Separation of a Density Current from the Bottom of a Continental Slope

Melvin E. Stern Department of Oceanography, The Florida State University, Tallahassee, Florida

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Abstract

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For O(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) offslope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important for the explanation of the full separation of the Gulf Stream from the bottom of the continental slope. The kinematics involved in this process should apply to a continuously stratified jet, as well as to other forcing mechanisms known to be of importance in continental boundary current separation.

Corresponding author address: Dr. Melvin E. Stern, Department of Oceanography (4320), The Florida State University, Tallahassee, FL 32306-4320.

Email: stern@ocean.fsu.edu

Abstract

Separation from the continental slope of stratified jets like the Gulf Stream involves the sliding of successive isopycnal layers from a nearly horizontal bottom to the adjacent offshore isopycnal in the deep ocean. One mechanism for producing such an effect is due to a downstream convergence of slope isobaths, as shown herein for a 1-layer density model. Upstream of the convergence, a geostrophically balanced jet is assumed with an inshore region of cyclonic vorticity resting on the continental slope and an offshore anticyclonic region resting on the isopycnal interface above heavier water. For O(1) Rossby number and cross-stream topographic variation, the steady transverse current displacements forced by slowly varying downstream topography are computed. For “supercritical” upstream flow (i.e., fast compared to free topographic waves) offslope displacements are produced by converging isobaths; extrapolation of the small amplitude result suggests that the mechanism is quantitatively important for the explanation of the full separation of the Gulf Stream from the bottom of the continental slope. The kinematics involved in this process should apply to a continuously stratified jet, as well as to other forcing mechanisms known to be of importance in continental boundary current separation.

Corresponding author address: Dr. Melvin E. Stern, Department of Oceanography (4320), The Florida State University, Tallahassee, FL 32306-4320.

Email: stern@ocean.fsu.edu

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 (η̂ŷ), such as are produced by a small (ε) increase in transverse continental slope at a downstream section () 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

Part of the steady upstream jet in Fig. 2 rests on a rigid bottom where the assumed depth and geostrophically balanced velocity are given respectively by
HŷH0ŷrL0UŷU0ŷL0
where rH0/L0 is the nondimensional slope (assumed constant for y ≥ −L0). Farther offshore (ŷ < −L0) the geostrophic current lies above a more dense (ρ + Δρ) layer, which is assumed to be at rest in the rotating coordinate system with Coriolis parameter f. Going downstream () the bottom depth profile changes to:
h(η̂) = H0(1 − η̂r(1 + ε))/L0
where η̂ measures the transverse displacement of a material column originating at ŷ, and ε varies parametrically with x. At any such section the η̂ = 0 origin is taken on the same isobath (H0) as occurs upstream at ŷ = 0. Note that the free streamline (Fig. 2) crosses isobaths and is displaced from η̂ = 0 at the downstream end. The parameters in Fig. 2 are assumed to be such that the Lagrangian column originating upstream at ŷ = −L0 is deflected off the bottom slope and onto the density interface; the dotted curve indicates the intersection of the density interface with the continental slope at any downstream position. Different material columns cross this curve at different downstream positions, causing the transport of the fluid remaining on the slope to decrease. The downstream component of current at all sections is essentially in geostrophic equilibrium (because ∂/∂ is small). The downstream velocities in the boundary current are assumed to vanish at ŷ = −∞ and η̂ = −∞, and the layer thickness is the same in these two regions.

