Idealized branch sections and hydrodynamic analysis characterize rainfall partitioning

Water flow on plant canopies determines the partitioning of intercepted rainfall between stemflow and throughflow, yet understanding of these flow processes remains minimally developed. Plant canopies may concentrate intercepted water into rivulets that flow beneath branches. If the rivulets remain attached to branches until they encounter the main tree trunk, they form stemflow. If the rivulets detach from branches, however, they form ‘pour points’, regions of concentrated throughfall. Here, rivulet flow below uniform cylinders was studied experimentally. Experimental observations were interpreted using hydrodynamic theory to predict the likelihood of a rivulet detaching based on a critical rivulet height ( λc$$ {\lambda}_c $$ ). Predictions of rivulet detachment were explored for a simplified case where water is introduced at known flow rates at a discrete point above a uniform, cylindrical ‘branch’. Where the branch was straight, the percentage of trials forming a rivulet, or the rivulet formation rate, increased with increasing branch inclination angle to the horizontal axis. Rivulet formation rates did not vary with flow rate. However, the volume of water collected at the end of the branch, analogous to stemflow, decreased with a higher flow rates due to the formation of flow instabilities. Rivulets forming on straight branches did not detach. Instead, the rivulet stream accelerated and reduced in height as it flowed along the cylinder. Rivulets formed on curved branches detached only when the branch axis after the curve approached the horizontal. Future theoretical and experimental studies can extend this work to improve understanding of more complex surfaces and branch architectures to mechanistically understand canopy interception.


| INTRODUCTION
Plants cover approximately 60% of the global land surface (Ritchie & Roser, 2013).Over these surfaces, rainfall is likely to be intercepted by plant canopies.The intercepted rain may pass through canopies as throughfall (TF) or stemflow (S) (Savenije, 2005), or be stored on the canopy and evaporate forming canopy interception losses (IL).
Evaporation of intercepted rainfall reduces the water available for streamflow, groundwater recharge, and ecosystems.Typically, canopy interception loss represents 10%-30% of gross rainfall (Zhong et al., 2022), although observations at heavily vegetated sites may exceed this figure (Van Stan et al., 2020).Appropriate management of water resources in vegetated landscapes, therefore, relies on accurately quantifying canopy interception losses.
Typically, canopy interception losses are estimated from a canopy water balance: where GR is the gross rainfall falling on the canopy.The residual of the gross rainfall when compared to throughfall and stemflow yields an estimate of the interception loss.Any errors in the measured quantities therefore propagate to IL estimates.The estimates of interception loss from several storms can be generalized to all/any storm for the canopy studied using rainfall interception models (Muzylo et al., 2009).Again, measurement errors in the mass balance components would propagate as errors in the parameters of rainfall interception models.
GR is the greatest flux.It is easy to scale from measurements with rain gauges due to rain's relatively low spatial variability.TF is the next greatest flux, however, unlike gross rainfall, the spatial variability of throughfall distributions is significant, arising from water storage and transport mechanisms on the canopy (Keim et al., 2005;Lloyd et al., 1988).This makes scaling the throughfall measurement from rain gauges more challenging, and sampling the TF field requires complex statistical treatment (B.Zimmermann et al., 2010).S, which represents the flow which reaches the trunk of the plant, is typically less than throughfall.Unlike TF and GR, S does not need to be sampled from a spatial field but rather is a 'point' process at a catchment scale (Carlyle-Moses et al., 2018;Sadeghi et al., 2020).To scale the measurements of individual 'point' measurements of S, representative vegetation must be sampled.Consequently, the canopy interception literature to date has prioritized better sampling (A.Zimmermann & Zimmermann, 2014) over improved mechanistic understanding of rainfall partitioning.
Rainfall partitioning is the process by which GR is divided between IL, S, and TF.The canopy structure determines the partitioning of GR, via flow pathways, and consequently controls the heterogeneity in the TF that determines the sampling requirements (Kimmins, 1973).A pertinent example of the increased heterogeneity is the concentration of TF flux at specific locations.Concentrated TF fluxes have been referred to as 'woody surface drip points' (WSDP), when observed occasionally and temporarily under a branch, and as 'structurally-mediated' woody surface drip points (SM WDSP) on a branch if observed at fixed locations (Herwitz, 1987;Nanko et al., 2022).Kunadi et al. (2024) provided a detailed description characteristics and hydrological significance of the SM WSDP, naming them 'pour points' based on the volume of water received underneath these points.We shall use the term pour points to refer to SM WSDPs in this article.
