Preliminary Results of a Finite-Element, Multi-Scale Model of the Mahakam Delta (Indonesia)

Ocean Dynamics (2011) 61:1107–1120 DOI 10.1007/s10236-011-0410-y

Preliminary results of a finite-element, multi-scale model of the Mahakam Delta (Indonesia)

Benjamin de Brye · Sébastien Schellen · Maximiliano Sassi · Bart Vermeulen · Tuomas Kärnä · Eric Deleersnijder · Ton Hoitink

Received: 8 November 2010 / Accepted: 14 March 2011 / Published online: 5 May 2011 © Springer-Verlag 2011

Abstract The Mahakam is a 980-km-long tropical river high number of channels connected to the Makassar flowing in the East Kalimantan province (Borneo Strait. This article focusses on the flow in the delta Island, Indonesia). A significant fraction of this river is channels, which is characterised by a wide range of influenced by tides, the modelling of which is the main time and space scales. To capture most of them, the subject of this study. Various physical and numerical depth-integrated and the section-integrated versions of issues must be addressed. In the upstream part of the the unstructured mesh, finite-element model Second- domain, the river flows through a region of three lakes Generation Louvain-la-Neuve Ice-Ocean Model are surrounded by peat swamps. In the lowland regions, used. Unstructured grids allow for a refinement of the the river is meandering and its hydrodynamics is mostly mesh in the narrowest channels and also an extension influenced by tides. The latter propagate upstream of of the domain upstream and downstream of the delta the delta, in the main river and its tributaries. Finally, in order to prescribe the open-boundary conditions. the mouth of the Mahakam is a delta exhibiting a The Makassar Strait, the Mahakam Delta and the three lakes are modelled with 2D elements. The rivers, from the upstream limit of the delta to the lakes and the Responsible Editor: Phil Peter Dyke upstream limit of the domain, are modelled in 1D. This article is part of the Topical Collection on Joint Numerical The calibration of the tidal elevation simulated in the Sea Modelling Group Workshop 2010 Mahakam Delta is presented. Preliminary results on the B. de Brye (B) · S. Schellen · E. Deleersnijder division of the Eulerian residual discharge through the Institute of Mechanics, Materials and Civil channels of the delta are also presented. Finally, as a Engineering (IMMC), Université catholique de Louvain, first-order description of the long-term transport, the 4 Avenue G. Lemaître, 1348 Louvain-la-Neuve, Belgium age of the water originating from the upstream limit e-mail: [email protected] of the delta is computed. It is seen that for May and M. Sassi · B. Vermeulen · T. Hoitink June 2008, the time taken by the water parcel to cross Hydrology and Quantitative Water Management Group, the estuary varies from 4 to 7 days depending on the Department of Environmental Sciences, channel under consideration. Wageningen University, Droevendaalsesteeg 4, Wageningen, Gld, The Netherlands Keywords Finite element · Model · Mahakam River · B. de Brye · S. Schellen · T. Kärnä Multi-scale · Tide · Hydrodynamics Georges Lemaître Centre for Earth and Climate Research (TECLIM), Université catholique de Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium 1 Introduction E. Deleersnijder Earth and Life Institute (ELI), Georges Lemaître Most of aquatic ecosystems are changing due to Centre for Earth and Climate Research (TECLIM), Université catholique de Louvain, 2 Chemin du Cyclotron, anthropic pressures (Millenium Ecosystem Manage- 1348 Louvain-la-Neuve, Belgium ment 2005). Among the changing ecosystems, tropical 1108 Ocean Dynamics (2011) 61:1107–1120 land–sea continua are probably the most affected. The physical and biological processes driving ecosystems. Mahakam River is not an exception to the rule. This Moreover, the topography of the region of interest is 980-km-long tropical river flows in the East Kalimantan very intricate. It is appropriate to split the domain into province (Borneo Island, Indonesia) and discharges subdomains according to the typical time and space into the Makassar Strait at the border of Pacific and scales of motion: Indian Oceans (Fig. 1). A hydrodynamic model of the main part of the Mahakam River is being developed – The Makassar Strait, between the islands of Borneo with the objectives of studying the impact of external and Sulawesi: Its width varies between 200 and forcing factors such as sea-level rise, climate change, 250 km and its length is around 600 km. The upstream sediment impact, as well as human interfer- Makassar is an important strait in terms of the ence on past, present and future development of the global conveyor belt. According to Gordon et al. Mahakam delta over different timescales. (1999), the Makassar throughflow is, with an aver- Land–sea continua are particularly complex systems age of 9.2 Sv for the year 1997, the main pathway characterised by many physical processes interacting of water between the Pacific and Indian Ocean with each other over a wide range of time and space (Waworuntu et al. 2001; Susanto and Gordon 2005). scales. The need for a multi-scale approach is justified Because the Makassar Strait is surrounded by two by the differences between scales at which the ecosys- islands and is located around the equator, strong tem degradation is driven, scales at which the result- wind events are rare. Moreover, the impact of mon- ing change to ecosystem functioning has most impact, soonal winds is minimal as explained in Roberts and scales at which ecosystems are managed and of course Sydow (2003). The water level is therefore mainly scales at which we have the best understanding of the controlled by tides. The two main components are