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
i1520-0485-28-10-2040-e2-1a
where the modified Lagrangian displacement function ψ̂ is defined by
ŷ = η̂ + L0ψ̂/η̂.
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
i1520-0485-28-10-2040-e2-1c
Further analysis is now restricted to small ε, in which case LL0 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)
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
i1520-0485-28-10-2040-e2-3
where (υ1, u1) are the cross-stream and downstream () 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 η̂ŷ, we also have υ1 = U(ŷ)∂(η̂ŷ)/∂(), and the linearization of Eq. (2.1b) gives:
i1520-0485-28-10-2040-eq4
where η̂ŷ, and U(ŷ) = −U0ŷ/L0 have been used. The first equation in (2.3) then integrates to:
i1520-0485-28-10-2040-e2-4
and by substituting this in (2.2) we get
i1520-0485-28-10-2040-e2-5
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 δŷs. 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
δŷδŷs
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 Hs denotes the isobath (Fig. 3a) of the streamline at the inner edge of the baroclinic region −L + δŷs, then the integrals of the geostrophic balance equation across the δŷs = δŷ intervals yield
i1520-0485-28-10-2040-e2-8
The aforementioned streamline originating at ŷ = −L0 + δŷ, where H(ŷ) ≡ Hu, and conserving potential vorticity, increases its relative vorticity at η̂ = −L + δŷs by the amount
i1520-0485-28-10-2040-e2-8a
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 Hs, Eq. (2.8) becomes
i1520-0485-28-10-2040-eq5a
and then the geometrical relation (1 + r)H0Hu = H0rδŷ/L0 yields
i1520-0485-28-10-2040-e2-9
where
i1520-0485-28-10-2040-e2-10
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/=H0r/L0 of the bottom (Fig. 2) must exceed the slope (S) of the intersecting interface at ŷ = −L0, as given geostrophically by
SfU0g
and this requires
i1520-0485-28-10-2040-e2-11b
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 Hu isobath located at η̂ = −L + |δŷs|, while the outermost barotropic streamline (at η̂ = −L) originates offshore at ŷ = −L0|δŷ|, where Us > U0 and Hs > Hu. 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) = Usŷ(−L0 − |δŷ|)−1, where Us = U0(L0 + |δŷ|)L−10. With the definition ŷη̂ + L0ψ̂/η̂, it follows that
i1520-0485-28-10-2040-eq6
At η = −L, u = Us, H = h, and linearization yields
L0|δŷ|LL0 dψ̂/dη̂Oψ2
Since −L0 − |δŷ| ≈ −Lψ̂ 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
fU0δŷgHsrH0
and, when the downstream geometrical relation
i1520-0485-28-10-2040-eq9a
is used, the result is
i1520-0485-28-10-2040-eq10
From the conservation of potential vorticity at η = −L + |δŷs| we obtain the relative vorticity
i1520-0485-28-10-2040-eq11
and then the proceeding equation becomes
i1520-0485-28-10-2040-eq12
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 (), 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 − (1 + ε)] for the depth at the downstream section, while retaining H(ŷ) = H0[1 − ] for the upstream depth profile. Thus the potential vorticity at the downstream end is expanded as
i1520-0485-28-10-2040-eq13
and the linearized Eulerian steady-state conservation equation is
i1520-0485-28-10-2040-eq14
By using (2.3) and the long-wave approximation (∂υ1/∂ ≪ ∂u1/∂ŷ) the perturbation vorticity becomes
i1520-0485-28-10-2040-e2-12
and thus the previous equation can then be integrated in to give
i1520-0485-28-10-2040-eq15
When this is nondimensionalized using
i1520-0485-28-10-2040-e2-13
it becomes
i1520-0485-28-10-2040-e2-14
and the boundary condition (2.5) becomes
ψαψ
(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
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
i1520-0485-28-10-2040-eq16
and therefore (2.9) can be written as
i1520-0485-28-10-2040-e2-16
Instead of considering the inhomogeneous equation (2.14), a simplification can be achieved in terms of a solution ψ0(y) of the homogeneous equation
i1520-0485-28-10-2040-e2-17
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
i1520-0485-28-10-2040-eq18a
When the boundary condition (2.15a) for ψ′(1) is applied this becomes:
i1520-0485-28-10-2040-eq19
and thus the final expression of (2.16), or
i1520-0485-28-10-2040-e2-18
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
i1520-0485-28-10-2040-e3-1
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 Ay1/2J1(βy1/2), where J1 is the first-order Bessel function and A is an amplitude factor. Thus,
i1520-0485-28-10-2040-eq20
and the y = 1 boundary condition is satisfied if
i1520-0485-28-10-2040-eq21
or
i1520-0485-28-10-2040-eq22
Thus the complete solution for the quasigeostrophic streamfunction is
i1520-0485-28-10-2040-e3-2
The r → 0, ζ∗ → 0 limit of (2.16) is
i1520-0485-28-10-2040-eq23
and, when (3.2) is used, we get the explicit result
i1520-0485-28-10-2040-e3-3
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
i1520-0485-28-10-2040-eq24
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,
Eq. (2.17) becomes
i1520-0485-28-10-2040-e4-2
and (2.18) becomes
i1520-0485-28-10-2040-e4-3
where
i1520-0485-28-10-2040-e4-4
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
i1520-0485-28-10-2040-eq25

