## 1. Introduction

The interaction of a boundary current with bathymetric features such as a gap in the ridge or a strait between two islands is an important and interesting oceanographic problem. Examples include western boundary currents such as the Gulf Stream leaping from the Yucatan to Florida or the Kuroshio leaping from Luzon to Taiwan and shelfbreak currents such as the one leaping from the Scotian shelf to Georges Bank (Cho et al. 2002; the so-called Scotian Shelf Water crossover event). Multiple flow patterns (when flow either leaps across or penetrates through the gap forming a loop current and shedding rings) are known to exist in such systems. A number of theoretical works have been devoted to explaining various aspects of gap flow dynamics (e.g., Nof and Olson 1983; Simmons and Nof 2002; Johnson and McDonald 2004). Recently Sheremet (2001) considered a gap-leaping problem in the context of a western boundary current and, by using numerical analysis, found that the multiple steady states can be explained by variation in the balance between the inertia (which promotes the leaping state) and the *β* effect (which promotes the penetrating state). Moreover, it was found that in transition from one type of flow to another a hysteresis exists: the flow state depends on prior evolution and two different flow states are possible for exactly the same external governing conditions.

It is known that numerical methods sometimes exhibit spurious behavior, especially in the presence of sharp features in the boundary; for example, the notorious problem of the Gulf Stream separation from Cape Hatteras: certain ocean circulation models fail to properly represent separation and allow the current to hug the coast. To dismiss such a possibility we felt compelled to demonstrate the existence of multiple steady states and hysteresis in the gap-leaping problem using a laboratory model and thus confirm our earlier numerical findings.

In (Sheremet 2001) the western boundary current was assumed to be infinitely long, which is impossible to produce in the laboratory experiments. Instead we built a laboratory model in a circular tank and developed a numerical model that matches it as accurately as possible. To set up a gap-leaping current we use a circular tank with a sloping bottom (simulating the *β* effect). Traditionally in experiments simulating ocean circulation the flow was generated by a lid rotating at a slightly different angular velocity than the tank (Pedlosky and Greenspan 1967). Here we drive the flow using a new method of pumping fluid through sponges (thus generating a Sverdrup flow in the interior). The laboratory model is explained in detail in section 2. The mathematical formulation of the problem is presented in section 3 while details of the associated numerical model in curvilinear coordinates are discussed in the appendix. We use a bipolar coordinate system in order to accommodate a circular geometry of the tank and the ridge inside it. The results of the laboratory experiments along with the matching numerical solutions are shown in section 4. The last section presents the conclusions.

## 2. Laboratory model

The laboratory apparatus consists of a circular tank with a radius of *R* = 48.7 cm, which is placed in the center of a rotating table. The tank is divided into several compartments (Fig. 1). We introduce Cartesian coordinates (*x*, *y*) with the origin being at the center of the tank, *x* pointing “east” (right of the figure), and *y* pointing “north” (top of the figure). The analogs of geographic east, west, north, and south will become clear below. The “western” semicircle (*x* < 0) is a working area where the circulation is supposed to mimic the subtropical gyre, it is separated from the “eastern” semicircle (*x* > 0) by a sponge wall. The eastern semicircle is in turn divided by a solid barrier into the “northern” (*y* > 0) and “southern” (*y* < 0) sectors. The flow is driven by pumping water from the southern into the northern sector, lowering pressure in the former and raising it in the latter relative to the western compartment. As a consequence water starts to percolate through the sponges generating circulation in the working area.

