3.1 A simple example

As a very simple example for which it is possible to write down an explicit solution (Swaters (Reference Swaters2013, Reference Swaters2015a ) describes the nonlinear steady solution associated with a boundary abyssal height profile $h(x,y_{0})$ that possesses both upslope and downslope groundings) let us consider the case for which

where $\widetilde{h}_{0}>0$ and $\unicode[STIX]{x1D6FC}$ are constants. In this simple example, the bottom topography has constant slope and the water column deepens as $x$ increases. The abyssal water mass height along $y=y_{0}$ varies linearly in the cross-slope direction and has a single grounding located at $x=x_{g}\equiv \widetilde{h}_{0}/\unicode[STIX]{x1D6FC}$ , and the meridional velocity along $y=y_{0}$ is $v(x,y_{0})=-(1+\unicode[STIX]{x1D6FC})$ , which is independent of $x$ . There is no loss of generality in choosing $h_{B}^{\prime }=-1$ on account of the underlying non-dimensionalization scheme in (2.4). (Observe that if $\unicode[STIX]{x1D6FC}=-1$ then $h_{B}+h_{0}=\widetilde{h}_{0}$ , implying $v=0$ along $y=y_{0}$ .) The no-shock condition (3.9), in this example, reduces to simply

Substitution of (3.11) into (3.3) implies that the characteristics or geostrophic streamlines are the non-parallel lines given by (for constant $\unicode[STIX]{x1D70F}$ )

which if substituted into (3.2) yields

from which we see, of course, that $\unicode[STIX]{x1D70F}(x,y_{0})=x$ and $h(x,y_{0})=h_{0}(x)$ . It is only for this simple example that the solution for $h(x,y)$ has the separated form seen in (3.14) with respect to its $x$ and $y$ dependences. In general, it does not (see Swaters Reference Swaters2013, Reference Swaters2015a ).

Figure 2 is a contour plot of the characteristics or geostrophic streamlines $\unicode[STIX]{x1D70F}(x,y)$ as given by (3.13) for the parameter values $\widetilde{h}_{0}=1$ , $\unicode[STIX]{x1D6FD}=0.02$ , $y_{0}=5$ and $\unicode[STIX]{x1D6FC}=-0.5$ (corresponding to an equatorward speed of approximately $2~\text{cm}~\text{s}^{-1}$ ) for the region $-2\leqslant x\leqslant 2.5$ and $-5\leqslant y\leqslant 5$ for the selected contours $\unicode[STIX]{x1D70F}=-2.0$ , $-1.5$ , $-0.75$ , 0.0, 0.5, 1.25 and 2.0, respectively. The shaded region corresponds to where $h(x,y)>0$ . The flow is grounded along $x=x_{g}=-2.0$ and $h(x,y)>0$ for $x>-2.0$ . Other than the $\unicode[STIX]{x1D70F}=-2.0$ contour, all the other contours possess the property that $\text{d}y/\text{d}x>0$ for fixed $\unicode[STIX]{x1D70F}$ , implying a slight upslope velocity as the flow moves equatorward.

Figure 2.

Contour plot of the geostrophic streamlines $\unicode[STIX]{x1D70F}(x,y)$ for selected $\unicode[STIX]{x1D70F}$ contours.

Note that it follows from (3.14) that the location of the grounding is constant with respect to $y$ . This is a general result independent of the boundary condition (see Swaters Reference Swaters2013, Reference Swaters2015a ) that requires only that the topography be independent of  $y$ .

The meridional velocity $v$ associated with this simple example is independent of $x$ and is given by

for the region $-\unicode[STIX]{x1D6FD}^{-1}\ll y\leqslant y_{0}$ (focusing on the northern hemisphere). It follows from (3.15) that $v_{y}<0$ in this example. The increase in the equatorward speed is exactly compensated for by the decreasing height $h$ as $y$ moves equatorward so that the meridional volume transport is constant with respect to $y$ (see Swaters Reference Swaters2013, Reference Swaters2015a ). For example, if we denote the equatorward transport between the $\unicode[STIX]{x1D70F}=-2.0$ and $\unicode[STIX]{x1D70F}=3.0$ geostrophic streamlines as $T$ , it follows after a little algebra that

for this example.

