2 Linear theory

The system under consideration is shown schematically in figure 1(a,b) and consists of a rectangular $\unicode[STIX]{x1D6FD}$-plane basin with mean depth $D$ and zonal and meridional extent $L_{x}$ and $L_{y}$, respectively. In the laboratory context, $\unicode[STIX]{x1D6FD}=f_{0}s/D$ is provided by variable depth with constant bottom slope $s$ (Pedlosky & Greenspan Reference Pedlosky and Greenspan1967), where $f_{0}=2\unicode[STIX]{x1D6FA}$ is the Coriolis frequency and $\unicode[STIX]{x1D6FA}$ the frequency of tank rotation, anticlockwise when viewed from above. The basin is divided into western, $x_{w}\leqslant x\leqslant x_{1}$, and eastern, $x_{2}\leqslant x\leqslant x_{e}$, sub-basins by a meridional barrier extending from $x_{1}$ to $x_{2}$. The sub-basins are connected by two meridional gaps of length $d$ and the island barrier between the gaps extends from $y_{1}$ to $y_{2}$. We focus here on an island placed with north–south symmetry such that $y_{1}=L_{y}-y_{2}$, noting that our results may be straightforwardly generalized to asymmetric barrier geometries. We follow the approach outlined by Pedlosky (Reference Pedlosky2000b) to derive the linear theory for Rossby basin modes interacting with an island barrier, modifying the solution for the barrier geometry shown in figure 1(a). We summarize the essential aspects of the derivation here; interested readers are directed to Pedlosky (Reference Pedlosky2000b), as well as closely related work in Pedlosky (Reference Pedlosky2001) and Pedlosky & Spall (Reference Pedlosky and Spall1999), for more detail.

Figure 1.

(a,b) Geometry of basin, barrier and forcing for linear theory and laboratory experiments. (a) $xy$-plane. (b) $yz$-plane. (c) Velocity response along the eastern side of the barrier when $n$ is even and odd.

In keeping with the theoretical studies mentioned above, we model the flow using the linear barotropic quasi-geostrophic potential vorticity equation for the stream function $\unicode[STIX]{x1D6F9}(x,y,t)$, given in non-dimensional form by

(2.1)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6F9}_{t}+\unicode[STIX]{x1D6F9}_{x}=-r\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6F9}+A\unicode[STIX]{x1D6FB}^{4}\unicode[STIX]{x1D6F9}+W(x,y,t).\end{eqnarray}$$

Here $x$ and $y$ are scaled by $L_{y}$ and time $t$ by $(\unicode[STIX]{x1D6FD}L_{y})^{-1}$. The last term is a forcing function with dimensional amplitude $W_{0}$ and the stream function is scaled by $W_{0}L_{y}/\unicode[STIX]{x1D6FD}$. The coefficients $r$ and $A$ arise from bottom Ekman drag (and surface drag in the case of a no-slip rigid lid) and lateral friction, respectively.

It should be noted that while here we model the linearized barotropic problem using quasi-geostrophic dynamics, in the case of strong topographic slopes the rotating shallow water equations may be a more appropriate choice of model (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2002). However, for the problem described here, the simpler framework of the quasi-geostrophic model allows us to better illuminate the mechanism by which waves may be transmitted across the barrier, as described below. The theory outlined here could also be straightforwardly extended to include baroclinic modes (as in e.g. Pedlosky (Reference Pedlosky2000a)) or nonlinear effects.