*S*= 0.05 from north to south, in order to produce the

*β*effect. In a solid body rotation the surface of fluid assumes the shape of a paraboloid, hence the total depth of the fluid layer

*H*(

*x*,

*y*) in the working area iswhere

*H*

_{0}is the depth at the center of the tank in the nonrotating state, Ω = 1 s

^{−1}is the rotation rate of the table,

*r*= (

*x*

^{2}+

*y*

^{2})

^{1/2}is the radius, and

*g*= 980.3 cm s

^{−2}is the gravitational acceleration. The last term (½)

*R*

^{2}in (1) takes into account conservation of mass between the rotating and nonrotating states. The potential vorticity

*q*(

*x*,

*y*) = 2Ω/

*H*(

*x*,

*y*) is constant along geostrophic contours that have circular shapes by virtue of (1); they originate at the eastern boundary (the sponge wall

*x*= 0), curve “northward,” and impinge on the circumference of the tank (

*r*=

*R*) (Fig. 1). Because the fluid layer becomes shallower and hence the potential vorticity increases northward, the topographic Rossby waves in the tank would propagate “westward” in analogy to the planetary Rossby waves in the ocean.

Inserted within the working area is a narrow vertical impermeable ridge having a circular shape and passing through the points (*x*, *y*) = (0, −*R*); (−2*R*/3, 0); (0, *R*). In the middle the ridge has a gap of half-width *a* extending all the way to the bottom. When the pump is on, the flow incoming through the northern sponge (*x* = 0, *y* > 0) follows geostrophic contours, according to the Sverdrupian dynamics, until it impinges on the ridge and forms a western boundary current carrying water south along the ridge (Sverdrup 1947; Stommel 1948). Depending on the governing parameters, the boundary current can either leap across the gap for relatively strong inertia or in the case of weak inertia penetrate through the gap as a loop current pushed farther westward by the *β* effect (Sheremet 2001). Upon negotiating the gap the boundary current continues south along the ridge gradually feeding the eastward outflow, which follows the geostrophic contours back to the eastern boundary where its is sucked through the sponge (*x* = 0, *y* < 0).

*u*(0,

*y*) =

*U*

_{1}(0 <

*y*<

*R*) and outlet

*u*(0,

*y*) =

*U*

_{2}(−

*R*<

*y*< 0) velocities are uniform as well, except near the boundaries (

*y*= 0, ±1), provided that the sponges have uniform thickness. This was checked by rapidly adding and mixing dye in the source eastern sector and observing a straight dyed water front appearing and propagating away from the sponge in the working compartment. The constants

*U*

_{1}and

*U*

_{2}are constrained by the conditionwhere

*Q*is the volume pumping rate.

*S*= 0.05) to match that of the free surface at the northern end (

*x*= 0,

*y*=

*R*),the potential vorticity gradient vanishes there,also this point is a common center of all circular geostrophic contours [e.g., (1)]. In this way we can generate a western boundary current with nearly constant potential vorticity at its origin. There is a lot of evidence that oceanic western boundary currents have this property. In case of the Kuroshio in the western Pacific, Toole et al. (1990) measured that in the latitude range from 8° to 18°N the potential vorticity of the surface layer above the main thermocline is practically a constant 1 × 10

^{−7}m

^{−1}s

^{−1}with the accuracy better than 10%. We also note that the Kuroshio flows northward while in the laboratory experiment we model it by a southward-flowing boundary current. We do so in order to take advantage of nearly uniform potential vorticity in the northern part of the tank and also in order to have a longer span for the boundary current to travel once it formed since the geostrophic contours bend north.

## 3. Mathematical formulation

*h*= (

_{E}*ν*/Ω)

^{1/2}, where

*ν*is the kinematic viscosity of water. Furthermore, the Ekman layer correction decays exponentially fast outside the boundary layer. This is justified, if for the typical horizontal velocity

*U*, the Rossby number Ro =

*U*/(2Ω

*R*) is small. Then for the flow outside the Ekman layer the Navier–Stokes equations can be written only in terms of horizontal components of velocity

**u**= (

*u*,

*υ*) independent of vertical coordinate

*z*. And these equations can be regarded to be exact as far as the expansion in Ro is concerned. The horizontal momentum equations in a Gromeka–Lamb vector form

^{1}areand the continuity equation isThe subscript

*t*indicates differentiation with respect to time;