Consequently, they increase spatial heterogeneity in the throughfall distributions, leading to significant error and uncertainty between sampled throughfall from rain gauges and the actual quantity of throughfall reaching the ground (Lloyd et al., 1988;A. Zimmermann & Zimmermann, 2014;A. Zimmermann et al., 2009).Hydrologically, pour points fluxes may infiltrate deeper into the soil than other forms of TF (Kunadi et al., 2024), which may be comparable to the enhanced infiltration caused by stemflow (Johnson & Lehmann, 2006;Liang, 2020;Taniguchi et al., 1996).Increased infiltration at points within ecosystems is of considerable ecohydrological significance (Carlyle-Moses et al., 2018).Understanding when and where pour points occur and if they need to be included in interception measurements could thus address practical challenges associated with monitoring throughfall fields.
In this article, we explore a part of the rainfall partitioning process that creates pour points, TF spatial distributions, and S quantities, by developing a mechanistic understanding of the flow of water on canopy branches.There has been a sustained effort to understand the flow of water on plant canopies.For example, leaf biomechanical properties and tip structure have been shown to control drainage from the canopy (Ginebra-Solanellas et al., 2020;Wang et al., 2020).Tucker et al. (2020) pioneered a mechanistic approach to stemflow travelling through approximately-vertical furrows in tree bark.
Although important in some cases, this model neglected the effect of surface tension, which cannot be neglected when describing the flow of water below a branch.Whilst the development of physically-based models has been explored at the primary stage of leaf tips and the ultimate stage of stemflow drainage, the formation, transport, and partitioning between stemflow and throughfall of branch flows remain understudied.This paper aims to fill this gap in the literature by developing a mechanistic understanding of water flow on branches.
An immediate benefit of such understanding may be improved techniques to identify pour points in the field as an input to the design of throughfall measurements.A long-term benefit may be the identification of rainfall partitioning through mechanistic models of the canopy, with applications to upscaling, downstream hydrological predictions and to the biogeochemical role of throughfall drainage (Kuzyakov & Blagodatskaya, 2015;Levia et al., 2011).
To achieve the aim of a mechanistic understanding of water flow on branches, we focus on the hydrodynamics of that process.As Herwitz (1987) showed, intercepted water flows around a branch's profile, and collects beneath the branch as a 'pendent rivulet' (Indeikina et al., 1997;Pitts, 1973).These rivulets may flow beneath the branch until they reach the main stem of the tree where they form stemflow.
Alternatively, they may detach from branches, forming a permanent, concentrated flux of throughfall called a 'pour point'.
Stemflow and pour points are consequences of flow pathways in the canopy, which involve a rivulet, hanging below the branch surface.
Experimental studies have shown that stemflow on a cylindrical branch varies non-monotonically across different rainfall intensities (Dunkerley, 2014;Van Elewijck, 1989) and branch inclination angles (Levia & Herwitz, 2002;Van Elewijck, 1989).Herwitz (1987) observed the flow generated by rainfall over cylindrical branches and after sufficient wetting, a fixed location of stream detachment formed, particularly where the surface orientation of the branch changed rapidly (Herwitz, 1987).Our own work in rainfall simulator experiments suggested that branches with greater leaf area increased flow through pour points, in a hard-to-predict fashion (Kunadi et al., 2024).
Predicting whether rivulets form and ultimately produce stemflow or pour points requires understanding the fluid dynamics of rivulets and their detachment from branch surfaces (Levia & Germer, 2015).
Previous theoretical work had addressed the cross-sectional stability for liquid below a horizontal plane (Pitts, 1973), providing a theoretical framework that we could apply to the problem of rivulet detachment.Similarly, previous studies of rivulet flow on cylindrical surfaces (Alekseenko et al., 1996(Alekseenko et al., , 2008)), demonstrated that different flow regimes were produced by changing the rivulet flow rate Q and cylinder inclination angle α, and provided an experimental template for us to follow.Several studies have investigated drop fall-off from wave crests on pendent rivulets (Indeikina et al., 1997;Zhou & Prosperetti, 2022) We used a combination of experimental and theoretical approaches to answer two research questions: 1. What are the principles governing the partitioning of a rivulet?and 2. What are the implications of rivulet partitioning on the canopy water balance?
We confine ourselves to considering the simplified situation where water is introduced at the top of an 'idealized cylinder' branch at a discrete point, after which it flows to the base of the branch, forms a rivulet and flows along the branch, as illustrated in Figure 1a.
This situation is the one produced in our experiments, and allows a straightforward investigation of hydrodynamics along individual, idealized branch sections.

