Stratified Secondary Circulation in the Rio Magdalena, Colombia
M. Lindhart1,2, E. Mosselman2,3, and H. Avila4.1Technical University of Denmark, mathilde.lindhart@gmail.com. 2Deltares, the Netherlands
3Delft University of Technology, the Netherlands, erik.mosselman@deltares.nl. 4Universidad del Norte, Colombia, havila@uninorte.edu.co
1. Introduction
In classic theory of curvature-induced secondary circu-lation in river bends an equilibrium is assumed between the centrifugal acceleration, the barotropic pressure gradi-ent, and friction (Kalkwijk and Booij, 1986). Due to the depth-varying centrifugal acceleration, a circulation cell is set up, directed toward the outer bend in the upper wa-ter column, and toward the inner bend in the lower wawa-ter column. However, when considering an estuarine envi-ronment, baroclinic pressure gradients must be taken into account. In particular, the curvature-induced secondary circulation creates upwelling of saline water in the inner bend, and downwelling of fresh water in the outer bend, setting up a baroclinic pressure gradient towards the outer bend. This, in effect, works to set up a circulation cell of opposite orientation of the curvature-induced secondary circulation. The influence of stratification on transverse velocities in flow around a headland was studied by the seminal work of Geyer (1993), and Chant and Wilson (1997) documented the generation of a transverse baro-clinic pressure gradient in stratified estuarine flow to op-pose the effect of curvature. This was further observed by Lacy and Monismith (2001) and Nidzieko et al. (2009).
Figure 1: Mouth of the Rio Magdalena, Colombia. Sattellite image from Google Inc. (2015a). In this work the effect of stratified secondary circulation is investigated in the estuary of the Rio Magdalena in Colombia, using Delft3D. See Figure 1 for an overview of the estuary and the river bend studied and the location of the estuary in Colombia in 2. Through comparison of simulations with and without salinity stratification, this study seeks to characterise the effect stratification has on secondary circulation in the Rio Magdalena, as well as to describe the physical processes governing the local hydro-dynamics, and hence the morphohydro-dynamics, of the estuary.
Figure 2: Location in Colombia. Map from Google Inc. (2015b).
2. Method
A numerical 3D model of the Rio Magdalena estuary was set up in Delft3D, using 20 vertical, equidistant σ -layers. The model represents the lower reach of the river, as well as part of the coast and ocean boundary to enable the propagation of tides into the estuary. The simulations were run at a constant upstream boundary river discharge of Q = 3, 000 m3/s representing low river flow, and ocean water level boundaries were created using the TPX Global Inverse Tide model. Two scenarios were considered; one with a zero-value salinity on the entire model domain, and one with an upstream river boundary salinity of 0 ppt and ocean boundary salinity of 35 ppt. This allows for comparison of curvature-induced secondary circulation with and without the influence of salinity stratification. The month of January 2010 was simulated.
The resulting velocity profiles as well as each term in the cross-channel momentum balance are calculated to deter-mine the physical processes controlling secondary circu-lation in the estuary. The transverse momentum equation is given by Equation 1. ∂ un ∂ t = −us ∂ un ∂ s + u2s Rs −g∂ η ∂ n− g ρ0 Z 0 z ∂ ρ ∂ ndz− f us+ ∂ ∂ z Az ∂ un ∂ z (1) where s indicates the streamwise direction, n the
trans-verse direction, us (un) is the horizontal velocity in the
depth-averaged (transverse) direction, Rsis the radius of
curvature, ρ is the density, Azthe vertical eddy viscosity,
and η the surface water level.
3. Results
The Rio Magdalena is a microtidal estuary, with tidal amplitudes of ∼ 0.3 m in the simulated period, and a relatively large river discharge even during the dry sea-son. This creates highly stratified conditions throughout the period considered, while still allowing the salinity to intrude upstream of the river bend investigated in the study. The river bend is located approximately 4 -8 km into the estuary, while the salinity intrusion was approximately 11 km throughout the simulation, with limited spring/neap and ebb/flood variation.
