2.1. Rectangular channel theory

As in the Gill (Reference Gill1977) model, the potential vorticity $q^*={(f+{\partial v^*}/{\partial x^*}-{\partial v^*}/{\partial y^*})}/{d^*}$ is assumed to be constant, eliminating Rossby waves from consideration. The value of the potential vorticity may be obtained by evaluating the expression for potential vorticity in the stagnant region far to the east of the boundaries, where the dimension depth is $D_\infty$, and thus $q^*={f}/{D_\infty }$. The scale for potential vorticity is ${|f|}/{D}$.

The non-dimensional expression for the semigeostrophic potential vorticity q in a Southern Hemisphere channel is then

(2.1)\begin{equation} q=\frac{-1+\dfrac{\partial v}{\partial x}}{d}={-}\frac{D}{D_\infty}. \end{equation}

The semigeostrophic approximation also requires that the northward velocity is geostrophically balanced

(2.2)\begin{equation} v={-}\frac{\partial (d+h)}{\partial x}. \end{equation}

Assuming the bottom elevation $h(y)$ in the along-channel direction and eliminating v between these two yields a relationship for the cross-channel (x) structure of the layer thickness $d$

(2.3)\begin{equation} \frac{\partial^2 d}{\partial x^2} - |q|d ={-}1. \end{equation}

Since there is no upstream influence in our multi-passage system, the western boundary current can be defined independently and can solely represent the upstream condition. Assuming that the upstream basin is infinitely wide compared with the Rossby radius of deformation in the upstream basin $L_D$, the solution of flow along the western wall ($x=0$) in the upstream region may be written as

(2.4) \begin{equation} \left. \begin{array}{c@{}} d(x)=|q|^{{-}1}+[d(x=0)-|q|^{{-}1}]\,{\rm e}^{-|q|^{1/2}x},\\ v(x)=|q|^{1/2}[d(x=0)-|q|^{{-}1}]\,{\rm e}^{-|q|^{1/2}x}. \end{array}\right\} \end{equation}

The solutions of (2.3) in the channel are given by

(2.5) \begin{equation} \left. \begin{array}{c@{}} d(x)=|q|^{{-}1}+\hat{d}\dfrac{\sinh[|q|^{1/2}(x-x_0)]}{\sinh(\tfrac{1}{2}|q|^{1/2}w)} +(\bar{d}-|q|^{{-}1})\dfrac{\cosh[|q|^{1/2}(x-x_0)]}{\cosh(\tfrac{1}{2}|q|^{1/2}w)},\\ v(x)=|q|^{1/2}\left[\hat{d}\dfrac{\cosh[|q|^{1/2}(x-x_0)]}{\sinh(\tfrac{1}{2}|q|^{1/2}w)} +(\bar{d}-|q|^{{-}1})\dfrac{\sinh[|q|^{1/2}(x-x_0)]}{\cosh(\tfrac{1}{2}|q|^{1/2}w)}\right], \end{array}\right\} \end{equation}

where $x_0$ is at the centre of the channel, $w$ is the width of the channel,

(2.6a,b)\begin{equation} \bar{d}=\frac{d\left(x_0-\dfrac{w}{2}\right)+d\left(x_0+\dfrac{w}{2}\right)}{2}\quad {\rm and}\quad \hat{d}=\frac{-d\left(x_0-\dfrac{w}{2}\right)+d\left(x_0+\dfrac{w}{2}\right)}{2}, \end{equation}

represent the average and difference of the flow thickness at walls. The corresponding average and difference of $v$ at the walls are given by

(2.7) \begin{equation} \left. \begin{array}{c@{}} \bar{v}=\dfrac{v\left(x_0-\dfrac{w}{2}\right)+v\left(x_0+\dfrac{w}{2}\right)}{2}={-}|q|^{1/2}T^{{-}1}\hat{d},\\ \hat{v}=\dfrac{-v\left(x_0-\dfrac{w}{2}\right)+v\left(x_0+\dfrac{w}{2}\right)}{2}={-}|q|^{1/2}T(\bar{d}-|q|^{{-}1}), \end{array}\right\}, \end{equation}

where $T=\tanh (\frac {1}{2}|q|^{1/2}w)$.

We then define a transport streamfunction $\varPsi$ from the along-channel velocity $vd={\partial \varPsi }/{\partial x}$. Since the interior of the upstream basin is stagnant, the streamfunction there is constant. For convenience, we take $\varPsi =0$ in the interior basin ($x\gg 0$), which allows $\varPsi$ at any streamline to be given by the flow flux between that streamline and the stagnant basin interior. For example, the western wall can be treated as a single streamline extending from the upstream basin toward the channel because of the condition of no normal flux. Given the transport $Q$ of the western boundary current in the upstream basin, we find $\varPsi =-Q$ along the western wall. Here, $Q$ can be approximated by integrating $vd\equiv -d({\partial d}/{\partial x})$ from $x=0$ to a point in the basin interior where $d(x\gg 0)\approx |q|^{-1}$ and $v(x\gg 0)\approx 0$

(2.8)\begin{equation} Q\simeq{-}\frac{|q|^{{-}2}-d^2(x=0)}{2}.\end{equation}

From (2.4) and (2.8), we get $d(x=0)\simeq \sqrt {2Q+\!|q|^{-2}}$ and $v(x=0)\simeq |q|^{1/2}(\sqrt {2Q+\!|q|^{-2}} -\!|q|^{-1})$. For an inviscid flow that has a constant $q$, the Bernoulli function $B\!=\!{v^2}/{2}\!+d\!+h$ is conserved along streamlines (Crocco Reference Crocco1937)

(2.9)\begin{equation} q=\frac{\mathrm{d}B}{\mathrm{d}\varPsi}.\end{equation}

Integrating (2.9) from the basin interior where $B(x\gg 0)\approx |q|^{-1}$ and utilizing $\varPsi (x=0)=-Q$, the Bernoulli function at the western wall is determined by $Q$ and $q$

(2.10)\begin{equation} B(x=0)\simeq|q|Q+|q|^{{-}1}. \end{equation}

The no-normal-flux boundary condition also ensures that the island boundary is a streamline ($\varPsi =-\varPsi _0$). If the upstream inflow splits as it approaches the entrance of the channel, a streamline originating from the upstream western boundary current area would meet the island boundary at a stagnation point upstream of the sill ($P$ in figure 2a). The streamfunctions at that streamline and at the island boundary share the same value. It is worth noting that a stagnation point located downstream of the sill on the island boundary is also possible, in which case the flow immediately to the east of the splitting streamline has to turn around as it approaches the stagnation point. An upstream/downstream stagnation point corresponds to a uniformly northward flow/reversal flow in the eastern part of the channel. In regions upstream of and at the sill, where the flow cannot undergo a hydraulic jump, we may assume that $v$ and $d$ are continuous and compute the transport through the channel $Q_1$ by integrating $-d({\partial d}/{\partial x})$ between the two sidewalls of the channel

(2.11)\begin{equation} Q_1={-}2\hat{d}\bar{d}. \end{equation}

From definition, $\varPsi _0=Q-Q_1=Q+2\hat {d}\bar {d}$.

