Kyoungsik Chang1, and George Constantinescu21School of Mechanical Engineering, The University of Ulsan, Ulsan, South Korea2IIHRโHydroscience and Engineering and Department of Civil and Environmental Engineering, The University of Iowa, IA, USAAbstract3-D numerical simulations are used to investigate the formation, evolution and sediment entrainment capacity of planar, Boussinesq, compositional gravity currents propagating over a long, inclined surface reaching the free surface at a small angle. The generation of the gravity current is driven by surface cooling associated with the diurnal, periodic variation of the free surface total heat flux. Thermal convective plumes of cooler fluid generated along the free surface act as a source of buoyancy during the initial stages of propagation of the current. Simulations conducted with different magnitudes of the sinusoidal heat flux and bottom slopes show that gravity current head forms after 0.25-0.4T (T is the period of the oscillatory heat flux, time intervals are measured from the time when the heat flux becomes negative). The time interval for the total buoyancy of the current to peak is the largest for the case with a low bottom slope and a low magnitude of the heat flux. In all cases, the current reaches a regime where its front velocity is close to constant after 0.5-0.7T. Analysis of the bed shear stress distributions shows that the peak bed shear stresses do not generally occur beneath the head of the current and sediment entrained beneath the downstream part of the tail is the largest contributor to the total flux of sediment entrained from the inclined bed surface. Besides the fast fluid inside the gravity current, the interactions of the convective plumes with the inclined surface generate energetic vortical eddies and circulatory motions that also contribute to the amplification of the bed shear stress. The capacity of the current to entrain sediment peaks 0.25-0.3T after the gravity current enters the nearly constant front velocity regime. The flux of sediment entrained from the bed is increasing with both bed slope and magnitude of the heat flux, but the rates of increase are much higher when the magnitude of the heat flux is varied.1 INTRODUCTIONGravity currents forming in reservoirs, lakes and oceans are often characterized by a large capacity to erode the loose bed over which they propagate. Such gravity currents generally propagate over an inclined surface, at least until they reach the bottom part of the waterbody. For example, this is the case of turbidity currents forming on the inclined continental shelf of oceans (Birman et al., 2007). These currents are a main contributor to the formation of submarine canyons on the continental slope of oceans (Pratson & Coakley, 1996) and can carry a large amount of sediments into the deep parts of oceans. Saltwater intrusions into estuaries generate a compositional gravity current propagating over an inclined surface. As they propagate over a loose bed, compositional (e.g., salinity or temperature driven) and turbidity currents can induce lots of sediment erosion, carry large amounts of sediments over large distances and modify the local morphodynamics over time in regions where such currents form regularly.Such types of gravity currents propagating over a sloped bottom in reservoirs, lakes or oceans are generally investigated using lock-exchange laboratory experiments (e.g., Britter & Linden, 1980, Beghin et al.,1981; Maxworthy & Nokes, 2007; Maxworthy, 2010, Dai 2013b, 2013c, 2014; Dai & Huang, 2016; Martin et al., 2019; Zemach et al., 2019; De Falco et al., 2021; Maggi et al., 2023a, 2023b) in which the ambient fluid is either homogeneous (constant density) or linearly stratified. Given that the ambient fluid is at rest and the free surface is horizontal, the lock-exchange current advances in a positive mean background pressure gradient along the incline. The effect of earth rotation and associated Coriolis forces were investigated among others by Cenedese & Adduce (2008) and Negretti et al. (2021). One limitation of most of these laboratory studies was the limited length of the incline which allowed the study of the physics of such currents only during their initial stages Most numerical studies conducted using eddy-resolving techniques focused on the simpler case of a lock-exchange Boussinesq or non-Boussinesq current propagating in a channel with a free surface parallel to the bed (Blanchette et al., 2005, Birman et al., 2007; Dai, 2013a, 2013b, 2014, 2015) for which the background pressure gradient in the direction parallel to the incline is equal to zero. This case is more relevant for currents forming in rivers but there was no co-flow in the simulations. Steenhauer et al. (2017) used three-dimensional (3-D) large eddy simulation to simulate lock-exchange Boussinesq gravity currents propagating down a constant-slope incline based on the set up used in the experiments of Maxworthy (2010) conducted in a reservoir with a horizontal free surface. The angle of the incline was varied between 00 and 600 in these simulations which were conducted with a much longer length of the incline compared to the corresponding experiments.For the simplest case of a planar, finite-release lock-exchange compositional current propagating in a reservoir containing homogeneous ambient fluid at rest, experimental and numerical studies have shown that, following an acceleration phase, the front velocity starts decaying and the turbulent current transitions to a 2/3 power law regime that is similar to the buoyancy-inertia phase observed for currents propagating over a horizontal bed. At later times the current transitions to a 3/8 power-law regime corresponding to the buoyancy-viscous phase. Over this regime, the head is strongly distorted and raises from the bed such that the shear stresses beneath the bed are very small. The numerical study of Steenhauer et al. (2017) showed that the sediment entrainment capacity of such compositional currents peaks for slope angles of 300 to 400. These values are larger than the ones (200-300) corresponding to the fastest currents. For slope angles larger than 100, a large, intensified mixing vortex (IMV) forms at the back of the dissipative wake region. Its strength increases with increasing slope angle while the distance traveled by the current before this vortex forms decreases with increasing slope angle. For angles larger than 200, most of the buoyancy lost by the head is entrained into the core of the IMV vortex during the later stages of propagation of the current. The IMV has a larger capacity to entrain sediment