2.1 Constants of motion

Benjamin & Lighthill (Reference Benjamin and Lighthill1954) describe trains of steadily translating long gravity waves using three constants of motion: volume flux, alongshore momentum flux and total energy. The volume flux of river water is simply the parameter $Q_{0}$ , and similar expressions for the momentum flux and total energy can also be derived.

Multiplying the $x$ -momentum equation (2.1) by $h$ , the mass conservation equation (2.3) by $u$ , taking the sum of these and integrating from $y=0$ to $y=\infty$ gives

(2.16) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\int _{0}^{\infty }uh\,\text{d}y+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\int _{0}^{\infty }\left(hu^{2}+\frac{h^{2}}{2}\right)\text{d}y-u_{W}h_{W}v_{W}=\int _{0}^{\infty }vh\,\text{d}y.\end{eqnarray}$$

(Note that these terms are bounded because the fluid is stagnant as $y\rightarrow \infty$ .) The right-hand side of (2.16) can be computed by multiplying (2.3) by $y$ and integrating,

(2.17) $$\begin{eqnarray}\int _{0}^{\infty }vh\,\text{d}y=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\int _{0}^{\infty }yh\,\text{d}y+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\int _{0}^{\infty }yuh\,\text{d}y,\end{eqnarray}$$

so that (2.16) becomes

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}I}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}x}=u_{W}h_{W}v_{W},\end{eqnarray}$$

where

(2.19a,b ) $$\begin{eqnarray}I=\int _{0}^{\infty }(u-y)h\,\text{d}y,\quad S=\int _{0}^{\infty }\left(hu(u-y)+\frac{h^{2}}{2}\right)\,\text{d}y\end{eqnarray}$$

are the along-wall impulse and the flow force respectively. The quantity $u_{W}h_{W}v_{W}$ is the downstream momentum input at the wall, and vanishes away from the source region. The total downstream momentum added across the source in steady flow is thus

(2.20) $$\begin{eqnarray}S_{0}=\int _{-1}^{1}u_{W}Q^{\prime }(x)\,\text{d}x=\int _{0}^{Q_{0}}u_{W}\,\text{d}Q,\end{eqnarray}$$

giving a second constant of the motion. The momentum input term is discussed in greater detail for QG flow in Southwick et al. (Reference Southwick, Johnson and McDonald2017), where its presence is shown to resolve the momentum imbalance paradox of Pichevin & Nof (Reference Pichevin and Nof1997).

In steady flow, (2.3) allows a volume-flux streamfunction $\unicode[STIX]{x1D713}$ to be defined such that $vh=\unicode[STIX]{x1D713}_{x}$ and $uh=-\unicode[STIX]{x1D713}_{y}$ . The $x$ -momentum equation (2.1) can be integrated once in $x$ to give

(2.21) $$\begin{eqnarray}\frac{u^{2}}{2}-\unicode[STIX]{x1D713}+h=R.\end{eqnarray}$$

Since the $y$ -derivative of (2.1) vanishes, $R$ is constant throughout the flow, providing a third constant of motion for steady flow that can be found by evaluating (2.21) at the wall, where $\unicode[STIX]{x1D713}=Q$ .

2.2 Vortical and Kelvin-wave-driven flow

There are two principal mechanisms that drive the outflow, and it is instructive to see how their relative strengths affect the behaviour of solutions.

The first of these mechanisms is flow driven by a Kelvin wave, which propagates faster than the river water and thus transports both river and oceanic fluid (Helfrich et al. Reference Helfrich, Kuo and Pratt1999; Helfrich Reference Helfrich2006). To isolate the effects of this wave, consider the zero-PVa problem (that is, with $H=1$ ). The expression for the layer depth is now

(2.22) $$\begin{eqnarray}h=1+B(x,t)\text{e}^{-y}\end{eqnarray}$$

where $B$ satisfies the $x$ -momentum equation (2.1) at the wall,

(2.23) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}t}+(1+B)\frac{\unicode[STIX]{x2202}B}{\unicode[STIX]{x2202}x}=Q^{\prime }(x).\end{eqnarray}$$

This is the forced Hopf equation, which can be solved explicitly using the method of characteristics, and gives rise to a current that expands offshore and heads downstream, moving with speed $u_{KW}=-1+\surd (1+2Q_{0})$ at the wall. This is referred to hereafter as the Kelvin-wave flow. Non-trivial solutions of the homogeneous Hopf equation always develop shocks, which occur in the present problem in the oceanic (rather than river) water at the leading edge of the Kelvin-wave flow. These shocks are discussed in greater detail in § 3.5.

The other effect that drives the flow is image vorticity. Vortical columns expelled by the source stretch or squash as they adjust to match the depth of the oceanic layer, and this drives a change in relative vorticity, which in turn acts to send fluid downstream (vortex stretching, positive PVa) or upstream (vortex squashing, negative PVa). This effect can be isolated by considering the limiting case of rotating fluid ejected into a non-rotating background, and it is shown in Johnson & McDonald (Reference Johnson and McDonald2006) that the velocity of the current at the wall is $u_{V}=\pm \surd (Q_{0}|H-1|/H)$ where the sign depends on the sign of the PVa. Thus, for positive PVa both the vortical and Kelvin-wave effects drive fluid downstream, while for negative PVa the two effects are in opposition.

The relative strengths of these two effects can be measured by the ratio between the two speeds:

(2.24) $$\begin{eqnarray}a=\frac{|u_{V}|}{u_{KW}}=\frac{\sqrt{Q_{0}|H-1|}}{-\surd H+\surd H(1+2Q_{0})}.\end{eqnarray}$$

When the jump in PV is small, and the flow is quasi-geostrophic, JSM found that solutions can be fully characterised by the value of $a$ and the sign of the PVa, and one of the main goals of the present work is to assess to what extent $a$ can be used to categorise solutions to the semi-geostrophic problem at larger values of $Ro$ . It is expected, and shown below, that the results here reduce to those of JSM in the limit $Ro\rightarrow 0$ and $Q_{0}\rightarrow 0$ , with $a\sim \surd (Ro/Q_{0})$ fixed. In this limit, $a$ can also be identified as the ratio of the Rossby radius $L_{R}$ to the vortical width $L_{V}=\surd (Q_{0}H/|H-1|)$ (the width of the vortical current). The two interpretations are interchangeable in the quasi-geostrophic limit, but for $Ro=O(1)$ the definition (2.24) in terms of speed is more instructive.

