## 1. Introduction

[2] Historically, the interaction of river flow with tides in lowland rivers has been subject to investigation by oceanographers. In their studies on upriver tidal propagation, the river flow is generally treated as a constant, distorting the propagation of diurnal and semidiurnal tides [e.g., *Dronkers*, 1964; *Godin*, 1991; *Jay*, 1991]. Adopting the perspective of a hydrologist, at first glance tides may seem a periodic perturbation of the river flow. The interactions of tides with the river flow are, however, not all periodic. River-tide interaction creates steady as well as oscillatory gradients of the subtidal (averaged over a diurnal period) water surface, steepening the surface level profile up to the point of extinction of the tide [*LeBlond*, 1979; *Godin and Martínez*, 1994]. In flat areas, the region of influence of a permanent water level gradient and low-frequent surface level variations potentially reaches much further inland than diurnal and semidiurnal tidal motion [*Godin and Martínez*, 1994].

[3] The analysis of subtidal water level variation in response to river discharge waves requires long-term data series of discharge. Obtaining continuous discharge estimates has recently become facilitated by the development of techniques to convert data from horizontally deployed acoustic Doppler current profilers (H-ADCPs) to discharge [*Le Coz et al.*, 2008; *Nihei and Kimizu*, 2008; *Hoitink et al.*, 2009]. This paper provides an investigation of the sources of subtidal water level variation, using continuous series of discharge obtained from an H-ADCP in a relatively pristine tidal river in the tropics. The tidal river dewaters a relatively small catchment with a river discharge that varies, relatively rapidly, around an average of about 600 m^{3} s^{−1}.

[4] The generation of subtidal water level variation due to river-tide interactions can be captured in analytical one-dimensional models [*LeBlond*, 1979; *Kukulka and Jay*, 2003a; *Jay and Flinchem*, 1997]. In succession to an earlier paper on tidal river hydrodynamics [*LeBlond*, 1978], *LeBlond* [1979] derived subtidal balance equations of mass and momentum. These balances resulted after decomposing the cross-section averaged river velocity into a mean flow contribution, a contribution representing fortnightly variation and a contribution from diurnal and higher frequency modulations. After scaling, filtering, and retaining only the first order terms, the subtidal momentum balance revealed that fortnightly waves are forced in shallow tidal rivers. It showed that terms in the subtidal momentum balance other than those representing friction and the surface elevation gradient can be neglected. The magnitude of the subtidal friction term, in turn, strongly depends on the tidal range and on river flow velocity, explaining why at a constant river discharge the surface elevation gradient features oscillations with the frequency of a spring-neap cycle. If the mean sealevel is assumed steady, this translates into fortnightly river level oscillation.

[5] The approach of *LeBlond* [1979] presumes a spectral gap between the river flow variation and fortnightly tidal oscillations. That condition is usually only met in large catchments, where intramonthly river discharge variations are small, such as the St. Lawrence river that was investigated by *LeBlond* [1979].

[6] Further progress in understanding subtidal surface level variation was made by *Jay and Flinchem* [1997], who obtained an analytical expression showing how in a first order approximation the daily mean river level depends on river velocity, tidal velocity amplitude, drag coefficient, water depth, and parameters representing the rate of exponential decrease of depth and width, in upstream direction. It shows the river level to depend on the square of the river flow velocity, and on the square of the ratio of tidal velocity amplitude and river flow velocity scales. Analytical expressions were also obtained for diurnal, semidiurnal and quarterdiurnal tidal elevation amplitude.

[7] To validate the obtained expressions, *Jay and Flinchem* [1997] employed continuous wavelet transforms to decompose time series of surface elevation at several stations along the Columbia river and Estuary in unsteady low-frequent, diurnal, semidiurnal and quarterdiurnal components. They showed that analytical expressions using those decomposed time series captured the basic mechanisms of the river-tide interaction in the highly dynamic Columbia river, where discharge ranges between 2500 and over 16,000 m^{3} s^{−1}. *Jay and Flinchem* [1997] did not elaborate on the validation of the full expression for the daily mean river level. They emphasized that river stage varies with the square of river flow velocity, as in a uniform flow in which the effect of tides is accounted for by an elevated Chézy parameter.

