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Waters of rivers flowing into confluences can create prominent visual patterns of mixing when the incoming flows differ in colour or turbidity. Such patterns provide visible evidence of abiotic processes that influence water quality and biodiversity in fluvial ecosystems1,2 and define chemical transformations of transported substances downstream of confluences3,4. Complex patterns of turbulent flows associated with mixing also have important implications for channel morphology5,6,7, benthic fauna8,9 and habitat use by fish10.

Contrasts in the turbidity of confluent rivers are often visible on widely available remote-sensing images11,12, and such contrasts provide the basis for identifying mixing interfaces characterized by interpenetration of highly turbid and less turbid waters (Fig. 1). In some instances, mixing interfaces are narrow near the junction apex and gradually increase in width over distance (Fig. 1a). In other cases, the mixing interface originates downstream of a broad zone of stagnant water near the confluence apex and expands downstream rapidly (Fig. 1b).

Fig. 1: Patterns of mixing at the confluence of the rivers Danube and Save at Belgrade, Serbia.
figure 1

a, In the mixing-layer mode with higher momentum water from the Danube, 30 September 2017. b, In the wake mode, showing mixing of waters with almost equal momentum and turbulent vortices arranged into an unsteady wake pattern, 22 May 2014. Location: 44° 49′ 54‘ N, 20° 26′ 56‘ E. Credit: satellite images, Landsat/Copernicus, Google Earth © 2022 Maxar Technologies.

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Mixing along a mixing interface between incoming flows with different velocities has been a focus of field studies13,14,15,16,17,18,19,20,21 and laboratory22,23,24,25,26,27,28 and numerical29,30,31,32,33 experiments, which have attempted to characterize this mixing using classic mixing-layer theory—a process of mixing referred to hereafter as mixing-layer mode34,35,36,37. Hydrodynamic theory34,35,37 provides an asymptotic solution for this mode by relating the width of the mixing interface to the velocity difference between merging flows. When the velocity difference is small near the confluence apex, the theory37 predicts that the layer should not grow. However, recent field and numerical studies show that, in contradiction to classic mixing-layer theory, mixing interfaces for confluent flows with small velocity differences often grow rapidly over distance18,31,38,39. Constantinescu et al.39 proposed a mode-switching hypothesis according to which the dynamics of mixing interfaces can change to a wake mode when the velocity difference between tributaries is small and the stagnation zone induces an internal velocity deficit that acts much like a wake behind an obstacle18,39,40,41,42. Although hydrodynamic theory provides an asymptotic solution for wakes37,40, a critical need exists for a theory-based model that can accommodate both mixing-layer and wake modes given that elements of these modes can potentially co-exist either at a confluence at different times or in different parts of a confluence at the same time. The development of such a model would also be valuable for providing predictions that could be evaluated by coupling ground-based monitoring and remote sensing of mixing over a broad range of spatial scales12.

In this Article, we introduce a theoretical framework for confluence mixing based on depth-averaged two-dimensional shallow-flow equations that include terms representing advective and turbulent fluxes of momentum as well as bed friction43. We show that lateral advective fluxes of momentum induced by helical motion driven by the planar geometry of confluences and by turbulence associated with lateral shear can greatly exceed the inhibiting effect of riverbed friction and thereby substantially enhance mixing at river confluences.

Theoretical framework and experimental tests

The theoretical framework developed in this study is based on the mixing-length approach37, but in contrast to conventional use of single-velocity and length scales, we focus on the dynamics of mixing interfaces simultaneously controlled by multiple external and internal velocity and length scales. The difference in the mean velocities of the incoming tributaries, ΔUm, is the external velocity scale driving dynamics in the mixing-layer mode. The velocity deficit, ΔUw, induced by the stagnation zone near the confluence apex is an internal velocity scale related to differences in velocity between the stagnation zone and adjacent freestream flows; it drives the dynamics of a wake imbedded into the interface. Flow shallowness, the ratio of flow width (B) to flow depth (H), represents the stabilizing effect of bed friction. The converging planar morphology of a confluence is characterized by the junction angle θ, which, in turn, is related to the radius of curvature R of the flows entering the confluence—an external length scale controlling lateral advective fluxes of momentum. Our theoretical framework accounts for interactions between these multiple scales to predict mixing-interface width δ—the integral length scale of the mixing interface.

