A 3-D hydrodynamic model (Regional Ocean Modeling System) is used to investigate the dynamics and structure of hyperpycnal river plumes over sloping continental shelves. The focus is on the plume's response to varying slopes and settling velocities (ws). The idealized model is configured to represent small mountainous river systems during a flood event. A hyperpycnal sediment concentration of 60 g/L is specified at the river mouth such that the sediment–freshwater mixture is denser than the seawater, causing the plumes to traverse the shelves as undercurrents. A realistic range of shelf slope of 0.001–0.03 is chosen. The settling velocity is varied based on river's carrying capacity. The model-derived velocity profiles and the entrainment rate compare favorably against prior laboratory experiments. Both cross-shore and alongshore momentum balances are primarily between gravitational forcing and bottom friction. But, the Coriolis deflection is significant at the plume core in the alongshore momentum budget (i.e., Ekman balance). As the slope increases and settling velocity decreases, hyperpycnal plumes transition from depositional to autosuspending regime. An estimate of critical slope governed by a dimensionless parameter (qb is buoyancy input) reasonably captures the regime transition. In the depositional regime, the plume's runout (cross-shore penetration) scales with advective distance: increasing slopes and discharge enhance the gravitational forcing and plume velocity, leading to an increase in runout. In contrast, increasing settling velocity shortens the vertical settling time, thereby reducing the plume's horizontal footprint. For the range of parameters considered, the runout of depositional plumes is confined within 15 km from the mouth, whereas the penetration of autosuspending plumes is essentially unlimited.
 A hyperpycnal river plume is a negatively buoyant outflow that occurs when the suspended sediment concentration in the river is sufficiently high such that the density of the sediment–freshwater mixture exceeds the receiving ambient seawater. As the outflow enters the coastal ocean, due to the excessive density, the plume dives and moves along the sea floor as an undercurrent [Kassem and Imran, 2001]. For a typical range of temperature and salinity of seawater, the threshold of suspended sediment concentration for hyperpycnal discharge is 35–45 g/L [Mulder and Syvitski, 1995], although a few laboratory experiments suggest that hyperpycnal plume may form at a lower concentration in the presence of convective instabilities [Parsons, et al., 2001].
 Hyperpycnal conditions in rivers are not common. The high sediment concentration threshold requires large sediment supply and a turbulent lifting force of strong river flow [Winterwerp, 2001]. Therefore, hyperpycnal river discharge is usually short lived (lasting a few hours to days) and is more frequently generated by small- to medium-size rivers over mountainous terrain where passages of severe storms or typhoons provide large rainfall, triggering high sediment yields [Mulder et al., 2003; Milliman and Kao, 2005]. Using a 280 river global database, Mulder and Syvitski  identified 9 and 74 rivers that could produce multiple hyperpycnal discharge events at 1 year and 100 year return periods, respectively.
 Despite the infrequent occurrence, hyperpycnal discharge represents a significant delivery of terrestrial materials to the global oceans. Milliman and Syvitski  suggested that small mountainous river systems, such as high-standing islands in Asia and Oceania (e.g., Taiwan and Indonesia), are the major contributors to the global sediment flux. Six mountainous islands in the East Indies alone can account for over 20% of the global sediment flux [Milliman, 2001]. In these mountainous river systems, a significant portion of the cumulative sediment flux to the oceans could take a form of hyperpycnal discharge. For example, hyperpycnal discharge out of the Santa Clara River, a small river draining the Transverse Range in southern California, accounts for 75% of the cumulative sediment loads over a 50 year period while represents only 0.15% of cumulative time (∼30 days) [Warrick and Milliman, 2003]. Dadson et al.  estimated that, during the period of 1970–1999, 30–42% of the cumulative sediment discharge from Taiwanese rivers to the ocean occurs at hyperpycnal concentration (>40 g/L). Between 1980 and 2001, Choshui River of Taiwan, one of the nine “dirty” rivers identified by Milliman and Syvitski , discharged approximately 970 million tons of sediment to Taiwan Strait. Around half of this 22 year sediment load was delivered as hyperpycnal discharge in total of only 30 days during Typhoon events [Kao and Milliman, 2008].
 After entering the coastal oceans, the behavior and structure of hyperpycnal plumes are rarely documented. This is primarily due to the challenges of making direct field measurements: the plume is highly episodic and its massive sediment discharge could easily bury moored instruments [Liu and Lin, 2004; Warrick et al., 2008; Liu et al., 2012]. Some field observations were made at the mouth of Hunghe [Wright et al., 1990; Wang et al., 2010], where hyperpycnal discharge routinely enters the gentle shelf. Wang et al.  reported that Hunghe's hyperpycnal outflow was observed at the mouth during ebb tides but was destroyed during flood tides, presumably due to tidal mixing (see section 'Discussion and Summary').
 Our limited understanding of hyperpycnal plumes comes primarily from laboratory, theoretical, and numerical modeling studies, mostly on turbidity currents (i.e., interstitial fluid is salt water; see below). On inclined planes like continental shelves (Figure 1), the downslope motion of the undercurrents is driven by gravitational force acting on the excess density of sediment suspension. The gravitational force is countered by bottom friction and interfacial drag due to entrainment of low-momentum overlying fluid [Middleton, 1966; Komar, 1971, 1973; Wright et al., 1990; Mulder et al., 1998; Bonnecaze and Lister, 1999]. There are strong feedbacks between the flows and sediment suspension: turbulence generated by the undercurrent along the bottom boundary provides the lift to prevent particles from settling, acting to maintain the density anomaly that drives the undercurrents.
