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 identified, 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 flow 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 confined 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 significant 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 coefficient 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 inflow 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 coefficient in the Windward Islands passages increases from zero, South Atlantic water mass is partially deflected 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.
Whether models based on Sverdrup dynamics explain the circulation of the North Atlantic Ocean has been the subject of much controversy since Leetmaa et al.'s (1977) note that the observed magnitude of Florida Strait transport is consistent with that required to balance the southward Sverdrup transport in the ocean interior. Wunsch and Roemmich (1985) showed that the diagnosed northward heat transport for the North Atlantic is not consistent with a Sverdrup model in which the flow through Florida Strait is the return flow of the northern subtropical gyre. Also inconsistent with this simple type of Sverdrup model is evidence of South Atlantic water masses flowing through the passages of the Lesser Antilles (Wilson and Johns 1997), and as much as 11 Sv (Sv ≡ 106 m3 s−1) through Florida Strait (Schmitz and Richardson 1991). However, realistic GCM simulations [see, e.g., Maltrud et al.'s (1998) Plate 2] show that a Sverdrup model actually describes well the mean transports in the ocean interior of the North Atlantic over the latitudes of interest from 10° to 30°N.
Observations of South Atlantic water masses penetrating the Caribbean can be reconciled with Leetmaa et al.'s (1977) note if it is recognized that estimating transports from the Sverdrup balance does not necessarily imply anything about the water mass origin of the flow. The properties of the western boundary layer in which the Sverdrupian gyres close are crucial, as described in Wajsowicz (1999b) in the context of the Indonesian Throughflow. With this revised perspective, the wind-driven circulation within the North Atlantic and Caribbean is reexamined. Sverdrup dynamics are shown to be consistent with observations provided a thermohaline circulation, which is confined to the western boundary layer over the domain of interest, is included, as noted by Townsend et al. (2000). Also, the transports through certain straits need to be limited; frictional effects are assumed. The streamfunction on island boundaries, and so strait transports, are calculated using a frictional form of the multiple-island rule (Wajsowicz 1993). This is a quite powerful result, as it enables the wind stress bands determining a strait transport to be identified, and so in turn the sensitivity of the transport to changes in the wind stress and strength of the thermohaline overturning circulation to be better understood.
The basic multiple-island rule (Wajsowicz 1993) is recapped in section 2, and several extensions appropriate for the Caribbean derived. The wind-driven transports through the major passages of the Caribbean are calculated using the multiple-island rule and Hellerman and Rosenstein (1983) wind stress climatology in section 3. The cases of dynamically wide and narrow channels are considered. Determining strait transports by the multiple-island rule enables the effect of using different wind stress climatologies to be readily deduced, and the variety of behavior Townsend et al. (2000) found for 11 different wind stress climatologies is easily explained. These wind-driven multiple-island rule results are described in detail, as adding a thermohaline circulation does not alter the basic dependencies.
The model is extended to three layers in section 4 to include a meridional overturning thermohaline circulation. Restricting attention to the latitude band from 10° to 30°N enables the thermohaline circulation to be approximated by a western boundary current of specified transport. Once again, the cases of dynamically wide and narrow passages are considered. Including the upper limb of the thermohaline circulation corrects the direction of the Guyana Current and transport through Grenada Passage. South Atlantic water masses enter the Caribbean Sea through the passages of the Lesser Antilles and exit through the Florida Straits. The exact pathway for the South Atlantic water mass depends on the amount of frictional resistance in the passages. The results are summarized and discussed in section 5.
2. The multiple island rule
The island rule and multiple-island rule were described in general form in Wajsowicz (1993). Their derivation is recapped below in section 2b for the case of frictional effects confined to oceanic western boundary layers and assuming that all of the passages are dynamically wide and deep. In section 2c, modifications to the rules, assuming frictional effects are important, are described with emphasis on the influence of wind stresses for outside the latitude band of the island under consideration. Asymptotic solutions in the limit of small and large friction coefficient are presented for an arbitrary length island chain, assuming a simple northwest–southeast skew.
a. Equations of motion
To keep the discussion succinct, the description is given in terms of quasi-steady motion of an active layer above an inert deep layer in which all bottom topography is assumed confined; the only external forcing is due to surface wind stresses. Let ψ describe the transport streamfunction of the active layer so that the depth-integrated zonal and meridional velocities for the layer are given by u = −ψy, υ = ψx. Then, the depth-integrated momentum equations are
where f is the Coriolis parameter, P is the depth-integrated pressure, τ = (τx, τy) is the wind stress, ρo is the density of the layer, and F = (Fx, Fy) is the depth integrated friction term. The number of boundary conditions required depends on the form of F. However, the no-normal flow condition reduces to ψ = const on boundaries.
b. Dynamically wide passages
From Wajsowicz (1993), if it is assumed that frictional effects are unimportant outside western boundary layers, then the value of ψ on an island to the west of a landmass upon which ψ = 0, say, may be determined by integrating the momentum equations (2.1) along the closed path 𝒞0 shown in Fig. 1a. The resulting island value is
where Δf0 is the difference in Coriolis parameter between the northern and southern tips of the island, and 𝒞0 is the closed, anticlockwise path consisting of the lines of latitude from the northern and southern tips of the island to the continent, the western boundary of the island and the section of the landmass between the lines of latitude; see Fig. 1a (cf. Godfrey 1989).
