Channel geometry in tidally influenced river deltas can show a mixed scaling behavior between that of river and tidal channel networks, as the channel forming discharge is both of river and tidal origin. We present a method of analysis to quantify the tidal signature on delta morphology, by extending the hydraulic geometry concept originally developed for river channel networks to distributary channels subject to tides. Based on results from bathymetric surveys, a systematic analysis is made of the distributary channels in the Mahakam Delta (East Kalimantan, Indonesia). Results from a finite element numerical model are used to analyze the spatial variation of river and tidal discharges throughout the delta. The channel geometry of the fluvial distributary network scales with bifurcation order, until about halfway the radial distance from the delta apex to the sea. In the seaward part of the delta, distributary channels resemble funnel shaped estuarine channels. The break in morphology, which splits the delta into river- and tide-dominated parts, coincides with a break in the ratio between tidal to fluvial discharges. Downstream hydraulic geometry exponents of the cross-sectional area show a transition from the landward part to the seaward part of the delta. The numerical simulations show that the tidal impact on river discharge division at bifurcations increases with the bifurcation order, and that the variation of river discharge throughout the network is largely affected by the tides. The tidal influence is reflected by the systematic variation of downstream hydraulic geometry exponents.
 Hydraulic geometry (HG) is a set of empirically derived power law relations between the channels' width, mean depth, and mean flow velocity, and the discharge conveyed by the channel [Leopold and Maddock, 1953]. In river deltas, channel geometry scales according to the downstream HG relation logA ∼ β logQ, where Ais channel cross-sectional area,Q is water discharge conveyed by the channel, and β is an exponent typically lying in between 0.8 and 1.2 [Edmonds and Slingerland, 2007]. In tidal systems, the exponent β often shows the same range of variation, but the tidal prism or peak tidal discharge is used instead of a discharge with a constant frequency of exceedance [Friedrichs, 1995; Rinaldo et al., 1999; D'Alpaos et al., 2010]. Channel geometry in tidally influenced river deltas can show a mixed scaling behavior between that of river and tidal channel networks, as the channel forming discharge is both of river and tidal origin. As a consequence, tidal processes play a prominent role in the morphological evolution of tidally influenced river deltas [Geleynse et al., 2011], which may also have an impact on the response of the delta to permanent changes in river discharge [Edmonds et al., 2010] and ultimately on its evolutionary structure [Wolinsky et al., 2010]. Although tidal effects on delta morphology can be studied by adopting a process-based morphodynamic modeling approach [van der Wegen et al., 2011; Geleynse et al., 2011], studies on HG relations may help to acquire a synoptic insight into the morphology of delta channel networks affected by tides, and provide the basis for idealized models of delta evolution [e.g., Kim et al., 2009].
 In this paper we show that the traditional tool of hydraulic geometry can be used to map tidal hydrodynamic processes onto a delta network, which can bridge the gap between physical oceanographic research on tides and the geological literature on river deltas. Traditionally, researchers use tidal amplitude or tidal prism to quantify the tidal influence on a delta [e.g., Syvitski and Saito, 2007]. However, to asses the degree in which channel morphology is influenced by the tidal motion throughout a delta, which may be used to improve the common classification based on the tripartite division between river-, wave-, and tidally dominated deltas [Galloway, 1975], a more objective and reliable set of metrics is required. Hydraulic geometry provides a set of relations for describing the dominant controls over delta channels synoptically, linking explicitly process and form.
 Tidal rivers are intrinsically complex, as tidal propagation is influenced by river discharge and vice versa. Tidal waves propagating upstream become distorted and damped, which is caused both by bottom friction and by the river flow [Godin, 1999; Horrevoets et al., 2004; Buschman et al., 2009]. Adopting the hydrological perspective, tides impact the river flow by inducing fortnightly variations, which are generated by variation of the tidally averaged friction over a spring-neap cycle. At spring tide, high levels of tidally averaged friction act to block the river discharge, increasing water depth and allowing discharge waves to be admitted during neap tide. These effects of the spring-neap cycle can extend far upstream from the estuary [LeBlond, 1979; Godin, 1991].
 In tidal rivers, the along-channel tidally averaged friction is mainly balanced by a subtidal pressure gradient [Buschman et al., 2009]. As a consequence, river-tide interaction induces a water level setup, which becomes progressively larger in the upstream direction [LeBlond, 1979; Godin and Martinez, 1994]. At a bifurcation, a mismatch may occur between the water level setups that would develop in the two channels if they would have been disconnected. The water level setup in the channel where river-tide interaction and the associated propensity for water level setup is largest will promote the allocation of river discharge to the other channel.
 Tides can affect river discharge division in one other way. The Stokes transport, which is the drift associated with a traveling wave that can be calculated as the Lagrangian mass transport minus the Eulerian mean, can be different in two adjacent channels that join at a bifurcation. Based on an idealized model, Buschman et al. found that asymmetries in subtidal flow division at the apex of a tidal river splitting over two sea-connected branches were enhanced when one of the sea-connected branches is deeper or shorter. In their study, bed roughness differences resulted in the opposing effect.Sassi et al. [2011b] elaborated on the work by Buschman et al. , showing that differences in water level setup may play a key role in the division of discharge at tidal junctions. The division of river discharge at tidal junctions leads to variation of the HG exponents, and helps to explain the complexity in HG relations of mixed river-tide dominated deltas.
