2.1. Curved chute experiment

In order to investigate the process of longitudinal stripe formation, experiments are performed on a curved chute that initially accelerates the granular material downslope and then gradually brings it to rest to form a thin deposit, as shown in figures 1–4, as well as movies 1 and 2, available at https://doi.org/10.1017/jfm.2026.11311. The chute has no sidewalls and is made from a medium-density fibre board ( $2100\times 700\times 6.5$ mm), which can be bent into the required shape. The board was covered with double-sided sticky tape and then a monolayer of spherical turquoise glass ballotini (750–1000 $\unicode{x03BC}$ m) was applied. Over time, some of these beads have worn off (figure 1), which has improved the ability of the experiment to generate stripes, and so the chute has not been recoated. The granular material consists of 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) spherical glass ballotini of the same bulk density. The grains are released from a double-gated hopper at the top of the chute. The first gate is fixed to a set height of 25 mm, to partially control the mass flux, while the second gate is used to release a mass of approximately 2 kg. This finite mass ensures that stripes are preserved in a sizeable part of the deposit (figure 1).

Figure 1. A photograph of the curved experimental chute, which consists of a flexible plane bent to produce a gradual slope angle change from $\zeta =46^\circ$ to the horizontal at its steepest point to $\zeta =17^\circ$ at its shallowest point. A hopper and two-gate system controls the inflow, the first gate is set at 25 mm and the second gate is used to start the flow. The bed was originally roughened with a monolayer of spherical turquoise glass ballotini (750–1000 $\unicode{x03BC}$ m), which was stuck down with double-sided tape. Over time this has partially worn off to reveal the white tape. A deposit formed by the release of a 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) glass ballotini is shown on the chute. Movies 1 and 2, available in the online supplementary material, show the complete experiment in real time and slow motion.

Figure 2. Photos showing the time-dependent evolution of the curved chute experiment shown in figure 1. The images are taken at (a) $t=0.5$ , (b) $0.76$ , (c) $1.22$ , (d) $1.24$ , (e) $2.04$ and (f) $13.24$ seconds after the gate is opened. Movies 1 and 2 in the online supplementary material show the complete experiment in real time and slow motion.

Figure 3. (a) An overhead photograph of the deposit formed by the release of a 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) glass ballotini on the curved chute in figures 1 and 2. The deposit consists of alternating green and white surface stripes, which have high concentrations of large and small particles, respectively. The stripes are terminated by a large-rich front that has just started to finger before arresting. In the upper left side of (a) material has been scraped away to reveal the internal structure (b), which shows that the large- and small-particle-rich stripes extend all the way from the surface to the base of the deposit. (c) A close-up photo of the vertical structure of a single large stripe.

Figure 4. (a) A perspective photograph showing a similar deposit to that in figure 3, but with oblique lighting to highlight the subtly raised large-particle ridges in the deposit formed by the release of a 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) glass ballotini on the curved chute. (b) An obliquely lit photograph showing the deposited ridges in a monodisperse flow of large green ballotini on the same curved chute. Note that the oblique lighting makes the particles look grey.

The initial charge of grains is placed in the centre of the hopper and the second gate is then opened. Figure 2, as well as movies 1 and 2, show the time-dependent evolution of the flow down the curved chute. At $t=0.5$ s, when the grains are on the steepest 46 $^\circ$ section of the chute, the flow spreads both longitudinally and laterally, and the surface appears a uniform shade of greeny white. It is not until $t=0.76$ s that the first signs of the stripes emerge just behind the flow front as it transitions onto the shallow slope. The front itself appears slightly wavy and diffuse, and is whiter in colour suggesting that it may be richer in small particles. There are also a couple of stripes higher up, in the thinner regions at the sides, which initially spread out laterally. By $t=1.22$ s, long linear alternating bands of green and white particles (i.e. the stripes) extend from the front to the tail of the flow. Near the front, there is some evidence of stripe formation and merging, but once a distinct wavelength has developed the stripes can be traced all the way to the back of the flow. The front is still somewhat diffuse, wavy and whiter than the main body of grains.

At $t=1.24$ s the flow front reaches the decelerating part of the chute, where the slope angle reduces to $\zeta =17^\circ$ . In this region large grains are segregated towards the flow surface and then sheared towards the flow front (Gray & Kokelaar Reference Gray and Kokelaar2010b ; Johnson et al. Reference Johnson, Kokelaar, Iverson, Logan, LaHusen and Gray2012; Gray Reference Gray2018). There is therefore a rapid transition to the development of a large-rich flow front, which feels more resistance to motion than the finer grained material behind, and the flow begins to finger just before it stops (Pouliquen, Delour & Savage Reference Pouliquen, Delour and Savage1997; Woodhouse et al. Reference Woodhouse, Thornton, Johnson, Kokelaar and Gray2012; Baker, Johnson & Gray Reference Baker, Johnson and Gray2016b ). By $t=2.04$ s, the front of the avalanche has completely stopped and in the rear half of the flow there is a tiny shock wave that propagates back upslope bringing the remaining grains to rest. This thickens the flow and the depth of the large-rich surface layer, making the greener regions stand out more, which is consistent with the observations and model of Gray & Kokelaar (Reference Gray and Kokelaar2010a , Reference Gray and Kokelaarb ) (see § 4.2). During this slowing phase, it is likely that the vortical circulation shuts off and the flow begins to transition to an inversely graded segregation pattern, with large grains on top of small. However, the flow comes to rest so quickly, within the banded section, that not that much segregation/diffusion occurs, allowing the stripes to be well preserved.

In the final phase of flow, some of the slowly moving material upstream of the shock wave is able to break through it and initiate small erosion-deposition waves that propagate relatively easily on the existing deposit (Edwards & Gray Reference Edwards and Gray2015; Rocha, Johnson & Gray Reference Rocha, Johnson and Gray2019). In these waves, standard segregation is active and erosion-deposition wave fronts become rich in large particles, whilst the tails are almost pure fines. Note that, if the change in chute angle is too weak, or the flow volume is too large, erosion-deposition waves can completely rework the deposit, and destroy any evidence of stripe formation. In the current experiment the change in chute angle and the release volume have been optimised to ensure the stripes are preserved in the deposit. By $t=13.24$ s these slowly propagating erosion-deposition waves have also come to rest to reveal the final deposit. It consists of a large-rich slightly fingered front (which only formed in the later stages of motion), a large region of well-preserved stripes immediately behind the front, which extends a long way upstream, and finally a heavily reworked scalloped tail section in which the stripes have been destroyed by erosion-deposition waves.

Figure 3(a) shows a close-up overhead view of the large-rich fingered flow front and the striped deposit behind it. Qualitatively it looks very similar to the Sherman rock avalanche deposit illustrated in figure 4 of Shreve (Reference Shreve1966) and figure 6 of Post (Reference Post1967). The stripes are remarkably evenly spaced, with a wavelength of approximately 12–13 mm between the centres of the large-rich bands. The large-rich bands themselves are slightly diffuse, but have a typical width of 2–4 mm at the surface. The white bands are wider, which is a direct reflection of the initial 30 : 70 mixing ratio. The deposit has subtly raised ridges that coincide with the large-rich bands, as shown by the obliquely lit photograph in figure 4(a). Figure 4(b) shows that similar ridges form even for a monodisperse flow of large green ballotini. This suggests that particle-size segregation is not the underlying cause of the vortical motion, but instead helps to make its effects visible. In the upper left corner of figure 3(a) a ruler has been used to cut the flow vertical downwards and scrape the upstream material backwards and away from the deposit to reveal the internal structure (figure 3 b). The stripes extend all the way from the surface to the base of the flow, i.e. there are alternating vertical bands of almost pure large and pure small particles. The close-up view of a single large vertical stripe is shown in figure 3(c). The band is slightly wider at the top and bottom of the flow than in the middle. As shall be shown in this paper, such a particle-size distribution is consistent with existing mechanisms for particle-size segregation combined with streamwise secondary vortices in the bulk flow.