**∇**, rot, div are the vector operators involving only horizontal derivatives;

*p*is the pressure anomaly relative to a solid body rotation state divided by uniform density of water;

**k**is the vertical unit vector;

*e*is the kinetic energy divided by density of water; and

*ω*= rot

**u**is the vertical component of relative vorticity. According to the linear Ekman (1905) theory for a rapidly rotating fluid the flow divergence in the bottom boundary layer can be expressed in terms of the ambient relative vorticity:This expression gives the first-order contribution of vertical viscosity to dissipation, which is also called Ekman suction. More subtle details, such as the three-dimensional structure of the flow near lateral boundaries, can be found in (Pedlosky 1968). We neglect the contribution of the pressure anomaly to the total layer depth, therefore (6) is linear.

*ψ*, which combines the transport in the inviscid interior and the transport in the Ekman layer:Expanding (8) in

*h*/

_{E}*H*[

*O*(10

^{−2}) in the experiments] and keeping only leading terms results in the expression for the velocity outside the Ekman layer:Taking curl of (5) and expressing the velocity field in terms of the function of total transports leads towhere

*q*= (2Ω +

*ω*)/

*H*. These equations are valid for arbitrary large variations of depth

*H*, which is essential for the laboratory experiments and also take into account the flow divergence caused by Ekman suction. However, the dominant terms are the same as in the standard quasigeostrophic equations. We retained the evolutionary term in (10) for the sake of generality and to indicate a numerical way of finding steady solutions by the method of equilibration. The system in (10) and (11) will accurately describe transient motions only if the Rossby radius of deformation

*L*= (

_{R}*gH*

_{0})

^{1/2}/(2Ω) is much larger than the scale of the motion. In the experiments

*L*= 49.5 cm, thus, suggesting that the western boundary layer–type transient motions will be practically unaffected but that the basin-scale transients will be somewhat modified; however, both are irrelevant for the final steady states. The kinematic boundary conditions become

_{R}*ψ*= 0 at the solid walls; at the sponges, the total transport is prescribed in accordance to (2):with

*ψ*= −

*Q*at the center of the tank. The no-slip condition on the tangential velocity component is easy to implement by virtue of (9).

*β*effect against the bottom drag in (10) gives the Stommel boundary layer thickness

*L*:balancing the topographic

_{S}*β*effect against the lateral friction gives the Munk boundary layer thickness

*L*:and balancing the topographic

_{M}*β*effect against the advection of relative vorticity gives the inertial boundary layer thickness

*L*:In the experiments both

_{I}*L*and

_{S}*L*were fixed at about 1.0 cm while

_{M}*L*varied with

_{I}*Q*;

*L*= 1.0 cm for

_{I}*Q*= 5 cm

^{3}s

^{−1}.

## 4. Results of the experiments with varying flow rate

To study dependence of steady flow on parameters we conducted a series of laboratory experiments with fixed gap width 2*a* = 9.3 and 14.0 cm and a varyied pumping rate *Q* in the range from 0 to 60 cm^{3} s^{−1}. For the gap width 9.3 cm we saw almost monotonic transition from the penetrating (small *Q*) to gap-leaping (large *Q*) flow patterns near *Q* = 12 cm^{3} s^{−1} with some evidence of multiple states, but the difference was difficult to distinguish. However, for the gap width 14.0 cm we obtained dramatic evidence of multiple steady states and hysteresis in the range 23 ≤ *Q* ≤ 55 (cm^{3} s^{−1}), which is illustrated in Fig. 2.