 Studies of estuarine morphology tend to focus on overtide generation to describe the unique geomorphological characteristics of tidal environments [e.g., Wang et al., 1999]. This stems from the fact that landscape-forming discharges in tidal systems are highly influenced by peak discharges, which are typically controlled by tidal asymmetry [Lanzoni and Seminara, 2002; Fagherazzi et al., 2004]. Tidal asymmetry is generally attributed to non-linear interaction of the main semidiurnal tide with itself [Friedrichs and Aubrey, 1988]. Persistent asymmetrical tides may also be produced by the interaction of diurnal and semidiurnal constituents in tidal regimes where both semidiurnal and diurnal tides contribute significantly to the tidal motion [Hoitink et al., 2003]. Friedrichs  indicated that in stable channels the minimum shear stress τsnecessary to maintain a net zero gradient in the along-channel sediment transport leads to a convergence point, from whichτs decreases both seaward and landward. Downstream HG relations of the area of such channels will then exhibit exponents (β) greater than unity in flood-dominated systems, and values ofβsmaller than unity in ebb-dominated systems.Rinaldo et al.  suggested that ebb/flood transitions in tidal channels are marked by a break in the slope of the HG relation of the area, which may be interpreted as downstream variation in β. This notwithstanding, we argue that processes such as differential water level setup driven by river-tide interaction [Sassi et al., 2011b], which need weeks to develop rather than days, exert a strong control on the morphological evolution of tidally influenced river deltas, via the re-distribution of river discharge at bifurcations.
 Here we focus on the River Mahakam, which constitutes the major navigable river in East Kalimantan, Indonesia. The River Mahakam flows through a relatively flat basin characterized by a very mild slope. At about 150 km from the river mouth, an alluvial plain marks the transition to the upper reaches of the catchment, where a system of interconnected lakes with a total area of about 400 km2 is located. Water level fluctuation induced by the tide has been observed upstream of the lakes region [Hidayat et al., 2011]. It has been suggested that tidal processes may have a dampening effect on the fluvial dynamics of the delta region, causing a characteristic progradation pattern [Allen et al., 1977]. Recently, the absence of discharge peaks was also ascribed to the non-flooding discharge regime resulting from the buffering effect of the lakes [Storms et al., 2005; Hidayat et al., 2011]. While flood flows up to 5000 m3 s−1 can cause a rise in water level up to five meters in the upper reaches, flood surges are virtually damped by the buffering effect of the lakes [Storms et al., 2005; Hidayat et al., 2012], which eliminates sudden and large variations in river discharge in the lower reaches of the river. Indeed, the lack of channel migration [Allen et al., 1977] and the absence of channel avulsions [Storms et al., 2005] support these two hypotheses, rendering the Mahakam delta channel network virtually fixed. The relatively constant river discharge in the downstream reaches of the Mahakam allows us to investigate the subtle processes of river-tide interaction, which are often obscured by short-term and high-magnitude tidal processes, and/or fluvial instability. Since the Mahakam river discharge typically fluctuates at timescales longer than a fortnight, the system can adjust to subsequent quasi-equilibrium states (in the order of months), greatly simplifying the complex dynamics that river-tide interactions impose.
1.3. Objective and Structure of This Paper
 This contribution aims to develop a method to quantify the tidal signature on delta morphology by applying the hydraulic geometry concept to a delta channel network. We describe the study area, data collection methods and the hydrodynamical model of the delta in section 2. Section 3 presents the main results of the geomorphic analysis of the channel network and scaling of the hydrodynamics. Section 4 introduces the HG framework for channel networks affected by tides and presents the downstream HG relations based on selected cross sections in the delta. Section 5describes the mechanisms governing river discharge division, and the impact these mechanisms have on downstream HG relations of the cross-section area, and the mean flow velocity. We finalize this contribution with conclusions insection 6.
2. Mahakam Delta Channel Network
2.1. Site and Data Collection
 The River Mahakam debouches into the Makassar Strait, forming a regularly distributed, fan-shaped delta (Figure 1). The delta channel network exhibits a quasi-symmetric planform of rectilinear distributaries and sinuous tidal channels. The two main fluvial distributary systems are directed SE and NE, and comprise eight and four outlets to the coastal zone, respectively. The tide-dominated inter-distributary zone allocates many tidal channels, with tidal channels occasionally connected to the fluvial system. Due to the high river discharge, the study area is generally subject to freshwater conditions. During extremely low flows, which may be related to El Niño Southern Oscillation such as during the drought in 1997, salinity intrusion can reach beyond the delta apex. In general, however, salinity intrusion typically reaches to about 10 km seaward from the delta apex (or 30 km from the coast). Depending on the river discharge, the tidal wave can propagate up to 190 km from the river mouth. The tidal regime in the Mahakam delta is mixed, mainly semidiurnal.