The theoretical model is derived by integrating the two-dimensional shallow-flow equations of momentum in a steady river flow with a realistic schematization of confluence geometry (Supplementary Fig. 1). Lateral advective fluxes of momentum are represented by parameterized solutions for the dynamic components of the total pressure head (Supplementary equation (7)) using the analogy of back-to-back curved flows within the confluence (Supplementary equation (13)). Alternatively, these fluxes can be represented as additional advective coefficients of friction for the wake stability parameter complementing bed friction41 (Supplementary equation (9)). Because in river confluences advective fluxes greatly exceed turbulent fluxes18, they play a critical role in correct prediction of mixing.

Model predictions were tested using data obtained from field-based experiments. Most previous experimental research has examined flow structure in laboratory junctions with small B/H ratios and discordant channels5,23,24,27. Natural bed roughness and flow shallowness have not been previously considered in experimental contexts24,26,27,28 but are recognized as crucial for theoretical modelling of shallow flows34,35,36,40. The lack of suitable experimental data necessitated completion of a set of controlled field experiments with B/H ≥ 25 and natural riverbed roughness (Fig. 2 and Extended Data Figs. 1, 4 and 6).

Fig. 2: Patterns of mixing in experiments.
figure 2

a, Shallow mixing layer, Mr = 20. b, Shallow wake downstream of a circular obstruction with a diameter D = 2 m (rectangle is enlarged in d). c, Shallow mixing interface at 40° confluence, Mr = 2.2 (rectangle is enlarged in e). d, Patterns of large-scale lateral turbulence interacting with smaller-scale bed-friction-induced turbulence. Vertically oriented small vortices extract momentum from the large-scale lateral vortices and diffuse it in the ambient fluid through vortex stretching, which is visible through the feathering on edges of the large vortices. e, Pattern of mixing induced by streamwise-oriented vortical cells39. Lines 1–1ʹ and 2–2ʹ mark near-field boundaries; dashed lines are mixing-interface boundaries.

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Experiments for shallow mixing layers (ΔUw = 0) between parallel flows (R = 0) and shallow wakes (ΔUm ≈ 0, R = 0) in the lee of an obstacle allow rigorous testing of model predictions for interfaces with asymptotic behaviour, pure mixing-layer mode and pure wake mode (Fig. 2a,b and Supplementary Table 1). These experiments focus on the effects on mixing of riverbed friction in shallow flows—a well-understood context for interpreting deviations from expected outcomes34,35,40. In the experiments with angled tributary channels (θ = 40°and 70°, R > 0), we explore the dynamics of mixing interfaces in the intermodal state (ΔUm, ΔUw ≠ 0) for a wide range of momentum flux ratio values 2 ≤ Mr ≤ 100, where Mr = ρ1Q1U1/ρ2Q2U2, Q is the discharge, U is the flow velocity, ρ is the fluid density and subscripts 1 and 2 refer to each incoming flow42 (Fig. 2c and Supplementary Table 1). The experimental measurements provide the basis for comparing theoretical predictions with documented outcomes and with shallow-flow theories34,35,40 (Extended Data Figs. 3, 5a,c,d and 8c,d). The field-scale experiments are free of scaling effects present in most previous laboratory experiments22,23,24,25,26,27,28 and, contrary to many field studies17,18,38,44, were obtained in true experiments, in which certain controls were kept nearly constant while others were varied.

Asymptotic behaviour of a shallow mixing interface

Asymptotic behaviour for the theoretical framework corresponds to cases of free mixing layers and free wakes37 and, by adding scales for shallowness34,35,40, links the asymptotes to the cases of shallow mixing layers (Fig. 3a) and shallow wakes (Fig. 3b). Predicted lateral profiles of depth-averaged streamwise velocity u(y), where y is a lateral coordinate and Uc is the mean velocity in the mixing interface, match closely experimental data for the shallow mixing layer and shallow wake cases (Fig. 3a,b).