| THEORETICAL FRAMEWORK
Rivulet hydrodynamic studies initially considered rivulets flowing above an inclined surface, known as 'sessile' rivulets.For example, Towell and Rothfeld (1966) established solutions for laminar, sessile rivulet flow on an inclined plane.Branches, however, result in rivulet flow lying below a surface, so-called 'pendent' rivulets.
In this study, we apply a few assumptions to model pendent rivulet flow.First, we consider branches with a wide diameter such that the curvature of the branch is low relative to the width of a rivulet.
This lets us approximate the branch as a plane (Figure 1b).We also confine ourselves to consideration of steady flows, meaning that for a F I G U R E 1 Experimental coordinate system for pendent rivulet flow over an inclined cylinder.
constant density of water, at any point along the rivulet the flow satisfies a constant: where the volumetric flow rate Q is uniform along the rivulet, where U is the stream velocity, and A the rivulet cross-sectional area.When applying Equation (2), we assume that U is uniform over the rivulet cross-section.Finally, we assume that the geometry of the rivulet varies smoothly and gradually so that the stability of the cross-section represents the local stability of the rivulet itself.
A two-dimensional 'slice' of a pendent rivulet is acted on by surface energies at the fluid-air and fluid-solid interface, and by gravity, as indicated in Figure 1b.Here a coordinate system is defined so that z is parallel to the slope of the branch or cylinder (at an angle of α above the horizontal), x is the coordinate in the horizontal plane across the flow, and y is normal to z and x.The fluid-air interface, or the rivulet profile, at a given z depends on the contact angle, θ.The contact angle is specific to the fluid and surface interface; between water and PVC, it is assumed to be 80 (Emelyanenko et al., 2015;Loo et al., 2012).We experimentally verified this to be true for a drop hanging below the cylinder.The rivulet profile at any z can be described as a curve of the form x y ð Þ, which is assumed to only gradually change along z.Solving for the shape of this profile is a classical problem, which has attracted scientists since Carl Jacobi, and the analysis here follows that of Pitts (1973).Pitts (1973) used a variational method to identify the function x y ð Þ that minimizes the potential energy of (half of) the cross-section.The solution profile, x y ð Þ, can be parameterised using Jacobi elliptic integrals, and the height of the rivulet h can then be determined for a given contact angle θ.In turn, for a fixed contact angle, the cross-sectional area A can be computed, and represented as a function of the rivulet height.The variational method reveals that the rivulet profile is unstable as soon as the crosssectional area A decreases as a function of height h (when surface tension can no longer withstand gravity), as illustrated in Figure 1c.
For a given contact angle θ, the analysis summarized above determined that any cross-section is stable if the cross-sectional area A increases as height increases.For theoretical development, we nondimensionalize the rivulet cross-sectional area as σ in Equation ( 3) and we nondimensionalize the rivulet height as λ in Equation ( 4).
Where L c is a length scale known as the capillary length and defined by: where γ is the surface tension of water, ρ is the density of water, and g is acceleration due to gravity.
All theoretical solutions shown were developed using the parameters in Table 1.However, the use of a dimensionless parameter framework makes this study applicable to other values of these system variables.
As indicated in Figure 1c, beyond a certain 'critical' dimensionless rivulet height λ c (and critical cross-sectional area σ c ), σ no longer increases with an increasing λ, indicating that rivulets with geometries in this regime are unstable.Under this unstable condition, there will be loss of liquid from the rivulet (Pitts, 1973).Our approximation of the cylinder as a flat surface has the effect of reducing the stability of the rivulet where cross-sectional curvature is present.So, Figure 1c presents a 'worst-case' scenario of λ c , where branch curvature is infinite.
If flow conditions require the rivulet to adopt a geometry where λ > λ c then we expect that those flow conditions would be unstable and would not form a rivulet.Similarly, if, at a given location along a rivulet, changes in flow or surface conditions caused the rivulet geometry to change such that λ shifted from λ < λ c to λ > λ c , then the rivulet would become unstable and would detach.We hypothesise that such conditions are associated with the formation of pour points.Conversely, if these conditions do not occur along a branch, then water flowing in a rivulet would reach the end (stem) of the branch and form stemflow.
To determine if unstable rivulet conditions will arise along a branch, we consider a simplified situation where water at flow rate Q is injected at a single point at the up-gradient end of a branch and forms a rivulet, similar to Figure 1a.If the rivulet cross-sectional area σ or height λ can be estimated from Q, the flow in the rivulet could be related directly to its stability.
Several studies characterize λ from the flow rate Q, surface inclination angle below horizontal α and characteristics of the fluid and the surface.Typically these studies consider a two-dimensional problem with steady velocity, that is when frictional resistance balances gravitational acceleration, and do not account for non-uniformity of U along z.Kuibin (1996) derived an asymptotic solution for λ a rivulet flowing beneath an inclined cylinder (Equation ( 6)) by considering Pitts (1973)'s two-dimensional droplet profile for a cylinder, and solving the equations of motion as the profile moves along the z-direction, with the assumptions of a constant velocity along the z-direction and a 'thin' rivulet.His asymptotic model is as follows: here, R ¼ R=L c represents the dimensionless cylinder radius, where L c is the capillary length constant (5).Q ¼ Q=q c is the dimensionless flow rate, where q c is the capillary flow rate constant: Table 2 offers a comparison between Kuibin (1996)'s analysis of the flow problem and our estimation of σ based on Figure 1c and Equation ( 2).Whilst Kuibin (1996)'s model considers the variation in velocity of Pitts (1973)'s interface in the x and y-directions, the model does not account for a changing stream velocity along the z-direction.
Since we expect a region of acceleration before constant velocity is reached, we consider an average stream velocity using Equation ( 2), so that we account for a changing rivulet area as the stream moves down a branch.As a compromise, the mean velocity used here does not capture the variations due to shear flow along the x and ydirections.
By linking these definitions with the stability criteria, it is possible to determine the stability of a rivulet flowing at its steady-state velocity, as a function of the rivulet flow rate and branch angle.Such steady velocity solutions are illustrated in Figure 2, where the value of the flow rate is shown on the vertical axis, the branch angle on the horizontal axis, and the dimensionless height in colour.The contour for critical stability λ c ¼ 2:2 is labelled, which approximately follows a linear relationship for low inclination angles (α < 20 ), where Q c ≈ 106α (in the units of mL/s for Q and degrees for α).For flow rates Q < Q c , the rivulet cross-section should remain stable, however if Q > Q c , the rivulet is expected to enter an unstable regime.It is clear that if flow rate is increased, the rivulet profile becomes more unstable, and if the branch inclination is increased, the rivulet profile becomes more stable.
Figure 2 indicates that once a constant stream velocity is reached, a rivulet profile will remain stable unless the flow rate is large (Q > 100 mL/s) or the branch inclination is almost horizontal (α < 1 ).However, the formation of rivulets on canopies, and in the simplified scenario of water injection at a point, involve non-uniform velocities in the z-direction.If water initially moves onto a branch from the vertical direction, then only a component of its initial momentum is aligned along the branch in the z direction, so there may be changes in velocity of the stream as it changes direction to form a rivulet along the z-direction.Assuming that the initial stream velocity U 0 of a rivulet that has just formed is smaller than the steady state velocity, the rivulet will accelerate as it flows down along the branch.
This makes it useful to consider the impact of acceleration on the stability of the rivulet, and the stability of the rivulet at the point of formation.
Provided a stable rivulet forms and accelerates, Equation (2) implies that A (and σ) will decrease with distance along the branch.
For stable rivulets, a decrease in σ is associated with a corresponding decrease in λ (from Figure 1c).Thus, acceleration moves the rivulet further from the stability threshold λ c , causing the rivulets to become more stable as they accelerate.This implies that the most unstable conditions should occur when the rivulet is initiated, at z ¼ 0 cm, when the velocity U 0 is lowest.
T A B L E 2 Comparison between assumptions made by Kuibin (1996) and our analyses.