 Several water level gauges and two horizontally deployed acoustic Doppler current profilers (H-ADCPs) were operational for about 18 months at locations along the river and in the delta. The two H-ADCPs were deployed upstream of the lakes region and next to the delta apex. The gauges recorded one minute averages of water level fluctuations measured at 1 Hz every 15 min, whereas the H-ADCPs yielded a 10 minute average of flow velocity, also at 1 Hz, every 30 min. Array data of flow velocity collected with the H-ADCP were converted to river discharge using calibration data from conventional shipborne ADCP discharge measurements. Upstream of the lakes, where tidal influence was found to be negligible, eight 6 h ADCP campaigns covered a wide range of flow conditions [Hidayat et al., 2011]. Close to the delta apex, where tides dominate, seven 13 h ADCP campaigns were carried out spanning high- and low- flow conditions, during spring tide and neap tide [Sassi et al., 2011a].
 Cross-river depth profiles with an interspacing of about 200 m were obtained with a single-beam echo sounder at locations spanning the river, its tributaries, the three lakes and the delta region. A bathymetric map of the channels was produced by linear interpolation of the transect data of bed elevation, previously projected onto a curvilinear grid based on the channel centerline [Legleiter and Kyriakidis, 2007]. The resulting bathymetry has been simplified by omitting all tidal channels that were disconnected from the fluvial network. The bathymetry of the delta (Figure 1) shows channels with variable depths, ranging between 5 m and 15 m, with occasional deep spots usually located at bends, junctions and constrictions, and very shallow areas often situated in the regions around bifurcations. The distributaries become increasingly shallow seaward whereas the river has an average depth of around 15 m.
2.2. Hydrodynamical Model
 The hydrodynamics driven by river discharge and tides was simulated using a depth-averaged version of the unstructured mesh, finite element model SLIM (Second-generation Louvain-la-Neuve Ice-ocean Model,http://www.climate.be/slim). The Mahakam delta, the coastal zone and the lakes region were represented by a 2D computational domain, which was connected to a 1D computational domain representing the river and several tributaries. GEBCO (http://www.gebco.net) database information was used in the continental shelf and the Makassar Strait, whereas measured bathymetry was used in all other domains. Tides from the global ocean tidal model TPX07.1 (http://volkov.oce.orst.edu/tides) were used to force the model at open boundaries, located far away from the delta and stretching across the entire Makassar Strait. Discharge series obtained from H-ADCP velocity data were used to force the model at the upstream boundary. A rainfall-runoff model calibrated with discharge data from the main subcatchment provided discharge series at the boundaries where tributaries connect to the modeling domain. Extremely small river bed slopes in lowland areas cannot directly be retrieved from a bathymetric survey, because of a lack of an absolute vertical reference with a sufficiently high accuracy. Therefore, the slope of the river was obtained following the approach described inBuschman et al. ; when concurrent water level and discharge data are available, as in the case of the discharge monitoring station, the river bed slope can be inferred from conservation of momentum. A regional, along-channel momentum balance was set up for a control volume bounding the discharge station and a pressure sensor located further downstream. By collecting additional discharge observations at the downstream point and using the continuous estimates from the discharge station, the bottom slope was readily estimated.
 A calibration procedure in which the model domain was decomposed in three regions [de Brye et al., 2011], provided the bottom friction coefficients. Model calibration was performed by comparing model results with water level time series at three locations in the delta, and with flow measurements obtained at the downstream discharge station. As a form of model validation, we have compared model results with discharge division measurements at the two principal bifurcations in the delta, during spring tide and during neap tide [Sassi et al., 2011b]. More details of the model implementation can be found in de Brye et al.  and in Sassi et al. [2011b].
 The computational mesh of the Mahakam delta channel network contains approximately 70% of the elements in the model's computational domain. The model does not include intertidal storage areas such as tidal flats or salt marshes, which may potentially affect the estimation of peak flows. Remote sensing images show the total intertidal area in the Mahakam Delta is very limited. Intertidal areas that once existed, when the system was still natural, have been subjected to land reclamation and are now excluded from the channel network. This ensures that the omission of intertidal areas in the model has limited impact on the model calculations.
3. Scaling of the Channel Network
 Banklines were obtained from remote sensing images, from maps and from the bathymetric survey. Channel center lines were obtained as the mean location line between the bank lines (Figure 2) and re-sampled to achieve a constant resolution of 100 m. At each cross section, the width and mean water depth were computed. Tidal channels attached to the fluvial network were not considered in the analysis.Figure 3 shows that mean water depth is inversely correlated to width, for channels in distributary outlets going from North to South. The inverse relation weakens when going to the South, because channels in the South are used for navigation and are subject to continuous dredging activities. Despite some variability across the channels, the distributaries show a clear relationship between depth and width.
