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

Preliminary results of a finite-element, multi-scalemodel 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 riverflowing in the East Kalimantan province (BorneoIsland, Indonesia). A significant fraction of this river isinfluenced by tides, the modelling of which is the mainsubject of this study. Various physical and numericalissues must be addressed. In the upstream part of thedomain, the river flows through a region of three lakessurrounded by peat swamps. In the lowland regions,the river is meandering and its hydrodynamics is mostlyinfluenced by tides. The latter propagate upstream ofthe delta, in the main river and its tributaries. Finally,the mouth of the Mahakam is a delta exhibiting a

Responsible Editor: Phil Peter Dyke

This article is part of the Topical Collection on Joint NumericalSea Modelling Group Workshop 2010

B. de Brye (B) · S. Schellen · E. DeleersnijderInstitute of Mechanics, Materials and CivilEngineering (IMMC), Université catholique de Louvain,4 Avenue G. Lemaître, 1348 Louvain-la-Neuve, Belgiume-mail: benjamin.debrye@uclouvain.be

M. Sassi · B. Vermeulen · T. HoitinkHydrology and Quantitative Water Management Group,Department of Environmental Sciences,Wageningen University, Droevendaalsesteeg 4,Wageningen, Gld, The Netherlands

B. de Brye · S. Schellen · T. KärnäGeorges Lemaître Centre for Earth and ClimateResearch (TECLIM), Université catholique de Louvain,2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium

E. DeleersnijderEarth and Life Institute (ELI), Georges LemaîtreCentre for Earth and Climate Research (TECLIM),Université catholique de Louvain, 2 Chemin du Cyclotron,1348 Louvain-la-Neuve, Belgium

high number of channels connected to the MakassarStrait. This article focusses on the flow in the deltachannels, which is characterised by a wide range oftime and space scales. To capture most of them, thedepth-integrated and the section-integrated versions ofthe unstructured mesh, finite-element model Second-Generation Louvain-la-Neuve Ice-Ocean Model areused. Unstructured grids allow for a refinement of themesh in the narrowest channels and also an extensionof the domain upstream and downstream of the deltain order to prescribe the open-boundary conditions.The Makassar Strait, the Mahakam Delta and the threelakes are modelled with 2D elements. The rivers, fromthe upstream limit of the delta to the lakes and theupstream limit of the domain, are modelled in 1D.The calibration of the tidal elevation simulated in theMahakam Delta is presented. Preliminary results on thedivision of the Eulerian residual discharge through thechannels of the delta are also presented. Finally, as afirst-order description of the long-term transport, theage of the water originating from the upstream limitof the delta is computed. It is seen that for May andJune 2008, the time taken by the water parcel to crossthe estuary varies from 4 to 7 days depending on thechannel under consideration.

Keywords Finite element · Model · Mahakam River ·Multi-scale · Tide · Hydrodynamics

1 Introduction

Most of aquatic ecosystems are changing due toanthropic pressures (Millenium Ecosystem Manage-ment 2005). Among the changing ecosystems, tropical

1108 Ocean Dynamics (2011) 61:1107–1120

land–sea continua are probably the most affected. TheMahakam River is not an exception to the rule. This980-km-long tropical river flows in the East Kalimantanprovince (Borneo Island, Indonesia) and dischargesinto the Makassar Strait at the border of Pacific andIndian Oceans (Fig. 1). A hydrodynamic model of themain part of the Mahakam River is being developedwith the objectives of studying the impact of externalforcing factors such as sea-level rise, climate change,upstream sediment impact, as well as human interfer-ence on past, present and future development of theMahakam delta over different timescales.

