ (F , F ) is the depth wind stress climatologies to be readily deduced, and the integrated friction term. The number of boundary con- variety of behavior Townsend et al. (2000) found for ditions required depends on the form of F. However, 11 different wind stress climatologies is easily ex- the no-normal ¯ow condition reduces to ϭ const on plained. These wind-driven multiple-island rule results boundaries. are described in detail, as adding a thermohaline cir- culation does not alter the basic dependencies. b. Dynamically wide passages The model is extended to three layers in section 4 to From Wajsowicz (1993), if it is assumed that fric- include a meridional overturning thermohaline circu- tional effects are unimportant outside western boundary lation. Restricting attention to the latitude band from layers, then the value of on an island to the west of 10Њ to 30ЊN enables the thermohaline circulation to be a landmass upon which ϭ 0, say, may be determined approximated by a western boundary current of speci- by integrating the momentum equations (2.1) along the ®ed transport. Once again, the cases of dynamically closed path shown in Fig. 1a. The resulting island wide and narrow passages are considered. Including the C0 value is upper limb of the thermohaline circulation corrects the direction of the Guyana Current and transport through Ϫ1 Grenada Passage. South Atlantic water masses enter the 0 ϭ ´ dl, (2.2) ⌬ f0 ͶC o Caribbean Sea through the passages of the Lesser An- 0
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FIG. 1. The path C0 used in the island rule (2.2) in (a). The steady-state streamfunction is constant on landmasses and assumed to take the value 0 on the mass to the east, which spans the latitudes of the island and 0 on the island. The paths C0, C1 used in the multiple-island rule (2.3) are shown in (b), where an island upon which ϭ 1 lies to the west of the original island and partially overlaps its meridional extent. (c) A group of N ϩ 1 islands, which are increasingly distant from the landmass as the island index n, n ϭ 0,´´´,N increases. The meridional extent of an island, n, n ϭ 1,´´´,N is partially overlapped only by the island n Ϫ 1 immediately to the east. The value of
ϭ n on island n is given by (2.4). The thin dotted horizontal lines in each schematic represent isolines of f, and the difference between the northern and southern tips of an island n and between the northern and southern extents of its overlap with island n Ϫ 1 are identi®ed.
where ⌬f 0 is the difference in Coriolis parameter be- worthy that, if the eastern island lies within the latitudes tween the northern and southern tips of the island, and spanned by the western island, then it has no effect on
C0 is the closed, anticlockwise path consisting of the the circulation around the western island, which is given lines of latitude from the northern and southern tips of by the equivalent of (2.2). The result (2.3) can be ex- the island to the continent, the western boundary of the tended to a chain of N ϩ 1 sequentially overlapping island and the section of the landmass between the lines islands, see Fig. 1c. If takes the value n on the n ϩ of latitude; see Fig. 1a (cf. Godfrey 1989). 1th island, n ϭ 0, 1, . . . , N, then If this island partially shelters an island to the west n from the landmass, then the value of on the western ϭ a (n)I , (2.4a) nii island may be determined by integrating the momentum iϭ0 equations (2.1) around the closed path C1 shown in Fig. 1b, yielding where
n ⌬ f o1 1 ´ dl ␣o j, i ϭ 0,´´´,n Ϫ 1 10ϭ Ϫ , (2.3) jϭiϩ1 ⌬ f11⌬ f ͶC o a (n) ϭ (2.4b) 1 i 1 i ϭ n, where ⌬f 1 is the difference in Coriolis parameter be- tween the northern and southern tips of the western ␣oioϭ⌬f ii/⌬ f , i ϭ 0,´´´,n, and island, ⌬f o1 is the difference between the Coriolis pa- rameter at the northern and southern extents of the is- 1 ´ dl lands' overlap, and 0 is given by (2.2); a similar ex- Ii ϭϪ , i ϭ 0,´´´,n. (2.4c) ⌬ f ioͶC pression was derived in Wajsowicz (1993). It is note- i
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FIG. 2. Contour maps of the Caribbean and Gulf of Mexico from the ETOPO5 1/12Њϫ1/12Њ bathymetry dataset (NOAA 1988). Land resolved by the dataset is denoted by cross-hatching, and sea depths above (below) 200 m by white (grayscale) shading. The passages considered in the calculations are labeled (a), and the islands and sand banks of the western and eastern Caribbean in (b) and (c). The dotted lines in (b) and (c) denote the extent of the island, sand bank, and landmass as used in the island rule calculations.
In (2.4b), ⌬f i is the difference in Coriolis parameter c. Dynamically narrow passages between the northern and southern tips of i ϩ 1th island; 1) TWO ISLANDS ⌬f oi is the difference in Coriolis parameter between the northern and southern extents of the overlap between i Following Wajsowicz (1993), if the strait between the ϩ 1th and ith islands. The path Cn is as shown in Fig. islands in Fig. 1b is suf®ciently narrow, and/or shallow, 1c. The transport through the passage between the (n that frictional effects are important, then the value of ϩ 1)th and nth islands is the streamfunction on the eastern island is
Tnnϵ Ϫ1 Ϫ n 00ϭ I ϩ F 0ϵ 0ideal0ϩ F , (2.6) B nϪ1 where F0 ϭ #A F ´ dl/⌬f 0 depends on the form of friction ϭϪI ϩ (1 Ϫ ␣ ) a (n Ϫ 1)I . (2.5) noni i F speci®ed in the depth-integrated momentum equations iϭ0 (2.1), A → B is a clockwise line segment along the west The above expressions are valid for the surface of a coast of the island in the overlap region, and 0 ideal is sphere as well as a  plane. the value derived for a dynamically wide passage, name- Equations (2.2)±(2.5) express the in¯uence of differ- ly (2.2). The value of on the western island is obtained ent wind systems on the island circulation and transport by integrating around C1 in Fig. 1b, yielding through the passage between the islands. For example, ϭ I ϩ ␣Ϫ ␣ F , (2.7) assuming the landmass is a continent upon which ϵ 11 o10 10 0, then the circulation on the easternmost island is af- where ␣1 ϭ⌬f 0/⌬f 1. Substituting for 0 from (2.6) and fected only by wind stresses over latitudes spanned by denoting the value of 1 for a dynamically wide passage, the island to the east of the island's western boundary. that is, (2.3), by 1 ideal, then However, that on the westernmost island is affected by ϭ Ϫ (␣ Ϫ ␣ )F . (2.7a) wind stresses over its own latitudes as well as those 1 1 ideal 1 o10 over latitudes of partially overlapping islands, though The simplest form for F is given by assuming it is the in¯uence of wind stresses outside its own latitude proportional to the depth-integrated velocity, then F0 band likely diminishes rapidly with distance down the can be approximated by ϪrЈ0( 0 Ϫ 1)/⌬f 0 (Wajsowicz chain, because of the factor ai(n) Յ 1. Similar conclu- 1993). The coef®cient, rЈ0 ϭ LRa/W, where L, W are sions can be drawn for the strait transports. lengthscales representing the length and width of the
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FIG.2.(Continued)
