2.1 The zero PVa component

If the inlet and ambient upper-layer depths are equal then the PVa is zero, there is no vortex stretching or squashing and the expelled fluid has zero relative vorticity. The flow adjusts instantaneously, on the quasi-geostrophic time scale, to the solution of the linear problem given by field equation (2.1) with $\unicode[STIX]{x1D6F1}=0$ , subject to the boundary conditions (2.3) and (2.4). For the finite-width uniform source of § 4 the zero PVa component for a source of half-width unity can be written as

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D702}(x,y)=Q(x)\exp (-y/a)+\frac{2}{\unicode[STIX]{x03C0}}\int _{0}^{\infty }(l/\unicode[STIX]{x1D6FE}^{3})[\exp (-|x+1|/\unicode[STIX]{x1D6FE})-\exp (-|x-1|/\unicode[STIX]{x1D6FE})]\sin (ly)\,\text{d}l,\end{eqnarray}$$

for $\unicode[STIX]{x1D6FE}=(l^{2}+1/a^{2})^{1/2}$ , and rapidly evaluated by fast Fourier transforms. Downstream of the source the non-dimensional flow speed along the wall approaches $1/a$ , giving a dimensional speed of $Q_{0}\,f/c$ which, in the quasi-geostrophic limit, is small compared to the long-wave speed $c$ . The boundary ${\mathcal{C}}$ of expelled zero PVa fluid is advected as a passive material line by the steady KW flow and the nose of the expelled fluid moves downstream at constant dimensional speed $Q_{0}\,f/c$ . This flow remains as a component of the flow for non-zero PVa outflows and, for brevity, this KW-initiated, zero PVa component is referred to here simply as the KW flow. The parameter $a$ thus gives the ratio of the scale $Q_{0}/L_{v}$ for the speed of vorticity-driven flow to the KW flow speed, $Q_{0}\,f/c$ . Figure 3(a) shows contours of $\unicode[STIX]{x1D702}$ , and thus streamlines, of this steady component for $a=3$ , plotted for non-zero PVa with horizontal distances scaled here, as elsewhere, on the vortex length scale, $L_{v}$ . The flow is relatively broad and weak on the vortex scale. Figure 3(b) gives streamlines for $a=1/3$ , showing that the current is relatively narrow and strong (since both currents carry the same unit flux) on the vortex scale and far more asymmetrically turned downstream on this scale. Sections 4 and 5 discuss how the KW flow reinforces image vorticity advection for positive PVa outflows, described here as reinforcing dynamics, and competes with image vorticity in negative PVa flows, described as opposing dynamics.

Figure 3. Evenly spaced contours of surface elevation (and thus streamlines) for the zero PVa component of non-zero PVa flow for uniform outflow from an inlet of half-width unity, plotted with horizontal distances scaled on the vortex length scale, $L_{v}$ . The total efflux in each case is one. (a)  $a=3$ . For larger $a$ the flow is relatively broad and slow and closer to upstream–downstream symmetric vorticity-dominated flow. (b)  $a=1/3$ . For smaller $a$ the flow is relatively narrow and fast, turning rapidly downstream.

2.2 The long-wave equations

If the evolution is sufficiently gentle that the interface ${\mathcal{C}}$ does not overturn, ${\mathcal{C}}$ can be expressed as the curve $y=Y(x,t)$ for $Y$ a single-valued function of $x$ . The condition that the interface ${\mathcal{C}}$ between expelled fluid and the ambient is a material surface then can be written

(2.6a ) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x2202}_{t}+u\unicode[STIX]{x2202}_{x}+v\unicode[STIX]{x2202}_{y})(y-Y)=0,\quad \text{on}~y=Y(x,t), & & \displaystyle\end{eqnarray}$$

i.e.

(2.6b ) $$\begin{eqnarray}\displaystyle Y_{t}=[\unicode[STIX]{x1D702}(x,Y(x,t))]_{x}. & & \displaystyle\end{eqnarray}$$

Equation (2.6b ) is the equation for the conservation of mass for the outflow and consequently also for the conservation of PVa. For steady solutions (2.6b ) gives immediately that $\unicode[STIX]{x1D702}$ is constant on ${\mathcal{C}}$ , i.e. that the material line is a streamline as expected. As in contour dynamics, knowledge of $Y(x,t)$ gives the entire flow evolution. Now suppose that the interface ${\mathcal{C}}$ varies only slowly along the current compared to cross-current variations in $\unicode[STIX]{x1D702}$ and look for solutions of the form

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D702}(x,y,t;\unicode[STIX]{x1D716})=\unicode[STIX]{x1D702}^{(0)}(X,y,T)+\unicode[STIX]{x1D716}^{2}\unicode[STIX]{x1D702}^{(1)}(X,y,T)+O(\unicode[STIX]{x1D716}^{4}),\quad \text{where}~X=\unicode[STIX]{x1D716}x,T=\unicode[STIX]{x1D716}t.\end{eqnarray}$$

The speed ratio $a$ is taken to be of order unity as $\unicode[STIX]{x1D716}\rightarrow 0$ to retain the effects of both vortical and KW advection at leading order.

2.2.1 The non-dispersive, hydraulic limit

Substituting (2.7) in (2.1) gives the leading-order field equation

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{yy}^{(0)}-(1/a^{2})\unicode[STIX]{x1D702}^{(0)}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6F1},\quad & 0<y<Y,\\ 0,\quad & y>Y,\end{array}\right.\end{eqnarray}$$

subject to the boundary condition that $\unicode[STIX]{x1D702}^{(0)}(X,0)=Q(x)$ on the wall $y=0$ , thus implicitly requiring that $Q$ be a slowly varying function of $x$ . The current boundary $y=Y$ is a slowly varying function of $x$ and $t$ but for clarity this explicit dependence is suppressed here and subsequently. The leading-order (oceanic) flow outside the vortical layer (i.e. $y>Y$ ) can be written

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(0)}=Q_{e}\exp [-(y-Y)/a],\quad y\geqslant Y,\end{eqnarray}$$

giving the downstream current

(2.10) $$\begin{eqnarray}u^{(0)}=-\unicode[STIX]{x1D702}_{y}^{(0)}=(Q_{e}/a)\exp [-(y-Y)/a],\quad y\geqslant Y.\end{eqnarray}$$

Here $Q_{e}(x,t)$ gives the leading-order instantaneous downstream flux (of oceanic, zero PVa water) exterior to the vortical layer so $Q(x)-Q_{e}(x,t)$ gives the leading-order instantaneous downstream flux of vortical fluid at any station $x$ . In the irrotational flow region

(2.11) $$\begin{eqnarray}a\unicode[STIX]{x1D702}_{y}^{(0)}+\unicode[STIX]{x1D702}^{(0)}=0,\quad \forall y\geqslant Y,\end{eqnarray}$$

and so, by continuity of $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{y}$ across $y=Y$ , equation (2.11) gives the leading-order outer boundary condition at the edge of the vortical fluid.