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
i1520-0485-28-10-2040-eq26
and
i1520-0485-28-10-2040-eq27
is
i1520-0485-28-10-2040-e4-5
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 (), then
i1520-0485-28-10-2040-eq28
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
i1520-0485-28-10-2040-eq30
and therefore Eq. (4.3) then gives
i1520-0485-28-10-2040-eq31
As previously mentioned S/(−dH/) is less than unity, and for order of magnitude purposes we now assume that the ratio is 0.5, in which case
i1520-0485-28-10-2040-eq32
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.

Acknowledgments

I gratefully acknowledge the financial support for this work provided by the National Science Foundation under Grants OCE-9216319 and OCE-9529261.

REFERENCES

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Fig. 1.
Fig. 1.

The path of the mean Gulf Stream (dotted) plotted over the topography of the eastern coast of the United States. After Olsen et al. (1983).

Citation: Journal of Physical Oceanography 28, 10; 10.1175/1520-0485(1998)028<2040:SOADCF>2.0.CO;2

Fig. 2.
Fig. 2.

Perspective sketch of a current separating from a bottom whose slope increases in the downstream direction (). The upstream laminar velocity U(ŷ) is barotropic in ŷ > −L0, and at ŷ < −L0 the current lies on the density interface of a stagnant layer of resting heavy (ρ + Δρ) water. η̂ (ŷ) is the transverse coordinate of a material column originating at ŷ. As a result of the slow (ε ≪ 1) downstream variation of bottom depth [h(η̂) = H0(1 − η̂r(1 + ε))/L0)], columns of fluid are forced offshore at the separation curve (dotted). The η̂ = 0 origin at the downstream end is located on the same isobath as the ŷ = 0 origin, and u(η̂), h(η̂) denote the respective values of downstream current and depth in the long-wave theory. The layer thickness at ŷ = −∞ and η̂ = −∞ are equal.

Citation: Journal of Physical Oceanography 28, 10; 10.1175/1520-0485(1998)028<2040:SOADCF>2.0.CO;2

Fig. 3.
Fig. 3.

Plan view of streamlines and isobaths (dashed) near the outer edge of the barotropic region: (a) assuming the streamline originating at ŷ = −L0 is deflected offshore and onto the density interface, thereby (see text) conserving its thickness [(1 + r)H0] and speed (U0). At the outer edge η = −L + δŷs of the downstream barotropic region the streamline on isobath Hs originates upstream at Hu. The problem is to predict the (small) fraction δŷ of the current that separates as a function of ε, and other parameters; (b) assuming the displacement of the ŷ = L0 streamline is toward a shallower isobath Hu < (1 + r)H0, which is located at η = −L + |δys|. It is shown that for ε > 0 this only occurs for “subcritical” velocities relative to the speed of the free topographic waves.

Citation: Journal of Physical Oceanography 28, 10; 10.1175/1520-0485(1998)028<2040:SOADCF>2.0.CO;2

Fig. 4.
Fig. 4.

Plot of 8/β2 − 2J1(β)/J2(β) as a function of β = 2(r/ζ∗)1/2, giving the fractional offshore displacement δŷ/ε in the quasigeostrophic limit [Eq. (3.3)]. Stationary wave resonance occurs for β = 5.1, and Fig. 5 gives the condition when ζ∗ and r are O(1).