passage respectively, and Ra is the Rayleigh friction ⑀01ϭ 1 ϩ ␣ Ϫ ␣o1 (Ͼ1). coef®cient;rЈ0 is assumed constant. Substituting in (2.6) Therefore, as noted in Wajsowicz (1999a), for this iso- and (2.7) yields simultaneous equations for 0, 1, lated two-island system, friction serves only to reduce which can be solved to give the magnitude of the strait transport; it cannot change its direction. 0001ϭ I Ϫ rT, (2.8a) T T ϵ Ϫ ϭ 1ideal , (2.8b) 2) THREE ISLANDS 101(1 ϩ ⑀ r ) 00 Adding a third island, say to the northwest, which is where from (2.5), partially overlapped by the second island, introduces in¯uences from more northerly latitudes. The stream- T1 idealϭϪI 1 ϩ (1 Ϫ ␣o10)I , (2.8c) function 0 is given as before in (2.8a), but now T1 is and where r 0 ϭ rЈ0/⌬f 0(Ͼ0) and given by
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T101ϵ Ϫ The equations for T1, T 2 involve both T1 ideal, T 2 ideal, and the coef®cients multiplying them are positive. There- [(1 ϩ ⑀1r 1)T 1 idealϩ rT 1 2 ideal] ϭ , fore, if T1 ideal and T 2 ideal are of opposite sign, then in {1 ϩ ⑀00r ϩ ⑀ 11r ϩ [⑀ 1ϩ (⑀ 0Ϫ 1)␣ 2]rr 01} the presence of friction situations could exist where the transport in one of the straits is reversed. Wajsowicz (2.9a) (1999a) demonstrated the effect for the more geomet- where from (2.5), rically complex case of the Indonesian archipelago with T ϭϪI ϩ (1 Ϫ ␣ )[I ϩ ␣ I ], (2.9b) quite drastic changes in circulation resulting from sup- 2 ideal 2 o21 o10 posing transport through Torres Strait was completely and where ⑀1 ϭ 1 ϩ ␣ 2 Ϫ ␣o2 and T1 ideal is as given blocked due to frictional effects (the r → ϱ); for ex- in (2.8c). The transport in the other strait, between the ample, see Wajsowicz (1999a) Fig. 4. second and third islands, is
T212ϵ Ϫ 3) N ϩ 1 ISLANDS [(1 ϩ ⑀ r )T ϩ r (⑀ Ϫ 1)(⑀ Ϫ ␣ )T ] ϭ 0 0 2 ideal 0 0 1 2 1 ideal . {1 ϩ ⑀ r ϩ ⑀ r ϩ [⑀ ϩ (⑀ Ϫ 1)␣ ]rr} The above method can be extended to the (N ϩ 1)- 00 11 1 0 2 01 island system shown in Fig. 1c, the result is a system (2.9c) of N ϩ 1 simultaneous equations:
Ϫr01 ϩ (1 ϩ r 0) 0ϭ I 0ϩ (␣o 00ϩ ␣ rϪ1)a _
Ϫrnn ϩ1 ϩ (1 ϩ ␣nnr Ϫ1 ϩ rnn) Ϫ (␣ onnnϩ ␣ r Ϫ1)nϪ1 ϭ In _
(1 ϩ ␣NNr Ϫ1 ϩ rNN) Ϫ (␣ o NNNϩ ␣ r Ϫ1)NϪ1 ϭ INNbϩ r , (2.10)
where rn, n ϭ 0,´´´,N Ϫ 1 is the frictional constant 4) ASYMPTOTIC SOLUTION FOR SMALL FRICTION rЈn for the strait between the (n ϩ 1)th and (n ϩ 2)th Assuming that the friction coef®cients, r K 1, n islands divided by ⌬f n. In preparation for application n ϭ to the Caribbean in the next section, the island chain is Ϫ1, 0, 1, . . . , N, and a, b are O(1) or identically assumed to lie between two landmasses upon which zero, then (2.10) has the asymptotic solution is known, namely a on the southeastern mass and b on the northwestern mass. Wajsowicz (1996) described n ϭ Ϫ rT ϩ a (n)(⑀ Ϫ 1)rT analytical solutions for (2.10) with ®nite r for the special nnideal nnϩ1 ideal iiϪ1 iϪ1 i ideal iϭ0 case of the ®rst island representing Australia and the (N ϩ 1)th island representing the Asian continent with the ϩ O(r2), n ϭ 0,1,2,´´´,N, (2.11) islands in between representing the chain from Java to Timor with the simpli®cation that they all spanned the where n ideal, Tn ideal, ai(n) are as de®ned in (2.4)±(2.5), same latitudes. In general, the solution to (2.10) will be T 0 ideal ϭ a Ϫ 0 ideal, TNϩ1 ideal ϭ N ideal Ϫ b, and such that the circulation around an island and transport through its adjacent straits will depend on the wind 1 ϩ ␣ioϪ ␣ i, for i ϭ 1,2,´´´,N Ϫ 1 ⑀ ϭ stress over the latitudes of the whole island chain with iϪ1 Ά1 ϩ ␣0, for i ϭ 0. the possibility of reversed transports in straits due to frictional in¯uence. The corresponding strait transports are
nϪ1 T ϵ Ϫ ϭ rT ϩ (1 Ϫ ⑀ r )T ϩ (⑀ Ϫ ␣ ) a (n Ϫ 1)(⑀ Ϫ 1)rT ϩ O(r2), nnϪ1 nnnϩ1 ideal nϪ1 nϪ1 n ideal nϪ1 ni iϪ1 iϪ1 i ideal iϭ0
n ϭ 1,2,´´´,N. (2.12)
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To O(r), the streamfunction on an island boundary charts for the passages in the eastern Caribbean, and only depends on the values on overlapping islands to some small landmasses and banks are combined or ig- the east and the ®rst layer of overlapping islands to the nored. The boundaries actually used are denoted by dot- west. Whether the boundary value increases or decreas- ted lines in Figs. 2b,c. In applying the multiple-island es as the friction coef®cients increase from zero depends rule, the main task is to identify the layers of islands on the relative magnitude of rnTnϩ1 ideal to the sum term as de®ned by their degree of overlap. The ®rst layer in (2.11). In the next section, in application to the Lesser consists of islands wholly exposed to the interior At- Antilles, it is assumed that the straits are frictional, but lantic; the streamfunction on their boundaries is cal- that the islands do not directly overlap. In this case, the culated using the island rule (2.2). These are the Wind- island value n increases as the r increase from zero ward Islands, which are grouped as Grenada plus St. simply if ␣nrnϪ1Tn ideal Ͼ rnTnϩ1 ideal, and the strait trans- Vincent and the Grenadines, St. Lucia, Martinique, and port Tn increases if rnTnϩ1 ideal Ͼ (1 ϩ ␣n)rnϪ1Tn ideal. Dominica. The other nonoverlapping island groups are Guadeloupe, the Leeward Islands from Montserrat to Anguilla, Caicos Bank, and the Little and Great Bahama 5) ASYMPTOTIC SOLUTION FOR LARGE FRICTION Banks. The second layer consists of islands, whose
The limit of the friction coef®cients rn tending to boundary streamfunction values can be determined from in®nity is equivalent to the straits being wholly blocked, the multiple-island rule (2.3) once the ®rst layer values and for physically realistic situations b ϵ a. There- are known. These islands are Puerto Rico plus the Virgin fore, the zero-order solution is n ϳ a, Tn ϳ 0, n ϭ Islands, which is partially overlapped by the Montser- 0, 1, . . . , N. The next order in the expansion, n1, which rat±Anguilla group, Great Inagua, and Cay Sal Bank is O(1/r), has the form partially overlapped by Caicos Bank and Great Bahama Bank, respectively (see Figs. 2b,c). The third layer con- ϭ g (J ,´´´,J ; r ,´´´,r ) n1 n 0 nϪ10 nϪ1 sists of islands, whose boundary values can be deter- mined once the previous layer's values are known. This ϩ Gn(rϪ1,´´´,rnϪ101) , layer consists only of Hispaniola, which is partially n ϭ 1,2,´´´,N, (2.13) overlapped by Puerto Rico plus the Virgin Islands and where the Montserrat±Anguilla group. Actually, from (2.3), since Puerto Rico plus the Virgin Islands is wholly con-
JNNNϪ (␣ r Ϫ1 ϩ rNN)g ϩ ␣ NNrgϪ1 NϪ1 tained within the latitudes of Hispaniola, its boundary ϭ , 01 value does not need to be explicitly calculated to de- (␣NNr Ϫ1 ϩ rNN)G Ϫ ␣ NNrGϪ1 NϪ1 termine that on Hispaniola, which could be moved to JnnϭI ϩ (␣ onaϪ 1) , n ϭ 0,1,´´´,N, the second layer. The fourth and ®nal layer consists of and the functions g , G satisfy the recursive formulas Cuba and Jamaica. Jamaica is wholly overlapped by n N Hispaniola, and Cuba is partially overlapped by the Great Bahama Bank and Hispaniola; Caicos Bank and Jnnn␣ r Ϫ1 gnϩ1 ϭϪ ϩ 1 ϩ (gnnϪ g Ϫ1), Great Inagua lie wholly within its latitudes and thus do rr nn not need to be directly calculated.
