\documentstyle[11pt]{article} \oddsidemargin 0.0in \evensidemargin -0.5in \marginparwidth 0pt \marginparsep 0pt \topmargin -0.75in \headsep .5in \textheight 9in \textwidth 6.5in \setcounter{page}{1} \begin{document} % This and the \end{document} at end REQUIRED \normalsize \newpage \begin{center} {\large\bf ABSTRACT} \end{center} \baselineskip=18pt \rm A series of wind-driven isopycnic coordinate model experiments are described and analyzed. In five preliminary experiments, a progression is established from the simple calculations of part 1 of this paper (Bleck and Smith, 1990) to multilayer simulations incorporating realistic forcing in a framework of coarse-mesh Atlantic basin geometry and smoothed bottom topography. The primary experiment, a five-layer seasonally forced model run, simulates known features of the annual mean wind-driven circulation of the Atlantic, including a western boundary current separating from the coast near Cape Hatteras and a Labrador Current flowing past the Grand Banks. The model reproduces (though with reduced amplitude) the annual cycle of Florida Current transport, seasonal variations in the equatorial region (including seasonal heat transport) and seasonal sea level cycles in the vicinity of strong currents. \newpage \baselineskip=18pt \noindent \bf {\large\bf I. \hspace{.4cm}INTRODUCTION} \vskip 6pt \rm The large-scale circulation of the ocean is known to be driven both by wind and by buoyancy forces resulting from differential heating of the Earth's surface. In any attempt to explain aspects of that circulation, modelers have of necessity made simplifications and approximations to the complex physical laws involved. Early simulations such as the linear analytic models of {\em Stommel} [1948] and {\em Munk} [1950] provided solutions for flat-bottom, rectangular basins forced by wind only. {\em Veronis} [1966] and {\em Bryan} [1963] extended the work of Stommel and Munk by developing numerical models with the inclusion of time-dependent and nonlinear terms in the equations of motion. Succeeding wind-driven numerical studies focused on the various effects of lateral and bottom friction parameterization, boundary conditions, basin shape, and bottom topography. Baroclinic effects were included by {\em Holland} [1973] in a model with idealized forcing and simple topography, in order to verify the theory that transport is enhanced by the joint effect of baroclinicity and bottom relief [{\em Sarkisyan and Ivanov}, 1971]. The multilayer model developed at the Geophysical Fluid Dynamics Laboratory (GFDL), Princeton [{\em Bryan}, 1969; {\em Bryan and Cox}, 1972], incorporates both wind and thermohaline forcing into a framework which allows a generalized basin shape and bottom topography. Adaptations of this model have formed the basis for numerous experiments, both for the world ocean and for individual regions, e.g., the early Atlantic Ocean model of {\em Holland and Hirschman} [1972], the robust-diagnostic global model of {\em Sarmiento and Bryan} [1982], and the recent eddy-resolving global calculation of {\em Semtner and Chervin} [1988]. The Holland and Hirschman model and other Atlantic studies [{\em Friedrich}, 1970; {\em Anderson et al.}, 1979] seek to explain the discrepancy between observed Gulf Stream transport and the wind-driven transport predicted by the Sverdrup relation [{\em Leetmaa et al.}, 1977] and to investigate the extent of topographic control of the circulation. Recent models focusing on the Atlantic seasonal cycle include that of {\em Anderson and Corry} [1985a], who perform a wind-driven, two-layer calculation to examine the annual cycle of transport in the Florida Straits, and of {\em Greatbatch and Goulding} [1989], who use a one-layer, linear model to investigate Atlantic seasonal cycles in transport and sea level. {\em Sarmiento} [1986] has examined seasonal variations in heat transport using the GFDL model for the Atlantic, and a seasonally forced model by {\em Philander and Pacanowski} [1986a,b] has as its focus the annual cycle in the Atlantic equatorial region. Models based upon geophysical inverse theory [e.g., {\em Wunsch}, 1978; {\em Olbers et al.}, 1985] incorporate both observational data and imposed conservation laws in order to produce a physically realistic solution for the Atlantic circulation. The progression from simple models of idealized domains to complex numerical representations of three-dimensional ocean circulation has been made possible by the size and speed of modern computers. At the same time, the observational data base needed to design and test such models remains inadequate over many areas of the world ocean. One aim of the World Ocean Circulation Experiment (WOCE), a global survey of ocean currents and properties, is the compilation of a comprehensive data set which will both test and stimulate the development of models needed to establish the role of the ocean in climate change. The design of the experiment itself may in turn be improved by the use of models which help define the dynamically most interesting regions of the ocean. This is the second of a two-part set of papers describing an isopycnic coordinate primitive equation model being developed to study basinwide and global-scale ocean circulation driven by both wind and buoyancy forces. Here we report on the results of purely wind-forced experiments on an Atlantic Ocean basin from 30$^{\circ}$S to 62$^{\circ}$N. In a separate effort, {\em Bleck et al}.\ [1989] report on development and testing of a framework for the inclusion of buoyancy forcing in the model. Continued development and further advances in computer technology will allow expansion to the global ocean. The present model is based on the same physical assumptions as the GFDL model but differs in terms of numerical representation. An isopycnic coordinate model can be viewed as a stack of shallow fluid layers, each of which is characterized by a constant value of density and is governed by equations resembling the shallow-water equations. The layers interact through hydrostatically transmitted pressure forces. The analytical models of {\em Welander} [1966] and {\em Parsons} [1969], in which the vertical structure of the ocean is represented in terms of layers of constant density, can be considered prototypes of the model employed in this study. {\em Huang} [1986a] has recently extended the work of Welander and Parsons in numerical models with idealized forcing and basin geometry. The ventilated thermocline model of {\em Luyten et al.} [1983], which seeks to describe the vertical structure of a subtropical gyre, also represents that structure by means of constant-density layers. In contrast to multilevel models formulated in Cartesian coordinates, such as the GFDL model, an isopycnic coordinate model is able to alter the vertical resolution by dynamical distortion of the coordinate surfaces. In this way, it yields optimal resolution of the baroclinic structure of the water column at all times for a given number of layers. In addition, the isopycnic framework conforms to the accepted view that large-scale oceanic mixing processes take place primarily along surfaces of constant potential density [{\em Montgomery}, 1938]. In the particular case of a purely wind-driven calculation, an Eulerian model gradually smears out density variations present in the initial conditions while an isopycnic coordinate model retains the density contrast between water masses regardless of the length of time for which it is integrated. For long time scales, Eulerian models must of necessity be forced thermodynamically in order to maintain this contrast. These advantages have, until recently, been offset by the obvious numerical difficulties associated with coordinate surfaces coming together with each other, the ocean surface, the bathymetry, or sloping lateral boundaries. The {\em Bleck and Boudra} [1981] quasi-isopycnic model avoided such occurrences by locally reverting to an Eulerian coordinate system wherever required to maintain a minimum layer thickness. By contrast, the pure isopycnic model described by {\em Bleck and Boudra} [1986] (hereafter, BB86) and the present model allow the existence of massless layers at any location in the model domain. BB86 solved this numerical problem in a flat-bottom eddy-resolving framework by using the Flux-Corrected Transport (FCT) algorithm [{\em Boris and Book}, 1973; {\em Zalesak}, 1979] to solve the mass continuity equation. Extending the work of BB86, development and initial testing of the current model had the primary goal of allowing the incorporation of realistic basin geometry and topography into a pure isopycnic framework. One of the most challenging aspects of this task has been proper resolution of the interaction of the flow with bottom topography, a troublesome aspect of coarse resolution models, in which the bottom depth may vary by more than 1000\,m from one grid point to the next. A detailed description of the model formulation developed for this purpose is given by {\em Bleck and Smith} [1990] (hereafter, Part 1), which also presents the results of experiments on a rectangular basin with simple bottom topography which support the fidelity of the model results to accepted theory. The principal focus of these experiments was to demonstrate the complete spindown of the bottom layer flow in cases where the following conditions are satisfied: the layer is not directly forced by the wind; the wind forcing is constant; and flow instabilities do not occur, that is, the flow reaches a steady state. Two-layer experiments in which the top layer is very thick (1000\,m on average) indeed produced this outcome. Three-layer experiments were also conducted, in which the first layer is initially shallow enough that its lower interface eventually outcrops over a large portion of the subpolar gyre. One effect of this outcropping is to inhibit the numerical solution from reaching a steady state, possibly due to dynamical instabilities associated with the steep isopycnal interface. With unsteady near-surface flow being able to perform work on the bottom layer, a very small residual circulation is then maintained there. The prototype version of the model as described herein is configured with five constant-density layers. Both lateral and bottom friction are included. Interfacial friction, whose purpose might be to emulate the effects of mesoscale eddies, has not yet been incorporated. The five-layer calculations are extended to 60 years of integration and a suitable multiyear period near the end is chosen for averaging and analysis. Seasonally forced experiments are driven by the mean monthly