Are there internal Kelvin waves in Lake Tanganyika?



[1] It is generally believed that the Earth's rotation has negligible impact on the water circulation in basins which are very narrow or located near the Equator. However, herein evidence is presented of the influence of the Earth's rotation on the hydrodynamics of Lake Tanganyika, which is both very narrow (width/length ≈ 0.08) and located near the Equator. Numerical simulations exhibit small upwellings at the western shores as a result of the thermocline oscillations induced by the southeasterly winds of the dry season. These structures tend to propagate cyclonically around the lake similar to internal Kelvin waves. Numerical experiments in which f is varied concludes that internal Kelvin waves are present in Lake Tanganyika. It is also evidenced from this study that the internal Kelvin waves cannot be anticipated based on classic scaling arguments.

1. Introduction

[2] Lake Tanganyika (3°20′ to 8°45′ S and 29°05′ to 31°15′ E), is a large freshwater rift lake (Figure 1), whose mean width, length and mean depth are about 50 km, 650 km and 570 m, respectively. The lake is shared by 4 developing countries, i.e., Congo, Burundi, Tanzania and Zambia. The latter depend on it for food and drinking water, and discharge waste water in it. This is why it is important to understand the circulation and mixing in the near shore region.

Figure 1.

The map of Lake Tanganyika. The arc length shown by arrows around the lake used in Figure 2. The six points in each basin are used in Figures 35.

[3] In Lake Tanganyika, thermal stratification is well marked and varies seasonally above a permanently anoxic hypolimnion [Coulter, 1968]. The thermal structure and circulation in the lake depend largely upon the dry season (May–August/September) strong southeast winds [Coulter, 1968; Naithani et al., 2002, 2003]. The wind stress pushes surface water away from the southern end, thereby inducing upwelling in this region of cold and nutrient rich bottom water. While the thermocline becomes shallower in the southern part of the lake, it deepens at the opposite end. Superimposed on this movements are thermocline oscillations that are present all year round [Coulter and Spigel, 1991; Naithani et al., 2002, 2003].

[4] As the epilimnion is much shallower than the hypolimnion, the displacements of the thermocline may be studied in the framework of the so-called reduced-gravity model [Naithani et al., 2002, 2003]. The latter focuses on the upper layer and allows various types of waves, including internal inertia-gravity waves which are governed by inertia, Coriolis and gravitational forces. Internal Kelvin waves (hereafter Kelvin Waves), are those internal inertia-gravity waves which are trapped in the vicinity of the shore. For this type of motion, the cross-shore velocity is zero, the geostrophic equilibrium prevails in the cross-shore direction, and the amplitude of the motion decreases exponentially as the distance to the shore increases. The length scale in the cross-shore direction is known as the internal Rossby radius, which is defined as R = ∣f−1equation image, where f, ɛ, g and h denote the Coriolis factor, the relative density difference between the lower and upper water layer, the gravitational acceleration and the equilibrium depth of the pycnocline, respectively; the Coriolis factor is defined as f = 2Ω sin θ, where Ω ≈ 7.3 × 10−5 s−1 is the angular velocity associated with the rotation of the Earth and θ is the latitude, which is positive (negative) in the Northern (Southern) Hemisphere. In the Southern (Northern) Hemisphere, Kelvin waves tend to propagate along the boundary of a lake in the clockwise (counterclockwise) direction with a phase speed of the order of equation image.

[5] For motions that can be interpreted as Kelvin waves to develop in a lake, it is widely believed that its width has to be sufficiently larger than the Rossby radius [Mortimer, 1974; Csanady, 1982; Antenucci and Imberger, 2001]: according to Mortimer [1974] the width of the lake must be of the order of at least 5R for the rotational effects to become significant and greater than 20R for them to become dominant. The mean width of Lake Tanganyika is 50 km, while the Rossby radius is of the order of 20 and 70 km at the southern and northern ends, respectively. In other words, the internal Rossby radius is not much smaller than the width of the lake, so that the Earth's rotation should have no discernible influence. This is to say that according to the classic scaling arguments internal Kelvin waves cannot exist in Lake Tanganyika owing to its extreme narrowness and nearness to the Equator. This might be due to the fact that most of the work done towards the understanding of this phenomena pertains to temperate lakes and no great lake near the Equator is studied [Coulter and Spigel, 1991].