The zonal velocity $u$ associated with this simple example is given by

so that there is a $O(\unicode[STIX]{x1D6FD})$ upslope or westward velocity as the flow moves equatorward that is induced by the planetary vorticity gradient.

4.1 Linearly sloping bottom

In the approximation that

the nonlinear solution to the initial-value problem, although implicit, can be completely determined. Assuming (4.11), it follows from (4.4) and (4.6) that

which when substituted into (4.10) yields

We note that the initial conditions associated with the characteristic equations imply that $\unicode[STIX]{x1D702}(y,0)=y$ and $\unicode[STIX]{x1D707}(x,y,0)=x$ . Thus, given $(x,y,t)$ , $\unicode[STIX]{x1D702}$ is determined by (4.12) and $\unicode[STIX]{x1D707}$ implicitly by (4.13), and consequently $h(x,y,t)$ by (4.7). We now turn to two illustrative examples where a complete analytical description is possible.

4.2 Initial parallel along-slope flow with no cross-slope shear

As a very simple initial-value example for which it is possible to write down an explicit nonlinear time-dependent solution, let us consider the initial conditions

with $1+\unicode[STIX]{x1D6FC}>0$ (no-shock condition), corresponding to an initial parallel along-slope equatorward flow with no cross-slope shear in the along-slope velocity. Substitution of (4.14) into (4.13) implies

which if substituted into (4.7) yields the explicit nonlinear time-dependent solution for the abyssal height given by

Similar to the steady solution (3.14), it is only for this very simple example that the solution for $h(x,y,t)$ has the separated form with respect to its $x$ and $(y,t)$ dependences, respectively. In general, it will not. The no-shock condition $1+\unicode[STIX]{x1D6FC}>0$ ensures that the denominator in (4.16) is never zero for any $t\geqslant 0$ .

The solution (4.16) describes a time-dependent flow that evolves away from a parallel along-slope flow with no cross-slope shear. The abyssal current height is non-zero only for $x>\widetilde{h}_{0}/\unicode[STIX]{x1D6FC}=-2.0$ (i.e. there is a grounding along $x=\widetilde{h}_{0}/\unicode[STIX]{x1D6FC}$ which does not change in time (see Swaters Reference Swaters2013, Reference Swaters2015a )) and there is no abyssal current in the region $x<\widetilde{h}_{0}/\unicode[STIX]{x1D6FC}$ . The initial geostrophic pressure contours are parallel to the isobaths.

As $t$ increases, the geostrophic pressure contours become oriented in a (time-dependent) southwest-to-northeast configuration as a result of the $\unicode[STIX]{x1D6FD}$ -induced upslope deflection of the equatorward flow. Further, as $t$ increases, for fixed $x$ and $y$ , $h$ decreases towards zero. The net effect is for $(h+h_{B})_{x}$ to decrease. This corresponds to a ‘speeding up’ of the equatorward flow within the abyssal current for fixed $x$ and $y$ . It follows from (4.4) and (4.14) that the equatorward velocity increases as one moves equatorward. However, from (4.7), given an initial position $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D702}$ , $h$ monotonically decreases as one moves equatorward. Thus, the solution at position $(x,y)$ for a given $t$ is determined by data located at position $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D702})$ at $t=0$ , which is necessarily northward of $y$ . And since the initial data $h_{0}(x,y)$ are independent of $y$ , this means that $h$ continuously decreases for fixed $(x,y)$ as $t$ increases.

Although the formal limit $t\rightarrow \infty$ is not physically meaningful in the mid-latitude approximation examined here (see the comment made below), one can say that the trend in this solution as time increases, since $h$ is decreasing towards zero, is for the geostrophic pressure to approach $h_{B}(x)$ . That is, the flow is approaching a parallel along-slope flow in which the cross-slope velocity is zero and the equatorward speed (for fixed $y$ ) has increased towards $1/(1+\unicode[STIX]{x1D6FD}y)$ .