Land–sea continua are particularly complex systemscharacterised by many physical processes interactingwith each other over a wide range of time and spacescales. The need for a multi-scale approach is justifiedby the differences between scales at which the ecosys-tem degradation is driven, scales at which the result-ing change to ecosystem functioning has most impact,scales at which ecosystems are managed and of coursescales at which we have the best understanding of the

physical and biological processes driving ecosystems.Moreover, the topography of the region of interest isvery intricate. It is appropriate to split the domain intosubdomains according to the typical time and spacescales of motion:

– The Makassar Strait, between the islands of Borneoand Sulawesi: Its width varies between 200 and250 km and its length is around 600 km. TheMakassar is an important strait in terms of theglobal conveyor belt. According to Gordon et al.(1999), the Makassar throughflow is, with an aver-age of 9.2 Sv for the year 1997, the main pathwayof water between the Pacific and Indian Ocean(Waworuntu et al. 2001; Susanto and Gordon 2005).Because the Makassar Strait is surrounded by twoislands and is located around the equator, strongwind events are rare. Moreover, the impact of mon-soonal winds is minimal as explained in Roberts andSydow (2003). The water level is therefore mainlycontrolled by tides. The two main components are

Fig. 1 The Mahakam River(Borneo Island, Indonesia).The estuarine part of theriver forms a delta. Theriver in its upstream part issurrounded by three lakes

Ocean Dynamics (2011) 61:1107–1120 1109

semidiurnal. The amplitude of M2 and S2 reaches0.55 and 0.35 m, respectively, resulting in a well-marked fortnightly cycle. In addition, diurnal com-ponents are also present in the Mahakam.

– At the mouth of the Mahakam River lies a re-gion of freshwater influence (Simpson 1997) with ahigh salinity gradient. With an average discharge of2,000 m3/s, one could expect that nearly no salinewater enters the delta from the Makassar Strait.Nevertheless, as was seen by Storms et al. (2005),there is a salt wedge in some delta branches inthe early hours of ebb tides. However, the salinityremains below 10 PSU. This was also observedduring extremely dry periods with a discharge lowerthan 400 m3/s. Such events appear periodicallysince the island of Borneo is strongly affected bythe El Niño–Southern Oscillation (ENSO) cycles(Kishimoto-Yamada and Itioka 2008).

– Figure 1 shows that the lower estuarine part ofthe Mahakam River is a delta exhibiting a largenumber of channels. One can distinguish tidal chan-nels without any connection with the riverine partand channels which are influenced by the riverinedischarge and the tidal regime. The width of thechannels ranges from 10 m to 3 km. The deltaextends 40 km on the continental shelf creating ahabitat for a rich but weak ecosystem. The envi-ronment of the delta suffers from an inappropriatedevelopment: conversion of mangrove forest, ex-tensive aquaculture and fishing industry, dredgingactivities etc. These transformations combined withexternal factors bring extreme events such as watershortage, floods and salinity intrusions and leadto urban inundations and problems for agricultureand drinking water supply. Predicting water level iscrucial for the sustainable development of the localeconomy.

– In the lowland regions, the hydrodynamics of themeandering river is mostly influenced by tides. Thelatter propagate upstream of the delta, in the mainriver and its tributaries over a distance exceeding200 km.

– In the upstream part of the domain, the river flowsthrough a region including three lakes surroundedby peat swamps. Typical to tropical rivers, suchswamps act as a buffer for the highly variabledischarge regime of the river. The destruction ofpeat swamps due to industrial and agricultural us-ages may disrupt the discharge regimes (Hoekman2007).

The main challenge of dealing with such a multi-scale domain is to resolve all the different processes at

their specific scale. The timescale ranges from minutesfor the tidal influence in the narrowest channels of thedelta to years for the ENSO variability. The lengthscale may be as small as a few metres in the narrow-est channels and reach hundreds of kilometres in theMakassar Strait. To the best of our knowledge, the onlyprevious study of the domain of interest was achievedby Mandang and Yanagi (2008). They developed a two-dimensional barotropic hydrodynamical model basedon a structured grid, using the finite difference method.They used a C-grid with a resolution of 200 m. Thewater level and velocity in the delta channels werecomputed. With such a procedure, the resolution wasgenerally insufficient in the delta, while the MakassarStrait was over-resolved. Despite the fact that finitedifference methods are easily implemented, they sufferfrom a lack of spatial flexibility. In addition, Mandangand Yanagi (2008) imposed the tidal elevation at onlyfive points on the continental shelf relatively close fromthe delta. It is generally appropriate to extend the com-putational domain out of the continental shelf and toa location where tidal forcing can be easily prescribed.For structured grid models, such an extension increasesthe computational coast significantly. Grid nesting isoften used to improve the performance of the model,but this method faces some drawbacks like unphysicalreflections and perturbations.