␣nnr Ϫ1 ␣nnr Ϫ1 Gnϩ1 ϭ 1 ϩ GnnϪ G Ϫ1, rrnn b. Transports assuming dynamically wide passages n ϭ 0,1,´´´,N Ϫ 1, The results from applying (2.2)±(2.4) to the major islands, under the assumption that all of the passages and g ϭ 0, G ϭ 1. The solution cannot be written 0 0 are dynamically wide and deep, are given in Table 1 for down as compactly as in (4), as now the streamfunction Hellerman and Rosenstein's (1983) climatological an- on each island at ®rst order depends on the wind stress nual mean wind stress (henceforth referred to as HR over latitudes spanning the whole length of the island climatology). The transport through each passage, as chain. labeled in Fig. 2a, is also tabulated to enable comparison with recent observations, summarized in Table 1. The 3. The wind-driven circulation sign convention is northward and eastward transports are positive. Obvious discrepancies between the model a. The geometry and scheme and observations are signi®cant eastward versus west- The islands and passages, which are the object of this ward ¯ow in Grenada Passage, too strong ¯ows through 1 1 study, as resolved by the ETOPO5 ⁄12Њϫ ⁄12Њ bathym- Old Bahama Channel and Northwest Providence Chan- etry dataset (NOAA 1988), are shown in Fig. 2. The nel, and too weak ¯ows through Yucatan Channel and 200-m isodepth is typically taken as the boundary for the Straits of Florida. The total depth transport through application of the island rule, but this is modi®ed ac- the collection of passages from St. Vincent to Guade- cording to published cross sections from navigational loupe inclusive is 9.3 Sv, which is much stronger than
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC 980 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 W Њ 9.5 Ϫ thro 63.5 h 2.2 4.9 2.4 2.8 0.4 (Sv) Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð 15.5 16.5 Ϫ Ϫ Ϫ Ϫ Ϫ Observed 0±200 m transport f c a a a a a a a a e e g b d 2.5 5.7 2.9 1.5 1.6 1.1 3.1 2.6 1.2 1.8 6.9 1.9 Ð Ð 29.5 23.8 32.3 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ (Sv) Observed total transport 4.8 2.6 0.4 5.2 1.3 4.6 7.3 3.8 4.0 2.4 1.4 1.2 1.3 3.0 23.2 25.8 18.4 (Sv) Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Inferred total transport lb
am am ± gb gb ± ± gb lb ±
ma cs cs gi cu am an do lu ±
± ± ± cu N) N)
gu ± lu ± cu Њ Њ hi gb gr pr ± ± cu
gu ±
gr ± ma hi ± ± do
hi
an am ± pr
Grenada Anegada Yucatan Ch. St. Vincent St. Lucia Ch. Martinique Dominica Guadeloupe Mona Windward Great Inagua Old Bahama Ch. NW. Providence Ch. Santaren Ch. Nicholas Ch. Florida Str. (26 Florida Str. (27 1. Multiple-island rule plus HR wind stresses: all straits assumed dynamically wide and deep. 3.97 1.61 0.25 0.93 2.22 5.25 6.54 0.00 0.00 ABLE (Sv) Passage 15.86 23.23 25.79 11.09 14.86 23.68 14.81 18.44 T Ϫ Ϫ Ϫ ) pr ) ) )
) gr an gb lb
) ) ) cs ) gi ) cb ) gu ) ) ma hi
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) af cu
Roemmich (1981); inverse model of Caribbean, geostrophicAtkinson transport et relative al. to (1995). 1000 m. Larsen (1992) submarine cable. Johns et al. (1999). Johns et al. (2001). Sheinbaum et al. (2001); mean for SepLeaman 1999±Jun et 2000 al. inclusive. (1995). Niiler and Richardson (1973). a b c d e f g h Hispaniola ( Great Bahama Bank ( Cuba ( Puerto Rico±Virgin Is. ( Americas ( Grenada±St. Vincent ( Guadeloupe ( Caicos Bank ( Jamaica ( Martinique ( Cay Sal Bank ( St. Lucia ( Montserrat±Anguilla ( Little Bahama Bank ( Dominica ( Great Inagua ( Africa (
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC MARCH 2002 WAJSOWICZ 981 the 4.6 Sv over the upper 200 m inferred from Johns section. This latter circulation over the upper ocean is et al.'s (1999) observed transport through 63.5ЊW, but also needed to reverse the direction of the Guyana Cur- comparable with the 10.2 Sv over the total depth from rent, whose wind-driven component is southward as the recent compilation of passage measurements by shown in Fig. 3b, but observations (Johns et al. 1999) Johns et al. (2001). The inferred transports through indicate is northward. Mona Passage and Great Inagua (a proxy for Windward) The remaining islands of the Lesser Antilles provide Passage are in reasonable agreement with direct obser- little obstruction to the Sverdrup ¯ow, and the circu- vations, Johns et al. (1999) over the upper 200 m, and lation around each can be approximated by considering over the total depth, Johns et al. (2001). Transport the limit of (2.2) as the latitudinal extent of the island through Santaren Channel is in the wrong direction. tends to zero yielding Before considering how frictional effects could mod- 1 Island ify the above results, it is worth describing how robust ϭ curl ´ dl, (3.1) 0  ͵ the results are to the wind stress dataset used. The cal- Africa culations are not repeated for each of the more than 10 where curl ´ dl ϭϪk ´ curl R cosd; R is the radius climatologies described in Townsend et al. (2000, here- of the earth; , are the latitude, longitude; and  is after THH) and four described in Fanning et al. (1994), the planetary vorticity gradient. Equation (3.1) is just but rather general similarities and differences between the local value of the Sverdrup streamfunction, so the the wind stress curl patterns and how these affect the transport through the Windward Island passages, with transports are described. the exception of Grenada Passage, may be regarded as just a measure of the meridional gradient in Sverdrup 1) GENERAL GYRE CIRCULATION streamfunction, or alternatively wind stress curl. From THH and Fanning et al. (1994), an O(8 Sv) increase in The curl of the Hellerman and Rosenstein wind the Sverdrup streamfunction from Grenada up to An- stresses over the North Atlantic is shown in Fig. 3a, guilla is common to a wide range of wind stress cli- and the corresponding Sverdrup streamfunction taking matologies. Therefore, topographic blocking, and so into account island circulations in Fig. 3b. The gyre frictional effects, must be invoked to reduce the net circulation is closed in the western boundary layers by westward transport, which will be de¯ected northward. assuming that relative vorticity is destroyed at the lat- If the de¯ected transport, O(2 Sv) say, entered the Ca- itude of creation so that the streamfunction monoton- ribbean Sea through Anegada Passage, then its transport ically decreases or increases across the layer. Although would be in better agreement with observations. different wind stress climatologies differ in detail, they all display the same general structure over the North Atlantic, namely positive wind stress curl between the 3) CENTRAL CARIBBEAN TRANSPORTS equator and 10ЊN across the basin and extending north- The dependencies of the transports through Anegada ward up the African coast to midlatitudes. The wind and Mona Passages on wind stress, as given by the stress curl is then negative to the west over these lat- multiple-island rule assuming dynamically deep pas- itudes except over the southern Caribbean and Gulf sages, is summarized in Table 2 along with the depen- Stream, where it is positive again (see Fig. 3a). The dencies for neighboring islands. The convention used is result is a cyclonic Sverdrup gyre between about 5Њ that Iab is the wind stress integral as described in (2.10) and 15ЊN, whose northern limb extends into the south- for the island identi®ed by letters ab, as detailed in Table ern Caribbean with an additional cyclonic gyre in the 1. The contribution for each component, calculated for Panama±Columbia gulf. This latter feature is in agree- HR climatology, is given in the last column of Table 2. ment with drifter tracks (e.g., Leaman and Wilson For Anegada Passage, the idealized transport is weak 2000). The region from 15ЊN to about 35ЊN is ®lled as the contributions from the Puerto Rican wind stress with a basinwide anticyclonic gyre, whose southern integral (ϪIpr ) and Anguilla±Montserrat integral limb extends into the northern Caribbean. (0.27Ian) are similarly small and almost cancel. Different wind stress climatologies do not differ much over these latitudes between Africa and Puerto Rico, and so the 2) EASTERN CARIBBEAN TRANSPORTS transport would be similar, as found by THH. According From Stokes theorem, each line integral in (2.2)±(2.4) to the idealized multiple-island rule (MIR), transport can be reexpressed as an integral of the wind stress curl through Mona Passage is dependent on the wind stress over the area enclosed by the line integral. The tropical integrals for Anguilla±Montserrat, Puerto Rico, and His- cyclonic Sverdrup gyre produces a negative circulation paniola. From Fig. 3a, the HR climatology has a dipole around Grenada±St. Vincent with eastward ¯ow counter in the wind stress curl over the western edge of His- to observations through Grenada Passage. From (2.9), paniola with the positive pole to the north. In different for this simple con®guration, frictional effects cannot wind stress climatologies, the strengths of the two poles alter the sign of the transport through Grenada Passage; can differ considerably. For example, the positive pole a thermohaline must be invoked as discussed in the next is almost absent in the Isemer and Hasse (1987) cli-
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FIG. 3. The wind stress curl for Hellerman and Rosenstein (1983) annual mean climatology is contoured in (a). The corresponding Sverdrup streamfunction with values on islands calculated from the island rule assuming that all resolved passages are dynamically wide and deep is contoured in (b). The contour intervals are (a) 1 ϫ 10Ϫ8 dyn cmϪ3 and (b) 2 Sv. Positive contours are denoted by solid lines, negative by dashed lines, and the zero contour by a dotted line. The western boundary layer scale has been exaggerated for clarity.