Kuibin
Our analyses rivulet height (λ) at lengths beyond attainment of steady velocity (z > z SV ) (Equation ( 6)), considering that the critical height for rivulet stability in our system is λ c ¼ 2:2.
Several hypotheses can therefore be made based on this theoretical analysis: • If the angle and discharge conditions are suitable, a rivulet will form; • If a rivulet forms and accelerates with a steady flow rate, its height will decline with distance, approaching the asymptotic limits given by Equation 6; • Declining height in an already stable rivulet creates greater stability-thus once a rivulet forms and accelerates at a steady flow rate, it will remain stable and a pour point will not form.So, use of Figure 2 will overestimate the initial stability of a rivulet (when it is most unstable on a straight branch); • Therefore, pour points will form in a stable rivulet only if surface or geometry conditions change to cause instability, such as a change in branch inclination.
Our experimental methods enabled us to test and evaluate these hypotheses.

| EXPERIMENTAL METHODS
As outlined in the introduction, we conducted laboratory experiments using PVC cylinders as an idealized surface to analyse rivulet flow phenomena as observed on real branches (Figure 3).
In the experiments, we explore the simple situation where water is introduced at one point above a branch, flows to the underside of the branch, and then forms a rivulet.We aimed to address the research questions through four different experiments.Experimentally, we followed Alekseenko et al. (1996), using 1 m PVC cylinders Section 4), not all of which produced rivulets with a quantifiable flow rate.We therefore set criteria for determining trial validity as outlined in Table 5.This criteria was used for all types of experiments conducted.
We found that the likelihood of producing a valid trial varied with the flow rate and the distance between the syringe pump nozzle and the cylinder.Lifting the nozzle (to 4-5 cm above the cylinder) increased the number of valid trials at low flow rates (Q < 2 mL/s), whilst lowering the nozzle (to 1-1.5 cm above the cylinder) increased the rate of valid trials for higher flow rates (Q > 2 mL/s).In practice, for each α-Q combination we piloted different nozzle elevations, and then selected one for each flow rate on the basis of the observed stability of the rivulets produced.
For each trial, we recorded whether a rivulet, pour point or stemflow formed, using the definitions of each of these behaviours as per Table 4.The percentage of trials in which a behaviour was observed was called the 'formation rate'.We made multiple trials (up to 50) for each α-Q combination, until the formation rate stabilized.Discharged stemflow was collected in a beaker (Figure 4) allowing the stemflow volume to be identified as a percentage of the total injected volume.

| Initial velocity estimate to explain rivulet formation
Earlier, we noted that Kuibin (1996)'s model for λ does not consider rivulet profile before steady state velocity, and thus cannot accurately estimate the initial rivulet height λ 0 .We therefore utilize a simplistic model which assumes that U 0 scales as a function of the injected vertical stream velocity (U n ) and branch inclination angle α.From experimental observations at α ¼ 0 (horizontal cylinder), U 0 ¼ 0, since there is no flow along the cylinder, due to stream detachment.Conversely, for α ¼ 90 (vertical cylinder), U 0 ≈ U n .Considering these conditions, we assume that U 0 ≈ U n sinα at the point when the sub-streams rejoin, and flow starts along the z-direction.
For the experimental conditions shown in Figure 1a, the initial velocity can then be related to the flow rate, cross-sectional area of the injecting syringe nozzle (A n ), and the branch inclination, as: Given this approximation, the initial rivulet cross-sectional area can be estimated as: We estimate the initial dimensionless cross-sectional area σ 0 for the rivulet then scales as: Finally, from Figure 1c, we can compare this estimated σ 0 to σ c ¼ 1:54, to determine if the rivulet has the cross-sectional stability required to form.
Graphically, this model is displayed by the blue line in Figure 9a, representing the 'predicted' rivulet formation rate.This distribution was generated by first using Equation ( 10), where σ 0 ≈ 1:54 was used to determine the initial rivulet area A 0 as a function of α.Then, using Equation ( 9), a 'critical minimum initial velocity' (U 0 c ð Þ ), was calculated for each α, Q.Then, we estimated U 0 from Equation (8) to compare with U 0 c ð Þ .As a minimalistic model, we assume that the variation in U 0 might scale with U 0 itself, and we characterize this variability as Gaussian noise.Consequently, we estimate that the variability in U 0 has the SD defined by SD ¼ rU 0 .
Using a mean squared error method to best fit the experimental rivulet formation rate data, we obtained r ≈ 0:35, yielding a standard deviation of AE35% of U 0 .Based on this distribution, a cumulative probability function was generated representing the probability that ð Þ , resulting in σ < σ c , which is the condition for a stable rivulet to form.Note that the predicted rivulet formation rate curve is independent of Q, like Figure 2.
T A B L E 5 Key classifications of initial stream behaviour.