2.2. Straight chute experiment

Although the formation of stripes on the curved chute is striking, it is a complicated flow involving many processes. In order to simplify the system, experiments have also been performed on a straight 2.84 m long chute that is inclined at an angle $\zeta =34.5^\circ$ to the horizontal, and has glass sidewalls that are separated by a gap of 7.8 cm. The base is covered with the same spherical turquoise glass ballotini (750–1000 $\unicode{x03BC}$ m) as the curved chute (see right-hand side of figure 5 a). A well-mixed sample of the same 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) glass ballotini (of the same bulk density) is released from a two-gate system. The first gate is set at 10 mm and maintains a steady uniform-depth inflow until the hopper runs out of material.

Figure 5. Overhead photographs (a–e) taken in a window that lies 2.15–2.55 m down the chute, showing the flow of a 30 : 70 mix of large green (200–250 $\unicode{x03BC}$ m) and small white (60–150 $\unicode{x03BC}$ m) glass ballotini on a chute inclined at $\zeta =34.5^\circ$ to the horizontal. The photos are taken at $t=0.28$ , $1$ , $1.92$ , $5.94$ and $12.58$ s after the avalanche first enters the camera window. The complete time-dependent evolution is shown in real time and slow motion in movies 3 and 4 .

When the grains are released they flow rapidly down the chute as shown in the real time and slow motion overhead movies 3 and 4. The camera window is located between 2.15 and 2.55 m downslope. Stills of the movie are shown in figure 5 at a sequence of times measured from when the front first enters the camera window. In figure 5(a) the front is just propagating into view. It is somewhat diffuse, and there is a cloud of fine grained material at the front. The stripes appear just behind the front and are highly dynamic in their initial phase (figure 5 b), bending and merging as they try to equilibrate themselves. By $t=1.92$ s a quasi-steady-state striped pattern has established itself, with alternating bands of large (green) and small (white) material (figure 5 c). There are five fully formed stripes in the centre of the channel, measured from centre to centre of adjacent large-rich bands, with wider fines-rich bands adjacent to the chute walls. The stripes are evenly spaced and are approximately 10 mm wide, which is slightly narrower than on the curved chute. While there is strong variation in composition across the chute, downstream gradients are generally very weak. There is, however, some slow drift of the bands and occasional intense pulses (figure 5 d) with a sequence of waves that span the bands and travel rapidly down slope out of view. This generates a short-lived periodic flashing effect. As the material runs out of the hopper and the flow thins, the pattern changes from downstream oriented stripes to cross-slope oriented large-rich bands that propagate down slope. These are due to the formation of bidisperse roll waves (Viroulet et al. Reference Viroulet, Baker, Rocha, Johnson and Gray2018).

A space–time plot derived from movie 3 is shown in figure 6(a). It clearly shows the fines-rich front flowing down the empty chute (horizontal grey lines for $t\lt 0.4$ s), which is followed by the highly dynamic region where the stripes begin to form. Simultaneous laser height measurements (figure 6 b) indicate that this dynamic regime corresponds to when the flow is thickening. Between $t=0.4$ and $1.9$ s, when the flow is thinner, the surface images show that the flow appears to have more stripes, but as the flow thickens the bands coarsen and equilibrate (figure 6 a). There then follows a sustained period of approximately constant depth flow $t\in [1.8,7.7]$ s, with little variation across the chute, except near the sidewalls where the flow bulges slightly upwards (figure 6 b,c). During this phase of flow the stripes are relatively stable, with six large-rich flow bands that deviate only slightly in width and position. As the material runs out of the hopper, and the flow begins to thin again, between $t=7.7$ and $8.5$ s, the space–time plot shows that the flow tries to develop more stripes again (figure 6 a). However, the stripes become diffuse and peter out by $t=10$ s and there is a transition to the formation of bidisperse roll waves that propagate downslope, and have large-rich cross-slope bands that correlate with the peak flow heights as seen by Viroulet et al. (Reference Viroulet, Baker, Rocha, Johnson and Gray2018).

Figure 6. (a) An overhead space–time plot of the straight-chute experiment showing the development of the stripes. It is created from movie 3 by stacking a series of pixel columns taken at approximately 2.55 m down the chute. (b) A corresponding space–time plot of laser height measurements $H$ across the chute at approximately 2.6 m downslope. The superposed red line indicates the average height as a function of time using the colour bar as an axis. (c) The flow thickness across the chute between times $t=1$ and $7$ s when the quasi-steady stripes have formed. The grey region shows all the data, while the red line shows the average flow thickness.

The laser height measurements (figure 6 c) indicate that during the uniform-depth flow phase the average depth is approximately 2.7 mm and there are no obvious bulges or ridges that correlate with the position of the large- and small-particle bands, although the large-rich bands are subtly raised in the deposit (see figure 4 a). The absence of obvious ridges during flow may be due to (i) the relatively low inclination of the chute, (ii) differential dilation of the two species and/or (iii) simply that the ridges are of sufficiently small amplitude that they are lost in the measurement noise, especially if there is some lateral drift in their position. Note that in monodisperse flows, well-defined ridges during flow were observed at significantly higher inclinations and only for specific hopper openings (Börzsönyi et al. Reference Börzsönyi, Ecke and McElwaine2009). Our premise here is that segregation is helping to reveal the presence of secondary vortices, rather than being the root cause of them. This is supported by the fact that subtly raised ridges are observed in the deposit of a monodisperse flow on the same chute (see figure 4 b).

In our conceptual model for the flow, which will be discussed shortly in detail in § 3, it is argued that each stripe consists of two counter-rotating secondary vortices that are oriented in the downslope direction. Since the width of each stripe is approximately 10 mm, it follows that the height-to-width aspect ratio of each vortex is approximately 1.85. Surface velocities have also been measured during the uniform-depth phase by tracking individual tracer particles. Typical downslope velocities are of the order of 1.38 m s⁻¹, which are much larger than any cross-slope velocity components. The fact that the flow is rapid and dense, and the particles are of the same bulk density, suggests that the secondary vortices correspond to the dense regime observed in monodisperse flows by Börzsönyi et al. (Reference Börzsönyi, Ecke and McElwaine2009), i.e. particle-size segregation is not causing the secondary vortices, but responds sensitively to their presence and helps to reveal their existence.

4.1. The segregation-advection equation

Consider a bidisperse mixture of large and small particles that is flowing down a chute inclined at a fixed angle $\zeta$ to the horizontal. A Cartesian coordinate system $Oxyz$ is defined with the $x$ -axis pointing in the downslope direction, the $y$ -axis in the cross-slope direction and the $z$ -axis normal to the slope as illustrated in figure 7. All the particles are assumed to have the same intrinsic density and the prescribed bulk velocity field $\boldsymbol u=u\boldsymbol i+v\boldsymbol j+w\boldsymbol k$ has components $(u,v,w)$ in the directions of the downslope, cross-slope and normal unit vectors $\boldsymbol i$ , $\boldsymbol j$ and $\boldsymbol k$ . The large- and small-particle concentrations per unit granular volume are $\phi ^l$ and $\phi ^s$ , respectively. These sum to unity

(4.1) \begin{equation} \phi ^l+\phi ^s=1, \end{equation}

and satisfy a pair of segregation-advection equations (Gray & Thornton Reference Gray and Thornton2005; Gray & Ancey Reference Gray and Ancey2011; Gray Reference Gray2018)

(4.2) \begin{align}& \frac {\partial \phi ^l}{\partial t}+ \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,(\phi ^l\boldsymbol{u})+ \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,(f_{ls}\phi ^l \phi ^s \boldsymbol e) = 0, \end{align}

(4.3) \begin{align}& \frac {\partial \phi ^s}{\partial t}+ \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,(\phi ^s\boldsymbol{u})- \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,(f_{ls}\phi ^s \phi ^l\boldsymbol e) = 0, \end{align}

where $f_{ls}$ is the segregation velocity magnitude, $\boldsymbol e$ is a unit vector oriented in the direction that large particles segregate, $ \boldsymbol{\nabla}$ is the gradient operator and $ \boldsymbol{\cdot}$ is the dot product. Motivated by the relatively sharp segregation observed in experiments in § 2, diffusion is neglected in this simple formulation. The time rate of change of the large- and small-particle concentrations therefore evolves due to advection by the bulk flow field and particle segregation parallel to $\boldsymbol e$ . Note that when (4.2) and (4.3) are added together, the summation condition (4.1) implies that the bulk velocity field $\boldsymbol u$ is incompressible

(4.4) \begin{equation} \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,\boldsymbol u=0. \end{equation}

The summation constraint (4.1) can be used to eliminate one of the concentrations. For instance, eliminating the large-particle concentration implies that the uncoupled form of the small-particle segregation-advection equation, (4.3), is

(4.5) \begin{equation} \frac {\partial \phi ^s}{\partial t}+ \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,(\phi ^s\boldsymbol{u})- \boldsymbol{\nabla }\, \boldsymbol{\cdot }\,\Bigl (f_{ls}\phi ^s (1-\phi ^s)\boldsymbol e\Bigr ) = 0. \end{equation}

This is subject to a no-flux condition on the boundary of the avalanche. Gray & Thornton (Reference Gray and Thornton2005) showed that this is equivalent to the condition that either

(4.6) \begin{equation} \phi ^s = 0\quad \hbox{or}\quad \phi ^s = 1 \end{equation}

on the boundary of the avalanche.