4.3 Initial isolated grounded abyssal dome or pool

As another initial-value example for which it is possible to write down an explicit nonlinear solution, let us consider the case for which $h_{B}=-x$ with the initial abyssal height given by

(where $a>0$ ) with $0<\widetilde{h}_{0}<a/2$ (this ensures that the initial meridional velocity is everywhere equatorward, which in turn ensures that a shock does not develop in the solution in the region and over the time of interest), corresponding to an isolated abyssal dome or pool centred at $x=y=0.0$ with maximum height $\widetilde{h}_{0}>0$ . Isolated grounded abyssal domes or pools that survive for months have been observed on, for example, the North American continental slope (e.g. Armi & D’Asaro Reference Armi and D’Asaro1980; Houghton et al. Reference Houghton, Schlitz, Beardsley, Butman and Chamberlin1982; Ou & Houghton Reference Ou and Houghton1982). Isolated abyssal domes are also seen to emerge in baroclinic instability numerical simulations of unstable grounded abyssal currents (e.g. Swaters Reference Swaters1998). The nonlinear solution we present here is a non-steady meridionally travelling $\unicode[STIX]{x1D6FD}$ -plane generalization (within the context of the planetary geostrophic approximation) of the propagating ‘cold eddy’ solution described by Nof (Reference Nof1983). The solution described here is one of the few known exact nonlinear solutions for a propagating coherent isolated eddy that crosses the planetary vorticity gradient.

Substitution of (4.17) into (4.13) implies, after a little algebra, that

where

with

where $\unicode[STIX]{x1D702}(y,t)$ is given by (4.12). Finally, it follows from (4.7) that $h(x,y,t)$ will be given by

We note that the initial conditions associated with the characteristic equations imply $\unicode[STIX]{x1D6F9}(x,y,0)=x$ . In addition, note that $\unicode[STIX]{x1D6E4}(y,0)=0$ and that $\unicode[STIX]{x1D6E4}(y,t)<0$ for $t>0$ .

It is important to point out that, since these time-dependent solutions describe equatorward-propagating solutions on a mid-latitude $\unicode[STIX]{x1D6FD}$ -plane for which we assume $-\unicode[STIX]{x1D6FD}^{-1}\ll y\simeq O(1)$ (focusing on the northern hemisphere), there will be an upper limit in time for which these solutions are physically meaningful. This can be deduced from (4.12), which can be rearranged into the form

for $\unicode[STIX]{x1D702}\simeq O(1)$ . Consequently, these solutions are physically meaningful for no more than the time interval $0\leqslant t\ll 1/(2\unicode[STIX]{x1D6FD})\simeq 25$ (which corresponds to approximately 750 days based on our scalings) assuming the initial data are located where $y\simeq O(1)$ .

Figure 3(a) is a contour plot of the initial height $h(x,y,0)$ with $a=1$ and $\widetilde{h}_{0}=0.4$ (corresponding to maximum dimensional height of 224 m and a diameter of 200 km for the initial abyssal dome) for the contour intervals $0,0.08,0.16,\ldots ,0.4$ , respectively. The initial abyssal dome or pool is radially symmetric and is centred (for convenience) at $x=y=0$ . The volume of water within the abyssal dome (4.17) is given by $\unicode[STIX]{x03C0}\widetilde{h}_{0}\,a^{2}/2\simeq 0.63$ for these parameter values, which dimensionally corresponds to a volume of approximately $3520~\text{km}^{3}$ . The mass conservation equation (2.7) implies that volume is conserved as the abyssal dome propagates along the slope.

Figure 3(b) is a contour plot of the height $h(x,y,3)$ , i.e. when $t=3$ , corresponding dimensionally to approximately 90 days. As the abyssal dome propagates equatorward, the radial symmetry is lost (see figure 3 b). This is a consequence of (4.4), which has $|\text{d}y/\text{d}t|$ (the meridional speed of a point of constant phase within the abyssal dome, which is not to be confused with the Eulerian meridional fluid speed as determined by (2.5)) increasing equatorward for decreasing  $y$ . As a result, the equatorward ‘face’ travels slightly faster equatorward than the poleward ‘face’ of the abyssal pool and the initially radially symmetric abyssal dome deforms into an elliptical-like shape where the major axis is oriented in the along-slope direction. In addition, from (4.3), we see that the upslope motion is proportional to the height of the abyssal dome $h>0$ . Consequently, the point of maximum height drifts upslope from its initial ‘centred’ position at $x=0$ as $t$ increases at a faster rate than the upslope motion associated with other points within the abyssal dome. This induces further asymmetry within the abyssal dome in addition to the elliptical-like shape (i.e. the height contours are not ‘pure’ ellipses). The location and value of the point of maximum height will be explicitly determined later in this section.