Citation: Journal of Physical Oceanography 28, 10; 10.1175/1520-0485(1998)028<2040:SOADCF>2.0.CO;2

Fig. 5.
Fig. 5.

Plot of Eq. (4.4) as a function of slope (r) and c [Eq. (4.1)] for various Burger numbers (2.10). The dashed vertical lines give the values of a downstream uniform r for which a stationary free wave occurs. On the smaller side of each critical r the flow is subcritical, and for ε > 0 the positive δy corresponds to separation (Figs. 2, 3a). For c = 2.0 the value of the ordinate at r = 0.1 is 0.548, and this value of 1 + G(r, c) may be used over a much wider range of (r, c).

Citation: Journal of Physical Oceanography 28, 10; 10.1175/1520-0485(1998)028<2040:SOADCF>2.0.CO;2

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  • Haidvogel, D. B., J. C. McWilliams, and P. R. Gent, 1992: Boundary current separation in a quasi-geostrophic eddy resolving ocean circulation. J. Phys. Oceanogr.,22, 882–902.

  • Hughes, R. L., 1986: On the role of criticality in coastal flows over irregular topography. Dyn. Atmos. Oceans.,10, 129–147.

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  • Parsons, A. T., 1969: A two-layer model of Gulf Stream separation. J. Fluid Mech.,39, 511–528.

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  • Fig. 1.

    The path of the mean Gulf Stream (dotted) plotted over the topography of the eastern coast of the United States. After Olsen et al. (1983).

  • Fig. 2.

    Perspective sketch of a current separating from a bottom whose slope increases in the downstream direction (). The upstream laminar velocity U(ŷ) is barotropic in ŷ > −L0, and at ŷ < −L0 the current lies on the density interface of a stagnant layer of resting heavy (ρ + Δρ) water. η̂ (ŷ) is the transverse coordinate of a material column originating at ŷ. As a result of the slow (ε ≪ 1) downstream variation of bottom depth [h(η̂) = H0(1 − η̂r(1 + ε))/L0)], columns of fluid are forced offshore at the separation curve (dotted). The η̂ = 0 origin at the downstream end is located on the same isobath as the ŷ = 0 origin, and u(η̂), h(η̂) denote the respective values of downstream current and depth in the long-wave theory. The layer thickness at ŷ = −∞ and η̂ = −∞ are equal.

  • Fig. 3.

    Plan view of streamlines and isobaths (dashed) near the outer edge of the barotropic region: (a) assuming the streamline originating at ŷ = −L0 is deflected offshore and onto the density interface, thereby (see text) conserving its thickness [(1 + r)H0] and speed (U0). At the outer edge η = −L + δŷs of the downstream barotropic region the streamline on isobath Hs originates upstream at Hu. The problem is to predict the (small) fraction δŷ of the current that separates as a function of ε, and other parameters; (b) assuming the displacement of the ŷ = L0 streamline is toward a shallower isobath Hu < (1 + r)H0, which is located at η = −L + |δys|. It is shown that for ε > 0 this only occurs for “subcritical” velocities relative to the speed of the free topographic waves.

  • Fig. 4.

    Plot of 8/β2 − 2J1(β)/J2(β) as a function of β = 2(r/ζ∗)1/2, giving the fractional offshore displacement δŷ/ε in the quasigeostrophic limit [Eq. (3.3)]. Stationary wave resonance occurs for β = 5.1, and Fig. 5 gives the condition when ζ∗ and r are O(1).

  • Fig. 5.

    Plot of Eq. (4.4) as a function of slope (r) and c [Eq. (4.1)] for various Burger numbers (2.10). The dashed vertical lines give the values of a downstream uniform r for which a stationary free wave occurs. On the smaller side of each critical r the flow is subcritical, and for ε > 0 the positive δy corresponds to separation (Figs. 2, 3a). For c = 2.0 the value of the ordinate at r = 0.1 is 0.548, and this value of 1 + G(r, c) may be used over a much wider range of (r, c).

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