The general solution of the field equation (2.8) inside the vortical layer (i.e. $y<Y$ ) can be written

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(0)}=-a^{2}\unicode[STIX]{x1D6F1}+A(X,T)\exp (y/a)+B(X,T)\exp (-y/a),\quad y\leqslant Y,\end{eqnarray}$$

where $A$ and $B$ are determined by the boundary conditions. Substituting (2.12) into (2.11) gives $A$ with (2.3) determining $B$ so

(2.13a,b ) $$\begin{eqnarray}A(X,T)=(1/2)a^{2}\unicode[STIX]{x1D6F1}\exp (-Y/a),\quad B(X,T)=Q+a^{2}\unicode[STIX]{x1D6F1}-{\textstyle \frac{1}{2}}a^{2}\unicode[STIX]{x1D6F1}\exp (-Y/a),\end{eqnarray}$$

with the leading-order solution for the vortical current in $y\leqslant Y$ ,

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(0)}=-a^{2}\unicode[STIX]{x1D6F1}+{\textstyle \frac{1}{2}}a^{2}\unicode[STIX]{x1D6F1}\exp [(y-Y)/a]+[Q+a^{2}\unicode[STIX]{x1D6F1}-{\textstyle \frac{1}{2}}a^{2}\unicode[STIX]{x1D6F1}\exp (-Y/a)]\exp [-y/a].\end{eqnarray}$$

Evaluating (2.14) at the current edge $y=Y$ gives

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(0)}(X,Y)=Q_{e}(x,t)=-{\textstyle \frac{1}{2}}a^{2}\unicode[STIX]{x1D6F1}+(Q+a^{2}\unicode[STIX]{x1D6F1})\exp (-Y/a)-{\textstyle \frac{1}{2}}a^{2}\unicode[STIX]{x1D6F1}\exp (-2Y/a).\end{eqnarray}$$

Substituting (2.15) into (2.6b ) then gives a first-order quasi-linear hyperbolic inhomogeneous equation for the leading-order evolution of the interface displacement $Y(x,t)$ . As is typical for such equations, for some forcings the interface can overturn. At the order here such overturnings can then be incorporated by allowing jumps in $Y$ . These solutions are discussed in § 4 below.

2.2.2 Dispersive effects

The jumps present in the leading-order solutions are smoothed by dispersion at the next order. The field equation at $O(\unicode[STIX]{x1D716}^{2})$ becomes

(2.16) $$\begin{eqnarray}\unicode[STIX]{x1D702}_{yy}^{(1)}-(1/a^{2})\unicode[STIX]{x1D702}^{(1)}=-\unicode[STIX]{x1D702}_{XX}^{(0)},\end{eqnarray}$$

whose solution in $y>Y$ , bounded as $y\rightarrow \infty$ , can be written

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(1)}=(1/2)ay\exp (-y/a)[Q_{e}\exp (Y/a)]_{XX}+C(X,T)\exp (-y/a),\end{eqnarray}$$

where $C(X,T)$ remains to be determined. Now (2.17) satisfies

(2.18) $$\begin{eqnarray}a\unicode[STIX]{x1D702}_{y}^{(1)}+\unicode[STIX]{x1D702}^{(1)}=(1/2)a^{2}\exp (-y/a)[Q_{e}\exp (Y/a)]_{XX},\quad \forall y\geqslant Y,\end{eqnarray}$$

and so, again by continuity of $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D702}_{y}$ across $y=Y$ , equation (2.18) forms the boundary condition at $y=Y$ for vortical flow at this order.

The solution of (2.16) in $0\leqslant y\leqslant Y$ , vanishing on $y=0$ to satisfy (2.3), can be written

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D702}^{(1)}=-(1/2)ay\exp (y/a)A_{XX}+(1/2)ay\exp (-y/a)B_{XX}+D\sinh (y/a),\end{eqnarray}$$

where $D(X,T)$ is determined from substituting in (2.18) as

(2.20) $$\begin{eqnarray}\displaystyle D(X,T) & = & \displaystyle (1/2)a^{2}\exp (-2Y/a)[Q_{e}\exp (Y/a)]_{XX}\nonumber\\ \displaystyle & & \displaystyle +\,(1/2)a^{2}(1+2Y/a)A_{XX}-(1/2)a^{2}\exp (-2Y/a)B_{XX}.\end{eqnarray}$$

Summing (2.15), (2.19) and (2.20) then gives the governing equation for the current edge as (2.6b ) with $\unicode[STIX]{x1D702}(x,Y(x,t))$ given with error of order $\unicode[STIX]{x1D716}^{4}$ by

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}(x,Y(x,t)) & = & \displaystyle Q_{e}(x,t)-[(1/2)aY\exp (Y/a)-a(a+2Y)\sinh (Y/a)]A_{xx}\nonumber\\ \displaystyle & & \displaystyle -\,[(1/2)aY\exp (-Y/a)+(1/2)a^{2}\exp (-2Y/a)\sinh (Y/a)]B_{xx}\nonumber\\ \displaystyle & & \displaystyle +\,(1/2)a^{2}\exp (-2Y/a)[Q_{e}\exp (Y/a)]_{xx}.\end{eqnarray}$$

As is typical for these expansions (Johnson & Clarke Reference Johnson and Clarke1999, Reference Johnson and Clarke2001), $\unicode[STIX]{x1D716}$ no longer appears explicitly although (2.21) formally requires that variations in $x$ are slow. In practice the long-wave equations can be accurate even when variations are not slow, as demonstrated by comparison with numerical integrations of the full equations in § 5. The dispersive aspects of solutions of (2.21) follow closely those for the evolution of vortical currents past coastal perturbations in Clarke & Johnson (Reference Clarke and Johnson1997a ,Reference Clarke and Johnson b ) with shocks in the leading-order equations smoothed through spreading dispersive wave trains, as in § 5.

3.1 Steady hydraulic solutions

Before considering the unsteady evolutions it is useful to discuss the solutions of equations (3.2) that are steady across the source region as these appear later as asymptotically steady states in most parameter regimes. For steady solutions the exterior flux $Q_{e}$ is constant and (2.15) can be written, using (3.2), as

(3.8) $$\begin{eqnarray}f(\unicode[STIX]{x1D6FC},Z)=-2\unicode[STIX]{x1D6FC}_{e},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{e}=Q_{e}/a^{2}\unicode[STIX]{x1D6F1}$ . Figure 4 shows contours of $f(\unicode[STIX]{x1D6FC},Z)$ . Steady solutions are given by the contours joining $\unicode[STIX]{x1D6FC}=0$ (corresponding to $Q=0$ at $x=-W$ ) to $\unicode[STIX]{x1D6FC}=1/a^{2}\unicode[STIX]{x1D6F1}$ (corresponding to $Q=1$ at $x=W$ ). As is standard in steady hydraulic solutions, $x$ appears only parametrically, through $\unicode[STIX]{x1D6FC}$ which determines $Q(x)$ which, in turn, determines $x$ for a given $Z$ .