To deal with multi-physics and multi-scale models inspace and time, it is our conviction that unstructured-mesh modelling is a promising option. The mainadvantage is the spatial flexibility with a possiblerefinement in small channels, in shallow areas or acrossinclined bottom. The Second-Generation Louvain-la-Neuve Ice-Ocean Model (SLIM1) is able to copewith highly multi-scale applications (Deleersnijder andLermusiaux 2008; Lambrechts et al. 2008a) such as theGreat Barrier Reef (Lambrechts et al. 2008b) or theScheldt River Basin (de Brye et al. 2010). Therefore,SLIM is well suited to address the complexity of theMahakam River.

The next section presents the model setup: the de-scription of the domain, the equations, the forcings, thenumerical method and the grid. In Section 3, numericalresults of the model are presented. First the model iscalibrated against elevation data in the delta. Then, thedivision of the Eulerian discharge through the channelsof the delta is described. Finally, the age of the wateroriginated from the upstream limit of the estuary iscomputed. Concluding remarks are made in Section 4.

1www.climate.be/slim

1110 Ocean Dynamics (2011) 61:1107–1120

2 The model

2.1 The domain

Deltas, estuaries or bays exhibit open boundaries, re-quiring a relevant treatment. The positioning of theopen boundaries is crucial, and in many cases, thesimplest choice is not the most appropriate (de Bryeet al. 2010). For example, imposing boundary condi-tions near the mouth of the Mahakam Delta is nottrivial, and the extension of the domain far from themouth and up to the deep sea facilitates the task indifferent ways.

– In deep water, the equations are more linear andthe common problem of spurious local fluxes goinginto and out of the delta is generally avoided.

– External hydrodynamic data need to be imposedalong the open boundary. If no coarser coastalmodel is available for the region of interest, globaltidal model results may be resorted to.

– If the downstream boundary is sufficiently far fromthe domain of interest, the meteorological forcing(wind stress, atmospheric pressure, etc.) can beimposed as a surface flux only and be ignored inthe open boundary conditions.

– When modelling tracers such as salinity, prescrib-ing a concentration near the mouth often requiresrelaxation scheme that can be avoided if the bound-ary is far from the mouth.

Consequently, the downstream open boundaries areplaced in the Makassar Strait, one in the northern partand the other in the southern part (Fig. 3).

The computational domain is also extended up-stream. Firstly, imposing the discharges outside of thetidal dominance is simple since the flow is always di-rected downstream and daily-averaged discharge canbe imposed. Secondly, the three lakes (Fig. 1) mayact as a buffer and consequently affect the dischargeregime of the river.

The hydrodynamics of the Makassar Strait and theMahakam Delta is simulated in 2D (depth-averagedequations). The 2D approach may seem questionable inthe Makassar Strait, but the domain of interest remainsthe delta and the 2D shallow-water equations succeedin representing the propagation of the tide through thestrait. Unfortunately, the 2D hypothesis may introducelarger errors during strong wind conditions. Neverthe-less, due to its location, the impact of wind in theMakassar Strait is limited. Other components of theflow, such as the Makassar throughflow, are assumedto have negligible interactions with the delta. Up-stream of the delta, in the river and its tributaries, the

hydrodynamics is simulated by means of 1D, section-averaged equations. Finally, the three upstream lakesare modelled with the 2D depth-averaged equations.

2.2 The governing equations

In all equations, vectors are set in bold so as to dis-tinguish them from scalar variables. The equations tobe solved are the shallow-water equations in their 2Ddepth-integrated and 1D section-integrated versions.The equations and the parametrisations used in thisstudy are similar to those of de Brye et al. (2010) for theScheldt tidal continuum (The Netherlands/Belgium).The latter paper also describes the coupling betweenthe 2D part and the 1D part of the model as well asthe coupling of the different branches of the 1D rivernetwork.

2.2.1 The 2D equations

Let t be the elapsed time and ∇ the horizontal deloperator. The variables are the depth-integrated veloc-ity u = (u, v) and the free-surface elevation η. In itsconservative form, the continuity equation reads:

∂η

∂t+ ∇ ·

(Hu

)= 0 , (1)

where H is the total water depth, i.e. H = h + η whereh is the unperturbed water depth.