Note that the choice of $L_{R}$ for horizontal length scales ensures that the length over which the Kelvin wave decays is $O(1)$ , while vortical currents can be of arbitrary width. On the other hand, JSM scale horizontal lengths on $L_{V}$ , thus keeping the vortical current width at $O(1)$ and allowing the Kelvin-wave decay scale to be arbitrarily small as $a\rightarrow 0$ . Anticipating the importance of $a$ in determining the behaviour of the flow, we will briefly outline some results from the quasi-geostrophic limit.

When $a<1$ , the Kelvin-wave-driven flow is dominant: $|u_{V}|<u_{KW}$ and $L_{R}<L_{V}$ . For positive PVa, the Kelvin wave pulls ahead of the vortical flow, vortical turning is weak at the nose of the fluid and the outflow boundary joins smoothly to the coast. Depth perturbations decay quickly away from the wall before vortical effects can take hold. For negative PVa, the weak vortical flow is not sufficient to counteract the Kelvin wave, and the solution expands indefinitely away from the coast, with fluid spreading up and downstream.

When $a>1$ , vorticity dynamics dominate the outflow: $|u_{V}|>u_{KW}$ and $L_{R}>L_{V}$ . For positive PVa, vortex turning affects fluid closer to the wall, causing the current to overturn and a shock to form. As $a$ increases further, this happens over a shorter and shorter scale until, for $a=a_{m}\approx 1.868$ , the vortical current manages to adjust to the far-field condition within the expelled fluid and there is no disturbance in the oceanic fluid $y>w$ . In this case, the solution downstream is a current of constant width and no transport around the edge. For negative PVa, the relatively stronger vortical flow balances the Kelvin wave and the flow is steady in the source region.

3.1 Numerical results

Having now reduced the problem to two nonlinear partial differential equations (2.12) and (2.13) in $x$ and $t$ , solutions can be obtained numerically. Starting from the initial conditions $U=w=0$ , the equations are integrated away from $t=0$ using the Lax–Wendroff method with a nonlinear smoothing filter (Engquist, Lötstedt & Sjögreen Reference Engquist, Lötstedt and Sjögreen1989) to prevent overshoot in the region of a shock. To ensure that the numerical scheme explicitly preserves mass and momentum, the equations are solved in conservation form, i.e.

(3.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D719}_{1}\\ \unicode[STIX]{x1D719}_{2}\end{array}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(\begin{array}{@{}c@{}}U^{2}/2+\sqrt{H}U\\ h_{W}^{2}/2\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ Q^{\prime }(x)\end{array}\right),\end{eqnarray}$$

where

(3.2a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{1}=U-w,\quad \unicode[STIX]{x1D719}_{2}=\int _{0}^{w}h\,\text{d}y+H(U-w).\end{eqnarray}$$

This presents some difficulty, since the flux functions in (3.1) cannot be written explicitly in terms of $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ , and so $U$ and $w$ must be extracted at each grid point and each time step. This is done via Newton’s method, using as an initial guess the values of the solution at the previous time step. Applying homogenous Neumann boundary conditions at the upstream end of the spatial domain increases stability. Runs are conducted with grid spaces of $\unicode[STIX]{x0394}t=0.01$ or $0.005$ , and $\unicode[STIX]{x0394}x\approx 0.03$ as increasing the resolution further made no noticeable difference.

Figure 3.

The off-shore extent of the expelled fluid, $y=w(x,t)$ , for flows with positive PV anomaly. (a) $Q_{0}=1$ , $Ro=0.3$ and hence $a=0.66$ (Kelvin-wave-dominated flow) at $t=5,10,25,40$ . (b) $Q_{0}=0.4$ , $Ro=1$ and $a=1.31$ (vortically dominated flow) at $t=5,10,40,60$ . Vertical dotted lines mark the source region, and the dashed curve (red online) is the steady profile $w_{sp}$ computed in § 3.2.

Figure 4.

Outflow with positive PVa, $Q_{0}=0.4$ , $Ro=0.5$ and hence $a=1.07$ . (a) The current width $w$ and (b) the edge velocity $U$ are shown at times $t=4,9,12,40$ . The thick dashed curve (red online) in (a) gives the steady profile $w_{sp}$ , and thin dashed lines correspond to predictions from the theory of §§ 3.3–3.5. (c) Contours of $h-H$ , perturbations to the depth of the upper layer at $t=12$ . The thick curve gives $w$ at the same time, thus the shaded region is the extent of the expelled fluid.

Figure 3 shows numerically computed $w$ for two different sets of parameters, with each panel showing the solution at several times. Figure 4 shows $w$ , $U$ and contours of $h-H$ for a third set of parameters. In all plots, the source region is marked by vertical dotted lines, and theoretical results derived below are shown by dashed lines. From the two functions $w$ and $U$ , the depth and velocity profile at any point can be computed using (2.7).

In all solutions with positive PVa, a steady current forms across the source region. This current width increases monotonically across the source, and the fluid at the current edge has zero velocity, so the outer layer is undisturbed in regions where the flow is steady. Numerical results agree excellently with the theory (heavy dashed curve, red online) presented in § 3.2. Once the steady solution reaches all the way across the source, a steady current of constant width propagates downstream.

Downstream of the steady region, the current narrows and the velocity around the edge increases according to a similarity solution discussed in § 3.3. The parameters are chosen to illustrate the range of possible behaviours and, as in JSM, these correlate with the value of the speed ratio $a$ . For small $a$ , as in figure 3(a), the flow is dominated by the Kelvin wave and the downstream current joins smoothly to the wall. For larger $a$ , as in figures 3(b) and 4(a), vorticity is stronger and the current terminates in a shock. The width of the shock increases monotonically, until it reaches a maximum value that is set by the flow parameters. If $a$ is large enough, and vortical effects are strongly dominant, this maximum value can be the same as the width of the steady current at the downstream edge of the source region, in which case there is no rarefaction and the solution after a finite time is a ‘blunt-headed current’ (cf. figure 3 b).

In regions where $w=0$ , the governing equations are still valid but $U$ is now to be interpreted as the velocity at the wall, and the layer depth decays exponentially away from the coast like $U\exp (-y/\surd H)$ . At all times and for all parameters, a nonlinear Kelvin wave propagates ahead of the expelled fluid. That is, $U>0$ in some region where $w=0$ . The effects of this are most clearly seen in the contour plot of figure 4(c): the river water (shaded) has reached $x=4$ , but the wall depth is increased by 0.25 over the region $4<x<16$ , an interfacial disturbance travelling in the oceanic fluid ahead of the river water. In the quasi-geostrophic limit, the Kelvin wave travels at infinite speed in the outer layer and so is not seen explicitly in the solutions – although it still induces a flow of finite velocity in the river water. Moving away from the quasi-geostrophic limit and permitting arbitrary values of $H$ changes the scale to one where the Kelvin wave is clearly visible, and information propagates at finite speed in both the river fluid and the oceanic layer. This Kelvin wave is discussed in detail in § 3.5.

