2.1 The dynamic condition at a flow front

Along the leading flow front of the current, illustrated in figure 2(a), I impose the stress condition

where $x=X(y,t)$ is the frontal position and $\hat{\boldsymbol{n}}$ is the unit horizontal normal along the leading edge (Pegler & Worster Reference Pegler and Worster2012). Note that a positive frontal thickness ( $H>0$ ) allows for the cliff-like terminus typical along the calving front of an ice shelf. Integrating the $x$ component of (2.13) across the width of the flow, using (2.9b ) to evaluate $\unicode[STIX]{x1D707}$ and rearranging, I obtain

where the no-penetration condition (2.5b ) has been used to evaluate the boundary term involving $v$ . In obtaining (2.14), I have dropped the moduli in the viscosity (2.9b ) because the frontal rate of extension implied by (2.13) is always positive ( $\unicode[STIX]{x2202}\bar{u}/\unicode[STIX]{x2202}x>0$ ). Equation (2.14) represents a balance between the stress due to longitudinal extension and the discontinuity in depth-integrated hydrostatic pressure between the leading flow front and the ocean in front of it.

2.2 The dynamic condition at a channel exit

Once the layer extends beyond the exit of the channel ( $X(t)>l$ ), it forms a jet-like flow free of sidewall stresses, illustrated in figure 2(b) and later in the experimental photograph of figure 10. Assuming that the leading edge beyond the channel exit remains a free boundary (it does not contact a boundary downstream, say), the details of the flow downstream of the channel exit have no effect on the flow inside the channel and there is an effective boundary condition at the channel mouth that is independent of the flow beyond it. To develop this, I begin by rewriting (2.2) in the stress–divergence form

With reference to figure 2(b), I perform an integral of (2.15) over the horizontal area bounded between the line $x=l$ , denoted by $C_{E}$ in figure 2(b), and the curve of the fluid front $C_{F}$ , to obtain, with use of the divergence theorem,

where $\hat{\boldsymbol{n}}$ is the unit horizontal normal to $C_{F}$ , $\hat{\boldsymbol{x}}$ is the unit vector in the $x$ direction and d $s$ is the line element. In accord with (2.13), the integral over the fluid front $\int _{C_{F}}$ is zero. With the same width integration and imposition of zero lateral velocity used to obtain (2.14) above, equation (2.16) similarly reduces to

The specification of zero lateral velocity $v$ near the corners of the channel exit may not in fact hold to good approximation if the flow undergoes a die swell in front of the channel exit. This point will be discussed in the context of my laboratory experiments in § 4.2.

In conclusion, the stress conditions at a fluid front (2.14) and at a channel exit (2.17) are equivalent. That is, the boundary condition at the mouth of a channel (2.17) is equivalent to fixing a fluid front at $x=l$ . The equivalence arises because the pressure of the ‘ocean’ communicates directly through the detached region of the layer to the channel exit. This is similar to the way that an idealized one-dimensional ice shelf transmits forces directly to the grounding line of a marine ice sheet (Weertman Reference Weertman1974). By allowing fluid to extend freely downstream of the sidewalls, it is thus possible to simulate the conditions that apply at a fixed calving front experimentally (see § 4).

2.3 Evolution of the thickness and fluid front

With $\bar{u}$ obtained from an integration of (2.12), the evolution of the thickness $H$ can be determined using the width-integrated form of the continuity equation (2.3),

which is a one-dimensional advection equation. While not included here, the combined effects of snowfall accumulation and basal melting could be accommodated by including a distributed source (or sink) term of the form $a(x,t)$ on the right-hand side of (2.18). With the overdot used to denote $\text{d}/\text{d}t$ , I assume finally that the fluid front $X(t)$ evolves kinematically according to

3.1 Early-time flow and the intervention of sidewall stresses

As indicated by the second term in (2.12), transverse shear stresses are larger for smaller channel widths $w=2s$ . It is therefore reasonable to expect shear stresses to be negligible if the length of the flow is much shorter than the width ( $w=2s\gg X(t)\mathscr{L}$ ), as applies at sufficiently early times. In the absence of the shear-stress term $Hu$ , equation (3.4) reduces to

which is equivalent to the one-dimensional form of (2.2). A direct first integral of this equation subject to (3.7a ) yields

which implies that, with no sidewall stresses, the rate of horizontal extension is everywhere proportional to the thickness (not only at the front, as implied by (3.7a )). With reference to (3.4), this direct integration cannot be performed with sidewall stresses, as they form a component of the divergence of deviatoric stresses that is distinct from the gradient in pressure (an analogous situation occurs with viscous hoop stresses (see Pegler & Worster Reference Pegler and Worster2012)). Combining (3.5), (3.6) and (3.11), one can determine the analytical solution

(DiPietro & Cox Reference DiPietro and Cox1979; Robison et al. Reference Robison, Huppert and Worster2010). In this regime of purely extension-resisted flow, the current fills a steady, concave envelope (3.12a ) with a time-dependent frontal position (3.12b ). Figure 3(a) shows the envelope as dashed curves and confirms its role as an early-time regime for $D=6$ . The numerical solution begins to deviate from the envelope significantly by $t\approx 4$ as sidewall stresses intervene.