Fig. 3: Mixing interfaces with asymptotic behaviour: scales relations and comparison of predictions and measurements.
figure 3

a, Mixing layer scaled by hyperbolic tangent function37. b, Wake scaled by exponential function37. c, Mixing in mixing-layer mode (splitter plate; Mr = 20; dashed line is free mixing layer; solid lines are shallow mixing layer (blue, equation (7)), and accounting for lateral advection and turbulence (red, equation (6)). d, Mixing in wake mode (circular cylinder; Mr = 1; dashed line is a free wake; solid lines are shallow wake, equation (7) with (red) and without (blue) the effects of lateral advection and turbulence).

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At short distances (x < 10 m (x/H < 30, where x is a streamwise coordinate)), the mixing interface for the splitter-plate and wake experiments behaves as a free (deep) lateral shear flow, and experimental data collapse around the line of predicted values for such flows (dashed line) similar to findings of previous experimental34,35,37,40 and field studies36 (Fig. 3c,d and Extended Data Fig. 2). In both cases, the internal bed-generated boundary layer has not yet fully developed, and the effect of bed friction is minimal.

At long distances (x > 10 m), the data systematically deviate from the prediction for free shear flows (Fig. 3c,d), which agrees with results of previous studies33,34,35,40 and with flow visualizations (Fig. 2c). Moreover, model predictions that account for the effects of bed friction only systematically overestimate rates of mixing (Fig. 3c,d). The full model, which includes the effects of lateral fluxes of momentum by advection and turbulence (Extended Data Fig. 9a,b), yields predictions that correspond closely to measured experimental data on mixing-interface growth (Fig. 3c,d). Lateral fluxes of momentum reduce rates of downstream growth by increasing the exchange of momentum between the confluent flows, thereby reducing the velocity difference between these flows (Extended Data Figs. 3c, 5c and 8c).

Intermodal behaviour of mixing at angled confluences

Intermodal behaviour, or behaviour that includes mixing-layer and wake modes, is expected when a variety of scales substantially contribute to the dynamics of the mixing interfaces. In angled shallow confluences (R > 0), the stagnation zone near the confluence apex17,45 imposes a wake-like effect on the flow (ΔUm, ΔUw ≠ 0). To incorporate this effect into a multi-scale theoretical model, we introduce a lateral velocity profile in which the velocity scale ΔU is a sum of mixing-layer ΔUm = ϕmΔU and wake ΔUw = ϕwΔU contributions (ϕm + ϕw = 1). The shape of the composite velocity profile is determined by the ratio of external and internal velocity scales \(\varphi = \phi _{\mathrm{m}}/\phi _{\mathrm{w}}\), which provides a metric to characterize intermodal cases (Fig. 4a,b).

Fig. 4: Mixing interface with intermodal behaviour: scale relations and comparison of predictions and measurements.
figure 4

a, In mixing-layer mode with a small wake effect (solid blue line is equation (8); dashed line is hyperbolic tangent function37). b, In wake mode with a small mixing-layer effect. c, Mixing with a weak wake effect (40°, Mr = 5; solid red line is equation (6), red dashed line is free mixing layer, solid blue line is equation (7) and dashed blue line is free wake). d, Mixing with a weak mixing-layer effect (40°, Mr = 2, the same definitions for lines as in c).

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At short distances (x < 7 m (x/H < 20)), the mixing interface behaves as a free wake, which exceeds, even for a momentum flux ratio of Mr ≈ 5, the effect of the free mixing layer (Fig. 4c). For Mr = 2, the wake effect is quite strong, resulting in rapid initial growth of the mixing interface (Fig. 4d). Again, over these short distances from the junction apex, bed-friction effects are not pronounced (Extended Data Fig. 7). The predictions and patterns of experimental data also show how initial growth rates over distance associated with wake effects are much greater than those for the asymptotic mixing-layer case (Fig. 3c) and similar to those for the asymptotic wake case (Fig. 3d).

For x < 7 m, the mixing interface evolves in a wake mode that depends on Mr (Fig. 4c,d). The switch from wake mode to mixing-layer mode occurs when the velocity deficit associated with the wake becomes small (ϕw ≈ 0) and the mixing interface then behaves in a shallow mixing-layer mode. The distance from the apex at which this switch occurs determines whether the behaviour will be that of a free mixing layer or a shallow mixing layer, but in most cases it seems likely that this behaviour will be influenced to some extent by the stabilizing effects of lateral advective fluxes of momentum, turbulence and bed friction. Indeed, the experimental data conform well with predictions of the multi-scale model that includes these effects (Fig. 4c,d). Computation of the contributions of these different effects to exchange of momentum using data from the field experiments indicates that lateral advective fluxes contribute up to 70% of the exchange of momentum while only about 20% is contributed by bed friction—considerably less than previously indicated34,35,40 (Extended Data Figs. 7, 9 and 10).