 Based on the competition between upward turbulent lift and downward settling, hyperpycnal plumes may be classified into three regimes: decelerating, autosuspending, and accelerating (analogous to turbidity currents, based on a recent laboratory study by Sequeiros et al. ). The decelerating regime is associated with gentle slopes over which the turbulent lift is insufficient to overcome downward settling. In such conditions, the plume would lose its density contrast and thus the gravitational force. As the driving force decreases, there is less turbulent production, which in turn leads to even more settling. Therefore, the plume decelerates (i.e., dU/ds < 0, U is plume's velocity, s is distance along plume's path) and is depositional. When the turbulent lift exceeds settling (i.e., steep slopes), the undercurrent is self-sustaining, which is indicated by the lack of net deposition [Pantin, 2001]. Without entraining additional bed sediment, the undercurrent is in an autosuspending state [Bagnold, 1962] and likely travels at an approximately constant velocity (dU/ds ≈ 0). When provided with readily erodible bed, the undercurrent may reinforce its downslope forcing through erosion (i.e., ignition by Parker ) and thus accelerates along its path (dU/ds > 0). Parker  suggested that, for turbidity currents, the accelerating state may precede autosuspension as the self-accelerating processes allow the current to evolve with time until the bed is depleted. Nevertheless, autosuspension represents a lowest energy limit at which the undercurrent could be self supported. Criteria for autosuspension based on flow speed, settling velocity (i.e., particle size), and bottom slope have been proposed by Pantin , Parker , and Kao and Milliman  (see section 'Discussion and Summary').
 While hyperpycnal river plumes follow the same basic dynamics as turbidity currents [Meiburg and Kneller, 2010], it is important to note the differences in physical settings that may ultimately affect the plume's characteristics. Unlike the marine-originated turbidity currents, hyperpycnal plumes at sea are continuations of sediment-laden rivers, meaning that the interstitial fluid is low-density, riverine water [Mulder and Syvitski, 1995]. Thus, the plumes are driven by a comparably weaker gravitational force, given the same sediment suspension. Moreover, contrasting to coarse-grained turbidity currents, hyperpycnal plumes are mainly fine grained [Mulder et al., 2003; Warrick et al., 2008; Kao et al., 2008]. Therefore, as muddy hyperpycnal plumes traverse continental shelves, self-acceleration is unlikely to occur because shelves typically lack easily erodible mud due to wave sorting, reworking processes, and advection [Garrison, 2009; Wheatcroft et al., 1997; Wheatcroft, 2000; Harris and Wiberg, 2002].
 The aforementioned studies and others have established the fundamentals of the hyperpycnal plume dynamics. However, most of these works have focused on channelized cases and are often limited to low-Reynolds-number, lab-scale flows. Hence, the insight gained from two-dimensional, lab-scale flows may not translate directly to the fully turbulent, 3-D coastal ocean setting. Recently, there are increasing number of modeling studies that pay attention to the 3-D nature of the dense plumes at field scales. For example, Chao  used an ocean model to explore the circulation patterns associated with a transition from hyperpycnal to surface plumes as the undercurrent loses buoyancy to settling. Khan et al.  used depth-averaged Navier-Stokes equations to investigate the spreading and deposition patterns of a hyperpycnal plume, with an idealized domain configured to emulate Western Adriatic shelf (Italy). Huang et al.  further relaxed the depth-integration constraint. They numerically solved 3-D Reynolds-averaged Navier-Stokes equations to study the bed steepening and levee formation of a submarine canyon-fan system. Wang et al.  used an ocean model to simulate hyperpycnal discharge out of an idealized river mouth that represents the Huanghe (China). They found stability of hyperpycnal plume sensitive to riverine sediment concentration, settling velocity, and ambient tidal mixing.
 Despite the recent advancing in modeling work, there are fundamental questions yet to be addressed. For example, the influences of bathymetry such as bottom slopes and lateral (alongshore) expansion on the plume dynamics have, for the most part, not been systematically studied. Bonnecaze and Lister's  theoretical analyses of undercurrents on inclined planes presented scalings for plume width and runout distance (i.e., cross-shore penetration). However, at field scales, the momentum budgets of the plume and the parameter dependence of the plume structures have not been evaluated. Furthermore, except Sequeiros et al.'s  recent laboratory demonstration using turbidity currents, there seems to be lack of evidence for the occurrence of autosuspending hyperpycnal plumes. Conditions at which autosuspension happens remain largely uncertain.
 The primary objective of this study is to investigate the dynamics and 3-D structure of hyperpycnal plumes over sloping continental shelves using a field-scale numerical model. We concentrate on plume's responses to varying bottom slopes and settling velocities of riverine fine-grained sediment. The shelves are assumed to be fine-sediment-starved, meaning that self-accelerating processes are not considered here. We then identify the critical slopes that differentiate the decelerating (depositional) and autosuspending regimes, and a simple theory is proposed for estimating the critical conditions. Both cross-shore and alongshore momentum budgets are evaluated. Finally, the plume's characteristic length scales including runout, width, and thickness are quantified, and their sensitivities to external parameters are examined by comparing with Bonnecaze and Lister  scaling.
2.1. Numerical Model
 We use the Regional Ocean Modeling System (ROMS) to explore the dynamics and structure of hyperpycnal river plumes on sloping continental shelves. ROMS is a 3-D, hydrostatic, primitive equation ocean model that solves the Reynolds averaged form of the Navier-Stokes equations on a horizontal orthogonal Arakawa “C” grid and uses stretched terrain following coordinates in the vertical [Haidvogel et al., 2000; Shchepetkin and McWilliams, 2005]. The idealized model is configured to represent small mountainous river-shelf systems. It consists of a straight river channel (10 km long, 1 km wide, and 5 m deep) connected to a constant sloping continental shelf (Figure 1). The shelf slope is varied between 0.001 and 0.03, covering a wide range of observed values (Table 1, cases 6–11). There are 40 layers in the vertical. To resolve the undercurrents, the vertical grid is stretched such that high near-bottom resolution is maintained with increasing depth (Figure 1). For example, at 500 m water depth, the vertical grid spacing (Δz) is kept less than 1 m within 5 m above the bed. The mouth region where the plume plunges and spreads (10 km × 10 km black box in Figure 1) is highly resolved, with an averaged horizontal resolution of 140 m. Outside this area, the grid spacing increases linearly toward the open boundaries to obtain a big shelf, which helps minimize the influences of imperfect boundary conditions.