In order to explore when deviation from the purely extensional regime (3.12) occurs across the full range of $D$ , I calculate the time $t_{E}$ defined by

which represents when the frontal position $X$ first deviates from the position $X_{E}$ implied by (3.12b ) by 2 %. As shown in figure 4, $t_{E}$ increases with $D$ , implying that a larger input thickness favours the endurance of the extensional regime. However, $t_{E}\rightarrow 3.6$ remains finite as $D\rightarrow \infty$ , implying that, even when conditions ideally favour the endurance of the extensional regime, sidewall stresses intervene significantly by $t=3.6$ . At this time, the dimensional length is comparable to just one-quarter of a channel width ( $X\mathscr{L}\approx 0.24w$ ).

Figure 4.

Parameter–time regime diagram showing when the two possible early-time flow regimes of kinematic and extensional flow occur as a function of the dimensionless input thickness $D$ . Before the time $t_{E}$ , defined by (3.13) and shown by a solid curve, the flow is described to good approximation by the purely extensional flow regime (3.12). For $t>t_{E}$ , the flow has been influenced significantly by sidewall stresses. The earlier time $t_{K}$ defined by $X(t_{K})=1.02D^{-1}t$ represents the time up to which the flow is controlled kinematically by the input speed (3.6b ). Sidewall stresses always intervene significantly by $t\approx 3.6$ , even when conditions are optimized for the endurance of the extensional regime.

There is a further transition time $t_{K}$ , which I define by $X(t_{K})=1.02D^{-1}t_{K}$ , up to which the flow is simply controlled kinematically by the input flow speed (3.6b ). The time $t_{K}$ is shown by a dashed curve in figure 4. For $D>1$ , the buoyancy-driven contribution to (3.12b ), represented by the quadratic term $t^{2}/16$ , is only significant in the interval $t_{K}<t<t_{E}$ . For $D<1$ , this interval does not exist, implying that sidewall stresses intervene before buoyancy forces in these cases. The concave regime of (3.12) then never occurs, as evidenced by the example with $D=0.3$ in figure 3(b). In conclusion, the one-dimensional regime (3.12) breaks down over all parameter settings once lateral contact exceeds one-quarter of a flow width.

3.2 Steady state

At late times, the flows approach steady states, shown by blue dotted curves in figure 3(a,b). Using the steady form of (3.5) ( $Hu=1$ ) to eliminate $u$ in the second term of (3.4), I obtain the ordinary differential equation

where the dimensionless transverse shear stress, represented by the $-1$ on the left-hand side, is (favourably) uniform. Integration of (3.14) subject to (3.7b ) yields

which generalizes (3.11) to accommodate lateral stresses. This equation shows that the effect of lateral stress is to reduce the rate of extension in proportion to the distance from the channel mouth $(L-x)$ . In physical terms, the equation shows that the normal stress associated with driving fluid forwards through the channel (or back-force) at any point, represented by the left-hand side, is proportional to the distance downstream over which the flow contacts the sidewalls. Lateral stresses have no effect on the force balance at the channel mouth, $x=L$ , where the dominant balance is between buoyancy and extension alone in accord with (3.7b ).

Using the mass conservation equation $u=1/H$ to eliminate $u$ fully in (3.15), I integrate (3.15) further subject to the input condition (3.6a ) to obtain the exact analytical solution

which produces the blue dotted curves in figure 3. A suite of the steady solutions (3.16) is plotted in figure 5 for (a) a fixed input thickness $D=6$ and various flow lengths $L=1{-}5$ , and (b) a fixed flow length $L=5$ and various input thicknesses $D=0.2{-}\infty$ . Figure 5(a) illustrates the emergence of two characteristically different steady-state flow regimes, depending on whether the flow is sufficiently short or long.

Figure 5.

(a) The steady-state surface profiles predicted by (3.16) for various flow lengths $L=1{-}5$ and a fixed input thickness $D=6$ . The two possible flow regimes that occur depending on the flow length $L$ are illustrated. In the short case $L=1$ , the flow can be treated as effectively one-dimensional and approximated well by the envelope (3.17), which is shown for the lower surface by a red dotted curve. In the longer case $L=5$ , the prevailing flow is instead described by the universal profile (3.18) shown by a dashed blue curve for $L=5$ . (b) A suite of solutions for a range of input thicknesses $D=0.2{-}\infty$ and fixed length $L=5$ , illustrating their spatial adjustment from the input towards the universal profile (3.18).