The forcing is located in the eastern sub-basin along $x=x_{f}$ with structure

(2.2)$$\begin{eqnarray}W(x,y,t)=\unicode[STIX]{x1D6FF}(x-x_{f})\text{e}^{\text{i}\unicode[STIX]{x1D714}_{0}t}\mathop{\sum }_{n=1}^{\infty }W_{n}\sin n\unicode[STIX]{x03C0}y,\end{eqnarray}$$

in which $\unicode[STIX]{x1D6FF}(x)$ is the Dirac delta function and the coefficients $W_{n}$ for integer $n\geqslant 1$ specify the meridional structure of the forcing. Accordingly, we seek forced harmonic solutions to (2.1) of the form $\unicode[STIX]{x1D6F9}=\text{Re}[\unicode[STIX]{x1D713}(x,y)\text{e}^{\text{i}\unicode[STIX]{x1D714}_{0}t}]$. Solutions for $\unicode[STIX]{x1D713}(x,y)$ are found separately in each sub-basin and the two gaps and then matched along the island meridians. Generally $A\ll 1$ and the lateral friction is ignored in each sub-basin. However, as the relevant length scale within the gaps $d\ll L_{y}$, we retain lateral friction within the gaps. Bottom drag is allowed throughout the domain using the shifted frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{0}-\text{i}r$.

This full solution includes the unknown stream function on the island, $\unicode[STIX]{x1D6F9}_{I}=\text{Re}[\unicode[STIX]{x1D713}_{I}\text{e}^{\text{i}\unicode[STIX]{x1D714}_{0}t}]$, with $\unicode[STIX]{x1D713}_{I}$ constant. As in similar problems of large-scale circulation in multiply connected domains (Godfrey Reference Godfrey1989; Pedlosky et al. Reference Pedlosky, Pratt, Spall and Helfrich1997), $\unicode[STIX]{x1D6F9}_{I}$ is found though integration of the tangential component of the horizontal momentum equation along a closed contour $C_{I}$ that girdles the island (see figure 1c) giving a version of Kelvin’s circulation theorem

(2.3)$$\begin{eqnarray}\text{i}\unicode[STIX]{x1D714}\oint _{C_{I}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}l-A\oint _{C_{I}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}l=0,\end{eqnarray}$$

where $\boldsymbol{n}$ is the unit normal of $C_{I}$. Note that since lateral friction is neglected in the sub-basins, the second term in (2.3) only applies to the solution in the gaps, and viscous effects along the eastern and western edges of the barrier are neglected. When the solutions for $\unicode[STIX]{x1D713}$ in the basins and along the barrier are substituted into (2.3), an algebraic expression for this integral constraint,

(2.4)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}_{I}\left[\mathop{\sum }_{n=1}\frac{\unicode[STIX]{x1D707}_{n}g_{n}a_{n}\cos n\unicode[STIX]{x03C0}y_{1}}{n\unicode[STIX]{x03C0}}\frac{\sin a_{n}[L_{x}-l_{x}]}{\sin a_{n}(x_{2}-x_{e})\sin a_{n}(x_{1}-x_{w})}-\frac{2(1+q)l_{x}/d}{(1+\text{i})(1+q)-2\unicode[STIX]{x1D70C}(1-q)}\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-\text{i}\mathop{\sum }_{n=1}\frac{\unicode[STIX]{x1D707}_{n}W_{n}\cos n\unicode[STIX]{x03C0}y_{1}}{\unicode[STIX]{x1D714}n\unicode[STIX]{x03C0}}\exp \text{i}k(x_{2}-x_{f})\frac{\sin a_{n}(x_{f}-x_{e})}{\sin a_{n}(x_{2}-x_{e})},\end{eqnarray}$$

is found. Here, $a_{n}^{2}=k^{2}-n^{2}\unicode[STIX]{x03C0}^{2}$ with $k=1/(2\unicode[STIX]{x1D714})$, $l_{x}=x_{2}-x_{1}$ and $\unicode[STIX]{x1D707}_{n}=1-(-1)^{n}$. The coefficients $g_{n}$ correspond to the structure of $\unicode[STIX]{x1D713}$ along the barrier, with $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}_{I}g(y)=\unicode[STIX]{x1D713}_{I}\sum _{n}g_{n}\sin n\unicode[STIX]{x03C0}y$, and $\unicode[STIX]{x1D70C}$ and $q$ are related to the structure of the oscillatory boundary layer in the gaps (Batchelor Reference Batchelor2000; Pedlosky Reference Pedlosky2000b). Equation (2.4) is similar in structure to equation (2.18) of Pedlosky (Reference Pedlosky2000b); the different barrier geometry considered here is reflected in the $\cos n\unicode[STIX]{x03C0}y_{1}$ terms in (2.4) as well as in the details of the coefficients $g_{n}$. Importantly, if the barrier geometry and structure of the forcing are known, equation (2.4) can be solved for the island constant $\unicode[STIX]{x1D713}_{I}$, allowing for predictions of the transmission of wave energy across the barrier into the western sub-basin.