The average Bernoulli function in the channel

(2.12)\begin{align} \bar{B}&=\frac{B\left(x_0-\dfrac{w}{2}\right)+B\left(x_0+\dfrac{w}{2}\right)}{2}\nonumber\\ &=\frac{v^2\left(x_0-\dfrac{w}{2}\right)+v^2\left(x_0+\dfrac{w}{2}\right)}{4} +\frac{d\left(x_0-\dfrac{w}{2}\right)+d\left(x_0+\dfrac{w}{2}\right)}{2}+h, \end{align}

can be simplified by substituting ${(v^2(x_0-{w}/{2})+v^2(x_0+{w}/{2}))}/{2}$ with $\bar {v}^2+\hat {v}^2$ and employing (2.7)

(2.13)\begin{equation} \bar{B}=\tfrac{1}{2}|q|[T^{{-}2}\hat{d}^2+T^2(\bar{d}-|q|^{{-}1})^2]+\bar{d}+h. \end{equation}

Also, $\bar {B}$ can be rearranged as ${(B(x_0+{w}/{2})-B(x_0-{w}/{2}))}/{2}+B(x_0-{w}/{2})$. Since $q$ is constant, the first term can be written in terms of $Q_1$ and $q$ following (2.9)

(2.14)\begin{equation} \bar{B}={-}\frac{|q|Q_1}{2}+B\left(x_0-\frac{w}{2}\right),\end{equation}

where $B(x_0-{w}/{2})$ is conserved along the streamline at the western wall, and thus can be expressed by $Q$ and $q$ from (2.10).

Combining (2.13) and (2.14) by substituting (2.10) and (2.11) relates a single unknown flow property in the channel $\bar {d}$ to the local geometric parameters $h, w$ by the equation

(2.15)\begin{align} \mathcal{G}(\bar{d}\,|\,h;w;q;Q;Q_1) & = \frac{Q^2_1}{4T^2\bar{d}^2}+T^2(\bar{d}-|q|^{{-}1})^2+2|q|^{{-}1}\bar{d}\nonumber\\ &\quad -2|q|^{{-}1}(|q|Q+|q|^{{-}1}-h)+Q_1=0. \end{align}

We are primarily interested in flows that are hydraulically controlled by a sill (maximum bottom elevation) that lies within the channel. Such flows approach the sill in a subcritical state that allows upstream propagation by at least one of the waves permitted under the semigeostrophic approximation. Upon reaching the sill, the flow undergoes a transition to a supercritical state in which the same wave propagates downstream. At the transition the wave is stationary, meaning that it is technically possible to locally alter the steady flow without changing the upstream conditions. As shown by Gill (Reference Gill1977), this critical condition is given by ${\partial \mathcal {G}}/{\partial \bar {d}}|_c=0$ and leads here to

(2.16)\begin{equation} (1-T^2_c)|q|^{{-}1}+T^2_c\bar{d}_c=\frac{Q^2_{1_c}}{4T_c^2\bar{d}_c^3},\end{equation}

where the subscript $()_c$ denotes quantities evaluated at the critical section. Elimination of $\bar {d}_c$ between (2.15), evaluated at the critical section, and (2.16) determines the critical transport in the channel $Q_{1_c}$ in terms of the topography of the critical section $w_c$, $h_c$, and the upstream conditions $q$ and $Q$. In the dimensionalized form, the determination of $Q^*_{1_c}$ depends on $w^*_c$, $h^*_c$, $D_\infty$ and $Q^*$.

It is possible that the flow in the channel may become separated from the eastern wall and thereby take on a width $w_e$ that is less than the local channel width. In this case, it is easy to show that $Q_1=2\bar {d}^2$, so that $\bar {d}$ becomes fixed, and it is better to use $w_e$ as the primary variable characterizing the flow. Equation (2.15) is now replaced by

(2.17)\begin{align} \mathcal{G}(T_e\,|\,\bar{d};h;q;Q) & = T_e^2(\bar{d}-|q|^{{-}1})^2+T_e^{{-}2}\bar{d}^2+2\bar{d}^2+2|q|^{{-}1}\bar{d}\nonumber\\ &\quad -2|q|^{{-}1}(|q|Q+|q|^{{-}1}-h)=0, \end{align}

where $T_e=\tanh (|q|^{1/2}w_e/2)$. The width and the transport of a separated flow under hydraulic control can be solved by substituting the critical condition ${\partial \mathcal {G}}/{\partial T_e}|_c=0$ into (2.17).

2.2. Parabolic channel theory

The need to distinguish between separated and non-separated states is a consequence of the unrealistic assumption of vertical sidewalls and is eliminated by the use of a channel cross-section with continuously varying bottom elevation. A parabolic cross-section is almost invariably a better approximation to conditions in nature and has been considered by Borenäs & Lundberg (Reference Borenäs and Lundberg1986) within the context of uniform potential vorticity. We do, however, continue to assume that the upstream basin is bounded by vertical walls, and we employ the same streamline configuration as in the rectangular channel case. Therefore, the upstream condition is specified by the value of the potential vorticity and by the transport of the upstream western boundary current. Now we give the derivation of the analogue of (2.15) and (2.16) for a parabolic channel and the procedure for estimating the partitioning of inflow transport.

Consider a channel with parabolic cross-sectional bottom elevation, in dimensional variables

(2.18)\begin{equation} h^*=h^*_0+{\alpha}(x^*-x_0^*)^2,\end{equation}

where $h^*_0$ is the height of the channel bottom at its deepest point, and has value zero in the upstream bottom. Equation (2.18) is non-dimensionalized using $L_d=\sqrt {g'D}/|f|$ for the cross-sectional dimension $x$ and $D$ for the bottom topography $h$ to yield

(2.19)\begin{equation} h=h_0+\frac{(x-x_0)^2}{r}, \end{equation}

where $r={|f|^2}/{g'\alpha }$ is the non-dimensional shape parameter, large for a ‘wide’ channel (one for which the radius of curvature ${\gg }L_d$).

The non-dimensional governing equations for the flow in a parabolic channel are given by

(2.20) \begin{equation} \left. \begin{array}{c@{}} \displaystyle\dfrac{\partial^2 d}{\partial x^2} - |q|d ={-}1-2r^{{-}1},\\ \displaystyle v={-}\dfrac{\partial (d+h)}{\partial x}, \end{array}\right\} \end{equation}

which can be solved analytically given the boundary conditions $d(x_0-a)=d(x_0+b)=0$, where $x=x_0-a$ and $x=x_0+b$ are the locations where the interface intersects the bottom (figure 2d)

(2.21) \begin{equation} \left. \begin{array}{c@{}} d(x)=\dfrac{1+2r^{{-}1}}{|q|\sinh(|q|^{1/2}(a+b))}[\sinh(|q|^{1/2}(x-x_0-b))\\ \hspace{8pt}-\sinh(|q|^{1/2}(x-x_0+a))]+\dfrac{1+2r^{{-}1}}{|q|},\\ v(x)={-}\dfrac{1+2r^{{-}1}}{|q|^{1/2}\sinh(|q|^{1/2}(a+b))}[\cosh(|q|^{1/2}(x-x_0-b))\\ -\cosh(|q|^{1/2}(x-x_0+a))]-2r^{{-}1}(x-x_0). \end{array}\right\} \end{equation}

The volume flux $Q_1$ can be obtained by integrating $-d({\partial (d+h)}/{\partial x})$ across the channel

(2.22)\begin{align} Q_1&=(a-b)\frac{1+2r^{{-}1}}{|q|r}\{2|q|^{{-}1/2}[\sinh^{{-}1}(|q|^{1/2}(a+b))\nonumber\\ &\quad -\coth(|q|^{1/2}(a+b))]+(a+b)\}, \end{align}

and the wall-average Bernoulli function in the channel is again defined by

(2.23)\begin{equation} \bar{B}=\frac{B(x_0-a)+B(x_0+b)}{2}, \end{equation}

where $B(x_0-a)$ and $B(x_0+b)$ are Bernoulli functions at the western and eastern edges of the flow, respectively,