For positive PVa, i.e. $\unicode[STIX]{x1D6F1}=1$ , a unique contour, corresponding to $\unicode[STIX]{x1D6FC}_{e}=0$ and shown bold in figure 4, joins $\unicode[STIX]{x1D6FC}=0$ , $Z=1$ to any $\unicode[STIX]{x1D6FC}=1/a^{2}>0$ . The fluid outside the vortical current is stagnant and current width increases monotonically across the source from zero at $x=-W$ (where $\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}Z=0$ so the long-wave speed vanishes and the flow is critically controlled) to have width

(3.9) $$\begin{eqnarray}Y_{hp}^{d}=a\log (1+1/a^{2}+\sqrt{2/a^{2}+1/a^{4}}),\end{eqnarray}$$

at $x=W$ . As $a\rightarrow \infty$ , $Y_{hp}^{d}\rightarrow \surd 2$ , the non-dimensional current width in the vorticity-dominated limit (Johnson & McDonald Reference Johnson and McDonald2006).

For negative PVa, i.e. $\unicode[STIX]{x1D6F1}=-1$ , steady solutions require a contour joining $\unicode[STIX]{x1D6FC}=0$ to $\unicode[STIX]{x1D6FC}=-1/a^{2}<0$ . The contour crossing the line $\unicode[STIX]{x1D6FC}=0$ that extends furthest into negative $\unicode[STIX]{x1D6FC}$ is that originating from $\unicode[STIX]{x1D6FC}=0$ , $Z=0$ and so has $f(\unicode[STIX]{x1D6FC},Z)=1$ . The contour consists of the lines $Z=0$ and $Z=2(1+\unicode[STIX]{x1D6FC})$ , also shown bold in figure 4. Thus no steady solution can connect $\unicode[STIX]{x1D6FC}=0$ to $\unicode[STIX]{x1D6FC}\leqslant -1$ and so if $\unicode[STIX]{x1D6FC}\leqslant -1$ , i.e. $a\leqslant 1$ , the flow cannot be steady for negative PVa: no steady flow is possible for KW-flow-dominated, opposing dynamics. It is shown in §§ 3.2 and 4 that this corresponds to the current expanding monotonically in the neighbourhood of the source. For $a>1$ a continuum of contours crosses the source region. The narrowest current (that with largest $Z$ ) terminates in $\unicode[STIX]{x1D6FC}<0$ with $Q=1$ when $\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}Z=0$ and the long-wave speed vanishes, i.e. on the line $Z=1+\unicode[STIX]{x1D6FC}=1-1/a^{2}$ shown dashed in figure 4. This current is critically controlled at the downstream edge of the source, giving a flow with flux $Q_{e}=1-1/2a^{2}$ of zero PVa (oceanic) fluid external to the outflowing current. This external flux has been observed by McCreary et al. (Reference McCreary, Zhang and Shetye1997) in their numerical integrations, where they describe outflow water flowing upstream before reversing direction and, together with fresher oceanic water, flowing downstream as a current straddling the density front between the fresher and salty waters. Away from the source the upstream and downstream vortical flows have widths

(3.10a,b ) $$\begin{eqnarray}Y_{hn}^{u}=-a\log (1-\sqrt{2/a^{2}-1/a^{4}}),\quad Y_{hn}^{d}=-a\log (1-1/a^{2}),\end{eqnarray}$$

(with $Y_{hn}^{u}\rightarrow \surd 2$ in the vorticity-dominated limit) shown as functions of the speed ratio $a$ in figure 5. The critical solution is the steady state flow that evolves from the unsteady hydraulic initial value problem discussed below. The full CD integrations in § 5 show that for narrower sources, where along-shore variations become significant, other steady solutions can be selected.

Figure 5. The upstream current width $Y_{hn}^{u}$ (upper curve) and the downstream critical-point current width $Y_{hn}^{d}$ (lower) of the critical hydraulic solution for a negative PVa outflow as a function of speed ratio for $a>1$ .

3.2 Unsteady hydraulic solutions: simple rarefactions

The unsteady, lengthening tongue of expelled fluid visible in figure 1, and the corresponding regions in the full integrations of § 5, can be regarded as rarefactions of the coastal flow. A rarefaction in gas dynamics, where the density smoothly decreases from a region where the density is larger to one where it is smaller, lengthens when the local propagation speed, being a function of density, is faster at the leading edge of the rarefaction than at its trailing edge. In the flow here a rarefaction smoothly joins two regions of differing current width, again lengthening when its leading edge travels faster than its trailing edge. The simplest rarefaction solutions of (3.2) are the self-similar solutions in terms of the speed variable $s=x/at$ , with $s$ giving the rarefaction propagation speed non-dimensionalised on $(|\unicode[STIX]{x1D6F1}_{0}|D/f)c$ which, in the quasi-geostrophic limit, is small compared to the KW speed $c$ . These require $Q$ to be a function of $s$ alone and so for an impulsively switched on source require $Q(s)=\text{H}(s)$ , a Heaviside function, corresponding to a point source at the origin. Away from the origin $Q$ is a constant and the governing equation (3.2) becomes

(3.11) $$\begin{eqnarray}[-s+\unicode[STIX]{x1D6F1}Z(1+\text{H}(s)/a^{2}\unicode[STIX]{x1D6F1}-Z)](\text{d}Z/\text{d}s)=0.\end{eqnarray}$$

Thus either $\text{d}Z/\text{d}s=0$ and the current has constant width or

(3.12a ) $$\begin{eqnarray}\displaystyle Z^{2}-(1+\text{H}(s)/a^{2}\unicode[STIX]{x1D6F1})Z+\unicode[STIX]{x1D6F1}s=0, & & \displaystyle\end{eqnarray}$$

i.e.

(3.12b ) $$\begin{eqnarray}\displaystyle Z={\textstyle \frac{1}{2}}(1+\text{H}(s)/a^{2}\unicode[STIX]{x1D6F1})\pm [{\textstyle \frac{1}{4}}(1+\text{H}(s)/a^{2}\unicode[STIX]{x1D6F1})^{2}-\unicode[STIX]{x1D6F1}s]^{1/2}. & & \displaystyle\end{eqnarray}$$

For positive PVa ( $\unicode[STIX]{x1D6F1}=1$ ), reinforcing dynamics, image vorticity reinforces the KW flow, shown from (3.12b ) by noting that solutions with $0\leqslant Z\leqslant 1$ are possible only downstream where $Q=1$ . Introducing

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D706}=|1+1/a^{2}\unicode[STIX]{x1D6F1}|,\end{eqnarray}$$

and writing $Z=\unicode[STIX]{x1D706}\hat{Z}$ , $s=\unicode[STIX]{x1D706}^{2}{\hat{s}}$ , gives the universal form

(3.14) $$\begin{eqnarray}\hat{Z}=(1-\sqrt{1-4{\hat{s}}})/2,\end{eqnarray}$$

with $0\leqslant {\hat{s}}\leqslant 1/4$ . The maximum value of $Z=\unicode[STIX]{x1D706}/2$ , which gives the minimum current width, occurs when ${\hat{s}}=1/4$ . For $a<1$ , when the KW flow dominates, the minimum is negative and so the rarefaction terminates downstream against the wall, driven by the narrow, intense flow of figure 3(b). For $a>1$ image vorticity dominates the broad, weak KW flow of figure 3(a) decreasing the speed at the wall: the maximum value of $Z$ occurs for $Z<1$ and the downstream edge of the rarefaction does not reach the wall, with instead the rarefaction terminating in a shock. This is discussed in greater detail for the finite-width source in § 4.2 and these behaviours can be seen in the solutions for finite-width sources there with figure 8(a) showing a downstream rarefaction terminating at the wall and figure 8(b) showing a downstream rarefaction terminating at a shock.