3.2 Analysis across the source region

Since the PVa is positive, the image vorticity and the Kelvin wave both drive the flow downstream. System (2.12), (2.13) has two eigenvalues, $\unicode[STIX]{x1D706}_{R}$ and $\unicode[STIX]{x1D706}_{C}$ , with $\unicode[STIX]{x1D706}_{R}\leqslant \unicode[STIX]{x1D706}_{C}$ always (see appendix A for more details). Kubokawa & Hanawa (Reference Kubokawa and Hanawa1984a ) note that $\unicode[STIX]{x1D706}_{C}$ is a nonlinear coastal wave, and that $\unicode[STIX]{x1D706}_{R}$ is associated with disturbances at the current edge – across which the density jump from their model has been replaced by a PV jump in the present work.

When $w=U=0$ , one can show that $\unicode[STIX]{x1D706}_{R}=0$ and $\unicode[STIX]{x1D706}_{C}=\sqrt{H}$ . Thus, characteristics $\unicode[STIX]{x1D706}_{R}$ that start in $x<-1$ are unable to reach the source region. Further, it is impossible to satisfy $\unicode[STIX]{x1D706}_{C}=0$ or $\unicode[STIX]{x1D706}_{R}=0$ when $U>0$ and $H>1$ and so the flow is supercritical everywhere in $x>-1$ , with no information able to propagate upstream.

Since the characteristic speed $\unicode[STIX]{x1D706}_{R}$ vanishes at $x=-1$ and long-wave disturbances are stationary here, this is the control point for the steady flow that forms across the source region. The flow matches smoothly to the undisturbed outer layer at $x=-1$ , where $w=U=0$ .

Solving the steady versions of (2.12) and (2.13) gives

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle U=c_{1}, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle h_{W}=\sqrt{2Q(x)+c_{2}}, & \displaystyle\end{eqnarray}$$

where $c_{1}$ and $c_{2}$ are constants of integration determined by the requirement that $U$ and $h_{W}$ are continuous at $x=-1$ to be 0 and $H^{2}$ respectively. Downstream, the dimensional depth of the current at the wall is therefore

(3.5) $$\begin{eqnarray}h^{\ast }=\sqrt{\frac{2Q^{\ast }f}{g^{\prime }}+H^{\ast 2}},\end{eqnarray}$$

where $Q^{\ast }$ and $H^{\ast }$ are the (dimensional) mass flux and outer-layer depth respectively. The relationship (3.5) can be derived from a simple quasi-geostrophic scaling argument when $H=0$ (Chant Reference Chant, Wolanski and McLusky2011, p. 218).

Solving (2.8) for $w$ provides the steady solution

(3.6) $$\begin{eqnarray}w_{sp}(Q)=\operatorname{acosh}\left(\frac{\sqrt{2Q+H^{2}}-1}{H-1}\right),\end{eqnarray}$$

which is valid exactly when $H>1$ . The steady solution $w_{sp}$ is shown as a dashed curve (red online) in figures 3 and 4(a). Agreement with the numerical results is excellent in all cases. The steady current depends on $x$ only parametrically through $Q$ , and hence the downstream current width $w_{D}=w_{sp}(Q_{0})$ does not depend on the specific form of the outflow velocity. In the quasi-geostrophic limit, $Ro\rightarrow 0$ and $Q_{0}\rightarrow 0$ with $a^{2}=Ro/Q_{0}$ fixed,

(3.7) $$\begin{eqnarray}w_{D}\sim \log (1+1/a^{2}+\sqrt{1/a^{4}+2/a^{2}})+O(Ro)\quad \text{as }Ro\rightarrow 0,\end{eqnarray}$$

as in (3.9) of JSM, up to the use of $L_{V}$ , the vortex width, as the characteristic length for $y$ .

Figure 5.

The width of the steady current $w_{D}$ downstream of the source, as a function of the total mass flux $Q_{0}$ , for various values of the Rossby number $Ro$ .

The downstream width of the steady solution $w_{D}$ is shown as a function of $Q_{0}$ and $H$ in figure 5. Since the steady solution always has $U=0$ , the river water completely adjusts to the upper-layer depth over the vortex scale $L_{V}$ in the source region. Large $Q_{0}$ – which corresponds to small $a$ and large $L_{V}$ – thus gives rise to a wider current (growing logarithmically as $Q_{0}\rightarrow \infty$ ), while large $Ro$ gives small $L_{V}$ and a narrow current as the vortical turning happens closer to the wall. Because $w$ is scaled by the Rossby radius $L_{R}$ , the non-dimensional width of the steady current is equal to the Kelvin number $K$ , following Garvine (Reference Garvine1995). Figure 5 shows that $K>1$ in most flows and so the outflow behaviour is of the ‘large-scale’ type, i.e. significantly affected by rotation and geostrophically balanced.

The downstream constants of momentum and energy $S_{0}$ and $R$ can also be computed for steady solutions. From (2.20) and (2.10),

(3.8) $$\begin{eqnarray}S_{0}=\int _{0}^{Q_{0}}(H-1)\sqrt{\unicode[STIX]{x1D703}^{2}-1}\,\text{d}Q,\end{eqnarray}$$

where $\unicode[STIX]{x1D703}=\cosh (w_{sp})$ so that, from (3.6),

(3.9) $$\begin{eqnarray}(H-1)\sqrt{\unicode[STIX]{x1D703}^{2}-1}=\surd 2\sqrt{(H+Q-\surd (H^{2}+2Q))}=u_{W}.\end{eqnarray}$$

Analysis of the integral in (3.8) shows that $S_{0}$ grows like $Q_{0}^{3/2}$ as $Q_{0}\rightarrow \infty$ . Curves of $S_{0}$ for various parameter values are plotted in figure 6.

The total energy budget $R$ in the steady current is, from (2.21) and (3.9),

(3.10) $$\begin{eqnarray}R=\frac{u_{W}^{2}}{2}-Q+h_{W}=H,\end{eqnarray}$$

as expected to match with the outer layer at $x=-1$ .

Figure 6.