Phenomenon Abbreviation Classification
No join NJ If sub-streams do not re-join, the trial is rejected but if the streams reconnect within 3 s after injection, the trial is accepted as valid

| Rivulet height estimation experiments
In a separate experiment, a camera with an attached macro lens was positioned to video record the rivulet from the side at z ¼ 10 and z ¼ 50 cm.
A 0.5 mm-increment ruler was included in the frame beneath the rivulet.In each recording, the footage was paused at instants when the rivulet height was consistent (e.g., Figure 5a) and the rivulet height estimated using the ruler.Where the rivulet height fluctuated, an average of the maximum and minimum heights was reported.Rivulet heights were measured using α ¼ 8 ,9 ,12 ,20 with flow rates of Q ¼ 1:25,1:67,2:50 mL/s.The rivulet height, and thus λ, below α < 8 could not be captured due to fluctuations caused by flow instabilities.
Between 5 and 10 images were averaged to provide an estimate of rivulet height.The rivulet height was nondimensionalized to obtain λ.
Measuring λ at different lengths along the cylinder allowed us to test the hypothesis that an accelerating stream corresponds to decrease in λ, as predicted by our theoretical analysis.

| Dye experiments for velocity estimation
We used dye tracer experiments to determine if the rivulets were approaching a steady velocity were accelerating along the entire length of the straight cylinder.Dye was injected into rivulets formed for α ¼ 8 ,9 ,12 and Q ¼ 1:67 mL/s a few cm below the point of injection.The dye stream was filmed in slow motion from the side.
The velocity of the dye was estimated by identifying the time when it passed 5 cm increments marked along the cylinder (see Figure 5b).
We were able to make these measurements in the region between 15 and 40 cm along the cylinder.An average velocity for each 5 cm interval in this region was obtained from 5 trials for each condition.

| Flow below curved cylinders
The theoretical analysis suggested that pour point formation from a previously stable rivulet would require a change in conditions to destabilize the rivulet.One way to introduce such a change is to introduce curvature into the cylinder.If the curvature caused a steep branch angle to become shallow, the rivulet would decelerate, causing λ to increase and the rivulet to become less stable.
Two 1 m, 42.5 mm diameter PVC cylinders were bent by sealing an end of the cylinder, pouring boiling water into the other end, and then bending the hot cylinder.This produced a very gradual, continuous curve.We measure the curvature simply based on the difference in inclination between the extremities of the cylinder, Δα.We made two trials with Δα ¼ 13 and 20 (see Figure 5c).We then held the volumetric flow rate constant at Q ¼ 1:67 mL/s, and varied the inclination of the curved cylinders.We report the inclination of the downstream limb, α 2 (Table 6).We reported the pour point formation rate from an average of 20-25 trials for each cylinder and each α 2 .

| RESULTS
The introduction of water from the syringe pump and the possible outcomes are illustrated in Figure 6.After injection, the flow split into streams, which flowed along both sides of the cylinder (a).Typically, there was only one stream on either side of the cylinder (b), although sometimes the streams split (c) and two or more streams flowed along one side of the cylinder.Where only one stream formed on either side (b), they either re-joined at the bottom of the cylinder (d), forming a rivulet if cylinder inclination was sufficient (i), or remained separate (e).Bifurcated streams (c, f) often re-joined (h) and connected to form a rivulet (i).If the streams did not re-join after 3 s (e), the trial was deemed invalid, and usually no rivulet formed (g).Sometimes, however, a rivulet was formed from a single stream, which had not rejoined.However, as the flow rate of such rivulets was unknown, these trials were also considered invalid.
Other flow phenomena appeared after rivulet formation.For lower α and higher Q values, waves and meandering (Figure 7) were produced.These phenomena slowed the bulk flow rate, increasing

| Rivulet flow simulation experimental results
Rivulet formation beneath the straight cylinder was most dependent on the inclination of the cylinder, as illustrated in Figure 9a.As the inclination α increased, the fraction of trials forming rivulets also increased, until it was 100% for angles > 10 .Angles between 4 ≤ α ≤ 10 showed a transitional behaviour for rivulet formation.At the site of where the rivulet was initiated, a large bulge was produced in the stream before the height stabilized (Figure 8).This was most noticeable for rivulets produced at lower inclinations.It may be that the stability of this initial bulge is important in determining the formation of a rivulet.
The rivulet formation rate distribution predicted by the initial velocity estimate (Section 3.1.1)fitted the experimental data well.It was consistent with our hypothesis that rivulet formation would depend on the stability of the initial flow, governed by a velocity affected by cylinder angle and the injection conditions.Experimental variation in these conditions-potentially caused by small changes in cylinder angle or fluctuations in the injected stream velocity and cross-section-would cause the sharp transition predicted by theory to adopt the observed sigmoidal shape.
Whilst the influence of α on rivulet formation is apparent in Figure 9a, flow rate does not show a clear relationship to rivulet formation.This observation is consistent with Equation ( 10), which indicates that the initial rivulet stability is independent of the flow rate.Figure 9b The stemflow formation rate (as a percentage of trials), which is slightly lower than the rivulet formation for all flow rates.

Figure 9c
The stemflow volume collected (as a percentage of the total injected volume).
The rate of stemflow formation as a function of flow rate and angle is shown in Figure 9b.This figure shows that for a given cylinder angle and flow rate, the stemflow formation rate was slightly lower than the rivulet formation rate, especially at low inclination.These differences were not because of pour points, as no visible pour points formed in any of the straight cylinder experimental trials.Instead, some trials produced a rivulet but did not form stemflow since the propagation of the rivulet was too slow to reach the end of the cylinder within the duration of the trial, either due to a low inclination angle, or due to dispersed losses of water along the cylinder-for example as isolated drips caused by flow instabilities, in contrast to concentrated water loss in a pour point or stemflow.
Stemflow formation slightly varied with Q, with stemflow formation rates (at a given angle) decreasing for high flow rates, as seen in Figure 9b.However, there is a more noticeable variation between flow rate and stemflow volume (Figure 9c).This was due to increased meandering and wave formation at higher flow rates, which encouraged drop fall-off and reduced the propagation velocity of the bulk stream.Consequently, for higher flow rates, there fewer trials reached the end of the cylinder, and rivulets which formed stemflow transported a smaller proportion of the injected flow.
F I G U R E 6 Some typical flow phenomena associated with rivulet formation.Injected water splits and flows along each side of the cylinder (a, b), sometimes in complex flow patterns (c, f).
Where the split flows re-join below the cylinder, a rivulet can form (d, h, i) if cylinder inclination is sufficient, however if the split flow does not re-join beneath the cylinder (e) the rivulet either does not form, or forms from only part of the flow, resulting in an invalid trial (g).
F I G U R E 7 Development of rivulet flow instabilities at lower α and higher Q values.