The horizontal momentum budget equation read:

∂u∂t

+ u · (∇u) + f k × u + g∇η

= −g||u||uC2

h H+ τ s

ρH+ 1

H∇ ·

(Hνh (∇u)

), (2)

where f is the Coriolis parameter: f = 2ω sin φ, whereω is the Earth’s angular velocity and φ is the latitude.Next, k is the upward unit vector, g is the gravitationalacceleration, ρ is the water density (assumed constant)and νh is the horizontal eddy viscosity. For specifyingthe latter, the Smagorinsky (1963) parametrisation isresorted to. This parametrisation depends on the localmesh size � and the spatial derivatives of the velocity:

νh = (0.1�)2

√2

(∂u∂x

)2

+ 2

(∂v

∂y

)2

+(

∂u∂y

+ ∂v

∂x

)2

.

(3)

Ocean Dynamics (2011) 61:1107–1120 1111

The surface stress vector is denoted by τ s and is esti-mated by means of the formula of Smith and Banke(1975):

τ s = 10−3(0.630||w|| + 0.066||w||2) w , (4)

where w is the wind velocity vector at 10 m above thesea level. The parameter controlling the bottom frictionis the Chézy coefficient Ch:

Ch = H1/6

n, (5)

where n is the Manning coefficient, which depends onphysical properties of the seabed and ranges from 0.017to 0.023 s/m1/3(see Section 3). Friction along coasts andriverbanks is also taken into account by means of thefollowing formula:

νh∂ut

∂n= αut , (6)

where α is the slip coefficient (Haidvogel et al. 1991)and ∂ut

∂n is the normal derivative of the tangential veloc-ity ut. A value of α = 2.5 × 10−3 m/s is chosen here.

Fig. 2 Bathymetry of theMahakam Delta (in metres).The colorbar is cropped at16 m. Easting and northingcoordinates correspond toUTM50M. Red points givethe location of the water levelmeasurement stations

1112 Ocean Dynamics (2011) 61:1107–1120

Finally, the equation governing the evolution of atracer concentration C is :

∂

∂t

(HC

)+ ∇ ·

(HuC

)= ∇ ·

(Hκ∇C

), (7)

where κ is tracer diffusivity coefficient. In order totake into account the effect of the mesh size on thediffusivity, a parametrisation inspired by Okubo (1971)is used:

κ = 0.03�1.15m2/s . (8)

2.2.2 The 1D equations

The hydrodynamic equations solved in the 1D part ofthe domain are the section-integrated shallow-waterequations. The variables are the section-integrated ve-locity u and the section S. In its conservative form, thecontinuity equation read:

∂S∂t

+ ∂

∂x(Su) = 0 , (9)

where x is the along-river distance. The section S(x, t)depends on the position x and the elevation η in orderto take into account the shape of the river and is in-terpolated from fieldwork campaigns. The momentumbudget equation is:

∂u∂t

+ u∂u∂x

+ g∂η

∂x= −g|u|u

C2h H

+ 1

S∂

∂x

(νhS

∂u∂x

), (10)

where here H = S/b is the effective depth and b(x, t)is the width of the river. Finally, the equation governingthe evolution of a tracer concentration C reads:

∂

∂t

(SC

)+ ∂

∂x

(SuC

)= ∂

∂x

(Sκ

∂C∂x

). (11)

2.3 The model forcings

The data needed to model the Mahakam delta arethe bathymetry, the tidal forcing at downstream open

Fig. 3 Views of the mesh. Panel 1 shows the whole domain. Panel2 is a zoom on the Mahakam Delta. A closer view in the delta isshown in panel 3. Finally, the panel 4 is a zoom on the lake region.The dashed lines in panels 2 and 4 indicate the location of the

boundary between 1D and 2D meshes. The 1D mesh is presentedwith quadrangular elements instead of line elements in order todepict the width of the river. The mesh contains 60×103 trianglesand 3,700 lines segments

Ocean Dynamics (2011) 61:1107–1120 1113

boundaries, the river discharges at the upstream bound-aries and the wind forcing:

– The bathymetry of the model is composed ofdifferent bathymetries from various sources. Theglobal one arc-minute grid GEBCO2 is used forthe Makassar Strait. The resolution of the latterdoes not permit to model the Mahakam Delta.This is why this coarse bathymetry is replaced inthe delta by a much more accurate one obtainedfrom fieldwork campaigns carried out in 2008–2009(Fig. 2; Sassi et al. 2011). Wetting and dryingprocesses being rather unimportant in the delta,the bathymetry was cropped at 2 m everywhere inorder to avoid drying. To simulate the bufferingeffect of the lakes, the bathymetry of the latter wasassumed to be 5 m. During strong discharge condi-tions, the extension of lakes may increase consid-erably. Such a process can be modelled by wettingand drying algorithm but is not yet implemented inthis model.