If this island partially shelters an island to the west from the landmass, then the value of ψ on the western island may be determined by integrating the momentum equations (2.1) around the closed path 𝒞1 shown in Fig. 1b, yielding
where Δf1 is the difference in Coriolis parameter between the northern and southern tips of the western island, Δfoυ1 is the difference between the Coriolis parameter at the northern and southern extents of the islands' overlap, and ψ0 is given by (2.2); a similar expression was derived in Wajsowicz (1993). It is noteworthy that, if the eastern island lies within the latitudes spanned by the western island, then it has no effect on the circulation around the western island, which is given by the equivalent of (2.2). The result (2.3) can be extended to a chain of N + 1 sequentially overlapping islands, see Fig. 1c. If ψ takes the value ψn on the n + 1th island, n = 0, 1, … , N, then
In (2.4b), Δfi is the difference in Coriolis parameter between the northern and southern tips of i + 1th island; Δfoυi is the difference in Coriolis parameter between the northern and southern extents of the overlap between i + 1th and ith islands. The path 𝒞n is as shown in Fig. 1c. The transport through the passage between the (n + 1)th and nth islands is
The above expressions are valid for the surface of a sphere as well as a β plane.
Equations (2.2)–(2.5) express the influence of different wind systems on the island circulation and transport through the passage between the islands. For example, assuming the landmass is a continent upon which ψ ≡ 0, then the circulation on the easternmost island is affected only by wind stresses over latitudes spanned by the island to the east of the island's western boundary. However, that on the westernmost island is affected by wind stresses over its own latitudes as well as those over latitudes of partially overlapping islands, though the influence of wind stresses outside its own latitude band likely diminishes rapidly with distance down the chain, because of the factor ai(n) ≤ 1. Similar conclusions can be drawn for the strait transports.
c. Dynamically narrow passages
1) Two islands
Following Wajsowicz (1993), if the strait between the islands in Fig. 1b is sufficiently narrow, and/or shallow, that frictional effects are important, then the value of the streamfunction on the eastern island is
where ℱ0 = ∫BA F · dl/Δf0 depends on the form of friction F specified in the depth-integrated momentum equations (2.1), A → B is a clockwise line segment along the west coast of the island in the overlap region, and ψ0 ideal is the value derived for a dynamically wide passage, namely (2.2). The value of ψ on the western island is obtained by integrating around 𝒞1 in Fig. 1b, yielding
The simplest form for F is given by assuming it is proportional to the depth-integrated velocity, then ℱ0 can be approximated by −r′0(ψ0 − ψ1)/Δf0 (Wajsowicz 1993). The coefficient, r′0 = LRa/W, where L, W are lengthscales representing the length and width of the passage respectively, and Ra is the Rayleigh friction coefficient; r′0 is assumed constant. Substituting in (2.6) and (2.7) yields simultaneous equations for ψ0, ψ1, which can be solved to give
where from (2.5),
and where r0 = r′0/Δf0(>0) and
Therefore, as noted in Wajsowicz (1999a), for this isolated two-island system, friction serves only to reduce the magnitude of the strait transport; it cannot change its direction.
2) Three islands
Adding a third island, say to the northwest, which is partially overlapped by the second island, introduces influences from more northerly latitudes. The streamfunction ψ0 is given as before in (2.8a), but now T1 is given by
where from (2.5),
and where ε1 = 1 + α2 − αoυ2 and T1 ideal is as given in (2.8c). The transport in the other strait, between the second and third islands, is
The equations for T1, T2 involve both T1 ideal, T2 ideal, and the coefficients multiplying them are positive. Therefore, if T1 ideal and T2 ideal are of opposite sign, then in the presence of friction situations could exist where the transport in one of the straits is reversed. Wajsowicz (1999a) demonstrated the effect for the more geometrically complex case of the Indonesian archipelago with quite drastic changes in circulation resulting from supposing transport through Torres Strait was completely blocked due to frictional effects (the r → ∞); for example, see Wajsowicz (1999a) Fig. 4.
3) N + 1 islands
The above method can be extended to the (N + 1)-island system shown in Fig. 1c, the result is a system of N + 1 simultaneous equations:
where rn, n = 0, · · · , N − 1 is the frictional constant r′n for the strait between the (n + 1)th and (n + 2)th islands divided by Δfn. In preparation for application to the Caribbean in the next section, the island chain is assumed to lie between two landmasses upon which ψ is known, namely ψa on the southeastern mass and ψb on the northwestern mass. Wajsowicz (1996) described analytical solutions for (2.10) with finite r for the special case of the first island representing Australia and the (N + 1)th island representing the Asian continent with the islands in between representing the chain from Java to Timor with the simplification that they all spanned the same latitudes. In general, the solution to (2.10) will be such that the circulation around an island and transport through its adjacent straits will depend on the wind stress over the latitudes of the whole island chain with the possibility of reversed transports in straits due to frictional influence.
4) Asymptotic solution for small friction
Assuming that the friction coefficients, rn ≪ 1, n = −1, 0, 1, … , N, and ψa, ψb are O(1) or identically zero, then (2.10) has the asymptotic solution
where ψn ideal, Tn ideal, ai(n) are as defined in (2.4)–(2.5), T0 ideal = ψa − ψ0 ideal, TN+1 ideal = ψN ideal − ψb, and
The corresponding strait transports are
To O(r), the streamfunction on an island boundary only depends on the values on overlapping islands to the east and the first layer of overlapping islands to the west. Whether the boundary value increases or decreases as the friction coefficients increase from zero depends on the relative magnitude of rnTn+1 ideal to the sum term in (2.11). In the next section, in application to the Lesser Antilles, it is assumed that the straits are frictional, but that the islands do not directly overlap. In this case, the island value ψn increases as the r increase from zero simply if αnrn−1Tn ideal > rnTn+1 ideal, and the strait transport Tn increases if rnTn+1 ideal > (1 + αn)rn−1Tn ideal.
5) Asymptotic solution for large friction
The limit of the friction coefficients rn tending to infinity is equivalent to the straits being wholly blocked, and for physically realistic situations ψb ≡ ψa. Therefore, the zero-order solution is ψn ∼ ψa, Tn ∼ 0, n = 0, 1, … , N. The next order in the expansion, ψn1, which is O(1/r), has the form
and the functions gn, GN satisfy the recursive formulas
and g0 = 0, G0 = 1. The solution cannot be written down as compactly as in (4), as now the streamfunction on each island at first order depends on the wind stress over latitudes spanning the whole length of the island chain.