The direction of segregation $\boldsymbol e$ is usually assumed to be aligned with the direction of gravitational acceleration $\boldsymbol g$ . This is consistent with the idea that small particles percolate downwards under the action of gravity. However, the process in which large particles are squeezed towards the surface by force imbalances (Savage & Lun Reference Savage and Lun1988) may instead align with the pressure gradient, and $\boldsymbol e$ may therefore be normal to the chute. The precise orientation of the segregation direction is therefore still open to some interpretation, although it is not important for the purposes of this paper.

4.2. Uniform stripes in the downslope direction

The straight chute experiment in § 2.2 shows that a series of stripes form that are weakly dependent of the downstream coordinate $x$ . This suggests that a useful simplification is to assume that the large- and small-particle concentrations are spatially uniform in the downstream direction, i.e. $\phi ^l=\phi ^l(y,z,t)$ and $\phi ^s=\phi ^s(y,z,t)$ . Since, the velocity field $\boldsymbol u$ is also assumed to be independent of $x$ , the incompressibility condition (4.4) reduces to

(4.7) \begin{equation} \frac {\partial v}{\partial y}+\frac {\partial w}{\partial z}=0. \end{equation}

Assuming that the segregation direction $\boldsymbol e$ does not have a cross-slope component, the small-particle segregation-advection equation, (4.5), becomes

(4.8) \begin{equation} \frac {\partial \phi ^s}{\partial t}+\frac {\partial }{\partial y}(\phi ^s v)+\frac {\partial }{\partial z}(\phi ^s w)-\frac {\partial }{\partial z}\left ( q\phi ^s(1-\phi ^s) \right ) = 0, \end{equation}

where the normal component of the segregation velocity is

(4.9) \begin{equation} q=f_{ls}\,\boldsymbol e\, \boldsymbol{\cdot }\,\boldsymbol k. \end{equation}

The assumption of uniformity in the downslope direction implies that (4.8) appears to be independent of the downslope velocity $u$ . In general, the segregation velocity $q$ is linearly proportional to shear rate $\dot \gamma =2||\boldsymbol D||$ , inversely proportional to the pressure and has further dependencies on the particle sizes, the particle-size ratio and the local concentrations of the two particle species (Trewhela et al. Reference Trewhela, Ancey and Gray2021; Maguire et al. Reference Maguire, Barker, Rauter, Johnson and Gray2024). The downslope velocity therefore enters (4.8) indirectly through its gradient $\partial u/\partial z$ . Since $\partial u/\partial z$ is much larger than any of the other strain-rate components (related to the secondary recirculation), it provides the dominant contribution that sets the magnitude of $q$ . For simplicity $q$ is assumed to be a fixed constant in this paper, although, in general, there is some weak dependence of $q$ on $z$ in a Bagnold flow, which subtly changes the structure of the breaking-size-segregation waves that form (Edwards et al. Reference Edwards, Rocha, Kokelaar, Johnson and Gray2023). Assuming that $q$ is constant has the advantage that it allows us to investigate the qualitative dependence of the solutions over the complete parameter space, and derive exact solutions to the equations that yield considerable insight.

4.3. Parameterisation of the secondary vortices

The secondary vortices are not solved for in this paper. Instead they are parameterised in a very simple way, by assuming that they are rectangular in shape and have an incompressible velocity field of the form

(4.10) \begin{align}&\qquad v=\varOmega \sin {\left (\frac {\pi y}{W}\right )}(2z-H),\qquad 0 \leqslant z \leqslant H,\qquad m W\leqslant y\leqslant n W, \end{align}

(4.11) \begin{align}& w=-\left (\frac {\varOmega \pi }{W}\right )\cos {\left (\frac {\pi y}{W}\right )}z(z-H),\qquad 0 \leqslant z \leqslant H,\qquad m W\leqslant y\leqslant n W, \end{align}

where $\varOmega$ sets the rotation rate, $H$ is the cell depth and $W$ its width. The integers $m$ and $n\gt m$ determine the lateral extent of the cells. In general, the flow has $n{-}m$ vortices, with adjacent vortex cells that counter-rotate. Since each complete stripe consists of two counter-rotating vortices, the flow will have $(n{-}m)/2$ stripes, as shown in figure 7.

4.4. Non-dimensional variables

It is useful to introduce non-dimensional variables by using the scalings

(4.12a,b,c) \begin{equation} (y,z)=H(\tilde {y},\tilde {z}), \hspace {1cm} (v,w)=\varOmega H(\tilde {v},\tilde {w}), \hspace {1cm} t=(H/q)\tilde {t}, \end{equation}

where the non-dimensional variables are denoted by a tilde. Note, the velocity scale $\varOmega H$ is based on a typical rotational velocity magnitude, and the time scale $H/q$ is based on a typical time for segregation to occur. Substituting these implies that the incompressibility condition (4.7) and the small-particle segregation-advection equation, (4.8), become

(4.13) \begin{align}&\qquad\qquad\qquad\qquad\quad \frac {\partial \tilde {v}}{\partial \tilde {y}}+\frac {\partial \tilde {w}}{\partial \tilde {z}}=0, \end{align}

(4.14) \begin{align}& \frac {\partial \phi ^s}{\partial \tilde {t}}+\frac {1}{\varLambda }\frac {\partial }{\partial \tilde {y}}(\phi ^s \tilde {v})+\frac {1}{\varLambda }\frac {\partial }{\partial \tilde {z}}(\phi ^s \tilde {w})-\frac {\partial }{\partial \tilde {z}}\Bigl (\phi ^s(1-\phi ^s) \Bigr ) = 0, \end{align}

where the non-dimensional parameter

(4.15) \begin{equation} \varLambda =\frac {q}{H\varOmega }, \end{equation}

is the ratio of a typical segregation velocity $q$ to a typical secondary-vortex velocity $H\varOmega$ . The scalings (4.12) imply that the non-dimensional velocity field becomes

(4.16) \begin{align}&\qquad \tilde {v}=\sin {\left (\frac {\pi \tilde {y}}{\tilde {W}}\right )}(2\tilde {z}-1),\qquad 0 \leqslant \tilde {z} \leqslant 1,\qquad \tilde {W}m \leqslant \tilde {y}\leqslant \tilde {W} n, \end{align}

(4.17) \begin{align}& \tilde {w}=-\left (\frac {\pi }{\tilde {W}}\right )\cos {\left (\frac {\pi \tilde {y}}{\tilde {W}}\right )}\tilde {z}(\tilde {z}-1), \qquad 0 \leqslant \tilde {z} \leqslant 1,\qquad \tilde {W}m\leqslant \tilde {y}\leqslant \tilde {W}n, \end{align}

where $\tilde {W}$ is the non-dimensional vortex cell width. The non-dimensional velocity field in a single clockwise-rotating secondary vortex is plotted in figure 8. It uses the same contour scale as in figure 7(c,d), so the symmetry of the entire velocity field becomes apparent.