(2.24) \begin{align} \left. \begin{aligned} B(x_0-a)&=\dfrac{1}{2}\left\{-\dfrac{1+2r^{{-}1}}{|q|^{1/2}\sinh(|q|^{1/2}(a+b))}[\cosh(|q|^{1/2}(a+b))-1]+2r^{{-}1}a \right\}^2\\ &\quad +h_0+\dfrac{a^2}{r},\\ B(x_0+b)&=\dfrac{1}{2}\left\{-\dfrac{1+2r^{{-}1}}{|q|^{1/2}\sinh(|q|^{1/2}(a+b))}[1-\cosh(|q|^{1/2}(a+b))]-2r^{{-}1}b \right\}^2\\ &\quad +h_0+\dfrac{b^2}{r}. \end{aligned}\right\} \end{align}

As before, we invoke conservation of the Bernoulli function and volume flux between the upstream region, where ((2.10), (2.14)) are still valid, and the sill section of the parabolic channel. This connection is made for streamlines that originate upstream and that pass through the channel. Use of (2.22)–(2.24) leads to an equation for the width ($a+b$) of the current

(2.25) \begin{align} &\mathcal{G}((a+b)\,|\,h_0;r;q;Q;Q_1)\nonumber\\ &\quad = \frac{1}{2}\left\{\frac{(1+2r^{{-}1})[\cosh(|q|^{1/2}(a+b))-1]}{|q|^{1/2}\sinh(|q|^{1/2}(a+b))}\right\}^2\nonumber\\ &\qquad -\frac{(1+2r^{{-}1})[\cosh(|q|^{1/2}(a+b))-1]}{r|q|^{1/2}\sinh(|q|^{1/2}(a+b))}(a+b)\nonumber\\ &\qquad +\frac{r+2}{4r^2}[(a+b)^2+(a-b)^2]+\frac{|q|Q_1}{2}-|q|Q-|q|^{{-}1}+h_0=0, \end{align}

where $(a-b)$ is connected with $(a+b)$ via (2.22)

(2.26)\begin{align} (a-b)&=\frac{|q|rQ_1}{1+2r^{{-}1}}\{2|q|^{{-}1/2}[\sinh^{{-}1}(|q|^{1/2}(a+b))\nonumber\\ &\quad -\coth(|q|^{1/2}(a+b))]+(a+b)\}^{{-}1}. \end{align}

The dimensionalized form of (2.25) is $\mathcal {G^*}((a^*+b^*)\,|\,h^*_0;\alpha ;D_\infty ;Q^*;Q^*_1)=0$. The critical condition can be achieved by taking ${\partial \mathcal {G}}/{\partial (a+b)}|_c=0$

(2.27)\begin{align} &\frac{2(1+2r_c^{{-}1})^2}{|q|^2r_c^2}[{\rm csch}(|q|^{1/2}(a+b)_c)- \coth(|q|^{1/2}(a+b)_c)]\nonumber\\ &\qquad \times \{|q|^{{-}1/2}(r_c+2)\,{\rm csch}(|q|^{1/2}(a+b)_c)[{\rm csch}(|q|^{1/2}(a+b)_c)-\coth(|q|^{1/2}(a+b)_c)]\nonumber\\ &\qquad +|q|^{{-}1/2}+(a+b)_c\,{\rm csch}(|q|^{1/2}(a+b)_c)\}+\frac{(1+2r_c^{{-}1})^2}{|q|^2r_c^2}(a+b)_c\nonumber\\ &\quad =\frac{2{\rm csch}(|q|^{1/2}(a+b)_c)[{\rm csch}(|q|^{1/2}(a+b)_c)-\coth(|q|^{1/2}(a+b)_c)]+1}{\{2q^{{-}1/2}[{\rm csch}(|q|^{1/2}(a+b)_c)-\coth(|q|^{1/2}(a+b)_c)]+(a+b)_c\}^3}Q^2_{1_c}, \end{align}

where all variables with subscript $c$ denote quantities evaluated at the critical section. Solving the controlled volume flux $Q_{1_c}$ by eliminating $(a+b)_c$ between $\mathcal {G}((a+b)_c\,|\,h_{0_c};r_c;q;Q;Q_{1_c})$ and (2.27) is not easy. Instead, we solve the critical level of $(a+b)_c$ numerically by optimizing $Q_{1_c}$ while satisfying the constraints (2.25)–(2.27) using the interior-point algorithm (Byrd, Gilbert & Nocedal Reference Byrd, Gilbert and Nocedal2000).

2.3. Dependence of the flow variable on geometry

The theory described in §§ 2.1 and in 2.2 can be employed to predict the partitioning of the inflow transport $Q$ given the upstream conditions ($q$ and $Q$) for different sill topographic parameters ($w_c$ and $h_c$ for a rectangular cross-section or $r_c$ and ${h_0}_c$ for a parabolic cross-section). Furthermore, various topographic regimes can be characterized by the different features of the flow as shown in figures 3 and 4(a). Only results for the upstream conditions $q=-1$ and $Q=0.5$ are shown, which are selected to represent the inflow upstream of the Samoan Passage (see the Samoan Passage application in more detail in § 4.1). Different $q$ and $Q$ yield qualitatively similar regime boundaries.

Figure 3. The theoretical critical transport $Q_{1_c}$ in the rectangular channel is contoured as a function of the channel width $w_c$ and bottom height $h_c$ at the critical section (sill). The upstream condition is set by the potential vorticity ($q=-1$) and volume transport ($Q=0.5$) of the upstream inflow. The transport to the east of the island is $Q-Q_{1_c}$. Above the curve $Q_{1_c}=0$ the channel flow is topographically blocked; below the curve $Q_{1_c}=0.5$ the volume transport in the channel exceeds that of the upstream flow and the theory is considered invalid (dark shades). The flow is separated from the eastern wall to the right of the dashed curve (light shades). The triangles show the predicted $Q_{1_c}$ for the Samoan Passage topography.

Figure 4. (a) Value of $Q_{1_c}$ as a function of $r_c$ and ${h_0}_c$ for a hydraulically controlled flow in a parabolic channel from theory (contours). The inflow has a potential vorticity of $q=-1$, and a transport of $Q=0.5$. The flow tends to be blocked by the sill in the regime of large ${h_0}_c$. Below the $Q_{1_c}=0.5$ contour is the regime where the theory becomes no longer valid (dark shades). On the right of the dashed grey curve, the predicted channel flow at the western boundary is southward, suggesting a reversal circulation (light shades). Numbers in black and red are results from the theory and numerical model, respectively; ‘nan’ (dark shades) represents null values. Insets (b–i) show the time-mean ($100\leq t\leq 400$) northward velocity $v$ (red suggesting positive) and interface height $d+h$ (black contours) in the channel of selected topographic parameters from the numerical simulation.

Figure 3 suggests four separate regimes. Above the $Q_{1_c}=0$ contour, the flow is topographically blocked by the sill. In the regime below the $Q_{1_c}=0.5$ contour, the predicted critical transport $Q_{1_c}$ is larger than the inflow transport $Q$, meaning that part of the transport must be sourced from regions to the east or north of the island. If this is the case, the potential vorticity of the channel flow is partially set by downstream conditions, and the stagnation point (P in figure 2a) would no longer exist. We have elected not to venture into this regime since the conditions do not seem realistic. In the regime lying to the right of the dashed curve, the flow is separated from the eastern wall at the sill section. Therefore the flow is controlled only by the bottom elevation $h_c$ while $Q_{1_c}$ remains constant for an increasing $w_c$. However, historical laboratory experiments (Shen Reference Shen1981; Pratt Reference Pratt1987) and numerical simulations (Pratt, Helfrich & Chassignet Reference Pratt, Helfrich and Chassignet2000) have both suggested that hydraulic control of separated sill flows is very difficult to establish, so this regime may only exist in theory. In the regime lying between the $Q_{1_c}=0$ and $Q_{1_c}=0.5$ contours, and on the left of the separation curve, the flow in the channel is hydraulically controlled and $Q_{1_c}$ decreases with increasing $h_c$ and decreasing $w_c$.