wind stress data of {\em Hellerman and Rosenstein} [1983]. The analysis of our primary Atlantic experiment is preceded by description of a few simpler experiments in order to establish consistency with the results of Part 1 on this realistic basin geometry and to document the variability in the solution as lateral friction is decreased from the rather high value used in Part 1 ($10^5 \, \mbox{m}^2 \, \mbox{s}^{-1}$), as seasonal forcing is introduced, and as the number of model layers is increased. The analysis of the five-layer, seasonally forced primary calculation has been influenced by some of the WOCE objectives of defining the character of significant signals in the ocean and of producing output which can be compared to observations. For the North Atlantic, the seasonal cycle of the Florida Current provides a topic which is appropriate in both respects and which is suitable for examination by a wind-driven model with bottom topography. Other aspects of the Atlantic seasonal cycle, particularly in the equatorial region, are emphasized in this study, and the principal features of the mean flow are also described. Although thermohaline and eddy processes have not been incorporated, comparisons to observations and to the work of other modelers verify that the wind-driven isopycnic coordinate model reproduces major features of the seasonally varying North and Equatorial Atlantic circulation. \vskip 12pt \noindent {\large\bf II. \hspace{.4cm}DESCRIPTION OF THE MODEL} \vskip 6pt \rm The isopycnic coordinate model equations and details of the methods employed in their numerical solution are given in Part 1. Here we summarize the principal points of that description before outlining the particular model configuration used in the Atlantic basin calculations. In each layer of constant potential density, three equations are solved: the $ x $ and $ y $ momentum equations and the mass continuity equation, which may also be termed the layer thickness tendency equation. The advection terms in the horizontal momentum equations, omitted in the calculations of Part 1, are here retained for completeness. The continuity equation is solved by the Flux-Corrected Transport method which maintains the positive-definite character of the thickness field. The numerical integration of the momentum and continuity equations is carried out in split-explicit mode [{\em Gadd}, 1978] with separate, though interactive, integrations for the barotropic and baroclinic components of the prognostic variables. A specific advantage of this manner of dealing with the barotropic mode over using the traditional rigid-lid approximation is that sea surface elevation becomes an explicitly predicted variable, facilitating comparison with observed sea level and eventual assimilation of altimetry data into the model. For the Atlantic calculations, the model basin is artificially truncated by solid walls at $ 30.5^{\circ} $S and $ 62.5^{\circ} $N. No-slip boundary conditions are imposed on all lateral boundaries, including submerged sidewall boundaries. Horizontal resolution of the Mercator projection grid is uniformly $ 2^{\circ} $ in longitude while decreasing with the cosine of latitude so that grid boxes remain square. There are a maximum of 58 grid points in both north-south and east-west directions. Smoothed bottom topography data at half-degree intervals for the world ocean were made available by the ocean modeling group at GFDL, Princeton, and were averaged and interpolated to obtain values at the model grid points (Figure 1). The Gaussian smoothing function applied to the raw depth data had the effect of removing all islands from the basin. Shelf regions have been further deepened to a minimum of 200\,m. A notable absence is that of any representation of the Antilles island arc bordering the eastern Caribbean or of the Bahama Islands to the east of Florida. For the seasonally forced Atlantic experiments, the value of the wind stress at the sea surface is interpolated at each time step from {\em Hellerman and Rosenstein} [1983] monthly means. The wind-induced stress is assumed to linearly decrease to zero in the upper 100\,m of the water column. (Note that at any given grid point, more than one layer of constant density may lie within that 100\,m depth.) Bottom friction is incorporated by assuming a linear stress profile in a boundary layer of 10\,m thickness using a standard bulk formula with drag coefficient 0.003. Interfacial friction between layers of constant density is not applied in these experiments. A variable eddy viscosity coefficient, generally referred to as ``deformation-dependent" [{\em Smagorinsky}, 1963], is employed in the primary experiment to provide for large friction in regions of substantial velocity shear (e.g., western boundary currents) and weaker friction in the ocean interior. The viscosity coefficient is parameterized as $ \nu =0.4 \Delta x \Delta y | {\rm Def} | $, where $ \Delta x \Delta y $ is the size of the grid box and $ |{\rm Def}| $ is the absolute value of the total deformation of the velocity field: \[ |{\rm Def}| = \left[ \left( \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} \right)^2 +\left( \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y} \right)^2 \right]^{1/2} \] The proportionality constant 0.4 is in the high range of values suggested by {\em Deardorff} [1971] to suppress subgrid-scale activity. The vertical structure of the five-layer model is represented by layers with initial pressure thickness values (discounting topography) of 100, 200, 400, 800 and 4000 $ \times 10^4\,$Pa. (For convenience, we will refer to layer thickness in units of meters, with $ 10^4\, \mbox{Pa} \simeq 1\,$m.) The average potential temperatures at the interface between each pair of layers, computed from the atlas data of {\em Levitus} [1982] for the model domain, are then 17, 12, 7 and $ 4^{\circ} $C respectively, the temperatures chosen by {\em Worthington} [1976] for a five-layer representation of the thermal structure of the North Atlantic. The average potential density of the uppermost layer was determined from a standard Knudsen-Fofonoff equation of state with temperature and salinity data for the model domain from {\em Levitus} [1982]. The difference between the average potential densities of each pair of layers (referenced to the pressure depth of the interface) was then calculated and successively added to the surface value. The resulting $ \sigma_\theta $ values for the five layers are 25.36, 26.65, 27.23, 27.73 and 28.07. The output quantities for each layer, stored at 15-day intervals, are the total horizontal velocity and pressure thickness fields of the layer. The barotropic velocities and the barotropic pressure component are also stored in the file of model history data. A mass transport streamfunction is computed, for diagnostic purposes, by solving Poisson's equation relating the curl of the barotropic velocity field to the streamfunction. The elliptic equation is solved by the method of successive overrelaxation, taking into account the variable depth of the fluid. \vskip 12pt \noindent {\large\bf III. \hspace{.4cm}PRELIMINARY EXPERIMENTS} \vskip 6pt In this section, we wish to discuss the extent to which some of the basic results of Part 1, in which the forcing and domain geometry are highly idealized, hold in the case of the Atlantic Ocean coastal and bottom geometry. We are particularly concerned with forcing mechanisms for currents in the deep ocean and with the possibility that the model, perhaps due to numerical inconsistencies in the way steep bottom topography is handled, harbors spurious mechanisms for generating flows along the ocean bottom. As shown in Part 1, an isopycnic coordinate model offers a relatively easy framework in which to establish whether this is the case. In such a layered configuration, experiments in which steady forcing leads to a steady state should exhibit interior motion only in the layer(s) directly forced by the wind in the final state. Without thermohaline forcing, fluctuating wind forcing, interfacial friction (or diapycnal mixing) or dynamical instabilities, there is no means for kinetic energy to be transferred into the unforced layers. The so-called ``baroclinic compensation" [{\em Anderson and Killworth}, 1977] eventually eliminates the horizontal pressure gradient force in the unforced layers so that they come to rest. This also means that, whenever bottom topography is completely submerged below the layers receiving direct forcing, the final solution should greatly resemble that obtained by integrating the Sverdrup relation along lines of constant $f$ from the eastern boundary and then allowing currents along the northern, southern and western boundaries to connect the unclosed Sverdrup streamlines. The magnitude of the model viscosity coefficient will determine the distance from the lateral boundaries over which the Sverdrup balance is modified by friction and the degree to which the numerical solution is smoother or rougher than the Sverdrup contours. Wherever bottom topography protrudes into layers with motion, such as we expect to happen in regions of continental shelves and the Gulf of Mexico/Caribbean area, the solution will be locally modified by the topography. \vskip 10pt \underline{\large\bf EXP1---two layers, strong friction, annual mean winds} \vskip 6pt The configuration in which we test the model's fidelity to the above theoretical results is similar to that used in the two-layer experiments of Part 1. The initial depth of the interface is 1000\,m (wherever the bottom depth is greater than 1000\,m), $g' = g \Delta\rho/\rho$ is 0.02\,m\,s$^{-2}$, and the lateral viscosity coefficient is $ 10^5 \times \cos \varphi \, \mbox{m}^2 \, \mbox{s}^{-1}$. (The variation with latitude $\varphi$ results from our scaling of the eddy viscosity coefficient as the product of a constant diffusive transport velocity and the horizontal grid size.) For these parameters, the results of Part 1 indicate a final steady state devoid of any significant bottom layer flow. Here we use the Atlantic basin of Figure 1 and the annual mean of the {\em Hellerman and Rosenstein} [1983] monthly winds. The curl of this wind field (${\bf k} \cdot \nabla \times \mbox{\boldmath $\tau$} $) and the corresponding Sverdrup streamlines are shown in Figure 2. EXP1 was spun up from rest and integrated to 25 years, at which time the kinetic energy trace for the experiment (Figure 3) indicates an approximate steady state. Only during the first five years of this experiment is there any noticeable difference between the total basin-averaged kinetic energy and that in the top layer. In