 Given that the 3-D characteristics of hyperpycnal plume are not well understood, it seems reasonable to start by considering only the essential plume elements. We therefore focus on cases of dense plume empting into motionless shelves. The influences of ambient forcing such as tides and waves are left for future studies (see section 'Discussion and Summary'). At the landward boundary, we impose a 3 day river discharge event, typical of typhoon-induced floods in mountainous river systems [Dadson et al., 2005]. The discharge ramps up to its maximum over 1 day using a hyperbolic tangent function (based on observed hydrography in the study by Dadson et al. ) and stays steady for 2 days. The steady discharge, albeit unrealistic, allows investigations of fully developed plumes. The potential impacts of unsteadiness are evaluated in section 'Potential Influence of Variable Forcing'. Maximum river discharge Qr,max ranges between 1 and 2 × 104 m3 s−1, corresponding to 2–4 m s−1 river velocity. The reason for this extreme river velocity is related to the energy required to maintain a dense suspension, as explained below. A hyperpycnal sediment concentration Cmax of 60 kg m−3 is fixed at the river mouth. During the ramp-up, sediment concentration C follows the rating curve
where the empirical constant b equals to 1.0, consistent with Mulder and Syvitski  and Dadson et al. . The density of river-sediment mixture is computed as , where is the density of interstitial fluid and is the relative sediment excess density (ρs = 2650 kg m−3). Temperature is fixed at 10°C throughout the domain, thus making no contribution to the density anomaly. Salinity of ambient shelf water is 30 psu. The reduced gravity of the plume is computed as
where is the density of ambient shelf water and represents a scaled density difference between ambient and interstitial fluids . At peak discharge (C = Cmax), the plume density exceeds the ambient water, yielding a hyperpycnal plume with g′ of 0.133 m s−2.
 The k-ε turbulence closure [Jones and Launder, 1972] with a stability function by Kantha and Clayson  is used. The molecular diffusivities for sediment and salt are set as 5 × 10−6 m2 s−1. This choice does not affect the model results, as turbulent diffusion dominates over molecular processes. Suspended sediment influences turbulence through its impacts on density stratification. The k-ε closure has been shown capable of reproducing vertical structures of velocity and sediment suspension of lab-scale turbidity currents [Huang et al., 2007]. Bottom stress is computed by assuming a logarithmic current profile in the lowest computational cell and a roughness length of 1.0 mm (equivalent drag coefficient CD of 0.003 at 2 m reference height). The erosion/deposition formulations and a constant erosion rate of 5 × 10−5 kg m−2 s−1 follow Warner et al.  and Chen et al. . The settling velocities range from 0.05 to 0.5 mm s−1 (i.e., fine to medium silt-size particles). This range is chosen based on the carrying capacity of hyperpycnal discharge (see section 'Constraining Settling Velocities of Hyperpycnal Discharge'). The dynamics of aggregation and floc breakup are not explicitly considered for simplicity. To our knowledge, existing flocculation models require site-dependent calibration of empirical coefficients, which is not available for the present idealized study. We therefore focus on constant settling as a first step. The critical shear stress for erosion is set equal to 0.05 N m−2 [Madsen, 2002]. The model results are however insensitive to the choice of erosion rate and critical shear stress, as the incorporation of sediment effects on turbulence limits the maximum carrying capacity of the plume [see Winterwerp, 2001, section 'Constraining Settling Velocities of Hyperpycnal Discharge']. There is initially no sediment on the shelf (see justification in section 'Introduction'), leaving the sediment-laden river as the sole sediment and buoyancy source.
2.2. Tracer and Integrated Plume Properties
 We use a passive tracer to distinguish the plume water from the ambient [Legg et al., 2006]. The tracer concentration ϕ is unity in the river and is zero elsewhere. We normalize ϕ by the local maxima at each alongshore sections to account for dilution due to entrainment. A cutoff of normalized concentration is then used to mark the plume boundary, based on observations that the cutoff corresponds well with the sharp interface between the core of the plume (i.e., high momentum portion) and the ambient water (Figure 2). Using different cutoffs gives qualitatively similar results.
 We then compute the time-averaged, integral properties of the plume at each alongshore sections. The plumes usually reach a steady state within 0.5 day after the peak discharge. Once a steady state is reached, all of the quantities below are averaged over a 0.5 day period to obtain steady-state properties. The only exception is the run-out distance of autosuspending plumes. In such a case, the run-out distance grows with time, and a different metric will be used (see section 'Hyperpycnal Plume Regimes: Autosuspending Versus Decelerating (Depositional)').
 The plume cross-sectional area A, averaged layer thickness H, and width W are calculated as
where M is a plume mask using the tracer, the angle bracket represents average in alongshore direction, and the overbar indicates time average (days 2–2.5). The time-mean, sectionally averaged velocity, and reduced gravity are
 Two important dimensionless parameters that govern the plume dynamics can be readily obtained. They are Froude number and Ekman number . For various experiments considered, the averaged Fr of the plume is around 0.5–2.0 within 10 km from the mouth. The averaged Ek is around 3.5–10.3, i.e., friction dominated. The large Ek indicates that the principal flow direction is downslope [Cenedese and Adduce, 2010, equation (7), Figure 2]. Therefore, the integral over the alongshore sections represents most of the fluxes.
2.3. Sediment Mass Budget
 We partition the sediment mass into bed (mbed) and suspended (msus; vertically integrated) components, according to
where c is suspended sediment concentration, p (=0.5) is porosity, and D is bed thickness. The time variations of total mass integrated over the entire domain (x < 0 in Figure 1) is then
 We validate the model by comparing the predicted vertical structure of the undercurrent with laboratory experiments by Garcia  (i.e., his saline currents). The same 5° bottom slope is used. The cross-shore extension of the model shown in Figure 1 remains unchanged, but the depth is scaled accordingly and the alongshore domain is trimmed to emulate a straight laboratory flume. Particle settling velocity (ws) is set to zero, consistent with the laboratory settings of buoyancy conserving flows. Normalization of velocity, density (i.e., g′), and depth follows Garcia  and Choi and Garcia . Figure 3 shows that the normalized velocity and reduced gravity at various downstream locations collapse to a profile that agrees with Garcia's  experiments. The field scale model is shown to be capable of resolving a near-bed region where velocity profile is logarithmic and an upper region where velocity decays due to entrainment of overlying low-momentum water.