For a sufficiently short length $L$ , equation (3.16) recovers the one-dimensional steady-state profile of (3.12a ) asymptotically. This is demonstrated by taking the limit $L\rightarrow 0$ in (3.16), for which the exponentials can all be approximated as unity to leading order and (3.16) reduces asymptotically to

which is indeed equivalent to (3.12a ). The asymptotic extensional envelope (3.17) is shown as a red dotted curve in figure 5(a) and is confirmed to describe the short steady-state solutions with $L=1$ and 2 to good approximation.

For sufficiently large $L$ , equation (3.17) fails to provide a good approximation. Instead, a new regime arises, characterized by a rapid spatial adjustment from the input towards a prevailing region of relatively milder slope. The flow in this downstream region is of a universal dimensionless form up to translation with respect to the exit of the channel. Its universality is demonstrated in figure 5(b), where steady states over a large range of input thickness $D$ are shown to approach a single profile shown by the blue dashed curve. The universality is also consistent with the earlier observation in figure 3 that the steady-state frontal thicknesses in the two cases $D=6$ and 0.3 are effectively equal.

3.3 Universal downstream profile

By considering the limit $L\rightarrow \infty$ with $x\leqslant L$ , equation (3.16) reduces asymptotically to

where $\text{erfcx}\,\unicode[STIX]{x1D701}\equiv \text{e}^{\unicode[STIX]{x1D701}^{2}}(1-\text{erf}\,\unicode[STIX]{x1D701})$ is the scaled complementary error function. This expression represents the universal profile towards which the steady states adjust spatially, shown by the blue dashed curve in figure 5 for $L=5$ . The universal profile (3.18) varies only with distance backwards from the front $(L-x)$ . Therefore, with respect to the front, $x=L$ , it is of a universal dimensionless form. In particular, the thickness and flow rate at the front itself take the special values

respectively. The complete absence of any parametric dependence in these frontal values contrasts significantly with the dependences implied by the purely extensional solution (3.17) applicable for small $L$ , which predicts that $H(L)=(D^{-1}+L/4)^{-1/2}$ . By contrast, the predictions of (3.19) are entirely independent of $D$ and $L$ . These differences imply a fundamental alteration in the way that the flow structure of an ice shelf is determined between short and long flow lengths. For a short length, the profile (3.17) is determined causally from the input conditions, with no dependence on downstream conditions. It is sensitive in particular to the specific thickness imposed at the input $D$ . For long lengths, there is a reversal in the manner in which the profile is determined. Conditions at the downstream front (3.19) set the reference conditions from which the prevailing flow is determined. This flow is of a universal dimensionless form (up to scaling by (3.2)) that, in contrast to the purely extensional limiting form (3.17), is independent of the specified input thickness $D$ and depends only on the position of the downstream channel exit $L$ .

Figure 6.

The along-channel velocity profile $u_{\ast }=1/H_{\ast }$ of the universal downstream profile (3.18) (solid black curve). The plot illustrates its boundary-layer structure comprising an outer zone of transverse-shear-dominated flow described to leading order by (3.20b ) (dashed black curve), and the inner ‘extensional zone’ wherein the extensional stress at the flow front (3.7b ) has a significant effect and is responsible for regularizing the flow speed. The extensional zone has a characteristic length $\unicode[STIX]{x1D6E5}_{\ast }\approx 2.6$ . The analytical approximation to the velocity of the universal profile based on neglecting only the divergence of extensional stresses, given by $u_{\ast }=1/H_{\ast }$ , with $H_{\ast }$ approximated by (3.21), is shown by the solid blue curve.

Figure 7.

(a) An under-thick input thickness $D=1<H_{+}$ and (b) an over-thick input thickness $D=6>H_{+}$ , each for $L=5$ , where $H_{+}\approx 3.75$ is defined as the value of the universal profile (blue) extrapolated to the input (3.22). The transition length $\unicode[STIX]{x1D6E5}$ defined by (3.24) for $\unicode[STIX]{x1D716}=0.02$ is shown by a vertical dashed line in each case. (c) Plot of $\unicode[STIX]{x1D6E5}$ as a function of $D$ for $L=5$ , illustrating its increase with $|D-H_{+}|$ and maximal value of 1.4 for over-thick inputs.

3.4 The input transition zone

As one follows the universal profile (3.18) upstream, its thickness $H_{\ast }(x)$ will generally conflict with the thickness imposed at the input, $H_{\ast }(0)\neq D$ . This conflict is resolved within a short transition zone involving significant extensional stresses, wherein conditions at the input are matched to the universal profile (3.18).

As an indication of the degree of conflict between the input conditions and the universal profile, I compare $D$ to the thickness of the profile (3.18) extrapolated to the input, namely,

If $D>H_{+}$ , the flow is called over-thick and characterized by a spatial adjustment to the universal profile involving thinning and accelerated flow, as illustrated in figure 7(b). If $D<H_{+}$ , the flow is under-thick, as illustrated in figure 7(a).