Fig. 1 The Mahakam River (Borneo Island, Indonesia). The estuarine part of the river forms a delta. The river in its upstream part is surrounded by three lakes Ocean Dynamics (2011) 61:1107–1120 1109

semidiurnal. The amplitude of M2 and S2 reaches their specific scale. The timescale ranges from minutes 0.55 and 0.35 m, respectively, resulting in a well- for the tidal influence in the narrowest channels of the marked fortnightly cycle. In addition, diurnal com- delta to years for the ENSO variability. The length ponents are also present in the Mahakam. scale may be as small as a few metres in the narrow- – At the mouth of the Mahakam River lies a re- est channels and reach hundreds of kilometres in the gion of freshwater influence (Simpson 1997)witha Makassar Strait. To the best of our knowledge, the only high salinity gradient. With an average discharge of previous study of the domain of interest was achieved 2,000 m3/s, one could expect that nearly no saline by Mandang and Yanagi (2008). They developed a two- water enters the delta from the Makassar Strait. dimensional barotropic hydrodynamical model based Nevertheless, as was seen by Storms et al. (2005), on a structured grid, using the finite difference method. there is a salt wedge in some delta branches in They used a C-grid with a resolution of 200 m. The the early hours of ebb tides. However, the salinity water level and velocity in the delta channels were remains below 10 PSU. This was also observed computed. With such a procedure, the resolution was during extremely dry periods with a discharge lower generally insufficient in the delta, while the Makassar than 400 m3/s. Such events appear periodically Strait was over-resolved. Despite the fact that finite since the island of Borneo is strongly affected by difference methods are easily implemented, they suffer the El Niño–Southern Oscillation (ENSO) cycles from a lack of spatial flexibility. In addition, Mandang (Kishimoto-Yamada and Itioka 2008). and Yanagi (2008) imposed the tidal elevation at only – Figure 1 shows that the lower estuarine part of five points on the continental shelf relatively close from the Mahakam River is a delta exhibiting a large the delta. It is generally appropriate to extend the com- number of channels. One can distinguish tidal chan- putational domain out of the continental shelf and to nels without any connection with the riverine part a location where tidal forcing can be easily prescribed. and channels which are influenced by the riverine For structured grid models, such an extension increases discharge and the tidal regime. The width of the the computational coast significantly. Grid nesting is channels ranges from 10 mto3 km. The delta often used to improve the performance of the model, extends 40 km on the continental shelf creating a but this method faces some drawbacks like unphysical habitat for a rich but weak ecosystem. The envi- reflections and perturbations. ronment of the delta suffers from an inappropriate To deal with multi-physics and multi-scale models in development: conversion of mangrove forest, ex- space and time, it is our conviction that unstructured- tensive aquaculture and fishing industry, dredging mesh modelling is a promising option. The main activities etc. These transformations combined with advantage is the spatial flexibility with a possible external factors bring extreme events such as water refinement in small channels, in shallow areas or across shortage, floods and salinity intrusions and lead inclined bottom. The Second-Generation Louvain-la- to urban inundations and problems for agriculture Neuve Ice-Ocean Model (SLIM1) is able to cope and drinking water supply. Predicting water level is with highly multi-scale applications (Deleersnijder and crucial for the sustainable development of the local Lermusiaux 2008; Lambrechts et al. 2008a) such as the economy. Great Barrier Reef (Lambrechts et al. 2008b)orthe – In the lowland regions, the hydrodynamics of the Scheldt River Basin (de Brye et al. 2010). Therefore, meandering river is mostly influenced by tides. The SLIM is well suited to address the complexity of the latter propagate upstream of the delta, in the main Mahakam River. river and its tributaries over a distance exceeding The next section presents the model setup: the de- 200 km. scription of the domain, the equations, the forcings, the – In the upstream part of the domain, the river flows numerical method and the grid. In Section 3, numerical through a region including three lakes surrounded results of the model are presented. First the model is by peat swamps. Typical to tropical rivers, such calibrated against elevation data in the delta. Then, the swamps act as a buffer for the highly variable division of the Eulerian discharge through the channels discharge regime of the river. The destruction of of the delta is described. Finally, the age of the water peat swamps due to industrial and agricultural us- originated from the upstream limit of the estuary is ages may disrupt the discharge regimes (Hoekman computed. Concluding remarks are made in Section 4. 2007).