Because of a hysteresis we distinguished experiments with increasing and decreasing *Q*. In the increasing flow rate experiments, a 30-min spinup with the pump switched off led to a solid body rotation state; the spinup time scale was *T _{S}* =

*H*

_{0}/(

*h*Ω) = 100 s. Then we switched the pump on and gradually increased the flow rate

_{E}*Q*in small steps, typically Δ

*Q*= 5 cm

^{3}s

^{−1}(we used smaller Δ

*Q*= 1 cm

^{3}s

^{−1}near the critical values), and let the flow equilibrate to a new state while keeping

*Q*fixed for about 10–15 min, which is sufficiently longer than

*T*. Similarly, after the maximum flow rate had been reached and transition to the new state had occurred we started to gradually decrease the flow rate thus tracing a different branch of steady flow patterns. At some reference values of flow rate

_{S}*Q*= 10, 25, 40, 55, 60 cm

^{3}s

^{−1}we kept

*Q*fixed for an additional 15 min before starting flow visualization using a dye release method. It took approximately another 15 min for the dye to travel across the basin and we also waited to make sure that the streak line was steady. This ensures that the different flow patterns seen in the top versus bottom panels in Fig. 2 represent truly different steady states and not transients.

*x*= 0 and

*y*=

*R*/2 we could visualize whether the core of the current leaped across the gap or not.

We also obtained steady numerical solutions for the same values of *a* and *Q*. Some important details of the numerical model are given in the appendix. The patterns of the total transport *ψ* are shown in Fig. 3 using the same arrangement of images. Like in the laboratory experiments we had to trace different branches of steady solutions by increasing or decreasing *Q*. The numerical model predicts that two different steady solutions are possible within the range 25 ≤ *Q* ≤ 52 (cm^{3} s^{−1}). These values are just slightly off from those obtained from the laboratory data; therefore, the last image in the bottom panel in Fig. 3 corresponds to *Q* = 52 rather than 55 (cm^{3} s^{−1}). The small discrepancy could be caused by various uncertainties in the laboratory setup such as small variation in fluid viscosity with temperature, warping of the Plexiglas bottom slightly changing the *β* effect, and some errors in pumping velocities (about 2% for the Ismatec gear pump). In fact, in the experiment with *Q* = 22 cm^{3} s^{−1} we estimated the zonal flow in the Sverdrup interior based on consecutive photographs of dye propagation to be 0.0517 cm s^{−1} while the numerical model predicts it to be 0.0526 cm s^{−1} according to (2). However, overall, the numerical and laboratory results show very close agreement both in the shape of the loop current and in the critical values of *Q* between which the multiple states are possible.

*L*becomes comparable to the half-width of the gap

_{P}*a*. This follows from the balance between the zonal advection and the

*β*effect [see (19) of Sheremet 2001]. In the present notationwhere

*β*= 2Ω

*S*/

*H*

_{0}is the topographic

*β*effect. For the observed critical value

*Q*= 55 cm

^{3}s

^{−1},

*L*= 8.2 cm; thus, the nondimensional ratio

_{P}*L*/

_{P}*a*= 1.17 is as close to unity as expected.

*L*becomes comparable to

_{L}*a*, which follows from balancing the meridional advection and the

*β*effect [see (16) of Sheremet 2001]. In the present notationwhere

*L*is the width of the western boundary current. In (Sheremet 2001)

_{B}*L*was the Munk thickness

_{B}*L*; in the present case the width of the western boundary current is determined by a combination of

_{M}*L*,

_{M}*L*, and

_{S}*L*defined in (14), (13), and (15), respectively. If we assume that roughly

_{I}*L*=

_{B}*L*+

_{M}*L*+

_{S}*L*, for the critical we get

_{I}*Q*= 23 cm

^{3}s

^{−1}and

*L*= 13 cm and the ratio is

_{L}*L*/

_{L}*a*= 1.9.

Equations (17) and (18) should be understood only as scaling estimates. However, they suggest different dependence on the flow rate *Q*: the power of ⅓ versus 1. This explains the divergence between the critical values of *Q* as the gap width is increased: they should grow as ∼ *a*^{3} and ∼ *a*, respectively. That is why in the experiments with the gap width 2*a* = 9.3 cm, the hysteresis was hardly seen with critical values close to 12 cm^{3} s^{−1} while for larger gap width 2*a* = 14.0 cm the hysteresis was very dramatic, with multiple flow patterns existing within 23 ≤ *Q* ≤ 55 (cm^{3} s^{−1}). Thus, the experimental results are consistent with these scaling laws.