Figure 8. A single clockwise rotating steady-state secondary vortex in the $(\tilde {y},\tilde {z})$ plane of width $\tilde {W}=1.85$ . (a) A quiver plot of the cross-slope and normal velocities, (b) the associated streamfunction and (c,d) contour plots of the individual velocity components given by (4.16)–(4.17).

In order to solve for the small-particle concentration it is necessary to set the initial concentration

(4.18) \begin{equation} \phi ^s(\tilde {y},\tilde {z},0)=\phi ^s_0(\tilde {y},\tilde {z}), \qquad (\tilde {y},\tilde {z})\in [\tilde {W}m,\tilde {W}n]\times [0,1], \end{equation}

where $\phi ^s_0$ is the initial small-particle concentration distribution. Along the surface and base of the flow ( $\tilde {z}=0,1$ ) the no flux conditions (4.6) imply that

(4.19) \begin{equation} \phi ^s = 0,\quad \hbox{or}\quad \phi ^s = 1\quad \hbox{on}\quad \tilde z=0,1, \end{equation}

and the no-flux condition is trivially satisfied on the vortex interfaces.

5.1. Numerical method

The segregation-advection equation, (4.14), is solved with the high-resolution central scheme of Kurganov & Tadmor (Reference Kurganov and Tadmor2000) using a second-order Runge–Kutta time-stepping method. The non-dimensional width of the secondary vortices is assumed to be

(5.1) \begin{equation} \tilde {W}=1.85, \end{equation}

which is consistent with the experimental observations in § 2. The computations are performed in a single clockwise-rotating secondary-vortex cell defined in the region $(\tilde y,\tilde z)\in [0,1.85]\times [0,1]$ non-dimensional units. The numerical method implicitly assumes that $\partial \phi ^s/\partial y=0$ along the boundaries with the neighbouring vortices at $\tilde {y}=0,1.85$ . The non-dimensional velocity field (4.16)–(4.17) is prescribed (see figure 8).

5.2. Evolution from an inversely graded initial state

A 30 : 70 mixture of large and small particles is initially assumed to be sharply segregated with large particles on top of the small grains, as shown in figure 9(a). In the absence of secondary recirculation, this would be a natural stable steady-state solution. However, when the recirculation is non-zero, the velocity field rotates the interface clockwise with the flow, causing it to steepen. Large particles therefore become concentrated in the downwelling section on the right-hand side of the cell (as shown in figure 9 b), while small particles become concentrated in the upwelling region on the left. At approximately $\tilde {t}=1.5$ non-dimensional units, the interface between large and small particles becomes vertical in the interior of the flow, and breaks to form a new structure that is known as a breaking-size-segregation wave (Thornton & Gray Reference Thornton and Gray2008; Gray & Ancey Reference Gray and Ancey2009; Edwards et al. Reference Edwards, Rocha, Kokelaar, Johnson and Gray2023). This grows in size, until it occupies the full height of the cell (figure 9 c,d,e). The wave then oscillates backwards and forwards in time, eventually approaching a steady state, as shown in figure 9(e,f) and movies 5. Although the outer margins of the breaking-wave stabilise relatively quickly, the central eye of the wave has oscillations that persist for a very long time. The structure of the steady-state concentration distribution will be solved for explicitly in § 6.

Figure 9. The evolving small-particle concentration $\phi ^s$ at times (a) $\tilde {t}=0$ , (b) $1.5$ , (c) $3$ , (d) $5$ , (e) $160$ and (f) steady state $\tilde {t}=175$ for $\varLambda =3/2$ and a 30 : 70 mix. The clockwise-rotating secondary vortex is defined in $[0,1.85]\times [0,1]$ by the velocities (4.16) and (4.17). The complete time-dependent evolution of the interface is shown in movie 5.

The formation of a series of breaking-size-segregation waves that are aligned with the flow direction is critical for the formation of longitudinal stripes. This is because they allow large particles, which are drawn down to the base of the flow in the downwelling regions, to segregate upwards as soon as they encounter small particles above them. Conversely small particles, which rise to the surface in the upwelling sections, are able to percolate down to the base as soon as they experience large particles beneath them. In this way, both large and small particles can form their own interpenetrating steady-state recirculation loops within each secondary vortex, as will be explicitly shown in § 6.

Figure 10 and movie 6 show a perspective view of the simultaneous time-dependent development of two surface stripes and the internal particle-size distribution in the $(\tilde {y},\tilde {z})$ plane. The solution is constructed from the time-dependent solution in a single vortex cell (figure 9), by using the reflective symmetry along adjacent vortex cell boundaries. Initially the particles are inversely graded (i.e. large on top of small) so the free surface appears green. Even as the secondary recirculation steepens and breaks the internal interface between the large and small grains to form a breaking-size-segregation wave, the free surface remains green, i.e. large particles are still on the surface (see figure 10 a,b,c,d). It is only at approximately $\tilde {t}=6$ non-dimensional time units, that small particles first break onto the free surface in the centre of upwelling regions. The white region then rapidly widens, so that approximately 70 % of the free surface is occupied by small particles (figure 10 e). Below the free surface, a breaking-size-segregation wave develops, but it takes a very long time to settle down to steady state (figure 10 f). As a result, the width of the surface bands of large and small particles oscillates subtly in time, which can be seen in movie 6.

Figure 10. Perspective view of the time-dependent formation of two stripes for $\varLambda = 3/2$ and a 30 : 70 mix that is initially inversely graded. The solution spans four counter-rotating vortex cells. The small-particle concentration $\phi ^s(x,y,z,t)$ is shown at non-dimensional times (a) $\tilde {t}=0$ , (b) $1.5$ , (c) $3$ , (d) $5$ , (e) $160$ , and approximately at steady state (f) $\tilde {t}=175$ . The complete time-dependent evolution is shown in movie 6.

5.3. The effect of varying $\varLambda$

The effect of varying the ratio $\varLambda$ of a typical segregation velocity to a typical secondary-recirculation velocity is shown in figure 11. When there is no circulation (i.e. when $\varLambda =\infty$ ) the steady state corresponds to a inversely graded state with all the large particles separated from the small grains beneath by a concentration shock (figure 11 a). In the absence of diffusion, any amount of secondary recirculation is sufficient for the horizontal interface between large and small particles to be rotated around and overturn to form a breaking-size-segregation wave, in a similar manner to figures 9 and 10 and movies 5 and 6. If isotropic diffusion is included in the model, then the lateral diffusion may compete against the steepening of the interface and prevent the formation of breaking-size-segregation waves. This will be investigated in a subsequent paper (Pearse et al. submitted).

Figure 11. Approximate steady-state numerical solutions for the small-particle concentration $\phi ^s$ for (a) $\varLambda =\infty$ , (b) $100$ , (c) $10$ , (d) $3/2$ , (e) $1$ and (f) $0.5$ . The blue and turquoise lines show the steady-state exact solution for the shocks and lead characteristics of the expansion fans, respectively, for comparison. The central eye takes a long time to settle towards steady state. In (a) there is no secondary recirculation, and so the interface between small and large particles remains horizontal. In (b–f) secondary recirculation is present and a breaking-size-segregation wave forms, which becomes progressively larger as $\varLambda$ is decreased. Note that the surface concentration is only weakly dependent on the size of the breaking wave.

The focus of this paper is to understand the structure of the breaking-size-segregation waves, and how they are able to recirculate both the large and small particles in the flow to form stable stripes. Figure 11(b)–(f) shows that when $\varLambda \ne \infty$ steady states are formed with a pure phase of large particles in the downwelling section that is separated from a pure region of small particles in the upwelling section by a breaking-size-segregation wave. As $\varLambda$ is decreased, the width of the breaking wave increases. Note that in the limit as $\varLambda \rightarrow \infty$ from below, the breaking-size-segregation wave essentially collapses onto a vertical shock that lies at $\tilde {y}=0.7\tilde {W}$ for a 30 : 70 mix (figure 11 b). This lies perpendicular to the case $\varLambda =\infty$ (no secondary recirculation), i.e. the existence of secondary vortices is a singular perturbation to the problem. Although the width of the breaking-size-segregation waves are sensitively dependent on the value of $\varLambda$ , their general location is not. In particular, when viewed from above the apparent stripe width is insensitive to the choice of $\varLambda$ . The fact that the stripes are present across the full parameter space (figure 11 b–f), apart from when there is no secondary recirculation (figure 11 a), highlights the robustness of this striped segregation structure.