3.2. Entrainment Estimates
 We also evaluate the representation of entrainment in the model. An adequate representation is important because entrainment of ambient water generates interfacial drag acting on the plume and, at the same time, regulates plume's density anomaly through dilution and increasing salt content. To estimate entrainment, we use the default model setup as in Figure 1 and consider five bottom slopes (cases 1–5 in Table 1). Settling velocity is again set to zero for this test. Using tracer as a marker, we first quantify the time-averaged volume transport of the plume across a given section . We then choose a control volume seaward of the tip of the plunging point (e.g., red box in Figure 2). The mean entrainment coefficient of this control volume is estimated as
where S represents plume's surface area [Legg et al., 2006]. The size of the control volume is varied, with a cross-shore extension ranging from 0.1 to 3 km.
 In Figure 4, we plot the entrainment coefficient CE against Froude number Fr. CE and Fr are averaged over various control volume boxes. Variability is given by the error bars. Overlaid are three parameterizations based on laboratory investigations of entrainment by Ellison and Turner , Parker et al. , and Cenedese and Adduce . It is clear that the entrainment estimates are insensitive to the choices of control volume. As expected, the entrainment coefficient is larger on steeper slopes due to greater downslope gravitational forcing and thus velocity. For Fr ≥ 1, the model-derived CE is in good agreement with laboratory results. For Fr < 1, our estimates continue to roughly follow the parameterizations by Parker et al.  and Cenedese and Adduce  but significantly deviate from that by Ellison and Turner . The deviations, which suggest mixing occurred in subcritical Fr regime, have been found in many oceanic overflows (see Cenedese and Adduce  for a comprehensive review). Notwithstanding the uncertainty in the subcritical regime, the general agreement between model and laboratory experiments lends support to the validity of our modeling approach to investigate the dynamics of hyperpycnal plumes.
4. Constraining Settling Velocities of Hyperpycnal Discharge
 Based on energy constraints, hyperpycnal conditions are limited to small settling velocities. Winterwerp  argued that when a suspension of fine sediment exceeds the carrying capacity of a turbulent flow, the depositing sediment forms a concentrated mud layer near the bed. The sharp concentration gradient near the bed attenuates turbulence, which decreases the carrying capacity further, leading to a catastrophic collapse of turbulent field. However, a steady “saturation” condition can be maintained when the energy required to maintain the suspended load is exactly balanced by the turbulence. The limiting condition for concentration can be represented by the critical flux Richardson number Rif (≅0.15 based on Osborn ) using stratified flow theory:
 In equation (8), the numerator and denominator represent buoyancy destruction due to sediment-induced stratification and turbulent shear production, respectively. The nearly saturated transport condition is supported by observations in Huanghe (Yellow River), China [Winterwerp, 2006], and by k-ε turbulence modeling as well as Direct Numerical Simulation [Cantero et al., 2009].
 Integrating equation (8) over a bed layer, we obtain a relation between layer-averaged sediment concentration C, flow power U3, and particle settling velocity ws,
where a (= ; constant Ks depends on velocity structure) has a range of 0.1–0.5 for layer thickness of 2–100 m (see Winterwerp  for details). A value a = 0.16 is found for a plume thickness of 5 m (see section 'Plume Structure and Dynamics').
 Equation (9) represents an energy constraint that, for the same concentration, lifting particles with larger ws requires greater flow power. The river velocity thus determines an upper limit of settling velocity that can be delivered to the ocean in a form of hyperpycnal discharge. To illustrate this, equation (9) for three settling velocities is plotted in Figure 5. For ws = 1.5 mm/s, the carrying flow velocity needs to be extremely large (>3 m/s) to support the targeted hyperpycnal concentration of 60 g/L. Observations from Gaoping River, the second largest river in Taiwan in terms of sediment discharge, give a peak river velocity of 2.8 m/s during the Category 3 Typhoon Fanapi (Water Resources Agency of Taiwan 1990–2010). Velocities higher than 3 m/s for natural rivers would be unlikely, indicating that a steady hyperpycnal discharge with ws = 1.5 mm/s is also unlikely to occur. Lower settling velocities allow the occurrence of hyperpycnal conditions within the expected range of river velocities. Therefore, we adopt a narrower range of ws = 0.05–0.5 mm/s for the coastal settling considered here (i.e., a river velocity is 2 m/s for the basecase; Table 1). Note that this range is also consistent with observations from Choshui River that over 60% of sediment discharge during Typhoon Mindule is below 25 µm [Kao et al., 2008].
5. Plume Structure and Dynamics
 Here, we explore the responses of hyperpycnal plumes to external parameters, including bottom slope, settling velocities, and river discharge. The sectionally averaged plume properties defined in equations (3) and (4) are used. The momentum budgets, autosuspending conditions, and scaling laws for plume's runout distance are evaluated.
5.1. Plume Structure on Different Slopes
 We first evaluate the influences of bottom slopes on plume structure. Three cases representing typical hyperpycnal plumes (ws = 0.1 mm/s; Ur = 2 m/s) on gentle (sinθ = 0.001), medium (0.005), and steep (0.03) shelves are compared (cases 6, 8, and 11 in Table 1). Figure 6 shows the sediment mass distribution at the end of the discharge event. The top to bottom rows are total (msus + mbed; equation (5)), bed (mbed), and suspended (msus) mass, respectively. In each panel, a curved line indicates the trajectory of the center of mass. The trajectories are deflected northward due to the influence of Earth rotation. The total mass distribution usually consists of plume's main body (dark color) and a secondary cloud of mass (light color). The secondary cloud is a result of leakage of suspended mass after the plume terminates (Figures 6a and 6b) and/or due to instabilities along plume's lateral boundaries (Figure 6c, x < −10 km and y > 5 km). Because the secondary cloud has much less mass, the following analyses concentrate on plume's main body.
 Several key features are evident in Figure 6: First, the plume's runout distance (i.e., cross-shore penetration) increases as the slope increases. This is clearly shown by an increase in the length of the center-of-mass trajectories; Second, the width (i.e., alongshore expansion) of plume's main body decreases as the slope increases; Third, mass deposition decreases as the slope increases (Figures 6d, 6e, and 6f). Between the medium and steep slopes, the bed mass distribution changes abruptly from displaying a well-defined deposition footprint to nearly nondepositional. The transition implies a shift from depositional to autosuspending regimes. Each of these features is examined below.