When comparing the simulations with and without salinity, two particular phenomena are identified; the secondary circulation increases by an order of magnitude in the stratified estuary, and instead of a single circulation cell, multiple vertically stacked cells are observed. Both results deviate significantly from the predictions of classic theory of river bend flow.
3.1 Magnitude
Velocity profiles of usand un, the horizontal velocities in
the streamwise and transverse direction, are found in Fig-ure 2. In the simulations without salinity, us takes on a
classic logarithmic boundary layer profile. In the simula-tions with salinity, a shear flow is observed due to salinity stratification, with denser water intruding into the river in the lower water column (Figure 4, left panel).
Figure 3: Velocity profiles at the thalweg of section A, Figure 1. Left panel; Black/dark grey (blue/light grey) indicates mean and minimum/maximum values of us
simulation without (with) salinity. Middle panel; Black/dark grey indicates mean value and minimum/maximum of unwithout salinity. Red and blue
lines indicate analytical solutions from Kalkwijk and Booij (1986) with and without Coriolis. Right panel; Black/dark grey indicates mean value and minimum maximum of unwith salinity. Positive us(un) directed
towards the ocean (left, looking towards the ocean). In the simulation without salinity, the expected secondary circulation is observed, with a single circulation cell (Fig-ure 4, middle panel). These profiles correspond to the analytical solution of Kalkwijk and Booij (1986).
How-ever, when salinity is included, the secondary circulation increases by an order of magnitude (Figure 4, right panel). From evaluating the momentum terms, the centrifugal ac-celeration, proportional to u2s, is found to increase sig-nificantly in the stratified estuary as the stratification in-creases the shear in us. This increase in transverse forcing
produces larger transverse velocities. 3.2 Orientation
Figure 4, right panel, shows that the circulation pattern changes in the stratified estuary, roughly to two vertically stacked circulation cells. The upper cell has the expected counter-clockwise orientation, whereas the lower cell has a clockwise circulation, with velocities directed to the outer bend in the lower water column. Consider terms 2,3,4 and 5 of the right-hand side of Equation 1, found in Figure 4.
Figure 4: Vertical profiles of transverse momentum at section A, during spring flood. Terms 2-5 on the right-hand side of Equation 1. Positive usdirected
towards the left, looking towards the ocean. Evaluating the terms of the momentum balance gives in-sight into the underlying processes driving secondary cir-culation in the estuary. The secondary circir-culation creates upwelling of denser, saline water in the inner bend and downwelling of fresh water in the outer bend - setting up a baroclinic pressure gradient across the channel, with the same sign as the centrifugal acceleration. These are pri-marily balanced by the barotropic pressure gradient, as the viscosity term is found to be small in comparison, and Coriolis is negligible on the scale of the estuary. The com-bined forcing, with maximum in the upper and lower wa-ter column, creates the multi-layer flow observed in the velocity profiles.
4. Discussion
The results show that when salinity is included in the sim-ulation, secondary circulation increases, and the circula-tion pattern changes significantly. Although not included in this study, these effects are of relevance to studies of morphology and sediment transport in estuarine environ-ments. In particular, these effects should be taken into account in modelling estuaries, as assuming for example the parameterisation of secondary circulation when using a 2D horizontal model would not include the effects of stratification.
5. Conclusions
Simulations of the Rio Magdalena estuary comparing sec-ondary circulation with and without salinity stratification show that the stratification has two important effects; the magnitude of secondary circulation increases by an order of magnitude in the stratified estuary, and the circulation pattern changes from a single circulation cell to multi-ple, vertically stacked cells. The increase in magnitude is found to be due to larger transverse centrifugal accelera-tion as streamwise velocities increase under stratificaaccelera-tion. This sets up a baroclinic pressure gradient, which has a maximum in the lower water column, and forces the ve-locities along the river bed to the outer bend, creating a three-layer flow.
Acknowledgements
This study was carried out in collaboration between Deltares in the Netherlands and Universidad del Norte. The first author carried out this research for her final the-sis in partial fulfilment of the requirements for obtaining a master degree in environmental engineering at the Tech-nical University of Denmark.
References
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