| Height and velocity estimation using dyes and photographs
The dye experiments showed that velocity was constantly increasing along cylinder length within the measured region, as shown in Figure 10a.Measurements of the rivulet height at z ¼ 10 and z ¼ 50 cm indicated that λ decreased with distance along the cylinder for a flow rate of Q ¼ 2:5 mL/s (Figure 10b).The experimental data followed a similar shape to the theoretical curves when α varied, lying between the theoretical limits of the α-λ curve predicted by initial velocity estimate (10) at z ¼ 0 cm and the asymptotic α-λ model of Kuibin (1996) (at z ¼ z SV ).The average λ at z ¼ 50 cm remained larger than predicted by Kuibin (1996), which is consistent with the dye experiment observations that flow was continuing to accelerate at distance of z ¼ 50 cm down the cylinder.We were unable to experimentally verify the strongly nonlinear α-λ relation predicted for low inclination angles, as height was difficult to measure due to flow instabilities and less frequent rivulet formation.We also found that λ at a given α and location z increased with increasing Q (full data not shown in Figure 10b), as predicted by Kuibin (1996).

| Curved cylinder results
We were able to generate pour points on curved cylinders provided that α 1 > α 2 .Pour point locations initially migrated, but then became fixed during the trial (Figure 11).Pour points were generated independently of meandering and waves (Figure 7).
Pour point formation was maximized when α 2 was minimized.
Pour points formed at the expense of stemflow, so that pour point formation was inversely related to stemflow formation and volume.
Pour points did not form when α 2 ≥ 7 .For α 2 ≤ 1 , both curved cylinders always produced a pour point.The 20 curved cylinder, with a sharper bend than the 13 curved cylinder, had a more gradual transition between 0% and 100% pour point formation with decreasing α 2 (Figure 12).
The location of the pour point varied with the cylinder inclination, and was similar for repeated trials with a fixed α 2 .As α 2 declined towards a flat cylinder, the pour point location moved upstream from near the end of the cylinder to near to the bend, presumably because α 1 was lower, resulting a lower rivulet velocity before the bend and less deceleration needed to destabilise the rivulet after the bend.
F I G U R E 8 Initial bulge observed at the site of rivulet formation.
F I G U R E 9 (a) Rivulet formation rate (as a % of trials) at different α and Q.The fitted blue line shows the predicted fraction of stable rivulets based on Equations ( 8)-( 10) assuming that U 0 is normally distributed with a standard deviation of 0:35 Â U 0 .
5.1 | What are the principles governing the partitioning of a rivulet?
The theoretical framework and experimental results identified the height of the stream (λ) as the key variable which determines the local attachment or detachment of the rivulet.Under steady state velocity conditions, height may be related to the stream discharge and velocity (Kuibin, 1996).The cross-sectional profile developed by Pitts (1973) enables height to be related to the cross-sectional area (σ) of the rivulet.Photographic evidence shows a decrease in the rivulet height along the length of the branch.Additionally, once a rivulet formed, it continued to flow down a straight branch without detaching.This supports the principle that a stream flowing below a 'critical height' will remain stable and will not detach from the branch.
Dye experiments showed that as the height of the rivulet decreased along the branch, the velocity of the rivulet stream increased.Assuming that the mass flow rate of the stream is conserved from the point of rivulet formation (in absence of significant drop fall-off), an increase in the stream velocity must be associated with a lower cross-sectional area of the rivulet along the branch.
A lower cross-sectional area and thereby lower rivulet height, theoretically improves the stability of the stream.In principle, the stability of a rivulet at a certain distance may be related to its local stream velocity.
Since rivulet stability increases along the length of a straight branch, if detachment is to occur, it will happen at the most upstream point of the branch section, or with the introduction of more water.In our experiment, this point was directly below the injection site where the initial streams re-joined (at length z ¼ 0 cm).
Due to insufficient theoretical development and complex phenomena underpinning the stability of the re-joining region, such as chaotic stream movement (Figure 6) and the noticeable bulge (Figure 8), we made the simplistic assumption that the initial stream velocity after re-joining (U 0 ) was equal to the component of injected velocity aligned with the branch axis.Although the initial velocity estimate in Equation ( 10) was a simplistic assumption used for this experimental setup, the rivulet formation rate results indicate that in principle, U 0 may scale with α.
These principles can be directly applied to the understanding of flow within the canopy.For straight branch sections, an increase in inclination angle increased the likelihood of a rivulet forming, and there was a transition range of roughly 5 observed where a small change in α had a significant effect on the likelihood of whether a rivulet would form.The irrigation flow rate Q was not influential on the rivulet formation rate and thus did not shift the α transition range.
However, it was found that at higher flow rates, there was less stemflow volume collected due to prominent meandering and wave flow.
There was also a corresponding increase in the average λ for an increasing flow rate for a given inclination and length along the branch.On real canopies, the confluence of streams from converging leaf and branch stems may contribute to a sudden increase in the flow rate of the combined rivulet.
Although not observed on straight branches, pour points could be formed by reducing the cylinder angle through curvature.For sufficiently low angles, a rivulet that was previously stable may decelerate as the component of gravitational acceleration along z is reduced, and the viscous effects of friction dominate.For steady flows, any deceleration will increase the rivulet cross-section.Our observations remain consistent with the hypothesis that this increase causes λ > λ c , causing the rivulet to become unstable, to detach, and a pour point to form.
In this study, however, due to instrumental limitations, there was no stream velocity or λ estimations at different lengths along the curved cylinder, which would test the hypothesis that an unstable cross-sectional profile, governed by λ c , is the cause of a pour point detachment.Also, whilst deceleration of the stream is a possible explanation for producing a pour point on PVC, where the cylinder is bent smoothly with a large radius of curvature, branches exist with varying radii of curvature; some with sharp 'kinks'.For branches with a sharp, instantaneous change in inclination, even stable, fastmoving streams may detach as the surface tension is unable to withstand the sudden change in momentum of the stream.Although simplistic, the idealized experiments of this study allow us to infer that pour point formation on real canopies must be associated with changes to flow conditions-for example additional inputs of water-or changes in the homogeneity of the branch, such as the changes in branch angle tested here, or potential changes in surface properties, such as roughness or wetness.Separate observations made in the field and under rainfall simulators (Kunadi et al., 2024) suggest that pour points commonly form at the point where several branches converge (increasing the flow rate of the rivulet), where a branch orientation moves from steeply angled to flatter, and, on initially dry canopies, at transition points from wet to dry branch surfaces.These canopy features are all visually identifiable, suggesting that it may be possible to predict the formation of pour points, and estimate stemflow/throughfall partitioning from canopy properties.
For example, our findings suggest that rivulet partitioning may vary on plagiotropic (typically straight, horizontal shoots) and orthotropic branches (that typically curve vertically upward from the trunk) (Hallé et al., 2012), which could be a starting point for identifying pour point and stemflow behaviour in the field.