Momentum added by the source, $S_{0}$ , as a function of $Q_{0}$ and $Ro$ for positive PVa.

3.3 Downstream solution

Away from the source region the governing equations are unforced, and hence possess two Riemann invariants, i.e. there exist functions ${\mathcal{R}}_{C/R}(w,U)$ such that

(3.11) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}{\mathcal{R}}_{C/R}(w,U)=0\quad \text{on }\frac{\text{d}x}{\text{d}t}=\unicode[STIX]{x1D706}_{C/R}.\end{eqnarray}$$

Thus ${\mathcal{R}}_{C/R}$ are constant on each characteristic, with that constant determined by the values of $w$ and $U$ at the time when a given characteristic leaves the source region. For this system, the Riemann invariants cannot be found analytically and are obtained instead by solving the equivalent problem

(3.12) $$\begin{eqnarray}\frac{\text{d}U}{\text{d}w}=\unicode[STIX]{x1D6FC}_{C/R}(w,U)\quad \text{on }\frac{\text{d}x}{\text{d}t}=\unicode[STIX]{x1D706}_{C/R}\end{eqnarray}$$

as in Helfrich (Reference Helfrich2006). The functions $\unicode[STIX]{x1D6FC}_{C/R}$ are found by diagonalising the system (2.12)–(2.13), and are given explicitly in appendix A. The initial conditions for integrating (3.12) along a particular characteristic are given by the values of $w$ and $U$ at the point when that characteristic leaves the source region. In particular, once the steady solution reaches the downstream edge of the source region, the values of $w$ and $U$ are fixed and so all integrations start with the same conditions – the Riemann invariants are uniform.

The uniformity of one Riemann invariant is used in Stern & Helfrich (Reference Stern and Helfrich2002) to build similarity solutions $U(w)$ and describe the propagation of a coastal current with negative PVa. The same approach is applied here, whilst noting that uniform Riemann invariants only occur once the steady solution has been completely set up, and so the similarity solution is only valid after a finite time.

Immediately downstream of $x=1$ both Riemann invariants are uniform. Here, there are two universal relationships ${\mathcal{R}}_{C/R}(w,U)=m_{C/R}$ and so the only continuous solution has $w=w_{D}$ and $U=0$ (visible at late times in figures 3 and 4). The region of two uniform invariants is led by the slower characteristic $\unicode[STIX]{x1D706}_{R}$ , and so the speed at which the constant-width current spreads can be calculated by setting $w=w_{D}$ and $U=0$ in $\unicode[STIX]{x1D706}_{R}$ . Downstream of this constant-width region, only the $\unicode[STIX]{x1D706}_{C}$ characteristic is determined by the steady values of $w$ and $U$ and so only the ${\mathcal{R}}_{C}$ invariant is uniform. The similarity solution $U(w)$ can thus be found by integrating (3.12) away from the initial conditions $U=0$ , $w=w_{D}$ until the point $w=0$ , where we define $U(w=0)=U_{nose}$ . Similarity solutions for various source fluxes are shown for $Ro=0.5$ and $Ro=2$ in figure 7. The solutions are all rarefactions (also called wedge intrusions or expansion waves) since they all have the property that $U$ increases as $w$ decreases. Rarefaction solutions have been previously described for coastal currents driven by PV jumps in Helfrich (Reference Helfrich2006) and JSM.

Figure 7.

The velocity $U$ at the edge of the expelled fluid, as a function of the width $w$ , in the unsteady downstream rarefactions that lead the outflow. (a) $Ro=0.5$ , (b) $Ro=2$ . These relationships are valid once the steady solution has been set up across the source region.

3.3.1 Termination of the rarefaction

In regions where the similarity solution applies, the slope of the $\unicode[STIX]{x1D706}_{R}$ characteristics – those on which the Riemann invariant ${\mathcal{R}}_{R}$ is not uniform – can be computed as a function of $w$ only. This is done for various parameter values in figure 8.

Figure 8.

Characteristic slope $\text{d}x/\text{d}t=\unicode[STIX]{x1D706}_{R}$ as a function of $w$ in regions where the similarity solution $U(w)$ applies (downstream of the source). The characteristic slopes are computed over the range $0\leqslant w\leqslant w_{D}$ and (a) $Ro=0.3$ , (b) $Ro=1$ . The dashed curve (a)(iv) does not form a shock, and circular markers on other curves indicate the maximum value of $w$ immediately behind the shock for that outflow. Solid curves without a marker develop shocks that increase in width until $w=w_{D}$ (a ‘blunt current’).

Since the solution grows monotonically in the source region, characteristics with larger values of $w$ leave the source later. Shocks form after a finite time if $\unicode[STIX]{x1D706}_{R}$ is not a monotonic decreasing function of $w$ , since characteristics with larger $\text{d}x/\text{d}t$ catch those with smaller $\text{d}x/\text{d}t$ . Differentiating the characteristic speed with respect to $w$ and evaluating at the wall gives

(3.13) $$\begin{eqnarray}\left.\frac{\text{d}\unicode[STIX]{x1D706}_{R}}{\text{d}w}\right|_{w=0}=\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}_{R}}{\unicode[STIX]{x2202}w}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D706}_{R}}{\unicode[STIX]{x2202}U}\frac{\text{d}U}{\text{d}w}\right|_{w=0}=H-1+\frac{H-2}{\sqrt{H}}U_{nose}.\end{eqnarray}$$

A shock develops if (3.13) is positive. This occurs either if $H\geqslant 2$ , or if

(3.14) $$\begin{eqnarray}U_{nose}<\frac{\sqrt{H}(H-1)}{2-H}\end{eqnarray}$$

and $1<H<2$ . For large Rossby number ( $H>2$ ) a shock always forms, while for moderate or small Rossby number a shock forms only if the vortical current overturns at the wall and the nose velocity is sufficiently small. From the plots of $U_{nose}$ shown in figure 10, a slow nose velocity corresponds to smaller $Q_{0}$ (and bigger $a$ ) and so, as in the quasi-geostrophic limit, vortically dominated flows are more likely to develop a shock. The dashed curve in figure 8(a)(iv) is the only one presented that does not develop a shock – note that it has the smallest $a$ of all solutions shown. The values of $Q_{0}$ beneath which a shock forms are marked on curves with $1<H<2$ in figure 10 below.

