1. Introduction

The Gulf Stream emerging from the Straits of Florida consists of an inshore region with large cyclonic vorticity overlying a steep continental slope, the latter being intersected by isopycnals extending shoreward from the anticyclonic region of the offshore jet. As the stream flows northward (Olsen et al. 1983), there is a continual deflection of successive isopycnal layers off the slope and onto the isopycnals in the deeper ocean. Eventually all of the fluid over the slope separates completely from the bottom in a remarkably localized region north of Cape Hatteras, thereby forming a “free” jet in deep water.

Although the separation problem (Haidvogel et al. 1992; Özgökmen et al. 1997) has received considerable attention in the past, the focus has usually been on planetary-scale effects in models whose lateral boundary consisted of a vertical wall. One of these effects (with which the separation of the Gulf Stream must be consistent), is the downstream increase in wind-driven transport to higher latitudes, which leads to surfacing of isopycnals near the western wall (Parsons 1969; Morgan 1956; Charney 1955). Other known “global” constraints are associated with the termination of the subtropical gyre at the latitude of vanishing wind stress curl (Munk 1950), the “collision” of the subtropical gyre with the deep western boundary current (Agra and Nof 1993), and the abrupt change in direction of the coastline (Stern and Whitehead 1990). An indication of the important role of the bottom slope (in contrast to the vertical wall) appears in the barotropic models of Baines and Hughes (1996) and Becker and Salmon (1997). In both of these papers the depth-independent boundary current separates from the slope while remaining in contact with the bottom, whereas our focus will be on the separation of a baroclinic jet from the rigid bottom. Our viewpoint is also more local (f plane), more inertial (larger jet Rossby number), and starts with a specified upstream jet structure with a more realistic inshore (cyclonic) shear. It should be mentioned that there is no accepted explanation for the generation of this shear, as contrasted with the anticyclonic offshore shear.

In connection with Fig. 1, Olsen et al. (1983) mention that “the gradient of bottom topography increases by a factor of 2 around 29°N. At this point the stream leaves the shelf and enters deeper water.” More striking is the extreme convergence of the isobaths at 35.5°N; it is at this point, apart from the influence of small amplitude meanders, that the synoptic baroclinic jet separates completely from the continental slope.

We will investigate the topographic effect using a 1-layer model (Fig. 2) in which the upper layer of uniform density ρ rests partially on the bottom slope and partially on the interface above a stagnant layer of density ρ + Δρ; at all cross-stream positions ŷ the upstream flow is in geostrophic equilibrium. A steady flow is assumed farther downstream where the isobaths converge (or diverge) gradually compared to the cross-stream variation. Accordingly, the forced topographic response will be computed by a steady-state long-wave theory, which will eventually be linearized for the case of a small amplitude downstream variation in slope, but neither the cross-stream slope nor the Rossby number will be assumed small in the main calculation (section 4). Also noteworthy (Fig. 2) is the absence of an unrealistic slippery vertical wall; instead we have a free streamline with vanishing velocity, located in finite depth water on the bottom slope. This boundary condition and this model are complementary to those in a recent study (Stern 1997) of a free jet flowing over a sill with nonconvergent isobaths. Clearly, a combination of both kinds of topography, that is, variation in cross-stream curvature as well as isobath convergence, needs to be considered in general.

The most novel physical consideration (sections 2a–c) in this paper is the connection condition at the point where a material column [a streamline originating at ŷ (Fig. 2)] leaves the rigid bottom and moves onto the density interface; at any downstream section η̂ (ŷ) indicates the transverse displacement of any streamline from a designated isobath at ŷ = 0 (or η̂ = 0). The calculation is eventually limited to small cross-isobath displacements (η̂ − ŷ), such as are produced by a small (ε) increase in transverse continental slope at a downstream section (x̂) relative to the upstream section.