the bottom layer, the kinetic energy comes very close to zero within five years and remains there for the duration of the experiment. The first result which we have intended to verify with the model --- no bottom layer motion in the final steady state --- thus appears from this figure to be essentially satisfied. The final circulation pattern of EXP1 is illustrated in Figure 4 by the layer 1 streamfunction plot. This is essentially the total streamfunction, as a corresponding calculation for layer 2 has no values exceeding 1\,Sv (1\,Sv $\equiv 10^6 \, \mbox{m}^3 \, \mbox{s}^{-1}$). Comparing with Figure 2b, we note many of the expected similarities away from the northern, southern, and western boundaries. The numerical solution is visibly smoother than the Sverdrup streamlines, however, no doubt due to the large value used for the frictional coefficient. This is especially true off the U. S. east coast around $ 40^{\circ} $N where a cyclonic feature in the Sverdrup contours, deriving from a corresponding region of annual mean positive wind curl, has apparently been smoothed away. An appendage to the subpolar gyre (a weak Labrador Current), which turns around the Grand Banks toward the southwest, is apparently the model's only representation of this cyclonic feature in the Sverdrup streamlines. The model Gulf Stream separates from the coast in a broad flow near Cape Hatteras. One component becomes involved in recirculation in the western North Atlantic, while the other heads northeastward, gradually shedding streamlines to the south to fill out the subtropical gyre. The maximum transport in the EXP1 subtropical gyre near $30^{\circ}$N is slightly greater than 35\,Sv, very close to that predicted by linear Sverdrup dynamics using the annual mean wind curl. While the tightly packed Sverdrup streamlines in the subpolar gyre (Figure 2b) show a maximum value of $\sim $40\,Sv, the strong friction in the model coupled with the requirement to close the streamlines within the computational domain inhibits the maximum transport in the EXP1 subpolar gyre from becoming much greater than $\sim$18\,Sv. (We note the same tendency for reduced transport in the truncated South Atlantic subtropical gyre.) In the Gulf/Caribbean region, the streamlines are shaped by the bottom topography and coastal geometry, but elsewhere the model is relatively loyal to linear Sverdrup dynamics, indicating that the baroclinic compensation has isolated the circulation from the influence of topography. Another illustration of the final flow configuration for EXP1 is given in Figure 5, which shows detailed velocity vector plots for layers 1 and 2. This figure gives more definition to the current structure in layer 1, but it also shows that, primarily near the western boundary, the lower layer possesses non-negligible flows which bottom friction has not destroyed. According to {\em Anderson and Killworth} [1977], such flows should exist due to a difference in scale between the barotropic and baroclinic frictional western boundary layers. Whatever the reason for their existence, we feel that these boundary currents, which are in general in the same direction as the upper layer currents, are tolerable in view of their negligible contribution to bottom-layer kinetic energy and to the total transport. \vskip 10pt \underline{\large\bf EXP2---two layers, moderate friction, annual mean winds} \vskip 6pt In this experiment we decrease the lateral viscosity coefficient from the high value ($ 10^5 \times \cos \varphi \: \mbox{m}^2 \, \mbox{s}^{-1}$) used in EXP1, while keeping other parameters identical to those of EXP1. We first found that use of the deformation-dependent coefficient with a prescribed minimum value of $ 2 \times 10^4 \, \mbox{m}^2 \, \mbox{s}^{-1}$ led to the development of a vigorous and eventually unstable circulation in which the layer interface, initially at 1000\,m depth, steepened excessively, reaching the surface in the center of the subpolar gyre. Reasonable and numerically stable model solutions were then attained by choosing an intermediate range of values for the friction coefficient, $ 5 \times 10^4 \times \cos \varphi \: \mbox{m}^2 \, \mbox{s}^{-1}$. The circulation pattern of EXP2 after 40 years of model integration is shown in Figure 6. As in the case of EXP1, the layer 1 streamfunction is essentially the total streamfunction, as layer 2 transport remains less than 1\,Sv. Comparison of Figures 4 and 6 discloses features in the flow pattern of EXP2 which are attributable to the reduced friction --- streamlines that are rougher in appearance than those of EXP1, enhanced transport in the subpolar gyre ($\sim$22\,Sv versus $\sim$18\,Sv for EXP1) where the viscosity coefficient attains its minimum value, and an extension of the model Labrador Current further around the Grand Banks than in EXP1. In the subtropical gyre, the broad western boundary current of EXP1 appears in EXP2 as a more spatially concentrated flow, as {\em Munk's} [1950] theory would predict. However, little increase over the 35\,Sv of EXP1 is noted in the maximum transport of the gyre, as the latitude-dependent friction of EXP2 remains strong in the subtropical region. An arrow plot of the layer 2 velocities for EXP2 (not shown) is similar to that for EXP1 (Figure 5), with marginally higher velocities now found along the western and northern boundaries of the basin. As in the case of EXP1, these currents contribute negligibly to the transport and kinetic energy of the bottom layer. \vskip 10pt \underline{\large\bf EXP3---two layers, moderate friction, seasonal winds} \vskip 6pt In our final two-layer experiment, we introduce forcing by seasonal winds into a model configuration which is otherwise identical to EXP2. The transient motions reflected in the kinetic energy trace for the 40-year integration of EXP3 (Figure 7) are induced by seasonal wind curl variations which, in a linear model, would make no contribution to the annual mean circulation [{\em Veronis}, 1970]. In our experiments, the large horizontal scale and frictionally damped velocities imply a Rossby number sufficiently small so that the response is, in fact, essentially linear [see, for example, {\em Willebrand et al.}, 1980]. Consequently, we find that the mean circulation of the final year of EXP3 differs from the final circulation of the steadily forced EXP2 (Figure 6) only in a negligible increase in western boundary current velocity and width, as {\em Pedlosky's} [1965] analytic model would predict. We postpone discussion of seasonal variations in the model results until the analysis of our primary five-layer Atlantic calculation. \vskip 10pt \newpage \underline{\large\bf EXP4---five layers, moderate friction, annual mean winds} \vskip 3pt \underline{\large\bf EXP5---five layers, moderate friction, seasonal winds} \vskip 6pt Our three two-layer Atlantic experiments have established consistency with the results of Part 1 as realistic basin geometry and topography, decreased lateral friction, and seasonal forcing enter into the model representation. We now vary the vertical structure by introducing the configuration described in section 2 above, namely, five layers of initial thickness (discounting topography) 100, 200, 400, 800 and 4000\,m. The 60-year kinetic energy traces for experiments EXP4 and EXP5 (Figure 8), the five-layer analogs of EXP2 and EXP3 respectively, exhibit the multiyear vacillations and overall unsteadiness which characterized the three-layer experiments described in Part 1. In those experiments, the top layer was chosen sufficiently thin (300\,m) to allow outcropping of its lower interface over a large part of the cyclonic gyre. Here, in the five-layer configuration, the 300\,m initial thickness of layers 1 and 2 combined enables both the first and second layer interfaces to outcrop in the subpolar region. The point in time at which both interfaces have reached the surface is apparent near year 17 in both panels of Figure 8, when the kinetic energy of layer 3, then fully exposed to the wind near the center of the subpolar gyre, rises and remains above that of layer 2. The relatively high velocities in the thin upper layers, which lead to steepening and outcropping of the isopycnal surfaces, contribute to an overall higher level of kinetic energy for these experiments than for the two-layer runs (Figure 7), in which outcropping of the single, deeper interface did not occur. It is evident from Figure 8 that a residual circulation persists in the bottom layer for the entire length of both EXP4 and EXP5. In the case of our seasonally forced model runs (EXP3, EXP5) we would expect this result, i.e., non-negligible currents in deep water [{\em Veronis and Stommel}, 1956]. The existence of this bottom-layer circulation in a case with annual mean forcing (EXP4), however, points to the failure of the five-layer simulations to attain a steady state, like the three-layer experiments of Part 1. As in the two-layer cases discussed above, the layer 5 flow in these experiments is found primarily along the western boundary slope region, and the maximum transport within that layer remains less than 2\,Sv. To briefly examine the results of EXP4 and EXP5, we choose to average the 12-year period from years 49 through 60. The mean barotropic streamfunctions obtained for that time period for both EXP4 and EXP5 are shown in Figure 9. The analogous figures for EXP2 and EXP3 were found to be essentially identical to one another, and only minor differences are noted here for the two five-layer cases, again indicating little contribution to the annual mean circulation of EXP5 by seasonal wind curl variations. Given our intention to document variability in the solution as the model configuration is altered, a more relevant comparison is that between Figure 6 and Figure 9a (or Figure 9b), which show the barotropic streamfunctions derived from two-layer and five-layer experiments respectively. The single quantifiable difference between the two figures is the $\sim$4\,Sv increase in maximum subpolar gyre transport in the five-layer case, from 22 to 26\,Sv. This increase suggests that the vertically integrated frictional torque exerted by sidewalls on the gyre circulation may be reduced when increased vertical resolution and higher levels of kinetic energy drive the system toward the Sverdrup balance between planetary vorticity advection and wind stress curl. \vskip 12pt \noindent {\large\bf IV. \hspace{.4cm}THE PRIMARY ATLANTIC EXPERIMENT} \vskip 6pt In five preliminary experiments, our model configuration has changed from the simple steadily forced two-layer box of Part 1 to a seasonally forced