 Time evolution of sediment budget confirms the changes in mass partitioning with slopes (Figure 7). At any given time, plumes on steeper slopes support more mass in suspension (Msus; gray line in Figure 7). At slope of 0.03, the bed mass (dashed line) is close to zero during the entire discharge event, again suggesting that the plume is self-supported. The negligible deposition occurs near the plunging region (Figure 6f) where flow rapidly decelerates (see section 'Hyperpycnal Plume Regimes: Autosuspending Versus Decelerating (Depositional)'). On the gentle slope, Msus gradually levels off after day 2, indicating a balance between net deposition and the river sediment input. It can be shown that, for the gentle slope case, the adjustment time to the steady state for Msus is consistent with but slightly longer than the settling time (∼H/ws ∼ 0.6 days for H = 5 m, ws = 0.1 mm/s). As the slope increases, upward turbulent lift increases due to the strengthened undercurrent, which delays the settling and allows more mass in suspension.
5.2. Momentum Balance and Cross-Shore Variations
Wright et al.  approximated the cross-shore momentum balance of hyperpycnal plume as downslope gravitational force (first term) balancing bottom friction and interfacial drag due to entrainment of ambient water
where θ is the angle of bottom slope , H is layer thickness, U and G′ are section-averaged cross-shore velocity and reduced gravity, CD and CE are bottom drag and entrainment coefficients. Assuming interfacial drag is small compared to bottom drag, we can readily estimate the plume velocity with .
 Seaward the plunging point, the dominant cross-shore momentum balance of the hyperpycnal plumes is between gravitational force and bottom friction. The contribution of interfacial drag increases as the slope increases. As Figure 8a shows, UChezy generally agrees with the model-derived plume velocity U, suggesting a leading-order balance between gravitational forcing and bottom friction. Increasing slopes therefore generates greater downslope gravitational forcing that in turn drives stronger flow and a longer runout distance, as seen in Figure 6 (see section 'Decelerating (Depositional) Regime: Quantifying Plume's Runout Distance'). On the gentle slope, UChezy agrees with U, consistent with our entrainment estimate that CE (inverted triangle in Figure 4) is an order of magnitude smaller than CD. However, as the slope increases, differences between UChezy and U increase because interfacial drag becomes increasingly important. The overestimation of UChezy is particularly pronounced within 3 km from the mouth on the steep slope. This result is consistent with the strong Froude number dependence of entrainment.
 On gentle and medium slopes, the cross-shore velocity decelerates from the mouth and the plume terminates at a location where it loses density anomaly (i.e., G′ approaches zero; Figure 8b). The rapid deceleration indicates that the plumes are in the depositional regime. In contrast, on a steep slope, the plume velocity is relatively steady. In fact, when accounting for the alongshore velocity (V) driven by spreading and Coriolis deflection (Figure 6c), plume's velocity magnitude reaches a steady value in the far field (see section 'Hyperpycnal Plume Regimes: Autosuspending Versus Decelerating (Depositional)' and Figure 11), which implies an autosuspending state.
 The plume's reduced gravity G′ is larger on steeper slopes. This is because the plume velocity is stronger on steeper slopes, thereby supporting more sediment in suspension (Figures 6g, 6h, and 6i). Although there is more entrainment and dilution on steeper slopes, the entrained high salinity water increases the density of interstitial fluid, largely compensating the loss of buoyancy due to dilution.
 Both the width and thickness of the plume's core increase as the slope decreases (Figures 8c, 8d, and 6). This can be qualitatively understood by noting that, when the shelf is flat, a spreading plume would be axisymmetric [Bonnecaze et al., 1995]. As the slope increases, the primary flow direction is preferentially aligned downslope, thereby limiting the alongshore expansion (i.e., width). The thickness of the plume core is also reduced for steeper slopes as a result of volume conservation for the faster flow. In the limit of negligible entrainment (e.g., gentle and medium slope cases), the plume's cross-sectional area must decrease with increasing slopes to conserve the riverine volume fluxes. The dependence of width and thickness on slopes will be treated in section 'Hyperpycnal Plume Regimes: Autosuspending Versus Decelerating (Depositional)'.
5.3. Lateral (AlongShore) Momentum Budget
 The alongshore momentum balance of hyperpycnal plumes is primarily between pressure gradient force (PGF) and bottom friction. However, at the plume core, the contributions of Coriolis deflection are not negligible (i.e., Ekman balance). In Figure 9, the medium slope case is taken as a representative example. The momentum terms are layer-averaged using a tracer mask (equation (3)). Advective terms are combined. Layer integration of the viscous term leaves bottom and interfacial drag. The contribution of interfacial drag is small (∼10% of total drag). Within a day after peak discharge, the plume has reached a steady state, so unsteadiness is two orders of magnitude smaller than PGF . It can be seen that the PGF is directed away from plume's principal axis and has larger values near the boundaries (Figure 9a). The spatial distribution of bottom friction largely mirrors the PGF, suggesting the dominance of the frictional balance. The sign of the friction term indicates that lateral flow is directed outward, leading to the plume's alongshore expansion. The Coriolis term is large at the center where the cross-shore velocity peaks (Figures 9c and 10b). Distribution of the ratio of friction and Coriolis terms clearly shows that the influence of Earth rotation is significant at the plume center but is negligible along the plume periphery (Figure 9e). Seaward of the plunging zone (>1.5 km from the mouth), contributions of advection are small. But, at the plunging zone, PGF and advection are in the balance, implying that the outward PGF is mainly countered by centrifugal force along the curved streamlines (i.e., cyclostrophic balance).
 Lateral structure of density anomaly and velocity further reveal that the plume's gravitational forcing is responsible for the width expansion. The plume core is thicker than the periphery, creating a baroclinic PGF that drives the divergent lateral flows (Figure 10; looking seaward). We estimate the magnitude of lateral flows using a simplified balance of baroclinic PGF and bottom friction . A favorable comparison between the model-derived and estimated lateral velocity structure supports the gravity-friction balance away from the plume center (Figure 10c). The cross-shore velocity maxima are at the plume center. Cross-shore velocity decays upward due to entrainment and decreases toward the sides because of the reduction of downslope gravitational forcing (small H). Coriolis deflection shifts the plume core slightly northward and leads to the axial asymmetry in lateral flows (Figure 10c). This analysis indicates that the width expansion of hyperpycnal plumes is primarily controlled by frictional processes. Earth rotation appears to act only to deflect plume's trajectory.