[6] According to the discussion above, Kelvin waves should not be identified in Lake Tanganyika, except, perhaps, near the southern tip of the lake, where R is smallest. Surprisingly, numerical simulations of thermocline displacements carried out by means of a reduced-gravity model suggest that thermocline displacements are far from being homogeneous in the lateral direction. From time to time, structures exhibiting a crest or a trough at the shore tend to propagate along the boundary of the lake in the clockwise direction. In this study we plan to investigate if the Earth's rotation do play a role in the circulation of Lake Tanganyika and if the internal Kelvin waves do exist in the lake.

2. Model Description

[7] In using a reduced-gravity model, one assumes that the density stratification is much more important in determining the internal oscillations than the underlying bottom topography. In Lake Tanganyika, it is appropriate to have recourse to such a model since stratification is present all year round and the depth of the surface mixed layer is generally much smaller than the thickness of the hypolimnion. The effectiveness of a two-layer reduced-gravity model in explaining the seasonal and intraseasonal variability in Lake Tanganyika level, dynamic height, upwelling in the upper layer, thermocline oscillations and the origin of oscillations have been demonstrated earlier [Naithani et al., 2002, 2003]. This type of model has been proved to be useful in explaining many observed features in the upper layer of oceans and lakes, including upwelling [Busalacchi and O'Brien, 1981; Luthar and O'Brien, 1985; Luthar et al., 1990; de Young et al., 1993; Inoue and Welsh, 1993; Handoh and Bigg, 2001].

[8] The detailed description of the model can be found in [Naithani et al., 2003]. A brief model summary is given here. In the model domain, the x-axis is along the width of the lake while the y-axis is along the length of the lake. The governing equations of the reduced-gravity model are:

display math
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where ξ is the downward displacement of the thermocline; u and v are the depth-averaged velocity components in the surface layer in the x- and y-directions, κx and κy are the eddy viscosities in the x- and y-directions, respectively; τ is the along-lake wind stress. In certain model runs, the upward displacement of the thermocline can be equivalent to the unperturbed depth of the thermocline. To prevent the thermocline from outcropping, an elementary wetting-drying algorithm is implemented [Balzano, 1998], consisting in setting to zero the water fluxes crossing the boundaries of every grid box in which the actual water column depth would become — after one time step — smaller than a critical value, which is taken to be Hmin = 5 m herein.

[9] Equations (1)(3) are discretised on a Cartesian Arakawa's C grid, with Δx = 3 km and Δy = 10 km. The model uses the forward-backward time stepping, with a time increment of 5 minutes. At the initial instant the thermocline displacement and velocity components are taken to be zero. The temperature of the upper and lower layer are set to 26 and 23.5°C, respectively, implying that relative density difference is 6.3 * 10−4.

3. Results

[10] The model is forced with the along-lake wind stress calculated from the wind observed at Mpulungu (8°45′S; 31°6′E) for a one year period, from April 1993 until March 1994. To determine the direction and speed of propagation of structures believed to be Kelvin wave packets, the thermocline displacement at the shore is displayed as a function of time and arc length of the shore increasing in the clockwise direction (Figure 2a). In these diagrams, lines with a slope equal to equation image, the theoretical velocity of Kelvin waves, are also displayed. Kelvin waves should lead to isolines of the thermocline displacement that tend to be parallel to these lines. Figure 2a suggests that for some periods of time and along some portions of lake's shores, there are indeed signals propagating with the phase speed of Kelvin waves. If the width-averaged displacement of the thermocline is displayed in the same type of diagram (Figure 2b), there is virtually no isoline parallel to the lines with equation image slope, suggesting that the motions propagating with a phase speed equal to equation image are, like Kelvin waves, trapped at the coast. As the Rossby radius depends on f, which is a function of latitude, a sensitivity model run is made in which the lake is assumed to be on the Equator (f = 0). The Rossby radius became infinite with f = 0. In this case, the model results exhibit no features propagating with the appropriate phase speed along the shores of the lake (Figures 2c and 2d). By contrast, if the lake is shifted southward so that its center is at about 20°S (Figures 2e and 2f), thereby making the Rossby radius significantly smaller than it is in the actual lake, the structures propagating as Kelvin waves are more pronounced and pervasive. To test the impact of the variation of f with latitude, i.e., the so-called beta effect, f is kept constant throughout the lake, at the value corresponding to a latitude of 6°S. Then, the model yields results which are not much different from those obtained with the actual Coriolis force (Figures 2g and 2h), implying that the beta effect is not crucial for the phenomenon we are looking at. All the results discussed above are consistent with the existence of Kelvin waves in Lake Tanganyika.