matology, its center is displaced signi®cantly westward see, for example, THH. The impact on Ihi is typically in the FNMOC climatology (Hogan and Rosmond small, as the integral is dominated by the contribution 1991), and it extends over the whole of Hispaniola and from over the breadth of the Atlantic. However, for the to the east in the National Centers for Environmental NCEP climatology, the patch of positive wind stress Prediction (NCEP) climatology (Kalnay et al. 1996); curl is suf®cient to reduce the dominant contribution of
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TABLE 2. Contributions from overlapping islands. Contributions using HR wind stresses Island/passage MIR formula (Sv)
Puerto Rico±Virgin Island pr 0.73 Ian ϩ Ipr 3.82 ϩ 2.71
Hispaniola hi 0.39 Ian ϩ Ihi 2.04 ϩ 9.05
Great Inagua gi Igi ϩ 0.50 Icb 6.93 ϩ 0.22
Cuba cu 0.03 Ian ϩ 0.07 Ihi ϩ Icu ϩ 0.34 Igb 0.15 ϩ 0.67 ϩ 9.75 ϩ 7.87
Anegada Passage an±pr 0.27 Ian Ϫ Ipr 1.43 Ϫ 2.71
Mona Passage pr±hi 0.34 Ian ϩ Ipr Ϫ Ihi 1.78 ϩ 2.71 Ϫ 9.05
Windward Passage hi±cu 0.36 Ian ϩ 0.93 Ihi Ϫ Icu Ϫ 0.34 Igb 1.89 ϩ 8.38 Ϫ 9.75 Ϫ 7.87
Ihi to the Mona Passage transport, which explains THH's The changes to the transports, if the Bahaman chan- ®ndings of a much lower Mona transport for NCEP nels are frictional, are considered in section 3c, where climatology than the others examined. From Fig. 3, al- it is shown that the southward MIR-derived transport though the dipole makes only a small contribution to through Santaren Channel can be reversed by frictional
Ihi, it does affect the Sverdrup circulation to the west effects in the Straits of Florida. The low value obtained in the Caribbean. for the Florida Straits transport from applying the MIR can only be further reduced by frictional effects. As shown in section 4, a thermohaline-driven western 4) WESTERN CARIBBEAN TRANSPORTS boundary current is needed to increase the transport to Transport through Windward Passage is sensitive to observed values. wind stresses spanning latitudes from Montserrat to the Finally, a striking feature of Fig. 3b are the thin west- northern tip of the Grand Bahama Bank (see Table 2). ward zonal jets that cross the central Caribbean and the
The contributions from Icu and 0.34Igb are of similar oppositely directed neighboring jets in the Gulf of Mex- magnitude and of opposite sign to the contribution ico. These jets are a feature of the Sverdrup solution
0.93Ihi. In assessing results using the different wind assuming frictional effects are con®ned to western stress climatologies, the neighboring contributions from boundary layers. In practice, these jets are likely to be
Ihi and Icu may be regarded as canceling. Then, the trans- barotropically unstable, and so a source of mesoscale port through Windward Passage is effectively deter- eddies across the breadth of the Caribbean. The oppo- mined by wind stresses integrated over the latitudes of sitely directed jets obtained in the Gulf of Mexico are the Great Bahama Bank and to a lesser extent Mont- similar to those obtained between the tip of South Africa serrat±Anguilla. This is borne out by the calculations and Brazil by Godfrey (1989). Higher-order dynamics of THH, who found that wind stress datasets, which modify these jets to give the observed Loop Current produced strong (weak) Florida Strait transports (Igb ex- and its enormous variability. actly) also had strong (weak) Windward Passage trans- ports. There are considerable differences between the c. Transports assuming dynamically narrow passages wind stress curl climatologies around 25ЊN, mainly be- cause of the differing extents of positive curl along the If the islands and banks of the Caribbean from Gre- African and American coasts. From Table 1, just over nada to the Bahamas formed a solid barrier, then the half the transport through Windward Passage comes value of on the barrier is about 16 Sv, and the cir- through the Great Inagua Channel; the remainder comes culation consists of bidirectional ¯ow in Grenada Pas- southward as a western boundary current along Cuba, sage with a 22 Sv westward current entering against the see Fig. 3b. northern bank and a 6-Sv current exiting against the The wind-driven transport through Yucatan Channel is southern bank. The northern boundary current is fed by given by the MIR formula for Cuba (see Table 2). It is a southward western boundary current against the bar- dominated by the wind stress integrals Icu and 0.34Igb, rier. The bifurcation point for the interior ¯ow on the which are of similar magnitude for the HR climatology. barrier is at about the latitude of Caicos Bank. The north-
As for Windward Passage, the dependence on Igb means ward western boundary current against the barrier peaks that there will be signi®cant variation between the different at about 10 Sv at the tip of Little Bahama Bank and climatologies, though it is scaled by 0.34, and so reduced joins the 16 Sv exiting the Florida Straits. If ¯ow to a similar order to the differences expected for Icu. Fan- through Grenada Passage and/or Florida Straits exerted ning et al. (1994) show the difference in Sverdrup stream- frictional resistance on the barrier, then on the barrier function at the edge of the western boundary for four wind is reduced. The barrier's northward (southward) western stress climatologies; Icu differences are characterized by boundary current is stronger (weaker), and the bifur- those for about 20ЊN versus those for 25Њ±30ЊN. THH cation latitude is more southerly. This circulation pattern obtained wind-driven Yucatan Channel transports as high is obviously unrealistic, but it is the simplest illustration as 25.6 Sv for Isemer and Hasse (1987) climatology, but of regionwide frictional effects, and serves as a contrast in general they were O(20 Sv). to the ¯ow in Fig. 3b. In fact, the circulation in Fig. 3b
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FIG. 4. The variation in (a) island circulation i and (b) strait transport Ti with friction coef®cient rЈ for the eastern, central, and western Caribbean are shown in panels from top to bottom, respectively. The streamfunction
is set to zero on the American and African continents, i.e., am, af ϭ 0. The Yucatan Channel and Anegada and Windward Passages are assumed dynamically wide (their rЈϵ0). The friction coef®cients in all of the other passages are assumed equal to rЈ. Note, the rЈ scale is linear for 0 to 1 ϫ 10Ϫ5 s Ϫ1 and logarithmic for 1 ϫ 10Ϫ5 sϪ1 to 1 ϫ 10Ϫ3 sϪ1. is more realistic, which suggests that many of the pas- chain from Grenada to Anguilla is only affected by sages may be considered dynamically wide. wind stresses over its latitudes. Puerto Rico±Virgin For sake of argument in the following, passages with Islands and Hispaniola also form an isolated chain, and minimum sill depths greater than 1500 m, are consid- so are not affected by wind stresses to the north. The ered dynamically wide. These are Anegada and Wind- passages around Great Inagua Island are all greater ward Passages and Yucatan Channel. Therefore, the than 1000 m, so the Bahama Banks, Cuba, and Cay