– Tides are imposed at the shelf break using el-evation and velocity harmonics of the globalocean tidal model TPXO7.1. The latter producesthe best fits, in a least-squares sense, of theLaplace tidal equations and along-track averageddata from Topex/Poseidon and Jason satellites (onTopex/Poseidon tracks since 2002) obtained withOTIS (Egbert et al. 1994).3

– The meteorological forcings are wind fields at10 m above the sea level. These fields are fourtimes daily NCEP reanalysis data provided by theNOAA/OAR/ESRL PSD (Kalnay et al. 1996).4

– Discharge was estimated from a horizontal acousticDoppler current profilers (H-ADCP) deployed atthe riverbank. H-ADCPs measure water level incombination with flow velocity array data acrossa river section. Velocities measured with theH-ADCP were converted to river discharge usingconventional ADCP shipborne discharge measure-ments. Five 13 H-ADCP campaigns spanning awide range of flow conditions were used to calibrateand validate the discharge estimates. Details ofthe procedures to convert flow velocity across theriver section into water discharge can be found inHoitink et al. (2009) and Sassi et al. (2011).

2https://www.bodc.ac.uk/data/online_delivery/gebco/3http://www.oce.orst.edu/research/po/research/tide/index.html4http://www.cdc.noaa.gov/cdc/data.ncep.reanalysis.surfaceflux.html

2.4 The numerical method and grid

The model used in this paper is SLIM. Space derivativesare treated by means of the discontinuous Garlerkin(DG) finite-element method (FEM). The solution is ap-proximated by piecewise polynomial functions for bothelevation and velocity. For this study, linear polynomi-als are employed. Therefore, solution exhibits discon-tinuities along element boundaries, and the elementscommunicate by numerical fluxes computed with an ap-proximate Riemann–Roe solver (e.g. Roe 1981; Toro1997). Additional details about the numerical methodare given in Comblen et al. (2009, 2010). The DG-FEMallows one to use unstructured meshes, enabling largevariation of the space resolution. Figure 3 shows themesh used in this study. The latter was generated us-ing GMSH (Geuzaine and Remacle 2009; Lambrechtset al. 2008a). The domain is partitioned into 60 × 103

triangles (for the 2D depth-averaged equations) and3,700 line segments (for the 1D section-averaged equa-tions). The refinement criterion is primarily based onthe celerity of the long surface gravity waves (

√gh).

Preliminary runs show that the continental slope mustbe well represent. Indeed, as the depth varies from2,000 to 100 m in less than 30 km across the continentalslope, a resolution of 10 km used in the strait cannotbe maintained without obtaining too many disconti-nuities between the elements. Consequently, a secondrefinement taking into account the gradient of thebathymetry was added (Legrand et al. 2007). Finally,it was prescribed that each channel of the delta wasrepresented with at least four triangles across the width.

04−May−2008 11−May−2008 18−May−2008 25−May−2008−1.5

−1

−0.5

0

0.5

1

1.5

2Delta North

observedmodelled

Fig. 4 Time series of observed (blue curve) and modelled (redcurve) elevation at Delta North during May 2008

1114 Ocean Dynamics (2011) 61:1107–1120

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Am

plitu

de [m

]

observedmodelled

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

30

60

90

120

150

180

210

240

270

300

330

360

Pha

se [d

eg]

Delta NorthDelta North

observedmodelled

Fig. 5 Amplitudes (left panel) and phases (right panel) of theten major tidal components, at Delta North; dark green andlight green are associated with observations and modelled values,

respectively. The error bars give the error estimation associatedwith the harmonic decomposition

The maximal size of triangles is around 10 km in thedeepest part of the Makassar Strait while the size ofthe smallest triangles is small as 5 m in the narrowestbranches of the delta (Fig. 3). The resolution in the 1Dmodel is about 100 m. As regards the CPU time, it takesabout 2.5 days to simulate 1 month on the mesh of Fig. 3using 16 processors in parallel.