| Study limitations
The use of straight and curved PVC cylinders to represent the processes of rivulet flow and detachment beneath branches offered the advantage of control, reproducibility and replication, but at the expense of making several idealisations.Specifically, the wettability of the cylinder was very low due to the hydrophobicity of PVC, the micro-scale roughness (e.g., roughness on scales smaller than the depth of flow) of the cylinder was very low, and nor did cylinders The pour point formation rate at secondary inclinations for various cylinders used.
display any macro-scale roughness (e.g., roughness elements larger than the scales of flow).Real branches, conversely, often exhibit roughness on many scales, and exhibit a range of wettabilities, and storage capacities.Additionally, the injection of water at a single point at the upstream end of the branch is a discrete representation of water arriving continuously along the branch, as would be expected during a rainfall event.
Although the experiments simplify the features of branches and rainfall in this way, the theory that was developed and tested can accommodate several of these more realistic aspects with parameter changes or simple elaborations.For example, the effects of changes in surface wettability on rivulet stability can be explored by altering the capillary length constant and contact angle.Changing wettability may also capture some important effects of surface micro-scale roughness.
For example, rough bark surfaces can be non-wetting due to the difficulty of displacing air from within bark cracks and fissures (Cassie & Baxter, 1944;Petric & Oven, 2015).Similarly, whilst the experiments and tests addressed water input from a single point, extending this to a spatially continuous input of water, such as might be provided by rainfall, requires only coupling the current theoretical framework to a flow accumulation model.The stability dynamics revealed will be unchanged, even if the way that they vary along a rivulet's flow path will shift with progressive, downstream increases in flow volume.
One way to contextualize the contribution of this study is to consider it as a step forward in bridging knowledge of fundamental rivulet hydrodynamics (Kuibin, 1996;Towell & Rothfeld, 1966) with rainfall simulation and field studies describing water drainage along branches (Dunkerley, 2014;Herwitz, 1987;Levia & Herwitz, 2002;Van Elewijck, 1989).Figure 13 illustrates the idea of this bridge, where this study aims to extend the 'zone understood by theory' from a focus on planar, flat surfaces, to an idealization that isolates the effect of α,

(
marked with 5 cm dot increments) with a diameter of 42.5 mm to represent a branch, and injecting 25 mL of water vertically onto this cylinder using the Hamilton Microlab 500 Series syringe pump.We verified that the formation of stemflow was largely insensitive to the point at the top of the pipe where the water was injected onto the cylinder, and held this point constant at z ¼ 50 cm for all straight cylinder simulation experiments.The experimental apparatus is shown in Figure4.3.1 | Idealized rivulet flow simulation experiments (straight cylinder)In the first experiment, we used a straight PVC cylinder (with a constant inclination angle of α) to represent a uniform branch section, and varied α and volumetric flow rate Q as outlined in Table3.Cylinder inclinations were measured with a laser inclinometer.Flow rates exceeding 2.50 mL/s caused splash and spray at the cylinder surface, whilst the lowest flow rate the syringe pump could supply was 1.25 mL/s.The following definitions inTable 4 are used to classify the key behaviours in experiments.Not all experimental trials produced a 'valid' outcome.Vertical injection of water at the top surface of the cylinder typically caused the flow to split into a stream flowing along each side of the cylinder.Where these streams re-joined to make a single rivulet beneath the cylinder, a valid trial resulted.However, a broader range of phenomena was observed (described in greater detail in F I G U R E 3 Field observation of a pour point (Kunadi et al., 2024) and idealized lab observation of a pour point.F I G U R E 4 Experimental setup.(1) Syringe pump, (2) reservoir beaker, (3) glass rod to stabilize nozzle, (4) nozzle from which water is injected downwards, (5) PVC cylinder, (6) bucket for collecting falling liquid, (7) scale for measuring stemflow mass and (8) collection beaker.T A B L E 3 Flow conditions tested in straight cylinder simulation experiments.Angle α3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 15    Flow rate Q 1.25, 1.67, 2.08, 2.50 mL/s is at least 5 cm long (estimated by 5 cm dot increments), if not visibly moving (c) OR if satisfies (a) or (b) and rivulet stops, but re-starts again to satisfy (a) or (b) during the trial Pour point A stationary location where the rivulet detaches from the surface of the cylinder before reaching the end of the cylinder Stemflow A rivulet which has reached the end of the cylinder and poured any volume into the collection beaker