five-layer domain incorporating geometry and bottom topography appropriate to the Atlantic basin. While the magnitude of the lateral viscosity coefficient has been cut in half in the course of these experiments, the smoothness of the streamlines for EXP4 or EXP5 (Figure 9) suggests that the numerical solutions are dominated by viscous processes more than is essential. Therefore we wish to use a smaller value of lateral friction for our primary Atlantic experiment, if in fact it is possible to do so while maintaining numerical stability. Fortunately, use of the deformation-dependent viscosity coefficient with a prescribed minimum value of $ 2 \times 10^4 \, \mbox{m}^2 \, \mbox{s}^{-1}$ provides both sufficient numerical damping in the ocean interior and large friction in regions of substantial velocity shear. Our primary experiment, {\em EXP6}, is therefore defined to be a five-layer calculation incorporating the Atlantic geometry and bottom topography of Figure 1, forced by the mean monthly wind stress data of {\em Hellerman and Rosenstein} [1983], and employing deformation-dependent lateral friction with a minimum coefficient value of $ 2 \times 10^4 \, \mbox{m}^2 \, \mbox{s}^{-1}$. In Figure 10 we show time traces of both kinetic and available potential energy for the 60-year integration of EXP6. As in our previous model runs (first described in Part 1) in which outcropping of the thin upper layers occurs, a steady state is not attained in this experiment. (Note that the apparent difference in the amplitude of the fluctuations shown in Figures 10a and 10b results from a 1000-fold change in the ordinate scale between the two panels.) A significantly longer integration time and extensive further analysis would be required to statistically characterize this interannual variability in the model results. For purposes of the present study, it is only necessary that we select for averaging a period of time in which the energy levels are reasonably steady and representative of the post-spinup years of the experiment. We choose the 14-year period consisting of years 43 through 56, and we construct both a mean circulation for the entire 14 years and a mean annual cycle derived by averaging the model results month-by-month through all 14 years. We first describe the annual mean circulation before examining the principal seasonal variations in the model results. \vskip 12pt \noindent {\large\bf A. \hspace{.4cm}The annual mean circulation for EXP6} \vskip 6pt \noindent {\large\bf \hspace{.4cm}a. \hspace{.2cm}General description} \vskip 6pt In Figure 11 we show the barotropic streamfunction for the mean circulation of years 43-56 of EXP6 together with velocity vectors averaged over the upper 100\,m for the same period. In comparing Figure 11a to Figure 9b (for EXP5), we note the effects of reduced lateral friction in EXP6, for example, in the increased maximum transport of the subtropical gyre and its extension into the Caribbean, in the more tightly packed western boundary streamlines, and in the southwestward extension of the subpolar gyre. Overall, the EXP6 streamlines are rougher than those of EXP5 (Figure 9) and reflect the contours of topography (Figure 1) to a greater extent, particularly in the Gulf and Caribbean basins. Sign changes in the curl of the wind stress field (Figure 2a) dictate the pattern of alternating gyres seen in Figure 11. In the cyclonic subpolar gyre, the model Labrador Current flows to the south along the western boundary and continues down the coast along the eastern edge of the Grand Banks. In the anticyclonic subtropical gyre, the model's broad western boundary current separates from the coast at approximately the latitude of Cape Hatteras, and thereafter divides into two branches, one abutting the recirculating subpolar gyre and the other turning to the east and southeast before spreading into a weak clockwise flow around the Sargasso Sea. Southwestward velocities delineate a western boundary current recirculation cell east of Florida, centered on the region of maximum negative wind stress curl (Figure 2a) as in {\em Munk's} [1950] experiment. This is the expected result in a model of coarse horizontal mesh and high lateral friction, in contrast to an inertial recirculation in the northwest corner of the gyre such as that found in {\em Bryan's} [1963] nonlinear study. The model subtropical gyre appears to be extended into the Gulf and Caribbean basins as an effect of the smoothed bottom topography, which affords much less impedance to the flow than is provided in reality by the Antilles islands and the Yucatan Peninsula. In the equatorial region, counterrotating gyres are seen to the north and south of $ 5^{\circ} $N, the approximate mean annual position of the Intertropical Convergence Zone. The North Brazil Coastal Current turns east into the North Equatorial Countercurrent along that latitude, and a region of upwelling is found throughout the eastern tropics in the model results. This qualitative description conforms to the classical view of the Atlantic circulation as presented by {\em Sverdrup et al}.\ [1942], {\em Mann} [1967], and others. A quantitative comparison to experimental results is most easily made in the subtropical western boundary current region, where both the maximum surface current speed (35\,cm\,s$^{-1}$) and transport (38\,Sv) seen in Figure 11 fall far short of those found, for example, by {\em Halkin and Rossby} [1985] and {\em Stommel et al.} [1978]. While lower surface currents in the model results are attributable to the use of coarse horizontal resolution and large lateral friction, the transport difference may be primarily due to the lack of eddy and thermohaline processes [{\em Holland and Rhines}, 1980] and to the absence in the model of an inertial recirculation in the northwest corner of the subtropical gyre, the region in which {\em Stommel et al.} [1978] found transport in excess of 100\,Sv. At $ 30^{\circ} $N, however, the model transport of 38\,Sv compares favorably to the 37\,Sv found by {\em Richardson et al.} [1969] across the Florida Current, a value close to that predicted by the Sverdrup relation at that latitude (Figure 2b). In the Gulf and Caribbean sectors, model surface currents are generally in the observed (clockwise) direction, exiting in a northerly direction to the east of Florida, although the maximum transport of 22\,Sv is less than the average 30\,Sv reported in each basin [{\em Gordon}, 1967; {\em Hofmann and Worley}, 1986]. As noted earlier for the case of EXP1, the maximum subpolar gyre transport of 26\,Sv in the model results is lower than the 40\,Sv inferred from the Sverdrup relation (Figure 2b). Results in the equatorial region, which has significant seasonal variability, are discussed at length in a later section. \vskip 12pt \noindent {\large\bf \hspace{.4cm}b. \hspace{.2cm}Baroclinic structure} \vskip 6pt As the solution of an isopycnic coordinate model calculation evolves, its baroclinic structure remains easily defined in terms of the reshaping of initially flat layer interfaces and the related mass structure of each distinct layer. In general, the interface below each layer responds to the forcing which is imparted by the curl of the wind stress to one or more of the layers above the interface and subsequently transmitted to lower layers by means of hydrostatic pressure forces. Positive wind stress curl, as in the subpolar gyre, tends to raise an interface above its initial position, while negative curl leads to the depression of an interface, as in the subtropical gyre. Near the equator, where the magnitude of Ekman transport can become significant due to its inverse relation to the Coriolis parameter, surface convergence of fluid into a layer (or divergence from the layer) can contribute to seasonal changes in the thickness of the layer. In Figure 12 we show selected layer interface depth maps and individual layer streamfunctions for the annual mean solution of EXP6. Interface 1 (Figure 12a), initially at 100\,m depth, outcrops in the core of the subpolar gyre, approaches the surface in the eastern equatorial region, and reaches a maximum depth of near 500\,m within the subtropical gyre. The depth structure of the second layer interface (not shown) is similar to that of the first. On both of these interfaces, a sharp gradient in layer depth exists along the path of the model Gulf Stream, evidence of the tilting and packing of isopycnal surfaces associated with the frontal structure of the boundary current. We compare this structure to that found in {\em Parsons'} [1969] model, in which large layer depth gradients normal to the line of boundary current surfacing provide the pressure gradient force necessary to maintain the intense separated current. The position of that surfacing line is found by Parsons to depend primarily on integral properties of the applied wind stress. Both interface 3 (Figure 12b) and interface 4 (not shown) rise steeply in the subpolar gyre and bend downward in the subtropical gyre. They remain relatively flat throughout the tropics where, as a consequence of rapid baroclinic adjustment, interior velocities in the deepest three layers of the model solution are negligible. The mass transport streamfunction for layer 1 (Figure 12c) is confined mainly to the subtropical gyre due to the thinning of the layer elsewhere, as seen in Figure 12a. The intense gyre circulation in this layer, accounting for 31 of the 38\,Sv maximum transport seen in Figure 11a, may be attributed to the absence of eddy processes and interfacial friction from our calculation. The strength of the layer 1 circulation allows for only weak evidence of the subtropical gyre at the depth level of the second and third layers (Figures 12d and 12e). In regions outside of the subtropics, these layers are of sufficient thickness that even small velocities are able to produce discernible transport patterns, specifically the tropical gyres in layer 2 and the subpolar gyre in layer 3. The layer 4 streamfunction (not shown) resembles that of layer 3; as noted above, in these deeper layers there is essentially no flow in the tropics (nor in the Gulf and Caribbean basins), but a significant circulation exists in the subpolar region as the layer interfaces are pulled toward the surface under positive wind stress curl. Finally, as in the case of EXP4 and EXP5, the streamfunction for layer 5 (not shown) indicates maximum transport within the layer of less than 2\,Sv. Because the model is driven by wind alone, our results include neither the North Atlantic deep western boundary current [{\em Stommel and Arons}, 1960] nor the near-surface poleward