3. The wind-driven circulation
a. The geometry and scheme
The islands and passages, which are the object of this study, as resolved by the ETOPO5 1/12° × 1/12° bathymetry dataset (NOAA 1988), are shown in Fig. 2. The 200-m isodepth is typically taken as the boundary for application of the island rule, but this is modified according to published cross sections from navigational charts for the passages in the eastern Caribbean, and some small landmasses and banks are combined or ignored. The boundaries actually used are denoted by dotted lines in Figs. 2b,c. In applying the multiple-island rule, the main task is to identify the layers of islands as defined by their degree of overlap. The first layer consists of islands wholly exposed to the interior Atlantic; the streamfunction on their boundaries is calculated using the island rule (2.2). These are the Windward Islands, which are grouped as Grenada plus St. Vincent and the Grenadines, St. Lucia, Martinique, and Dominica. The other nonoverlapping island groups are Guadeloupe, the Leeward Islands from Montserrat to Anguilla, Caicos Bank, and the Little and Great Bahama Banks. The second layer consists of islands, whose boundary streamfunction values can be determined from the multiple-island rule (2.3) once the first layer values are known. These islands are Puerto Rico plus the Virgin Islands, which is partially overlapped by the Montserrat–Anguilla group, Great Inagua, and Cay Sal Bank partially overlapped by Caicos Bank and Great Bahama Bank, respectively (see Figs. 2b,c). The third layer consists of islands, whose boundary values can be determined once the previous layer's values are known. This layer consists only of Hispaniola, which is partially overlapped by Puerto Rico plus the Virgin Islands and the Montserrat–Anguilla group. Actually, from (2.3), since Puerto Rico plus the Virgin Islands is wholly contained within the latitudes of Hispaniola, its boundary value does not need to be explicitly calculated to determine that on Hispaniola, which could be moved to the second layer. The fourth and final layer consists of Cuba and Jamaica. Jamaica is wholly overlapped by Hispaniola, and Cuba is partially overlapped by the Great Bahama Bank and Hispaniola; Caicos Bank and Great Inagua lie wholly within its latitudes and thus do not need to be directly calculated.
b. Transports assuming dynamically wide passages
The results from applying (2.2)–(2.4) to the major islands, under the assumption that all of the passages are dynamically wide and deep, are given in Table 1 for Hellerman and Rosenstein's (1983) climatological annual mean wind stress (henceforth referred to as HR climatology). The transport through each passage, as labeled in Fig. 2a, is also tabulated to enable comparison with recent observations, summarized in Table 1. The sign convention is northward and eastward transports are positive. Obvious discrepancies between the model and observations are significant eastward versus westward flow in Grenada Passage, too strong flows through Old Bahama Channel and Northwest Providence Channel, and too weak flows through Yucatan Channel and the Straits of Florida. The total depth transport through the collection of passages from St. Vincent to Guadeloupe inclusive is 9.3 Sv, which is much stronger than the 4.6 Sv over the upper 200 m inferred from Johns et al.'s (1999) observed transport through 63.5°W, but comparable with the 10.2 Sv over the total depth from the recent compilation of passage measurements by Johns et al. (2001). The inferred transports through Mona Passage and Great Inagua (a proxy for Windward) Passage are in reasonable agreement with direct observations, Johns et al. (1999) over the upper 200 m, and over the total depth, Johns et al. (2001). Transport through Santaren Channel is in the wrong direction.
Before considering how frictional effects could modify the above results, it is worth describing how robust the results are to the wind stress dataset used. The calculations are not repeated for each of the more than 10 climatologies described in Townsend et al. (2000, hereafter THH) and four described in Fanning et al. (1994), but rather general similarities and differences between the wind stress curl patterns and how these affect the transports are described.
1) General gyre circulation
The curl of the Hellerman and Rosenstein wind stresses over the North Atlantic is shown in Fig. 3a, and the corresponding Sverdrup streamfunction taking into account island circulations in Fig. 3b. The gyre circulation is closed in the western boundary layers by assuming that relative vorticity is destroyed at the latitude of creation so that the streamfunction monotonically decreases or increases across the layer. Although 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 equator and 10°N across the basin and extending northward up the African coast to midlatitudes. The wind stress curl is then negative to the west over these latitudes except over the southern Caribbean and Gulf Stream, where it is positive again (see Fig. 3a). The result is a cyclonic Sverdrup gyre between about 5° and 15°N, whose northern limb extends into the southern Caribbean with an additional cyclonic gyre in the Panama–Columbia gulf. This latter feature is in agreement with drifter tracks (e.g., Leaman and Wilson 2000). The region from 15°N to about 35°N is filled with a basinwide anticyclonic gyre, whose southern limb extends into the northern Caribbean.
2) Eastern Caribbean transports
From Stokes theorem, each line integral in (2.2)–(2.4) can be reexpressed as an integral of the wind stress curl over the area enclosed by the line integral. The tropical cyclonic Sverdrup gyre produces a negative circulation around Grenada–St. Vincent with eastward flow counter to observations through Grenada Passage. From (2.9), for this simple configuration, frictional effects cannot alter the sign of the transport through Grenada Passage; a thermohaline must be invoked as discussed in the next section. This latter circulation over the upper ocean is also needed to reverse the direction of the Guyana Current, whose wind-driven component is southward as shown in Fig. 3b, but observations (Johns et al. 1999) indicate is northward.
The remaining islands of the Lesser Antilles provide little obstruction to the Sverdrup flow, and the circulation around each can be approximated by considering the limit of (2.2) as the latitudinal extent of the island tends to zero yielding
where curl τ · dl = −k · curlτ R cosθdλ; R is the radius of the earth; θ, λ are the latitude, longitude; and β is the planetary vorticity gradient. Equation (3.1) is just the local value of the Sverdrup streamfunction, so the transport through the Windward Island passages, with the exception of Grenada Passage, may be regarded as just a measure of the meridional gradient in Sverdrup streamfunction, or alternatively wind stress curl. From THH and Fanning et al. (1994), an O(8 Sv) increase in the Sverdrup streamfunction from Grenada up to Anguilla is common to a wide range of wind stress climatologies. Therefore, topographic blocking, and so frictional effects, must be invoked to reduce the net westward transport, which will be deflected northward. If the deflected transport, O(2 Sv) say, entered the Caribbean Sea through Anegada Passage, then its transport would be in better agreement with observations.