6.1. Quasi-linear form and its transformation to streamfunction coordinates

At steady state, the non-dimensional incompressibility condition (4.13) can be used to reduce the segregation-advection equation, (4.14), to the quasi-linear form

(6.1) \begin{equation} \tilde {v}\frac {\partial \phi ^s}{\partial \tilde {y}}+\tilde {w}\frac {\partial \phi ^s}{\partial \tilde {z}}+\varLambda (2\phi ^s-1)\frac {\partial \phi ^s}{\partial \tilde {z}}=0. \end{equation}

Following Gray & Thornton (Reference Gray and Thornton2005) and Thornton & Gray (Reference Thornton and Gray2008), it is useful to transform (6.1) into streamfunction coordinates $(\xi ,\psi )$ by defining the mapping

(6.2a,b) \begin{equation} \xi =\tilde {y},\qquad \psi =\int ^z_0 \tilde {v}(\tilde {y},\tilde {z}')\, \mathrm{d}\tilde {z}', \end{equation}

where $\psi$ , in this case, is now the streamfunction associated with the two-dimensional projected velocity field $\boldsymbol{ \tilde {v}}={\tilde {v}}\boldsymbol{j}+\tilde {w}\boldsymbol k$ in the $(\tilde {y}, \tilde {z})$ plane. In these new coordinates the derivatives transform as

(6.3a,b) \begin{equation} \frac {\partial }{\partial \tilde {y}}= \frac {\partial }{\partial \xi }-\tilde {w}\frac {\partial }{\partial \psi },\qquad \frac {\partial }{\partial \tilde {z}}= \tilde {v}\frac {\partial }{\partial \psi }, \end{equation}

and hence (6.1) transforms to the reduced quasi-linear form

(6.4) \begin{equation} \frac {\partial \phi ^s}{\partial \xi }+\varLambda (2\phi ^s-1)\frac {\partial \phi ^s}{\partial \psi }=0, \end{equation}

provided $\tilde {v}\ne 0$ . However, the cross-slope velocity (4.16) is equal to zero in the centre of the domain at $\tilde {z}=1/2$ . The solution is therefore constructed in two transformed domains that lie above and below the no-mean-flow line $\tilde {z}=1/2$ . For the non-dimensional velocity field (4.16)–(4.17) the streamfunction is

(6.5) \begin{equation} \psi =\sin \left (\frac {\pi \tilde {y}}{\tilde {W}}\right )(\tilde {z}^2-\tilde {z}), \end{equation}

which is illustrated in figure 12(a). This is quadratic in $\tilde {z}$ , so given a mapped position $(\xi ,\psi )$ it maps to the physical point $(\tilde {y},\tilde {z})$ by setting

(6.6a,b) \begin{equation} \tilde {y}=\xi ,\qquad \tilde {z}=\frac {1}{2}\left (1\pm \sqrt {1+\frac {4\psi }{\sin \left (\frac {\pi \xi }{\tilde {W}}\right )}}\right )\!, \end{equation}

Figure 12. A schematic diagram showing (a) the streamfunction coordinates, (b) the steady-state breaking-size-segregation wave and (c) the large- and small-particle paths for $\varLambda =3/2$ and a 30 : 70 bidisperse mixture of large and small particles. The exact concentration solution (b) consists of two expansion fans, centred at $A$ and $C$ , which intersect the surface and base of the flow at points $B$ and $D$ , respectively. There are two shocks $BC$ and $DA$ , and a central eye of constant concentration that lies between the dot-dashed lines. A single small-particle path is also illustrated in (b). It starts in the pure phase of small particles at point 0 on the $\tilde {z}=1/2$ line and then intersects with 1 the lead characteristic of the top fan, 2 the top eye, 3 the $\tilde {z}=1/2$ line for the second time, 4 the lower eye, 5 the lower shock and then reconnects at 0 to form a closed loop. (c) Shows a series of both small-particle paths (black) and large-particle paths (green). These allow for the simultaneous steady recirculation of both species within a single secondary vortex.

where the positive root lies in the upper domain, above $\tilde {z}=1/2$ , and the negative root lies in the lower domain. Note that, on the top, bottom and sides of the secondary vortices $\psi =0$ , while on the $\tilde {z}=1/2$ line its value is given by

(6.7) \begin{equation} \psi =\psi _m(\xi )=-\frac {1}{4}\sin \left (\frac {\pi \xi }{\tilde {W}}\right )\!. \end{equation}

Since, by definition

(6.8) \begin{equation} \frac {\partial \psi }{\partial \tilde {z}}=\tilde {v}, \end{equation}

the height $\tilde {z}=1/2$ marks the points where $\partial \psi /\partial \tilde {z}=0$ , and the streamfunction reaches a local maximum, or minimum, dependent on whether the cell is rotating clockwise, or anticlockwise, for a fixed value of $\xi$ . The global maximum and minimum of the streamfunction is

(6.9) \begin{equation} \psi _{\textit{max/min}}=\frac {(-1)^{n+1}}{4},\quad \hbox{at}\quad \tilde {y}=\frac {2n+1}{2}\tilde {W},\quad n=\ldots , -1,0,1,2,\ldots .\end{equation}

The mapping (6.2) has the advantage that the transformed segregation-advection equation, (6.4), is completely independent of the velocity field, allowing it to be solved in a generic way. The velocity field only re-enters the problem in defining the streamfunction coordinate (6.5) and its return mapping (6.6).

6.2. Solution by the method of characteristics

Suppose that $\lambda$ parameterises the path of a characteristic curve $(\xi (\lambda ),\psi (\lambda ))$ , then the rate of change of the concentration along the characteristic curve

(6.10) \begin{equation} \frac {\mathrm{d}\phi ^s}{\mathrm{d}\lambda }=\frac {\partial \phi ^s}{\partial \xi }\frac {\mathrm{d}\xi }{\mathrm{d}\lambda }+\frac {\partial \phi ^s}{\partial \psi }\frac {\mathrm{d}\psi }{\mathrm{d}\lambda }. \end{equation}

Comparing (6.4) with (6.10) implies that the characteristic equations are

(6.11a,b,c) \begin{equation} \frac {\mathrm{d}\phi ^s}{\mathrm{d}\lambda }=0,\qquad \frac {\mathrm{d}\xi }{\mathrm{d}\lambda }=1,\qquad \frac {\mathrm{d}\psi }{\mathrm{d}\lambda }=\varLambda (2\phi ^s-1). \end{equation}

Using the chain rule to eliminate $\lambda$ between equations (6.11a ) and (6.11b ), and then solving the resulting ordinary differential equation (ODE), it follows that the concentration is equal to a constant $\phi ^s_\lambda$ (say) along the characteristic curve $\lambda$ , i.e.

(6.12) \begin{equation} \phi ^s=\phi ^s_\lambda ,\quad \hbox{on characteristic}\ \lambda . \end{equation}

Using this, and the chain rule, equations (6.11b , c ) imply that the shape of the characteristic curve $\lambda$ is given by solving the ODE

(6.13) \begin{equation} \frac {\mathrm{d}\psi }{\mathrm{d}\xi }=\varLambda (2\phi ^s_\lambda -1). \end{equation}

Assuming that the characteristic starts at $(\xi ,\psi )=(\xi _\lambda ,\psi _\lambda )$ , then integrating (6.13) shows that the characteristics are straight lines

(6.14) \begin{equation} \psi =\psi _\lambda +\varLambda (2\phi ^s_\lambda -1)(\xi -\xi _\lambda ), \end{equation}

in streamfunction coordinates. The method of characteristics allows the solution to be constructed within the smoothly varying parts of the flow. However, the numerically computed steady-state solutions, shown in figure 11, contain two concentration shocks. These have to be solved for using jump conditions (Chadwick Reference Chadwick1976; Gray & Thornton Reference Gray and Thornton2005; Thornton, Gray & Hogg Reference Thornton, Gray and Hogg2006).