Figure 3.

(a) Contour plot of the height $h$ of the time-dependent abyssal dome solution (4.21) at $t=0$ . The contour interval is approximately 0.08. (b) Contour plot of the height $h$ of the time-dependent abyssal dome solution (4.21) at $t=3$ . The contour interval is approximately 0.08.

The evolution of the boundary of the abyssal dome into an elliptical-like shape as it propagates equatorward can be exactly determined. The boundary of the abyssal pool as a function of time is given by

which implies, as a result of (4.21), that $h(x,y,t)=0$ . However, since $h_{0}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D702})=0$ on the abyssal dome boundary, it follows from (4.13) that $\unicode[STIX]{x1D707}=x$ exactly on the boundary (this implies that the groundings located at $x=\pm a$ do not shift up- or downslope as the dome propagates equatorward) so that substituting in (4.12) implies that the along-slope position of the abyssal dome boundary, denoted as $y_{b}(x,t)$ , is given by

which can be rearranged into

with $-a\leqslant x\leqslant a$ .

Unlike the internal spatial structure of the abyssal dome, its boundary remains symmetrical about $x=0$ , i.e. $y_{b}(-x,t)=y_{b}(x,t)$ . The along-slope velocity of a point on the abyssal dome boundary can be written in the form

from which we conclude that the along-slope velocity of a point on the boundary is the Nof velocity (Nof Reference Nof1983) understood in the context of a mid-latitude $\unicode[STIX]{x1D6FD}$ -plane. Exploiting the ‘smallness’ of $\unicode[STIX]{x1D6FD}$ , it follows that

The first two terms of this asymptotic relation are the unsteady along-slope translation for the boundary that are independent of $x$ and, in the present context, the third term is responsible for an $O(2\unicode[STIX]{x1D6FD}t)$ increase, as $t$ increases, in the along-slope distance between the poleward and equatorward points on the boundary for fixed $x$ , i.e.

It is this $\unicode[STIX]{x1D6FD}$ -induced stretching in the distance (for the reasons given above) that results in the elliptical-like shape developing in the abyssal dome boundary as it propagates equatorward with the major axis forming in the along-slope direction. In the absence of $\unicode[STIX]{x1D6FD}$ , the along-slope distance between the poleward and equatorward points on the boundary for fixed $x$ would be just $2\sqrt{a^{2}-x^{2}}$ , which would correspond to the dome boundary being a circle of radius $a$ , as it is initially.

The maximum height $h_{max}(t)$ of the abyssal dome and its coordinate location $(x_{max}(t),y_{max}(t))$ are determined by

which, after a little algebra, can be shown to result in

where $\unicode[STIX]{x1D701}(t)$ is the continuous simple real root of the fourth-order polynomial

where $\unicode[STIX]{x1D701}(0)=1~\Longrightarrow ~x_{max}(0)=y_{max}(0)=0$ .

Perhaps surprisingly, given the quasi-linearity of the underlying dynamics, the along-slope position of the point of maximum height $y_{max}(t)$ is independent of the abyssal dome height parameter $\widetilde{h}_{0}$ . However, the cross-slope position of the point of maximum height $x_{max}(t)$ depends linearly (as might be expected) on the abyssal dome height parameter $\widetilde{h}_{0}$ . Since it follows from (4.20) that $\unicode[STIX]{x1D6E4}(y_{max},t)\leqslant 0$ , we have $x_{max}(t)\leqslant 0$ from (4.31) (necessarily $|\unicode[STIX]{x1D702}|\leqslant a$ ) so that the point of maximum height moves upslope from its initial location at $x_{max}(0)=0$ as the dome propagates equatorward.