[8] *Godin* [1999] provided an elaborate analysis of the friction term in the momentum balance, yielding a generic overview of periodicities involved in river-tide interactions. Crucial in his approach is the notion that the product *U*∣*U*∣ can unconditionally be approximated by two terms including the product of two constants with the first and third order terms of the nondimensionalized velocity [*Doodson*, 1924]. The constants can be calculated by expanding *U*∣*U*∣ to Chebyshev polynomials [*Dronkers*, 1964]. This allows to evaluate the subharmonics that can be expected to develop if the amplitudes of the tidal constituents at the estuarine or oceanic boundary of the tidal river are known. Also, the potential zero-frequency (permanent) steepening of the water level surface, as observed by *Godin and Martínez* [1994] in results from a numerical model, can be explained analytically from forcing conditions.

[9] *Godin* [1999] further showed that in a pragmatic approach low-passed water levels can be regressed with tidal range and river discharge, which already can yield satisfactory results. *Gallo and Vinzon* [2005] used specific cases of the overview of *Godin* [1999], to analyze how the MSf tide develops in the Amazon river.

[10] *Kukulka and Jay* [2003b] and *Kukulka and Jay* [2003a] elaborated on the work of *Jay* [1991], studying the nonlinear interactions of river flow and tides in the upriver stretches of the Columbia river at diurnal, semidiurnal, quarterdiurnal and subtidal frequencies. To do so they decomposed the friction term using the Chebyshev polynomial approach into four contributions as in the work of *Dronkers* [1964], and retained the one contribution that is dominant upriver during high flow periods, where tidal currents are weak. Assuming that the subtidal water surface gradient is constant over the investigated Columbia river reach, this allowed to obtain an analytical solution of the subtidal momentum balance, showing how water levels depend on river discharge and tidal discharge amplitude when the ratio of river discharge and tidal discharge amplitude is high. Remarkably, the obtained expression agreed well with observations, even at seaward stations close to the mouth of the Columbia river, where tidal discharge amplitudes exceed the river discharge. *Kukulka and Jay* [2003a] further pointed out that it may be necessary to account for atmospheric forcing of the subtidal water level variation, impacting especially seaward stations by about −10^{−2} m per mbar pressure increase.

[11] In this contribution a new method is presented to analyze subtidal water level dynamics in tidal rivers, and applied to the Berau river (East Kalimantan, Indonesia). The method decomposes time series of discharge and water levels into diurnal, semidiurnal, quarterdiurnal and mean flow components, using wavelet transforms as in the work of *Jay and Flinchem* [1997]. Using an approximation of the friction term provided by *Godin* [1999], a new expression for subtidal friction is derived. This new expression is used to decompose the subtidal friction into contributions from river flow, asymmetry of the tidal flow and river-tide interactions.

[12] The method proposed herein provides insight into subtidal water level dynamics generated by river and tidal flows and their interactions. This insight can be used to attribute the commonly observed rise of subtidal water levels at spring tide, with respect to neap tide, to river flow, river-tide interaction and tidal asymmetry. Based on the theoretical results as presented, a regression model can be developed relating subtidal water levels to the aforementioned contributions, aiming to predict subtidal water levels in case of peak discharges. Predicting water levels under extreme river discharge conditions will be feasible provided that the damping of diurnal and semidiurnal tides as a function of river discharge can be quantified. Such predictions are relevant for designing flood protection measures along tidal rivers.

[13] This paper continues with a description of the *tidal dynamics in the river Berau*. In section 3 a *local subtidal momentum balance* is set up for a cross section in the Berau river about 60 km from the coast, using discharge and water level data that spanned over several months. From this momentum balance equation the local subtidal water level gradient will be solved, and compared with subtidal water level differences between neighboring stations to investigate the degree in which local subtidal water level gradients represent the regional subtidal behavior. In the subsequent section, *sources of subtidal friction* are analyzed for the station where the H-ADCP is located, aiming to distinguish between contributions by river flow, by diurnal, semidiurnal and quarterdiurnal tidal velocity and by interactions of the two. The latter section is followed by a *summary and conclusions*.