The aforementioned connection condition provides one boundary condition for the well-known linearized potential vorticity equation (section 2d), which applies to the barotropic fluid on the slope; the second boundary condition requires vanishing velocity on the inshore free streamline. The solution of the resulting inhomogeneous ordinary differential equation (section 2) gives the fraction δŷ of the upstream current on the slope, which appears downstream on the density interface for a given topographic slope change ε.

For the quasigeostrophic range of topographic, Rossby, and Burger numbers, Eq. (3.3) or Fig. (4) gives an analytic result for δŷ, and section 4 gives the numerical result for O(1) Rossby number and topographic variation. The extrapolation of this δŷ/ε result, given in the conclusion, provides an order of magnitude estimate of the topographic effect for the Gulf Stream.

2. Derivation of linear equations

a. Outer boundary condition for offshore displacement

For gradual changes in bottom slope, a Lagrangian material column (ŷ ≥ −L₀) with downstream velocity u(η̂), and thickness h(η̂), satisfies mass conservation in the sense of the long-wave approximation:

uh dη̂ = UH dŷ = −U₀ŷH dŷ/L₀

where H(ŷ) is the upstream depth of this column. Before applying this to ŷ > −L₀ and before turning to the potential vorticity equation, we consider the novel boundary condition required at the edge of the barotropic region. First, the previous equation is rewritten as

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where the modified Lagrangian displacement function ψ̂ is defined by

ŷ = η̂ + L₀ψ̂/η̂.(2.1b)

The geostrophic transport between two streamlines located completely in the deep water is proportional to the square of the difference of the respective layer thickness and, since H(−∞) = h(−∞), it follows that the streamline originating at the upstream edge ŷ = −L₀ of the slope and terminating offshore at η̂ = −L in the baroclinic region must satisfy h(−L) = H(−L₀). Moreover the Bernoulli invariant on this streamline implies u(−L) = U₀. Evaluating (2.1a) and (2.1b) on ŷ = −L₀ then gives

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Further analysis is now restricted to small ε, in which case L − L₀ and ψ̂ are correspondingly small. Therefore, when the previous equation is substituted into (2.1c) and the small quadratic terms are neglected, we get a linearized boundary condition:

ψ̂(−L₀) + L₀ψ̂′(−L₀)(2.2)

for ψ̂ in the barotropic region ŷ > −L₀. Although the density is not explicitly involved in this relation, its validity clearly depends upon the fact that the outer part of the current originating on the slope lies on the density interface. The explicit importance of Δρ will appear subsequently.

The novel Lagrangian function ψ̂ in (2.2) will now be related to the conventional Eulerian mass transport function M defined by

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where (υ₁, u₁) are the cross-stream and downstream (x̂) perturbation velocities, respectively. [N.B., no confusion should arise here or in section 2d by the use of ŷ as transverse Eulerian coordinate for M(x̂, ŷ)]. On a steady Lagrangian streamline, transversely displaced by an amount η̂ − ŷ, we also have υ₁ = U(ŷ)∂(η̂ − ŷ)/∂(x̂), and the linearization of Eq. (2.1b) gives:

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where η̂ ≈ ŷ, and U(ŷ) = −U₀ŷ/L₀ have been used. The first equation in (2.3) then integrates to:

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and by substituting this in (2.2) we get

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Equation (2.5) provides a boundary condition for the M(ŷ) satisfying the linearized Eulerian potential vorticity equation appearing in section 2d.

b. The offshore transport relation

The main calculation pertains to the fraction δŷ > 0 (Fig. 3a) of the upstream slope current, which separates from the bottom and which appears on the density interface in an interval δŷ_s. At the outermost limit (η̂ = −L) of this interval, the layer thickness h∗ and the velocity u∗ must equal the corresponding upstream values on the (1 + r)H₀ isobath (for “transport” and “Bernoulli” reasons previously mentioned). Correct to first order, the conservation of mass in the intervals then requires

δŷδŷ_s(2.7)

This will be computed by relating it, as follows, to the vorticity perturbation in the barotropic region, and the latter quantity will be related to ε in the Eulerian calculations of section 2d.