3) Central Caribbean transports
The dependencies of the transports through Anegada and Mona Passages on wind stress, as given by the multiple-island rule assuming dynamically deep passages, is summarized in Table 2 along with the dependencies for neighboring islands. The convention used is that Iab is the wind stress integral as described in (2.10) for the island identified by letters ab, as detailed in Table 1. The contribution for each component, calculated for HR climatology, is given in the last column of Table 2. For Anegada Passage, the idealized transport is weak as the contributions from the Puerto Rican wind stress integral (−Ipr) and Anguilla–Montserrat integral (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 transport would be similar, as found by THH. According to the idealized multiple-island rule (MIR), transport through Mona Passage is dependent on the wind stress integrals for Anguilla–Montserrat, Puerto Rico, and Hispaniola. From Fig. 3a, the HR climatology has a dipole in the wind stress curl over the western edge of Hispaniola with the positive pole to the north. In different wind stress climatologies, the strengths of the two poles can differ considerably. For example, the positive pole is almost absent in the Isemer and Hasse (1987) climatology, its center is displaced significantly westward in the FNMOC climatology (Hogan and Rosmond 1991), and it extends over the whole of Hispaniola and to the east in the National Centers for Environmental Prediction (NCEP) climatology (Kalnay et al. 1996); see, for example, THH. The impact on Ihi is typically small, as the integral is dominated by the contribution from over the breadth of the Atlantic. However, for the NCEP climatology, the patch of positive wind stress curl is sufficient to reduce the dominant contribution of Ihi to the Mona Passage transport, which explains THH's findings of a much lower Mona transport for NCEP climatology than the others examined. From Fig. 3, although the dipole makes only a small contribution to Ihi, it does affect the Sverdrup circulation to the west in the Caribbean.
4) Western Caribbean transports
Transport through Windward Passage is sensitive to wind stresses spanning latitudes from Montserrat to the northern tip of the Grand Bahama Bank (see Table 2). The contributions from Icu and 0.34Igb are of similar magnitude and of opposite sign to the contribution 0.93Ihi. In assessing results using the different wind stress climatologies, the neighboring contributions from Ihi and Icu may be regarded as canceling. Then, the transport through Windward Passage is effectively determined by wind stresses integrated over the latitudes of the Great Bahama Bank and to a lesser extent Montserrat–Anguilla. This is borne out by the calculations of THH, who found that wind stress datasets, which produced strong (weak) Florida Strait transports (Igb exactly) also had strong (weak) Windward Passage transports. There are considerable differences between the wind stress curl climatologies around 25°N, mainly because of the differing extents of positive curl along the African and American coasts. From Table 1, just over half the transport through Windward Passage comes through the Great Inagua Channel; the remainder comes southward as a western boundary current along Cuba, see Fig. 3b.
The wind-driven transport through Yucatan Channel is given by the MIR formula for Cuba (see Table 2). It is dominated by the wind stress integrals Icu and 0.34Igb, which are of similar magnitude for the HR climatology. As for Windward Passage, the dependence on Igb means that there will be significant variation between the different climatologies, though it is scaled by 0.34, and so reduced to a similar order to the differences expected for Icu. Fanning et al. (1994) show the difference in Sverdrup streamfunction at the edge of the western boundary for four wind stress climatologies; Icu differences are characterized by those for about 20°N versus those for 25°–30°N. THH obtained wind-driven Yucatan Channel transports as high as 25.6 Sv for Isemer and Hasse (1987) climatology, but in general they were O(20 Sv).
The changes to the transports, if the Bahaman channels are frictional, are considered in section 3c, where it is shown that the southward MIR-derived transport through Santaren Channel can be reversed by frictional effects in the Straits of Florida. The low value obtained 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 boundary current is needed to increase the transport to observed values.
Finally, a striking feature of Fig. 3b are the thin westward zonal jets that cross the central Caribbean and the oppositely directed neighboring jets in the Gulf of Mexico. These jets are a feature of the Sverdrup solution assuming frictional effects are confined to western boundary layers. In practice, these jets are likely to be barotropically unstable, and so a source of mesoscale eddies across the breadth of the Caribbean. The oppositely directed jets obtained in the Gulf of Mexico are similar to those obtained between the tip of South Africa and Brazil by Godfrey (1989). Higher-order dynamics modify these jets to give the observed Loop Current and its enormous variability.
c. Transports assuming dynamically narrow passages
If the islands and banks of the Caribbean from Grenada to the Bahamas formed a solid barrier, then the value of ψ on the barrier is about 16 Sv, and the circulation consists of bidirectional flow in Grenada Passage with a 22 Sv westward current entering against the northern bank and a 6-Sv current exiting against the southern bank. The northern boundary current is fed by a southward western boundary current against the barrier. The bifurcation point for the interior flow on the barrier is at about the latitude of Caicos Bank. The northward western boundary current against the barrier peaks at about 10 Sv at the tip of Little Bahama Bank and joins the 16 Sv exiting the Florida Straits. If flow through Grenada Passage and/or Florida Straits exerted frictional resistance on the barrier, then ψ on the barrier is reduced. The barrier's northward (southward) western boundary current is stronger (weaker), and the bifurcation latitude is more southerly. This circulation pattern is obviously unrealistic, but it is the simplest illustration of regionwide frictional effects, and serves as a contrast to the flow in Fig. 3b. In fact, the circulation in Fig. 3b is more realistic, which suggests that many of the passages may be considered dynamically wide.