For negative PVa ( $\unicode[STIX]{x1D6F1}=-1$ ), opposing dynamics, image vorticity opposes the KW flow. The form of the downstream rarefaction (with $Q=1$ ) depends on the value of $a$ . For $a=1$ , $Z=\surd s$ with the range $0\leqslant s\leqslant 1$ covering all possible values of current width and so joining any current smoothly to the wall. For $a<1$ , KW flow dominated, the rarefaction has universal form

(3.15) $$\begin{eqnarray}\hat{Z}=(-1+\sqrt{1+4{\hat{s}}})/2.\end{eqnarray}$$

Since $Z\rightarrow 0$ as $s\rightarrow 0$ and $Z\rightarrow \infty$ as $s\rightarrow \infty$ , the rarefaction can join any current smoothly to the wall, as can be seen in the downstream rarefaction for finite-width sources in figure 9(a). For $a>1$ , image vorticity dominated, equation (3.12b ) becomes

(3.16) $$\begin{eqnarray}\hat{Z}=(1+\sqrt{1+4{\hat{s}}})/2,\end{eqnarray}$$

with $Z\rightarrow \infty$ as $s\rightarrow \infty$ and $Z=\unicode[STIX]{x1D706}$ when $s=0$ , giving a current of width fixed for all time at the downstream edge of the source region precisely equal to $Y_{hn}^{d}$ of (3.10), the width at the downstream edge of the source region of the unique hydraulic solution that is critically controlled when $Q=1$ . Thus the rarefaction smoothly joins this controlled hydraulic solution to the wall for all time, as can be seen for finite-width sources in figure 9(b,c). For negative PVa, upstream rarefactions (with $Q=0$ ) are also possible with

(3.17) $$\begin{eqnarray}Z=(1-\sqrt{1+4s})/2,\end{eqnarray}$$

giving a universal form defined for $-{\textstyle \frac{1}{4}}\leqslant s\leqslant 0$ . Figure 6 shows the current widths of upstream and downstream negative PVa rarefactions from (3.15), (3.17). The universal forms (3.14)–(3.17) may prove useful in collapsing time-dependent data from experiments and observations onto a single curve.

Figure 6. The width $Y/a=-\text{log}\,Z$ of a self-similar rarefaction in a negative PVa current as a function of the speed variable $s=x/at$ . (a) The upstream rarefaction in $x<0$ . This is the universal form for all $a$ . (b) A downstream rarefaction in $x>0$ . This form is not universal in these variables and is given here for $a=0.5$ .

4.1 The source region

For the uniform source (4.1), the right-hand side of (4.2) is piecewise constant and, in particular, can be integrated directly in $|x|\leqslant W$ to give the scaled current width across the source region,

(4.3a,b ) $$\begin{eqnarray}Z_{S}(t)=1/(1+t/2aW),\quad \text{i.e.}~Y_{S}(t)=a\log (1+t/2aW),\end{eqnarray}$$

on applying the initial condition that $Z_{S}=1$ when $t=0$ , shown in figure 7. Note that $Y_{S}\rightarrow t/2W~\text{as}~t\rightarrow 0$ as fluid in the source region is swept uniformly away from the wall at speed $1/2W$ . The integrations of the full equations in § 5 indicate that the growth of the current in the source region is relatively insensitive to the precise exit profile and so formula (4.3) will closely estimate the current width for other profiles provided the scale $W$ of the source width is chosen appropriately. The current width for this exit profile is the same on all characteristics issuing from the source region and so is constant (in $x$ ) across the source region.

Figure 7. The temporal development of the scaled width $Y_{S}/a$ of the constant (in $x$ ) width current in the neighbourhood of the source forced by the uniform discharge (4.1) as a function of the scaled time $t/2aW$ , as given by the formula (4.3). For positive PVa and negative PVa with $a\geqslant 1$ , the expansion terminates in the formation of a steady hydraulic solution. For negative PVa with $a<1$ , so KW flow dominates, the expansion continues indefinitely but slows dramatically at larger times.

Figure 8. The evolution of the boundary of a positive PVa outflow, reinforcing dynamics, from a uniform source occupying the region $|x|<1$ . In this and subsequent figures the coast $y=0$ is shown as a thick line and the edges of the source region as dashed lines with the thin line indicating the locus in time of the junction between the rarefaction and its leading jump. Each panel shows the outflow boundary at times $t=2$ , 5 and 12. (a) For speed ratio $a=1$ , typical of all KW-flow-dominated evolutions with $a\leqslant 1$ . (b) For $a=1.6$ so $1<a<a_{m}$ . (c) For $a=5$ , so $a>a_{m}$ , typical of image-vorticity-dominated evolutions.

Figure 9. As in figure 8 but for a negative PVa, opposing dynamics outflow. (a) For speed ratio $a=0.75$ , typical of all KW-flow-dominated evolutions with $a\leqslant 1$ , at times $t=2$ , 5 and 15. (b) For $a=1.8$ , so $1<a<a_{m}$ , at times $t=1$ , 5 and 15. (c) For $a=5$ , so $a>a_{m}$ , at times $t=2$ , 5 and 8. In (b), (c), as in all vortically dominated flows, the maximum width of the downstream current is determined by hydraulic control at the downstream edge of the source.

For positive PVa, reinforcing dynamics, § 3.1 shows that, for all $a$ , a steady, critical, hydraulic solution, controlled from the upstream edge ( $x=-W$ ), set up by characteristics spreading downstream from $x=-W$ , extends across the source region. Within the source region the unsteady constant-width current expands until it meets this steady solution as can be seen in each frame of figure 8. The controlled solution is established across the whole source region by time $t_{2p}$ , given from (3.9) and (4.3) by $Y_{S}(t_{2p})=Y_{hp}^{d}$ .