Lateral advection and turbulence drive mixing at confluences

This study advances knowledge of the mixing process at river confluences by showing how this process can be understood theoretically and experimentally as consisting of two intrinsic modes of behaviour—wake mode and mixing-layer mode. We demonstrate that intermodal behaviour, involving switching from one mode of behaviour to the other, is the generic mechanism governing dynamics of the mixing interface at confluences. Moreover, advective effects, particularly those associated with streamline curvature caused by the angled planform geometry of confluences and the need for incoming flows to reorient in direction, play an important role in momentum exchange that drives these dynamics.

The study provides the theoretical framework that consistently addresses findings of field research and numerical modelling of natural confluences, which has emphasized strong wake-like flow associated with flow stagnation and has shown that advective lateral fluxes of momentum can greatly exceed effects of shear-driven turbulence and bed friction18,36,41 (Extended Data Fig. 9). By incorporating a metric that accounts for wake and mixing-layer effects on the transverse pattern of depth-averaged velocity within a confluence, our theoretical approach introduces a schematization that couples the effects of momentum ratio with wake effects, thereby accommodating a mode-switching mechanism to account for a wide range of intermodal growth dynamics of the mixing interface. The study shows clearly that treating confluence flows as similar to flow downstream of a thin splitter plate separating parallel flows34,35,36 is overly simplistic because it does not consider the effects of flow stagnation and curvature within confluences.

For testing of our theoretical model, we draw on a dataset obtained in field experiments. These experiments examined the asymptotic and intermodal behaviour of the mixing interface in a shallow (B/H > 25), hydraulically rough open-channel Y-shaped confluence with a naturally formed gravel bed (H/d50 = 17) in a wide range of momentum ratios (2 < Mr < 100) for three junction angles (0°, 40° and 70°). Comparison of theoretical predictions with the results of field-based experiments shows that dynamics of mixing interface modelled using only bed friction systematically underestimate the stabilization effects related to lateral advective and turbulent fluxes of momentum—effects that are accommodated within our model (Fig. 3).

Comparison of experimental data with theoretical predictions shows that accounting for mode-switching mechanism together with the correct representation of lateral advective and turbulent fluxes of momentum provides accurate prediction of mixing with fast and slow rates (Fig. 4). These findings agree with the results reported for a large shallow river19 (B/H ≈ 470) that a fast rate of mixing (Mr = 1.8) is attributed to the development of channel-scale lateral circulation that was not observed at a slow mixing rate (Mr = 3.6). The results agree also with studies on small rivers15,17,21 reporting faster mixing than would be expected from conventional transverse diffusion models14,46 that relate transverse mixing distance Ly ≈ 0.18UB2/K to the effect of bed friction alone (for example, \(K = 0.42HU_ \ast\), where \(K\) is the dispersion coefficient and \(U_ \ast\) is the shear velocity).

The theoretical framework highlights the value of near-field processes in the vicinity of the confluence apex on the mixing in the far field because at short distances the wake sets the initial width of the interface in a weak mixing-layer mode (Fig. 4c,d). Although the stagnation zone has been recognized as an important element in conceptual models6,22,45,47, it remains comparatively less studied, and our work suggests that understanding of stagnation-zone hydrodynamics can substantially improve predictions of mixing. Furthermore, our theoretical framework provides promising opportunities for predicting rates of mixing from remotely sensed images and should enhance capabilities to analyse mixing on the basis of geographical information systems12.

Although our theoretical framework captures many relevant aspects of mixing, additional factors require further experimental and theoretical assessment. Past work has shown that flow at confluences may not be characterized only by wake and mixing-layer behaviour18 but also by jet-like behaviour44, which may be especially pronounced in confluences with discordant beds15,16,23. Cross-flow driven by density differences may also be important where incoming flows differ greatly in temperature, salinity or suspended load20,48. Unravelling the importance of these additional effects requires a firm understanding of how advective and diffusive processes affect mixing under less complicated conditions, and the present study sheds new light on how mixing occurs under such conditions.