Figure 2.

Thermocline displacement at the shore vs time and arc length of the shore. a–b, actual lake position. c–d, Coriolis force = 0. e–f, lake shifted to 20°S. g–h, Coriolis force as 6°S for the whole lake. The white lines represent the slope equal to equation image, the theoretical velocity of Kelvin waves.

[11] To further investigate if the oscillations are indeed internal Kelvin waves, the width averaged displacement was subtracted from each point and the time series at several cross sections (width-wise) along the length of the lake was studied. At each cross-section three points near each coast were selected (as shown in Figure 1). This is to see the variation of the amplitude of the waves inward from the coast. Figure 3 presents the time series of the subtracted displacement at six points along the east-west cross-section at two places. In the southern basin (Figure 3a) the amplitude of oscillations are larger near the west coast than near the east coast. In the northern basin (Figure 3b), the amplitude of the oscillations is equivalent for both coasts. However, throughout the year for both coasts, the amplitude of oscillations decreases as the distance to the coast increases.

Figure 3.

Time series of the thermocline displacement. a, the southern basin. b, the northern basin. S and N represent the southern and northern ends, while E and W represent the east and west shores, respectively. Width averaged thermocline displacement has been removed from each point.

[12] The time series of subtracted displacement for the lake straddling the Equator (Figure 4) shows no east-west contrast in the amplitude. Here the movements are essentially the north-south thermocline oscillations, which are independent of Earth's rotation; the small coastal upwellings disappear with f set to zero. The time series of subtracted displacement for the lake shifted to around 20°S (Figure 5) show that the amplitude of the oscillations decrease remarkably as the distance to the shore increases. The same figure plotted with simulations with f as 6°S for the whole lake shows variations (not shown) similar to the original rotation case, as expected. This again shows that the earths rotation do play a role in modifying the thermocline oscillations in Lake Tanganyika.

Figure 4.

Same as Figure 3 for Coriolis force = 0°C.

Figure 5.

Same as Figure 3 for the lake shifted to 20°S.

[13] Field data about the hydrodynamics of Lake Tanganyika are scarce, and Kelvin waves were never sought. There is however one set of measurements that may be consistent with the existence of Kelvin waves. A general north to south current in the upper 5 m about 6 km from the east coast near the north basin has been observed [van Well and Chapman, 1976]. Other observations that point to the importance of the Coriolis force are the westward swing of the Lufubu River as it enters the south of the lake [Coulter, 1968] and the deposition of sand on the northwest side of the harbour at Bujumbura at the north [United Nations Food and Agricultural Organization, 1978]. The satellite images of lake surface temperature show that the temperature is not homogeneous along the width of the lake and shows some structures along the west coasts (not shown).

[14] In summary this study demonstrates the influence of Coriolis force in modifying the dynamics of the internal gravity waves in Lake Tanganyika and that the internal Kelvin waves do exist in this lake. Small coastal upwellings were seen from time to time propagating clockwise around the western boundaries of the lake, which are the internal Kelvin wave packets. Lake Tanganyika is one of the deepest stratified lake of the world and is considered to be keeping the record of the past climate in its sediments. The knowledge of the presence of internal Kelvin waves and small coastal upwellings could be of use in deciding the sites of economically important sedimentary deposits. Due to the strong upwelling-productivity relationship, the knowledge of Kelvin waves may be important in the shoreline areas where fisheries are concentrated since this region of the lake is the major source of sediment-nutrient recycling during stratification or in the stratified lakes [Horne and Goldman, 1994]. Tracking internal waves is important to the fishing industry and for understanding pollution dispersal and to be able to predict how quickly they mix.


[15] This work was carried out for the project, ‘Climate Variability as Recorded by Lake Tanganyika’ (CLIMLAKE), funded by the Belgian program of Sustainable Development under contract EV/10/2D (Federal Office for Scientific, Technical and Cultural Affairs (OSTC), Prime Minister's Office). We thank the FAO/FINNIDA project GCP/RAF/271/FIN for the data used in this study. Useful suggestions were made by Jean-Marie Beckers and Eric Wolanski, whose help is gratefully acknowledged. Eric Deleersnijder is a Research Associate with the Belgian National Fund for Scientific Research (FNRS).