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Sal Bank form a frictional group in¯uenced by Florida function on Hispaniola decreases as the contribution to the west. from Anegada decreases, and that on Puerto Rico in- creases toward that of Hispaniola, as the friction co- ef®cient for the ¯ow in Mona Passage increases; for 1) FRICTIONAL EFFECTS IN THE EASTERN sake of argument, the coef®cient is assumed to have CARIBBEAN the same value as that for the more eastern passages. The effect of frictional blocking of the passages from The streamfunction on Puerto Rico and Hispaniola Grenada Passage to Guadeloupe Passage on the circu- tends to the island rule value assuming the islands form lation around the neighboring islands is described by a single block, which is about 9 Sv for HR climatology. (2.10) with the overlap coef®cients set to zero. If the This is the limiting value of the Anegada transport; see friction coef®cients rЈ of the passages are assumed middle panel of Fig. 4b. The transport in Mona Passage equal, then the resulting behavior as a function of rЈ is tends to zero as its rЈ → ϱ. shown in the top panels of Fig. 4; for a strait of equal length and width, assuming a bottom friction spindown time of 5 days, yields an rЈϳ0.2 ϫ 10Ϫ5 sϪ1. The 3) FRICTIONAL EFFECTS IN THE WESTERN streamfunction on each of the island boundaries tends CARIBBEAN to the value on the American continent, that is, zero, as rЈ → ϱ, and the transports in each of the eastern passages As the Windward Passage is assumed dynamically also tends to zero. The transports in the middle passages, wide, the islands and sand banks of the western Ca- St. Lucia Channel and Martinique and Dominica Pas- ribbean do not experience frictional rubbing directly sages, increase as rЈ increases from zero, and then de- by Hispaniola, but do know of frictional effects in the creases. This behavior is expected from the asymptotic more easterly passages from island overlap. The equa- behavior described in section 2c(4). As rЈ increases from tions for the island circulations can be derived in a zero, the streamfunction around St. Lucia and Marti- similar manner to (2.10) and are given in the appendix. nique tends toward that on Grenada, and those on Gua- The effect of friction in the passages of the western deloupe and Dominica tend to that on the Montserrat± Caribbean is shown in the bottom panel of Fig. 4. For Anguilla group. Dominica then exerts a stronger in¯u- simplicity in Fig. 4, the friction coef®cients rЈϭsi , ence on Martinique than St. Lucia, and the sign of as de®ned in the appendix (A2), are assumed equal and on its boundary changes sign before tending to zero. equal to the coef®cients in the more easterly frictional The westward ¯ow, which no longer passes through the passages. eastern passages, is mainly de¯ected into Anegada Pas- The streamfunction on the islands and sand banks sage as it is assumed dynamically wide. tends to that on the American continent, that is, zero, If one of the passages in the eastern Caribbean chain as rЈ → ϱ. Transports in all of the passages of the is dynamically wide, then the above behavior is altered. western Caribbean tend to zero except for that of Wind- The streamfunction on the islands between the dynam- ward Passage, which reverses as friction increases in ically wide passage and the American continent tends the passages to the west, and tends to ϳ 9Sv,asrЈ to zero again, as do the passage transports, as r → . hi Ј ϱ → ϱ (see Fig. 4). In this limit, there is very weak cir- However, for the islands between the dynamically wide culation in the Gulf of Mexico and Caribbean Sea north passage and Anegada Passage, their values tend to the of 20ЊN, as it is driven by the local wind stress curl value given by the island rule (2.2) assuming the group only. forms a single island. For example, if Martinique Pas- Increasing friction also reverses the transport direc- sage is dynamically wide, then as rЈ → ϱ, the transport in Martinique Passage tends to 3.8 Sv westward, the tions in Old Bahama, Santaren, and Nicholas Channels. limiting southward value in Anegada Passage is reduced It is possible to tune the resistances in each strait to to 6.7 Sv, and the return 10.5 Sv northward ¯ow exits give the observed directions and right order of mag- via Windward Passage. nitude since from (2.8) the rЈ are a function of geometry of the passage. However, the fundamental problem with this purely wind-driven model, namely that the western 2) FRICTIONAL EFFECTS IN THE CENTRAL boundary currents to the south of the Caribbean are in CARIBBEAN the wrong direction, remains uncorrected. As a conse- Once again, the equations can be derived from quence, the water masses ¯owing through the Caribbean (2.10). Assuming Anegada Passage is dynamically are always North Atlantic in origin in contrast with ob- wide, then the effect of the changes in the circulation servations of signi®cant South Atlantic water masses. around the Montserrat±Anguilla group is felt by the This problem is addressed in the next section by intro- islands to the west because of overlap. However, the ducing a thermohaline circulation. It is noteworthy that overlap contribution is minor for most of the islands introducing a thermohaline circulation does not affect (see Table 2). The effect on Puerto Rico and Hispaniola the basic dependencies described in this section and in is shown in the middle panel of Fig. 4a. The stream- section 2.
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FIG. 5. The transport streamfunction in the top layer of a model driven by Hellerman and Rosenstein (1983) wind stress climatology and incorporating an additional 15 Sv northward western boundary current transport against the American continent representative of the upper limb of the thermohaline circulation. All of the passages are assumed dynamically wide and deep. The pattern is the same as the wind-driven Sverdrup circulation in Fig. 3b, but with the condition that the streamfunction equals Ϫ15 Sv on the American continent rather than zero, the value on the African continent. The linestyle for the contours is as in Fig. 3b, but the negative streamlines, which originated in the Southern Hemisphere are contoured using a dot±dash line. The western boundary layer scale has been exaggerated for clarity.