Methods

We consider that the two confluent flows are shallow (B/H > 10), concordant and steady. Theoretical analysis focuses on the post-junction shallow flow (B/H > 20) downstream of the junction apex (Fig. 2a–c, lines 1–1ʹ). The coordinate system consists of a streamwise axis x with the origin at the downstream edge of a junction apex, a lateral coordinate y with the origin at the centreline and a vertical coordinate z with the origin at the riverbed and directed upwards.

Following the mixing-length approach, the dynamics of the mixing interface are modelled as37

$$\frac{{{{{\mathrm{d}}}}\,\delta }}{{{{{\mathrm{d}}}}\,x}} = \alpha \frac{{{\Delta}U}}{{U_{\mathrm{c}}}},$$
(1)

where α is the spreading coefficient. The average values of the spreading coefficient are 0.1135,36,37 in free (for example, away from boundaries/deep flow) mixing layers and 0.08 in a plane wake of a cylinder37.

A single-scale approach assumes that the velocity differential is constant along the mixing layer (ΔU/Uc = const) and that it decays over the distance in a plane wake as ΔU/Uc ≈ x–1/2. Integration of equation (1) yields a linear growth of δ ≈ x for free mixing layers and δ ≈ x1/2 for free plane wakes.

In mixing interfaces affected by multiple processes and scales, the velocity differential changes due to additional exchange of momentum caused by external (flow curvature) and internal (riverbed roughness) processes and scales. This exchange of momentum directly affects the dynamics of the mixing interface. Specifically in shallow flows, the growth of the mixing interface is suppressed by friction on a solid boundary34,35,40.

Two velocity difference functions are obtained by integrating the two-dimensional shallow-water equations (Supplementary equations (1–9))

$${\Delta}U = {\Delta}U_0 - \beta x,\;{{{\mathrm{and}}}}$$
(2)
$${\Delta}U = {\Delta}U_0\;\exp \left( { - \lambda x} \right),$$
(3)

where β and λ are decay rates of velocity difference and ΔU0 = ΔU (x = 0). In our multi-scale theoretical framework, the velocity decay rates are functions of flow bathymetry, curvature and bed friction. In the theory of shallow flows34,35,40, the effects of flow bathymetry and curvature are neglected, and conventionally, only equation (3) is used with λ representing bed friction as

$$\lambda = c_{\mathrm{f}}/H$$
(4)

in shallow mixing layer35,36, and

$$\lambda = 0.5\;C_{\mathrm{w}}/D$$
(5)

in shallow wake40, where cf is the bed-friction factor, Cw is the wake stability parameter and D is the diameter of the obstacle producing wake. The wake stability parameter is also obtained for shallow wakes considering only bed friction40. In our study, we demonstrate that in equations (2)-(5), the decay rates λ should be calculated using the total resistance factor ct, which includes contributions of lateral advective and turbulent fluxes of momentum, bed friction and pressure difference terms (Supplementary equation (9)).

Although equations (2) and (3) describe the same processes, their integration yields different solutions for the mixing interface width

$$\delta \left( x \right) = \alpha \frac{{{\Delta}U_0}}{{U_{c_0}}}x\left( {1 - \frac{\beta }{{2{\Delta}U_0}}x} \right) + \delta _0,$$
(6)

and

$$\delta \left( x \right) = \alpha \frac{{{\Delta}U_0}}{{U_{c_0}}}\lambda ^{ - 1}\left( {1 - \exp \left( { - \lambda x} \right)} \right) + \delta _0,$$
(7)