When $n$ is even, $\unicode[STIX]{x1D713}_{I}=0$ for the $y$-symmetric island geometry described here. The meridional velocity along the eastern side integrates to zero as illustrated in figure 1(c). The integral (2.3) indicates that no response in the western basin is required. However, when $n$ is odd the velocity along the eastern side is generally single signed so that integral along the eastern side is finite, requiring a similar magnitude response on the western side of the island in order to satisfy the integral constraint. This is the origin of a finite $\unicode[STIX]{x1D713}_{I}$, hence large transmission even for small gaps. Kelvin’s theorem effectively turns the island into an antenna. It should be noted that this result is not restricted to the linearized governing equation (2.1); integration around the island would still imply some response on the western side of the island to satisfy the integral constraint even if considering the full nonlinear equations of motion. However, in the case of highly nonlinear flows, this response may not necessarily correspond to transmission of the large-scale Rossby basin modes (as in the linear system).

For an isolated island, the linear theory predicts a resonant response at approximately the normal-mode frequencies associated with the full basin in the absence of the barrier as well as those for each individual sub-basin (Pedlosky & Spall Reference Pedlosky and Spall1999; Pedlosky Reference Pedlosky2000b). In non-dimensional form these frequencies are

(2.5)$$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{1}{2\unicode[STIX]{x03C0}\sqrt{n^{2}+(m/l_{b})^{2}}},\end{eqnarray}$$

in which the integers $m$ and $n$ refer, respectively, to the mode number in the $x$ and $y$ directions and $l_{b}$ is the zonal extent of either the sub-basin or the full domain (scaled by $L_{y}$). For the largest-scale basin modes ($m=n=1$) and the experimental geometry considered here (see table 1), these frequencies are $\unicode[STIX]{x1D714}_{F}=0.1125$, $\unicode[STIX]{x1D714}_{E}=0.0846$ and $\unicode[STIX]{x1D714}_{W}=0.0548$ for the full, eastern and western basins, respectively.

Table 1.

Laboratory parameters corresponding to figure 1(a,b).

Figure 2.

(a) Magnitude of theoretical island constant $|\unicode[STIX]{x1D713}_{I}|$ as a function of the forcing frequency $\unicode[STIX]{x1D714}_{0}$ and bottom friction $r$ for a barrier at $(y_{2}-y_{1})/L=1/2$. The top axis shows the equivalent dimensional forcing frequencies for the laboratory experiments described in later sections. (b–e) Spatial structure of $\unicode[STIX]{x1D713}(x,y)$, predicted by the linear theory, $\unicode[STIX]{x1D714}_{0}=0.01$ (non-dimensional, different $r$, and different barrier lengths: (b) $r=0.0001$, $(y_{2}-y_{1})/L=1/2$; (c) $r=0.005$, $(y_{2}-y_{1})/L=1/2$; (d) $r=0.0001$, $(y_{2}-y_{1})/L=5/6$; (e) $r=0.0001$, $(y_{2}-y_{1})/L=1/6$. Note that the colour bars have different limits in each panel.