If no shock forms then the rarefaction is propagated by the $\unicode[STIX]{x1D706}_{R}$ characteristic, which reduce to $\unicode[STIX]{x1D706}_{R}=U$ when $w=0$ . The expelled fluid thus travels along the wall with velocity $U_{nose}$ and the fluid velocity is equal to the propagation velocity. If, on the other hand, a shock does form, then applying conservation of mass moving at the shock speed $V$ gives

(3.15) $$\begin{eqnarray}\int _{0}^{w}(u-V)h\,\text{d}y=0\end{eqnarray}$$

or

(3.16) $$\begin{eqnarray}V=\frac{(1+(-1+H+\sqrt{H}U)\cosh w+U\sinh w)^{2}-(H+\sqrt{H}U)^{2}}{2(w-U+U\cosh w+(-1+H+\sqrt{H}U)\sinh w)},\end{eqnarray}$$

where $U$ and $w$ are evaluated just behind the shock.

Because shocks form after a finite time, jumps in $w$ and $U$ are initially small and increase in magnitude until one of two things happens. If the shock velocity $V$ becomes equal to the characteristic speed $\unicode[STIX]{x1D706}_{R}$ , then at this point the shock and characteristic are tangential in the $(x,t)$ plane and no more information can enter the shock. That is, there exists some value $w_{M}$ such that $\unicode[STIX]{x1D706}_{R}(w_{M},U(w_{M}))=V(w_{M},U(w_{M}))$ . The solution is thus a rarefaction terminating in a shock, where the width of the shock approaches $w_{M}$ as $t\rightarrow \infty$ . The circular markers on characteristic curves in figure 8 indicate the value of $w_{M}$ for that particular $Q_{0}$ and $H$ . An example of such a solution is displayed in figure 4(a), where the value of $w_{M}$ is marked as a horizontal dashed line. Alternatively, the characteristic speed $\unicode[STIX]{x1D706}_{R}$ can be greater than $V$ for all $0<w<w_{D}$ . In this case, the shock keeps expanding until the maximum width of the current is reached, as seen in figure 3(b) where at late times the solution is simply a blunt current of width $w_{D}$ . This situation occurs when vortical effects are dominant and $a$ is large, as in the unmarked solid curves of figure 8. Note that a full-width shock will only develop if $\unicode[STIX]{x1D706}_{R}(w_{D})$ is greater than $V(w_{D},0)$ , so the qualitative behaviour of the solution as $t\rightarrow \infty$ can be determined by comparing these two values.

3.4 Interpretation in terms of the speed ratio $a$

From § 3.3, the qualitative behaviour of outflows can be categorised according to the form of the downstream current as a rarefaction joining smoothly to the wall, a rarefaction terminating in a shock, or a blunt current. The space of parameters $(Q_{0},H)$ can be divided into three regions, the boundaries of which can be computed efficiently by integrating the characteristic equation (3.12) and then applying the two tests described above to determine whether a shock forms and, if it does, whether the maximum jump magnitude is less than $w_{D}$ .

Figure 9.

For positive PVa, the $(Q_{0},Ro)$ parameter space is divided into three regions (marked by thick lines, red online). Solution type one joins smoothly to the wall, solution type two is a rarefaction terminating in a shock, and solution type three is a full-width shock. Also shown are contours of the speed ratio $a$ . The inset details the behaviour in the quasi-geostrophic limit.

Figure 9 compares the numerically determined boundaries of these regions (numbered ‘1’, ‘2’ and ‘3’ respectively) with contours of $a$ , to illustrate the extent to which the flow behaviour depends on $Q_{0}$ and $H$ independently or whether, as in the quasi-geostrophic limit, the value of $a$ alone is sufficient to categorise the solution. In the quasi-geostrophic limit (inset) the region boundaries coincide precisely with the contours $a=1$ and $a=a_{m}\approx 1.87$ given in JSM. For larger $Ro$ and $Q_{0}$ , however, the region boundaries deviate from these values of $a$ , although the progression from no shock to partial-width shock and full-width shock with increasing $a$ remains.

3.5 Disturbance in the oceanic fluid

On the long-wave scale, the solution depends on $x$ through $U$ and $w$ alone. Information from the expelled fluid travels directly offshore into the oceanic fluid. This is at odds with, for example, the shallow-water numerical results of Helfrich (Reference Helfrich2006) (their figure 10) which shows waves that break ‘backwards’, and is a limitation of the long-wave assumption made here. From (2.7), this also means that regions where $U=0$ do not disturb the oceanic fluid, as shown in figure 4(c).

The similarity solutions obtained by integrating (3.12) give $U_{nose}$ : the value of $U$ at the point where $w=0$ , i.e. the fluid velocity at the leading edge of the expelled fluid. The horizontal dashed line in figure 4(b) marks the predicted value of $U_{nose}$ , and the figure also indicates that $U=U_{nose}$ in front of the expelled fluid, and thus that $U_{nose}$ is the coastal velocity of the Kelvin wave that travels ahead of the river water. This follows by noting that the similarity solution is valid anywhere that the ${\mathcal{R}}_{C}$ Riemann invariant is uniform; that is, at any point in the $(x,t)$ -plane occupied by a $\unicode[STIX]{x1D706}_{C}$ characteristic that left the source region after the steady solution was set up. In particular, this region extends beyond the nose of the river water and into the oceanic fluid. As discussed below, this then creates a shock in the oceanic fluid (the Kelvin-wave shock) and so all remnants of the initial conditions are erased with, at late times, ${\mathcal{R}}_{C}$ uniform everywhere that the fluid is disturbed. Note also that in vorticity-dominated flows where a shock develops in the river water, this is through the collision of $\unicode[STIX]{x1D706}_{R}$ characteristics and so ${\mathcal{R}}_{C}$ is conserved across the shock (El Reference El2005). The similarity solution thus remains valid at the nose of river water, and $U=U_{nose}$ ahead of the expelled fluid. Curves showing $U_{nose}$ as a function of the total mass flux are presented in figure 10 for various $Ro$ , along with the speed of the Kelvin-wave-driven flow $u_{KW}$ (dashed).

Figure 10.

Nose speed for various $Ro$ as a function of the source mass flux $Q_{0}$ . The ‘nose speed’ refers to the along-wall velocity of fluid in the Kelvin wave that propagates ahead of the river water. Circles mark the value of $Q_{0}$ below which a shock forms in the expelled fluid, and curves without circles always form a shock. Also shown (dashed) is the wall velocity of the zero PVa solution.