For negative PVa, opposing dynamics, a steady, critical, hydraulic solution, controlled from the downstream edge ( $x=W$ ) of the source region, extends across the source region provided $a>1$ , so vorticity dynamics dominates the KW flow. As for positive PVa, the unsteady constant-width current expands until it meets the steady solution as in figure 9(b,c). The controlled solution originates from $x=W$ and $y=Y_{hn}^{d}>0$ and is established over times $t>t_{1n}$ , where $Y_{S}(t_{1n})=Y_{hn}^{d}$ , by characteristics spreading upstream from $x=W$ , with all characteristics that exit downstream (and form the rarefaction described in § 4.2) having done so by time $t_{1n}$ . The controlled solution is fully established by time $t_{2n}$ when $Y_{S}(t_{2n})=Y_{hn}^{u}$ . For a negative PVa outflow with $a\leqslant 1$ , so KW flow dominates, all characteristics exit downstream establishing the rarefaction of § 4.2, there is no steady hydraulic solution crossing the source region and the constant-width current grows indefinitely as in figure 9(a), where $a=0.75$ .

4.2 The downstream ( $x>W$ ) region for positive PVa

The upstream and downstream currents for both positive and negative PVa outflows are led by rarefactions similar to those of § 3.2. Equations (4.2) show that characteristics for $|x|>W$ , outside the source region, are straight lines along which $Z$ remains constant. On each characteristic the value of the constant is given by the value of $Z$ on the characteristic when it crosses $x=\pm W$ . For the uniform source here the value of $Z$ on all characteristics in $|x|<W$ is simply $Z_{S}(t)$ and so the value of $Z$ on any characteristic in $|x|>W$ is determined by the time $t$ when the characteristic crosses $x=\pm W$ .

For positive PVa, consider a characteristic in $x>W$ and suppose that it crossed $x=W$ at a crossing time, $t=t_{c}$ . Then on this characteristic $Z(x,t)=Z_{S}(t_{c})=Z_{c}$ (say). Since the speed is constant on the characteristic (4.2) gives, equating expressions for the characteristic slope,

(4.4) $$\begin{eqnarray}(x-W)/a(t-t_{c})=\unicode[STIX]{x1D706}Z_{c}-Z_{c}^{2}.\end{eqnarray}$$

Inverting (4.3) for the crossing time $t_{c}$ in terms of $Z_{c}$ and substituting in (4.4) gives the implicit equation for the boundary of the downstream rarefaction as

(4.5) $$\begin{eqnarray}x=W+a(t-t_{c})Z_{c}(\unicode[STIX]{x1D706}-Z_{c})=W+aZ_{c}(\unicode[STIX]{x1D706}-Z_{c})t-2a^{2}W(1-Z_{c})(\unicode[STIX]{x1D706}-Z_{c}).\end{eqnarray}$$

The solution of this quadratic equation for $Z_{c}$ gives the scaled width $Z_{c}$ in terms of $x$ and $t$ . Equation (4.4) can also be regarded as giving the parametric representation

(4.6) $$\begin{eqnarray}(x_{c},Z_{c})=(W+a(t-t_{c})Z_{c}(\unicode[STIX]{x1D706}-Z_{c}),Z_{c}),\end{eqnarray}$$

with $Z_{c}=Z_{S}(t_{c})$ and the rarefaction at any time $t$ given by taking all values of $t_{c}$ in the interval $0\leqslant t_{c}\leqslant t$ , i.e. over all earlier crossing times. For $t\leqslant t_{2p}$ these rarefactions join smoothly to the unsteady growing current in the source region as can be seen in figure 8. For $t\geqslant t_{2p}$ , form (4.6) continues to give the rarefaction provided the parameter $t_{c}$ is restricted to the interval $0\leqslant t_{c}\leqslant t_{2p}$ , so the minimum value of $Z_{c}$ is $Z_{c}=Z_{S}(t_{2p})$ , giving a maximum current width of $Y_{c}=Y_{hp}^{d}$ . The rarefaction joins smoothly to the downstream end of the constant-width, hydraulically controlled current growing from the downstream edge ( $x=W$ ) of the source region at times $t\geqslant t_{2p}$ , visible in all panels of figure 8. For $a\leqslant 1$ , where the KW flow reinforces but dominates the vortical flow, as in figure 8(a) where $a=1$ , this completes the solution.

The slope of the nose of the rarefaction follows from differentiating (4.5) with respect to $x$ at fixed time and setting $Z_{c}=1$ to give the value of the slope at the wall as

(4.7) $$\begin{eqnarray}\text{d}Y/\text{d}x=-(a/Z_{c})(\text{d}Z_{c}/\text{d}x)=-a^{3}/[2a^{2}W+at(1-a^{2})].\end{eqnarray}$$

The initial slope is $-a/2W$ and remains constant for $a=1$ as in figure 8(a). For $a<1$ , KW flow dominated, the nose slope decreases (in magnitude) as $t$ increases but for $a>1$ , vortically dominated flow, the slope becomes progressively more negative, with the front steepening and becoming vertical when $t=t_{b}=2aW/(a^{2}-1)$ . At any subsequent time, $t>t_{b}$ , as for the positive PVa point source of § 3.2, the maximum downstream extent of the rarefaction occurs at $Y>0$ , away from the wall. The downstream edge of the rarefaction does not reach the wall and instead terminates in a shock as in figure 8(b,c). Let the shock have position $x_{J}(t)$ and the current there have width $Y_{J}(t)$ . Then, by conservation of mass, the shock moves at speed

(4.8) $$\begin{eqnarray}\text{d}x_{J}/\text{d}t=(1-Q_{e})/Y_{J}=-(1/2)a(1-Z_{J})(1+2/a^{2}-Z_{J})/\log Z_{J},\end{eqnarray}$$

the change in downstream flux divided by the current width, where, as usual, $Z_{J}=\exp (-Y_{J}/a)$ . Differentiating (4.5) with respect to $t$ and setting $Z_{c}=Z_{J}$ gives a second expression for the speed of the downstream edge of the rarefaction. Equating this with (4.8) gives the ordinary differential equation for $Z_{J}(t)$ ,

(4.9) $$\begin{eqnarray}[(\unicode[STIX]{x1D706}-2Z_{J})t+2a(1+\unicode[STIX]{x1D706}-Z_{J})]\frac{\text{d}Z_{J}}{\text{d}t}=-\frac{(1-Z_{J})(1+2/a^{2}-Z_{J})}{2\log Z_{J}}-Z_{J}(\unicode[STIX]{x1D706}-Z_{J}).\end{eqnarray}$$

This equation is to be solved subject to the condition of a jump initiated at $t=t_{b}$ , i.e. $Z_{J}(t_{b})=1$ . The jump height $Z_{J}(t)$ can be obtained as an implicit function of $t$ but it is more straightforward to compute $Z_{J}(t)$ directly by numerically integrating (4.9). The locus of the point $(x_{c},Z_{c})$ where the downstream rarefaction meets the jump is shown as the thin line in each of figure 8(b,c), growing from $x\sim 2.2$ in figure 8(b) where $a=1.6$ and almost immediately in figure 8(c) where $a=5$ . The jump can grow until the right-hand side of (4.9) vanishes, so that $\text{d}Z_{J}/\text{d}t=0$ . This occurs for a maximum jump height $Y_{m}=-a\log Z_{m}$ where $0\leqslant Z_{m}\leqslant 1$ solves