Exploiting the ‘smallness’ of $\unicode[STIX]{x1D6FD}$ , an asymptotic analysis of (4.33) shows that

implying that

and

To the level of approximation in (4.35)–(4.37), the abyssal dome radius $a$ has no effect on the propagation characteristics of the point of maximum height. Detailed calculation based on (4.30)–(4.33) shows that the equatorward and upslope motion in $y_{max}$ and $x_{max}$ , respectively, is marginally slowed, and the reduction in $h_{max}$ is marginally retarded, as $a$ increases, but the effect is very weak. For example, doubling the abyssal dome radius to $a=2$ led to less than a $5.3\times 10^{-4}\,\%$ variation in $x_{max}$ , $y_{max}$ and $h_{max}$ as compared to the values for  $a=1$ .

Figures 4(a) and 4(b), respectively, show the maximum height $h_{max}(t)$ and its coordinates $(x_{max}(t),y_{max}(t))$ versus $t$ for the interval $0\leqslant t\leqslant 3$ as determined from (4.30) to (4.33). Figure 4(a) shows the approximately linear decrease in $h_{max}(t)$ from its initial value of 0.4 to its value at $t=3$ of approximately 0.375. The decrease in $h_{max}(t)$ as the abyssal dome propagates equatorward is a consequence of the conservation of potential vorticity and the decreasing value of the Coriolis parameter as $y$ decreases. Figure 4(b) shows the equatorward and slightly upslope approximately linear trajectory of $(x_{max}(t),y_{max}(t))$ from its initial position of $(0,0)$ to the location at $t=3$ with coordinates approximately $(-0.025,-3.09)$ . It is noted that the asymptotic relations (4.35)–(4.37) very accurately reproduce the exact results for $(x_{max},y_{max})$ and $h_{max}$ at $t=3$ . The equatorward motion of $h_{max}(t)$ is primarily a consequence of the underlying geostrophic balance for the grounded abyssal dome on the sloping bottom, which holds even if $\unicode[STIX]{x1D6FD}=0$ (i.e. on a $f$ -plane). The upslope trajectory in $h_{max}(t)$ is, physically, purely a consequence of the Coriolis effect associated with $\unicode[STIX]{x1D6FD}$ , i.e. the Coriolis parameter decreasing as $y$ decreases.

Figure 4.

(a) Graph of the maximum height $h_{max}(t)$ for the propagating abyssal dome for the time interval $0\leqslant t\leqslant 3$ . (b) Graph of the path $(x_{max}(t),y_{max}(t))$ for the propagating abyssal dome for the time interval $0\leqslant t\leqslant 3$ .

5.1 Linearly sloping bottom

In the approximation that $h_{B}=-x,$ the nonlinear solution to the time-dependent boundary-value problem, although implicit, can be completely determined. Assuming $h_{B}=-x$ , it follows from (5.4) and (5.6) that

which when substituted into (5.9) yields

We note that the initial conditions associated with the characteristic equations imply $\unicode[STIX]{x1D702}(y_{0},t)=t$ and $\unicode[STIX]{x1D707}(x,y_{0},t)=x$ . Thus, given $(x,y,t)$ , $\unicode[STIX]{x1D702}$ is determined by (5.10) and $\unicode[STIX]{x1D707}$ implicitly by (5.11), and consequently $h(x,y,t)$ by (5.7). While there are any number of time-dependent boundary conditions one can consider, here we examine two idealized but illustrative examples where a complete analytical description is possible.

5.2 Time-dependent boundary condition with upslope and downslope groundings

As a time-dependent boundary-value example for which it is possible to write down an explicit nonlinear solution, let us consider the linearly sloping bottom $h_{B}=-x$ with the abyssal height boundary profile along $y=y_{0}$ given by

with $0<\widetilde{h}_{0}(t)<a(t)/2$ (ensuring that the meridional velocity is strictly equatorward along $y=y_{0}$ , which in turn ensures that a shock does not develop in the solution in the region and over the time of interest). The abyssal height (5.12) corresponds to a ‘parabolic’ cross-slope boundary profile with time-dependent half-width $a(t)>0$ and maximum height $\widetilde{h}_{0}(t)>0$ located at $x=b(t)$ , which possesses time-dependent upslope and downslope groundings located at $x=b(t)\pm a(t)$ . In the present circumstance, $a(t)$ , $b(t)$ and $\widetilde{h}_{0}(t)$ are assumed given.