5.4. Hyperpycnal Plume Regimes: Autosuspending Versus Decelerating (Depositional)
 From Figures 6 and 7, we have seen that, between the medium and steep slopes, the plume's bed mass distribution changes abruptly from displaying a well-defined deposition footprint to nearly nondepositional, suggesting that the plume transitions to a self-sustaining, autosuspending state. We can quantify the critical slope over which autosuspension occurs. Taking experiments with ws = 0.1 mm/s and Ur = 2 m/s as examples (cases 6–11 in Table 1), in Figure 11 we plot the ratio of bed mass to the total mass (Mbed/(Mbed + Msus); equation (6)) against bottom slopes. The envelope indicates temporal variability, with the thick line denoting the time mean. It can be seen that the bed mass fraction decreases monotonically with slope, as stronger flow on a steeper shelf can support more sediment in suspension. At slope of 0.03 (case 11), the bed mass fraction approaches zero, indicating that the plume has reached autosuspension. Another indication of autosuspension is the steady velocity. The inset of Figure 11 shows the cross-shore profile of velocity magnitude for the gentle and steep slope cases. It is clear that, on the steep slope, plume travels at an approximately constant speed in the far field (30–45 km from the mouth), consistent with the autosuspending criterion used by Sequeiros et al. . In contrast, over the gentle slope, the velocity of the depositional plume decreases to zero rapidly. Note that the bed mass fraction of the autosuspending plume (i.e., steep slope) does not go to zero because flow deceleration at the plunging region results in finite deposition (Figure 6f). We use a cutoff of bed mass fraction < 0.05, which accounts for the deposition off the mouth, to diagnose autosuspension. When the deposition near plunging is removed, the bed mass fraction reduces to 0.01.
 Following the above analyses, we can find critical slopes for autosuspension under different settling velocities. In Figure 12a, ws of 0.05, 0.1, 0.2, and 0.3 mm/s are evaluated. For each settling velocity, we hold discharge and inlet sediment concentration constant while varying the slope until both a bed mass fraction < 0.05 and a steady plume velocity are satisfied. The condition ws > 0.3 mm/s is not considered because the resulting critical slope over 0.1 is unrealistic as compared to the observed range of shelf slopes (<0.01; Garrison ). Experiments with channelized flow, denoted by gray circles in Figure 12a, are included for comparison.
 Figure 12a shows that the critical slope for autosuspension increases monotonically with settling velocity. This is because particles with larger ws need stronger flow and thus a steeper bottom slope to attain autosuspension. Besides, the critical slopes for channelized cases are lower than those under fully 3-D settings. Channelization, like conditions in submarine canyons, does not allow width expansion, thereby maintaining a thicker hyperpycnal layer and greater gravitational forcing that lead to a less strict slope requirement for autosuspension. However, the channelized and fully 3-D cases tend to converge as ws increases, because 3-D plumes exhibit progressively less width expansion at steep slopes (Figure 6) and large ws (see below).
 We may estimate the critical slope using a simple theory. To make the problem analytically tractable, we begin from channelized flow. The estimated critical slope could serve as a lower bound for the fully 3-D cases and provide insights into its parameter dependence. Combining the energy constraint of equation (9) and the cross-shore momentum balance of equation (10) with buoyancy conservation,
and entrainment parameterization by Parker et al.  (Figure 4)
we obtain a set of four balance equations that governs the dynamics of autosuspending hyperpycnal plume. Key elements and assumptions of this set of equations are: first, for autosuspending plumes that tend to occur on steep slopes, entrainment must be retained in the momentum budget. Second, the energy equation (9) is invoked to evaluate whether the flow power is sufficient to maintain dense suspension. Third, buoyancy is conserved, as there is negligible sediment loss to deposition when autosuspension is reached (in equation (11), qb is buoyancy flux per unit width); fourth, width is assumed constant based on the justifications given above.
 Using equations (2) and (11), we can rewrite the energy equation (9) and express the plume velocity U in terms of buoyancy flux qb,
where qb is given by the inlet boundary condition and qr (=UH) is the volume transport per unit width. Note that the term arises because the interstitial fluid of hyperpycnal plume is fresher than the ambient seawater (see equation (2)). We may approximate with the river mouth value by noting that the scaled density difference decreases as qr increases due to entrainment. Finally, from the momentum budget (equation (10)), we write the critical slope for autosuspension in terms of Frounde number Fr
 The critical slope predicted by equations (14) and (15) agrees reasonably well with the model results. depends only on external variables, settling velocity ws and buoyancy input qb, and can be readily calculated. The theory (solid line in Figure 12a) captures the increasing tendency of critical slope with settling velocity. As expected, the 2-D theory lies closer to the channelized cases, but the predicted and model-derived under fully 3-D settings generally agree within a factor of 2. The plumes within this parameter space can then be separated into two regimes. On slopes larger than , the plumes are autosuspending (upper left corner). Otherwise, they are depositional (see discussion). Besides, can be shown to scale with , where n is between 0.25 and 0.55. In Figure 12a, the model-derived critical slopes are indeed largely confined by these two limits (gray dashed lines).
 The theory can be used to elucidate the processes that control the criticality. Figure 12b shows the sensitivities of the theoretical critical slope (equation (14)) to varying settling velocity ws and buoyancy input qb. It is clear that the critical slope decreases as ws decreases and as qb increases, consistent with the dependence. Decreasing settling velocity lowers the requirement of flow power and therefore slope to sustain autosuspension. Increasing buoyancy input increases the density contrast and/or the plume thickness, which in turn strengthens the downslope gravitational forcing and broadens the conditions for autosuspension.