(4.10) $$\begin{eqnarray}(1-Z_{m})(1+2/a^{2}-Z_{m})=-2Z_{m}(\unicode[STIX]{x1D706}-Z_{m})\log Z_{m}.\end{eqnarray}$$

The root is simple and thus the jump asymptotes to its maximum height as a decaying exponential. As $a$ increases, $Y_{m}$ increases monotonically from zero when $a=1$ , until $Y_{m}=Y_{hp}^{d}$ , when $a=a_{m}\sim 1.8684$ , given by substituting (3.9) in (4.10). In figure 8(b), $1<a<a_{m}$ so $0<Y_{m}<Y_{hp}^{d}$ and the jump asymptotes downstream to height $Y_{m}$ , terminating the rarefaction at all times. The jump speed is faster than the speed of the junction with the constant-width current at the rear of the rarefaction and so the rarefaction elongates and flattens with increasing time. In figure 8(c), typical of all vortically dominated reinforcing flows, $a>a_{m}$ so $Y_{m}>Y_{hp}^{d}$ . The jump speed is slower than the speed of the junction with the constant-width current and so the rarefaction contracts with increasing time until it is extinguished at time $t_{J}$ when $Y_{J}=Y_{hp}^{d}$ , with the current then continuing downstream with constant width $Y_{hp}^{d}$ terminating in a full-width jump.

4.3 The downstream ( $x>W$ ) region for negative PVa

For negative PVa, the downstream rarefaction follows similarly to (4.5), equation (4.6) with (4.2) giving the slope-speed equation

(4.11) $$\begin{eqnarray}(x-W)/a(t-t_{c})=-\unicode[STIX]{x1D706}Z_{c}+Z_{c}^{2},\end{eqnarray}$$

and so the implicit equation for the boundary of the rarefaction as

(4.12) $$\begin{eqnarray}x=W-a(t-t_{c})Z_{c}(\unicode[STIX]{x1D706}-Z_{c})=W-aZ_{c}(\unicode[STIX]{x1D706}-Z_{c})t+2a^{2}W(1-Z_{c})(\unicode[STIX]{x1D706}-Z_{c}),\end{eqnarray}$$

and the parametric representation

(4.13) $$\begin{eqnarray}(x_{c},Z_{c})=(W-a(t-t_{c})Z_{c}(\unicode[STIX]{x1D706}-Z_{c}),Z_{c}),\end{eqnarray}$$

with the parameter $t_{c}$ covering the interval $0\leqslant t_{c}\leqslant t$ . Differentiating (4.12) and setting $Z_{c}=1$ shows that, for all $a$ , the slope of the nose of the rarefaction decreases with increasing $t$ and so no shock forms at the leading edge of the downstream rarefaction for negative PVa. Thus (4.13) gives the solution for all times for $a\leqslant 1$ , giving a rarefaction that matches smoothly to the indefinitely widening current in the source region, as in figure 9(a). For $a>1$ the source region flow becomes controlled from its downstream edge ( $x=W$ ) at times $t\geqslant t_{1n}$ . Form (4.13) continues to describe the rarefaction at times $t\geqslant t_{1n}$ provided the parameter $t_{c}$ is restricted to the interval $0\leqslant t_{c}\leqslant t_{1n}$ , giving a maximum rarefaction width of $Y_{c}=Y_{hn}^{d}$ which joins smoothly to the controlled solution in the source region as in figure 9(b,c).

4.4 The upstream ( $x<-W$ ) region for negative PVa

The upstream current for negative PVa shares many properties with the downstream current for positive PVa and so simply the differences are noted here. The slope-speed equation

(4.14) $$\begin{eqnarray}(x+W)/a(t-t_{c})=-Z_{c}+Z_{c}^{2},\end{eqnarray}$$

gives the implicit equation for the boundary the rarefaction as

(4.15) $$\begin{eqnarray}x=-W-a(t-t_{c})Z_{c}(1-Z_{c})=-W-aZ_{c}(1-Z_{c})t-2a^{2}W(1-Z_{c})^{2},\end{eqnarray}$$

and the parametric representation

(4.16) $$\begin{eqnarray}(x_{c},Z_{c})=(-1-a(t-t_{c})Z_{c}(1-Z_{c}),Z_{c}),\end{eqnarray}$$

with the parameter $t_{c}$ covering the interval $0\leqslant t_{c}\leqslant t$ . For all $a$ , for $t<t_{2n}$ , the downstream edge ( $x=-W$ ) of the rarefaction joins smoothly to the growing constant-width current in the source region. For $a\leqslant 1$ the source region width grows indefinitely and so to does the rarefaction width, as in figure 9(a). For $a>1$ steady flow, hydraulically controlled from the downstream edge, is established across the whole source region by $t=t_{2n}$ . Form (4.16) continues to describe the rarefaction at times $t\geqslant t_{2n}$ provided the parameter $t_{c}$ is restricted to the interval $0\leqslant t_{c}\leqslant t_{2n}$ , giving a maximum rarefaction width of $Y_{c}=Y_{hn}^{u}$ which joins smoothly to the controlled solution in the source region as in figure 9(b,c). Differentiating (4.15) and setting $Z_{c}=1$ shows that, for all $a$ , the rarefaction immediately has infinite slope at $x=-W$ , $t=0$ . The breaking time $t_{b}$ for negative PVa outflows is thus zero for all $a$ in contrast to positive PVa outflows where $t_{b}>0$ for $a>1$ and rarefactions do not break for $a\leqslant 1$ . A shock thus leads the rarefaction upstream for all $t>0$ . Similar considerations to those giving (4.9) give the governing equation for the jump height,

(4.17) $$\begin{eqnarray}[(2Z_{J}-1)t+4a(1-Z_{J})]\text{d}Z_{J}/\text{d}t=(1/2)(1-Z_{J})^{2}/\log Z_{J}+Z_{J}(1-Z_{J}),\end{eqnarray}$$

subject to the initial condition $Z_{J}(0)=1$ . Again a numerical solution is more convenient than the implicit analytical solution. The locus of the point $(x_{c},Z_{c})$ where the upstream rarefaction meets the jump is shown as the thin line in figure 9. The jump can grow until the right side of (4.17) vanishes, giving a maximum jump height $Y_{m}$ satisfying

(4.18) $$\begin{eqnarray}Y_{m}/a=-\log Z_{m}=(1-Z_{m})/2Z_{m},\end{eqnarray}$$

i.e. $Y_{m}\approx 1.25643a$ , with the jump asymptoting exponentially with increasing $-x$ or $t$ to its maximum height. The maximum increases linearly with $a$ until $Y_{m}=Y_{hn}^{u}$ which occurs first when

(4.19) $$\begin{eqnarray}a_{m}=[(1+\sqrt{1-b^{2}})/b^{2}]^{1/2}\approx 1.82206,\quad b=1-\exp (-Y_{m}/a).\end{eqnarray}$$

If $a\leqslant a_{m}$ , as in figure 9(a,b), then the shock height asymptotes upstream to $Y_{m}$ with the rarefaction lengthening and flattening as for the positive PVa outflow of figure 8(b). If $a\geqslant a_{m}$ , as in figure 9(c), $Y_{m}>Y_{hn}^{u}$ and, as in figure 8(c), the rarefaction is eventually extinguished with the upstream-flowing controlled current led by a full-width jump.