flows [{\em Roemmich and Wunsch}, 1985] which complete the vertical-meridional cell of the Atlantic circulation. \vskip 12pt \noindent {\large\bf \hspace{.4cm}c. \hspace{.2cm}Potential vorticity} \vskip 6pt In the absence of wind-induced or viscous stress curls, potential vorticity is a conserved quantity which follows the motion in the manner of a passive tracer and can thus provide information about flow paths in the model ocean interior. Here we define potential vorticity as $ Q \equiv (\zeta + f)/\Delta z $, where $ \zeta$ is relative vorticity and $\Delta z$ is the thickness of a layer expressed in meters. In the first and second layers (not shown), wind-forced over much of their extent, relatively homogeneous {\em Q} is found only in the maximum-thickness region of the subtropical gyre. A strong gradient of {\em Q} indicates the thinning of these layers along the northern and eastern rim of the gyre, and, in the case of layer 1, in the eastern equatorial region. A strong gradient is also seen in the subpolar gyre region of layer 3 (Figure 13a) where wind curl forcing provides an external {\em Q} source, pulling the layer's lower interface toward the surface. The strong sidewall friction applied throughout the depth of this layer in response to velocity shear within the western boundary current provides an additional source of {\em Q}, visible in Figure 13a as a tongue of high potential vorticity diffusing into the interior from the boundary north of $30^{\circ}$N. {\em Huang} [1988] points out that strong convective adjustment apparently precludes the existence of such high {\em Q} tongues in the real ocean, where thermally driven processes play a vital role in determining the distribution of potential vorticity. Both in layer 3 and in layer 4 (Figure 13b), the tropics are characterized by {\em Q} contours which approximate latitude lines, consistent with the relatively motionless state of these layers and their correspondingly flat upper and lower interfaces in that area (Figure 12). North of the tropics in layer 4, and particularly in the subpolar gyre, winding {\em Q} lines indicate regions of flow induced by pressure forcing from layers above in which the circulation is directly driven by the wind. The strong gradient of {\em Q} visible along the continental slope in layers 3 and 4 indicates regions where layer thickness approaches zero due to the steep topography. In order to conserve its high potential vorticity, water which is initially on the slope must remain in that region. A column of water attempting to move off the slope will be subjected to stretching, and the consequent generation of positive relative vorticity (i.e., counterclockwise rotation) will drive the fluid back onto the slope. Thus the high {\em Q} water remains confined to the topographic slope and can spread into the deep interior only if its potential vorticity is modified by friction or by the effect of the wind. We discuss here those aspects of potential vorticity dynamics which are related to forcing by wind alone. Isolines of {\em Q}, parallel to latitude circles in a resting fluid, are reshaped by the wind and, under steady-state conditions, coincide with paths of flow along surfaces of constant density wherever stresses are negligible. Potential vorticity injected at boundaries or along isopycnal outcrops is advected into the interior along flow lines and dissipated by friction. From the perspective of the ventilated thermocline theory of {\em Luyten et al}.\ [1983] (hereafter, LPS), motion can be induced in layers below the surface, sheltered from direct forcing, wherever ventilated fluid (i.e., fluid exposed to the wind at the surface) is subducted into them. The three {\em Q} regimes on constant density surfaces postulated by LPS -- the ventilated, pool and shadow zones -- have been analyzed in numerical models by {\em Cox and Bryan} [1984] and {\em Cox} [1985]. Their solutions illustrate the ventilated zone in the eastern region of a subtropical gyre, emanating from the outcropping line of a density surface, the unventilated pool zone surrounded by closed flow lines within the core of the gyre, and, toward the equator, the unventilated shadow zone where velocities are near zero. Analogous zones of potential vorticity may be seen in the contours shown in Figure 13a for layer 3, which differs from the layers above it in the sense that it is wind-forced only within a definable region north of the subtropical gyre (as indicated by the dashed line), i.e., the region in which the interface above the layer reaches depths shallower than 100\,m. Ventilation of the eastern part of the gyre is evidenced by the lines of flow (Figure 12e) and contours of {\em Q} which extend into that zone from the wind-forced region. South of $\sim 10^{\circ}$N, the layer's shadow zone conforms to the LPS description of an unventilated region of low potential vorticity, lying within the lowest ventilated layer of the subtropical gyre, and bounded below by a flat isopycnal surface (Figure 12b). The unventilated pool zone corresponding to that of the {\em Cox and Bryan} [1984] model is seen in our case near the western boundary, where {\em Q} contours thread into and out of the separating boundary current. (Note that this description differs from that of {\em Rhines and Young} [1982], who define the pool zone as a plateau of homogeneous potential vorticity within the subtropical gyre.) We have already commented on the advection of high {\em Q} into the gyre by the separating western boundary current, which is absent from the LPS theory. {\em Ierley and Young} [1983] and {\em Huang} [1986b] note that this frictional regime may influence the {\em Q} distribution in the gyre interior as a consequence of the passage of streamlines through the viscous boundary layer. {\em McDowell et al}.\ [1982] present maps of North Atlantic potential vorticity on constant density surfaces which indicate uniform {\em Q} marking the core of the subtropical gyre to mid-thermocline level, while at deeper levels open contour lines reach into the interior from regions of outcropping. Below $\sigma_\theta = 27.0$, the {\em Q} contours in the McDowell et al.\ maps approximate latitude lines equatorward of the subtropical gyre and the isopycnal surfaces are relatively flat. We note similar features in our model results for layers 3 and 4 (Figures 12b, 13a, and 13b), for which $\sigma_\theta = 27.2$ and $27.7$ respectively. The results of our wind-driven calculation fail to include another prominent feature of the North Atlantic {\em Q} distribution which McDowell et al.\ describe, namely the low {\em Q} ($18^{\circ}$) water injected into the subtropical gyre as a result of deep winter convection at higher latitudes. \vskip 12pt \noindent {\large\bf B. \hspace{.4cm}Seasonal variations} \vskip 6pt In order to identify regions of the Atlantic basin in which the most significant seasonal variations in the EXP6 annual cycle occur, standard deviations from the EXP6 mean circulation were calculated both for the layer interface depths and for the layer and total streamfunctions. Two well-documented regions of Atlantic seasonal variability, namely the equatorial zone and the region of the Florida Current and Antilles Current [see, for example, {\em Philander and Pacanowski}, 1986a,b; {\em Anderson and Corry}, 1985a,b], are found to be the primary sites for such variations in our experiment as well. In the tropics, the seasonal cycle is primarily a baroclinic response to seasonal changes in the strength and position of the trade wind systems. The associated relocation of regions of positive and negative wind stress curl near the equator causes changes in the depth of isopycnal surfaces through Ekman pumping. Rapid equatorial thermocline adjustment also is made possible at annual period by the relatively high phase speed of Rossby waves in low latitudes. In contrast, the Antilles region seasonal variation occurs at latitudes where years or decades may be required for adjustment by baroclinic Rossby waves. At annual period the variation in this area is primarily barotropic and therefore strongly affected by topography. We examine first the Florida Current seasonal cycle in the EXP6 results and second the cycle in the equatorial region, and we will then discuss two additional manifestations of annual variation in the model results, namely the seasonal cycles of heat transport and of sea level. \vskip 12pt \noindent {\large\bf \hspace{.4cm}a. \hspace{.2cm}The Florida Current/Antilles Current seasonal cycle} \vskip 6pt The seasonal cycle of transport through the Florida Straits exhibits a summer maximum and fall minimum with amplitude $\pm$3\,Sv [{\em Schott and Zantopp}, 1985], as opposed to the winter maximum and $\pm$15\,Sv amplitude predicted by integrating the curl of monthly wind stress values at that latitude according to the the Sverdrup relation. {\em Anderson and Corry} [1985a] (hereafter, AC) conclude that the nontopographic Sverdrup balance should be expected to hold for the Florida Straits transport only for periods sufficiently long (years to decades) to allow compensation for the effects of topography by the passage of baroclinic Rossby waves across the Atlantic basin. At annual period, the response is essentially barotropic and therefore strongly modified by the blocking effect of topography. In Figure 14 we compare the seasonal cycle of transport at the latitude of Miami in the EXP6 results both to the AC results and to the Sverdrup transport at that latitude. The cycle of EXP6 transport integrated from the coast to $77^{\circ}$W (which can be considered as the eastern edge of the continental shelf) and the Florida Straits transport calculated by AC both exhibit a double maximum (summer and winter) as well as the fall minimum. The two-layer wind-driven AC model, with $1^{\circ}$ horizontal resolution, includes an island representing the Bahamas, so that the location of their Florida Straits is well defined. At that location they obtain a mean transport of 30.5\,Sv, in agreement with observations [{\em Niiler and Richardson}, 1973; {\em Schott and Zantopp}, 1985], while their calculated annual peak-to-peak variation of $\sim3.5$\,Sv is too small by roughly a factor of two. The amplitude and phase of the EXP6 annual transport cycle integrated to $77^{\circ}$W agree well with the AC Florida Straits cycle, although the EXP6 mean value is $\sim$4\,Sv less than that of AC. {\em Greatbatch and Goulding} [1989] (hereafter, GG), whose $ 1^{\circ}$-resolution barotropic model also employs smoothed bottom topography, report a similar Florida Current cycle in their results, i.e., agreement