The main challenge of dealing with such a multi- scale domain is to resolve all the different processes at 1www.climate.be/slim 1110 Ocean Dynamics (2011) 61:1107–1120

2 The model hydrodynamics is simulated by means of 1D, section- averaged equations. Finally, the three upstream lakes 2.1 The domain are modelled with the 2D depth-averaged equations.

Deltas, estuaries or bays exhibit open boundaries, re- quiring a relevant treatment. The positioning of the 2.2 The governing equations open boundaries is crucial, and in many cases, the simplest choice is not the most appropriate (de Brye In all equations, vectors are set in bold so as to dis- et al. 2010). For example, imposing boundary condi- tinguish them from scalar variables. The equations to tions near the mouth of the Mahakam Delta is not be solved are the shallow-water equations in their 2D trivial, and the extension of the domain far from the depth-integrated and 1D section-integrated versions. mouth and up to the deep sea facilitates the task in The equations and the parametrisations used in this different ways. studyaresimilartothoseofdeBryeetal.(2010) for the Scheldt tidal continuum (The Netherlands/Belgium). – In deep water, the equations are more linear and The latter paper also describes the coupling between the common problem of spurious local fluxes going the2Dpartandthe1Dpartofthemodelaswellas into and out of the delta is generally avoided. the coupling of the different branches of the 1D river – External hydrodynamic data need to be imposed network. along the open boundary. If no coarser coastal model is available for the region of interest, global tidal model results may be resorted to. 2.2.1 The 2D equations – If the downstream boundary is sufficiently far from the domain of interest, the meteorological forcing Let t be the elapsed time and ∇ the horizontal del (wind stress, atmospheric pressure, etc.) can be operator. The variables are the depth-integrated veloc- imposed as a surface flux only and be ignored in ity u = (u,v) and the free-surface elevation η.Inits the open boundary conditions. conservative form, the continuity equation reads: – When modelling tracers such as salinity, prescrib- ing a concentration near the mouth often requires ∂η + ∇ · Hu = 0 , (1) relaxation scheme that can be avoided if the bound- ∂t ary is far from the mouth. = + η Consequently, the downstream open boundaries are where H is the total water depth, i.e. H h where placed in the Makassar Strait, one in the northern part h is the unperturbed water depth. and the other in the southern part (Fig. 3). The horizontal momentum budget equation read: The computational domain is also extended up- ∂u stream. Firstly, imposing the discharges outside of the + u · ∇u + f k × u + g∇η tidal dominance is simple since the flow is always di- ∂t rected downstream and daily-averaged discharge can g||u||u τ s 1 =− + + ∇ · Hν (∇u) , (2) be imposed. Secondly, the three lakes (Fig. 1) may 2 ρ h Ch H H H act as a buffer and consequently affect the discharge regime of the river. where f is the Coriolis parameter: f = 2ω sin φ,where The hydrodynamics of the Makassar Strait and the ω is the Earth’s angular velocity and φ is the latitude. Mahakam Delta is simulated in 2D (depth-averaged Next, k is the upward unit vector, g is the gravitational equations). The 2D approach may seem questionable in acceleration, ρ is the water density (assumed constant) the Makassar Strait, but the domain of interest remains and νh is the horizontal eddy viscosity. For specifying the delta and the 2D shallow-water equations succeed the latter, the Smagorinsky (1963) parametrisation is in representing the propagation of the tide through the resorted to. This parametrisation depends on the local strait. Unfortunately, the 2D hypothesis may introduce mesh size and the spatial derivatives of the velocity: larger errors during strong wind conditions. Neverthe- less, due to its location, the impact of wind in the Makassar Strait is limited. Other components of the 2 2 2 ∂u ∂v ∂u ∂v flow, such as the Makassar throughflow, are assumed ν = (0.1)2 2 + 2 + + . h ∂x ∂y ∂y ∂x to have negligible interactions with the delta. Up- stream of the delta, in the river and its tributaries, the (3) Ocean Dynamics (2011) 61:1107–1120 1111

The surface stress vector is denoted by τ s and is esti- where n is the Manning coefficient, which depends on mated by means of the formula of Smith and Banke physical properties of the seabed and ranges from 0.017 (1975): to 0.023 s/m1/3(see Section 3). Friction along coasts and riverbanks is also taken into account by means of the −3 2 τ s = 10 0.630||w|| + 0.066||w|| w , (4) following formula: where w is the wind velocity vector at 10 m above the ∂u ν t = αu , (6) sea level. The parameter controlling the bottom friction h ∂n t is the Chézy coefficient Ch: where α is the slip coefficient (Haidvogel et al. 1991) 1/6 ∂ H ut = , and ∂n is the normal derivative of the tangential veloc- Ch (5) −3 n ity ut. A value of α = 2.5 × 10 m/s is chosen here.