As shown in figure 4(a), there are four topographic regimes for flow in the channel with a parabolic cross-section. Similar to the rectangular cross-section case, the flow is likely to be blocked by the sill in the regime of large ${h_0}_c$ (especially when ${h_0}_c\geq 1.4$), where the theory predicted $Q_{1_c}$ is very close to zero. In the topographic regime below the $Q_{1_c}=0.5$ contour, the predicted $Q_{1_c}$ is larger than $Q$, and the theory is again considered invalid. Different from the rectangular cross-section case, there is no boundary separation for a flow in a channel with varying topography. However, for a wide channel with $r_c$ lying on the right of the dashed curve, the theory produces a southward flow at the western boundary of the channel, suggesting a reversal circulation. One may imagine that a stagnation point must be located upstream of the sill at the western boundary, which connects the streamline following the western boundary and the streamline that constitutes the other boundary of the reversal circulation. In the final regime (left of the dashed curve, above $Q_{1_c}=0.5$ and moderate ${h_0}_c$), the predicted channel flow is hydraulically controlled.

3.1. Numerical method

The numerical model used for this study was first described in Helfrich, Kuo & Pratt (Reference Helfrich, Kuo and Pratt1999) in a study of the nonlinear Rossby adjustment problem in a rotating channel. The model solves the non-dimensional shallow-water equations in flux form

(3.1)$$\begin{gather} \frac{\partial}{\partial t}(ud)+\frac{\partial}{\partial x}\left(u^2d+\frac{1}{2}d^2\right)+\frac{\partial}{\partial y}(uvd)-{\rm sign}(f)vd+d\frac{\partial h}{\partial x}={-}\lambda{ud}, \end{gather}$$

(3.2)$$\begin{gather}\frac{\partial}{\partial t}(vd)+\frac{\partial}{\partial y}\left(v^2d+\frac{1}{2}d^2\right)+\frac{\partial}{\partial x}(uvd)+{\rm sign}(f)ud+d\frac{\partial h}{\partial y}={-}\lambda{vd}, \end{gather}$$

(3.3)$$\begin{gather}\frac{\partial d}{\partial t}+\frac{\partial }{\partial x}(ud)+\frac{\partial }{\partial y}(vd)=0, \end{gather}$$

where $\lambda$ is the Rayleigh friction coefficient. Here, we choose to work with a linear parameterization of the bottom friction (or the ‘Rayleigh friction’ representing the momentum loss in a bottom Ekman layer) because it is more numerically stable than the more commonly used quadratic bottom drag. The model allows for ‘dry regions’ (defined by layer thickness $d<10^{-4}$) and permits the formation of sharp hydraulic jumps and bores, at the same time ensuring that the proper shock-joining conditions on mass and momentum flux are satisfied. The model assumes constant background rotation ($f$-plane physics) and the $x$-direction (cross-channel) and $y$-direction (along-channel) continue to act as eastward and northward coordinates. The non-dimensional variables in the model have been formed using slightly different scaling than that given in the theory. In particular (i) lengths in the $x$-direction and the $y$-direction are both scaled by the Rossby radius of deformation $L_d$ and therefore $u$ and $v$ are both scaled by the free long gravity wave speed $\sqrt {g'D}$ (i.e. non-semigeostrophic; $|v/u|\sim O(1)$); (ii) the time is scaled by $1/|f|$ (i.e. time dependent); (iii) $\lambda$ is scaled by $|f|$.

As shown in figure 5(a), a channel with a parabolic cross-section is introduced between the island and the western boundary of the model domain (figure 5b). The channel contains an obstacle having a Gaussian shape in the along-channel direction (figure 5c)

(3.4)\begin{equation} h(x,y)=\left[h_s+\frac{(x-x_s)^2}{r_s}\right]{\rm e}^{-{(y-y_s)^2}/{2\sigma^2_s}},\quad 0\leq x\leq 2x_s, \end{equation}

where $(x_s,y_s)$ and $(r_s,h_s)$ denote the location and the topographic parameters of the topographic saddle point (sill), respectively, and $\sigma _s$ indicates the channel length. The sloping western sidewall in the channel joins continuously to the vertical sidewalls of the western boundary north and south of the channel. A logistic function (figure 5b) makes up the sloping eastern boundary of the island, which gradually joins the sea bottom

(3.5)\begin{equation} h(x,y)=\frac{1}{1+{\rm e}^{{(x-x_i)}/{\sigma_i}}}h(0,y),\quad x>2x_s, \end{equation}

where $x_i$ and $\sigma _i$ indicate the starting point of the island's eastern slope and the span of the slope, respectively.

Figure 5. (a) Model domain and topography. The initial layer surface ($d+h=1$) is represented by a light blue sheet. (b) The cross-channel topography at $y_s=35$, looking towards downstream. The channel is parabolic. (c) The side view of a section across the deepest point in the channel, showing the along-channel Gaussian topography. The topographic parameters $\sigma _s$, $\sigma _i$ and $x_i$ are set to 5, 5 and 30, respectively.

We consider a set of simulations in which the flow in the interior of the domain is initially at rest, and the interface height $d+h$ is set to unity. At $t=0$, a steady, northward, geostrophically balanced flow with total volume flux $Q=0.5$ and with potential vorticity $q=-1$ is introduced across the southern boundary of the model domain. The profiles of northward velocity and layer thickness take the form of the boundary current described by (2.4). The boundary conditions at the western and eastern walls of the domain are free slip and no normal flux, and radiation conditions (Orlanski Reference Orlanski1976) are used at the northern boundary of the domain. The numerical domain is 60 units in both the $x$-direction and the $y$-direction, with spacing in the $y$-direction of $\Delta y=0.1$. The cross-channel grid spacing and time steps range from ($\Delta x=0.1$ and $\Delta t=0.01$) for wide channels with $r_s>1$ to ($\Delta x=0.05$ and $\Delta t=0.002$) for narrow channels with $r_s<1$. The validity of the numerical results was tested by reducing $\Delta x$, $\Delta y$ and $\Delta t$ by one half and by doubling the range of $x$ and $y$ while maintaining the same channel and island geometry. The former set of experiments can better resolve features in regions of hydraulic transients (see § 3.3) but also show more oscillations near discontinuities. Besides these small oscillations, all test runs show essentially the same overall flow patterns as models used for the analysis.

3.2. Friction and no-friction experiments

Several friction and no-friction numerical runs were made for comparison with results from the theory and to explore phenomena such as hydraulic transitions and signal transmissions from downstream that are not addressed by the theory. In a no-friction experiment, the Rayleigh friction coefficient $\lambda$ is zero everywhere in the model domain. In a friction experiment, $\lambda$ is set to zero between the western wall ($x=0$) and $x=15$, which includes the channel and much of the western boundary layer to keep the calculations close to the inviscid theories, whereas $\lambda =0.2$ is added to $x>15$, which includes the boundary layer on the east coast of the island.

Numerical results from two sets of topographic parameters ($r_s=1.3$, $h_s=0.6$) and ($r_s=0.2$, $h_s=0.6$), which represent the Samoan Passage (see details in § 4.1) and a nominal narrow channel, respectively, are examined closely. Time series of the volume transport in the channel ($Q_1$) and east of the island ($Q_2$) were calculated at $y_s$=35, the latitude of both the sill and the minimum width in the channel. From inviscid theories, this topographic saddle point coincides with the critical section (i.e. $r_c=r_s$, ${h_0}_c=h_s$) had the flow becomes hydraulically controlled in the channel (Pratt & Whitehead Reference Pratt and Whitehead2008, § 1.4).