Figure 2 shows the resulting magnitude of the island constant calculated from (2.4) as a function of the forcing frequency, $\unicode[STIX]{x1D714}_{0}$, for $A=10^{-6}$ and several values of the linear friction $r$. Also shown are the locations of $\unicode[STIX]{x1D714}_{F}$, $\unicode[STIX]{x1D714}_{E}$ and $\unicode[STIX]{x1D714}_{W}$. Resonant peaks are apparent in the inviscid and low-friction cases, albeit shifted to lower frequencies for the geometry considered here. As $r$ is increased and Ekman friction plays a larger role, the peak amplitudes decrease until eventually there is no clear signal of resonance. However, it should be emphasized that the predicted $|\unicode[STIX]{x1D6F9}_{I}|$ is still non-zero between the resonant peaks and for large $r$. That is, while the linear theory predicts an enhanced response near particular resonant forcing frequencies, the transmission of energy across the barrier is not itself a resonance phenomenon: some amount of wave energy is predicted to transmit from the eastern to the western sub-basin even for forcing frequencies away from the resonant peaks.

Also shown in figure 2 is the spatial structure of $\unicode[STIX]{x1D713}(x,y)$ predicted by the linear theory for different barrier lengths and values of the linear friction $r$. The overall spatial structure of $\unicode[STIX]{x1D713}$ is clearly dependent on both $r$ and the geometry of the barrier, particularly in the vicinity of the barrier itself. For example, stronger flows (corresponding to stronger gradients in $\unicode[STIX]{x1D713}$) are observed closer to the gaps. It is clear that while increasing the linear friction $r$ leads to a weaker response overall, it does not prevent the transmission of wave energy from the eastern to the western sub-basin (figure 2b,c) – there is a strong signal in the predicted stream function on both sides of the barrier. On the other hand, the barrier geometry is expected to impact the amount of wave energy transmitted between sub-basins, consistent with the dependence on $y_{1}$ in (2.4). Barriers with longer islands and shorter peninsulas lead to a strong response in both sub-basins. Conversely, when the gaps are very close together and the barrier is short (figure 2e), the linear theory predicts very little transmission of wave energy across the barrier. This is not surprising, given that we expect that in the limit of a vanishingly short island we expect that the incident wave will only ‘see’ a single gap, and thus will behave like waves in the classical diffraction problems discussed in § 1 with very little wave energy transmitted across the barrier.

3 Laboratory set-up

The experimental set-up is shown in figure 1(a,b), with the corresponding parameters listed in table 1. From this point forward, all results will be in dimensional form. The laboratory apparatus consists of a square tank with a sloping bottom on a rotating table to create a laboratory analogue to the $\unicode[STIX]{x1D6FD}$-effect (Pedlosky & Greenspan Reference Pedlosky and Greenspan1967). It should be noted that, for the parameters considered here, the change in depth between the north and south boundaries is not small compared to the overall water depth, and so strictly speaking our topographic $\unicode[STIX]{x1D6FD}$ is not constant (van Heijst Reference van Heijst1994). However, the bottom slope of $s=0.133$, while not tiny, is also not $O(1)$, suggesting that the quasi-geostrophic approach outlined in the previous section may still be viable. Additionally, our experimental results show that the variable topographic $\unicode[STIX]{x1D6FD}$ does not have a leading-order impact on the resulting flow dynamics – the essential features of the flow are still captured. The short peninsulas and island are constructed from 0.32 cm thick acrylic sheeting that extend over the full depth of the fluid. The tank is fitted with a rigid lid to eliminate surface gravity–capillary modes in the system. A 45 cm long, 0.32 cm wide acrylic paddle is periodically forced by a stepper-motor driven scotch yoke mechanism at radian frequencies $O$(0.10) s^-1 and amplitudes up to 3 cm.

To examine the behaviour of the Rossby waves as they interact with the barrier, we perform a series of particle image velocimetry (PIV) experiments. Saltwater with density $\unicode[STIX]{x1D70C}\approx 1.020~\text{g}~\text{cm}^{-3}$ is used as the working fluid and seeded with 50𝜇 plastic particles with density $\unicode[STIX]{x1D70C}\approx 1.016~\text{g}~\text{cm}^{-3}$. A pulsed (1–10 Hz) 532 nm green laser is used to illuminate the flow and horizontal velocities are obtained using LaVision’s DaVis software (v7.2). Velocities are measured in a horizontal plane approximately 3 cm below the rigid lid, with a spatial resolution of approximately 4 mm in both the $x$ and $y$ directions.