Figure 8.

Regime diagram for steady states over the full parameter space $(L,D)$ . Colour indicates the dimensionless thickness at the flow front, $H(L)$ . White represents the special value associated with the universal profile (3.19a ), $H_{c}=2\unicode[STIX]{x03C0}^{-1/4}\approx 1.50$ . White regions indicate regimes for which a transition towards the universal profile completes fully. Red represents an ‘over-thick’ case defined by a frontal thickness greater than the special value, $H(L)>H_{c}$ , and blue represents an ‘under-thick’ case defined by $H(L)<H_{c}$ . The critical line separating the two is shown by a thick curve. The off-white regions (blue and red) indicate ‘short’ flows for which the adjustment towards the universal profile is incomplete. The contours $H(L)=(1\pm 0.1)H_{c}$ represent the boundary between ‘short’ and ‘long’. Insets illustrate the side-view profiles for the four parameter settings at the black circles, with the downstream universal profile (3.18) shown by the blue dashed curves in each inset and the purely extensional solution (3.12) by the red dashed curves.

The transition zone at the input can be interpreted as a boundary layer in which the information associated with the input thickness $D$ is dissipated and the flow transitions to the outer region represented by the universal profile (3.18) (this boundary-layer structure is distinct from that of the universal profile itself discussed in § 3.3.1). In the limit $L\rightarrow \infty$ , equation (3.18) reduces asymptotically to

where $H_{+}^{2}\sim 2L$ , which describes the transition zone as a perturbation from the outer solution of the universal profile (3.20a ). The perturbation, represented by the second term, decays exponentially over a distance proportional to the dimensionless length scale $2L^{-1}$ and, as expected, increases with the absolute difference between $D$ and  $H_{+}$ .

To determine more precisely the size of the input transition zone, I define the transition length $\unicode[STIX]{x1D6E5}=\unicode[STIX]{x1D6E5}(L,D)$ as the position $x=\unicode[STIX]{x1D6E5}$ at which

where $\unicode[STIX]{x1D716}$ is a small parameter taken as 0.02. The position $x=\unicode[STIX]{x1D6E5}$ is indicated by a vertical dashed line in figure 7(a,b) and plotted as a function of $D$ in figure 7(c). The plot confirms that $\unicode[STIX]{x1D6E5}$ increases with the difference between $D$ and $H_{+}$ . If $D\approx H_{+}$ , there is no appreciable transition zone, $\unicode[STIX]{x1D6E5}\approx 0$ , and the whole flow is described uniformly by the universal profile (3.18). As $D\rightarrow \infty$ , $\unicode[STIX]{x1D6E5}\rightarrow 1.4$ remains finite, implying that any over-thick input, $D>H_{+}$ , results in a transition to the universal profile within half a channel width ( $1.4\mathscr{L}\approx 0.40w$ ). The transition length can become larger in under-thick cases $D<H_{+}$ . Indeed, for $D<0.3$ the (hypothetical) transition length exceeds the flow length $\unicode[STIX]{x1D6E5}>L=5$ and the transition to the universal profile remains incomplete. The dependences of $\unicode[STIX]{x1D6E5}$ on $D$ and $H_{+}(L)$ are encapsulated by the analytical approximation

obtained by using (3.20a ) and (3.23) to evaluate $H_{\ast }$ and $H$ in (3.24), respectively. The result (3.25) implies that $\unicode[STIX]{x1D6E5}\rightarrow 4H_{+}^{-2}\ln (1/2\unicode[STIX]{x1D716})$ as $D\rightarrow \infty$ , which confirms that $\unicode[STIX]{x1D6E5}$ remains finite in this limit. This asymptotic prediction is within 6 % of the numerically determined value. The outlying multiplicative factor $\propto H_{+}^{-2}$ in (3.25) implies that a larger matching thickness $H_{+}$ (a thicker downstream flow) results in a shorter transition length $\unicode[STIX]{x1D6E5}$ . This is because a thicker downstream flow exerts a larger back-force, which resists flow in the input transition zone and causes the transition to take place over a shorter distance.

Figure 9.

(a) Horizontal cross-section and (b) plan view of my experimental set-up.

Figure 10.

Plan-view photograph of a fully developed experiment showing the floating film of particle-laden syrup fed from the left and propagating to the right between sidewalls of length 16 cm and width 6 cm. The die swell beyond the channel exit is illustrated.

Figure 11.

Sequence of side-view photographs of an experiment at times $t=10$ , 20, 40 and 80 s after the initialization of the input, illustrating the development of the fluid layer and its under-thick input transition. The upper layer of water used to reduce the effects of surface tension in my other experiments was not included in this purely illustrative experiment in order to avoid optical distortion.

Figure 12.