4. Including a thermohaline-driven circulation a. Upper-layer circulation assuming dynamically The system is extended to three layers with the top wide passages layer containing the wind-driven circulation and upper The interior gyre and island circulations are exactly limb of the thermohaline-driven circulation. The bottom as in Fig. 3b. The effect of including the upper limb of layer contains the lower limb of the thermohaline cir- the thermohaline circulation is con®ned to the western culation. Interaction between the top and bottom layer, boundary layer against the American continent and the for example, upwelling and deep convection, is assumed zonal jets between peninsulas on the continent (see Fig. to be con®ned to latitudes outside the domain of interest. 5). The direction of the Guyana Current and transport Therefore, the middle layer is considered to be at rest. through Grenada Passage are reversed, now in agree- As before, bottom topography is assumed to be con®ned ment with observations. The transport in the Guyana beneath the wind-driven layer. A similar con®guration Current is about ϪTH, that is, 15 Sv at 4ЊN, but reduces was considered recently by Nof and Van Gorder (1999). to about 5 Sv at 10ЊN before increasing again. The As there is no explicit forcing of the thermohaline cir- strength of the Grenada Passage transport, given by sub- culation within the domain of interest, it effectively en- tracting 15 Sv from the value in Table 1, is now 11 Sv, ters and exits the domain as a western boundary current. which is much larger than the observed 4.9 Sv over the Therefore, a similar set of equations to (2.1) holds for upper 200 m (Johns et al. 1999). The transports through the upper layer in the domain of interest, but now the Yucatan Channel and the Florida Straits are increased no-normal ¯ow boundary condition on the American by ϪTH to 33.4 Sv and 38.2, 40.8 Sv (26ЊN, 27ЊN), continent is ϭ TH, a constant, where TH is a measure respectively. The other transports listed in Table 1 are of the strength of the thermohaline overturning circu- unchanged. The above passage transports can be re- lation in the domain of interest. In the following, TH duced by frictional effects as described in the next sec- is set to Ϫ15 Sv, consistent with estimates from Schmitz tion. and Richardson (1991). Details of the bottom-layer cir- Figure 5 also distinguishes the streamlines originating culation are not needed to solve for the top-layer cir- from the Southern Hemisphere. In the absence of the culation, and so will not be considered further. thermohaline circulation, the alternating cyclonic/anti-
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC MARCH 2002 WAJSOWICZ 987 cyclonic wind-driven gyres are contained between zero eastern Caribbean, is de¯ected northward, and enters streamlines reaching almost zonally from the African to the Caribbean Sea through the dynamically wide Ane- American continents. Hence, the upper-layer water mas- gada Passage. Hence, although the behavior for small ses circulating through the Caribbean are all of North values of the friction parameter (see middle panels of Atlantic origin, as shown in Fig. 3b. Including a ther- Fig. 6) is similar to that shown in the middle panels of mohaline circulation enables South Atlantic water mas- Fig. 4 in the absence of a thermohaline circulation, the ses to enter the region within the western boundary limits as rЈ → ϱ are different. The Anegada transport current against the American continent. In Fig. 5, about tends to ϪTH ϩ 3.2 Sv ϭ 18.2 Sv, where 3.2 Sv is the 5 Sv continues along the western boundary through the value of on Hispaniola and Puerto Rico, assuming Caribbean Sea, around the Gulf of Mexico, and exits they form a single island with dynamically wide chan- via the Florida Straits. A further 10 Sv from the South nels on either side, that is, as rЈ → ϱ in Mona Passage. Atlantic separates from the coast, and circulates around This latter limit is different from the limit for on a the cyclonic tropical gyre before entering the Caribbean conjoined Hispaniola and Puerto Rico in the absence of Sea in the passages from Grenada Passage to Martinique a thermohaline circulation. From Table 2, due to over- Passage. It crosses the Caribbean Sea in a series of zonal lap, the value is a function of the limiting value on the jets, and then joins the western boundary current con- Montserrat±Anguilla group, that is, TH. There are very tinuing northward to the Florida Straits, and onward few direct or inferred observations of transport in Ane- along the North American coast. Figure 5 indicates that gada Passage. However, Johns et al. (1999) estimate the water masses of the Peruvian±Columbian gyre are only 2.4 Sv passes southward over the upper 200 m isolated. However, their northern edge is cut by the zonal through the passage with a similar value for Mona Pas- jet crossing from the tip of Venezuela to Nicaragua, so sage; Johns et al. (2001) estimate similar values for the mixing of the water masses may be expected. total-depth-integrated transport based on mainly spring/ summer measurements. This suggests that a suitable rЈ b. Upper-layer circulation assuming dynamically for Mona Passage is ϳ0.1 ϫ 10Ϫ5 sϪ1 yielding an upper- narrow passages layer transport in each passage of just over 3 Sv. As If the passages of the Caribbean are frictional, then the transports in Mona and Anegada Passages rapidly Ϫ5 Ϫ1 including the upper limb of the thermohaline circulation diverge as rЈ increases/decreases from 0.1 ϫ 10 s affects the circulation on all of the islands, and so the (see Fig. 6b), if another common transport value were streamfunction to the west. chosen for the passages, then friction would need to be invoked in Anegada Passage to de¯ect some of its trans- port into Mona Passage. 1) FRICTIONAL EFFECTS IN THE EASTERN CARIBBEAN If the passages from Grenada Passage to Guadeloupe 3) FRICTIONAL EFFECTS IN THE WESTERN Passage are assumed frictional and governed by the CARIBBEAN same friction coef®cient rЈ, then the behavior of the The effect of including frictional effects in the chan- streamfunction on island boundaries and transport nels to the west of Windward Passage in the presence through the passages as rЈ increases from zero is shown of a thermohaline circulation are shown in the bottom in Fig. 6 (cf. Fig. 4, in the absence of a thermohaline panels of Fig. 6 (cf. bottom panels of Fig. 4 in its ab- circulation). According to (2.11), the value on all but sence). The island boundary decrease more rapidly Dominica and Guadeloupe decreases monotonically as as rЈ increases from zero, as the limiting value is TH rЈ increases from zero. The result is that, in contrast to ϭϪ15 Sv as rЈ → ϱ. The strait transports, besides Fig. 4b, the westward transport in St. Vincent Passage → those associated with the western boundary current increases initially. As rЈ ϱ, the island values tend to against the American continent, are little changed from TH and the transports tend to zero. In terms of an ap- those in Fig. 4b. The exception is Windward Passage, propriate value of the friction coef®cient so that the whose transport changes from southward to northward upper-layer transports match observations, it is required for r Ͼ 0.4 ϫ 10Ϫ5 sϪ1, which is less than half that that O(10 Sv), since the observed transport an Ϫ TH ϳ found in Fig. 4b. The transports in the western passages across 63.5ЊW is estimated at 9.5 Sv over the upper 200 tend to zero as rЈ → ϱ except that in Windward Passage, m (Johns et al. 1999) and 15.9 Sv over the total depth, which tends to the reverse of Anegada Passage. Match- (Johns et al. 2001). The transport through Grenada Pas- ing the results in the bottom panel of Fig. 6b to obser- sage ϳO(5 Sv), Johns et al. (2001). These suggest an vations given in Table 1 indicates that friction coef®- rЈ in the range 2±5 (ϫ 10Ϫ5 sϪ1). cients O(10Ϫ6 sϪ1) are appropriate. An example is given in Table 3 (on the rhs of each 2) FRICTIONAL EFFECTS IN THE CENTRAL column) of a possible combination of friction coef®cients CARIBBEAN for the passages, which results in island streamfunction The upper-layer transport, which in the absence of values that give reasonable agreement with observed trans- friction would have ¯owed through the passages of the ports. To help link with the dependencies displayed in Fig.
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FIG. 6. The same as Fig. 4 but for the model including a 15 Sv northward western boundary current along
the American continent representing the upper limb of the thermohaline circulation, so am ϭϪ15 Sv, and af ϭ 0. Transport through the Yucatan Channel, and through the Florida Straits at 26Њ and 27ЊN, has been scaled
to ®t on the plot; 15 Sv, i.e. Ϫam, needs to be added to the values plotted.