Using the laboratory parameters in table 1, we can estimate several important quantities describing our experimental set-up. The Ekman layer depth, relative to the mean depth $D$, is $\unicode[STIX]{x1D6FF}_{Ek}/D=\sqrt{2\unicode[STIX]{x1D708}/f_{0}}/D\sim 4\times 10^{-3}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. We can then estimate the non-dimensional linear friction drag in (2.1) as $r=\unicode[STIX]{x1D6FF}_{Ek}\,f_{0}/\unicode[STIX]{x1D6FD}DL_{y}$; for our set-up, $r\sim 0.01$ (where we have taken into account the presence of the rigid lid). The linear friction drag can also be expressed as $r=2\unicode[STIX]{x1D6FF}_{S}/L_{y}$, where $\unicode[STIX]{x1D6FF}_{S}=\sqrt{2\unicode[STIX]{x1D708}f_{0}}/2D\unicode[STIX]{x1D6FD}$ is the Stommel boundary layer thickness scale, here equal to approximately 3 mm. Finally, the Munk scale $\unicode[STIX]{x1D6FF}_{M}=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D6FD})^{1/3}$ is approximately 8 mm for the experimental parameters here. This allows us to estimate the non-dimensional lateral friction from (2.1) as $A=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D6FD}L_{y}^{3}=(\unicode[STIX]{x1D6FF}_{M}/L_{y})\sim 2\times 10^{-6}$. Additionally, the Munk scale relative to the gap width is $\unicode[STIX]{x1D6FF}_{M}/d<0.25$, preventing the boundary layers from overlapping and significantly impeding flow through the gap.

In each experiment, the tank is initially spun up for 20 min. The forcing is then turned on and allowed to run for at least an additional 20 min (or longer for the slowest forcing frequencies) in order to allow initial transients to die out before taking measurements. In the majority of our experiments, both of the barrier gaps shown schematically in figure 1(a) are left open; however, we also perform additional experiments with one or both of the gaps closed.

5 Discussion and conclusions

Here we have presented laboratory experiments corresponding to the theory of Pedlosky (Reference Pedlosky2000b) and Pedlosky (Reference Pedlosky2001), which predicts that barriers with more than one small gap may be quite inefficient in preventing the transmission of Rossby waves through the barriers. This result, arising from the requirement that circulation be conserved around individual segments of the barrier, is in contrast to classical diffraction problems in which only a small fraction of wave energy may be transmitted through a single gap when the incident wave is of much larger scale than the gap. Our experiments confirm the primary prediction of the linear inviscid quasi-geostrophic theory in a real fluid: while barriers with zero or only one small gap successfully prevent the transmission of large-scale wave modes from one sub-basin to the other, when two small gaps are open $O(1)$ transmission of basin-scale Rossby waves across the barrier is possible.

Comparisons between the linear inviscid theory and the experimental results indicate that the theory captures the large-scale structure of the flow; however, preliminary analysis of the results points to the importance of additional physics not captured by the linear theory in its current form. Both nonlinearity and viscous effects appear to play a role in the flow structure as the waves pass from the eastern to the western sub-basin, resulting in the formation of vortices and strong boundary currents in the vicinity of the barrier. When considering the integral constraint (2.4) derived from Kelvin’s circulation theorem, an imbalance in the circulation around the island is found, suggesting that these physical features (neglected in the theory) may play an important role in setting the response in the western sub-basin. However, even in these highly nonlinear cases, significant transmission of wave energy across the barrier is still observed. That is, the nonlinear features alone are insufficient to satisfy Kelvin’s circulation theorem around the barrier, necessitating the wave response in the western sub-basin.