Fig. 2 Bathymetry of the Mahakam Delta (in metres). The colorbar is cropped at 16 m. Easting and northing coordinates correspond to UTM50M. Red points give the location of the water level measurement stations 1112 Ocean Dynamics (2011) 61:1107–1120

Finally, the equation governing the evolution of a where x is the along-river distance. The section S(x, t) tracer concentration C is : depends on the position x and the elevation η in order ∂ to take into account the shape of the river and is in- HC + ∇ · HuC = ∇ · Hκ∇C , (7) ∂t terpolated from fieldwork campaigns. The momentum budget equation is: where κ is tracer diffusivity coefficient. In order to take into account the effect of the mesh size on the ∂u ∂u ∂η g|u|u 1 ∂ ∂u diffusivity, a parametrisation inspired by Okubo (1971) + u + g =− + ν S , (10) ∂ ∂ ∂ 2 ∂ h ∂ is used: t x x Ch H S x x κ = . 1.15 2/ . 0 03 m s (8) where here H = S/b is the effective depth and b(x, t) is the width of the river. Finally, the equation governing 2.2.2 The 1D equations the evolution of a tracer concentration C reads: The hydrodynamic equations solved in the 1D part of ∂ ∂ ∂ ∂C SC + SuC = Sκ . (11) the domain are the section-integrated shallow-water ∂t ∂x ∂x ∂x equations. The variables are the section-integrated velocity u and the section S. In its conservative form, the continuity equation read: 2.3 The model forcings ∂ S ∂ + (Su) = 0 , (9) The data needed to model the Mahakam delta are ∂t ∂x the bathymetry, the tidal forcing at downstream open

Fig. 3 Views of the mesh. Panel 1 shows the whole domain. Panel boundary between 1D and 2D meshes. The 1D mesh is presented 2 is a zoom on the Mahakam Delta. A closer view in the delta is with quadrangular elements instead of line elements in order to shown in panel 3. Finally, the panel 4 is a zoom on the lake region. depict the width of the river. The mesh contains 60×103 triangles The dashed lines in panels 2 and 4 indicate the location of the and 3,700 lines segments Ocean Dynamics (2011) 61:1107–1120 1113 boundaries, the river discharges at the upstream bound- 2.4 The numerical method and grid aries and the wind forcing: The model used in this paper is SLIM. Space derivatives – The bathymetry of the model is composed of are treated by means of the discontinuous Garlerkin different bathymetries from various sources. The (DG) finite-element method (FEM). The solution is ap- global one arc-minute grid GEBCO2 is used for proximated by piecewise polynomial functions for both the Makassar Strait. The resolution of the latter elevation and velocity. For this study, linear polynomi- does not permit to model the Mahakam Delta. als are employed. Therefore, solution exhibits discon- This is why this coarse bathymetry is replaced in tinuities along element boundaries, and the elements the delta by a much more accurate one obtained communicate by numerical fluxes computed with an ap- from fieldwork campaigns carried out in 2008–2009 proximate Riemann–Roe solver (e.g. Roe 1981;Toro (Fig. 2; Sassi et al. 2011). Wetting and drying 1997). Additional details about the numerical method processes being rather unimportant in the delta, are given in Comblen et al. (2009, 2010). The DG-FEM the bathymetry was cropped at 2 m everywhere in allows one to use unstructured meshes, enabling large order to avoid drying. To simulate the buffering variation of the space resolution. Figure 3 shows the effect of the lakes, the bathymetry of the latter was mesh used in this study. The latter was generated us- assumed to be 5 m. During strong discharge condi- ing GMSH (Geuzaine and Remacle 2009; Lambrechts tions, the extension of lakes may increase consid- et al. 2008a). The domain is partitioned into 60 × 103 erably. Such a process can be modelled by wetting triangles (for the 2D depth-averaged equations) and and drying algorithm but is not yet implemented in 3,700 line segments (for the 1D section-averaged equa- this model. tions). The refinement criterion is primarily based on – Tides are imposed at the shelf break using el- the celerity of the long surface gravity waves ( gh). evation and velocity harmonics of the global Preliminary runs show that the continental slope must ocean tidal model TPXO7.1. The latter produces be well represent. Indeed, as the depth varies from the best fits, in a least-squares sense, of the 2,000 to 100 minlessthan30 km across the continental Laplace tidal equations and along-track averaged slope, a resolution of 10 km used in the strait cannot data from Topex/Poseidon and Jason satellites (on be maintained without obtaining too many disconti- Topex/Poseidon tracks since 2002) obtained with nuities between the elements. Consequently, a second OTIS (Egbert et al. 1994).3 refinement taking into account the gradient of the – The meteorological forcings are wind fields at bathymetry was added (Legrand et al. 2007). Finally, 10 m above the sea level. These fields are four it was prescribed that each channel of the delta was times daily NCEP reanalysis data provided by the represented with at least four triangles across the width. NOAA/OAR/ESRL PSD (Kalnay et al. 1996).4 – Discharge was estimated from a horizontal acoustic Doppler current profilers (H-ADCP) deployed at the riverbank. H-ADCPs measure water level in Delta North 2 combination with flow velocity array data across observed modelled a river section. Velocities measured with the 1.5 H-ADCP were converted to river discharge using