5
Various experiments conducted.(a) Side view of rivulet captured for height estimation, edited to isolate the light rivulet layer against the darker PVC.(b) Dye experiment for α ¼ 12 .(c) Identical PVC cylinders were curved by filling with boiling water and manually bending.T A B L E 6 Angle range tested in curved cylinder simulation experiments.incidental dripping along the cylinder, and reduced the stemflow volume collected.

F
I G U R E 1 0 (a) Estimated stream velocity as inferred from dye experiments (at a flow rate of Q = 1.67 mL/s), and (b) theoretical and experimental λ's at different lengths along the cylinder and different α (at a flow rate of Q = 2.5 mL/s).F I G U R E 1 1 Pour point observed on the 20 curved cylinder inclined at α 2 ¼ 1 .

5. 2 |
What are the implications of rivulet partitioning on the canopy water balance?The dimensionless rivulet height λ, identified as a theoretical rivulet stability metric, is derived from the capillary length L c (4), which is affected by the surface tension (γ) between the stream and the branch.With a varying branch uniformity and greater surface roughness, like observed on real canopies, λ may be a more volatile parameter in the time domain and along the length of the branch, than observed in our experiments.In the theoretical framework of our experiment, L c was held constant and γ was quite low in comparison to real woody surfaces.Discounting the effects of surface roughness and geometry, when γ and thus L c , is greater, λ is lower, and so the rivulet should become more stable.On plant canopies, the introduction of a stream to a branch generally occurs by the 'sessile' funnelling of water collected by leaves above the branch, unlike the simple injection done in our experiments.This process is complex, and could vary with different types and shapes of leaves, and with different branch structures(Ginebra- Solanellas et al., 2020;Wang et al., 2020).So, rivulet formation on natural canopies may not be traced to a critical initial velocity U 0 but rather a change in leaf or branch inclination, or a sudden increase in the flow rate.
on rivulet dynamics on cylinders.Future experiments that investigate more complex phenomena and a broader range of experimental controls can build on the present study to further extend theoretical understanding of canopy flows.The findings with idealized experiments highlighted the stability of rivulets on smooth, straight branches, suggesting that pour point formation is likely caused by structural or chemical changes in a branch.Whilst the experiment has not explored these changes, it has illuminated when pour points will not form, and why.Further experimentation on real branches is merited to broaden and contextualize the findings of the present study, with the ultimate goal of better identifying pour point locations, and improving the quantification of interception losses.Further experimentation could also explore phenomena beyond the purely hydrodynamic within rivulets.For example, the concentration of throughflow into rivulets and pour points has the potential to generate locally elevated concentration of solutes transported from leaf and branch surfaces(Tucker et al., 2020), and further study into solute fate and transport in rivulets would be a useful avenue for further research.FI G U R E 1 3 A summary of some rainfall partitioning studies, categorized by complexity of flow type and surface type used in experimental or theoretical analyses.6| CONCLUSIONThe mechanistic underpinning of canopy interception and water redistribution remains a challenging open question in hydrology.Mechanistic understanding of these processes may, however, enable the development of improved 'lumped' models at canopy or ecosystem scales to predict interception rates and losses.This experiment has pointed to the importance of the inclination of branch elements in the canopy in controlling the partitioning of intercepted water between concentrated stemflow and concentrated throughfall fluxes.There remain many further potential elaborations on the question of stemflow-pour point partitioning, including the implications of continuous water input to a rivulet along the flow path, changes in surface roughness and uniformity, changes in cylinder diameter, as well as discrete changes in water and branch inclination as might occur at branch confluences in tree canopies, or at wound sites or kinks on tree surfaces.Similar investigations into how rivulet flow is initiated on real leaf and branch surfaces, and into how such flows can be routed along a branch network would also be helpful.Whilst this study only begins to explore the hydrodynamics of canopy flows, it demonstrates that such flows are amenable to mechanistic investigation and treatment.Further linking theoretical and experimental fluid mechanical investigations with the processes of interception and canopy routing is likely to yield new insights into this still challenging aspect of land surface hydrology.
, yet to date, hydrodynamic studies of rivulet flow have not explicitly considered rivulet formation and detachment.Field observations suggested that rivulets and pour points are conspicuous on tree species with smooth, non-absorbant bark surfaces, such as the (Kunadi et al., 2024)s macrophylla), smooth-barked eucalyptus species, such as the lemon gum (Corymbia citriodora) and banksia species, such as the firewood and candle banksias (Banksia menziesii and B. attenuata)(Kunadi et al., 2024).These branches can be idealized as a Parameter values used in theoretical development.
T A B L E 1