3.2.1. Persistent hydraulic control in the friction experiments

As shown in figure 6(c,d,g,h), beginning at $t\approx 50$, $Q_1$ from the simulation becomes quite close to the critical value ${Q_1}_c$ predicted by the theory for ($r_c=r_s$, ${h_0}_c=h_s$), suggesting that the hydraulic control is established after the adjustment of Kelvin waves and frontal waves to the topography of the channel (Pratt et al. Reference Pratt, Helfrich and Chassignet2000). The constant $Q_1$ after $t\approx 50$ persists until $t\approx 250$, then $Q_1$ from the no-friction experiments starts to drop, while $Q_1$ from the friction experiments remains relatively unchanged. After $t\approx 400$, the transport calculated at the southern boundary of the model domain $Q$ starts to show irregular variations, which may be due to the contamination with Kelvin waves that are initialized by the errors of the radiation boundary condition at the northern boundary of the domain and propagate along the domain's eastern boundary into the upstream section. The small decrease in $Q_1$ after $t\approx 400$ in figure 6(c) is likely due to the same process. Aside from these numerical artifacts, only the friction experiments yield persistent hydraulic control, while the no-friction experiments show a continuous transport adjustment. In fact, $Q_1$ in figure 6(g,h) continues to decrease towards zero as we run the simulation until $t=2000$ (not shown). We have run experiments with $\lambda =0.2$ everywhere of the model domain and also found a persistent hydraulic control in the channel, except for the magnitude difference in $Q_1$ due to the friction-induced flow spindown (not shown).

Figure 6. Numerical results from the (a–d) friction experiments (Rayleigh friction added to $x>15$) and the (e–h) no-friction experiments. (a,e) Flow speed $\sqrt {u^2+v^2}$ at $t=800$, with colour bar shown at the upper right corner of (a). The interface heights $d+h$ of 1.2 and 1.4 are marked with thick black contours. The bathymetry contours $h=0.6$ along the western boundary of the channel and around the island are shown by the cyan and green dashed curves respectively in (a) and partly in (b). (b, f) A magnified view of the modelled flow in the channel ($x\in [0, 6]$). Meridional velocity $v$ is shown in colours (red for northward flow and blue for southward flow, with colour bar shown at the lower left corner of (b)). The $d+h$ contours are shown at 0.1 intervals. The grey lines indicate the location of the sill. Panels (a,b,e, f) are all for topographic parameters ($r_s=1.3$, $h_s=0.6$). Volume transport time series for parameters ($r_s=1.3$, $h_s=0.6$) (c,g) and ($r_s=0.2$, $h_s=0.6$) (d,h). Here, $Q$, $Q_1$ and $Q_2$ represent the transport of the flow in the upstream basin, in the channel and along the eastern flank of the island, respectively. Theory predictions ${Q_1}_c$ for a hydraulically controlled channel flow at ($r_c=r_s$, ${h_0}_c=h_s$) are indicated by the dashed lines.

3.2.2. Transport adjustment by frontal waves in the no-friction experiments

In order to examine the role that transients may play in the transport adjustment, we show Hovmöller diagrams of the interface height $d+h$ along two circuits: one following the western boundary of the channel (the cyan dashed curve in figure 6a,b) and the other circling the island (the green dashed curve in figure 6a,b). Both of the circuits are isobaths with $h=0.6$. The coastal trapped waves that propagate along these circuits see zero layer thickness along the respective in-shore edges and are similar to the frontal waves described by Stern (Reference Stern1980). Like southern hemisphere Kelvin waves, they tend to propagate with the boundary to their left, at least in the absence of a mean flow. The Hovmöller diagrams for the west circuit figure 7(a,c), show that a set of these waves arrives in the channel prior to $t\approx 50$, carrying with them the initial surge in transport and establishing hydraulic control at the sill. In the friction experiments, after the hydraulic control is established, $d+h$ in regions upstream of and at the sill remains relatively constant with time. The closed contours downstream of the sill denote eddies (also seen in figure 6b). However, in the no-friction experiments, $d+h$ changes significantly in regions upstream of and at the sill after $t\approx 250$ and eddy formation ceases downstream of the sill.

Figure 7. Time evolution of the interface height ($d+h$) at the bathymetry contour $h=0.6$ (a,c) along the western boundary of the channel (west circuit: cyan dashed curve in figure 6a) and (b,d) around the island (east circuit: green dashed curve in figure 6a). Panels (a,b), (c,d) show results from the friction and no-friction experiments, respectively. Both experiments use topographic parameters ($r_s=1.3$, $h_s=0.6$). The $y$ coordinate in (a,c) is the distance from the most upstream point of the west circuit, in (b, d) is the distance from P1 around the island in a counter-clockwise direction. The contour interval is 0.05. The frontal waves that propagate counter-clockwise along the island and penetrate the channel from downstream are highlighted in a grey arrow in (d) with the phase speed marked ($c=0.3$).

The evolving interface elevation $d+h$ along the east circuit is shown in figures 7(b) and 7(d). For the case without friction, figure 7(d), the leading edge of frontal waves circles the island (counter-clockwise along segment P1–P2) with a speed of $c\approx 0.3$ and penetrates back up into the channel at $t\approx 300$ with a seemingly higher speed (segment P2–sill–P1). Thereafter, $d+h$ gradually increases between P2 and the sill, leading to flooding of the hydraulic control at the sill. Comparison with figure 6(e, f), reveals the presence of a circum-island flow, in particular, some southward penetration of this flow back into the channel from the north. In the friction experiments, the frontal waves along the east circuit disperse due to the frictional drags and do not appear to enter the channel (figure 7b). This may be explained by the diffusive boundary layer as a result of bottom friction derived by Pratt (Reference Pratt1997), whose width expands proportionally with the square root of the distance from the flow source.

3.2.3. Comparing theory with model

Here, we confine attention to the friction model output, since the no-friction model does not support a steady-state hydraulic control in the channel. Time-averaged $Q_1$ from $t=300$ to $t=400$ has been calculated at selected topographic parameters (triangles in figures 4a and 8a), and the numerical results are compared with ${Q_1}_c$ predicted by the theory at ($r_c=r_s$, ${h_0}_c=h_s$). Horizontal and cross-sectional flow structures are shown in the insets in figures 4 and 8, respectively. The most notable difference in flow structures between model and theory is in the regime to the right of the dashed curve in figures 4(a) and 8(a), where the theory predicts a southward flow at the western boundary of the channel (red arrows in figure 8g,h,i) while the model produces a jet-like flow positioned in the centre of the channel (blue arrows in figure 8g,h,i). Besides these differences, the model and theory produce similar $Q_1$ above the ${Q_1}_c=0.5$ isocontour (light shaded area). Below the ${Q_1}_c=0.5$ isocontour (dark shaded area), where the theory is formally invalid, the predicted ${Q_1}_c$ from theory is much larger than the inflow transport of 0.5 but the model yields $Q_1$ values close to 0.5.

Figure 8. Same as in figure 4, but insets for $v$ and $d+h$ at the sill ($y_s$). Red vectors and curves are analytical solutions from the theory. Blue vectors and areas are results from numerical runs averaged over $100\leq t\leq 400$.

3.3. Hydraulic transition in the friction experiments

Hydraulic jumps in rotating systems have been studied primarily in channels with rectangular cross-sections. A variety of structures have been found, including jumps that resemble a stationary Kelvin bore (Pratt Reference Pratt1983; Pratt et al. Reference Pratt, Helfrich and Chassignet2000). If the approaching supercritical flow is separated from one of the sidewalls of the channel, the jump can take the form of a sudden expansion in the flow width, so that the downstream (subcritical state) has finite depth all across the channel (Pratt Reference Pratt1987; Pratt et al. Reference Pratt, Helfrich and Chassignet2000; Pratt, Riemenschneider & Helfrich Reference Pratt, Riemenschneider and Helfrich2007). In the latter case, the abrupt change in fluid depth that typically occurs across a jump is absent: depth changes may occur but they are more gradual. Here we will show that supercritical-to-subcritical transitions occur in the friction experiments, the novel feature being that they occur in a channel with a rounded cross-section. In this subsection, numerical results are averaged from $t=100$ to $t=800$.