The motion in the oceanic fluid ahead of the river fluid is governed by (2.12) with $w=0$ ,

(3.17) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}t}+(U+\sqrt{H})\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}=0\end{eqnarray}$$

(cf. the zero-PVa equation (2.23)). The propagation speed ahead of the river water is therefore the sum of $\sqrt{H}$ , the non-dimensional speed of the linear Kelvin wave, and a nonlinear advection at speed $U$ . For a fixed volume flux $Q_{0}$ , increasing the oceanic layer depth, and so increasing $Ro$ , corresponds to lower fluid velocity $U$ . This decrease in $U$ dominates the increase in $H$ , causing the nose speed $U_{nose}$ to decrease as seen in figure 10.

Solutions to (3.17) always develop a shock when $U$ is a decreasing function of $x$ , as seen in the development of figure 4(b). Once a shock forms, the jump height increases until $U=U_{nose}$ and the shock propagates into the resting fluid at a speed determined by the relevant Rankine–Hugoniot condition to be

(3.18) $$\begin{eqnarray}V_{KW}=\frac{U_{nose}}{2}+\sqrt{H}.\end{eqnarray}$$

The horizontal vertical lines in figure 4(b) are separated by $3V_{KW}$ , and match well with the solutions at $t=9,12$ even though $U$ has not quite reached $U_{nose}$ . A full three-dimensional model of such shocks is considered in Fedorov & Melville (Reference Fedorov and Melville1996), and the propagation speed here agrees with their relation (7.20).

4.1 Numerical results

Figure 11.

An outflow with negative PVa, $Ro=0.4$ , $Q_{0}=0.7$ and $a=1.24$ at times $t=20,30,40$ . (a) The current width $w$ , which continues increasing across the source region (marked by vertical dotted lines). The dotted curve (blue online) gives the curve $y=\tilde{w}$ along which $u=0$ (at $t=40$ ). (b) The edge velocity $U$ at $t=20,30,40$ .

An example of an unsteady, widening current is shown in figure 11 where $Ro=0.4$ , $Q_{0}=0.7$ and so $a=1.24$ . Since $a>1$ the outflow is vortically dominated and the corresponding quasi-geostrophic flow would evolve to become steady across the source region. Here, however, $Ro$ is sufficiently large that the current width continues to grow. Some expelled fluid travels downstream, joining smoothly to the wall in a rarefaction with, as for positive PVa, a Kelvin wave travelling ahead in the outer layer. The rest of the expelled fluid travels upstream. The boundary between fluid moving upstream and fluid moving downstream, where $u=-h_{y}=0$ (and so the outflow depth is at a maximum), is denoted $y=\tilde{w}(x,t)$ and shown dotted (blue online) in figure 11(a). Fluid thus travels upstream along the wall, before turning and heading downstream around the edge of the current, taking oceanic water with it as $U>0$ . Thus one important difference between solutions with positive and negative PVa is that outflows with negative PVa transport oceanic fluid from upstream of the source region to downstream. Computation of the off-shore velocity $v$ reveals that fluid travels at infinite velocity around the upstream boundary of the outflow before heading downstream. This failure of the semi-geostrophic equations is due to the short length scales in the shock region, as noted by Helfrich (Reference Helfrich2006).

Figure 12.

Solutions to the outflow problem with negative PVa, $Ro=0.5$ , $Q_{0}=0.2$ and $a=2.44$ at times $t=20,40,120$ . (a) The current width $w$ , which is steady across the source region (marked by vertical dotted lines). The (green) dashed line at $x\approx -6$ indicates the location of the transect used to compare with experimental solutions in § 5. (b) The velocity $U$ at the current edge. The thick dashed curves (red online) are the steady solutions $w_{sn}$ and $U_{c}$ from § 4.2, the blue dotted line is $y=\tilde{w}$ , and the black dashed lines are derived from the theory of § 4.3 and discussed there. (c) Contours of the depth perturbation $h-H$ at $t=40$ , where the shaded region indicates the expelled fluid.

Figure 12 shows an example of a steady current where $Ro=0.5,Q_{0}=0.2$ and $a=2.44$ . In steady solutions, the curve $y=\tilde{w}$ (dotted, blue online) touches the wall at $x=1$ , thus $u_{W}=0$ here and all the outflow first heads upstream before a portion recirculates to form the downstream current, which again is a rarefaction. In all solutions with negative PVa, condition (3.14) cannot be satisfied with $U>0$ and so the downstream rarefaction joins smoothly to the wall, demonstrating the importance of vortical flow in the formation of leading shocks. The steady solution is very similar to that of McCreary et al. (Reference McCreary, Zhang and Shetye1997) (their figure 9) where upstream flows are driven by geostrophic adjustment of a density front. The upstream solution here is similar to the downstream solution for positive PVa: a constant-width region near the source, then a rarefaction terminating in a leading shock. Figure 12(b) shows the edge velocity $U$ , and the ripples visible at late times are the result of numerical dispersion. Figure 12(c) shows contours of the layer-depth perturbation $h-H$ at $t=40$ . The river fluid is shaded, with the region where $u<0$ shaded darker. The Kelvin wave is again seen to travel ahead of the river water and disturb the oceanic fluid. The contour closest to the coast has a negative value: the outflow has caused a shallowing of the layer upstream of the source.

4.2 Steady solutions across the source region

Since the oceanic layer is disturbed both upstream and downstream of the source, there are no matching conditions to determine the constants of integration and obtain a steady solution as in § 3.2. Instead, specifying the constants requires two additional relations, which are based on quasi-geostrophic theory and later verified against numerical results to the initial-value problem.

First, note that one family of characteristic slopes $\unicode[STIX]{x1D706}_{C}$ are always positive (see appendix A), while the other set $\unicode[STIX]{x1D706}_{R}$ vanishes whenever $u_{W}=0$ , i.e. $U=U_{c}$ , say, where

(4.1) $$\begin{eqnarray}U_{c}=\frac{1-H}{\coth (w)+\sqrt{H}}.\end{eqnarray}$$

Now $U_{c}<0$ when $H>1$ but $U_{c}>0$ when $H<1$ , allowing $\unicode[STIX]{x1D706}_{R}$ to send information upstream if the PVa is negative, and thus the $\unicode[STIX]{x1D706}_{R}$ characteristic is associated with the vortical flow, and $\unicode[STIX]{x1D706}_{C}$ with the Kelvin-wave-driven flow, as noted for the special case $H=0$ in Kubokawa & Hanawa (Reference Kubokawa and Hanawa1984a ).