If H_s denotes the isobath (Fig. 3a) of the streamline at the inner edge of the baroclinic region −L + δŷ_s, then the integrals of the geostrophic balance equation across the δŷ_s = δŷ intervals yield

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The aforementioned streamline originating at ŷ = −L₀ + δŷ, where H(ŷ) ≡ H_u, and conserving potential vorticity, increases its relative vorticity at η̂ = −L + δŷ_s by the amount

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where the basic upstream vorticity is

ζU₀L₀

Equation (2.8a) is the vorticity perturbation along a streamline, but because the cross-stream gradient of undisturbed vorticity vanishes, Eq. (2.8a) also equals (to leading order) the perturbation vorticity along a line that is at equal transverse distances from the H₀ reference isobath at all downstream distances x̂.

When (2.8a) is used to eliminate H_s, Eq. (2.8) becomes

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and then the geometrical relation (1 + r)H₀ − H_u = H₀rδŷ/L₀ yields

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where

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and F is the inverse ratio of the radius of deformationdivided by L₀. The perturbation vorticity ζ(−L₀) computed in sections 3 and 4 will be substituted in (2.9) to obtain the fraction of the fluid deflected offshore for any ε.

It is important to note that the slope −dH/dŷ=H₀r/L₀ of the bottom (Fig. 2) must exceed the slope (S) of the intersecting interface at ŷ = −L₀, as given geostrophically by

SfU₀g(2.11a)

and this requires

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Since δŷ > 0 has been assumed here, Eq. (2.9) requires that the sign of ε be such that ζ(−L₀) > 0.

c. Onshore deflection (Fig. 3b)

Although this case is not considered further here, it is of interest to derive the connection condition when the parameters are such that the streamline at ŷ = −L₀ is deflected toward the shallower H_u isobath located at η̂ = −L + |δŷ_s|, while the outermost barotropic streamline (at η̂ = −L) originates offshore at ŷ = −L₀ − |δŷ|, where U_s > U₀ and H_s > H_u. As before, mass conservation requires uh dη̂ = UH dŷ. If the linear part of the upstream shear flow U(ŷ) extends somewhat beyond ŷ = −L₀, then U(ŷ) = U_sŷ(−L₀ − |δŷ|)⁻¹, where U_s = U₀(L₀ + |δŷ|)L⁻¹₀. With the definition ŷ ≡ η̂ + L₀ψ̂/η̂, it follows that

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At η = −L, u = U_s, H = h, and linearization yields

L₀|δŷ|LL₀ dψ̂/dη̂Oψ²

Since −L₀ − |δŷ| ≈ −L − ψ̂ follows from the definition of ψ̂, it is now apparent that (2.2) also applies to this case, and likewise for (2.6).

It only remains to show that (2.9)–(2.10) also applies, provided δŷ is replaced by |δŷ|. The geostrophic velocity integral across the upstream baroclinic |δŷ| interval is

fU₀δŷgH_srH₀

and, when the downstream geometrical relation

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is used, the result is

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From the conservation of potential vorticity at η = −L + |δŷ_s| we obtain the relative vorticity

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and then the proceeding equation becomes

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which is the same as (2.9) except that the perturbation vorticity must be negative for this case.

d. Barotropic perturbation equation for slow downstream variation

In the following consideration of the Eulerian perturbation equations at any downstream section (x̂), the symbol for the previously used Lagrangian coordinate (η̂) will be replaced by ŷ, this being the transverse distance from the local isobath H(0) = H₀. With this understanding we may write H₀[1 − rŷ(1 + ε)] for the depth at the downstream section, while retaining H(ŷ) = H₀[1 − rŷ] for the upstream depth profile. Thus the potential vorticity at the downstream end is expanded as

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and the linearized Eulerian steady-state conservation equation is