Experimental data for the (plan-view) two-dimensional velocity field obtained using PIV on the late-time steady flow of experiment B. (a) The dimensionless longitudinal velocity $u/\mathscr{U}$ and (b) the transverse velocity $v/\mathscr{U}$ , plotted in the dimensionless coordinate system $(x/\mathscr{L},y/\mathscr{L})$ . The data show a transition from a qualitatively parabolic shear flow with negligible transverse velocity inside the channel to a plug-like extensional flow beyond the channel exit. Significant transverse velocities at the corners of the channel mouth drive a die swell beyond the channel mouth.

3.5 Regime classification

The results above show that a transition towards the universal profile completes for all $D>0.3$ if $L=5$ . With the shorter length $L=1$ , adjustment does not occur nearly as widely. In order to measure the degree of transition over the full range of $D$ and $L$ , I compare the frontal thickness $H(L)$ with the special value $H_{c}=2\unicode[STIX]{x03C0}^{1/4}\approx 1.50$ associated with a complete transition (3.19a ). In figure 8, I have shown a colour plot of $H(L)$ over the full parameter space $(L,D)$ , where the special value $H_{c}$ is indicated by white. Red represents an over-thick flow, for which $H(L)>H_{c}$ . Blue represents an under-thick flow, for which $H(L)<H_{c}$ . The boundary between over-thick and under-thick cases, represented by the curve $D=H_{+}(L)$ with $H_{+}$ given by (3.22), is shown as a thick curve. Near-white represents flows that undergo a complete transition towards the universal profile. The contours $D=(1\pm 0.1)H_{\ast }(0)$ , shown by thinner curves, represent the boundary between the parameter settings that produce flows long enough for the transition to the universal profile to complete, referred to as ‘long’, and others, referred to as ‘short’, for which the transition remains incomplete and the flow is characteristically similar to the extensional regime (3.17). The plot indicates that a transition towards the universal profile occurs readily if $L>2$ . If $L<2$ , complete transition is limited to a very narrow set of values around $D\approx 1.50$ , for which the input thickness is automatically close to the special value $H_{c}$ . For $L>2$ , all over-thick cases ( $D>H_{+}(L)$ ) undergo a complete transition to the universal profile. In other words, if the flow length is longer than just 60 % of the width ( $l=2\mathscr{L}=0.58w$ ), then an over-thick adjustment to the universal profile is guaranteed to complete.

4.1 Experimental results

The experiments listed in table 1 spanned aspect ratios $l/w$ from 2 to 4, reduced gravities $g^{\prime }$ from 15.8 to $29.7~\text{cm}~\text{s}^{-2}$ and volumetric fluxes per unit width $Q$ from 0.037 to $0.052~\text{cm}^{2}~\text{s}^{-1}$ . The longitudinal and transverse components of the velocity field $(u,v)$ obtained using PIV of the images are shown in figure 12. The data indicate a qualitatively parabolic shear profile across the interior of the channel, with very slight transverse flow $v\approx 0$ . On reaching the mouth of the channel, the layer detaches from the sidewalls with a significant increase in lateral velocity $v$ reminiscent of a die swell from a tube (Tanner Reference Tanner2000). Downstream of the channel exit, the lateral velocity $v$ decays, the transverse profile becomes plug-like, and the flow continues to accelerate. Note that an apparent reduction in $u$ very close to the rim of the detached region is not physical and can be attributed to a failure of the PIV to resolve flow in a small region where the rim optically obscures the particles.

The final width-averaged longitudinal velocity along the channel $\bar{u}(x)$ is plotted as a function of the horizontal position with respect to the channel front, $x-l$ , for all three experiments in figure 13. The velocity and length are each scaled by the intrinsic scales $\mathscr{U}$ and $\mathscr{L}$ defined by (3.2). There is a good collapse of the data, indicating that the downstream flow is indeed universal up to scaling by the parameter groupings (3.2). The only significant differences between the sets of data occur near the backwall, where the flow is affected by the input. Overlaid on the data in figure 13 is the prediction of the universal profile $\hat{u} _{\ast }=1/{\hat{H}}_{\ast }$ (black solid curve), where ${\hat{H}}_{\ast }$ is the analytical solution (3.18). The prediction matches the data very well to within a scaled distance of two dimensionless units or about half a channel width ( $2\mathscr{L}\approx 0.6w$ ) upstream of the channel exit. A discrepancy with the data of approximately 20 % occurs at the channel exit itself.

4.2 Revised theory accommodating back-stress from the die swell

A striking feature of the experiment is the sharp increase in lateral velocity $v$ at the corners of the channel exit, which drives the die swell. This indicates a significant rate of transverse extension $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}y$ at the channel mouth. In the context of radially flowing extensional flows, the hoop stress due to lateral (azimuthal) extension of this kind was found to resist flow and generate back-stress (Pegler & Worster Reference Pegler and Worster2012). I anticipate that a similar back-stress arises here as a result of lateral extension at the channel exit and may explain the discrepancy between theory and experiment noted above.