conventional ADCP shipborne discharge measure- 1 ments. Five 13 H-ADCP campaigns spanning a

wide range of flow conditions were used to calibrate 0.5 and validate the discharge estimates. Details of

the procedures to convert flow velocity across the 0 river section into water discharge can be found in

Hoitink et al. (2009) and Sassi et al. (2011). −0.5

−1

−1.5 − − − − − − − − 2https://www.bodc.ac.uk/data/online_delivery/gebco/ 04 May 2008 11 May 2008 18 May 2008 25 May 2008 3 http://www.oce.orst.edu/research/po/research/tide/index.html Fig. 4 Time series of observed (blue curve) and modelled (red 4http://www.cdc.noaa.gov/cdc/data.ncep.reanalysis.surfaceflux.html curve) elevation at Delta North during May 2008 1114 Ocean Dynamics (2011) 61:1107–1120

Delta North Delta North 0.7 observed 360 observed modelled modelled 0.6 330 300 0.5 270 240 0.4 210 180 0.3 Phase [deg]

Amplitude [m] 150 120 0.2 90

0.1 60 30 0 0 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6

Fig. 5 Amplitudes (left panel) and phases (right panel)ofthe respectively. The error bars give the error estimation associated ten major tidal components, at Delta North; dark green and with the harmonic decomposition light green are associated with observations and modelled values,

The maximal size of triangles is around 10 km in the Figure 4 shows the time series of the observed and deepest part of the Makassar Strait while the size of modelled elevation for the station of Delta North. The the smallest triangles is small as 5 m in the narrowest spring-neap tidal cycle is clearly visible and well repre- branches of the delta (Fig. 3). The resolution in the 1D sented by the model. Indeed, the errors on the M2 and model is about 100 m. As regards the CPU time, it takes S2 components of the tide are smaller than 3 cm for the about 2.5 days to simulate 1 month on the mesh of Fig. 3 amplitudes and smaller than 10◦ for the phases (Fig. 5). using 16 processors in parallel. The other components of the tide shown on Fig. 5 are also represented with an acceptable accuracy. Part of the error can be explained by the absence in the mesh (Fig. 3) of the channel containing Delta North (Fig. 2) 3Results

3.1 Calibration Delta South 2 The calibration of the model is based on the compar- observed modelled ison in May 2008 of modelled and observed temporal 1.5 series of free surface elevation at three stations (Fig. 2). Two of them are located at the mouth of the delta 1 (namely Delta North and Delta South). The third one is located near the upstream limit of the delta, which 0.5 corresponds to the connection between the 1D and the 2D meshes. The calibration consists in adjusting 0 the Manning coefﬁcient in order to represent the tidal signal correctly. The comparison of the tidal part of the −0.5 observed and modelled elevation is rather easy if its based on harmonical decomposition. The latter allows −1 one to compare the amplitudes and phases of each tidal component separately. The tidal decomposition −1.5 − − − − − − − − was performed using the T_TIDE routine (Pawlowicz 04 May 2008 11 May 2008 18 May 2008 25 May 2008 et al. 2002), which includes an error estimation for the Fig. 6 Time series of observed (blue curve) and modelled (red analysed components. curve) elevation at Delta South during May 2008 Ocean Dynamics (2011) 61:1107–1120 1115