with observations in phase though not in amplitude. While the coarse resolution and lack of islands in our model make a direct comparison of Florida Straits transport calculations difficult, evidence of the blocking effect of topography can be seen here as well as in the results of AC. At annual period, a western boundary current having the seasonal transport cycle predicted by the Sverdrup relation is unable to ride up onto the continental shelf east of Florida. Further to the east, however, away from coastal topography, the EXP6 transport cycle (integrated to $73^{\circ}$W, Figure 14) agrees in phase with the Sverdrup transport cycle, both cycles exhibiting a primary maximum in winter and secondary maximum in summer. Both the AC model results and the EXP6 results also include a strong (order 10\,Sv) seasonal signal in the so-called Antilles Current region, with transport to the north in winter and to the southeast in fall, agreeing in phase with the Sverdrup transport annual cycle. At annual period, when realistic bottom topography is included, this signal remains to the east of the Bahamas and cannot penetrate to the Florida Straits. Observations in this region indicate only a weak northward transport with mean value 4\,Sv [{\em Olson et al}., 1984; {\em Schott and Zantopp}, 1985]; no such mean Antilles Current transport is found in the AC results, nor in EXP6. The seasonal variations which AC display in their Figure 3 therefore represent total transport in this region. In Figure 15 we display monthly minus annual mean transport for February and October from the EXP6 results, showing the reversal in transport direction which corresponds to the Antilles Current cycle calculated by AC. The seasonal Antilles variation is also clearly seen in Figure 5 of GG. {\em Schott et al}.\ [1988] point out that transport of 10-12\,Sv in this region, as the models predict, could exist as a result of wide-area barotropic currents of less than 1\,cm\,s$^{-1}$ which have not yet been measured. The Antilles Current seasonal cycle may be attributable to changes in the wind stress curl which varies, east of the Bahamas, from strong negative curl in winter to positive curl for two months in early fall. A rapid (barotropic) response to such changes is possible in that region, which lies away from continental shelf topography. The Florida Current cycle, however, is more likely forced in the vicinity of the Straits, by one or more of several mechanisms: along-channel wind stress [{\em Lee and Williams}, 1988], passage of a coastal Kelvin wave over topography [AC], or upstream wind stress curl [{\em Schott and Zantopp}, 1985]. \vskip 12pt \noindent {\large\bf \hspace{.4cm}b. \hspace{.2cm}The seasonal cycle in the equatorial ocean} \vskip 6pt The seasonal cycle in the equatorial Atlantic owes its existence to the annual migration of the trade wind systems. In the early part of the year, when the southeast trades are weak, the Intertropical Convergence Zone (ITCZ) lies near the equator and winds along that line are slack. Equatorial surface currents are weak and to the west, with the North Brazil Coastal Current (NBCC) flowing to the northwest across the equator, along the coast of Brazil and then into the Caribbean. In the western basin, along the equator, the thermocline is at its shallowest depth in April. Beginning in May, the southeast trades intensify and the ITCZ moves to the north. The NBCC veers offshore at $ \sim5^{\circ}$N, under the influence of negative wind stress curl to the south of the ITCZ, and feeds the eastward North Equatorial Countercurrent (NECC) from June through late fall. Along the equator, the thermocline deepens in the western basin and shoals in the east as the trade winds strengthen. The deepening in the west continues until the winds relax in September, but the shoaling in the east ceases in July, when the westward winds weaken to the east of $ 30^{\circ}$W [{\em Busalacchi and Picaut}, 1983]. The ocean response in the equatorial Atlantic is largely in phase with the seasonal changes in forcing. To the west of $ 30^{\circ}$W, vertical excursions of the thermocline along the equator are highly correlated with variations in the intensity of the local westward winds, and the dominant signal is annual. East of $30^{\circ}$W, a double maximum in the east-west component of the wind stress contributes to a semiannual oscillation in thermocline depth, with maxima both in October and February. The February maximum occurs just prior to the shoaling of the thermocline in the east which is associated with the spring strengthening of the southeast trades in the west, while the October maximum is attributed to the intensity at that time of an eastward component in the winds near the African coast. The primitive equation model of {\em Philander and Pacanowski} [1986a] (hereafter, PPa), which is forced by the mean monthly wind stress data of {\em Hellerman and Rosenstein} [1983] and by heat flux at the ocean surface, reproduces this vertical movement of the equatorial thermocline as well as the pronounced seasonal cycle of the NECC. That current is intense and eastward in the PPa model results from May to December along $6^{\circ}$N, while from January to April the current reverses in the surface layers and the flow is westward, west of $ 30^{\circ}$W. {\em Garzoli and Katz} [1983], analyzing the seasonal cycle of the NECC on the basis of ship drift data and hydrographic records, report eastward velocities year round between $3^{\circ}$N and $10^{\circ}$N, east of $25^{\circ}$W, while to the west of that meridian the current reverses in spring. The seasonal variation of the NECC in the western basin is associated with a change in direction of the meridional tilt of the thermocline across $6^{\circ}$N. In October, the thermocline shoals to the north of that latitude, and deepens to the south, in geostrophic balance with the eastward-flowing NECC. In April, data show that the thermocline tilt across $6^{\circ}$N is relatively shallow in comparison to that in October, and is in the opposite direction, that is upward to the south. In the eastern basin, where the NECC is present year round, the thermocline across $6^{\circ}$N tilts upward to the north during all seasons, being deepest in October and shallowest in April. The spring deepening of the thermocline at $10^{\circ}$N in the western basin can be explained as a local Ekman response to an anticyclonic perturbation to the wind stress curl, related to the southward displacement of the ITCZ. In fall, the ITCZ is shifted to the north of its mean position and the thermocline then deepens at $3^{\circ}$N, again in response to anticyclonic forcing [{\em Busalacchi and Picaut}, 1983; {\em McCreary et al.}, 1984]. Because the curl of the wind stress is the driving force behind these seasonal oscillations, the results of our wind-driven experiment EXP6 correspond well to those of PPa and to observations. The annual cycle of zonal velocities across the EXP6 model basin at $6^{\circ}$N, the approximate axis of the NECC, agrees qualitatively with the cycle calculated by PPa, although the NECC velocities in EXP6 are smaller by a factor of three than those in PPa owing to the coarse horizontal resolution and large lateral friction of EXP6 and to the distribution of the wind forcing over the upper 100\,m of the water column. However, {\em Richardson and Philander} [1987] report that the PPa model velocities, particularly in the western basin, are approximately twice as large as those indicated by ship drift data from the region. The seasonal cycle of the NECC is also reflected in the tilt of the first layer interface across the axis of that current in the EXP6 results. In Figure 16, solid (dashed) lines indicate that the interface shoals to the north (south) of the NECC axis. The figure thus reveals the variation in interface tilt which occurs during the year in the western basin, but not in the east, in geostrophic balance with changes in the direction and strength of the flow along $ \sim6^{\circ}$N [{\em Garzoli and Katz}, 1983]. The most pronounced shoaling of the interface to the north of the NECC axis occurs in October in the western basin, lagging the time of maximum eastward velocities by approximately two months. This time lag is in agreement with observations [{\em Richardson and McKee}, 1984], which also indicate a more extreme interface tilt, and correspondingly stronger NECC velocities, than are found in EXP6. Along the equator, seasonal variations in the depth of the second layer interface across the EXP6 model basin conform to the variations in thermocline depth discussed by PPa. In the western basin, the spring minimum and fall maximum in depth are visible in Figure 17, while in the east, a semiannual cycle is suggested by local maxima in both March and November. Although the second interface in EXP6 is on average 200\,m deeper across the basin than the PPa model thermocline, it exhibits these wind-forced variations in depth due to the thinning of the uppermost layer along the equator and the consequent extension of wind curl forcing into the second layer. While the NECC seasonal cycle appears to be well reproduced in EXP6, the broader configuration of surface currents in the equatorial region shows areas of disagreement with observations and with the results of PPa. In October (Figure 18a), a weak coastal NBCC turns into the strong eastward NECC, and flow along the equator in the western basin is to the west, as in PPa. However in April (Figure 18b), when PPa report continuing westward velocities (though weakened in comparison to October), equatorial currents in our results are eastward, and the NBCC fails to flow to the northwest across the equator. The lack of westward equatorial surface flow during part of the year, as well as the absence of an Equatorial Undercurrent which balances that flow, may be due to the exclusion of vertical momentum mixing from our experiment. {\em McCreary} [1981] points out that in the absence of any vertical redistribution of momentum, equatorial flows are confined to the surface layers, with zonal pressure gradients balancing the driving wind in the case of steady forcing. In addition, it is difficult to isolate the thin undercurrent, which is $\sim$100\,m thick in the central equatorial Atlantic [{\em Moore et al}., 1978], within the coarse vertical structure of our layered model. \vskip 12pt \noindent {\large\bf \hspace{.4cm}c. \hspace{.2cm}Seasonal variations in heat transport} \vskip 6pt Poleward of approximately $15^{\circ}$ latitude, seasonal variations in heat storage in the Atlantic are primarily due to changes in