 In the depositional regime, plumes decelerate and continue to lose gravitational forcing as they propagate offshore. Their deposits thus exhibit a finite runout (i.e., cross-shore penetration; Figures 6a and 6b). Quantifying and understanding processes controlling the runout distance are important because the runout determines whether the river-borne materials can traverse continental shelves to reach the deep oceans.
 We use a scaling theory to guide the parameter-dependence exploration of plume's runout. Bonnecaze and Lister  obtained the characteristic length scales for depositional plumes, using the following set of simplified balance with an assumption of negligible entrainment:
 Equation (16) represents gravitational–frictional force balance in both cross-shore and alongshore momentum budgets (consistent with the results in sections 'Momentum Balance and Cross-Shore Variations' and 'Lateral (AlongShore) Momentum Budget'), volume conservation, and the plume width expressed as lateral velocity multiplied by a characteristic timescale T. T is assumed equal to particle settling time . Note that the main dynamical distinctions between depositional (equation (16)) and autosuspending (equations (9)-(12)) plume regimes lie in that the contribution of entrainment to momentum deficit for depositional plumes are small compared with bottom friction (Figure 8a) and the length scales of depositional plumes are limited by vertical settling. From equation (16), the scales of runout (Xp = UT; advective distance), plume thickness Hp, and width Wp can be found:
 In the above equation, the subscript p denotes predicted scales, represents the initial (mouth) value of section-averaged reduced gravity, , and is approximated to unity for small angles considered here.
 We then examine how plume's runout responds to varying slopes, settling velocities, and river discharge (cases 6–19 in Table 1). The correspondence between the runout and the advective length scaling Xp is evaluated in Figure 13. The runout is quantified as the cross-shore distance of the center of mass from the river mouth at the end of the discharge event (e.g., trajectories in Figure 6). It is important to note that only cases 10, 11, and 15 (i.e., steep slope and small ws) are in or near the autosuspending state (Figure 12a). We thus anticipate these three cases to exhibit a longer runout than the scaling of equation (17) predicts.
 It is evident in Figure 13 that plume's runout largely follows an advective scaling of for all of the depositional model runs. The runout for autosuspending plumes is as expected to be significantly longer than the scaling but is still finite because the offshore propagation of the center of mass is limited by the event duration (instead of vertical settling). For depositional runs, the runout increases with increases in slope due to enhanced plume velocity (cases 6–9). Increasing discharge results in expansion of runout (cases 16–18) and width (data not shown) because increased discharge pushes the plunging point seaward, leading to a thicker hyperpycnal layer and thus a stronger gravitational force. On the contrary, increasing settling velocity shrinks the plume's horizontal footprint (cases 12–14) by shortening the settling time. Overall, for the range of parameters considered, the runout of depositional plumes is largely confined within 15 km from the mouth. However, when the autosuspending state is reached, the plume can travel beyond 15 km and a different length scale is imposed. Note that the scaling factor of 0.3 is chosen to match the linear trend in the data set (R2 = 0.91). This factor is not of order one, due in part to the fact that the runout is quantified using the center of mass and the cross-shore mass distribution is skewed toward the mouth. Nevertheless, our intention is to demonstrate the differences in runout between decelerating and autosuspending regimes (i.e., two clustering of data), which is insensitive to the choice of the scaling factor.
6. Discussion and Summary
6.1. Potential Influence of Variable Forcing
 With an objective of establishing baseline understanding of hyperpycnal plume dynamics, we have focused on cases of dense plume empting into motionless shelves. Unsteady river discharge and ambient shelf conditions such as tides, waves, and bed sediment source are not considered. Here, we discuss the potential impacts of these conditions.
 Plumes over gentle slopes are expected to be more susceptible to ambient forcings. The influences of shelf tides may be evaluated using a simple theory. Assuming no alongshelf phase variations, Battisti and Clarke  found the magnitude of depth-averaged tidal currents as (η is tidal amplitude, ω is M2 tidal frequency). The inverse proportion to the slope arises because, for a given η and cross-shelf distance, a steeper shelf has a larger cross-sectional area to accommodate the volume flux, thus yielding weaker barotropic tidal currents. Using UChezy to represent hyperpycnal plume velocity, we may express the ratio of tidal and plume velocities as
 The strong negative sensitivity to the slope is clear. Shelf tides are then anticipated to have stronger effects on bottom plumes under conditions of gentler shelf slopes, larger tidal amplitude, and weaker density contrasts. We may estimate that the critical slope over UT and UChezy are comparable. Applying the parameters used in this study ( and thickness ∼ 5 m) and η of 2 m for accessing the upper bound of tidal effects, the critical slope is estimated to be around 8 × 10−4, which is at the lower end of the typical shelf slopes. The above exercise suggests that, on typical shelves, tidal modulations of hyperpycnal plumes are likely to be small. However, if a plume plunges in the river channel or near the mouth, the impacts of tides could be greatly enhanced, due to the strengthened tidal flow through constriction. Such a case was demonstrated by Wang et al.  from Huanghe (China). They found that a thin hyperpycnal undercurrent established at late ebbs was destroyed during floods by tidal mixing.
 Over shallower shelves, surface waves are expected to show greater impacts on undercurrents because of less attenuation of wave orbital velocity with depth. However, the impact zone may be near the plunging points, rather than plume's main body. Traykovski et al.  and Wright et al.  have shown that strong wave forcing acting on river-derived mud deposits over inner shelves could trigger gravity currents that move the sediment offshore. In these cases, wave-induced suspension plays a key role in maintaining a thin turbid layer. For a fully developed hyperpycnal plume, the dense suspension is inherited from turbid rivers and is maintained by plume's velocity, meaning that wave forcing is not a prerequisite. Besides, the plume is most likely thicker than the wave boundary layer (∼10 s cm). Thus, while wave actions are expected to enhance the near-bed turbulence and help maintain the dense suspension, they might not significantly alter the structure of a well-developed hyperpycnal layer. Near plume's plunging region, by contrast, the water depth is shallower. Strong wave-induced mixing could dilute the density contrasts, thereby preventing the initial development of an undercurrent.