6.3. Jump conditions across the concentration shocks

Given that the non-dimensional small-particle segregation-advection equation, (4.14), is in conservative form, the associated jump condition is

(6.15) \begin{equation} \left [\!\!\left[ \frac {1}{\varLambda }\phi ^s\,\boldsymbol{\tilde {v}}\,\boldsymbol {\cdot }\,\boldsymbol n-\tilde v_n\right ]\!\!\right] = \left [\!\left[ \phi ^s(1-\phi ^s)\boldsymbol k\, \boldsymbol{\cdot }\,\boldsymbol n\right ]\!\right]\!, \end{equation}

where $\boldsymbol n$ is the two-dimensional unit normal to the shock within the cross-slope plane, the jump bracket $[\![ f ]\!] =f_+-f_-$ is the difference of the enclosed quantity on the forward and rearward sides of the discontinuity and $\tilde v_n$ is the non-dimensional shock propagation speed. At steady state the shock does not move and $\tilde v_n=0$ . For a shock lying at height $\tilde {z}=\tilde {s}(\tilde {y})$ , with unit normal $\boldsymbol n=(-\mathrm{d}\tilde {s}/\mathrm{d}\tilde {y},1)/(1+(\mathrm{d}\tilde {s}/\mathrm{d}\tilde {y})^2)^{1/2}$ , the jump condition therefore reduces to

(6.16) \begin{equation} -\tilde {v}\frac {\mathrm{d}\tilde {s}}{\mathrm{d}\tilde {y}}+\tilde {w}=\varLambda (1-\phi ^s_+-\phi ^s_-),\quad \hbox{at}\quad \tilde {z}=\tilde {s}. \end{equation}

Note, by definition of the streamfunction coordinates (6.2a,b )

(6.17) \begin{equation} \frac {\mathrm{d}}{\mathrm{d}\xi }(\psi (\tilde {s}))=\frac {\mathrm{d}}{\mathrm{d}\tilde {y}}\left (\int _0^{\tilde {s}(\tilde {y})} \tilde {v}(\tilde {y},\tilde {z}')\,\mathrm{d}\tilde {z}'\right )\!. \end{equation}

Using Leibniz’ rule to exchange the order of differentiation and integration it follows that

(6.18) \begin{equation} \frac {\mathrm{d}}{\mathrm{d}\xi }(\psi (\tilde {s}))=\int _0^{\tilde {s}(\tilde {y})} \frac {\mathrm{d}}{\mathrm{d}\tilde {y}}(\tilde {v}(\tilde {y},\tilde {z}'))\,\mathrm{d}\tilde {z}' + \tilde {v}(\tilde {y},\tilde {s}(\tilde {y}))\frac {\mathrm{d}\tilde {s}}{\mathrm{d}\tilde {y}}. \end{equation}

The integral on the right-hand side can be performed directly by substituting the incompressibility condition (4.13) and using the fact that the normal velocity along the base is equal to zero $\tilde {w}(\tilde {y},0)=0$ to obtain

(6.19) \begin{equation} \frac {\mathrm{d}}{\mathrm{d}\xi }(\psi (\tilde {s}))=-\tilde {w}(\tilde {y},\tilde {s}(\tilde {y}))+ \tilde {v}(\tilde {y},\tilde {s}(\tilde {y}))\frac {\mathrm{d}\tilde {s}}{\mathrm{d}\tilde {y}}, \end{equation}

and hence that in streamfunction coordinates the shock condition (6.16) reduces to

(6.20) \begin{equation} \frac {\mathrm{d}}{\mathrm{d}\xi }(\psi (\tilde {s}))=\varLambda (\phi ^s_+ + \phi ^s_- -1),\quad \hbox{on}\quad \psi =\psi (\tilde {s}). \end{equation}

6.4. Solving for the breaking-size-segregation wave

The steady-state breaking-size-segregation waves in figure 11 do not move and are aligned with the downslope coordinate. They are therefore perpendicular to the breaking-size-segregation waves found by Thornton & Gray (Reference Thornton and Gray2008), Gray & Ancey (Reference Gray and Ancey2009) and Gray & Kokelaar (Reference Gray and Kokelaar2010b ) which travelled downslope at a constant speed. Figure 12(b) shows a schematic diagram of the breaking-wave structure in a clockwise rotating cell (figure 8). Unlike earlier breaking-size-segregation waves, the ones that form between the stripes extend through the whole flow depth.

The solution is started at an arbitrary point $A$ along the $\tilde {z}=1/2$ line as shown in figure 12(b). This point has coordinates $(\xi _A,\psi _A)$ , where $\psi _A=\psi _m(\xi _A)$ and the function $\psi _m$ , defined in (6.7), determines $\psi$ at an arbitrary point on the $\tilde {z}=1/2$ line. The solution has an expansion fan centred at $(\xi _A,\psi _A)$ , which propagates upwards into the upper half domain. The concentration within the fan is found by solving for $\phi ^s$ in the characteristic (6.14) to give

(6.21) \begin{equation} \phi ^s=\frac {1}{2}\left ( 1+\frac {\psi -\psi _A}{\varLambda (\xi -\xi _A)}\right )\!. \end{equation}

The lead characteristic AB, on which $\phi ^s=1$ , defines the outer boundary of the breaking-size-segregation wave

(6.22) \begin{equation} \psi =\psi _A +\varLambda (\xi -\xi _A). \end{equation}

This outer boundary rises until it intersects with the free surface at point $B$ , which lies at

(6.23a,b) \begin{equation} \xi _B=\xi _A-\frac {\psi _A}\varLambda ,\qquad \psi _B=0. \end{equation}

A concentration shock forms along $BC$ . It has large particles on one side ( $\phi ^s_+=0$ ) and the expansion fan (6.21) on the other. The shock condition (6.20) therefore implies

(6.24) \begin{equation} \frac {\mathrm{d}{\psi }}{\mathrm{d}\xi }=\frac {1}{2}\left ( \frac {\psi -\psi _A}{\xi -\xi _A} -\varLambda \right )\!. \end{equation}

This can be solved, subject to the initial condition that $\psi =\psi _B$ at $\xi =\xi _B$ , to give the shape of the upper shock

(6.25) \begin{equation} \psi =\psi _A-\varLambda (\xi -\xi _A)+2\sqrt {-\psi _A \varLambda (\xi -\xi _A)}. \end{equation}

This shock reaches the $\tilde {z}=1/2$ line at point $C$ . Its position $(\xi _C,\psi _C)$ can be found by iteratively solving for $\xi _C$ with the equation

(6.26) \begin{equation} \psi _C=\psi _m(\xi _C)=\psi _A-\varLambda (\xi _C-\xi _A)+2\sqrt {-\psi _A \varLambda (\xi _C-\xi _A)}, \end{equation}

where the function $\psi _m=\psi _m(\xi )$ is defined in (6.7). Here the shock breaks to form another expansion fan that propagates downwards into the lower domain. The fan is centred at $(\xi _C,\psi _C)$ , and the concentration within it is given by

(6.27) \begin{equation} \phi ^s=\frac {1}{2}\left ( 1+\frac {\psi _C-\psi }{\varLambda (\xi _C-\xi )}\right )\!. \end{equation}

The boundary between the breaking-size-segregation wave and the region of large particles is given by the $\phi ^s=0$ characteristic

(6.28) \begin{equation} \psi =\psi _C +\varLambda (\xi _C-\xi ). \end{equation}

This reaches the base of the flow at

(6.29a,b) \begin{equation} \xi _D=\xi _C+\frac {\psi _C}{\varLambda },\qquad \psi _D=0, \end{equation}

where a shock $DA$ is generated between the small particles and the lower expansion fan (6.27). The shock condition (6.20) implies that

(6.30) \begin{equation} \frac {\mathrm{d}\psi }{\mathrm{d}\xi }=\frac {1}{2} \left ( \varLambda +\frac {\psi _C-\psi }{\xi _C-\xi } \right )\!, \end{equation}

which can be solved, subject to the condition that $\psi =\psi _D$ at $\xi =\xi _D$ , to give the shape of the lower shock

(6.31) \begin{equation} \psi =\psi _C-\varLambda (\xi _C-\xi )+2\sqrt {-\psi _C\varLambda (\xi _C-\xi )}. \end{equation}

In order for the breaking wave to form a closed structure, the lower shock (6.31) must intersect the original point $(\xi _A,\psi _A)$ as it breaks to form the upper expansion fan (6.21).