Substitution of (5.12) into (5.11) implies, after a little algebra, that

where

with

where $\unicode[STIX]{x1D702}(y,t)$ is given by (5.10). Again, we note that $\unicode[STIX]{x1D6F9}(x,y_{0},t)=x$ , $\unicode[STIX]{x1D6E4}(y_{0},t)=0$ and that $\unicode[STIX]{x1D6E4}(y,t)>0$ for $y<y_{0}$ . Substitution of (5.10) and (5.12)–(5.15) into (5.7) determines $h(x,y,t)$ .

5.3 Time-limited surge in the equatorward transport along the poleward boundary

Here, we describe the model prediction associated with the effect that a time-limited upstream pulse or surge in the equatorward transport along the poleward boundary has on the grounded abyssal current in the mid-latitude domain. Physically, this scenario could correspond to a situation where, for example, seasonal variations have led to increased time-limited deep-water formation in high latitudes poleward of our domain. (We could equally well have examined the opposite effect, that is, a reduction in equatorward transport along $y=y_{0}$ . How the results would qualitatively differ from those described here will be self-evident.) Again, we note that the solution described here is one of the few known exact nonlinear solutions for a time-dependent oceanographic flow that crosses the planetary vorticity gradient.

To make our description concrete, we set

In the absence of any time variation, the upslope and downslope groundings associated with the boundary condition $h_{0}(x,t)$ are located at $\pm 1$ , which implies an abyssal current width along $y=y_{0}$ of 2 (or dimensionally 200 km), and where the maximum non-dimensional height above the topography is 0.45 (or dimensionally approximately 250 m) and is located at $x=0$ . At the time of maximum transport surge along $y=y_{0}$ , which occurs at $t=4$ (or dimensionally approximately 120 days after $t=0$ when the surge ‘just starts’ to occur), the abyssal current width is increased by 30 % to 260 km, and the maximum height above the topography is increased by approximately 43 % to approximately 360 m. These idealized parameter values ensure that $v(x,y_{0},t)<0$ for all $(x,t)$ , which ensures that no shock will form in the solution in the region of interest. In addition, choosing $b(t)=0$ implies that there is no cross-slope movement in the point of maximum height along $y=y_{0}$ . We will examine a situation, corresponding to the formation of a downslope loop or plume along the downslope grounding, where $b$ varies in time, later in this section.

Figure 5(a,b) are graphs of the abyssal current height on top of the sloping topography versus $x$ along $y=y_{0}$ , i.e. $h_{0}$ , in the absence any time variation in $a$ and $\widetilde{h}_{0}$ , and at $t=4$ (corresponding to the time of maximum transport surge), respectively. In figure 5(b) the downslope and upslope groundings in the abyssal current height are displaced symmetrically in the downslope and upslope directions by 0.3, respectively, and the maximum height is increased by 0.195, as compared to that in figure 5(a).

Figure 5.

(a) Graph of the abyssal current height $h_{0}$ on the sloping bottom along $y=y_{0}$ prior to the onset of the surge in volume transport. (b) Graph of the abyssal current height $h_{0}$ on the sloping bottom along $y=y_{0}$ at the time of maximum transport surge $(t=4)$ .

The non-dimensional meridional transport along $y=y_{0}$ , denoted by $T_{0}(t)$ , will be given by