3 Results

3.1 Calibration

The calibration of the model is based on the compar-ison in May 2008 of modelled and observed temporalseries of free surface elevation at three stations (Fig. 2).Two of them are located at the mouth of the delta(namely Delta North and Delta South). The third oneis located near the upstream limit of the delta, whichcorresponds to the connection between the 1D andthe 2D meshes. The calibration consists in adjustingthe Manning coefficient in order to represent the tidalsignal correctly. The comparison of the tidal part of theobserved and modelled elevation is rather easy if itsbased on harmonical decomposition. The latter allowsone to compare the amplitudes and phases of eachtidal component separately. The tidal decompositionwas performed using the T_TIDE routine (Pawlowiczet al. 2002), which includes an error estimation for theanalysed components.

Figure 4 shows the time series of the observed andmodelled elevation for the station of Delta North. Thespring-neap tidal cycle is clearly visible and well repre-sented by the model. Indeed, the errors on the M2 andS2 components of the tide are smaller than 3 cm for theamplitudes and smaller than 10◦ for the phases (Fig. 5).The other components of the tide shown on Fig. 5 arealso represented with an acceptable accuracy. Part ofthe error can be explained by the absence in the mesh(Fig. 3) of the channel containing Delta North (Fig. 2)

04−May−2008 11−May−2008 18−May−2008 25−May−2008−1.5

−1

−0.5

0

0.5

1

1.5

2Delta South

observedmodelled

Fig. 6 Time series of observed (blue curve) and modelled (redcurve) elevation at Delta South during May 2008

Ocean Dynamics (2011) 61:1107–1120 1115

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [m

]Delta South

observedmodelled

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

30

60

90

120

150

180

210

240

270

300

330

360

Pha

se [d

eg]

Delta South

observedmodelled

Fig. 7 Amplitudes (left panel) and phases (right panel) of theten major tidal components, the Delta South; dark green andlight green are associated with observations and modelled values,

respectively. The error bars give the error estimation associatedwith the harmonic decomposition

The errors obtained for the station of Delta South(Figs. 6 and 7) are quite similar to those obtained forthe station of Delta North. The main difference liesin the overestimation of the M2 amplitude at DeltaNorth while the M2 amplitude was underestimated atDelta South. The main part of the errors obtainedfor both downstream stations can be explained by twofactors. Firstly, the GEBCO bathymetry may not beaccurate enough in the vicinity of the mouth. Secondly,the model domain in the delta only includes the dis-tributaries. The tidal channels that do not convey thedischarge are not taken into account in this preliminarystudy though they may influence the tidal motion.

Representing the tide at the station Delta Apex ismore difficult than for the two downstream stations.Indeed, the tidal wave observed at Delta Apex propa-gates through the delta and undergoes larger deforma-tions due to the higher complexity of the morphologyand the increased non-linearity of the physics in tidalchannels. Nevertheless, the model seems to be in goodagreement with the observations (Fig. 8). The erroraffecting the amplitude of the M2 tide increases to5 cm while the error on the phase remains around10◦ (Fig. 9). For the S2 tide, the opposite is observed:The amplitude is very well represented while the erroron the phase reaches 15◦. Again due to the increasednon-linearities, the higher harmonical components ofthe tide become apparent (M4 and M6). The modelseems to have difficulties to represent the M4 tidewhich can be explained by resolution limitations ofthe mesh, bathymetry inaccuracies and bed roughness

variation. Furthermore, the bathymetry being shallowand varying rapidly in the delta channels, the errorson the advection and friction terms again increase theerrors on the non-linear tidal components.