 A representative channel geometry was computed as the average depth and width over all channels at a given radial distance from the delta apex, and the cross-sectional area and aspect ratio derived from those parameters. Variability across the channels is more apparent near the delta apex than near the coast, because the number of averaged channels increases from the apex to the sea.Figure 4shows the spatial development of the representative channel geometry from the delta apex toward the sea, as a function of the normalized along-channel distance,s/L, where srepresents the along-channel distance from the delta apex andLis the distance to the sea along the longest distributary. The representative channel width shows two well-defined regions; a third region arises as the link between these two regions. In the region bounded by an arc with a radius of about 10% of the total radial distance to the apex, the width varies little around a mean value of 1000 m. Within a distance roughly between 10 to 50% of the total radial distance between the apex and the coast, the width oscillates between 400 m and 1000 m. In the remainder of the delta, the width features an increasing trend toward the sea. The representative channel depth oscillates around an average value of about 7 m over the first half of the total radial distance, whereas it shows a decreasing trend closer to the coast. The representative channel area decreases from the delta apex up to about the central radius in the delta, and increases seaward beyond that arc. The best-fit line to the representative channel area, drawn inFigure 4(bottom left), is obtained from two linear functions in semi-logarithmic space [Guo, 2002]. The representative aspect ratio oscillates around a constant value in the landward half of the delta, and increases seaward in the remaining part.
 In the landward half of the Mahakam delta, the scaling behavior of the representative channel geometry coincides with the scaling observed in river deltas [Edmonds and Slingerland, 2007]. When made dimensionless, the representative channel area as a function of the bifurcation order follows the same trend as the data presented by Edmonds and Slingerland . In the seaward part of the delta, the representative channel geometry resembles that of funnel shaped estuarine channels [Davies and Woodroffe, 2010], reflecting the importance of the tidal discharge relative to the river discharge in channel forming processes [Fagherazzi and Furbish, 2001]. The Mahakam delta reveals a sharp transition between the river-dominated and tide-dominated domains (seeFigure 4). This regime change occurs at s/L= 0.55, which is based on matching two linear functions in log-space.
 A Continuous Wavelet Transform, using a Morlet mother wavelet, was applied to the modeled time series of water discharge obtained at cross sections selected in both downstream branches of each bifurcation in the delta (see Figure 2). Amplitudes of quarterdiurnal, semidiurnal and diurnal fluctuations were readily obtained, since spectrograms generally feature a well-defined gap between the tidal and subtidal band [Sassi et al., 2011b]. The concept of tidal species is introduced, to denote a group of tidal constituents with frequencies corresponding to a frequency band, as opposed to a unique frequency [Jay, 1997]. To distinguish between fortnightly fluctuations associated with river-tide interaction [Buschman et al., 2009] and monthly or seasonal fluctuations, we isolated the fortnightly variation by delimiting the fortnightly frequency domain in the normalized global wavelet power spectrum. Hence, fortnightly amplitudes corresponded to wavelet power concentrated in a band with periods roughly in between 10 to 20 days.
 Accordingly, water discharge Q at each cross section can be decomposed as:
where Qr is the river discharge and Qt is the tidal discharge. The tidal discharge is defined here as:
where i is the imaginary unit, the subscript 1/14 stands for a fortnightly period of the tide and 1, 2 and 4 denote diurnal, semidiurnal and quarterdiurnal tides, respectively, w is the angular frequency corresponding to a diurnal tide with an exact period of 24 h, ϕ represents the phase and Pl is the tidal discharge amplitude. We define P, the maximum tidal discharge amplitude at a given time, such that
Acknowledging that the tidal discharge at a given point and time is actually smaller, P can be considered to be a surrogate of the maximum astronomical tidal prism. The tidal prism is defined here as the volume of water between mean high tide and mean low tide. All quantities are averaged over the entire simulation period.
 Both P and Qr show a seaward decrease in magnitude (Figure 5). The scatter increases seaward because the flow is obtained at a limited number of cross sections, which span a confined range of the spatial extent of the delta. The decrease in Qr with distance to the delta apex reflects the partitioning of river discharge at the bifurcations. Conversely, the landward increase in P reflects the combination of tidal discharges at the bifurcations. Qr decreases with distance to the delta apex faster than P decreases; the exponents in a power law relation between Qr and s/L and P and s/L are equal to −3.15 ± 0.53 and −1.56 ± 0.5, respectively, where the variability is given by the standard error in the linear regression. Mean flow velocity U, defined as the ratio between Q and A, and the ratio between tidal discharge and river discharge P/Qr, remain nearly constant until s/L = 0.58 and s/L= 0.65, respectively, based on the intersection of two best-fit linear functions in log-space. From that point seaward,U and P/Qrdepict a decrease and increase in magnitude, respectively. Note that the location of the break-point in the hydrodynamics coincides fairly well with the location of a shift in scaling behavior of the channel network, separating the Mahakam delta into a river delta part and a coastal margin featuring funnel-shaped estuaries.
Figures 4 and 5 also show the cluster means of the data, binned as a function of s/L. Bins were uniformly distributed between 0 and 1 with a bin-size equal to 0.2. The general trend visible in the clustered values was not sensitive to the exact value of the bin size. Increasing the bin size may slightly shift the location of the break in morphology and hydrodynamics, as a result of the reduction of resolution. Decreasing the bin size increases the number of degrees of freedom in the linear regression, but the number of data points on each bin reduces accordingly. Our choice of bin size coincides with the minimum number of bins that still reproduced the general trend.