To model this force, I retain the term involving transverse extension $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}y$ in the width integration of (2.13) to obtain the generalized frontal stress condition

With (4.2), the integration of the dimensional form of the steady-state equation (3.14) yields

where $v_{0}\equiv v(l,s)$ . This is the same as the dimensional form of (3.15) but with the new, third term on the right-hand side representing the added resistive stress generated by lateral extension near the channel mouth. The lateral velocity $v_{0}$ is a new parameter, which I evaluate empirically as $v_{0}=\hat{v}_{0}\mathscr{U}$ with $\hat{v}_{0}\approx 0.22$ using the velocimetric data shown in figure 12(b). Using $u=Q/H$ to eliminate $u$ from (4.3) and then recasting it in terms of the dimensionless variables (3.3), I obtain the ordinary differential equation

I solved this equation subject to the matching condition (3.20a ) using a Runge–Kutta scheme in which ${\hat{H}}(L)$ is treated as a shooting parameter. The resulting velocity profile, $\hat{u} =1/{\hat{H}}$ , is plotted as a dashed curve in figure 13. The new prediction shows better agreement with the experimental data, indicating that the stresses due to lateral extension near the channel exit do indeed contribute to an appreciable reduction in the flow rate. According to the revised theory, the inclusion of the added resistive stress results in an 11 % reduction in the frontal flow rate from $\hat{u} (L)\approx 0.67$ given by (3.19b ), to $\hat{u} (L)\approx 0.60$ .

Figure 14.

The surface profiles given by the solution to the steady-state system defined by (5.2) and (5.3) in the shear-thinning case corresponding to Glen’s law, $n=3$ , with channel length $L=5$ and a range of input thicknesses $D=0.2{-}\infty$ . Like the Newtonian case, the profiles adjust towards a universal profile (dashed blue curve) obtained by solving (5.2) numerically subject to (5.3b ) and the matching condition (5.5).

5.1 Solutions for the universal profile

In the Newtonian case $n=1$ , I determined the universal profile in § 3.3 using an asymptotic reduction of the analytical solution (3.16). With no exact solution of this kind generally available for $n\neq 1$ , I determine the universal profile in a different way by solving (5.2) numerically subject to (5.3b ) with the frontal value $u_{c}$ treated as a shooting parameter that is tuned to satisfy a condition of zero stress upstream, $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x\rightarrow 0$ as $x\rightarrow -\infty$ . This zero-stress condition is equivalent to matching the flow to an outer region of drag-dominated flow, as was shown to be applicable to the Newtonian case in § 3.3.1. By neglecting the first term in (5.2), integrating it and then neglecting the constant of integration in the limit $L-x\gg 1$ , the outer solution is found to be

where $\unicode[STIX]{x1D6FC}\equiv (n+1)/n$ , which generalizes (3.20b ) to non-Newtonian cases.

The numerically obtained universal velocity profile $u_{\ast }(x)$ for $n=3$ is shown as a solid curve in figure 15(a) and is compared alongside solutions for $n=1$ and $n=\infty$ (perfectly plastic) in figure 15(b). The profile is qualitatively similar to the Newtonian case shown earlier in figure 6, but the dimensionless speed at the channel exit $u_{c}\approx 0.305$ is just under half the value found for the Newtonian case, 0.666. The boundary of the extensional zone, defined as where $u_{\ast }$ deviates from the outer solution (5.5) by more than 10 %, is shown by a vertical dashed line in figure 15(a). It has the remarkably long extent $\unicode[STIX]{x1D6E5}_{\ast }\approx 31.2$ , corresponding to approximately 7 flow widths ( $\unicode[STIX]{x1D6E5}_{\ast }\mathscr{L}\approx 7w$ ). That is, the frontal discontinuity in depth-integrated pressure has a significant impact on flow 7 channel widths upstream of the front. This is much longer than the corresponding distance of just three-quarters of a channel width found for the Newtonian case (§ 3.3.1). One contributing factor to the significant difference is that $H_{\ast }(L)$ is more than twice as large with $n=3$ , indicating that shear thinning creates a larger frontal pressure discontinuity. The case $n=\infty$ is given by the exact solution $H=-4W_{-}(-2\text{e}^{x-2})$ , where $W_{-}$ is the lower branch of the Lambert $W$ function (the inverse of the function $f(z)=z\text{e}^{z}$ for $z\leqslant -1$ ). The dimensionless frontal speed $u_{c}$ reduces with $n$ , as illustrated by the inset of figure 15(b). For $n=\infty$ , (5.3b ) implies that $u_{c}=1/8$ directly without any need to solve (5.2). In this case, the viscous stress at the channel front simply equals the yield stress specified by the constitutive law (2.4), and the frontal thickness $H_{\ast }(L)=8$ creates the discontinuity in depth-integrated frontal pressure needed to balance this uniform magnitude of stress.