3.3.1. A subcritical-to-supercritical-to-subcritical transition of the channel flow

In order to assess the flow criticality within the channel, we have computed the eigenmodes (the cross-stream structures of wave amplitudes) and corresponding propagation speeds of waves (eigenvalues) at positions along the channel following Pratt et al. (Reference Pratt, Riemenschneider and Helfrich2007). Each mode corresponds to a pair of waves, and we are particularly interested in long frontal waves, in which the two members of a pair are trapped to opposite edges. The linear wave mode calculation has been done by assuming a parallel flow in a steady state and including a small viscous term in the perturbation equations as detailed in Appendix A. The background flow velocity $V(x)$ is the time-averaged meridional velocity from various numerical runs and the background profile of the flow thickness $D(x)$ is computed by integrating $-V(x)-h_x$, in accordance with the semi-geostrophic balance relations (2.2). This is to avoid the large $D_x$ at the edges of the flow due to the numerical scheme in which a very thin layer of water ($d=0.0001$) was placed on top of the ‘dry’ areas.

As shown in figure 9, the lowest modes (0th modes: modes with no zero crossings) correspond to a pair of frontal waves that decay from the western ($\hat {v}_+$ in figure 9b,e,h) and eastern ($\hat {v}_-$ in figure 9c, f,i) edges of the flow. These waves tend to propagate in the downstream and upstream directions, respectively, provided that the background flow is stagnant. If that was the case, the flow is subcritical everywhere in all presented simulations. However, with the presence of the background flow, some of these waves may become stationary or propagate backward, thus hydraulic transitions may occur. In the no-friction experiments, the two frontal waves are found to propagate in opposite directions throughout the channel (figure 9g), indicating subcritical flow throughout the channel, at least with respect to the lowest modes. In the friction experiments, for topographic parameters representing the Samoan Passage ($r_s=1.3$, $h_s=0.6$), a transition from subcritical to critical to supercritical occurs near the sill ($34< y<35.5$) (figure 9a). The supercritical flow then terminates in an abrupt transition back to subcritical flow ($y\approx 36$). In the narrow-channel experiments ($r_s=0.2$, $h_s=0.6$), the transition from subcritical to supercritical occurs more upstream of the sill and the supercritical flow occupies a larger range in $y$. A transition back to subcritical flow occurs further downstream of the sill (figure 9d). Nevertheless, this hydraulic transition of flow confirms that the channel flow in the friction experiments is, in fact, hydraulically controlled, an essential assumption of the theory proposed in § 2.

Figure 9. The left (a–c), middle (d–f) and right (g–i) panels show modal results calculated from time-mean model output from the friction experiment, friction experiment for a narrow-channel case and no-friction experiment, respectively. Here, $h_s=0.6$ for all three experiments. The top panels show the phase speeds ($c_r$, the subscript $()_r$ denotes the real component) for the lowest modes along the channel. The bottom panels show the cross-sectional structure of the eigenfunction $\hat {v}$ normalized by its maximum for the lowest mode. The solid black curves, thick grey curves and dashed black curves represent modes at an upstream section $y=32$, the sill $y=35$ and a downstream section $y=38$, respectively (see a, d, g). For a pair of frontal wave modes, $\hat {v}_+$ and $\hat {v}_-$ correspond to waves that are downstream propagating and upstream propagating relative to the background flow, respectively.

3.3.2. The change in energy and potential vorticity through the hydraulic transition

The supercritical-to-subcritical hydraulic transition that occurs immediately downstream of the sill in the friction experiments shows very different characteristics from hydraulic jumps in rotating channels with rectangular cross-sections in that (i) streamlines over the sill do not show a sudden expansion in width (figure 10a); (ii) the interface height of flow does not show an abrupt ‘jump’ (figure 10b). Despite the gradual variations in flow characteristics across the hydraulic transition, it is of interest to see how much energy dissipation occurs in the transition and to further assess the implications for the distribution of potential vorticity in the flow. We will demonstrate in § 3.4 that the persistent hydraulic control in the friction experiments is a result of the frictional dissipation along the island balancing the energy dissipation in the hydraulic transition.

Figure 10. Time-averaged results from the friction experiment for the Samoan Passage topographic parameters ($r_s=1.3$, $h_s=0.6$). (a) Transport streamlines with $\varPsi$ ranging between $-$0.5 and $-$0.42 (dashed black contours) and between $-$0.38 and $-$0.28 (solid black contours). The contour interval is 0.02. (b) The magnitude of transport $|\boldsymbol {u}d|$ (colours) and the interface height $d+h$ (thin black contours). The thick dashed black curves and solid black curves in (a) and (b) represent streamlines $\varPsi =-0.46$ and $\varPsi =-0.3$, respectively. Flow properties along the two streamlines are shown in (c–f). The grey horizontal lines in (a,b) and vertical lines in (c–f) mark the location of the sill.

We have based our theory in § 2 on the criteria that for an inviscid, steady flow, $B$ is conserved along a streamline and $q$ remains constant (2.9). As shown in figure 10(e, f), $B$ and $q$ indeed remain constant along the two selected streamlines shown in figure 10(a,b) until slightly downstream of the sill. Note that the results are from the friction experiments where friction is added only to the east of the island and the flow is inviscid in the channel. We know from the wave mode analysis that a transition of flow from subcritical to critical to supercritical takes place near the sill, which is consistent with the descending flow surface (figure 10c) and the flow acceleration (figure 10d) along streamlines. As the flow passes the sill and across the region of hydraulic transition from supercritical to subcritical, both $B$ and $q$ decrease along streamlines. The acceleration ceases while the flow continues to descend. Since only frontal waves that decay from the eastern edge of the flow change directions across the hydraulic transition and thus contribute to the hydraulic transition, it is not surprising that changes in $B$ and $q$ along the streamline near the eastern edge of the flow exceed those along the streamline near the western boundary of the channel.

To assess the distribution of potential vorticity in the flow, we show in figure 11(b) $q$ at sections upstream of the sill, at the sill and downstream of the sill, respectively. In the channel with smoothly varying cross-section, $d$ goes to zero at the flow edges and can result in extreme $q$ and $B$ (figure 11a–c). Away from the flow edges, $q$ remains at its initial value imposed at the southern boundary of the model domain upstream of and at the sill, but it is reduced significantly after the supercritical-to-subcritical transition is passed. The change in $q$ across a hydraulic jump in a rotating channel can be calculated from the energy dissipation from shock theory (Pratt Reference Pratt1983)

(3.6)\begin{equation} \langle q \rangle=\frac{{\rm d}\langle B \rangle}{{\rm d}\varPsi}, \end{equation}

where $\langle ()\rangle =\lim _{\varepsilon \to 0} [()_2-()_1]$ represents the change in the indicated property across a shock. Although our model does not show a typical shock, we can still quantitatively estimate the change in $q$ by applying (3.6) to the region containing the hydraulic transition. From model output, we estimate $\langle B \rangle$ and $\langle q \rangle$ from the differences between a downstream section $y=40$ and the sill $y=35$ (‘$\times$’ symbols in figure 11d,e), and between the sill and a upstream section $y=30$ (‘$+$’ symbols), respectively. Over $35\leq y\leq 40$, $\langle B \rangle$ and $\langle q \rangle$ both show a large magnitude at larger $\varPsi$, which corresponds to streamlines near the eastern edge of the flow; $\langle q \rangle$ calculated from (3.6) overestimates but agrees with the order of magnitude of change (lower solid curve in figure 11d). On the contrary, $\langle q \rangle$ computed over $30\leq y\leq 35$ and predicted from theory (upper solid curve in figure 11d) are both near zero. This further confirms the existence of a dissipative hydraulic transition downstream of the sill, across which the potential vorticity decreases. Since the friction model contains no explicit friction in this region, the dissipation must arise from implicit dissipation due to the numerical scheme. The same is true for in past simulations using the same model, e.g. Pratt et al. (Reference Pratt, Helfrich and Chassignet2000).