4. Downstream Hydraulic Geometry
 Here we use the numerical model to simulate scenarios for alternative river discharge and tidal forcing conditions. Simulations include a constant river discharge and a tidal time series in which the diurnal tides and semidiurnal tides amplify each other maximally during spring tide. Simulated scenarios include high and low river discharge input, with and without the tidal forcing. Model runs span two months, and all quantities are averaged over the entire simulation period.
4.1. Including Tides in the Hydraulic Geometry Concept
 Consider the hydraulic geometry (HG) relation of the area of a channel network in morphological equilibrium that conveys both river and tidal discharge:
where α and β are two coefficients, Qr denotes the river discharge and Pis the maximum tidal discharge amplitude associated with the tidal prism. At-a-site HG refers to the temporal co-variation ofA with both Qr and P at a specific cross section, whereas downstream HG describes the spatial variation in Afor a constant, channel-forming discharge, such as bankfull discharge (Qbf). Here, we are concerned with the latter, and to keep the analysis simple we assume the bankfull discharge in the river is also formative in the channel network.
 If Qr/P < 1, retaining the first two terms of the binomial series expansion of A yields:
When Qr and Pare correlated, a relation in log-space betweenA and Pwill feature limited ambiguity and a well-resolved exponentβcan be obtained from the best-fit line.P can be expressed as a power law of the form
where c and d are two coefficients, and d < 1 (see Figure 5). The expression is valid only when considering spatial variations in P and Qr at a constant frequency of exceedance, since at a given cross section these two are typically inversely correlated. The exponent d controls how the tidal prism is accommodated throughout the fluvial network for a given input bankfull discharge. We expect d to remain constant for different values of Qbf, since the channel network is expected to be in morphological equilibrium. Equation (5) can be rewritten as:
Equation (7) shows the form of the relation between A and Qrin tidally influenced deltas. It provides an explanation why log-log plots ofA versus Qrdo not show a linear relation. In addition to the effect of the tidal prism, mechanisms of river-tide interaction causeQr to be directly impacted by the tide, provoking mass transport as a result of Stokes drift and differential water level setup [Buschman et al., 2010; Sassi et al., 2011b].
4.2. Simplifying Tidal Hydrodynamics
 To investigate the downstream HG relations in the delta, we ran the model imposing a constant bankfull river discharge Qbf = 2500 m3 s−1 at the upstream boundary. This estimate was obtained from combined flow velocity and water level measurements at the upstream discharge station [Hidayat et al., 2011]. Discharge in the tributaries was obtained by applying a rainfall-runoff model calibrated with discharge data from the main river. The contribution of bankfull discharge in the tributaries to the total bankfull discharge in the river was obtained from a lagged non-linear regression between the rainfall-runoff estimates and the discharge measured upstream in the river.
Figure 6a shows that channels conveying high river discharge also convey a high tidal discharge. The exponent d in equation (6) was estimated to be 0.7 ± 0.04, where the variability is given by the standard error in the linear regression. To assess the applicability of equation (6) for varying discharge conditions, we ran the model imposing a very low constant input river discharge at the upstream boundary (Qin = 250 m3 s−1), yielding d = 0.68 ± 0.04. This indicates that the exponent in equation (6) is largely independent of Qin. Contributions to P from the four tidal species are all positively correlated with the river discharge conveyed by the channel (Figure 6b), with similar slopes for all species. The semidiurnal species feature the largest tidal prism, followed by the diurnal, quarterdiurnal and fortnightly species.
Figure 7a shows the same data as Figure 6a, for simulations with Qbf, but coded for incremental ranges of s/L. The cluster means of P and Qr (Figure 7b), which remove the spreading for a given bin of s/L, suggest a nonlinear relation between P and Qr. The figure also shows a seaward decrease of the exponent d in equation (6). The same occurs for the low input river discharge (not shown). The exponent that results from fitting a line through all clustered data points coincides with that derived from a best-fit line through values ofP and Qr (d = 0.46 ± 0.04, see Figure 5). The reduction of d is due to the effect the spreading for a constant value of s/Lhas on the parameter estimation in a log-log plot [e.g.,Asselman, 2000; Packard and Birchard, 2008]. Since the spreading increases seaward, values of P and Qrtypically plot in the lower portion of the log-log plot, becoming influential in the parameter estimation [Cook and Weisberg, 1982] and hiding possible structures in the relations. Therefore, the reduction in the exponent dpartially reflects the variability across distributary branches rather than the inherent variation along the radial or along-channel dimension. The estimation of the exponents can be performed for cross sections downstream of each distributary channel, or by binning all information from channels corresponding to equal radial distances. We adopt the latter approach, which allows us to quantify the variation of the downstream HG exponent with radial distance to the delta apex.
