1. Introduction

A well-observed feature of the Antarctic Circumpolar Current (ACC) is the zonation of the flow field. In Drake Passage this zonation takes the form of three distinct regions separated by transition zones associated with the ACC current cores (Nowlin and Klinck 1986; see also Nowlin and Clifford 1982). There is some evidence that suggests that the zonation is circumpolar in extent, although possibly varying in the number of zones and their widths (Hofmann 1985; Orsi et al. 1995).

A banded structure in the velocity field is also seen in the results from numerical models of the Southern Ocean. The numerical solutions obtained by the Fine Resolution Antarctic Model (FRAM Group 1991) exhibit a flow structure that is dominated along the track of the ACC by elongated filaments of high velocity embedded in slower moving fluid. This is illustrated in Fig. 1, a plot of the eastward component of velocity u versus depth at 270°E. The jet structure is coherent down to depths of about 1500 m. Typical velocities are O(10 cm s⁻¹). The meridional scale of variation is of order a few hundred kilometers. The time dependent behavior of the multiple jets is illustrated in Fig. 2, which shows a contour plot of instantaneous near-surface (32.5 m) u velocity against time and latitude (i.e., a Hovmöller diagram) at 270°E. The flow consists of a number of jets correlated over time and space. The jets meander to the north and south (timescale ∼200 days). Some peter out or amalgamate with their neighbors and some are spontaneously generated. The Los Alamos Parallel Ocean Program (POP) model (Maltrud et al. 1998) displays similar behavior and will be compared with FRAM in section 4.

The formation of multiple jets in models of the ACC is reminiscent of results obtained by Rhines (1975), Williams (1978, applications to gas giant planets), and Pannetta (1993, applications to atmospheric jets) using quasigeostrophic models, which indicate that jet structure can arise from eddy–mean flow interaction. Rhines (1975) investigated the behavior of freely decaying turbulence on a barotropic β plane. Observing that the inverse cascade of energy to low wavenumbers in the turbulent flow would be halted when the eddy spatial scale reaches the resonant Rossby wave scale, he argued that such a flow organizes itself into zonal jets on long timescales. This yields a meridional scale for the long-term mean flow:

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where k_c is the dominant meridional wavenumber (corresponding wavelength λ_c), β is the (linear) variation of Coriolis parameter with latitude, and U is a characteristic eddy velocity.

Using a quasigeostrophic model with an imposed spatially uniform vertical shear and domains many times larger than the Rossby radius, Pannetta (1993) obtained time-mean flows consisting of several zonal jets where the scale of meridional variation was governed by (1).

Subsequently, Treguier and Pannetta (1994) discovered that zonation can arise naturally in a wind-forced baroclinic quasigeostrophic model provided the meridional extent of the domain is sufficient and the wind stress is of adequate strength. Meridionally oriented ridges or random topography with a specified height spectrum disrupt the jet structure and change the dynamical regime to one where the flow is controlled by topographically generated standing eddies. Transient eddies arising from baroclinic instability of the mean flow are believed to be responsible for maintaining the zonation in flat-bottomed cases, although the mechanism whereby this is achieved is still unclear. Clearly, it is also important to establish the role of lateral boundaries in restricting the number and spacing of multiple jets.

In this paper we consider the effect of meridional gradients of bottom topography. A meridional gradient of topography can act as a physical analog of the β effect, for example, a linear topographic slope in the meridional (y) direction induces planetary waves in a similar manner to the β effect, with the role of β played by the (constant) topographic slope α, scaled by the Coriolis parameter f₀ divided by the total ocean depth H. Like the β effect, a meridional topographic gradient inhibits meridional movement of water and thus, we suggest, stabilizes any evolving jets and additionally may alter jet spacing by locally changing the Rhines scaling parameter. In baroclinic flow, or with topography that does not vary linearly with latitude, there is no exact analogy with the β effect, and it becomes necessary to use models to examine the extent to which meridional topographic gradients influence the stability and spacing of jets.

To put the preceding ideas in quantitative form, we incorporate the scaled topographic slope into the β parameter resulting in a modified parameter, β_t, and postulate a new meridional scale:

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for which β_t represents the sum of all physical processes, for example, baroclinicity, topography, and boundary effects: the simplest hypothesis is that it will depend on the topographic slope (β_t = β + f₀α/H), or in the case of more complicated topography, some average of the slope. We shall refer to (2) as the topographically modified Rhines scaling. The validity of (2) will be examined for quasigeostrophic flow (sections 2 and 3) and the FRAM/POP solutions (section 4). The paper concludes with a discussion on the significance of the results (section 5).