Figure 11. (a–c) Flow thickness $d$, potential vorticity $q$ and Bernoulli function $B$ at the southern boundary of the model domain $y=0$ (dotted curves), the upstream channel entrance at $y=30$ (solid black curves), the sill at $y=35$ (grey thick curves) and a downstream section at $y=40$ (dashed black curves). Triangles and circles mark locations of streamlines $\varPsi =-0.46$ (thick dashed curves in figure 10a,b) and $\varPsi =-0.3$ (thick solid curves in figure 10a,b), respectively. (d,e) Along-streamline change in $q$ and $B$ between the sill and the upstream section (‘$+$’ symbols) and between the downstream section and the sill (‘$\times$’ symbols), respectively. Results are from the time-averaged friction model output for topographic parameters ($r_s=1.3$, $h_s=0.6$), except that the solid curves in (d) are predictions from shock theory.

3.4. The hydraulics of splitting flow

The circulation integral (1.1) expresses the need for friction to act on the east side of the island in order to support a steady, hydraulically controlled flow in the channel. We now return to the steady form of this integral as it applies in the numerical model

(3.7)\begin{equation} \oint_{C_l}(f+\zeta)\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d}s =\delta{B}-\oint_{C_l}\lambda\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{l}\,\mathrm{d}s, \end{equation}

and contrast the various terms and balances that arise in the friction and no-friction experiments. In both model settings, the island topography varies smoothly. Therefore, at the edge of the flow, the layer thickness vanishes and values of velocity are particularly susceptible to numerical noise. With this in mind, we have opted to consider an alternative contour $C_l$ that lies slightly offshore of the island where the layer depth is finite and noise is a minor issue. The constructed $C_l$ consists of a streamline that coincides with the eastern edge of the flow in the channel $S_4$, a streamline around the island $S_2$, a meridional segment $S_1$ and a zonal segment $S_3$ that connect $S_2$ and $S_4$ at the entrance and downstream of the channel, respectively (figure 12a). The transport streamfunction and all the variables in (3.7) are computed from the friction model output averaged over $100\leq t\leq 800$. The Rayleigh friction coefficient $\lambda$ was set to zero within the channel and 0.2 at $x>15$ for the friction experiments, therefore the integrations of the friction term along $S_1$ and $S_4$ are both zero. Equation (3.7) can then be re-written as

(3.8)\begin{equation} -\int_{S_1}q(ud)\,\mathrm{d} y+\int_{S_3}q(vd)\,\mathrm{d}\kern0.7pt x =\delta{B}-\int_{S_2}\lambda\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{l}\,\mathrm{d}s+\int_{S_3}\lambda u\,\mathrm{d}\kern0.7pt x. \end{equation}

Figure 12. (a) An example of an integration contour for Kelvin's circulation theorem analysis: $S_1$ along $x=2.1$, $S_2$ the streamline of $\varPsi =-0.1$, $S_3$ along $y=45.9$ and $S_4$ the streamline of $\varPsi =-0.3$. (b) Terms of the simplified Kelvin circulation theorem for a steady flow as expressed in (3.8). Results are calculated from different choices of integration circuits and shown with black vertical lines as $\pm$ one standard deviation about the mean. Also shown are the sum of terms on the left-hand side (LH) of (3.8), the sum of the right-hand side (RH) terms and the difference between the two terms (LH-RH). (a) and (b) are both from the time-averaged friction model output. (c) Same as in (b) but for the terms calculated from the no-friction model output averages. Topographic parameters for both experiments are ($r_s=1.3$, $h_s=0.6$).

We calculate the terms in (3.8) using several, slightly different, choices of the integration contour; $S_2$ and $S_4$ are defined to coincide with $\varPsi$ ranging from $-$0.1 to $-$0.02 and $-$0.38 to $-$0.24, respectively; $S_1$ and $S_3$ are chosen to lie in the range of $1\leq x\leq 3$ and $44\leq y\leq 50$, respectively. Bringing together the terms in (3.8) calculated from all the choices of integration circuits shows that the left and right sides of the equation are nearly balanced (figure 12b). In particular, the positive Bernoulli difference between upstream and downstream of the hydraulic transition is primarily balanced by the negative bottom friction integral.

For comparison, we compute the transport streamfunction from the time-averaged no-friction numerical model results and calculate all the terms in (3.8) (figure 12c). Choices for the integration contour are the same as for the friction model results, except that we restrict $S_4$ to a more limited range of $\varPsi$, $-$0.38 to $-$0.32 due to the large noise near the eastern edge of the flow. For the no-friction experiments, the frictional dissipation term is exactly zero. The gain of potential vorticity is also very close to zero with very small variance. A positive time evolution term $({\partial }/{\partial t})\oint _{C_l}\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {l}\,\mathrm {d}s$ is needed to balance the small but positive $\delta {B}$, which explains the accelerating counter-clockwise circulation around the island shown in figure 6(e).

4.1. Samoan Passage transport from theory

In the non-dimensional version of our theory, the potential vorticity $q$ and the upstream inflow volume flux $Q$ are upstream conditions required to estimate the critical transport through the channel ${Q_1}_c$. In the dimensionalized form, the potential depth $D_\infty$ serves as one upstream condition required for our theory. The interface of the Samoan Passage overflow is chosen to be the 1 $^\circ$C isotherm, the most commonly accepted upper boundary of the DWBC in the Southern Pacific Ocean (Reid & Lonsdale Reference Reid and Lonsdale1974; Rudnick Reference Rudnick1997). Based on this, $D_\infty =1000$ m is determined according to the layer thickness below the 1 $^\circ$C isotherm at the entrance to the passage (bottom depth $\sim$5200 m taken from figure 3 in Pratt et al. Reference Pratt, Voet, Pacini, Tan, Alford, Carter, Girton and Menemenlis2019 and figure 2 in Voet et al. Reference Voet, Girton, Alford, Carter, Klymak and Mickett2015). For the other upstream condition, we took the total transport below the 1.0 $^{\circ }$C isotherm ($Q^*=9.9$ Sv) from Roemmich et al. (Reference Roemmich, Hautala and Rudnick1996) based on hydrographic measurements along a section crossing the Samoan Passage and the Manihiki Plateau. Note that we ignored the existence of the Robbie Ridge, which contains a transport of 1.1 Sv in Roemmich et al. (Reference Roemmich, Hautala and Rudnick1996), and assumed that the total transport of 9.9 Sv is partitioned between the Samoan Passage and east of the Manihiki Plateau. Values of $g'$ and $f$ are also required for the transport estimation, which are taken from Girton et al. (Reference Girton2019) ($4\times 10^{-4}$ m s$^{-2}$) and estimated at 8 $^\circ$S ($2\times 10^{-5}$ s$^{-1}$), respectively. Given a generic depth scaling in the channel $D=1000$ m yields $L_d\approx 31$ km and dimensionless upstream conditions $q=-1$ and $Q\approx 0.5$. Note that the resulting ${Q^*_1}_c$ in the dimensional form does not depend on the value of $D$.