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By using (2.3) and the long-wave approximation (∂υ₁/∂x̂ ≪ ∂u₁/∂ŷ) the perturbation vorticity becomes

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and thus the previous equation can then be integrated in x̂ to give

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When this is nondimensionalized using

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it becomes

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and the boundary condition (2.5) becomes

ψαψ(2.15a)

(Note that this streamfunction is not the same as the ψ̂ used in section 2a.) On the free streamline, with unknown but small y ordinate both velocity components vanish, and therefore to leading order, we require υ(0) = 0, M(0) = 0, and therefore

ψ(2.15b)

This boundary condition will provide the only solution of (2.14), which is regular near y = 0. The quantity of interest for Eq. (2.9) is

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and therefore (2.9) can be written as

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Instead of considering the inhomogeneous equation (2.14), a simplification can be achieved in terms of a solution ψ₀(y) of the homogeneous equation

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which satisfies

ψ₀

Now multiply Eq. (2.17) by ψ(y), multiply Eq. (2.14) by ψ₀, subtract the results, and then integrate to get

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When the boundary condition (2.15a) for ψ′(1) is applied this becomes:

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and thus the final expression of (2.16), or

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only requires the solution of (2.17).

3. The quasigeostrophic limit r = 0(ζ∗) → 0

Before proceeding to the relevant Gulf Stream case, where both the Rossby number ζ∗ and r are O(1), it is instructive to consider the quasigeostrophic limit for which analytical results are obtainable. In the latter case r and ζ∗ are both small to the same order, and Eqs. (2.14), (2.15), (2.17) reduce to

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Direct substitution reveals that a particular solution of (3.1) is (y² − 8y/β²)(4/β²), and the only regular homogeneous solution (Abromowitz and Stegun 1970) is Ay^1/2J₁(βy^1/2), where J₁ is the first-order Bessel function and A is an amplitude factor. Thus,

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and the y = 1 boundary condition is satisfied if

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or

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Thus the complete solution for the quasigeostrophic streamfunction is

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The r → 0, ζ∗ → 0 limit of (2.16) is

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and, when (3.2) is used, we get the explicit result

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wherein Eq. (2.11b) requires 4F²/β² < 1. Equation (3.3), plotted in Fig. 4, shows that the deflection into deep water (δŷ > 0) occurs when β = 2(r/ζ∗)^1/2 is less than β = 5.1 [i.e., J₂(β) = 0], at which value resonance occurs. We note in passing that there are free (ε = 0) stationary waves at these values of J₂(β) = 0, and for such conditions at a point on the continental slope an upstream–downstream hydraulic transition (Hughes 1986) could occur. For β < 5.1 the flow is said to be supercritical (“fast”) with respect to the topographic wave speed, and we conclude that such a flow will produce displacements into deeper water if the isobaths converge downstream.

A final qualitative point may be noted before turning to larger ζ∗, and r. Using Eq. (3.2), we see that the vorticity perturbation, as given by the first term in (3.1) or by y − (β²/4y)ψ(y), has a value near y = 0 equal to

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which is negative for β smaller than the first zero of J₂(β) = 0 since the expansion of J₂(β) = β²/8 − · · · is less than β²/8. Thus we conclude that for supercritical flow the vorticity at y = 0 is negative so that here the streamlines are deflected into shallower water and its shear decreases (while the velocity remains zero). Near y = 1, on the other hand, the vorticity is proportional to 1 − β²/4ψ(1), and the discussion following (3.3) shows that this is positive. Although the streamlines near y = 1 are deflected toward the H₀ isobath, the isobaths near y = 1 are displaced by a larger amount, and this explains why the y = 1 streamline crosses its upstream isobath and emerges into deeper water (where its vorticity increases).

4. Numerical calculations for finite r, ζ∗

With the transformation

zry,ψ₀yϕzζζc,(4.1)

Eq. (2.17) becomes

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and (2.18) becomes

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where

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The evaluation of this requires a calculation of ϕ(z) up to z = r.