that the head. In the case of constant-flux current propagating in a homogeneous ambient fluid, Middleton (1966a, 1966b) found that the current reaches a nearly-constant front velocity regime that varies little with the change in the slope angle. Martin et al. (2020) provide a detailed discussion of the structure of such currents.The formation mechanism for gravity currents propagating over a sloped bottom surface can be much more complex in natural environments. For example, katabatic winds form as air situated closed to a sloped bottom boundary is cooled by radiative processes near the ground (Manins and Sawford, 1979). The negative heat flux at the ground surface results in the formation of a certain amount of cool air that starts propagating along the inclined surface in fairly similar way to a lock-exchange current. However, the extent of the region containing cool air is controlled by the heat flux at the ground surface. As opposed to a lock-exchange current where a finite amount of cool air is released instantaneously, in the case of katabatic winds there is a certain amount of time needed for the cool air to accumulate and then to start propagating along the inclined ground surface as a gravity-current-like flow.Winter cascading of cold-shelf waters in a deep lake or over the continental shelves bordering the ocean basins also generates gravity-current-like flows (Hill et al., 1998; Fer et al., 2002a). Similar to the aforementioned example of katabatic winds, radiative processes control the formation of the region containing cooler, heavier fluid near the shoreline. However, in this case the driving radiative process is due to diurnal variations in the heat flux at the surface of the waterbody that induces differential cooling between the shallow and deep regions (Wells & Sherman, 2001). Shallow regions near the shoreline cool more rapidly than the deeper regions, which results in the formation of a gravity current containing colder, denser water that propagates over the inclined bottom of the waterbody. The cooling of the water along the free surface generates unstable stratification during the times the water loses heat to the atmosphere. The water column near the free surface overturns as the denser surface water starts sinking and convective plumes are generated (Marshall & Schott, 1999) that move toward the sloping bottom. Some of these plumes are also a source of buoyancy for the gravity current. In lakes, basins and reservoirs with a sloped bottom, this can lead to stratification in the lower layer, at the base of the deeper zones, and upwelling of cold water by the convective motions in the surface mixed layer, as well as strong horizontal circulation (Monismith et al., 1990; Wells & Sherman, 2001). The circulation induced by diurnal surface cooling plays an important role in nearshore dynamics (Monismith et al, 1990; Sturman et al., 1999). Laboratory experiments were conducted to understand the relationship between the volume flux of the gravity current and the surface cooling rates (Finnigan & Ivey, 1999, 2000). Experiments have also shown that an initial period of disorganized motion driven by convection from the free surface is present over the shallower region before the gravity current forms (Sturman & Ivey, 1998).Directly relevant for the present study, Fer et al. (2001, 2002a, 2002b) found evidence of the formation of such gravity currents in Lake Geneva in response to diurnal cooling and heating occurring at the free surface. These studies measured the mean velocity of propagation of the gravity current, its thickness and time of passage at several locations situated away from the shoreline and observed that several regions of higher thickness and velocity (e.g., pulses) were present behind the head of the current. Using quantitative analysis, these studies proved that the circulation associated with the passage of such gravity currents is a main mechanism for flushing nearshore shelf water during the winter season. It was estimated that the mean net volume flux carried by these gravity currents is more than one order of magnitude larger than the mean flux into Lake Geneva from rivers during the winter season. Moreover, some of the collected data suggested that the passage of these gravity currents can erode and transport important quantities of sediment into the deeper regions of the lake as well as affect the transport of dissolved oxygen, nutrients and contaminants present in the shallower regions of the lake, thus ventilating and oxygenating the deeper water regions (Fer et al, 2002b). This has important ecological and biological consequences for the lake ecosystem. Constant-flux plunging flumes over a sloping bed in Lake Geneva were studied experimentally and numerically by Shi et al. (2022). The mechanisms for the formation of pulses in constant-flux plunging flows into lakes and reservoirs are discussed in detail by Kostaschuk et al. (2018).The present numerical study uses 3-D large eddy simulations to study the physics of gravity currents forming in a triangularly-shaped reservoir (Figure 1) due to a sinusoidal heat flux of period T applied at the top boundary that approximates the diurnal total heat flux at the free surface. The temperature of the fluid is uniform inside the reservoir at the start of the simulation when the free surface heat flux per unit surface becomes negative (free surface is cooling) and stays negative over the first half period. Similar to the bathymetry of Lake Geneva in a section perpendicular to the shoreline, a short region with a bed slope angle of 1.30 is followed by the main sloped region (Figure 1). The slope of the main sloped region is constant and equal to ฮธ. Its length is sufficiently large such that the front of the gravity current does not reach the end of this region after one period T. This allows studying the evolution of such currents under idealized conditions where the gravity current does not interact with the heavier fluid below the thermocline, generally present in lakes during the winter season. For such conditions, a surface mixed layer develops over the deeper layer containing heavier fluid in the deep region of the lake (Wells & Sherman, 2001) which induce large-scale circulatory motions affecting the evolution of the gravity current propagating over the non-uniformly sloped bottom of the lake. The constant slope of the main part of the inclined bottom and unimodal (e.g., sinusoidal) variation of the surface heat flux allow reducing the number of geometrical and flow variables affecting the evolution of such currents in the present numerical experiments.