In the absence of any time variation, $T_{0}\simeq -0.55$ (or dimensionally approximately $-1.22~\text{Sv}$ ; $1~\text{Sv}=10^{6}~\text{m}^{3}~\text{s}^{-1}$ ). At the time of maximum transport surge along $y=y_{0}$ , $T_{0}(4)\simeq -1.02$ (or dimensionally approximately $-2.28~\text{Sv}$ ), corresponding to an 87 % increase in equatorward transport along $y=y_{0}$ . For $t>4$ , the surge in the equatorward transport diminishes along $y=y_{0}$ . It is important to emphasize that this is an idealized scenario meant to elucidate the dynamics rather than to model a specific oceanographic event (although that would be interesting as well). These transport values are reasonably consistent with observations of the ‘deep’ transport associated with ‘overflow/lower deep water’ associated with the DWBC near Cape Cod (e.g. Joyce et al. Reference Joyce, Dunworth-Baker, Pickart and Waterman2005). Figure 6, which is a graph of $T_{0}(t)$ versus $t$ for $0\leqslant t\leqslant 10$ , shows the increase in the equatorward transport along $y=y_{0}$ from its value at $t=0$ , when the surge is just starting to occur, to its maximum when $t=4$ , and its subsequent decline back to its value before the surge occurs, which for all practical calculations is the case when $t=10$ .

Figure 6.

Graph of the meridional volume transport $T_{0}(t)$ along the poleward boundary $y=y_{0}$ versus $t$ for $0\leqslant t\leqslant 10$ .

Figures 7(a) and 7(b) are, respectively, contour plots of $h(x,y,t)$ for $t=0$ and $t=10$ , corresponding, respectively, to a time that is just immediately before the surge in transport occurs along $y=y_{0}$ and a time for which the anomaly associated with the transport surge has propagated well into the interior of the mid-latitude domain. Figure 7(a) is qualitatively similar to the steady nonlinear solution described in Swaters (Reference Swaters2015a ). Figure 7(b) illustrates the spatially limited and equatorward-propagating varicose-like anomaly within the grounded abyssal current that is generated by the time-limited transport surge along the poleward boundary $y=y_{0}$ . As the anomaly propagates equatorward, the point of maximum height decreases and shifts upslope as a consequence of the planetary vorticity gradient. To leading order, the along-slope position of the point of maximum height is determined by, essentially, the geostrophic (Nof Reference Nof1983) balance (understood in the context of a mid-latitude $\unicode[STIX]{x1D6FD}$ -plane).

Figure 7.

(a) Contour plot of the height $h$ prior to the onset of the surge in transport along the poleward boundary. The contour interval is approximately 0.05. (b) Contour plot of the height $h(x,y,10)$ when the transport surge has fully propagated into the domain. The contour interval is approximately 0.12.

The point of maximum height $h_{max}(t)$ of the anomaly associated with the transport surge as illustrated in figure 7(b) and its coordinate location $(x_{max}(t),y_{max}(t))$ are determined by

which, after a little algebra, can be shown to result in

where $y_{max}(t)$ is a continuous simple real root of

and, consequently, where

Exploiting the ‘smallness’ of $\unicode[STIX]{x1D6FD}$ , an asymptotic analysis of (5.20) shows that

and

The leading-order linear decrease in $y_{max}$ in (5.22) is independent of $\unicode[STIX]{x1D6FD}$ and is a consequence of the $f$ -plane geostrophic balance between the downslope gravity-driven acceleration of a grounded dense water mass (as compared to the ambient) sitting directly on a sloping bottom and the Coriolis effect (i.e. the Nof (Reference Nof1983) balance). The upslope drift in $x_{max}$ in (5.23) is purely a consequence of differential rotation. The $O(\unicode[STIX]{x1D6FD})$ linear decrease in $h_{max}$ in (5.24) is a consequence of the conservation of potential vorticity on the mid-latitude $\unicode[STIX]{x1D6FD}$ -plane. Note that to leading order $y_{max}\simeq y_{0}$ , $x_{max}\simeq 0$ and $h_{max}\simeq 0.645\equiv 0.45+0.195$ when $t=4$ , corresponding to the time when the transport surge is maximum along $y=y_{0}$ .

Figures 8(a) and 8(b), respectively, show the maximum height $h_{max}(t)$ and its coordinates $(x_{max}(t),y_{max}(t))$ versus $t$ for the interval $4\leqslant t\leqslant 10$ as determined from (5.19)–(5.21). Figure 8(a) shows the approximately linear decrease in $h_{max}(t)$ from its initial value of 0.645 at $t=4$ to its value at $t=10$ of approximately 0.58. Figure 8(b) shows the equatorward and slightly upslope approximately linear trajectory of $(x_{max}(t),y_{max}(t))$ from its leading-order position at $(0,y_{0})$ at $t=4$ to the location at $t=10$ with coordinates approximately $(-0.07,-0.67)$ .