Delta South Delta South 0.6 observed 360 observed modelled modelled 330 0.5 300 270 0.4 240 210 0.3 180 Phase [deg]

Amplitude [m] 150 0.2 120 90 0.1 60 30 0 0 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6

Fig. 7 Amplitudes (left panel) and phases (right panel)ofthe respectively. The error bars give the error estimation associated ten major tidal components, the Delta South; dark green and with the harmonic decomposition light green are associated with observations and modelled values,

The errors obtained for the station of Delta South variation. Furthermore, the bathymetry being shallow (Figs. 6 and 7) are quite similar to those obtained for and varying rapidly in the delta channels, the errors the station of Delta North. The main difference lies on the advection and friction terms again increase the in the overestimation of the M2 amplitude at Delta errors on the non-linear tidal components. North while the M2 amplitude was underestimated at The calibration was first performed with a constant Delta South. The main part of the errors obtained Manning coefficient which did not allow to represent for both downstream stations can be explained by two the tide correctly. With the constant value, both the factors. Firstly, the GEBCO bathymetry may not be downstream stations can be represented correctly while accurate enough in the vicinity of the mouth. Secondly, the amplitudes are underestimated (see below) at the the model domain in the delta only includes the dis- upstream station. The Manning coefficient seems lower tributaries. The tidal channels that do not convey the discharge are not taken into account in this preliminary study though they may influence the tidal motion. Delta Apex Representing the tide at the station Delta Apex is 2 more difficult than for the two downstream stations. observed modelled Indeed, the tidal wave observed at Delta Apex propa- 1.5 gates through the delta and undergoes larger deforma- tions due to the higher complexity of the morphology 1 and the increased non-linearity of the physics in tidal channels. Nevertheless, the model seems to be in good 0.5 agreement with the observations (Fig. 8). The error affecting the amplitude of the M2 tide increases to 0 5 cm while the error on the phase remains around ◦ 10 (Fig. 9). For the S2 tide, the opposite is observed: −0.5 The amplitude is very well represented while the error ◦ on the phase reaches 15 . Again due to the increased −1 non-linearities, the higher harmonical components of the tide become apparent (M4 and M6). The model −1.5 04−May−2008 11−May−2008 18−May−2008 25−May−2008 seems to have difficulties to represent the M4 tide which can be explained by resolution limitations of Fig. 8 Time series of observed (blue curve) and modelled (red the mesh, bathymetry inaccuracies and bed roughness curve) elevation at Delta Apex during May 2008 1116 Ocean Dynamics (2011) 61:1107–1120

Delta Apex Delta Apex 0.6 observed 360 observed modelled modelled 330 0.5 300 270 0.4 240 210 0.3 180 Phase [deg]

Amplitude [m] 150 0.2 120 90 0.1 60 30 0 0 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6 M2 S2 N2 K2 K1 O1 P1 Q1 M4 M6

Fig. 9 Amplitudes (left panel) and phases (right panel)ofthe respectively. The error bars give the error estimation associated ten major tidal components, at the Delta Apex; dark green and with the harmonic decomposition light green are associated with observations and modelled values,

in the delta than in the open sea. Therefore, a constant value of 0.023 was used in the Makassar Strait. Then, the applied Manning coefficient decreases linearly to a value of 0.017 at the upstream part of the Delta. This variation takes into account in a simple way the gradual transition between marine and riverine environments. The value of 0.017 was also chosen for the Manning coefficient in the 1D riverine part.

3.2 Division of the Eulerian residual discharge though the channels

In order to quantify the residual discharge through the different channels of the delta, a simulation over May and June 2008 was performed during which the transport was integrated through sections located in each channel (which are numbered in Fig. 10). The computed discharge was then averaged over the 2 months considered giving the Eulerian residual discharge for the different channels. Figure 11 shows the division of the discharge crossing the first section (number 1 in the figure). After crossing the first section, the water enters channel 2 or 3 with 40% and 60% of the residual discharge (respectively). Since the discharges were integrated over 1 months only and due to the long-term motions and associated storage, small corrections of the order of 1% were applied to the unbalanced percent- ages. The number of branches directly connected to the sea amounts to 18. Only three branches exhibit a residual discharge proportion larger than 10%(chan- Fig. 10 Figure showing the numbers attributed to arbitrary cross nels 8, 11 and 37). Channels 8 and 11 correspond to sections. One cross section is defined in each channel Ocean Dynamics (2011) 61:1107–1120 1117