the flux of heat across the air-sea interface. In the equatorial Atlantic, however, seasonal variations in surface heat gain are relatively small and cannot account for the amplitude of the measured annual cycle of ocean heat content in that region. That cycle is due instead to the horizontal redistribution of heat within the fluid, which is determined principally by the divergence of the vertically integrated oceanic transport of heat [{\em Merle}, 1980]. The redistribution is affected by the relatively rapid adjustment of the baroclinic velocity field to wind forcing in low latitudes. In addition, the Ekman transport, which is modified by seasonal variations in the wind field, is inversely related to the Coriolis parameter and thus increases (for a given wind stress) with proximity to the equator. Our model does not incorporate surface heat fluxes and therefore exhibits no mean annual transport of heat. However, model seasonal variations in heat transport in the equatorial region do exist and can be calculated. These variations are directly related to the wind-forced seasonal oscillations in equatorial currents and thermocline depth which are described in the preceding section. A deepening thermocline is associated with the convergence of heat transport and consequent storage of heat; a shoaling thermocline conversely implies the divergence of heat transport and release of heat [{\em Merle}, 1980]. In order to carry out a heat transport calculation for the EXP6 results, a temperature must be assigned to each of the five model layers. Since the density (and, by inference, the temperature) of each layer remains constant throughout the model integration, it would be reasonable to use those temperatures which correspond to the initial, at-rest depth range of each layer over the entire model basin. For the purpose of the heat transport calculation, however, a more realistic choice of temperatures is one based upon the average depth range of each layer in the EXP6 solution, in the region of the basin for which the calculation is to be performed. The temperature values thus determined are 25.2, 16.5, 8.3, 4.5 and $3.0^{\circ}$C, for layers of depth 0-41\,m, 41-322\,m, 322-689\,m, 689-1409\,m, and 1409\,m-bottom, respectively. These values are the average potential temperatures for the stated depth intervals in the region $20^{\circ}$S-$20^{\circ}$N, as determined from the {\em Levitus} [1982] atlas data for the Atlantic. For simplicity in the heat transport calculation, the layer temperatures are held constant throughout the year. {\em Levitus} [1984] reports seasonal ocean temperature variations of less than $1.5^{\circ}$ in the equatorial region, and the calculation was found to be relatively insensitive to changes of that order. The vertically and zonally integrated net meridional heat transport across a circle of latitude is defined as: \[ c_p \: g^{-1} \: \int_\lambda \int_p \: v \: {\theta} \: dp \: d\lambda \] where $ \lambda $ is longitude, $ p $ is pressure, $ c_p $ is heat capacity, $ g $ is the acceleration due to gravity, and $ v,{\theta}$ (functions of $ p $) are respectively the meridional velocity and potential temperature in each model layer. The integration in $ \lambda $ is performed across the model basin at a fixed latitude and the integration in $ p $ is performed over the depth of the fluid at a given longitude. The calculated seasonal cycle of heat transport for EXP6 is shown in Figure 19a, and the corresponding figure from the model of {\em Philander and Pacanowski} [1986b] (hereafter, PPb) is shown in Figure 19b. The heat transport values given by the two figures cannot be compared directly, since the PPb model is forced by surface heat flux and by Newtonian damping towards climatology outside the tropics, as well as by mean monthly wind stress data, and therefore exhibits a mean annual transport of heat which is not found in our wind-driven model results. The mean annual transport is northward in both hemispheres [{\em Bryan}, 1982] and varies over the PPb model region from approximately $0.6 \times 10^{15}$ to $1.0 \times 10^{15}$\,W. The two figures are therefore intended for qualitative comparison only. The notable feature in both Figure 19a and Figure 19b is a change in the direction of heat transport during the year across $\sim8^{\circ}$N. The PPb results indicate southward heat transport across that latitude in September, corresponding to a deepening of the model thermocline to the south of $8^{\circ}$N. The opposite situation, that is a deepening thermocline and release of heat to the north of $8^{\circ}$N, prevails in the PPb results for winter and spring. This change in heat transport direction across $8^{\circ}$N is associated with the seasonal cycle of the North Equatorial Countercurrent (NECC), which we have noted is well reproduced in the EXP6 results. The seasonally reversing tilt of the first layer interface across the EXP6 NECC axis, as described in the preceding section (Figure 16), is reflected in the seasonal reversal of heat transport direction which is apparent in Figure 19a. (Seasonal variations in the depth of layer interfaces below the first were found to have relatively minor impact on the pattern displayed in that figure.) The interface tilt reverses during the year in the western equatorial region, where {\em Merle} [1980] finds large seasonal variability in ocean heat content. The EXP6 results are also qualitatively consistent with those of {\em Sarmiento} [1986], whose model employs the wind data of {\em Hellerman and Rosenstein} [1983] (as do EXP6 and PPb) in addition to thermohaline forcing. Sarmiento finds small values of southward heat transport across $ \sim8^{\circ}$N in August-September, when eastward NECC velocities are at a maximum and the thermocline shoals to the north of that latitude. In winter and spring, the North Brazil Coastal Current is directed to the northwest across the equator, and the NECC is replaced by westward surface flow in the western equatorial region. The Sarmiento results then indicate northward heat transport across $8^{\circ}$N. The results of EXP6, PPb and {\em Sarmiento} [1986] are all indicative of the strong connection among seasonal oscillations of the NECC (and the associated thermocline tilt), the seasonal cycle of heat transport in the equatorial region, and seasonal variations in forcing by the curl of the wind stress. \vskip 12pt \noindent {\large\bf \hspace{.4cm}d. \hspace{.2cm}Seasonal variations in sea surface height} \vskip 6pt Because of the split barotropic-baroclinic calculation described in Part 1, sea surface height is an explicitly predicted variable of our model. The model-generated surface elevation is shown in Figure 20 for the EXP6 annual mean. A sharp gradient in sea level across the model Gulf Stream is apparent, with a maximum rise of less than 1\,m from the coast to the subtropical gyre interior, somewhat lower than values given by {\em Stommel et al}.\ [1978] and {\em Wunsch} [1981]. Relative to the subtropical lens, the surface is depressed over the subpolar gyre and along the eastern perimeter of the subtropical gyre. The model sea level rises to the west along the equator, but the east-west basinwide difference in sea surface height is $\sim12$\,cm in the EXP6 annual mean solution (Figure 20) as opposed to the average 20\,cm difference reported by {\em Merle and Arnault} [1985]. As discussed earlier in connection with Figure 17, the wind stress along the model equator is distributed through the first and second layers, and a weakened pressure gradient force balances that stress. The reduction in the magnitude of sea level tilt along the model equator, in comparison to observations, is consistent with the reduction in that pressure gradient force. Seasonal fluctuations in sea level on a global scale have been found to be primarily isostatic, i.e., strongly correlated with seasonal changes in atmospheric pressure and expansion or contraction of the water column due to surface heating or cooling [{\em Pattullo et al}., 1955; {\em Gill and Niiler}, 1973]. These effects cannot be studied here, since our model is neither coupled to the atmosphere nor thermodynamically forced. In regions of strong currents, however, seasonal oscillations in sea surface height reflect the variability of the surface geostrophic flow and provide an indirect measure of current velocity [{\em Maul et al}., 1985]. Two such areas which can be examined in the EXP6 results are the Florida Current/Gulf Stream system and the region spanned by the North Equatorial Countercurrent. {\em Maul et al}.\ [1985] find that seasonal fluctuations in surface current velocity and transport in the Florida Straits are correlated both with sea level differences across the Straits and with sea level variations on the western side of the Straits alone. The most extreme tilt of the sea surface across the Florida Current from Miami to Cat Cay, as well as the minimum sea surface height at Miami, are found in July, the time of maximum transport through the Straits. Conversely, at the time of minimum transport in October, Miami sea level reaches its annual maximum and the cross-stream surface tilt is then moderated. The steric effect is uniform across the channel [{\em Blaha}, 1984]; thus the oscillations in sea surface height difference across the Straits arise simply from a geostrophic balance with the northward-flowing current through the Straits. At annual period, Miami sea level (when corrected for steric effects) contains all the variation apparent in the cross-stream difference, while the Cat Cay side shows almost no response [{\em Schott and Zantopp}, 1985]. Values of sea level at Miami, and cross-stream sea level differences between Miami and Cat Cay, are shown in Figure 21a. The EXP6 results, interpolated at those locations from the calculated model sea level, are compared in this figure to data obtained from F. Schott and R. Zantopp (personal communication, 1987); the annual means have been removed from both sets of values. The annual cycles agree well in phase, but the amplitude of the model seasonal sea level variations in this region is reduced in comparison to observations. We noted in an earlier section that the amplitude of the seasonal cycle of Florida Current transport in EXP6 is similarly less than observed. {\em Blaha} [1984] finds that the annual cycle of summer minimum/fall maximum in sea surface height, as observed at Miami, becomes progressively more semiannual in character as measured along the coast to Norfolk. Figure 21b compares the Blaha data (which has been corrected for atmospheric and steric effects) to results from EXP6 at three locations along the coast. Again the agreement in phase is good, while the model values are reduced in amplitude. A similar result is reported by GG, who also compare their model sea level to the Blaha coastal records. The summer minimum in sea level observed at Miami is found as far north as Charleston in the Blaha data, implying that the summer maximum seen in Florida Straits transport continues to exist in the annual cycle of the Gulf Stream well north of the Florida Channel. The seasonal cycle of sea level difference across the Gulf Stream, east of Cape Hatteras, has been investigated by {\em Fu et al}.