6, results are also given (on the lhs of each column) as- not add to that into Windward Passage, whose transport suming Anegada and Windward Passages and the Yucatan is actually decreased due to the introduction of a modest Channel are dynamically wide, that is, their rЈ Ӎ 0. It is amount of friction in the Windward Passage and Yucatan noteworthy that, of the 4.6 Sv that is blocked from ¯owing Channel. Instead, it increases the transport through Old through Anegada Passage due to the introduction of fric- Bahama and Northwest Providence Channels and in the tion in the passage, 1.5 Sv is redistributed among the boundary current to the east of Abaco Island. The rela- Windward Islands passages to the south, 1.4 Sv adds to tively large friction coef®cients needed to give realistic the ¯ow into Mona Passage. The remaining 1.7 Sv does transports through the Windward Island passages are con-
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC MARCH 2002 WAJSOWICZ 989 9.12 7.17 5.45 3.93 2.54 1.39 2.63 6.23 9.94 0.00 15.86 15.37 17.69 14.86 12.76 13.28 15.00 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ
(Sv) 9.44 7.67 6.14 4.84 3.70 2.86 5.78 7.94 0.00 15.86 15.89 17.94 14.86 13.83 11.65 15.34 15.00 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ ) pr ) ) )
) gr an gb lb
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)
) af cu
Puerto Rico±Virgin Is. ( Grenada±St. Vincent ( Americas ( Guadeloupe ( Great Bahama Bank ( Jamaica ( Cuba ( Africa ( St. Lucia ( Dominica ( Montserrat±Anguilla ( Caicos Bank ( Hispaniola ( Great Inagua ( Martinique ( Little Bahama Bank ( Cay Sal Bank ( 8.6:0.0 13.3:15.0 2.1:0.0 2.3:0.0 2.6:0.0 0.5:0.0 12.8:15.0 15.4:15.0 17.7:15.0 0.0:5.9 0.0:1.9 0.0:1.7 0.0:1.5 0.0:1.4 0.0:1.2 2.6:1.4 3.6:0.0 7.0:0.0 NA:SA* Island Water mass ratio 6.9:0.0 0.5:0.0 2.0:0.0 2.1:0.0 1.5:0.0 0.0:5.6 0.0:1.8 0.0:1.5 0.0:1.3 0.0:1.1 0.0:0.8 5.8:2.9 2.2:0.0 7.4:0.0 15.3:15.0 13.8:15.0 15.9:15.0 17.9:15.0 2.1 2.3 2.6 0.5 5.9 1.9 1.7 1.5 1.4 1.2 4.0 3.6 7.0 8.6 27.8 30.4 32.7 28.3 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ (Sv) transport 5.6 1.8 1.5 1.3 1.1 0.8 8.6 2.2 7.4 6.9 0.5 2.0 2.1 1.5 30.3 28.8 30.9 32.9 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Inferred upper-layer ) 1 4.0 5.0 1.0 0.0 0.5 3.0 3.0 3.0 3.0 0.5 2.0 0.5 Ϫ 15.0 35.0 35.0 35.0 35.0 35.0 s 6 Ϫ 10 ϫ 3. Multiple-island rule plus HR wind stresses plus 15 Sv northward thermohaline circulation: frictional straits. Resistance ( 0.0 5.0 0.0 0.0 0.0 3.0 3.0 3.0 3.0 0.5 2.0 0.5 15.0 35.0 35.0 35.0 35.0 35.0 ABLE T am lb am ± am ±
cs gb gb
± ± gb lb ±
ma cs cs gi cu am an do lu ±
± ± ± cu N) N)
gu ± lu ± cu Њ Њ hi gb gr pr ± ± cu
gu ±
gr ± ma hi ± ± do
hi Passage an am ± pr
* North Atlantic: South Atlantic. Great Inagua Grenada St. Vincent St. Lucia Ch. Martinique Dominica Guadeloupe Anegada Mona Windward Yucatan Ch. Old Bahama Ch. NW. Providence Ch. Santaren Ch. Nicholas Ch. Florida Str. (Cay Sal) Florida Str. (26 Florida Str. (27
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FIG. 7. As in Fig. 5 but now the passages are assumed dynamically narrow. The values of the friction coef®cients, island circulations, and net strait transports are given in Table 3 (rhs columns). The western boundary layer scale has been exaggerated for clarity, and a (1:2:1) meridional smoother has been applied to the Sverdrup streamfunction to enable the water masses to be better discerned. sistent with observations and a ®ne resolution model of diverted into the Windward Islands passages (cf. ex- North Brazil Current rings (Johns et al. 2000). The rings amples in Table 3). Indeed, if the thermohaline circu- are about 400 km in diameter and are shallow (Ͻ200 m) lation is ϳ15 Sv and a net ϳ15 Sv crosses 63.5ЊW and or deep (Ͼ1000 m). The deeper rings sense the 800±900 2.5 Sv enters through Anegada Passage, as recently es- m sills and are de¯ected northward along the outer arc. timated by Johns et al. (2001), no South Atlantic water The shallow rings tend to break up on encountering the mass is expected to be found around the outskirts of the islands. Caribbean north of Anegada Passage. In Fig. 7, all of the streamlines from the South At- c. Water mass pathways lantic pass through the Florida Straits, yielding a frac- tion similar to the 45% estimated by Schmitz and Rich- The basinwide transport streamfunction for the upper ardson (1991). None is found east of the Bahama Banks. layer corresponding to the results in the rhs columns of As the South Atlantic streamlines have values from Ϫ15 Table 3 is shown in Fig. 7, which should be contrasted Sv to zero, it is not possible to divert the streamlines with Fig. 5 where all of the passages are dynamically into this region without limiting the Florida Straits trans- wide. The most obvious difference is the increase in port to less than 15 Sv; see Wajsowicz (1999b). Mass strength of the western boundary currents against the continuity requires that the 26 Sv from the closure of southern half of Cuba and against the Grand and Little the Sverdrup wind-driven gyre and 15 Sv from the upper Bahama Banks. Also, the South Atlantic streamlines are limb of the thermohaline circulation must pass either to no longer con®ned to the southern Caribbean Sea and the east or west of the Bahama Banks. If the Florida western boundary current against the American conti- Current is about 32 Sv, then continuity requires that 9 nent, but are swept northward, and enter the Caribbean Sv pass to the east of Abaco Island. This is considerably Sea through all of the passages from Grenada Passage larger than the estimated 5 Sv by Lee et al. (1996) from to Anegada Passage inclusive for the particular choice current meter moorings. However, if the thermohaline of frictional parameters given in Table 3. Interestingly, circulation were 2 Sv weaker, and the interior Sverdrup increasing frictional effects in Anegada and Mona Pas- streamfunction 1 Sv weaker, then the numbers would sages does not necessarily divert the South Atlantic be in much better agreement given the considerable tem- streamlines even farther northward into Mona and poral variability in currents observed off Abaco Island Windward Passages, as some of the excess transport is (Lee et al. 1990).
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5. Summary and discussion ence the streamfunction value on an island boundary. The asymptotic solution in the limit of large friction It has long been considered coincidence that the trans- gave that wind stresses over latitudes spanning the port measured through the Florida Straits is approxi- whole length of the island chain in¯uence the boundary mately that necessary to balance the southward Sverdrup value on an island. Frictional effects were incorporated transport across the ocean interior. The presence of South in their simplest form as a linear drag. More complex Atlantic water masses in the Windward Island passages forms require numerical solution and would yield dif- (Wilson and Johns 1997) and Florida Current (Schmitz ferent sensitivities. However, the fundamental notion of and Richardson 1991) is contradictory to the ¯ow path eastward and westward in¯uence of the island chain is in a classic Sverdrup model. However, high-resolution, unchanged. global ocean GCMs (e.g., Maltrud et al. 1998) show that In section 3, the multiple-island rule was shown to be the mean barotropic streamfunction over the ocean in- a very powerful tool in diagnosing the mean circulation terior between the latitudes spanned by the Caribbean is of the Caribbean because it gives in closed, analytical very similar to the Sverdrup streamfunction. The di- form the dependency between the island streamfunction chotomy is resolved by recognizing that a simple model value and hence strait transport, and the wind stress ®eld. of the upper-layer circulation can be constructed in which For example, the mean transport through Windward Pas- the wind-driven component W satis®es the Sverdrup bal- sage was found to depend on the wind stress integrated ance Wx ϭ curl/o over the ocean interior subject to along island-rule paths for the Great Bahama Bank and the no-normal ¯ow boundary condition W ϭ 0onthe Cuba, and only very weakly on wind stresses outside African and American continents, and the thermohaline these latitudes; Table 2 summarized the more complicated component T is represented by a western boundary cur- overlapping island dependencies. It also enables the fea- rent within the domain of interest, so that Tx ϭ 0 in the tures of the wind stress ®eld in different wind stress interior, and T ϭ TH, constant on the African and Amer- climatologies, which cause strait transports to differ in ican continents, respectively. The treatment of western an operational GCM, say, to be readily identi®ed [cf. boundary layers and islands depends on the level of so- Townsend