 Unsteadiness is an important characteristic of hyperpycnal discharge that has not been systematically examined. Typhoon (Storm)-induced flood hydrography typically exhibits a steep ramp-up (i.e., 1 day in this study) and a more gradual ramp-down (not considered here; see Dadson et al. ). Taking the steep-slope case as an example, we compare the plume structures in response to a discharge event with and without a 2.5 day ramp-down (data not shown). With a ramp-down, the period of hyperpycnal condition is shortened. In the far field (e.g., 10 km away from the mouth), the plume's mass and velocity distribution of the unsteady case (i.e., with a ramp-down) are similar to the steady case, and the plume remains to be nondepositional. The similarities are the results of identical discharge forcing during the ramp-up, as the far field corresponds to the initial forcing period. By contrast, in the near field, the plume in the unsteady case has less total mass (msus + mbed), which reflects the reduction of sediment input during the ramp-down. Above all, the plume's footprints in these two cases are nearly identical, suggesting that plume's characteristic scales are likely set by the peak discharge. It should be noted that the above comparisons assume a motionless shelf sea. We expect the plumes to be significantly altered when the unsteady discharge interacts with time-dependent ambient forcing. For example, the temporal coherence between the peak discharge and peak wind/wave and tidal forcing is likely to be important [Wheatcroft2000; Kniskern et al., 2011] and needs further investigations.
 In this study, bed sediment source is not considered, which excludes the possibility of self-accelerating hyperpycnal plumes. The neglect of fine-grained bed is justified, as energetic inner and midshelves are typically fine-sediment-starved [Garrison, 2009]. Here, we evaluate the potential influences of a sandy bed source on plume dynamics using two bottom boundary layer (BBL) models (Glenn and Grant  against van Rijn  models; see equations (15), (16) and (19), (20) in Cacchione et al. ). Taking the steep slope case as an example, we may estimate the maximal sand entrainment into the plume with the following parameters: very fine sand with D50 of 63 µm, ws of 3 mm/s, critical shear stress of 0.15 Pa, a maximum empirical constant γ0 of 2 × 10−4 for reference concentration, and a Rouse profile referencing to the lowest grid point at z = 0.2 mab [see Cacchione et al., 2008]. For the near-field plume region, two BBL models yield comparable area-mean sand concentrations of 3.6 and 2.9 kg/m3, respectively. While these concentration estimates would constitute significant sand transport, they are small in comparison with plume's area-mean concentration of 40.0 kg/m3 for fine-grained suspension. The comparisons therefore suggest that the presence of a sandy bed likely has a secondary effect on the fine-grained hyperpycnal plumes.
 Another simplification of this modeling study is the use of single-grain-sized hyperpycnal discharge. Although hyperpycnal discharge typically shows a very poorly sorted grain size distribution that are dominated by fine-grained sediment [Warrick et al., 2008; Kao et al., 2008], fraction of terrestrial sand is still present. The sand transported with hyperpycnal flows may escape the coastal littoral cells and thus lead to long-term storage on the shelves [Warrick and Barnard2012]. Besides, the above discussion focuses only on a few important ambient conditions. Other processes such as internal wave generation associated with dense plume propagation [White and Helfrich, 2008], freshwater release by the outflows [Hurzeler et al., 1996; Chao, 1998], and the formation of flood deposits (e.g., hyperpycnites) with mixed grain-size river input and sea bed [Mulder et al., 2003] are not addressed and merit future studies.
6.2. Implications of Autosuspension Criterion
 The autosuspension criteria proposed by prior studies on turbidity currents can be recovered from the theory presented here. The criteria documented in the literature are often expressed in terms of settling velocity ws, turbidity current velocity U, and slope
 Combining equations (9), (10), and (2) ( for turbidity currents), we recover a similar form
 Using a wide range of parameters with a = 0.1–0.5 (for H = 2–100 m), CD = 1–5 × 10−3 (factor of 103 roughness variations), and CE = 0.002–0.03 (for Fr = 1–3), we find to be bounded between 0.002 and 0.1. Without invoking additional constraint of riverine buoyancy input (equation (11)), this range is already fairly close to Stacey and Bowen  and Pantin . The apparent discrepancy with Parker  may be attributed to different natures of suspension. Parker  addressed dense suspension of sand particles (i.e., turbidity currents), and Bagnold type of lifting force provided by available stream power argument is used. In contrast, we focus on mud suspension, and Winterwerp's treatment based on energetics of two-layer fluid is adopted (equations (8) and (9); see Winterwerp ). Note however that Parker's criteria was applied for Fr < 1 and CE = 0 (see Winterwerp's  equation (19)). Setting CE = 0 in (20) increases the upper bound of to 0.3, which approaches Parker's criteria.
 It is important to note the difference between prior criteria and our theory. Prior criteria (equation (19)) are diagnostics, in the sense that the current velocity must be specified. The autosuspension condition presented here (equations (14) and (15)), in contrast, is prognostic, as the critical slope is determined by external variables of settling velocity ws and riverine buoyancy input qb. Using the predicted critical slopes, we may infer the likelihood of occurrence of autosuspending hyperpycnal plumes in several mountainous river systems. In Figure 12b, the critical slope with outflow concentration of 150 g/L and velocity of 4 m/s may represent a minimum requirement for autosuspension, based on observed conditions of hyperpycnal discharge by Warrick and Milliman . Slopes of continental shelves are typically below 0.01 [Garrison, 2009]. It is then clear in Figure 12b that autosuspending hyperpycnal plume is unlikely to form over typical shelves, except for very fine sediment (ws < 0.1 mm/s) during extreme events. In steeper submarine canyons that connect to river mouths such as Gaoping and Sepik River systems, autosuspending plumes are more likely to occur but are still limited to fine-grained, large discharge event [ws < 1 mm/s; Liu et al., 2012]. Overall, the analyses suggest a rather stringent requirement for an autosuspending hyerpycnal plume in the coastal oceans.
 S.N.C. is supported by Taiwan National Science Council grants NSC 100-2199-M-002-028 and NSC 101-2611-M-002-006-MY2. The computing resource was provided by the College of Science and the Taida Institute for Mathematical Sciences at National Taiwan University. W.R.G. and Hsu are supported by NSF grant OCE-0926427. S.N.C. thanks John Trowbridge for helpful discussion. Helpful comments provided by Jon Warrick (USGS) and an anonymous reviewer are appreciated.