For sake of argument in the following, passages with minimum sill depths greater than 1500 m, are considered dynamically wide. These are Anegada and Windward Passages and Yucatan Channel. Therefore, the chain from Grenada to Anguilla is only affected by wind stresses over its latitudes. Puerto Rico–Virgin Islands and Hispaniola also form an isolated chain, and so are not affected by wind stresses to the north. The passages around Great Inagua Island are all greater than 1000 m, so the Bahama Banks, Cuba, and Cay Sal Bank form a frictional group influenced by Florida to the west.
1) Frictional effects in the eastern Caribbean
The effect of frictional blocking of the passages from Grenada Passage to Guadeloupe Passage on the circulation around the neighboring islands is described by (2.10) with the overlap coefficients set to zero. If the friction coefficients r′ of the passages are assumed equal, then the resulting behavior as a function of r′ is 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 r′ ∼ 0.2 × 10−5 s−1. The streamfunction on each of the island boundaries tends to the value on the American continent, that is, zero, as r′ → ∞, and the transports in each of the eastern passages also tends to zero. The transports in the middle passages, St. Lucia Channel and Martinique and Dominica Passages, increase as r′ increases from zero, and then decreases. This behavior is expected from the asymptotic behavior described in section 2c(4). As r′ increases from zero, the streamfunction around St. Lucia and Martinique tends toward that on Grenada, and those on Guadeloupe and Dominica tend to that on the Montserrat–Anguilla group. Dominica then exerts a stronger influence on Martinique than St. Lucia, and the sign of ψ on its boundary changes sign before tending to zero. The westward flow, which no longer passes through the eastern passages, is mainly deflected into Anegada Passage as it is assumed dynamically wide.
If one of the passages in the eastern Caribbean chain is dynamically wide, then the above behavior is altered. The streamfunction on the islands between the dynamically wide passage and the American continent tends to zero again, as do the passage transports, as r′ → ∞. However, for the islands between the dynamically wide passage and Anegada Passage, their values tend to the value given by the island rule (2.2) assuming the group forms a single island. For example, if Martinique Passage is dynamically wide, then as r′ → ∞, the transport in Martinique Passage tends to 3.8 Sv westward, the limiting southward value in Anegada Passage is reduced to 6.7 Sv, and the return 10.5 Sv northward flow exits via Windward Passage.
2) Frictional effects in the central Caribbean
Once again, the equations can be derived from (2.10). Assuming Anegada Passage is dynamically wide, then the effect of the changes in the circulation around the Montserrat–Anguilla group is felt by the islands to the west because of overlap. However, the overlap contribution is minor for most of the islands (see Table 2). The effect on Puerto Rico and Hispaniola is shown in the middle panel of Fig. 4a. The streamfunction on Hispaniola decreases as the contribution from Anegada decreases, and that on Puerto Rico increases toward that of Hispaniola, as the friction coefficient for the flow in Mona Passage increases; for sake of argument, the coefficient is assumed to have the same value as that for the more eastern passages. The streamfunction on Puerto Rico and Hispaniola tends to the island rule value assuming the islands form a single block, which is about 9 Sv for HR climatology. This is the limiting value of the Anegada transport; see middle panel of Fig. 4b. The transport in Mona Passage tends to zero as its r′ → ∞.
3) Frictional effects in the western Caribbean
As the Windward Passage is assumed dynamically wide, the islands and sand banks of the western Caribbean do not experience frictional rubbing directly by Hispaniola, but do know of frictional effects in the more easterly passages from island overlap. The equations for the island circulations can be derived in a similar manner to (2.10) and are given in the appendix. The effect of friction in the passages of the western Caribbean is shown in the bottom panel of Fig. 4. For simplicity in Fig. 4, the friction coefficients r′ = si, as defined in the appendix (A2), are assumed equal and equal to the coefficients in the more easterly frictional passages.
The streamfunction on the islands and sand banks tends to that on the American continent, that is, zero, as r′ → ∞. Transports in all of the passages of the western Caribbean tend to zero except for that of Windward Passage, which reverses as friction increases in the passages to the west, and tends to ψhi ∼ 9 Sv, as r′ → ∞ (see Fig. 4). In this limit, there is very weak circulation in the Gulf of Mexico and Caribbean Sea north of 20°N, as it is driven by the local wind stress curl only.
Increasing friction also reverses the transport directions in Old Bahama, Santaren, and Nicholas Channels. It is possible to tune the resistances in each strait to give the observed directions and right order of magnitude since from (2.8) the r′ are a function of geometry of the passage. However, the fundamental problem with this purely wind-driven model, namely that the western boundary currents to the south of the Caribbean are in the wrong direction, remains uncorrected. As a consequence, the water masses flowing through the Caribbean are always North Atlantic in origin in contrast with observations of significant South Atlantic water masses. This problem is addressed in the next section by introducing a thermohaline circulation. It is noteworthy that introducing a thermohaline circulation does not affect the basic dependencies described in this section and in section 2.
4. Including a thermohaline-driven circulation
The system is extended to three layers with the top layer containing the wind-driven circulation and upper limb of the thermohaline-driven circulation. The bottom layer contains the lower limb of the thermohaline circulation. Interaction between the top and bottom layer, for example, upwelling and deep convection, is assumed to be confined to latitudes outside the domain of interest. Therefore, the middle layer is considered to be at rest. As before, bottom topography is assumed to be confined beneath the wind-driven layer. A similar configuration was considered recently by Nof and Van Gorder (1999). As there is no explicit forcing of the thermohaline circulation within the domain of interest, it effectively enters and exits the domain as a western boundary current. Therefore, a similar set of equations to (2.1) holds for the upper layer in the domain of interest, but now the no-normal flow boundary condition on the American continent is ψ = ψTH, a constant, where ψTH is a measure of the strength of the thermohaline overturning circulation in the domain of interest. In the following, ψTH is set to −15 Sv, consistent with estimates from Schmitz and Richardson (1991). Details of the bottom-layer circulation are not needed to solve for the top-layer circulation, and so will not be considered further.