Figure 8.

(a) Graph of the maximum height $h_{max}(t)$ versus $t$ for $4\leqslant t\leqslant 10$ as determined by (5.21). (b) Graph of the path $(x_{max}(t),y_{max}(t))$ versus $t$ for $4\leqslant t\leqslant 10$ as determined by (5.19) and (5.20).

5.4 Formation of a downslope plume or loop in the abyssal current along the poleward boundary

Here, we describe the resulting flow associated with a time-limited downslope shift in the cross-slope position of the boundary abyssal current’s centre of mass, or, equivalently in our model, the cross-slope position of the point of maximum height along $y=y_{0}$ . The resulting flow in the interior of the domain resembles a sinuous-like downslope loop or plume anomaly that propagates equatorward. It is remarked that the unstable modes associated with the baroclinic instability of grounded abyssal currents (see figure 9 in Swaters (Reference Swaters1991) and plate 1 in Swaters (Reference Swaters1998)) are along-slope propagating meanders that preferentially amplify along the downslope grounding growing into downslope plumes. It is suggested that the dynamics described here may be of relevance to the nonlinear along-slope propagation of these finite-amplitude plumes once they have formed. Again, we note that the solution described here is one of the few known exact nonlinear solutions for a time-dependent oceanographic flow that crosses the planetary vorticity gradient.

To make our description concrete, we set

These choices correspond to holding fixed in time the width of the abyssal current along $y=y_{0}$ to be 2 (or dimensionally approximately 200 km) and its maximum height to be 0.45 (or dimensionally approximately 250 m) but allowing the cross-slope position of the maximum height to shift 1 non-dimensional unit (or approximately 100 km) downslope (when $t=4$ ).

Figure 9(a) is a contour plot of the abyssal current height $h(x,y,t)$ for $t=10$ when the downslope plume or loop has propagated into the interior of the mid-latitude domain ( $-5\leqslant y\leqslant 5$ ). Prior to the formation of the downslope plume or loop $(t\ll 4)$ , a contour plot of $h(x,y,t)$ for our domain would look exactly like that shown in figure 7(a) (note, however, that figure 9(a) contains a larger range of $x$ -values as compared to figure 7(a)). The resulting anomaly in the abyssal current resembles a sinuous-like downslope finite-amplitude plume or loop. Owing to the conservation of potential vorticity, and as predicted by (5.7), the abyssal current height decreases as $y$ decreases.

Figure 9.

(a) Contour plot of the height $h(x,y,10)$ when the downslope plume or loop has fully propagated into the domain. The contour interval is approximately 0.05. (b) Graph of the along-slope position $y_{max}(t)$ of the point of maximum downslope extent, or crest, associated with the downslope plume as seen in figure 9(a) versus $t$ for $0\leqslant t\leqslant 14$ .

The plume or loop propagates equatorward with equatorward speed increasing as $y$ decreases. It follows from (5.7), (5.10), (5.12) and (5.25) that the along-slope position of the crest, denoted by $y_{max}(t)$ , of the downslope plume or loop, i.e. its furthermost downslope extent, is given by

which can be rearranged into

from which it follows that

Equation (5.28) is, again, the statement that the equatorward speed of propagation of the downslope loop or plume is the Nof velocity (Nof Reference Nof1983) understood in the context of the mid-latitude $\unicode[STIX]{x1D6FD}$ -plane we are working with. Clearly, as $y_{max}$ decreases, $|\text{d}y_{max}/\text{d}t|$ increases (albeit gradually since $\unicode[STIX]{x1D6FD}\simeq 0.02$ for our scalings). Figure 9(b) is a graph of $y_{max}(t)$ versus $t$ for $4\leqslant t\leqslant 14$ . Figure 9(a) depicts a more or less linear equatorward translation of $y_{max}(t)$ as $t$ increases (the apparent linearity is a consequence of $\unicode[STIX]{x1D6FD}\simeq 0.02$ ).