While extensional stresses at the front of the channel have the potential to impact flow over a significant distance, its divergence in the governing equation, given by the first term on the left-hand side of (5.2), may be neglected to excellent approximation. This was demonstrated for a Newtonian fluid in § 3.3.1. To confirm it in non-Newtonian cases, I neglect the first term in (5.2) and integrate it to obtain

is an integration constant representing the speed at the flow front. I set the value of $u_{c}$ as the value obtained from my numerical solution. Alternatively, one could use the analytical approximation (5.6b ), which is obtained by substituting (5.6a ) into the frontal condition (5.3b ). The inset of figure 15(b) shows (5.6b ) as a dashed green curve, and agrees to good approximation with the numerically determined values of $u_{c}$ , as well as reproducing the convergence $u_{c}\rightarrow 1/8$ as $n\rightarrow \infty$ . The approximation (5.6a ) is shown as a blue curve alongside each of the three solutions in figure 15(b), showing good agreement in all cases. The approximation is especially accurate in the case of Glen’s law, $n=3$ , for which the maximum deviation from the exact solution is just 2 %, occurring at $x\approx L-5.1$ . In conclusion, while the balance of extensional stress at the flow front has a far-reaching effect on the upstream flow, the divergence of extensional stresses along the channel plays an insignificant role and the analytical approximations based on its neglect (5.6) are almost exact.

Figure 16.

Speed map of the Amery Ice Shelf (data from Rignot et al. Reference Rignot, Mouginot and Scheuchl2011). The white arrows indicate the three primary glaciers feeding the ice shelf.

6.1 Classification of ice-shelf regimes

In applying the theoretical results to ice shelves, I begin by estimating parameters. In accord with Glen’s law, the viscosity coefficient and power-law exponent are

respectively, where $A(T)$ is a temperature-dependent function (Cuffey & Paterson Reference Cuffey and Paterson2010). With the value $A\approx 3.8\times 10^{-25}~\text{Pa}^{-n}~\text{s}^{-1}$ suggested by Cuffey & Paterson (Reference Cuffey and Paterson2010), $\unicode[STIX]{x1D707}_{0}\approx 7\times 10^{7}~\text{Pa}~\text{s}^{1/n}$ . Given the density of glacial ice $\unicode[STIX]{x1D70C}\approx 917~\text{kg}~\text{m}^{-3}$ and ocean water $\unicode[STIX]{x1D70C}_{w}\approx 1027~\text{kg}~\text{m}^{-3}$ (Griggs & Bamber Reference Griggs and Bamber2011), the reduced gravity $g^{\prime }\equiv (\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{w})g\approx 1.05~\text{m}~\text{s}^{-2}$ .

The estimates of $\unicode[STIX]{x1D707}_{0}$ , $n$ , $\unicode[STIX]{x1D70C}$ and $g^{\prime }$ given above are characteristic of most ice shelves. Substituting these estimates into the dimensionless parameters (5.4a,b ), I obtain

respectively. The grounding-line thickness $d$ , flow width $w$ , flow length $l$ and flux per unit width $Q$ can differ significantly between ice shelves. Using the velocimetric and topographic data from Griggs & Bamber (Reference Griggs and Bamber2011) and Rignot et al. (Reference Rignot, Mouginot and Scheuchl2011), I estimated the values of $d$ , $w$ , $l$ and $Q$ shown in table 2 for five ice-shelf configurations. Note that the Ross Ice Shelf is partitioned near its eastern edge by Roosevelt Island. To account for this, the ice shelf is considered in two components: its smaller eastern part, referred to as Ross (east); and its significantly larger western part, referred to as Ross (main). Since the flow width $w$ and grounding-line thickness $d$ are not all uniform for any given ice shelf, ranges for $w$ and $d$ are provided. The flux per unit width $Q$ was measured by evaluating the width average $Q=\int Hu\,\,\text{d}y/w$ over a series of nodes spanning each calving front.

Figure 17 shows the theoretically predicted dimensionless frontal thickness ${\hat{H}}(L)$ , obtained from my numerical solutions for $n=3$ in § 5, as a colour map over the parameter space $(L,D)$ , creating the Glen’s law analogue of figure 8. Again, white represents the special value associated with the universal profile, given by ${\hat{H}}_{c}\approx 3.28$ for $n=3$ , blue represents under-thick cases and red over-thick; the contour ${\hat{H}}(L)=(1\pm 0.1){\hat{H}}_{c}$ separates ‘long’ flows for which the transition to the universal profile completes and ‘short’ for which it does not. A long over-thick shelf generally occurs for $L>1$ which, in reference to (6.2b ), corresponds to lengths greater than one-third of a channel width, $l>w/3$ .