a. Upper-layer circulation assuming dynamically wide passages
The interior gyre and island circulations are exactly as in Fig. 3b. The effect of including the upper limb of the thermohaline circulation is confined to the western boundary layer against the American continent and the zonal jets between peninsulas on the continent (see Fig. 5). The direction of the Guyana Current and transport through Grenada Passage are reversed, now in agreement with observations. The transport in the Guyana Current is about −ψTH, that is, 15 Sv at 4°N, but reduces to about 5 Sv at 10°N before increasing again. The strength of the Grenada Passage transport, given by subtracting 15 Sv from the value in Table 1, is now 11 Sv, which is much larger than the observed 4.9 Sv over the upper 200 m (Johns et al. 1999). The transports through Yucatan Channel and the Florida Straits are increased by −ψTH to 33.4 Sv and 38.2, 40.8 Sv (26°N, 27°N), respectively. The other transports listed in Table 1 are unchanged. The above passage transports can be reduced by frictional effects as described in the next section.
Figure 5 also distinguishes the streamlines originating from the Southern Hemisphere. In the absence of the thermohaline circulation, the alternating cyclonic/anticyclonic wind-driven gyres are contained between zero streamlines reaching almost zonally from the African to American continents. Hence, the upper-layer water masses circulating through the Caribbean are all of North Atlantic origin, as shown in Fig. 3b. Including a thermohaline circulation enables South Atlantic water masses to enter the region within the western boundary current against the American continent. In Fig. 5, about 5 Sv continues along the western boundary through the Caribbean Sea, around the Gulf of Mexico, and exits via the Florida Straits. A further 10 Sv from the South Atlantic separates from the coast, and circulates around the cyclonic tropical gyre before entering the Caribbean Sea in the passages from Grenada Passage to Martinique Passage. It crosses the Caribbean Sea in a series of zonal jets, and then joins the western boundary current continuing northward to the Florida Straits, and onward along the North American coast. Figure 5 indicates that the water masses of the Peruvian–Columbian gyre are isolated. However, their northern edge is cut by the zonal jet crossing from the tip of Venezuela to Nicaragua, so mixing of the water masses may be expected.
b. Upper-layer circulation assuming dynamically narrow passages
If the passages of the Caribbean are frictional, then including the upper limb of the thermohaline circulation affects the circulation on all of the islands, and so the streamfunction to the west.
1) Frictional effects in the eastern Caribbean
If the passages from Grenada Passage to Guadeloupe Passage are assumed frictional and governed by the same friction coefficient r′, then the behavior of the streamfunction on island boundaries and transport through the passages as r′ increases from zero is shown in Fig. 6 (cf. Fig. 4, in the absence of a thermohaline circulation). According to (2.11), the value on all but Dominica and Guadeloupe decreases monotonically as r′ increases from zero. The result is that, in contrast to Fig. 4b, the westward transport in St. Vincent Passage increases initially. As r′ → ∞, the island values tend to ψTH and the transports tend to zero. In terms of an appropriate value of the friction coefficient so that the upper-layer transports match observations, it is required that ψan − ψTH ∼ O(10 Sv), since the observed transport across 63.5°W is estimated at 9.5 Sv over the upper 200 m (Johns et al. 1999) and 15.9 Sv over the total depth, (Johns et al. 2001). The transport through Grenada Passage ∼O(5 Sv), Johns et al. (2001). These suggest an r′ in the range 2–5 (× 10−5 s−1).
2) Frictional effects in the central Caribbean
The upper-layer transport, which in the absence of friction would have flowed through the passages of the eastern Caribbean, is deflected northward, and enters the Caribbean Sea through the dynamically wide Anegada Passage. Hence, although the behavior for small values of the friction parameter (see middle panels of Fig. 6) is similar to that shown in the middle panels of Fig. 4 in the absence of a thermohaline circulation, the limits as r′ → ∞ are different. The Anegada transport tends to −ψTH + 3.2 Sv = 18.2 Sv, where 3.2 Sv is the value of ψ on Hispaniola and Puerto Rico, assuming they form a single island with dynamically wide channels on either side, that is, as r′ → ∞ in Mona Passage. This latter limit is different from the limit for ψ on a conjoined Hispaniola and Puerto Rico in the absence of a thermohaline circulation. From Table 2, due to overlap, the value is a function of the limiting value on the Montserrat–Anguilla group, that is, ψTH. There are very few direct or inferred observations of transport in Anegada Passage. However, Johns et al. (1999) estimate only 2.4 Sv passes southward over the upper 200 m through the passage with a similar value for Mona Passage; Johns et al. (2001) estimate similar values for the total-depth-integrated transport based on mainly spring/summer measurements. This suggests that a suitable r′ for Mona Passage is ∼0.1 × 10−5 s−1 yielding an upper-layer transport in each passage of just over 3 Sv. As the transports in Mona and Anegada Passages rapidly diverge as r′ increases/decreases from 0.1 × 10−5 s−1 (see Fig. 6b), if another common transport value were chosen for the passages, then friction would need to be invoked in Anegada Passage to deflect some of its transport into Mona Passage.
3) Frictional effects in the western Caribbean
The effect of including frictional effects in the channels to the west of Windward Passage in the presence of a thermohaline circulation are shown in the bottom panels of Fig. 6 (cf. bottom panels of Fig. 4 in its absence). The island boundary ψ decrease more rapidly as r′ increases from zero, as the limiting value is ψTH = −15 Sv as r′ → ∞. The strait transports, besides those associated with the western boundary current against the American continent, are little changed from those in Fig. 4b. The exception is Windward Passage, whose transport changes from southward to northward for r > 0.4 × 10−5 s−1, which is less than half that found in Fig. 4b. The transports in the western passages tend to zero as r′ → ∞ except that in Windward Passage, which tends to the reverse of Anegada Passage. Matching the results in the bottom panel of Fig. 6b to observations given in Table 1 indicates that friction coefficients O(10−6 s−1) are appropriate.