As in the case of the breaking wave constructed by Gray & Ancey (Reference Gray and Ancey2009), the solution has a central eye of constant concentration

(6.32) \begin{equation} \phi _{\textit{eye}}^s=\frac {1}{2}\left ( 1+\frac {\psi _C-\psi _A}{\varLambda (\xi _C-\xi _A)} \right )\!, \end{equation}

which is bounded above and below by the characteristics

(6.33) \begin{align} \psi _{\textit{eye}}^{\textit{upper}}&=\psi _A+(2\phi ^s_{\textit{eye}}-1)\varLambda (\xi -\xi _A), \end{align}

(6.34) \begin{align} \psi _{\textit{eye}}^{\text{lower}}&=\psi _C-(2\phi ^s_{\textit{eye}}-1)\varLambda (\xi _C-\xi ). \end{align}

These characteristics are shown in figure 12(b), and emanate out of, and connect between, the fans centred at $(\xi _A,\psi _A)$ and $(\xi _C,\psi _C)$ , i.e. they both imply

(6.35) \begin{equation} \psi _C=\psi _A+(2\phi ^s_{\textit{eye}}-1)\varLambda (\xi _C-\xi _A), \end{equation}

consistent with (6.32). In particular, substituting for $\psi _C$ in (6.34) from (6.35) implies that the boundary of the lower eye can equivalently be expressed as

(6.36) \begin{equation} \psi _{\textit{eye}}^{\text{lower}}=\psi _A+(2\phi ^s_{\textit{eye}}-1)\varLambda (\xi -\xi _A), \end{equation}

which in streamfunction coordinates is the same as the boundary (6.33) for the upper eye. This fact will be useful later in solving for the particle paths, although it should be stressed that the upper and lower boundaries of the eye lie in the upper and lower streamfunction domains, respectively, so (6.33) and (6.36) represent different physical boundaries. Note that the solution in the upper domain (6.21)–(6.23), (6.25) and (6.33) can be transformed to the solution in the lower domain (6.27)–(6.29), (6.31) and (6.34) by the mapping

(6.37a–f) \begin{equation} \xi \mapsto -\xi ,\quad \psi \mapsto \psi ,\quad \phi ^s\mapsto 1-\phi ^s,\quad A\mapsto C,\quad B\mapsto D, \quad C\mapsto A. \end{equation}

This will be useful in Appendices A and B for tracking the large- and small-particle paths (figure 12 b,c). The combination of the two expansion fans, two shocks and the central eye of constant concentration, forms a steady-state breaking-size-segregation wave (Thornton & Gray Reference Thornton and Gray2008; Gray & Ancey Reference Gray and Ancey2009; Johnson et al. Reference Johnson, Kokelaar, Iverson, Logan, LaHusen and Gray2012; Edwards et al. Reference Edwards, Rocha, Kokelaar, Johnson and Gray2023). In contrast to previous solutions, the wave extends through the full depth of the flow, is spatially uniform in the downslope direction and connects the pure region of large grains to the pure region of fines, allowing stripes to be seen at the surface (as shown in figures 7 and 10). A comparison between the computed steady-state and the exact solution for a range of $\varLambda$ is shown in figure 11. The position of the lead expansions, the shocks and hence the overall size of the steady-state breaking-size-segregation waves is in almost exact agreement with the computed solutions. However, the central eye of constant concentration takes a very long time to equilibrate, and this aspect of the numerical solution has not yet reached steady state.

7.1. Three-dimensional bulk, small- and large-particle paths

The particle paths in the bulk flow field, and of the large and small grains, are given by solving the differential equations

(7.1a–e) \begin{equation} \frac {\mathrm{d}\tilde {x}}{\mathrm{d}\tilde {t}}=\tilde {u},\qquad \frac {\mathrm{d}\tilde {y}}{\mathrm{d}\tilde {t}}=\tilde {v},\qquad \frac {\mathrm{d}\tilde {z}}{\mathrm{d}\tilde {t}}=\tilde {w},\qquad \frac {\mathrm{d}\tilde {z}^l}{\mathrm{d}\tilde {t}}=\tilde {w}^l,\qquad \frac {\mathrm{d}\tilde {z}^s}{\mathrm{d}\tilde {t}}=\tilde {w}^s, \end{equation}

where the non-dimensional large- and small-particle velocities

(7.2a,b) \begin{equation} \tilde {w}^l=\tilde {w}+\varLambda \phi ^s,\qquad \hbox{and}\qquad \tilde {w}^s=\tilde {w}-\varLambda (1-\phi ^s), \end{equation}

follow from (4.2), (4.3) and the scalings (4.12) (Gray & Thornton Reference Gray and Thornton2005; Thornton et al. Reference Thornton, Gray and Hogg2006; Gray & Ancey Reference Gray and Ancey2009). Under the assumption of spatial uniformity in the downslope direction, the non-dimensional downslope velocity uncouples from the segregation problem, but $\tilde {u}$ is needed to solve for the particle paths. In (7.1) the downslope velocity and downslope length are assumed to scale as

(7.3a,b) \begin{equation} u = u^*\tilde {u},\qquad x=\left (\frac {u^*H}{q}\right )\tilde {x}, \end{equation}

where, recall from (3.1), $u^*$ is a typical downslope surface velocity magnitude. Since the downslope velocity is typically a lot larger than the segregation velocity ( $u^*\gg q$ ), it follows that a typical downslope length scale $L=u^*H/q$ is much larger than $H$ . Using the scalings (4.12) and (7.3) it follows that the non-dimensional downslope velocity is

(7.4) \begin{equation} \tilde {u}=1-(1-\tilde {z})^{\frac {3}{2}}. \end{equation}

Equations (7.1), (7.2) and (7.4) form three sets of coupled ODEs that can be solved numerically for the bulk, small- and large-particle paths assuming the exact small-particle concentration solution derived in § 6.4.

7.2. The case of a 30 : 70 mix

Figure 13 shows a three-dimensional perspective view of the solution surfaces that are generated for a 30 : 70 mix with $\varLambda =3/2$ , by initially releasing a series of particles along the $\tilde {x}=0$ , $\tilde {z}=1/2$ (orange) line and integrating their trajectories up until $\tilde {t}=20$ to find their final positions (red line). Individual paths are shown by the magenta, yellow, cyan, white and green lines. The solution surfaces are complex even for the prescribed bulk flow field (figure 13 a). In it, the magenta line starts closest to the cell boundary and describes a looping path that takes it through both high and low downslope velocity regions, near the surface and base of the flow, respectively. Particles on the magenta line therefore make most of their progress downslope when they are near the surface of the flow. In contrast, the green particle path starts near the centre of the vortex cell close to $\tilde {y}=\tilde {W}/2$ , and describes small oscillations around the centre of the vortex moving at almost constant speed downslope. After twenty non-dimensional time units the green trajectory has moved slightly further downslope than the magenta one. Intermediate trajectories describe loops of increasing amplitude the further away they start from the centre of the vortex $(\tilde {W}/2,1/2)$ , and the greatest downstream motion is achieved along one of these intermediate paths. When viewed down the axis of the vortex (figure 14 a) the bulk particle paths form closed loops when they are projected onto the $(\tilde {y},\tilde {z})$ plane. In Appendix A it is shown that these bulk particle paths correspond to isolines of the streamfunction. By $\tilde {t}=20$ non-dimensional time units the bulk particles that started along the $\tilde {x}=0$ , $\tilde {z}=1/2$ line, end up along the spiralling red line, and the green path has performed nearly four oscillations, whilst the magenta path has only performed approximately 1.75 oscillations. The difference determines the number of loops in the red spiral. The time taken to complete a loop of a projected bulk particle path as a function of the starting position $\tilde {y}$ along the $\tilde {z}=1/2$ line is shown in figure 15. There are two singularities at $\tilde {y}=0$ and $\tilde {y}=\tilde {W}=1.85$ , which correspond to the exterior boundary of the cell. Particles on the $\psi =0$ path therefore take an infinite time to complete a circuit, which is because both $\tilde {v}$ and $\tilde {w}$ tend to zero in the four corners of the cell. The minimum time to complete a circuit is achieved in the centre of the cell at $\tilde {y}=\tilde {W}/2$ .