2. The quasigeostrophic model

We begin with a process study of Rhines’ scaling using a model that solves the quasigeostrophic equations in Cartesian coordinates on a β plane in a zonally oriented reentrant (periodic) channel of length L_x, width L_y, and depth H. Variable bottom topography of height b above the bottom is included. The x, y, and z axes are oriented eastward, northward, and vertically upward, respectively. The flow is constrained by rigid boundaries to the north and south. It is forced by a zonal wind stress τ [τ_x = τ₀ sin(πy/L_y), τ_y = 0], while dissipation is effected by a linear bottom friction law (friction coefficient ν) and a biharmonic lateral friction operator (friction coefficient A₆). In the vertical direction a modal formulation is employed (Flierl 1978); that is, the solution to the full quasigeostrophic potential vorticity equation is expressed as the weighted sum of a complete set of orthonormal functions F_i(z). The equations are briefly reviewed below; see Flierl (1978) for a rigorous derivation and full discussion of the relationship between layer and modal models. Our choice of a modal model rather than the frequently used layer approach lets us test the robustness of multiple jet formation across various model formulations. Furthermore the modal formulation is the more consistent one to employ when more than two physical mechanisms are included (e.g., baroclinicity, wind forcing, nonlinearity, and bottom friction as in the present case). In the following, f₀ is the Coriolis parameter at a reference latitude θ₀ (=2Ω sinθ₀, where Ω is the earth’s rotation rate) and N²(z) is the stratification parameter [=(−g/ρ₀)dρ₀(z)/dz, where g is the acceleration due to gravity, ρ₀ the mean density, and ρ₀(z) the background density field]. We first define the geostrophic velocity field in terms of the geostrophic streamfunction Ψ,

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where

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The functions F_i(z) are defined by

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For the purposes of the paper we employ only the barotropic [i = 1, λ₁ = 0, F₁(z) = 1] and baroclinic (i = 2) modes and consider the simplest situation when baroclinicity is present. Here F₂(z) and λ₂ are derived numerically using N²(z) = N²₀ exp(2.8z/H) with N₀ = 2 × 10⁻³ s⁻¹. For the stratification considered, the internal Rossby radius, (λ₂)^−1/2, is 36 km.

The prognostic equation for each modal streamfunction ϕ_k is

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where ∇², ∇⁶ are Laplacian and biharmonic operators and J the Jacobian operator [J(A, B) = (∂A/∂x)(∂B/∂y) − (∂A/∂y)(∂B/∂x)]. The ξ_ijk are constants defined as ξ_ijk = (1/H) ∫⁰_−H F_i(z)F_j(z)F_k(z) dz. The appropriate boundary conditions are periodic in x and ∂ϕ_k/∂x = ∇²ϕ_k = ∇⁴ϕ_k = 0 on the rigid boundaries. Finally, additional area and line integrals are required to make the set of model equations consistent and solvable (McWilliams 1977; the present paper adapts these integrals for a modal model). The unchanging parameters of the model are listed in Table 1.

The model equations were discretized for numerical solution using standard second-order centered differences in space on an Arakawa-C-type grid (Arakawa 1966). Nonlinear terms were formulated using an energy and enstrophy conserving Jacobian (Arakawa and Lamb 1977). In time a leapfrog scheme was employed with an occasional forward time step to avoid development of a computational mode (Richtmyer and Morton 1967). The time step was 1.5 h and the horizontal resolution 20 km, sufficient to resolve the internal Rossby radius.

Four useful diagnostics from the model are the finite difference analogs of kinetic energy per unit area for each mode

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the total potential energy per unit area

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and the mass transport

TtHϕ₁L_ytϕ₁t

In each of the experiments the model was spun up to a state of statistical equilibrium such that E₁(t), E₂(t), P(t), and T(t) began to fluctuate about mean values with no discernible trend and was then integrated for a further period (typically 12 model years) in order to obtain mean streamfunctions (ϕ₁, ϕ₂) and velocites by averaging over the integration period. By analogy with the above equations for instantaneous energy, mean-flow energies (denoted by E₁, E₂, P) can be defined based on the mean streamfunctions. The difference between the instantaneous energy averaged over the integration and the mean-flow energy is defined as the eddy energy (denoted by E^′₁, E^′₂, P′).

A total of 11 quasigeostrophic simulations were performed with different combinations of channel width and topographic configuration (Table 2). Generally, experiments were conducted with a channel width of 3000 km (∼83 Rossby radii). Two experiments were conducted with channel widths of 1500 km (∼42 Rossby radii) and 4500 km (∼125 Rossby radii), respectively. The latter width is an extreme case and is physically at the limit of consistency with the β-plane approximation for the earth; therefore the results must be viewed with caution. However, it was useful to run the model with this channel width to ascertain the effects of boundaries in constraining the flow. Flat-bottomed runs (b = 0) in narrow and wide channels were conducted to provide reference solutions. Experiments were conducted with linear slopes of varying steepness (∂b/∂y = α) and with zonally constant (i.e., not varying in x) ridges centered at the midpoint of the channel, the cross section being either Gaussian, b = b₀ exp[−(y − L_y/2)²/a²], or harmonic, b = b₀(1 − cos2πy/L_y)²/4, in shape with maximum height b₀ = 750 m. In the case of Gaussian topography the width a was given the value 150 km, providing a strong contrast to the harmonic case where the width is of the order of half-channel width. A zonally constant form for the bottom topography was used in order to isolate the response to the augmentation of the β effect from other physical processes such as topographic form drag and upstream/downstream influences that may play a role in jet splitting (Wolff et al. 1991;Treguier and Pannetta 1994).