Figure 9(b) gives the current boundary for $a=1.8$ . The initial evolution follows that of figure 9(a) for $a\leqslant 1$ with an expanding current in the source region bordered by upstream and downstream rarefactions with a jump terminating the upstream-propagating rarefaction. However (4.3) shows that at time $t_{1}=2a[\exp (-Y_{hn}^{d})-1]$ the current width has grown to equal $Y_{hn}^{d}$ . All characteristics that exit the source region downstream have done so by time $t_{1}$ and characteristics spread upstream from the point $(t,x)=(t_{1},W)$ to give the hydraulic solution (3.8) controlled from the downstream edge of the source. This form of solution holds until time $t_{2}=2a[\exp (-Y_{hn}^{u})-1]$ when the hydraulic solution is established across the whole source region and the constant-width current in the source region has disappeared. The region of controlled flow then expands upstream at constant speed $aZ_{hn}^{u}(1-Z_{hn}^{u})$ , led by the rarefaction.

Figure 9(c) gives an example of the current boundary in this case for $a=5$ , so $a>a_{m}$ . The evolution initially follows that shown in figure 9(a,b) for evolutions with $a<a_{m}$ . A constant-width current in the source region expands until time $t_{1}$ when it becomes progressively squeezed from downstream as controlled flow is established in $x<W$ from the critical point at $x=W$ . By time $t_{2}$ this controlled flow is established across the whole source region and expands at constant speed into $x<-W$ as a constant-width current led by a rarefaction terminated by a growing jump. However when $a>a_{m}$ the limiting height of this jump exceeds the maximum width of the controlled solution. The junction between the downstream-edge-controlled solution and the rarefaction is non-dissipative and so moves at the controlled solution speed until it overtakes the slower-moving dissipative jump that terminates the rarefaction upstream. Thus at some finite time $t_{3}>t_{2}$ the rarefaction disappears and the hydraulic solution simply terminates at a dissipative jump.

4.5 Terminal speeds of the leading edges of the anomalies

The speeds of the leading edges of the PV anomalies are of observational interest. The solutions here show that these speeds vary as the flow evolves but asymptote to a steady terminal velocity, $u_{e}$ (say), at large times. Figure 10(a) shows $u_{e}$ of the downstream edge for positive PVa as a function of $a$ . For $a<1$ , KW flow dominated, this is the speed of the leading edge of the rarefaction and is given by substituting $Z_{c}=1$ into (4.4) as $u_{e}=a(\unicode[STIX]{x1D706}-1)=1/a$ , precisely the speed, downstream from the source, of the zero PVa component of § 2.1. For $1<a<a_{m}$ , moderately vorticity-dominated flow, the current is led by a jump of height $Y_{m}$ at speed given by substituting $Z_{m}$ from (4.10) into (4.4). With increasing $a$ the speed decreases as the width $Y_{m}$ increases. For $a\geqslant a_{m}$ , strongly vortex-dominated flow, the current width is the controlled width $Y_{hp}^{d}$ and so $u_{e}=1/Y_{hp}^{d}$ . In the limit $a\rightarrow \infty$ , $Y_{hp}^{d}\rightarrow \surd 2$ and $u_{e}\rightarrow 1/\surd 2$ , as in the vorticity-dominated limit (Johnson & McDonald Reference Johnson and McDonald2006). The speeds, here and elsewhere, are non-dimensionalised on $Q_{0}/L_{v}=(Q_{0}|\unicode[STIX]{x1D6F1}_{0}|D)^{1/2}$ , appropriate for vorticity-dominated flows. The dashed curve in figure 10(a) shows $au_{e}$ , the speed non-dimensionalised on $Q_{0}\,f/c$ , appropriate for the KW flow. In both cases the speed is proportional to the area flux $Q_{0}$ divided by an appropriate length, related to the current width, where the length is the dimensional Rossby radius in KW flow and the vortex scale $L_{v}$ in vorticity-dominated flow, with the non-dimensional parameter $a$ giving the ratio of the velocity scale for vorticity-driven flow to that for the KW flow. In the quasi-geostrophic limit both velocity scales are small compared to the long-wave speed $c$ .

Figure 10. The terminal speeds, $u_{e}$ , of the leading edges of the anomalies as a function of $a$ . (a) Positive PVa, $\unicode[STIX]{x1D6F1}>0$ , so KW flow and image vorticity reinforce. The solid line gives $u_{e}$ normalised on $(Q_{0}|\unicode[STIX]{x1D6F1}_{0}|D)^{1/2}$ , the vortex velocity scale, and approaches $1/\surd 2$ as $a\rightarrow \infty$ . The dashed line gives $au_{e}$ , the speed normalised on $Q_{0}\,f/c$ , the KW flow velocity scale, and so approaches 1 as $a\rightarrow 0$ . (b) Negative PVa, $\unicode[STIX]{x1D6F1}<0$ , so KW flow and image vorticity oppose, with speeds normalised on the vortex velocity scale, as elsewhere. The dashed line gives the downstream speed which is precisely the KW flow speed $1/a$ and the solid line gives the upstream speed which approaches $1/\surd 2$ as $a\rightarrow \infty$ .

For negative PVa outflows the terminal speed of the downstream leading edge is the speed of the rarefaction at the wall and so follows from substituting $Z_{c}=1$ into (4.11) yielding, once again, $u_{e}=1/a$ , precisely the speed, downstream from the source, of the zero PVa component of § 2.1, shown dashed in figure 10(b). The terminal speed of the downstream leading edge is given by the jump speed. For $a<a_{m}$ , KW flow dominated and also moderately vortically dominated flow, the terminal jump height is given by (4.18) with the terminal speed following from substituting $Z_{m}$ into (4.11). For $a\geqslant a_{m}$ , strongly vortex-dominated flow, the current width is the controlled width $Y_{hn}^{u}$ and so $u_{e}=Q_{e}/Y_{hn}^{u}=(1-1/2a^{2})/Y_{hn}^{u}$ , from § 3.1. In the limit $a\rightarrow \infty$ , $Y_{hn}^{u}\rightarrow \surd 2$ and $u_{e}\rightarrow 1/\surd 2$ , as in the vorticity-dominated limit (Johnson & McDonald Reference Johnson and McDonald2006). The solid line in figure 10(b) shows this upstream terminal speed, which is small compared to the downstream KW flow speed when vortex effects are weak ( $a\lesssim 2$ ) but exceeds the KW flow speed for strong vortex effects ( $a\gtrsim 2$ ).