The calibration was first performed with a constantManning coefficient which did not allow to representthe tide correctly. With the constant value, both thedownstream stations can be represented correctly whilethe amplitudes are underestimated (see below) at theupstream station. The Manning coefficient seems lower

04−May−2008 11−May−2008 18−May−2008 25−May−2008−1.5

−1

−0.5

0

0.5

1

1.5

2Delta Apex

observedmodelled

Fig. 8 Time series of observed (blue curve) and modelled (redcurve) elevation at Delta Apex during May 2008

1116 Ocean Dynamics (2011) 61:1107–1120

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [m

]Delta Apex

observedmodelled

M2 S2 N2 K2 K1 O1 P1 Q1 M4 M60

30

60

90

120

150

180

210

240

270

300

330

360

Pha

se [d

eg]

Delta Apex

observedmodelled

Fig. 9 Amplitudes (left panel) and phases (right panel) of theten major tidal components, at the Delta Apex; dark green andlight green are associated with observations and modelled values,

respectively. The error bars give the error estimation associatedwith the harmonic decomposition

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

3.2 Division of the Eulerian residual dischargethough the channels

In order to quantify the residual discharge through thedifferent channels of the delta, a simulation over Mayand June 2008 was performed during which the trans-port was integrated through sections located in eachchannel (which are numbered in Fig. 10). The com-puted discharge was then averaged over the 2 monthsconsidered giving the Eulerian residual discharge forthe different channels. Figure 11 shows the division ofthe discharge crossing the first section (number 1 inthe figure). After crossing the first section, the waterenters channel 2 or 3 with 40% and 60% of the resid-ual discharge (respectively). Since the discharges wereintegrated over 1 months only and due to the long-termmotions and associated storage, small corrections of theorder of 1% were applied to the unbalanced percent-ages. The number of branches directly connected tothe sea amounts to 18. Only three branches exhibit aresidual discharge proportion larger than 10% (chan-nels 8, 11 and 37). Channels 8 and 11 correspond to

Fig. 10 Figure showing the numbers attributed to arbitrary crosssections. One cross section is defined in each channel

Ocean Dynamics (2011) 61:1107–1120 1117

1–100%

2–40%

3–60%

4–13%

5–27%

6–29%

7–16%

8–11%

9–10%

10–1%

11–18%

12–1%

13–9%

14–20%

15–40%

16–4%

17–16%

18–1%19–3%

20-0%21–1%

22–8%

23–5%

24–3%

25–15%

26–25%

27–11%

28–4%

29–8%

30–3%

31–1%

32–24%

33–5%

34–10%

35–14%

36–24%

37–13%

38–2%

39–9%

40–2%

41–7%

4729

20

50

100

500

1000

2000

3000

4000

5000

Fig. 11 Figure representing the splitting of the discharge (com-puted as averaged flow during 2 months) through the delta.Figure 10 shows the number attributed to transects located ineach channel. The vertical bars indicate the part of the discharge

crossing each section. The f irst bar (on the left) contains obvi-ously 100% of the discharge. Then flow splits into the branches 1and 2 with 40% and 60% of the discharge, respectively, etc. Thedischarges values of each channels are given by the left bar

the largest branches in the northern part of the delta,whereas channel 37, located in the southern part, is notas wide as channels 8 and 11. A thorough analysis of thesubtidal discharge distribution between the differentbranches of the delta is under preparation.

3.3 Age of the upstream water

For environmental applications, one usually focusseson the long-term transport. There are different waysto quantify long-term transport processes. One of themis to compute the Eulerian and Lagrangian resid-ual transport (e.g. Wei et al. 2004; Liu et al. 2007;Muller et al. 2009). Alternatively one can resort to theConstituent-oriented Age and Residence time Theory(www.climate.be/CART) to compute at any time andposition timescales such as the residence time, theexposure time and the age of water. Contrary to theresidence time and the exposure time, for which anadjoint model is needed (Delhez et al. 2004), the ageis quite easy to compute (Deleersnijder et al. 2001).

The age of a water parcel is the time elapsed sincethe considered water parcel leaves a region where itsage is prescribed to be zero. To quantify the rate atwhich water parcels flow through the delta, the age ofthe water leaving the apex of the delta was computed.In practice, since the connection of the 1D and the2D model is located near the apex of the delta, theage of the water exiting the 1D part of the domain toenter the 2D part of the domain was computed (Fig. 3).Technically, the age of the water mass is computed byresolving two advection–diffusion–reaction equations.