 The transition between river- to tide- dominated hydrodynamics, as reflected in the downstream variation of the exponentd, may be primarily governed by the increasing number of bifurcations. The number of bifurcation nodes in the radial segments a through e in Figure 2 is 1, 2, 4, 8 and 8, and shows a spatial increase of the number of bifurcations only in the landward section up to s/L = 0.5. In the landward part of the delta, discharge is divided over a progressively larger number of channels in the downstream direction, but the absolute number of channels remains relatively small. Downstream changes in Qr and P occur concomitantly and the exponent d tends to unity. In the seaward part, the number of channels is much larger but the number of bifurcations remains constant. Then the exponent d tends to zero, so that P becomes almost independent of Qr.
4.3. Linking Tidal Hydrodynamics to Channel Morphology
Figure 8shows log-log plots of the cross-sectional areaA and the discharge conveyed by the channel using (a) river discharge Qr (equation (7)), (b) the maximum tidal discharge amplitude P (equation (5)), (c) the total discharge Qr + P (equation (4)) and (d) the river discharge scaled with the bifurcation order ϕ + 1. The bifurcation order ϕ + 1 is defined as the number of bifurcations preceding a particular cross section. The relation between A and Qr shows a larger spreading and reduced slope when compared to the relation between A and P.Qr + P shows a nearly unambiguous relation with A. To arrive at a similarly clear relation between A and Qr, we multiplied Qr with ϕ + 1 (Figure 8d). This scaling behavior can be simply explained by the downstream reduction in river discharge when the number of branches increases [see Edmonds and Slingerland, 2007]. Table 1presents a summary of the exponents obtained by fitting a line in log-log space. The close relation between the exponents found forQr + P and (ϕ + 1)Qr suggests the ratio of tidal discharge to river discharge scales with bifurcation order:
This is in qualitative agreement with the results presented in Figure 5.
Table 1. Summary of Parameters in Log-Log Plots of the Cross-Sectional AreaA of Selected Cross Sections and the Discharge Conveyed by the Channel, As Indicateda
α ± δα
β ± δβ
βland ± δβland
βsea ± δβsea
The variability is given by the standard error in the linear regression. The goodness of fit is represented by the coefficient of determination R2. The exponent βobtained with the best-fit to clustered data is denoted with the subscriptland and the subscript sea.
 The cluster means of A and discharge, binned as a function of s/L, depict a non-linear relation which can be approximated by two linear functions (Figures 9a through 9d). The two linear relations correspond to the landward part of the delta in the domain, up to s/L= 0.5, and with the seaward remainder (respectively). Best-fit lines through the clustered values yield constant slopes in the landward part of the delta, and differ in the seaward part (seeTable 1). These slopes can be interpreted as downstream HG exponents, whose variation across an arc with constant distance to the delta apex shows a transition from the landward to the seaward part of the delta. Negative values of β for the seaward part of the delta stem from the fact that at a given cross section, channel area scales with the volume of water (the tidal prism) conveyed by the channel. The tidal prism invariably increases in the downstream direction. The trends in β values are the same for both parts of the delta: the absolute value of β is highest for P and lowest for Qr.
 With the best-fit relation between cross section area ands/L, represented by the full line in Figure 4 (bottom left), characteristic relations can be calculated, which are shown as the continuous curved lines in Figure 9. The two linear functions approximate the characteristic relations, which supports the approach to establish these relations in Figure 9. The characteristic relations show a good agreement with the cluster values in the landward part of the delta, whereas some divergence occurs in the seaward part associated with the variation among the distributaries. The results indicate that βin the downstream HG relation of cross-sectional area exhibits an abrupt change along the normalized along-channel distance. In the landward partβ ≥ 1 whereas in the seaward part β < 1. The shift in βcoincides with the transition from river- to tide-dominated hydrodynamics.Table 1shows the exponents for binned and un-binned data usingQr + P and (ϕ + 1)Qr relations are very close to each other. The binning procedure was introduced only to improve the readability of the figures.
 The results presented above can be partly understood from existing downstream HG studies in tidal channels. In a single tidal channel, the minimum shear stress τsneeded to maintain a net zero gradient in the along-channel sediment transport results in downstream HG relations with values ofβgreater than unity in flood-dominated systems and smaller than unity in ebb-dominated systems [Friedrichs, 1995]. The abrupt change in the downstream HG exponent may be partly related to an overall transition from flood to ebb dominance in the main distributaries of the Mahakam delta channel network. Toward the delta apex, the decrease in intertidal storage area and deepening of the channels become two competing factors, which may exert a variable influence on the ebb or flood dominance. In the seaward part of the delta, the absence of tidal flats in the numerical model, and the shallowing of the channels, explain flood dominance. However, rather than being driven purely by tides, the ebb/flood transition may likely be associated to the relative increase in the P/Qr ratio. Since Qrdecreases seaward with each successive bifurcation, the occurrence of longer flood-flows relative to the tidally averaged discharge increases with bifurcation order. The negative values of the exponents can partly be related to the fact thatP is not the tidal prism, but the maximum tidal discharge amplitude.