Figure 13. Three-dimensional surfaces formed by the trajectories of (a) bulk, (b) small and (c) large particles that are released in a single clockwise rotating secondary vortex with $\varLambda =3/2$ . The trajectories assume a steady-state particle-size distribution for a 30 : 70 mix of large and small particles (shown on the base, side and rear walls). Particles that are released along the orange line (along $\tilde {x}=0$ , $\tilde {z}=1/2$ ) are transported downstream and by $\tilde {t}=20$ non-dimensional times units lie along the red line. Individual trajectories are shown with the magenta, yellow, cyan, white and green lines. At $\tilde {t}=0$ the magenta line is close to the exterior of the cell and the green line is close to (a) $\tilde {W}/2$ , (b) $\tilde {y}_A$ and (c) $\tilde {y}_C$ . The surface colour indicates the flow depth. A Bagnold velocity (7.4) is assumed in the $\tilde {x}$ direction, so that grains that are higher in the flow move faster down slope. Movies 7, 8 and 9 show animated flybys of each of the surfaces.

Figure 14. View down the secondary-vortex axis showing the (a) bulk, (b) small- and (c) large-particle paths in figure 13 for $\varLambda =3/2$ and a 30 : 70 mix of large and small particles. Note that, when projected on to the $(\tilde {y},\tilde {z})$ plane, the looping magenta, yellow, cyan, white and green particle paths form closed loops. In addition, particles that were initially released from various positions along the $\tilde {x}=0$ , $\tilde {z}=1/2$ line, end up on the spiralling red line after $\tilde {t}=20$ non-dimensional times units.

Figure 15. The time taken to complete a loop of the bulk (blue line), small (black line) and large (green line) projected particle paths at different starting positions $\tilde {y}$ along the $\tilde {z}=1/2$ line for (a) a 30 : 70 and (b) a 50 : 50 mix of large and small particles. The dashed lines indicate the positions of $\tilde {y}_A$ and $\tilde {y}_C$ where there are zeros and singularities. The dot-dash line marks the centre of the cell.

Figure 13(b) shows a similar surface swept out for a series of small-particle paths that initially start in the pure small phase on the $\tilde {x}=0$ , $\tilde {z}=1/2$ line. The magenta line is again close to the cell boundary, while this time the green path is close to the point $\tilde {y}_A$ . Looking down the axis of the breaking-size-segregation wave (figure 14 b) the green trajectory stays entirely within the region of small particles, so it rotates at the same speed as the bulk flow. Conversely, the white, cyan, yellow and magenta paths all pass through the breaking-size-segregation wave, and form closed loops of increasing amplitude in the projected plane. Exact solutions for the projected small-particle paths, illustrated in figure 12(b,c), are given in Appendix A. Figure 15(a) shows the time taken for small particles to perform a loop of the projected particle paths. For a 30 : 70 mix there is a region around the centre of the cell $\tilde {y}\in [\tilde {W}-\tilde {y}_A,\tilde {y}_A]$ , where the small particles perform loops entirely within the small-particle region. In this region the time taken for a small particle to complete a loop is the same as for the bulk flow. Outside this region small particles pass through the both the small-particle region and the breaking-size-segregation wave, i.e. for $\tilde {y}\in [0,\tilde {W}-\tilde {y}_A]\cup [\tilde {y}_A,\tilde {y}_C]$ . In the limit as $\tilde {y}\rightarrow 0$ or $\tilde {y}\rightarrow \tilde {y}_C$ the time taken to do a loop tends to infinity. This is again because the path lies on the outer boundary $\psi =0$ for at least some of the time, where both the cross slope and normal velocity components tend to zero in the cell corners. For intermediate starting positions just outside the central core there is a significant speed up in the time that it takes to do a loop, which is associated with the rapid percolation of the small particles through the breaking-size-segregation wave. Viewed up the vortex axis, the final small-particle positions initially appear to spiral outwards in a clockwise sense, before switching direction to spiral outwards in an anticlockwise sense (figures 13 b and 14 b). As a result particles perform more rotations as they are swept downslope than the bulk flow (figures 13 a and 14 a).

The circuits performed by the large particles are dramatically different to the bulk and small particles as shown in figures 13(c) and 14(c). This is because, as well as there being singularities at $\tilde {y}=\tilde {y}_A$ and $\tilde {y}=\tilde {W}$ , the time taken to perform a circuit tends to zero as $\tilde {y}\rightarrow \tilde {y}_C$ (figure 15 a). The surface swept out by the initial line of particles therefore wraps around itself very fast as one approaches $\tilde {y}_C$ , and the three-dimensional perspective view in figure 13(c) and movie 9 has the appearance of a filo pastry. This switch in behaviour is due to the existence of a central core of small particles that do not enter the breaking-size-segregation wave, and therefore forces the small particles to spend a considerable amount of time in the pure phase of small particles. The large particles near $\tilde {y}=\tilde {y}_C$ , on the other hand, only have to travel a short distance through the pure large phase before being re-entrained into the breaking-size-segregation wave. In some sense this is similar to the mixing of differently coloured particles in a rotating drum, which also switches behaviour dependent on the existence, or not, of a central slowly rotating core (Gray Reference Gray2001).

7.3. The case of a 50 : 50 mix

The case of a 50 : 50 mix of large and small particles is interesting because there is neither a slowly rotating core of pure large or pure small particles. The time taken for large particles to perform circuits is broadly similar to the 30 : 70 case, with singularities at $\tilde {y}=\tilde {y}_A$ and $\tilde {y}=\tilde {W}$ , and a zero at $\tilde {y}=\tilde {y}_C$ . However, in the 50 : 50 case, the small particles now have a similar structure to the large particles, with singularities at $\tilde {y}=0$ and $\tilde {y}_C$ , and a zero at $\tilde {y}=\tilde {y}_A$ as shown in figure 15(b). The existence of the two zeros now implies that the small-particle paths wrap tightly around $\tilde {y}_A$ , while the large-particle paths wrap tightly around $\tilde {y}_C$ as before. As a result the three-dimensional perspective view of the large- and small-particle paths both have the appearance of filo pastries as shown in figures 16(b,c) and 17(b,c) as well as movies 11 and 12.

Figure 16. Three-dimensional surfaces formed by the trajectories of (a) bulk, (b) small and (c) large particles that are released in a single clockwise rotating secondary vortex with $\varLambda =3/2$ . The trajectories assume a steady-state particle-size distribution for a 50 : 50 mix of large and small particles (shown on the base, side and rear walls). Particles that are released along the orange line (along $\tilde {x}=0$ , $\tilde {z}=1/2$ ) are transported downstream and by $\tilde {t}=20$ non-dimensional times units lie along the red line. Individual trajectories are shown with the magenta, yellow, cyan, white and green lines. At $\tilde {t}=0$ the magenta line is close to the exterior of the cell and the green line is close to (a) $\tilde {W}/2$ , (b) $\tilde {y}_A$ and (c) $\tilde {y}_C$ . The surface colour indicates the flow depth. A Bagnold velocity (7.4) is assumed in the $\tilde {x}$ direction, so that grains that are higher in the flow move faster down slope. Movies 10, 11 and 12 show animated flybys of each of the surfaces.

Figure 17. View down the secondary-vortex axis showing the (a) bulk, (b) small- and (c) large-particle paths in figure 16 for $\varLambda =3/2$ and a 50 : 50 mix of large and small particles. Note that, when projected on to the $(\tilde {y},\tilde {z})$ plane, the looping magenta, yellow, cyan, white and green particle paths form closed loops. In addition, particles that were initially released from various positions along the $\tilde {x}=0$ , $\tilde {z}=1/2$ line, end up on the spiralling red line after $\tilde {t}=20$ non-dimensional times units.