As in outflows with positive PVa, steady currents are controlled at the edge of the source region. In particular there exists a separatrix, i.e. a characteristic $\unicode[STIX]{x1D706}_{R}$ that has a turning point at $x=1$ . Characteristics that start to the right of the separatrix end up downstream of the source, and those that start to the left go upstream. There is thus a region in the wake of the turning point of the separatrix where characteristics that originate at $t=0$ cannot penetrate, and characteristics instead propagate out from the turning point, where (4.1) is satisfied. The steady solution thus spreads upstream across the source region, starting at the downstream edge where it is controlled by the $\unicode[STIX]{x1D706}_{R}$ characteristic. Physically, this sets $u_{W}=0$ at the downstream edge of the source region and shows that, for steady solutions, all of the outflow must first head upstream driven by image vorticity, as seen in figure 12(c). There are, however, an infinite number of steady solutions that satisfy this control condition, and a further equation is needed to close the system.

Note that the curve $y=\tilde{w}$ , on which $u=0$ , can be computed from (2.7) as

(4.2) $$\begin{eqnarray}\tilde{w}=w-\log \left(\frac{\sqrt{H-1-U+\surd HU}}{\sqrt{H-1+U+\surd HU}}\right)\end{eqnarray}$$

so that $w-\tilde{w}$ is constant for steady solutions. Using (4.2), the system can be closed with one further relationship between $w$ and $\tilde{w}$ . This is equation (21) in McCreary et al. (Reference McCreary, Zhang and Shetye1997),

(4.3) $$\begin{eqnarray}\tilde{w}_{\infty }={\textstyle \frac{1}{2}}\log (2\text{e}^{w_{\infty }}-1),\end{eqnarray}$$

where the subscript refers to values evaluated far upstream at late times, after a steady current has completely developed. In McCreary et al. (Reference McCreary, Zhang and Shetye1997), (4.3) is derived by taking the first term in an asymptotic expansion of a mass-balance equation similar to (3.15), which effectively requires that the coastal depth change is small relative to $H$ . As such, (4.3) is also the relationship between $w$ and $\tilde{w}$ in the quasi-geostrophic solutions of JSM and so must hold for small $Ro$ and $Q_{0}$ in the semi-geostrophic problem. We thus develop here a theory assuming that (4.3) holds for all values of $Q_{0}$ and $Ro$ , and then verify numerically that this asymptotic approximation captures the qualitative behaviour of the outflow even at moderate $Q_{0}$ and $Ro$ .

Given the conditions (4.1)–(4.3), the relevant solution to the steady equations (3.4) can be most succinctly described as

(4.4) $$\begin{eqnarray}w_{sn}(Q)=\log \left(\frac{r(Q)+\sqrt{r(Q)^{2}+k^{2}-1}}{1+k}\right),\end{eqnarray}$$

where

(4.5a,b ) $$\begin{eqnarray}k=\frac{U}{H-1+\sqrt{H}U},\quad r(Q)=\frac{\sqrt{2(Q+c)}-1}{H-1+\sqrt{H}U}\end{eqnarray}$$

and $c$ and $U$ are determined by applying (4.1)–(4.3). The heavy dashed curves (red online) in figure 12 give $w_{sn}$ and $U_{c}$ , both of which give good agreement with the numerical results even at $Ro=0.5$ .

Figure 13 shows the downstream width of the steady current $w_{sn}(Q_{0})=w_{D}$ . As for positive PVa, for a given $Ro$ the downstream current width increases with increasing mass flux. For fixed $Q_{0}$ , the width does not increase monotonically with $Ro$ . As the magnitude of the PVa is increased, the current first contracts (in line with the results for positive PVa) before expanding again after $Ro\approx 0.6$ . Numerical computations show that for a given $Ro$ there is a maximum flux $Q_{0}$ for which both (4.1) and (4.3) can be solved simultaneously, and thus an upper bound on the mass flux that supports a steady solution (marked by a dot in figure 13). The limited capacity of coastal currents to transport fluid with negative PVa was remarked on in Kubokawa (Reference Kubokawa1991) and observed in laboratory experiments in Horner-Devine et al. (Reference Horner-Devine, Fong, Monismith and Maxworthy2006). Figure 14(a) shows this maximum flux, which is not a monotonic function of $Ro$ – a result discussed in more detail in § 4.4. Also marked (dashed) in figure 14(a) is the line $Q_{0}=Ro$ , which is the maximum flux that supports a steady solution in the quasi-geostrophic limit.

Figure 13.

The steady solution width at the downstream edge, $w_{D}$ , as a function of the flux $Q_{0}$ . The largest $Q_{0}$ for which a steady solution exists at a given $Ro$ is marked by a dot.

In addition to the upper bound on $Q_{0}$ at fixed $Ro$ , there is an absolute maximum of $Q_{0}$ over all $Ro$ that supports a steady current. For real solutions, (4.4) requires that $r$ is positive. Expression (4.1) shows that $U<1-\sqrt{H}$ and so the denominator of $r$ is negative, and so therefore the numerator must also be negative, giving $Q+c<1/2$ . As $Q$ and $c$ are non-negative, they must both individually be less than $1/2$ . Thus no controlled steady solution exists when $Q_{0}>1/2$ , as in the similar relation obtained in Whitehead & Salzig (Reference Whitehead and Salzig2001) for flow out of a wide basin.

For negative PVa, only a portion of the outflow recirculates and so the downstream flux $Q_{d}$ , given by

(4.6) $$\begin{eqnarray}Q_{d}=\int _{0}^{w_{D}}uh\,\text{d}y=\left.\frac{h^{2}}{2}\right|_{y=w_{D}}^{y=0},\end{eqnarray}$$

is less than the total flux $Q_{0}$ . Profiles of $Q_{d}$ as a function of $Q_{0}$ are shown for various $Ro$ in figure 14(b). The curve for $Ro=0.6$ lies above that for $Ro=0.5$ , in accord with the non-monotonic behaviour of the total flux shown in (a).

Figure 14.

Mass flux for steady solutions with negative PVa. (a) The maximum mass flux supported by a given $Ro$ . (b) The downstream mass flux $Q_{d}$ as a function of total mass flux $Q_{0}$ , for various $Ro$ .