5. Tidal Impact on HG Relations
 The downstream HG relations established in the previous sections may depend on the division of river discharge at bifurcations in the delta. The present section investigates the division of water discharge over bifurcates, aiming to isolate the tidal influence on the variation in β as much as feasible. A factor separation analysis [Stein and Alpert, 1993] was applied to the time series of water discharge. Consequently, we ran the model with two sets of forcing conditions: 1) at the upstream boundary, bankfull river discharge is set to 2500 m3 s−1, and at the marine boundary the water level is set to equilibrium (no tides); and 2) the model is forced with the same bankfull river discharge at the upstream boundary and with the tidal boundary conditions described in section 2.2.
For an equal discharge division, the discharge asymmetry index Ψ is zero; when discharge in the southern channel is larger, Ψ is positive up to a value of one when river discharge is carried completely by the southern channel, and vice versa. We split the discharge asymmetry index in two components such that
where Ψr denotes the asymmetry in the discharge division from simulations forced with river flow only and Ψrt denotes the asymmetry in the discharge division from simulations forced with river flow and tides.
 The relative difference ratio Ψ/Ψrquantifies the tidal impact on river discharge division, as it increases with increasing contributions from tides and river-tide interaction. The tidal impact increases seaward (Figure 10), as Ψ/Ψr attains values of around −0.3. For s/L < 0.5, the tidal impact on river discharge division is low, as values of Ψ/Ψr remain below 0.1 in all distributaries. Because Ψrt is predominantly smaller than Ψr, Ψ/Ψr is typically negative, and we can conclude that the effect of tides is generally to counteract asymmetry in the division of river discharge at the bifurcations in the Mahakam delta.
Figure 11 shows spatial distributions of the quarterdiurnal (D4), semidiurnal (D2), diurnal (D1) and fortnightly (D1/14) contributions to surface level variation, averaged over the entire simulation period. In general, both D4 and D1/14 amplify toward the apex whereas D2damps out. Tidal damping of the main tidal species is generally attributed to frictional forces. Part of the semidiurnal tidal energy is transferred to the overtide and compound tide frequency bands, predominantly via non-linear interaction in the bottom friction term in the momentum balance. Spatial variations inD4 and D1/14 resemble each other. In strongly convergent channels, the effects of convergence can surpass the effect of bottom friction, causing tidal amplification [Lanzoni and Seminara, 1998]. D2 depicts relatively large values in several tidal channels and in sections of distributaries which show to be strongly convergent. The diurnal species D1 damp out in the southern branch, whereas it remains relatively constant in the northern branch. The outlets of the northern and southern branches show a different response to diurnal tidal forcing, which may be associated to D1 variation along the coastline. North of the delta, the continental shelf adjacent to the delta is relatively narrow. Inside the delta, small length differences can be the cause of substantial differences in reflection of diurnal tidal energy. Discrepancies in tidal amplitudes (and phases) between channels may lead to a net flux that eventually steers more discharge to a one particular branch. Figure 11 suggests that damping of the main tidal species mostly occurs in the radial direction, showing limited variability across channels along arcs with a constant distance to the delta apex. Only a few (mostly sinuous) channels depict large differences in amplitudes between bifurcating channels.
5.1. River-Tide Interaction
Figure 12 shows the mean water surface topography based on 〈η〉, where the brackets denote averaging over the entire simulation period (Figure 12, left), and the fortnightly contribution to water surface variation D1/14 (Figure 12, middle). The two panels show good resemblance. D1/14can be regarded as a surrogate for the strength of river-tide interaction [Buschman et al., 2009]. Since river-tide interactions not only generateD1/14but also result in a steady (non-periodic) rise in elevation [Godin, 1999; Buschman et al., 2009], the resemblance in the spatial distributions of 〈η〉 and D1/14for a constant discharge suggests that river-tide interaction contributes substantially to the water surface topography.Sassi et al. [2011b]showed that for a seaward bifurcation in the Mahakam delta, the difference in rise of water surface elevation induced by river-tide interaction at a bifurcation led to a substantial reduction of the asymmetry in the division of river discharge, compared to the case without tides. Since the water surface topography 〈η〉 shows a clear relation to D1/14, the present contribution generalizes this result, showing differential water level setup to be the dominant mechanism by which tides impact the division of river discharge at these bifurcations. This conclusion is supported by the fact that the ratio D1/14/〈η〉 (Figure 12) depicts a very close relation to the ratio Ψ/Ψr (Figure 10), when both are plotted as a function of normalized along-channel distances/L.
Figure 13 shows the downstream HG relations of the area A and the mean flow velocity U, for simulations forced with river discharge only (Figure 13, left) and simulations forced with river discharge and tides (Figure 13, right), using (ϕ + 1)Qr and Qr + P, respectively, as a measure of the discharge conveyed by the channel. Note that for the simulations without tides, U is a mean flow velocity, whereas for simulations with tides, U represents a mean flow velocity plus the amplitude of the tidal species. The values of β and γin the insets show the slopes of the best-fit line to values ofA and U before binning, respectively. Since U is computed as the ratio between (ϕ + 1)Qr (Figure 13, left) or Qr + P (Figure 13, right) and A, the sum of the exponents in the area (β) and flow velocity (γ) relations equals unity. Table 2shows the estimates of the downstream HG exponents, separating the river-dominated part of the delta (0 < s/L ≤ 0.5) from the remaining seaward part. Regarding the linear relations, the sums of the exponents obtained with A and Udo not necessarily equal unity, because data has been clustered as a function of the normalized along-channel distance. When tides are included, variation of the exponents are more apparent in the seaward part of the delta than in the landward part. The inclusion of the tides leads to an increase inβsea and to a decrease in γsea. The same occurs when the procedure is applied to the data before binning. In the landward part of the delta, tides have a limited influence on the exponents. This can be linked to the division of river discharge at the bifurcations, which experiences a significant tidal impact only in the seaward part of the delta beyond s/L = 0.5.