5.1 Positive PVa outflows

The full problem (2.1)–(2.4) can be solved numerically with extreme accuracy, using the CD implementation of Dritschel (Reference Dritschel1988) and simply replacing the logarithmic Green’s function in Johnson & McDonald (Reference Johnson and McDonald2006) by the appropriate Green’s function for (2.1), i.e. $\text{K}_{0}([(x-x_{0})^{2}+(y-y_{0})^{2}]^{1/2}/a)$ where $\text{K}_{0}$ is the modified Bessel function of the second kind of order zero. The analysis of § 4 formally requires that the source width is large, i.e. $W\gg 1$ , but, as suggested above, this is overly restrictive in practice. Figure 11 compares CD integrations of the unapproximated problem with long-wave solutions for a point source of positive PVa, obtained by setting $W=0$ in § 4.2. Even for this most extreme case, the agreement is remarkable. The exit velocity is infinite so the controlled solution is set up instantaneously giving a crossing time of $t_{c}=0$ . For $a<1$ and $t>0$ , equation (4.7) shows that both the rarefaction and its nose flatten as they propagate downstream. For $a=1$ the rarefaction again flattens as it propagates downstream with however the nose meeting the wall at right angles, as in figure 11(a), similarly to figure 8(a). In figure 11(b), $1<a<a_{m}$ , as in figure 8(b) so the leading jump in the long-wave solution asymptotes to a value less than the constant width of the hydraulically controlled current. Dispersion in the full problem means that the nose of the flow is rounded and waves appear on the boundary of the controlled current. In figure 11(c), $a>a_{m}$ , as in figure 8(c) so the constant-width, hydraulically controlled current is set-up immediately. The flow development closely follows the vorticity-dominated solution of Johnson & McDonald (Reference Johnson and McDonald2006), corresponding to $a\gg 1$ . Rossby waves on the current boundary originate from small fluctuations near the source and propagate downstream, with high PV to their right, towards the nose of the current. The speed of the waves decreases as they propagate towards the nose and so their amplitude increases to conserve wave action flux. Johnson & McDonald (Reference Johnson and McDonald2006) give exact solutions for linear waves on the steady asymptotic flow in the vorticity-dominated limit and the increase in amplitude is discussed there and in Haynes et al. (Reference Haynes, Johnson and Hurst1993). The finite amplitude waves are governed by (2.6b ) and (2.21). The development of a leading eddy in the vorticity-dominated limit is discussed in Stern & Pratt (Reference Stern and Pratt1985). Note that the hydraulic solution does not exceed $Y_{hp}^{d}$ and so ${\mathcal{F}}^{\prime \prime }(Z)$ remains single-signed along the entire current boundary. The disturbance speed is a monotonic function of the current width throughout the evolution and so only classical shocks and rarefactions are present in positive PVa outflows. For all $a$ , from (3.5), throughout the evolution of the current the injected momentum flux in the hydraulic limit is positive, $\unicode[STIX]{x0394}M_{s}>0$ , and the current turns downstream.

Figure 11. CD integrations for the full problem (thin lines, blue online) and the analytical long-wave solutions (thick lines, red online) for positive PVa outflows from a point source at times $t=10$ , 50, and 100 for non-dimensional Rossby radii of (top to bottom) (a)  $a=1$ , (b)  $a=1.3$ , (c)  $a=2$ .

5.2 Negative PVa outflows

Figure 12 compares CD integrations with long-wave solutions for a point source of negative PVa, obtained by setting $W=0$ in §§ 4.3 and 4.4. As for the positive PVa point source, the exit speed is infinite and so in the long-wave solution the asymptotic values on $x=0$ are set up instantly. For figure 12(a), where $a=0.75$ , so $a\leqslant 1$ as in figure 9(a), this means that the upstream and downstream rarefactions have infinite width at $x=0$ . Away from the origin the downstream rarefaction of the long-wave solution is indistinguishable from the full integration and the upstream rarefaction and jump model the full solution, where again sharp changes in the boundary are smoothed by dispersion. In figure 12(b), $1<a<a_{m}$ , as in figure 9(b) so the long-wave solution asymptotes to a jump of height less than the constant width of the hydraulically controlled current. The long-wave solutions capture much of the flow behaviour but differences are visible. The large exit velocity at the origin means that the full solution overshoots the minimal controlled solution. The full solution adjusts to a steady solution with upstream and downstream heights linked through (3.8) but the upstream current is wider and the downstream narrower than for the minimal solution. The downstream rarefaction joins to a constant-width current issuing from the downstream control and the upstream rarefaction joins to a constant-width current terminating in a jump. Both these effects are the result of the non-convexity of the flux function ${\mathcal{F}}(Z)$ .

Figure 12. As for figure 11 but for a negative PVa outflow. (a)  $a=0.75$ at times $t=10$ , 20 and 100. (b)  $a=1.3$ at times $t=30$ , 150 and 500.

Figure 13. As for figure 12 but for a uniform source of half-width $W=20$ . (a)  $a=1.3$ at times $t=30$ , 150 and 500. (b)  $a=2$ at times $t=10$ , 50 and 200.

The question arises as to how much of the deviation seen in figure 12 is due to choosing the extreme of point-source forcing $(W=0)$ . Figure 13 compares full integrations with long-wave solutions for a uniform source of negative PVa of width $W=20$ . The agreement is much closer, particularly at early times when the full and analytical solutions are almost indistinguishable. As in § 4.1, the current width across the source region grows uniformly until it reaches the controlled solution, which is established from the downstream edge, $x=W$ . At later times, as for positive PVa, boundary Rossby waves originating from small fluctuations near the source propagate, here upstream with low PVa to the left, towards the nose of the current, again slowing and increasing in amplitude. Lengthening the source region means that the hydraulic theory becomes more accurate near the source but the introduction of a definite length scale, and thus a particular value of $\unicode[STIX]{x1D716}$ , introduces dispersive waves that are absent in the point-source solutions of figure 12. At the latest times, a small shelf grows from the downstream edge, a manifestation of the non-convex flux function.

In figure 12(a) $a=0.75$ and so (3.7) suggests that throughout the evolution of the current the injected momentum flux in is positive, $\unicode[STIX]{x0394}M_{s}>0$ , and the current should turn predominantly downstream, dominated by the KW flow. With increasing $a$ , as in figure 12(b), the injected momentum flux decreases, becoming negative, and the current turns upstream, dominated by image-vorticity driving. This upstream current with weak downstream KW flow is clearly visible in the full-physics integrations in figure 3 of McCreary et al. (Reference McCreary, Zhang and Shetye1997).