## 5. Conclusions

Both the laboratory and numerical experiments presented in this paper dramatically show the existence of multiple states and hysteresis in the gap-leaping problem. The dominant variation in the flow pattern is concentrated near the gap while the upstream and downstream conditions are almost not affected. This indicates that it is the differences in separation and reattachment near the gap that lead to the multiple states similar to Sheremet (2001). Such behavior is well known, there are many examples of multiple states and hysteresis in fluid dynamics: flow past a wing, flow past a veer (Coanda effect), flow past a cavity; however, it is the first laboratory example where the *β* effect is the crucial player. Without the *β* effect no westward intensification (Sverdrup 1947; Stommel 1948) occurs and no western boundary current forms. The flow goes from the source to sink as a broad current without entering the gap at all. The idealized geometry of the present problem makes it an excellent test case for numerical models having geophysical applications. The laboratory results also warrant a more scrutinized search for the observational evidence of multiple states and hysteresis in strait flows.

## Acknowledgments

This work was generously supported by the Office of Naval Research Grant N00014-02-1-0472 and by the National Science Foundation Grant OCE-0351518. We thank two anonymous reviewers for their valuable comments.

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## APPENDIX

### Numerical Model

*R*to nondimensionalize the horizontal scales) in the complex plane (

*x*,

*y*) into an infinite strip (−

*π*/2 <

*σ*<

*π*/2, −∞ <

*τ*< ∞) in the complex plane (

*σ*,

*τ*). The circular ridge is then transformed into a line

*σ*= const. In terms of real functions the transformation isHowever, a numerical grid based on this conformal mapping is not convenient because it has higher density of nodes near the Poles (

*x*= 0,

*y*= −1) and (

*x*= 0,

*y*= 1) where it is not needed. Therefore, in the second step we stretch individual variables:The unit circle in (

*x*,

*y*) becomes a square −1 <

*ξ*< 1, −1 <

*η*< 1. A uniform grid in (

*ξ*,

*η*) will have a uniform step in

*x*along

*y*= 0 due to (A4) and will have a uniform step in

*ϕ*along the circumference

*r*= 1 due to (A5), where

*x*+

*iy*=

*r*exp(

*iϕ*) (see Fig. A1). The ridge will be located at

*ξ*= −⅔. Further stretching can be applied in the third step:

*ξ*=

*ξ*(

*ξ*′),

*η*=

*η*(

*η*′) if one desires increased resolution near the ridge or the boundaries. An example of such stretching can be found in Sheremet (2001). Another option is to use stretching based on the collocation points of Chebyshev polynomials. Here for simplicity we omit that step. The Lamé coefficients of the orthogonal curvilinear system (

*ξ*,

*η*) arewhere

*h*is the scalar factor of the conformal mapping (A1):and

*σ*and

*τ*should be expressed in terms of

*ξ*and

*η*according to (A4) and (A5).

The expressions for the differential operators present in the system of equations in (10) and (11) in a general orthogonal curvilinear coordinate system can be found, for example, in Batchelor (2000, his appendix 2). We approximate all terms in the equations using standard centered differences on a uniform grid with *N _{ξ}* ×

*N*cells covering the region −1 <

_{η}*ξ*< 0, −1 <

*η*< 1. We used a grid (90 × 190) for most of the calculations and checked convergence by conducting selected calculations on a (180 × 380) grid. Steady solutions were obtained by the method of equilibration (as the limiting solutions at large time).

^{1}

Professor I. S. Gromeka (1851–89) was a distinguished Russian physicist on the faculty at Imperial Kazan University and was known for his fundamental works in hydromechanics—in particular, Gromeka (1885).