Table 2. Summary of Exponents in Downstream HG of Cross-Sectional AreaA and Flow Velocity Ua
βland ± δβland
βsea ± δβsea
The exponents β and γwere obtained with the best-fit to clustered data and denoted with the subscriptland and the subscript sea. For simulations forced with river discharge only ((ϕ + 1)Qr), and simulations forced with river discharge and tides (Qr + P). The variability is given by the standard error in the linear regression.
(ϕ + 1)Qr
Qr + P
(ϕ + 1)Qr
Qr + P
1.1 ± 0.07
1.06 ± 0.17
−0.34 ± 0.09
−0.48 ± 0.12
γland ± δγland
γsea ± δγsea
(ϕ + 1)Qr
Qr + P
(ϕ + 1)Qr
Qr + P
−0.03 ± 0.05
−0.03 ± 0.11
1.27 ± 0.09
1.05 ± 0.08
 The effect of tides on the downstream HG relations of the Mahakam delta is, in general, to increase β and to decrease γin the seaward part of the delta. For constant river discharge, cross section areas are larger in the case of tides, due the steady water level setup induced by river-tide interaction. The surface slope steepens toward the sea [Godin and Martinez, 1994]. In the landward part of the delta, the surface slope approaches asymptotically the surface slope of the river. Differential water level setup induced by river-tide interaction in the Mahakam generally acts to reduce the asymmetry in discharge division, distributing the flow more equally over the downstream branches. This reduces the subtidal peak flow velocities. By counteracting asymmetry in discharge division and reducing peak flow velocity, tides act to stabilize channel morphology.
 A method to quantify the tidal signature on delta morphology has been developed, by applying the hydraulic geometry (HG) concept to a delta channel network. Downstream HG of distributaries in the Mahakam delta features distinct characteristics in two zones. From the delta apex to about half the radial distance to the sea, channel geometry scales similarly to alluvial channel geometries in river deltas, with cross-sectional area and width decreasing gradually in downstream direction, and depths fluctuating around a constant value. In the sea-connected remainder of the delta, distributaries resemble funnel-shaped estuaries, which become increasingly wide and shallow toward the channel mouths. The ratio of maximum tidal discharge, here quantified by the sum of the mean fortnightly, diurnal, semidiurnal and quarterdiurnal tidal discharge amplitudes, and bankfull river discharge scales with bifurcation order. Based on a simple theoretical analysis of HG equations and using a hydrodynamic model, we derive an equation that shows how tides modify downstream HG relations. By clustering selected cross sections in bins of radial distance to the delta apex, a non-linear HG relation arises from regression analyses of the clustered data. The non-linear HG relation can be approximated by two asymptotic lines. The HG relations in the river-dominated part of the delta are robust because the line determined in a log-log plot of clustered values (Figure 9) coincides with the analysis using the un-binned data (Figures 4 and 5). In the tide-dominated part of the delta, the large variation in channel geometry between distributaries and the increased scatter found in cross sections may mask the general trends. Recent studies on the physical mechanisms governing the division of discharge over downstream branches help to explain the HG complexity in the seaward portion of the delta. Nonlinear interaction of river discharge with the tidal motion creates a water level setup, which depends on channel geometry. The net effect of the tides in the Mahakam delta is to reduce the inequality in discharge division, especially at channel junctions near the sea. This effect, and the fact that part of river discharge is conveyed by tidal channels occasionally connected to the distributary network, exert a strong influence on the downstream HG of distributaries in mixed river-tide dominated deltas such as the Mahakam.
 This study is part of East Kalimantan Programme, supported by grant WT76-268 from WOTRO Science for Global Development, a subdivision of the Netherlands Organisation for Scientific Research (NWO). E. Deleersnijder is a Research Associate with the Belgian Fund For Scientific Research (F.R.S. - FNRS). His contribution to the present study, as well as that of B. de Brye, was carried out in the Framework of the project “Taking up challenges of multiscale marine modeling”, funded by the Communauté Française de Belgique under contract ARC 10/15-028. We thank Hidayat for providing the rainfall-runoff simulation results to force the hydrodynamic model, Bart Vermeulen for contributing to the setup of the model in its initial stage, and Rory Dalman for providing constructive comments on an earlier version of the manuscript. Doug Edmonds, Andrew Ashton, Alexander Densmore and two anonymous reviewers are gratefully acknowledged for their detailed and positive criticism on the draft of this paper.