12–1% 47295000 4000 4–13% 9–10% 13–9% 3000 10–1% 6–29% 2000 2–40% 7–16% 11–18%

5–27% 1000

8–11%

18–1% 21–1%20-0% 500 16–4% 19–3%

22–8% 1–100% 14–20% 17–16% 23–5% 24–3%

29–8% 27–11% 25–15% 100 3–60% 30–3% 28–4% 33–5% 31–1%

15–40% 34–10% 50 37–13%

26–25% 32–24% 36–24% 38–2% 35–14% 40–2% 39–9% 41–7% 20

Fig. 11 Figure representing the splitting of the discharge (com- crossing each section. The f irst bar (on the left) contains obvi- puted as averaged flow during 2 months) through the delta. ously 100% of the discharge. Then flow splits into the branches 1 Figure 10 shows the number attributed to transects located in and 2 with 40% and 60% of the discharge, respectively, etc. The each channel. The vertical bars indicate the part of the discharge discharges values of each channels are given by the left bar

the largest branches in the northern part of the delta, The age of a water parcel is the time elapsed since whereas channel 37, located in the southern part, is not the considered water parcel leaves a region where its as wide as channels 8 and 11. A thorough analysis of the age is prescribed to be zero. To quantify the rate at subtidal discharge distribution between the different which water parcels flow through the delta, the age of branches of the delta is under preparation. the water leaving the apex of the delta was computed. In practice, since the connection of the 1D and the 2D model is located near the apex of the delta, the 3.3 Age of the upstream water age of the water exiting the 1D part of the domain to enter the 2D part of the domain was computed (Fig. 3). For environmental applications, one usually focusses Technically, the age of the water mass is computed by on the long-term transport. There are different ways resolving two advection–diffusion–reaction equations. to quantify long-term transport processes. One of them Figure 12 gives the average over May and June 2008 is to compute the Eulerian and Lagrangian resid- of the age of the water originating from the upstream ual transport (e.g. Wei et al. 2004; Liu et al. 2007; limit of the delta. This figure reveals that the time Muller et al. 2009). Alternatively one can resort to the taken by water parcel to reach the sea varies from 4 to Constituent-oriented Age and Residence time Theory 7 days depending on the channel. It is not a surprise that (www.climate.be/CART) to compute at any time and the largest channels are associated with the smallest position timescales such as the residence time, the mean ages. The channel 31 (Fig. 10) exhibits a different exposure time and the age of water. Contrary to the behaviour. The residual current in this channel is so residence time and the exposure time, for which an small that the mean age increases to a maximum value adjoint model is needed (Delhez et al. 2004), the age of 7 days with the downstream direction. Then the is quite easy to compute (Deleersnijder et al. 2001). mean age decreases up to the next bifurcation. This 1118 Ocean Dynamics (2011) 61:1107–1120

Fig. 12 Two-month averaged -0.3 age of the water originating from the upstream limit of the Mahakam Delta. The color bar is cropped at -0.35 7 days in order to focus on the delta variations -0.4

-0.45

-0.5

-0.55

-0.6

-0.65

-0.7

-0.75

-0.8

-0.85

-0.9

-0.95 117.2 117.3 117.4 117.5 117.6 Age [days] 0 3.5 7

behaviour is due to the mixing of the water with sections 1–3–15–26–35–39–41, the mean age does only different mean ages at the downstream splitting bifur- vary between 0 and 1.5 days during the first 32 km cation. A maximum is also observed in the channel (from the apex to the section 35). This reveals an 33 (Fig. 10). Here the reason seems to be the alter- important residual velocity in the upstream part of the nating upstream and downstream current due to the delta. Downstream of the km 32, the residual velocity tide that can transport younger water upstream. In the decreases and the mean age gradient increases. Indeed, largest discharging channels, the mean age gradient is along the last 15 km separating the section 35 from the not constant. For example, in the channel crossing the mouth, the mean age varies from 1.5 to 5 days. Ocean Dynamics (2011) 61:1107–1120 1119