Interestingly, although the relatively strong topographic slope considered here might suggest that a shallow water model would be more suitable in describing the dynamics, the experimental results agree well with the main predictions of the quasi-geostrophic theory. Indeed, quasi-geostrophic theory has been found to work well outside of parameter regimes where it is formally applicable in other laboratory and numerical studies (e.g. Williams, Read & Haine Reference Williams, Read and Haine2010), although the reasons for this good agreement are not necessarily well understood.

Here we have considered a range of forcing frequencies, and three forcing amplitudes. Further extending the range of $A_{forcing}$ to both higher and lower values would help to further elucidate the roles of viscosity and nonlinearity. For example, smaller forcing amplitudes would decrease the relative importance of nonlinear effects and therefore help to clarify the role played by viscosity alone. At the other extreme, increasing the forcing amplitude, and thereby the influence of nonlinear effects on the flow, would also be of interest in future studies. For example, in simulations of the flow around isolated islands due to localized surface wind stress forcing, Pedlosky et al. (Reference Pedlosky, Pratt, Spall and Helfrich1997) found that vortices were able to form, detach and propagate away from the barrier. They also observed meanders forming due to detaching western boundary currents. Examining whether these effects might occur for the barrier geometry and forcing considered here at higher values of $A_{forcing}$ would be of great interest as they would allow for an additional means by which wave energy may be transmitted into the western sub-basin.

A further extension of the problem considered here would be to consider alternate geometries of both the forcing and the barrier. For example, by considering a forcing structure in which $n$ is even would allow for confirmation of the prediction in (2.4) that the symmetric two-gap barrier is opaque to wave modes with $n$ even. While the lack of transmission of symmetric modes has been observed in forced linearized numerical model results (Pedlosky & Spall Reference Pedlosky and Spall1999), the addition of nonlinear and viscous effects would be of interest to explore experimentally. Moreover, as Pedlosky & Spall (Reference Pedlosky and Spall1999) describe (see their figure 7), additional gaps in the barrier allow for different ways in which the waves may be transmitted through the barrier; comparing these scenarios with laboratory results could reveal further interesting features of the flow. Shifting the location of the gaps along the barrier would also help to more clearly separate the effects of the wave transmission from the formation of the strong meridional flows described in § 4.3, allowing for both phenomena to be studied.

Previous nonlinear numerical studies of forced Rossby waves interacting with topography with two or more small gaps have shown the ability of the waves to transmit across the barrier into the neighbouring sub-basin, consistent with the predictions from applying an integral constraint around the topography. However, these studies were either adiabatic (Pedlosky & Spall Reference Pedlosky and Spall1999) or considered large scales (Spall & Pedlosky Reference Spall and Pedlosky2005) where viscous effects were not expected to be of leading-order importance. As such, numerical simulations corresponding to the laboratory length scales would also be of great interest. For example, while it is difficult to compute contributions due to viscosity from the measured velocities, owing to the noisy nature of the data and the higher derivatives that appear in viscous terms, these terms could be more easily computed from numerical data and thus further clarify the details of the integral constraint. Additionally, numerical simulations may allow for a broader range of forcing amplitudes to be considered (not being limited by either the ability to measure very small velocities nor the physical set-up of the forcing paddle) and a wider range of forcing geometries, including ones in which $n$ is strictly even or further isolated in space (i.e. localized in the meridional, as well as the zonal direction).

Finally, while we have focused here on the case of Rossby waves, the concept of applying a circulation integral to the flow around a barrier island is much broader. We would expect that the requirement that such an integral constraint be satisfied, leading to barriers which are surprisingly inefficient at blocking the transmission of wave energy, to hold for any vorticity-containing wave for which Kelvin’s theorem is not automatically satisfied. Additionally, the requirement that the circulation constraint be satisfied is not limited to wave motions alone. For example, island chains may be only weak barriers to eddy propagation, even when eddy patch radii are much larger than the gaps in the barrier (Johnson & McDonald Reference Johnson and McDonald2005). As such, the fact that Kelvin’s theorem may require a flow response on both sides of an island may lead to surprisingly transparent barriers in a wide variety of flow scenarios.