where δ0 is the initial thickness of the mixing interface and the constant of integration. These two equations generally provide the same solutions for certain flow conditions, but their specific use can have some advantages and disadvantages. The parabolic solution (equation (6)) predicts both growth and decay of the mixing interface when velocity difference becomes small, while the exponential function (equation (7)) predicts only asymptotic infinite growth of the mixing interface. This difference has important implications for predicting mixing far away from the confluence and predictions of complete mixing. Equation (2) can also predict an increase of velocity difference where the river bottom is steeply sloping, as for example at discordant confluences. However, the consideration of such cases is beyond the scope of the present study. Equation (7) has the advantage that it universally applies to the mixing interface in both mixing-layer and wake modes, and the change of mode requires only a change in parameters (Fig. 4c,d). Furthermore, parameterization of λ in our model provides a theoretical background for previous empirical classification of shallow wakes40. This classification shows that a shallow wake behaves as (1) vortex street for Cw < 0.2, (2) unsteady wake for 0.2 < Cw < 0.5 and (3) steady wake for Cw > 0.5. We demonstrate that Cw increases with increasing lateral advective and turbulent fluxes of momentum (Extended Data Fig. 9a–c).

For intermodal behaviour of the mixing interface, we propose a semi-empirical model for a lateral depth-averaged profile of mean streamwise velocity

$$\frac{{u - U_{\mathrm{c}}}}{{{\Delta}U}} = \frac{1}{2}\phi _{\mathrm{m}}\tanh \zeta - \phi _{\mathrm{w}}^{1/2}\exp \left( { - 4\ln 2\,\zeta ^2} \right),\;\phi _{\mathrm{m}} + \phi _{\mathrm{w}} = 1,\;\xi = 2\left( {y - y_{\mathrm{c}}} \right)/\delta ,$$
(8)

which is based on hyperbolic tangent and exponential solutions of mixing layers and wakes37. Equation (8) assumes that the wake is imbedded into the mixing layer with external velocities matching velocities of merging tributaries and that the velocity difference is composed of mixing-layer mode and wake contributions (Fig. 4a,b). Comparison of equation (8) with measured profiles in our experimental studies shows a good agreement (Extended Data Fig. 8a,b).

The field experiments reported in this study were completed in a side branch of the Tagliamento River in northeastern Italy in a relatively straight and shallow part of the channel located at 46° 12ʹ 9.45ʺ N, 12° 58ʹ 14.55ʺ E. This river reach is characterized by a stable hydraulic regime during the summer period when the flow is 30–40 cm deep and 25–30 m wide. Bed material, which is mostly immobile during summer, ranges from medium (d16 = 14 mm) to coarse (d84 = 38.7 mm) gravel with d50 = 22.5 mm. Experiments with shallow mixing layers and angled confluences were carried out during June–August 201549. The study of shallow wakes was performed during June–September 201950. In the experiments with shallow mixing layers and shallow wakes, straight 40 and 50 m long and 10 m wide rectangular in-stream flumes were built in the central part of the side branch (Supplementary Table 1). The models of angled confluences consisted of symmetrical Y-shaped junctions with 5 m wide and 10 m long tributary channels. Experiments were conducted for models with junction angles of 0°, 40° and 70° with Mr ranging from 2 to 100 (Supplementary Table 1). Confluence angles of 45° are characteristic of natural junctions in arid landscapes while 72° angles are typical of junctions in humid environments42.

The experimental programme consisted of detailed measurements of the instantaneous flow velocities, topography of the free surface and visualization of flow patterns. Velocity measurements were performed with arrays of five (mixing layers) to seven (wakes) acoustic Doppler velocimeters (Vectrino +, Nortek AS). Surveys of the free surface topography were carried out with a laser total station Elta 55 (Zeiss). Visualizations of flow patterns were performed using uranine (green) and rhodamine (red) dyes (Fig. 2) and recorded with Phantom II and Mavic Pro quadrocopters. Velocities were collected at sampling frequencies of 10 and 25 Hz during the periods of 240 s on high-resolution spatial mesh of sampling locations (10–28 lateral transects with 10–14 sampling locations). Topographical surveys of the free surface were completed at short ranges (less than 30 m), enabling the vertical precision of elevation measurements within 1 mm. Visual patterns of dye tracers were recorded at the altitude of 40 m with a resolution of 2,704 × 1,520 pixels per frame at a rate of 25 frames per second for a period of about 10 min. Post-processing of velocity measurements was performed with the software package ExploreV (Nortek AS). The topographical surveys were post-processed to obtain the spatial maps of free surface topography. Post-processing of video records included rectification of images and geo-referencing to the local system of coordinates using the benchmarks on the walls of in-stream flumes. Further detailed information on the experiments and data post-processing is available from our recent publications49,50.