An example is given in Table 3 (on the rhs of each column) of a possible combination of friction coefficients for the passages, which results in island streamfunction values that give reasonable agreement with observed transports. To help link with the dependencies displayed in Fig. 6, results are also given (on the lhs of each column) assuming Anegada and Windward Passages and the Yucatan Channel are dynamically wide, that is, their r′ ≃ 0. It is noteworthy that, of the 4.6 Sv that is blocked from flowing through Anegada Passage due to the introduction of friction in the passage, 1.5 Sv is redistributed among the Windward Islands passages to the south, 1.4 Sv adds to the flow into Mona Passage. The remaining 1.7 Sv does not add to that into Windward Passage, whose transport is actually decreased due to the introduction of a modest amount of friction in the Windward Passage and Yucatan Channel. Instead, it increases the transport through Old Bahama and Northwest Providence Channels and in the boundary current to the east of Abaco Island. The relatively large friction coefficients needed to give realistic transports through the Windward Island passages are consistent with observations and a fine resolution model of North Brazil Current rings (Johns et al. 2000). The rings are about 400 km in diameter and are shallow (<200 m) or deep (>1000 m). The deeper rings sense the 800–900 m sills and are deflected northward along the outer arc. The shallow rings tend to break up on encountering the islands.
c. Water mass pathways
The basinwide transport streamfunction for the upper layer corresponding to the results in the rhs columns of Table 3 is shown in Fig. 7, which should be contrasted with Fig. 5 where all of the passages are dynamically wide. The most obvious difference is the increase in strength of the western boundary currents against the southern half of Cuba and against the Grand and Little Bahama Banks. Also, the South Atlantic streamlines are no longer confined to the southern Caribbean Sea and western boundary current against the American continent, but are swept northward, and enter the Caribbean Sea through all of the passages from Grenada Passage to Anegada Passage inclusive for the particular choice of frictional parameters given in Table 3. Interestingly, increasing frictional effects in Anegada and Mona Passages does not necessarily divert the South Atlantic streamlines even farther northward into Mona and Windward Passages, as some of the excess transport is diverted into the Windward Islands passages (cf. examples in Table 3). Indeed, if the thermohaline circulation is ∼15 Sv and a net ∼15 Sv crosses 63.5°W and 2.5 Sv enters through Anegada Passage, as recently estimated by Johns et al. (2001), no South Atlantic water mass is expected to be found around the outskirts of the Caribbean north of Anegada Passage.
In Fig. 7, all of the streamlines from the South Atlantic pass through the Florida Straits, yielding a fraction similar to the 45% estimated by Schmitz and Richardson (1991). None is found east of the Bahama Banks. As the South Atlantic streamlines have values from −15 Sv to zero, it is not possible to divert the streamlines into this region without limiting the Florida Straits transport to less than 15 Sv; see Wajsowicz (1999b). Mass continuity requires that the 26 Sv from the closure of the Sverdrup wind-driven gyre and 15 Sv from the upper limb of the thermohaline circulation must pass either to the east or west of the Bahama Banks. If the Florida Current is about 32 Sv, then continuity requires that 9 Sv pass to the east of Abaco Island. This is considerably larger than the estimated 5 Sv by Lee et al. (1996) from current meter moorings. However, if the thermohaline circulation were 2 Sv weaker, and the interior Sverdrup streamfunction 1 Sv weaker, then the numbers would be in much better agreement given the considerable temporal variability in currents observed off Abaco Island (Lee et al. 1990).
5. Summary and discussion
It has long been considered coincidence that the transport measured through the Florida Straits is approximately that necessary to balance the southward Sverdrup transport across the ocean interior. The presence of South Atlantic water masses in the Windward Island passages (Wilson and Johns 1997) and Florida Current (Schmitz and Richardson 1991) is contradictory to the flow path in a classic Sverdrup model. However, high-resolution, global ocean GCMs (e.g., Maltrud et al. 1998) show that the mean barotropic streamfunction over the ocean interior between the latitudes spanned by the Caribbean is very similar to the Sverdrup streamfunction. The dichotomy is resolved by recognizing that a simple model of the upper-layer circulation can be constructed in which the wind-driven component ψW satisfies the Sverdrup balance βψWx = curlτ/ρo over the ocean interior subject to the no-normal flow boundary condition ψW = 0 on the African and American continents, and the thermohaline component ψT is represented by a western boundary current within the domain of interest, so that ψTx = 0 in the interior, and ψT = ψTH, constant on the African and American continents, respectively. The treatment of western boundary layers and islands depends on the level of sophistication desired.
In the study herein, the circulation around islands is described using a frictional form of the multiple-island rule (Wajsowicz 1993). If frictional effects are confined to western boundary layers and all of the passages are assumed dynamically wide and deep, then the effect of the thermohaline circulation is confined to the western boundary layer against the American continent. The transports through Grenada Passage, the Yucatan Channel, and the Florida Straits are the return Sverdrup value plus the strength of the thermohaline circulation. The transport streamfunction elsewhere and the circulation around islands are simply those obtained from a purely wind-driven model. If the passages of the Caribbean are frictional, then the pressure gradient associated with the thermohaline circulation may be partially distributed along the island chain, thus affecting all of the passage transports.
To appreciate the underlying sensitivities in the frictional, wind- and thermohaline-driven model, as described by Fig. 7 and Table 3, the model was built up step-by-step from the simple Sverdrup model. The multiple-island rule, as described in Wajsowicz (1993), was recapped in section 2. Exact solutions for two- and three-island groups were derived to show how friction simply reduces the “ideal” transport for a single isolated strait, but may reverse the transport in one of the straits for a dual-strait system under certain conditions. The rule was extended to describe an arbitrary-length chain of overlapping islands. The asymptotic solution in the limit of small friction gave that wind stresses over latitudes of immediately adjacent islands, even if these islands lie to the west or do not directly overlap, influence the streamfunction value on an island boundary. The asymptotic solution in the limit of large friction gave that wind stresses over latitudes spanning the whole length of the island chain influence the boundary value on an island. Frictional effects were incorporated in their simplest form as a linear drag. More complex forms require numerical solution and would yield different sensitivities. However, the fundamental notion of eastward and westward influence of the island chain is unchanged.