\ [1987] on the basis of altimetric measurements. A maximum in sea level difference across the current (south minus north) is found in April, corresponding to the Gulf Stream transport maximum found in this region in late winter and early spring. The transport maximum may be related to winter cooling which deepens the subtropical thermocline and thus strengthens the baroclinicity of the current [{\em Worthington}, 1972]. However, the seasonal variability of wind stress curl over the North Atlantic also affects the annual cycle of Gulf Stream transport and sea level difference. Fu et al.\ show that the seasonal cycle of wind stress curl averaged along $35^{\circ}$N qualitatively agrees with the measured seasonal cycle of sea level difference across the Gulf Stream, with sea level lagging 2-3 months behind the wind. Figure 21c compares the annual variation of sea level difference across the Gulf Stream (south minus north) in the EXP6 results, averaged over $70^{\circ}-75^{\circ}$W, to the results of Fu et al.\ for the same region. The model qualitatively reproduces the spring maximum seen in the data, but fails to reproduce the distinct December minimum. GG find that they are unable to duplicate the Fu et al.\ cycle, either in phase or in amplitude, an inability which they attribute to the absence of baroclinic processes in their model. Thermally driven processes, which our calculation lacks, may be important in this region as well. In the equatorial Atlantic, the mean sea surface slopes upward to the west and exhibits zonally oriented ridges and troughs which are associated with the equatorial system of currents (Figure 20). Using data obtained from inverted echo sounders deployed in the equatorial Atlantic, {\em Katz} [1987] analyzes seasonal variations of the sea surface ridge and trough corresponding to the North Equatorial Countercurrent (NECC). As noted earlier, the seasonal variation of this current in the western equatorial region is associated with a change in direction of the meridional tilt of the thermocline across $6^{\circ}$N; in the eastern basin, no such change is found [{\em Garzoli and Katz}, 1983] (Figure 16). {\em Katz} [1987] finds the analogous cycle in sea surface height across the NECC axis, with the highest sea level (and deepest thermocline) found at $3^{\circ}$N in the western basin in October. The ridge at $3^{\circ}$N becomes a trough for intervals each spring, while in the eastern basin it flattens out almost entirely. A figure displaying the annual cycle of sea surface tilt across the NECC in EXP6 for the width of the basin is virtually identical in form to Figure 16 and therefore is not shown. As in the case of NECC velocities and the associated interface tilt (Figure 16), the model sea level differences across the NECC are smaller in magnitude than those found in the observations but are in qualitative agreement with {\em Katz's} [1987] results. \vskip 12pt \noindent {\large\bf V. \hspace{.4cm}SUMMARY AND OUTLOOK} \vskip 6pt We have presented the results of a series of experiments which progress from the simple calculations of Part 1 to a five-layer, seasonally forced model run incorporating geometry and bottom topography appropriate to the Atlantic basin. Our emphasis has been on the step-by-step development of an isopycnic coordinate model capable of simulating major features of the wind-driven circulation of the North and Equatorial Atlantic. The preliminary experiments, EXP1--EXP5, are designed to establish consistency with the results of Part 1 as the Atlantic framework is incorporated and to document the variability in the model solution as lateral friction is decreased, as seasonal forcing is introduced, and as the number of layers is increased. From these experiments we find: (1) the interior solution is governed primarily by linear Sverdrup dynamics, with the magnitude of the frictional coefficient determining boundary layer width as well as smoothness of appearance; (2) the seasonally forced runs, while exhibiting time-dependent transient motions, produce an annual mean solution which agrees with the solution of steadily forced runs; and (3) the five-layer experiments in which the upper layers outcrop do not attain a steady state, but the residual circulation in the lowest layer does not contribute significantly to the total transport. The primary Atlantic experiment, EXP6, is a seasonally forced, five-layer simulation employing deformation-dependent lateral friction. The annual mean solution exhibits a subtropical gyre/Gulf Stream system which includes a broad western boundary current separating from the coast near Cape Hatteras and a western boundary recirculation regime of the linear (Munk) type. Maximum interior transport conforms to that predicted by the Sverdrup relation but transport in the model Gulf Stream region falls far short of that observed, owing to the absence of inertial, eddy and thermohaline effects in our calculation. The subtropical gyre curves into the Gulf of Mexico and Caribbean basins where the model solution is not expected to be realistic, given the absence of the Antilles Islands from the smoothed bottom topography. The subpolar gyre in the EXP6 annual mean solution is truncated by an artificial wall at the northern boundary and exhibits maximum transport less than the Sverdrup transport. An appendage of this gyre represents a weak Labrador Current in the model results, flowing to the southwest along the North American coast. In the tropics, counterrotating gyres surround the approximate mean annual position of the Intertropical Convergence Zone. The vertical structure of the EXP6 annual mean shows the distortion of layer interfaces resulting from direct wind curl forcing and hydrostatic pressure forcing transmitted from layers above. Potential vorticity analysis approaches this reshaping of isopycnal surfaces from the standpoint of surface conditioning and ventilation of constant-density layers. Examination of the vertical structure makes clear the absence in our results of the thermohaline-driven vertical-meridional circulation of the Atlantic. Seasonal variations in the EXP6 results are found primarily in the Florida Current region and in the tropics. The model reproduces, although with reduced amplitude, the seasonal cycle of Florida Current transport as well as the Antilles Current cycle reported in the model of {\em Anderson and Corry} [1985a]. In the equatorial region, seasonal variations of the North Equatorial Countercurrent (NECC) and of the thermocline tilt along the equator are reproduced but surface currents along the equator are unrealistic, most likely due to the absence of vertical mixing, and the model fails to reproduce the Equatorial Undercurrent. Although our model exhibits no mean annual transport of heat, the seasonal variations in heat transport in the tropics calculated from the EXP6 results compare well to those calculated by {\em Philander and Pacanowski} [1986b], whose model is forced by surface heat fluxes as well as by wind. Finally, we examine seasonal variations in sea surface height, an explicitly predicted variable of the model, at various locations on the eastern U. S. coast, across the Florida Current/Gulf Stream and across the NECC. In most of these cases, the calculated phase of the annual cycle agrees with observations while the amplitude, in common with that of the Florida Current annual cycle, is consistently low. The $ 1^{\circ}$-resolution models of {\em Anderson and Corry} [1985a] and of {\em Greatbatch and Goulding} [1989] report a similar outcome, leading us to suggest that the discrepancy between observations and model results deserves further study in a framework of greater horizontal resolution. We believe that these results provide evidence of the potential value of an isopycnic coordinate model in large-scale oceanographic studies. This wind-driven coarse resolution model is to be considered the prototype for a model capable of representing the major features of the three-dimensional oceanic circulation. To expand this prototype, we might first consider an improvement in the horizontal resolution from $2^{\circ}$ to $1^{\circ}$. This would allow the inclusion of islands in the bottom topography, as for example in the {\em Anderson and Corry} [1985a] model. The location of the Florida Straits would then be reasonably well defined, and the solution in the Gulf and Caribbean basins would be far more realistic. In order to include the effects of mesoscale eddies in the solution, however, a further four- to fivefold refinement in the horizontal grid spacing would be required. A calculation of this kind would be 100 times as costly as the present experiment. We intend instead to modify the model, as it now exists, by incorporating a temperature/salinity version of the mixed-layer algorithm proposed by {\em Bleck et al}.\ [1989] to accommodate thermodynamic forcing. We then expect the model to reproduce known features of the thermohaline-driven Atlantic circulation such as the deep western boundary current (and its interaction with topography), the mean annual transport of heat, the buildup and destruction of the seasonal thermocline and the ventilation of the permanent thermocline. The development of a model in isopycnic coordinates which incorporates both wind and thermohaline forcing will provide a valuable instrument for the study of seasonal and interannual variability of the ocean climate and its interaction with the atmosphere. \vskip 3cm \parindent 0pt {\em Acknowledgments}: Support for this work by the National Science Foundation under grants OCE-8600593 and OCE-8812185 is gratefully acknowledged. Computations were carried out at the National Center for Atmospheric Research, sponsored by the National Science Foundation. Computer time was also made available in 1985 by the Geophysical Fluid Dynamics Laboratory, Princeton. We thank Claes G. H. Rooth, Donald B. Olson, Friedrich Schott, and Julian P. McCreary, Jr., for their continuing interest in this work. \end{document}