The momentum and energy constants $S_{0}$ and $R$ can also be computed for steady solutions. Since $u_{W}<0$ throughout the source region, the momentum input (2.20) across the source is negative:

(4.7) $$\begin{eqnarray}S_{0}=-\sqrt{2}\int _{0}^{Q_{0}}\sqrt{(Q-Q_{0}+\sqrt{2(Q_{0}+c)}-\sqrt{2(Q+c)})}\,\text{d}Q.\end{eqnarray}$$

The energy constant $R$ can be determined by evaluating (2.21) at the point $x=1,y=0$ . Here $u=0$ and $\unicode[STIX]{x1D713}=Q_{0}$ , so

(4.8) $$\begin{eqnarray}R=\sqrt{2(Q_{0}+c)}-Q_{0}.\end{eqnarray}$$

Figure 15 shows both $S_{0}$ and $R$ for various steady solutions. For small $Ro$ or $Q_{0}$ , $R\approx H$ , replicating the result for positive PVa. The momentum flux $|S_{0}|$ increases with the volume flux $Q_{0}$ , with larger $Ro$ corresponding to larger momentum flux.

Figure 15.

Constants of motion for steady solutions with negative PVa. (a) The momentum injected by the source, $S_{0}$ . (b) The total energy $R$ .

4.3 Downstream rarefaction

Once the steady solution has been determined, analysis via Riemann invariants can be applied by integrating (3.12) away from the point $U(w_{D})=U_{c}$ . In particular, figure 16 shows $U_{nose}$ as a function of $Q_{0}$ for various $Ro$ , allowing direct comparison with figure 10. As with positive PVa, increasing the mass flux for a fixed $Ro$ increases the nose speed. However, in contrast to figure 10, fixing $Q_{0}$ and increasing $Ro$ now increases the nose speed. That is, the fluid speed in the nonlinear Kelvin wave becomes slower and slower as the outer-layer depth is decreased, irrespective of the sign of the PVa. A similar relationship between $U_{nose}$ and layer depth is shown in Stern & Helfrich (Reference Stern and Helfrich2002, figure 2), and in Martin, Smeed & Lane-Serff (Reference Martin, Smeed and Lane-Serff2005, figure 3). The thin dashed horizontal line in figure 12(b) shows the value of $U_{nose}$ predicted from the analysis of the Riemann invariants, and is in good agreement with the numerical solution. When the current width vanishes, i.e. $w=0$ , the characteristic speed $\unicode[STIX]{x1D706}_{R}$ reduces to $U$ and so, since the rarefaction joins smoothly to the coast, $U_{nose}$ is also the propagation velocity of the expelled fluid at the farthest downstream edge. This is confirmed by the two vertical lines in figure 12(a), which are separated by $20U_{nose}$ corresponding to the interval between solutions at $t=20$ and $t=40$ .

Figure 16.

The nose speed $U_{nose}$ for a controlled rarefaction, as a function of $Q_{0}$ . The dashed line shows the wall velocity of the zero PVa solution.

The upstream rarefaction for steady solutions with negative PVa can also be described by Riemann invariants, although in this case it is ${\mathcal{R}}_{R}$ that is uniform. Applying a similar test to (3.13) shows that a shock always forms upstream for negative PVa, and the same ideas about matching shock and characteristic velocities can be used to determine the maximum width of this, which increases as vortical effects become more important.

4.4 Range of behaviours for negative PVa

For negative PVa, solutions can be categorised according to whether they are steady in the source region. In the quasi-geostrophic limit, JSM found that the speed ratio $a$ was again of crucial importance, with $a>1$ being necessary for a steady solution to form. Figure 17 shows the type of solution that occurs in semi-geostrophic flow based on the values of the two physical parameters $Ro$ and $Q_{0}$ . The solution type was determined by attempting to compute a steady solution that satisfies both (4.1) and (4.3). If, for a given $Q_{0}$ and $Ro$ , such a solution is possible, then these values permit a steady solution (region two). A fine sweep of the quasi-geostrophic limit of small $Q_{0}$ and $Ro$ shows that the boundary between flows that are steady and those that grow (region one) aligns with the contour $a=1$ as expected. In addition to this test, which is based on the asymptotic theory of § 4.2, the initial-value problem was solved over the full parameter range, using a grid that was refined close to the apparent boundaries between regions. Although there were some quantitative differences between theoretical and numerical results (these being larger far away from the QG limit) the test based on (4.3) and (4.1) was found to correctly describe the qualitative behaviour of all solutions.

Figure 17.

Solution type for negative PVa, compared with curves of constant speed ratio $a$ . Solution type one grows indefinitely in the source region, solution type two is steady and solution type three is believed to separate from the wall. The inset shows that the boundary between regions one and two aligns with the curve $a=1$ in the quasi-geostrophic limit.

As for positive PVa, away from the quasi-geostrophic limit region boundaries can differ from contours of $a$ , and so the system depends on $Q_{0}$ and $Ro$ independently, although it is still the case that solutions with small $a$ (Kelvin-wave dominated) are of the growing type. The shape of region two reflects the results of figure 14(a), where increasing $Ro$ allows a larger flux to be supported by steady solutions until, at some cut-off point around $Ro=0.6$ , the maximum permitted flux starts to decrease again. Increasing $Ro$ past this point is associated with a rapid increase in the value of $a$ – for flows that are strongly dominated by vorticity, steady solutions occur only at small fluxes.

Parameter values in the region marked ‘3’ in figure 17 give rise to complex solutions to (4.1) and (4.3). It appears that as the current width grows, the coastal layer depth $h_{W}$ decreases to zero and the expelled fluid separates from the coast. For negative $u$ , $\unicode[STIX]{x2202}h/\unicode[STIX]{x2202}y$ is positive and the expelled fluid is shallower at the coast than immediately offshore. If the initial outer layer depth is small (large $Ro$ ) and the magnitude of the wall velocity is sufficiently large, this suggests that the layer depth shrinks to zero and the expelled fluid is no longer attached to the coast upstream of the source.

Separation is explicitly excluded from the model presented here by the assumption that the expelled fluid lies in $0<y<w$ . Preliminary results from adapting the model to remove this restriction suggest that the fluid tries to separate in the source region, as discussed for separated flow out of a basin in, for example, Gill (Reference Gill1977) and Helfrich et al. (Reference Helfrich, Kuo and Pratt1999), as well as for laboratory experiments in Horner-Devine et al. (Reference Horner-Devine, Fong, Monismith and Maxworthy2006). For the present model, separation in the source region modifies the outflow profile $Q(x)$ and thus alters the problem beyond the scope considered here.

If for a given $Q_{0}$ and $Ro$ the solution starts to separate ( $h_{W}=0$ ), the initial-value problem is halted and the parameter pair is included in region three. It appears that the boundary between solutions which grow and those which separate is sharp in terms of $Ro$ : if the Rossby number is increased beyond some critical value of $Ro\approx 0.4$ then all unsteady flows eventually separate from the wall.