In section 3, the multiple-island rule was shown to be a very powerful tool in diagnosing the mean circulation of the Caribbean because it gives in closed, analytical form the dependency between the island streamfunction value and hence strait transport, and the wind stress field. For example, the mean transport through Windward Passage was found to depend on the wind stress integrated along island-rule paths for the Great Bahama Bank and Cuba, and only very weakly on wind stresses outside these latitudes; Table 2 summarized the more complicated overlapping island dependencies. It also enables the features of the wind stress field in different wind stress climatologies, which cause strait transports to differ in an operational GCM, say, to be readily identified [cf. Townsend et al.'s (2000) study of 11 wind stress climatologies]. The rule also can be used to ascertain conditions under which measurements of transport in Great Inagua Passage are a good proxy for that in Windward Passage, and when they may be misleading due to a significant western boundary current along Cuba.
The purely wind-driven Sverdrup model, in which all of the passages are dynamically wide and deep, has numerous discrepancies with observations, as described in section 3 and summarized in Table 1. Some (e.g., weakness in Yucatan Channel and the Florida Straits transport, and wrong direction for Grenada Passage transport and Guyana Current) can only be corrected by including a thermohaline transport. Others (e.g., too strong transport in Old Bahama Channel, too weak western boundary current east of Abaco Island, and wrong direction in Santaren Channel) can be corrected by including frictional effects. A demonstration of the simplest way in which the Caribbean islands interact under frictional effects, given by assuming that the frictional coefficients of the narrower, shallower passages are all equal to r′, was shown in Fig. 4; the Yucatan Channel and Windward and Anegada Passages were assumed dynamically wide and deep. A remarkable feature of Fig. 4 is the smallness of the friction parameter, r′ ∼ 0.8 × 10−5 s−1 ∼ 1/(1.4 days), required in the shallower passages to reverse the transport in Windward Passage. A typical value for r′ is (L/W)/(5 days), where L, W are lengthscales representative of the length and width of the strait, respectively.
The modification to the water mass distribution provided by including the upper limb of the thermohaline circulation as a northward western boundary current though the region was described in section 4. Frictional effects must be invoked in the passages to redistribute the extra transport among the passages, and so obtain reasonable agreement with observations. The required coefficient in the Windward Island passages is an order of magnitude larger than needed in the passages of the western Caribbean, as summarized in Table 3. Figure 7 confirms the “coincidence” that the Florida Straits transport matches the value of the Sverdrup streamfunction at the interior edge of the western boundary layer at 26°–27°N. In this simple model, frictional effects determine the fraction of the thermohaline transport plus return Sverdrup transport, which passes to the west of Great Bahama Bank. However, the amount of South Atlantic water mass passing through the Florida Straits is robust. As long as the Florida Straits transport is greater than the magnitude of the thermohaline circulation, all of the South Atlantic water mass passes to the west rather than east of Great Bahama Bank; see Wajsowicz (1999b) for more explanation on this type of model.
Choosing friction parameters, so that strait transports match observations, results in almost 10 Sv of South Atlantic water mass being diverted northward along the outer arc of the Lesser Antilles. It is difficult to prevent all of it from entering the Caribbean Sea before or at Anegada Passage. Indeed, this model shows that the Anegada Passage may be the favored entry passage for the Caribbean Sea rather than Windward Passage. In Fig. 6, both passages are assumed dynamically wide, but whereas the transport in Anegada is always southward, and increases toward the magnitude of the thermohaline circulation as r′ → ∞; that in Windward Passage switches to northward flow for an even smaller value of r′ than in Fig. 4. Sensitivity in Windward Passage transport direction has been noted recently in very high resolution (1/12° × 1/12°) simulations of the North Atlantic circulation (E. P. Chassignet and Z. D. Garraffo 2000, personal communication). There is northward flow of 1.2 Sv, averaged over years 15–21 of their integration, through Windward Passage when their model is forced by the Comprehensive Ocean–Atmosphere Data Set climatological surface fluxes (da Silva et al. 1994). However, southward flow of 2.1 Sv, averaged from years 0 to 6 of their integration, is obtained when the model is forced by mean wind stresses from the European Centre for Medium-Range Weather Forecasts 1979–99 reanalysis.
This research was conducted under ONR Grant N000149610611.
Frictional Multiple-Island Rule for the Western Caribbean
In section 2, the system of simultaneous equations describing the circulation around overlapping islands in a chain of arbitrary length was presented. This system can be applied with appropriate definition to the chain of islands from Grenada to Hispaniola. The geometry of the western Caribbean is slightly more complex, but the same principles may be applied. The resulting system of equations (using the notation of Table 1), assuming that Windward Passage and the passages around Great Inagua and Caicos Bank connecting to Cuba and Great Bahama Bank are dynamically wide, is
where the friction coefficients, equivalent to the r′, are defined for the straits as follows:
The wind stress integrals Iab, and Coriolis parameter differences Δfab, follow previous definitions. The overlap parameters Δfovcu1, Δfovcu2 are the difference in Coriolis parameter between the extreme overlap latitudes of Cuba and the Great Bahama Bank, and Cuba and Hispaniola, respectively.
Additional affiliation: Earth System Science Interdisciplinary Center, University of Maryland at College Park, College Park, Maryland
Corresponding author address: Roxana Wajsowicz, Dept. of Meteorology, University of Maryland, 3433 Computer and Space Science Bldg., College Park, MD 20742-2425. Email: email@example.com