For r < 1, Eq. (4.4) can be computed by means of a power series solution of (4.2) in the interval of convergence |z| < 1, and the result is

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In order to obtain solution for r > 1, this power series was first used to compute ϕ(z) at z = 0.5, and then the values of ϕ(0.5), ϕ′(0.5) were used in a second-order Runge-Kutta integration to continue the solution to z > 1 and r > 1. The resulting values of 1 + G, or Eq. (4.3), are plotted in Fig. 5.

For r → 0 and z → 0 the asymptotic value of (4.4), as obtained from

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and

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is

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For ζ∗ = 1, c = 2 the value of (4.5) is 1 + G = 5/9, and this not only agrees with the numerical calculation (Fig. 5) for r → 0, but is a fair approximation up to r = 5.0 where 1 + G = 0.69. The most noteworthy numerical variation in 1 + G [or Eq. (4.4)] at smaller ζ∗ (larger c) is the appearance of a resonance amplification due to the decrease of the current relative to the topographic slope r; this allows stationary topographic Rossby waves in the basic state. It is also clear that for certain values, for example, r = 2, c = 6, an onshore deflection occurs even for convergent isobaths (ε > 0). This isunlikely to occur, however, if the inshore shear is less than ζ = f/2 (c = 3). If the isobaths are divergent (ε < 0) and ζ = 0.2f (c = 6.0), then an onshore deflection occurs for r < 1.5.

5. Conclusions

The separation of a baroclinic boundary jet from the continental slope requires fluid in contact with the nearly horizontal rigid bottom to be displaced onto an isopycnal surface in the deep ocean. The geostrophic kinematics (Figs. 2 and 3) involved in this process have been applied to a topographic forcing mechanism acting on a “fast” (supercritical) boundary jet. It has been shown that isobaths converging downstream (ε > 0) produce offshore deflection of the slope current at a rate (δŷ/ε) given analytically by Eq. (3.3) for a quasigeostrophic jet, and by Fig. 5 when r and ζ are O(1), as is the case for the Gulf Stream. Onshore deflection for convergent isobaths occurs when the jet is subcritical (cf. Fig. 5, lower curves). Conditions for stationary waves, indicative of hydraulic control, have also been given for a jet with a free streamline on a uniformly sloping continental boundary.

The foregoing results may be used to estimate the magnitude of the effect on the Gulf Stream. If dr/dx̂, dL/dx̂ denote the respective change in slope and width of the overlying current per unit downstream distance (x̂), then

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is given by Eq. (4.3). Since the typical nondimensional Gulf Stream cyclonic vorticity lies between ζ∗ = 0.5 and 1.0, the asymptotic limit

Gr, c

is an acceptable approximation for Eq. (4.3), as indicated at the end of section 4. From ζ∗ = U₀/(fL₀), (2.10), and (2.11a), we have

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and therefore Eq. (4.3) then gives

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As previously mentioned S/(−dH/dŷ) is less than unity, and for order of magnitude purposes we now assume that the ratio is 0.5, in which case

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Thus the width L of the current on the slope decreases by 50% if the cross-stream slope (r) increases by a factor of 2 from the upstream to the downstream section.

This suggests that the geographical localization of the synoptic Gulf Stream separation point frequently observed at Cape Hatteras is due to the extreme convergence of the slope isobaths at this point. This does not preclude an important role for the other mechanisms cited in the introduction, but these provide“large”-scale and necessary climatalogical conditions in which the inertial–synoptic separation event occurs. Although the foregoing numerical conclusion is based on an extrapolation of small amplitude theory and is applied to simplified baroclinic current (Fig. 2), this model contains important and realistic physical features not found in previous theories; these features include a full jet with both cyclonic and anticyclonic shear and with a free streamline overlying a sloping bottom from which separation occurs. The explicit demonstration of the way in which particles separate from a rigid bottom is perhaps the most important fluid dynamical result.