et al.'s (2000) study of 11 wind stress cli- phistication desired. matologies]. The rule also can be used to ascertain con- In the study herein, the circulation around islands is ditions under which measurements of transport in Great described using a frictional form of the multiple-island Inagua Passage are a good proxy for that in Windward rule (Wajsowicz 1993). If frictional effects are con®ned Passage, and when they may be misleading due to a to western boundary layers and all of the passages are signi®cant western boundary current along Cuba. assumed dynamically wide and deep, then the effect of The purely wind-driven Sverdrup model, in which all the thermohaline circulation is con®ned to the western of the passages are dynamically wide and deep, has boundary layer against the American continent. The numerous discrepancies with observations, as described transports through Grenada Passage, the Yucatan Chan- in section 3 and summarized in Table 1. Some (e.g., nel, and the Florida Straits are the return Sverdrup value weakness in Yucatan Channel and the Florida Straits plus the strength of the thermohaline circulation. The transport, and wrong direction for Grenada Passage transport streamfunction elsewhere and the circulation transport and Guyana Current) can only be corrected by around islands are simply those obtained from a purely including a thermohaline transport. Others (e.g., too wind-driven model. If the passages of the Caribbean are strong transport in Old Bahama Channel, too weak west- frictional, then the pressure gradient associated with the ern boundary current east of Abaco Island, and wrong thermohaline circulation may be partially distributed direction in Santaren Channel) can be corrected by in- along the island chain, thus affecting all of the passage cluding frictional effects. A demonstration of the sim- transports. plest way in which the Caribbean islands interact under To appreciate the underlying sensitivities in the fric- frictional effects, given by assuming that the frictional tional, wind- and thermohaline-driven model, as de- coef®cients of the narrower, shallower passages are all scribed by Fig. 7 and Table 3, the model was built up equal to rЈ, was shown in Fig. 4; the Yucatan Channel step-by-step from the simple Sverdrup model. The mul- and Windward and Anegada Passages were assumed tiple-island rule, as described in Wajsowicz (1993), was dynamically wide and deep. A remarkable feature of recapped in section 2. Exact solutions for two- and Fig. 4 is the smallness of the friction parameter, rЈϳ three-island groups were derived to show how friction 0.8 ϫ 10Ϫ5 sϪ1 ϳ 1/(1.4 days), required in the shallower simply reduces the ``ideal'' transport for a single isolated passages to reverse the transport in Windward Passage. strait, but may reverse the transport in one of the straits A typical value for rЈ is (L/W)/(5 days), where L, W are for a dual-strait system under certain conditions. The lengthscales representative of the length and width of rule was extended to describe an arbitrary-length chain the strait, respectively. of overlapping islands. The asymptotic solution in the The modi®cation to the water mass distribution provid- limit of small friction gave that wind stresses over lat- ed by including the upper limb of the thermohaline cir- itudes of immediately adjacent islands, even if these culation as a northward western boundary current though islands lie to the west or do not directly overlap, in¯u- the region was described in section 4. Frictional effects
Unauthenticated | Downloaded 10/02/21 04:15 PM UTC 992 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 32 must be invoked in the passages to redistribute the extra of islands from Grenada to Hispaniola. The geometry transport among the passages, and so obtain reasonable of the western Caribbean is slightly more complex, but agreement with observations. The required coef®cient in the same principles may be applied. The resulting sys- the Windward Island passages is an order of magnitude tem of equations (using the notation of Table 1), as- larger than needed in the passages of the western Carib- suming that Windward Passage and the passages around bean, as summarized in Table 3. Figure 7 con®rms the Great Inagua and Caicos Bank connecting to Cuba and ``coincidence'' that the Florida Straits transport matches Great Bahama Bank are dynamically wide, is the value of the Sverdrup streamfunction at the interior (⌬ f ϩ s ϩ s ) Ϫ s edge of the western boundary layer at 26Њ±27ЊN. In this lb l b lb b gb simple model, frictional effects determine the fraction of ϭ Ilb⌬ f lbϩ s l TH the thermohaline transport plus return Sverdrup transport, which passes to the west of Great Bahama Bank. However, Ϫsblbgbbgscgbccuscs ϩ (⌬ f ϩ s ϩ s ϩ s ϩ s ) Ϫ s Ϫ s the amount of South Atlantic water mass passing through ϭ I ⌬ f ϩ s the Florida Straits is robust. As long as the Florida Straits gb gb b TH transport is greater than the magnitude of the thermohaline Ϫ(⌬ fovcu1cgbcunccuncsϩ s ) ϩ (⌬ f ϩ s ϩ s ) Ϫ s circulation, all of the South Atlantic water mass passes to ϭ I ⌬ f ϩ⌬f the west rather than east of Great Bahama Bank; see Waj- cu cu ov cu2 hi sowicz (1999b) for more explanation on this type of model. Ϫ(⌬ fcsϩ s s) gbϪ s n cuϩ (⌬ f csϩ s sϩ s nϩ s f) cs Choosing friction parameters, so that strait transports match observations, results in almost 10 Sv of South ϭ Ics⌬ f csϩ s f TH, (A1) Atlantic water mass being diverted northward along the where the friction coef®cients, equivalent to the rЈ, are outer arc of the Lesser Antilles. It is dif®cult to prevent de®ned for the straits as follows: all of it from entering the Caribbean Sea before or at
Anegada Passage. Indeed, this model shows that the sl: between Little Bahama Bank and Florida Anegada Passage may be the favored entry passage for s : NW Providence Channel the Caribbean Sea rather than Windward Passage. In b Fig. 6, both passages are assumed dynamically wide, sg: between Great Bahama Bank and Florida but whereas the transport in Anegada is always south- ss: Santaren Channel ward, and increases toward the magnitude of the ther- s : Nicholas Channel mohaline circulation as rЈ → ϱ; that in Windward Pas- n sage switches to northward ¯ow for an even smaller sc: Old Bahama Channel value of rЈ than in Fig. 4. Sensitivity in Windward Pas- s : between Cay Sal Bank and Florida. (A2) sage transport direction has been noted recently in very f high resolution (1/12Њϫ1/12Њ) simulations of the North The wind stress integrals Iab, and Coriolis parameter Atlantic circulation (E. P. Chassignet and Z. D. Garraffo differences ⌬f , follow previous de®nitions. The over- 2000, personal communication). There is northward ab lap parameters ⌬f ovcu1, ⌬f ovcu2 are the difference in Cor- ¯ow of 1.2 Sv, averaged over years 15±21 of their in- iolis parameter between the extreme overlap latitudes tegration, through Windward Passage when their model of Cuba and the Great Bahama Bank, and Cuba and is forced by the Comprehensive Ocean±Atmosphere Hispaniola, respectively. Data Set climatological surface ¯uxes (da Silva et al. 1994). However, southward ¯ow of 2.1 Sv, averaged from years 0 to 6 of their integration, is obtained when REFERENCES the model is forced by mean wind stresses from the Atkinson, L. P., T. Berger, P. Hamilton, E. Waddell, K. Leaman, and European Centre for Medium-Range Weather Forecasts T. N. Lee, 1995: Current meter observations in Old Bahama 1979±99 reanalysis. Channel. J. Geophys. Res., 100, 8555±8560. da Silva, A., C. Young, and S. Levitus, 1994: Atlas of Surface Marine Data 1994, Vol. 1, Algorithms and Procedures, NOAA Atlas Acknowledgments. This research was conducted un- NESDIS 6, U.S. Govt. Printing Of®ce, Washington DC, 83 pp. der ONR Grant N000149610611. Fanning, A. F., R. J. Greatbatch, A. M. Da Silva, and S. Levitus, 1994: Model-calculated seasonal transport variations through the APPENDIX Florida Straits: A comparison using different wind-stress cli- matologies. J. Phys. Oceanogr., 24, 30±45. Godfrey, J. S., 1989: A Sverdrup model of the depth-integrated ¯ow Frictional Multiple-Island Rule for the for the World Ocean allowing for Island Circulation. Geophys. Western Caribbean Astrophy. Fluid Dyn., 45, 89±112. Hellerman, S., and M. Rosenstein, 1983: Normal monthly wind stress In section 2, the system of simultaneous equations over the world ocean with error estimates. J. Phys. Oceanogr., 13, 1093±1104. describing the circulation around overlapping islands in Hogan, T. F., and T. E. Rosmond, 1991: The description of the Navy a chain of arbitrary length was presented. This system Operational Global Atmospheric Prediction System's Spectral can be applied with appropriate de®nition to the chain Forecast Model. Mon. Wea. Rev., 119, 1786±1815.
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