Figure 12 gives the average over May and June 2008of the age of the water originating from the upstreamlimit of the delta. This figure reveals that the timetaken by water parcel to reach the sea varies from 4 to7 days depending on the channel. It is not a surprise thatthe largest channels are associated with the smallestmean ages. The channel 31 (Fig. 10) exhibits a differentbehaviour. The residual current in this channel is sosmall that the mean age increases to a maximum valueof 7 days with the downstream direction. Then themean age decreases up to the next bifurcation. This

1118 Ocean Dynamics (2011) 61:1107–1120

Fig. 12 Two-month averagedage of the water originatingfrom the upstream limit ofthe Mahakam Delta. Thecolor bar is cropped at7 days in order to focuson the delta variations

117.2 117.3 117.4 117.5 117.6

-0.95

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

0 3.5 7

Age [days]

behaviour is due to the mixing of the water withdifferent mean ages at the downstream splitting bifur-cation. A maximum is also observed in the channel33 (Fig. 10). Here the reason seems to be the alter-nating upstream and downstream current due to thetide that can transport younger water upstream. In thelargest discharging channels, the mean age gradient isnot constant. For example, in the channel crossing the

sections 1–3–15–26–35–39–41, the mean age does onlyvary between 0 and 1.5 days during the first 32 km(from the apex to the section 35). This reveals animportant residual velocity in the upstream part of thedelta. Downstream of the km 32, the residual velocitydecreases and the mean age gradient increases. Indeed,along the last 15 km separating the section 35 from themouth, the mean age varies from 1.5 to 5 days.

Ocean Dynamics (2011) 61:1107–1120 1119

4 Conclusion

A 2D/1D model of the Mahakam Delta was presentedin this paper. The specificity of this model is the widespectrum of length scales taken into consideration. Toimpose the tide correctly, the computational domainwas extended through the Makassar Strait. This exten-sion allows the model to be forced by tidal componentsobtained from a global ocean model, thereby avoidingissues classically associated with open boundaries lo-cated too close to the actual domain of interest. TheMahakam Delta is prolonged upstream by a river thateventually connects to three lakes and other tributaries.The computational domain was also extended to thisupstream river by connecting the 2D part of the model(delta and strait) to a 1D river network. Part of thebranches of this network is finally connected to lakeswhich are modelled in 2D. By extending the domain sofar upstream, daily-averaged discharges can be imposedat upstream boundaries that are not affected the tidalmotion.

To our knowledge, this is the first time that such amulti-scale model is developed for the Mahakam land–sea continuum. The previous 2D model of Mandangand Yanagi (2008) only deals with the delta and its nearcoastal zone. Furthermore, its resolution of 200 m doesnot permit to model the channels in great detail. Theresolution of SLIM for some of the channels falls is asfar as 5 m since a minimum of four triangles per crosssection was required while generating the mesh.

The model was found to be able to represent ratherwell water elevations at three stations in the delta.For example, the error for the amplitude of the M2

component of the tide varies between 3 and 5 cm. Theerror on the phase of the M2 tide does not exceed10◦. This notwithstanding, larger errors may arise atstations further upstream, primarily due the constantfriction coefficient assumed all over the river and theoversimplification of the bathymetry in the lakes.

Two preliminary results permitting to quantify thelong-term transport through the different channels ofthe delta were presented. The splitting of the residualdischarge through the delta was presented as well asthe age of the water originating from the upstreamlimit of the delta. Although residual transport exhibitsits largest values (> 10%) in the widest channels, dis-crepancies may arise due to the storage associated withlong-term motions, as suggested by the age of the watermass in channel 31 (Fig. 12).

Although the lakes are included in the mesh, moredevelopments are needed to represent them correctly.Firstly, a comprehensive bathymetry of the lakes mustbe implemented. Then, the bottom friction coefficient

in the lakes (as well as in the rivers) must be calibrated.Finally, the wetting–drying described in Kärnä et al.(2010) must be introduced in the computation in orderto take into account the varying extension of the lakes.

Acknowledgements The present study was carried out in theframework of the project “Taking up the challenges of multi-scale marine modelling”, which is funded by the CommunautéFrançaise de Belgique under contract ARC 10/15-028 (Actionsde recherche concertées) with the aim of developing and usingSLIM (www.climate.be/slim). Eric Deleersnijder is a researchassociate with the Belgian National Fund for Scientific Research(F.R.S-FNRS).

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