4 Conclusion in the lakes (as well as in the rivers) must be calibrated. Finally, the wetting–drying described in Kärnä et al. A 2D/1D model of the Mahakam Delta was presented (2010) must be introduced in the computation in order in this paper. The specificity of this model is the wide to take into account the varying extension of the lakes. spectrum of length scales taken into consideration. To impose the tide correctly, the computational domain Acknowledgements The present study was carried out in the was extended through the Makassar Strait. This exten- framework of the project “Taking up the challenges of multi- sion allows the model to be forced by tidal components scale marine modelling”, which is funded by the Communauté obtained from a global ocean model, thereby avoiding Française de Belgique under contract ARC 10/15-028 (Actions issues classically associated with open boundaries lo- de recherche concertées) with the aim of developing and using SLIM (www.climate.be/slim). Eric Deleersnijder is a research cated too close to the actual domain of interest. The associate with the Belgian National Fund for Scientific Research Mahakam Delta is prolonged upstream by a river that (F.R.S-FNRS). eventually connects to three lakes and other tributaries. The computational domain was also extended to this upstream river by connecting the 2D part of the model (delta and strait) to a 1D river network. Part of the References branches of this network is finally connected to lakes which are modelled in 2D. By extending the domain so Comblen R, Lambrechts J, Remacle J-F, Legat V (2010) Prac- far upstream, daily-averaged discharges can be imposed tical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations. Int at upstream boundaries that are not affected the tidal J Numer Methods Fluids 63(6):701–724. http://dx.doi.org/ motion. 10.1002/ﬂd.2094 To our knowledge, this is the first time that such a Comblen R, Legrand S, Deleersnijder E, Legat V (2009) A finite multi-scale model is developed for the Mahakam land– element method for solving the shallow water equations on the sphere. Ocean Model 28:12–23. sea continuum. The previous 2D model of Mandang de Brye B, de Brauwere A, Gourgue O, Kärnä T, Lambrechts J, and Yanagi (2008) only deals with the delta and its near Comblen R, Deleersnijder E (2010) A finite-element, multi- coastal zone. Furthermore, its resolution of 200 m does scale model of the Scheldt tributaries, river, estuary and not permit to model the channels in great detail. The ROFI. Coastal Eng 57(9):850–863 Deleersnijder E, Campin JM, Delhez EJM (2001) The concept of resolution of SLIM for some of the channels falls is as age in marine modelling: I. Theory and preliminary model far as 5 m since a minimum of four triangles per cross results. J Mar Syst 28(3–4):229–267 section was required while generating the mesh. Deleersnijder E, Lermusiaux P (eds) (2008) Multi-scale model- The model was found to be able to represent rather ing: Nested-grid and unstructured-mesh approaches. Ocean Dyn 58:335–498 (Special Issue) well water elevations at three stations in the delta. Delhez EJM, Heemink AW, Deleersnijder E (2004) Residence For example, the error for the amplitude of the M2 time in a semi-enclosed domain from the solution of an ad- component of the tide varies between 3 and 5 cm. The joint problem. Estuar Coast Shelf Sci 61(4):691–702 Egbert G, Bennett A, Foreman M (1994) TOPEX/Poseidon tides error on the phase of the M2 tide does not exceed ◦ estimated using a global inverse model. J Geophys Res 10 . This notwithstanding, larger errors may arise at 99(C12):24,821–24,852 stations further upstream, primarily due the constant Geuzaine C, Remacle J-F (2009) GMSH a three-dimensional friction coefficient assumed all over the river and the finite element mesh generator with built-in pre- and post- oversimplification of the bathymetry in the lakes. processing facilities. Int. J Numer Methods Eng 79(11):1309– 1331 Two preliminary results permitting to quantify the Gordon A, Susanto R, Ffield A (1999) Throughflow within long-term transport through the different channels of Makassar Strait. Geophys Res Lett 26:3325–3328 the delta were presented. The splitting of the residual Haidvogel DB, McWilliams JC, Gent PR (1991) Boundary cur- discharge through the delta was presented as well as rent separation in a quasigeostrophic, eddy-resolving ocean circulation model. J Phys Oceanogr 22(8):882–902 the age of the water originating from the upstream Hoekman DH (2007) Satellite radar observation of tropical peat limit of the delta. Although residual transport exhibits swamp forest as a tool for hydrological modelling and en- its largest values (> 10%) in the widest channels, dis- vironmental protection. Aq Cons Mar and Freshw Ecosyst crepancies may arise due to the storage associated with 17(3):265–275 Hoitink A, Buschman F, Vermeulen B (2009) Continuous long-term motions, as suggested by the age of the water measurements of discharge from a horizontal acoustic mass in channel 31 (Fig. 12). Doppler current profiler in a tidal river. Water Resour Res Although the lakes are included in the mesh, more 45(11):W11406 developments are needed to represent them correctly. Kalnay E, Kanamitsua M, Kistlera R, Collinsa W, Deavena D, Gandina L, Iredella M, Sahaa S, Whitea G, Woollena Firstly, a comprehensive bathymetry of the lakes must J, Zhua Y, Leetmaaa A, Reynoldsa R, Chelliahb MW, be implemented. Then, the bottom friction coefficient Ebisuzakib HW, Janowiakb J, Mob KC, Ropelewskib C, 1120 Ocean Dynamics (2011) 61:1107–1120

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