Experiments F1, F2, and F3 demonstrate the β/2U scaling for a flat-bottomed ocean; L1–L6 form a parameter study of the effect of linear meridional variation of topography on the flow structure, while T1 and T2 consider the more complicated Gaussian and harmonic topographic shapes.

The method of analysis was to calculate the meridional scale of variation of the flows from the time- and zonally averaged u-velocity profiles and compare them with the predicted scale from (2) using U obtained from the eddy kinetic energy for the simulation and β_t obtained from the topographic slope as discussed in section 1.

3. Results from the quasigeostrophic model

4. FRAM and POP results

The results from two ocean GCMs have been analyzed, FRAM (FRAM Group 1991) and POP (Maltrud et al. 1998). The model configurations differ in a number of aspects. Here we detail those aspects that are most relevant to our study. FRAM has a horizontal resolution of approximately 28 km with a domain from the Antarctic coast to 24°S. The topography has been smoothed and the ocean forced by seasonal winds and a constant buoyancy term. POP, on the other hand, is almost global in extent, reaching to 78°N. It uses a Mercator grid that has a resolution of approximately 16 km at 60°S and unsmoothed topography. The forcing is by three-day averages of ECMWF winds and a seasonal heat flux.

FRAM generally possesses less eddy energy than quasigeostrophic models (cf. Kruse et al. 1990). The region of large mean currents is confined to a band of width 20°–25° latitude, though the band is centered on different latitudes in the various regions. The multiple jet structure is superimposed on a large-scale shear that varies on a length scale of >1000 km. The Hovmöller diagram (Fig. 2) shows that the flow consists of a number of jets meandering to the north and south over time. The velocity filaments remain well correlated over fairly large longitudinal ranges (typically 30° of longitude ∼1600 km at the latitude of the ACC).

The main topographic features along the track of the ACC are the midocean ridge systems, Kerguelan Island, and the Crozet Plateau. We consider three regions in the Indian Ocean and West and East Pacific Ocean (designated with a single letter a–c), respectively, where the bottom topography is relatively zonal (Table 5). The topography and the time-mean near-surface (32.5 m for FRAM, 12.5 m for POP) u velocity (based on the last five years of integration for both FRAM and POP) were longitude-averaged over the ranges (typically 30°–40°) specified in Table 5 (Figs. 7 and 8). In the cases selected a single more-or-less zonally oriented hill or slope dominates the bottom relief. The smoothed, slightly shallower topography in FRAM contrasts with the rougher, slightly deeper topography of POP. It should be noted that the large-scale zonal averaging tends to smooth out the velocity field. In addition to eddy–mean flow interaction the shape of the mean velocity profile will be affected by other factors such as topographic steering. It is hoped that these other effects will be small due to the relatively small departures of the topography from zonality.

We first consider Fig. 7 in detail. In case a, the mean flow is enhanced over a topographic ridge of height approximately 1000 m centered at about 50°S and of total width about 20° of latitude. This ridge is bounded to the north and south by very steeply varying topography that effectively confines the mean flow to the region above the ridge. On the north side of the ridge the topography is in such a direction as to enhance the planetary vorticity gradient, while on the south side there should be a weakening of the β effect, a situation analogous to the quasigeostrophic simulation T2. Observation of the velocity field above shows more structure on the south side of the ridge than on the north. Case b is dominated by depth increasing in the south–north direction, over which most of the mean flow is located, and would thus be expected to enhance the β effect. The flow above the topography is dominated by fairly broadscale jets, although there is smaller-scale variation that splits the tips of the jets. Finally, in case c the main topographic feature exhibits depth increasing in the southward direction, opposing the β effect. However, in all cases there is a region of steep topography to the extreme south oriented to strongly enhance β. Note that typical velocities in all regions are of order 10 cm s⁻¹ (the range is 7.5–12.5 cm s⁻¹)—much smaller than the equivalent quasigeostrophic experiments.