The parameter ranges estimated for each ice shelf are shown by the black lines in figure 17. All the ice shelves are found to lie firmly within the over-thick and long regions of the parameter space, for which they are predicted to contain prevailing regions of flow described by the universal profile. These results indicate that the universal profile is likely to arise widely in nature.

6.2 Comparison with the prediction for frontal flow rate

Based on the classification of the ice shelves in figure 17 as long, all can be expected to involve a complete adjustment to the universal profile. To test this, I compared the width-averaged frontal flow rates of the ice shelves listed in table 2 with the prediction associated with the universal profile, determined in § 5 as

where the units are $\text{m}~\text{s}^{-1}$ and I have substituted the common values of $\unicode[STIX]{x1D70C}$ , $g^{\prime }$ and $\unicode[STIX]{x1D707}_{0}$ estimated in § 6.1 to yield the expression (6.4), which is dependent only on the frontal flow width $w$ and the input flux per unit width $Q$ . The prediction (6.4) is compared with observed values in the scatter plot of figure 18 for the five ice-shelf systems listed in table 2. In cases where the flow lies in lateral contact all the way up to the calving front, I used the transverse width at the calving front $w$ given by the parenthetical values in table 2. In the case of the Amery Ice Shelf, I instead used the value of $w$ measured at the point at which the ice shelf detaches from its sidewalls; the prediction is compared with the observed width-averaged flow rate at that location. There is generally good agreement, with excellent agreement in the cases of the Amery and Ronne Ice Shelves (a less than 4% discrepancy). A relatively larger overprediction of 19% occurs in the case of the Filchner Ice Shelf. The good agreement indicates that these ice shelves have all approached the universal profile obtained theoretically in § 5 and that (6.4) is describing their frontal flow rates correctly over a range of parameters.

6.3 Discussion regarding the input transition zone

The classification as over-thick is consistent with the morphology of most (if not all) ice shelves, which thin along their full length. Thickness data for the Ronne Ice Shelf shows that it thins from approximately 2000 m at the grounding line of a number of its feeding glaciers to 1000 m over a length scale of only a few tens of kilometres (see figure 3 of Griggs & Bamber Reference Griggs and Bamber2011). That is, the ice shelf halves in thickness within just 1 % of its total length. This provides some evidence for a localized region of significant spatial variation near the grounding line, which is consistent with an over-thick input transition zone.

Given the theoretical possibility of an under-thick regime, the fact that no ice shelves lie within that category is indicative of an underlying physical cause. I explain this observation on the basis that a grounding line is a dynamically free boundary that moves in response to thickening or thinning of the ice sheet. Its position is not prescribed at a fixed location as I have assumed in (3.1). An under-thick profile requires a relatively larger force to input (there is a larger back-stress exerted by the ice shelf in this case) and it is thus feasible that, were an under-thick flow to form, it would stimulate grounding-line migration to a new position in deeper water where an over-thick ice shelf forms more stably. That is, the grounding settles more stably at positions where the thickness of the ice sheet is closer to the separation between under-thick and over-thick situations (the thick curve in figure 17). Such a situation would be consistent with a mode of grounding-line migration controlled by the lateral drag exerted on the ice shelf (Pegler Reference Pegler2012; Pegler et al. Reference Pegler, Kowal, Hasenclever and Worster2013; Kowal, Pegler & Worster Reference Kowal, Pegler and Worster2016).

While migration of the grounding line may reduce the development of an input transition zone in a marine ice sheet, the imposition of a fixed source in other physical contexts may sustain a transition zone more readily. For example, the introduction of fluid over a steep weir, incline or backwall (as in my experiments) forces fluid to detach and float along a fixed position. This allows for an under-thick input if the thickness at which fluid is introduced is sufficiently small, as evidenced by my experiments (see figure 11). A mode of input of this kind corresponds more closely to that of molten glass during the float-glass process.

6.4 Model limitations

The dimensionless solutions I have obtained describe a broad range of aspect ratios and fluid-mechanical parameters. However, realistic ice shelves can involve more complicated sidewall structures than the rectangular geometry I have assumed. They can also be significantly affected by snowfall accumulation and basal melting, which alter the mass of the ice shelf along its length. These effects could be accommodated within the model of § 2 and would probably create a different profile from that determined in § 5. Variations in temperature and grain size also create inhomogeneities in the coefficient of viscosity (6.1a,b ). Another limitation is my assumption of a source that is evenly distributed across the width of the channel. Large ice shelves are typically fed by localized ice streams, which could result in more complex two-dimensional transitional flow in front of the grounding line. There is also the potential for error caused by the ad hoc nature of the parametrization of lateral stress (2.10). I have provided some validation of this parametrization for a Newtonian fluid by comparison with my experiments. Further validation could be obtained by comparison with the predictions of the full two-dimensional extensional flow equations (2.2)–(2.4) or data from a laboratory experiment involving shear-thinning fluid.