Additional to upstream conditions, geometric parameters of the critical section are needed to estimate ${Q^*_1}_c$. Here, we assume that the critical section lies at the sill according to the inviscid hydraulic theory. Observations have shown that the transport in the Samoan Passage splits into two major pathways, each with multiple contractions and sills (Voet et al. Reference Voet, Girton, Alford, Carter, Klymak and Mickett2015; Girton et al. Reference Girton2019). From a visual inspection of figure 3 in Girton et al. (Reference Girton2019), the 1.0 $^{\circ }$C isotherm is located near 4000 m and spans over a distance of approximately 20 and 40 km at the western path sill (P2) and the eastern path sill (P4), respectively (also see in figure 1c,d). The two sills are approximately 600 and 400 m above the sea bottom depth at the entrance to the Samoan Passage. When determining the topographic parameters representing the Samoan Passage, we treated the two pathways as one channel with a sill height of 600 m and a width of 50 km, and further assumed that the contraction in the channel width is located at the sill. Values of $h^*_c=600$ m and $w^*_c=50$ km are considered as geometric parameters at the critical section and used in estimating the overflow transport from theory. Non-dimensional versions are $h_c=0.6$ and $w_c\approx 1.6$, respectively. For comparison, we also estimated the transport partitioning for the narrow- and shallow-channel limit ($h_c=0.6$, $w_c=0.6$) and the wide- and deep-channel limit ($h_c=0.4$, $w_c=1.6$). Considering that our estimate of the local geometry is rather subjective, the range of estimates can be seen as an uncertainty estimate to some extent. We then fit the Samoan Passage to a parabola by assuming $w^*_c$ to be the width of the overflow interface and $d^*_c=500$ m to be the thickness of flow between the deepest point of the parabola and the interface. The latter value is taken from figure 3 in Girton et al. (Reference Girton2019). The curvature parameter is $\alpha \approx 8\times 10^{-7}$ m$^{-1}$, or in the non-dimensional form $r_c\approx 1.3$, and the non-dimensional sill height at the bottom of the parabola is ${h_0}_c=0.6$. The two limits for sensitivity test are (${h_0}_c=0.6$, $r_c=0.2$) and (${h_0}_c=0.4$, $r_c=1.3$), respectively.

Our theory for a rectangular channel yields a transport in the Samoan Passage of ${Q^*_1}_c=7.4$ Sv (${Q_1}_c\approx 0.38$), and the sensitivity test yields transports of 4.3 and 10.2 Sv (${Q_1}_c\approx 0.22$ and 0.52) for geometric parameters under narrow-channel limit and wide-channel limit, respectively. To compare, the parabolic channel formulation predicts ${Q^*_1}_c=6.0$ Sv (${Q_1}_c\approx 0.3$), and results from the sensitivity test are ${Q^*_1}_c=3.3$ and 8.5 Sv (${Q_1}_c\approx 0.17$ and 0.43). Transports estimated from the parabolic channel theory are smaller than the estimates for a rectangular channel. As shown in figures 3 and 4(a), except for the wide-channel limit, our selected topographic parameters are within the regime of a hydraulically controlled flow. Overall, the theory-based transport estimates lie reasonably close to the time-mean estimates of 5.4 Sv (Voet et al. Reference Voet, Alford, Girton, Carter, Mickett and Klymak2016) and 6.0 Sv (Rudnick Reference Rudnick1997) obtained from moored measurements.

4.2. Two major pathways of flow in the Samoan Passage

Let us now consider a case where there are two narrow parabolic channels to the west of the island, which is a more realistic representation for the Samoan Passage. The western channel is narrower and shallower at the sill ($w_W^*=20$ km, ${h_0}_W^*=600$ m) and the eastern channel (note the difference from the ‘eastern passage’, which refers to the vast passage to the east of the island; the ‘eastern channel’ is part of the Samoan Passage and thus is narrow) is wider and deeper at the sill ($w_E^*=40$ km, ${h_0}_E^*=400$ m) (figure 13a,e). Dimensionless topographic parameters for the two channels are ($r_W\approx 0.18$, ${h_0}_W=0.6$) and ($r_E\approx 0.72$, ${h_0}_E=0.4$), respectively. Whitehead (Reference Whitehead2003) proposed an analytical solution to the partitioning of inflow among multiple channels by assuming such a scenario that the meridional inflow tends to pass through the westernmost channel, where the hydraulic control is preferentially established. The residuals of the inflow, as a consequence, make their way to the second westernmost passage. If the flow in the second westernmost channel is also hydraulically controlled and the critical transport combined is less than the transport of the inflow, the residuals will enter the next passage to the east, and so on for the rest of the eastern passages. In other words, the hydraulic control in a western channel is independent of processes within the passages to its east. This scenario has been adopted by Girton et al. (Reference Girton2019) to estimate the transport through the Samoan Passage, although many historical numerical simulations (e.g. Helfrich & Pratt Reference Helfrich and Pratt2003) as well as ours suggest that, even when the inflow approaches along the western wall, it veers offshore and enters the channel from the eastern side, which is due to the topographic beta effect.

Figure 13. Numerical results from the friction experiments but with two channels. (a) Magnified view of the northward velocity $v$ (colour) and the interface height $d+h$ (contours) in the channel at $t=400$. Panels (b) and (c) show the predicted transport (solid curves) and transport calculated from numerical simulations (dots) given different widths and sill heights of the eastern channel, while keeping other topographic parameters unchanged. Panels (a,d,e) are from the same model topographic parameters as indicated above (d). (d) Time series for the transport of the inflow $Q$, total transport of the two channels $Q_{1W}+Q_{1E}$, transport in the western channel $Q_{1W}$, transport in the eastern channel $Q_{1E}$ and transport along the eastern flank of the island $Q_2$. Numerical results and analytical solutions for the critical transport are shown in solid lines and dashed lines, respectively. (e) Values of $v$ and $d+h$ at the sill (grey line in (a)). Red vectors and curves are analytical solutions from the theory. Blue vectors and areas are results from numerical runs averaged over $t=100\text{--} 400$.

Despite these flaws, here we give an estimate of transport in the two channels within the Samoan Passage based on Whitehead (Reference Whitehead2003). We then compare the results with the transport computed from the numerical simulations to evaluate the validity of the theory. To implement the theory, first we compute the controlled transport in the western channel using upstream conditions and the topographic features of the sill. Then we compute the Bernoulli function along the eastern boundary of the western channel, which would be the same as the Bernoulli function along the western boundary of the eastern channel in the scenario suggested by Whitehead (Reference Whitehead2003). The along-boundary Bernoulli function combined with sill topography provides enough information to compute the critical transport of the eastern channel. Using this methodology, we compute a series of critical transports in the two channels by changing the width (figure 13b) or height (figure 13c) of the eastern channel while keeping other topographic parameters unchanged.

Theory predicts transports of ${Q_{1W}}_c\approx 0.17$ and ${Q_{1E}}_c\approx 0.28$, or dimensionally, 3.3 Sv and 5.6 Sv for the western channel and eastern channel in the Samoan Passage, respectively. However, numerical results are both smaller than the theory predictions (${Q_{1W}}\approx 0.13$, ${Q_{1E}}\approx 0.26$; or dimensionally, 2.5 and 5.0 Sv), especially for the transport in the western channel. By changing the topography of the eastern channel while leaving the western channel unchanged, we find the difference between model results and theoretical predictions is most evident when the eastern channel is wide and deep. Both theory and model assume inviscid flow in the two channels, which allows frontal waves to propagate along the boundary of the separation land of the two channels and connect the two channels. In this case, the western channel is not independent from processes in the eastern channel. However, since the mismatch between the theory and model is reasonably small, especially when the eastern channel is of modest width and height, one may still use the theory to estimate the transport.