The POP model (Fig. 8) produces multiple jets similar to FRAM, an interesting result in itself given the differences in model configuration, although the magnitudes of the mean velocities are slightly larger, typical peak velocities being about 15 cm s⁻¹, and the length scales are clearly smaller in general. Also the regions of strongest mean currents differ from those in FRAM. In case a there is a clearly defined difference in flow on each side of the ridge with multiple jets being confined to the southern side. The jet structure is much more pronounced than in FRAM. Case b shows multiple jet structure over a similar region to the FRAM model; however as already noted, the length scale of the jets appears to be smaller. Case c shows well-developed jet structure over the major topographic slope centered at about 55°S with two distinct length scales: broader (narrower) to the north (south) of 65°S.

As in the quasigeostrophic case we redefine the variation in background potential vorticity gradient to include topographic effects [Eq. (2)]. As two estimates of β_t we choose the mean and rms values across the domain (calculated for latitudes south of 40°S since this is the region of strong flow in all three cases) of the effective β [β + f₀(∂b/∂y)/H]. The values of U are obtained from the average near-surface eddy energy in each of the three regions over the final five years of both FRAM and POP simulations. Here U is of order 5–7 cm s⁻¹ for FRAM and 12 cm s⁻¹ for POP, well below the quasigeostrophic results. The observed meridional scale is much harder to judge by eye for the FRAM and POP results than for the quasigeostrophic results due to the presence of a broadscale flow on which the multiple jets are superimposed. In order to remove this feature each u-velocity profile in Figs. 7 and 8 was filtered using a running mean (51-point for FRAM and 61-point for POP equivalent to averaging on scales of ∼1500 km) and the result subtracted from the original profile to highlight the multiple jet structure. The midpoints of successive zero-crossings were taken as the centers of the jets allowing the number and average spacing of the jets to be calculated (Fig. 9). We restrict ourselves to latitudes south of 40°S as explained above. It is useful to invert Eqs. (1) and (2) in order to derive the effective β required to produce the observed scale of variation

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Tables 6 (FRAM) and 7 (POP) display values of observed wavelength, λ_obs, U, β_e, and the number of jets followed by the two estimates of β_t and the corresponding estimates of the Rhines scaling λ_c. In the case of FRAM there is excellent agreement (∼10% error) between observed and predicted length scales with the mean and rms providing high and low estimates, respectively. The results from POP present a much less consistent picture. Case a shows the best agreement with similar accuracy to FRAM. On the other hand, case b exhibits very short length scales (∼50% of the calculated values) and the Rhines scaling cannot be of major importance in determining this scale. Case c is an intermediate case. The presence of rough topography in the POP model supports the possibility that topographic control by standing eddies (Treguier and Pannetta 1994) is of greater importance compared with FRAM.

5. Conclusions

The various models with differing physics, resolution, and forcing considered in this study all produce a multiple jet flow and indicate that the presence of zonal jets is a robust feature of Southern Ocean–like flows. We have demonstrated that a major factor controlling the scaling of the jets in quasigeostrophic flow is the β effect coupled with eddy–mean flow interaction. In a zonal channel configuration the stratified flow has a strong barotropic component and hence “feels” the bottom. The effective β restricting the meridional extent of the flow is thus a function of the topography. While it is possible to confidently predict the scaling arising in flat-bottomed cases or cases with linearly varying topography, the effects of more complicated topography, although conforming to the same general principles, cannot necessarily be predicted in advance with accuracy. Preliminary simulations with bottom roughness present indicate that for roughness on certain scales at least, the large-scale meridional slope rather than the roughness still determines the meridional scaling [but Treguier and Pannetta (1994) show that other scales of roughness may have more radical effects]. Clearly, our understanding of the effects of rough topography would benefit from a detailed parameter study.

Despite their many differences in configuration, the ocean GCMs FRAM and POP produce a very similar streaky, jetlike flow. There is reasonable agreement between the observed length scales and the topographically adjusted Rhines scaling in all three regimes studied in FRAM and in one to two out of three regimes studied in POP. Because both FRAM and POP possess similar amounts of eddy energy, there is no great difference in the jet spacing observed between these cases (the range is 250–500 km). There is no agreement with the topographically adjusted Rhines scaling in one of the regimes studied in POP, and it is likely that other effects are of importance in determining the jet structure, in particular topographic control by standing eddies generated from nonzonal topography as in Treguier and Pannetta (1994). In support of this, it is important to note that the POP bathymetry is significantly rougher than that in FRAM.

That the ACC consists of multiple jets or current cores of fairly limited meridional scale is well known. The present study quantifies the expected scale of variation of the jet structure and highlights the factors determining both the scaling and the location of the multiple jets relative to the large-scale topography. Further studies are required to investigate the effects of isolated topographic features and smaller-scale roughness on the structure of the flow. It would also be of interest to make comparisons with zonal or near-zonal current systems, for example, the Azores Current (eastern North Atlantic, ∼34°N; see Pingree 1998).