Bankfull hydraulic geometry of submarine channels created by turbidity currents: Relations between bankfull channel characteristics and formative flow discharge

Authors


Corresponding author: Jessica Zinger, Department of Geography and Geographic Information Science, University of Illinois Urbana-Champaign, 220 Davenport Hall, 607 S. Mathews, Urbana, Illinois 61801-3637, USA. (zinger1@illinois.edu)

Abstract

[1] Detailed bathymetric surveys of the seafloor have enabled the identification and analysis of submarine channels worldwide. Previous authors have remarked on the morphologic similarity of submarine channels and rivers, and have identified a number of similarities and differences in processes of flow and sedimentation. In this study, we compare the width, depth, and slope of 177 submarine channel cross-sections to that of 231 river cross-sections. The results indicate that submarine channels have cross-sectional dimensions that can exceed the dimensions of the largest rivers on earth by an order of magnitude. For rivers and submarine channels with similar width or depth, the slope of submarine channels can be up to two orders of magnitude greater than the slope of rivers. An analysis of trends in driving force vs. channel size suggests that a reasonable estimate of the volumetric sediment concentration of channelized turbidity currents lies in the range C = 0.2% to 0.6%. Bankfull turbidity current velocities are estimated using this range in concentration. Friction coefficients are based on values identified for large rivers and a modified Chezy equation. These velocities are then used in a classic hydraulic geometry analysis of the submarine channels, which shows that submarine channels and rivers follow similar power law trends in width, depth, and velocity as functions of bankfull discharge.

1 Introduction

[2] Sediment-laden density flows, or turbidity currents, constitute a key mechanism for the transport and deposition of sediment on submarine fans and the continental slope. Turbidity currents play an important role in shaping the morphology of the seafloor, and frequently form channel systems. Submarine channels are not only distinctive morphologic features; their deposits also play an important role in the architecture of the sedimentary record, and hold economic value as a common type of hydrocarbon reservoir [Clark and Pickering, 1996; Weimer et al., 2000; Abreu et al., 2003]. Submarine channels have been studied extensively in outcrop exposures, echosounder images, laboratory experiments, and computer models. However, direct field measurements of turbidity current velocity, sediment concentration, and grain size, necessary to constrain laboratory and numerical models, are difficult to obtain due to the intermittent nature of turbidity current events and the difficulties of data collection in the submarine environment. One of the few field sites where such measurements have been conducted is Monterey Submarine Canyon off California [Xu et al., 2004; Xu, 2010]. Thus, there remains a “process gap” [McHargue et al., 2011] in our understanding of the formation of submarine channels and their associated deposits.

[3] Previous studies have shown that rivers and submarine channels share many similar morphologic elements, such as levees, meandering planforms, point bars, and scroll bars [Normark, 1970a, 1970b; Chough and Hesse, 1976; Lonsdale and Hollister, 1979; Flood and Damuth, 1987; Clark et al., 1992; Hagen et al., 1994; Pirmez and Flood, 1995; Klaucke et al., 1998; Nakajima et al., 1998; Schwenk et al., 2003; Fildani and Normark, 2004]. Unlike rivers, the width, depth, and levee thickness of submarine channels tend to decrease rapidly downstream [Flood and Damuth, 1987; Pirmez and Imran, 2003]. Still, submarine channels and rivers show similar relationships between meander bend wavelength and channel width, bend wavelength and radius of curvature, and channel sinuosity and slope [Clark et al., 1992; Pirmez and Imran, 2003; Straub et al., 2007]. Bend cutoffs, a common feature of meandering rivers, have also been identified on channels found on the Amazon Fan [Pirmez and Flood, 1995] and the Bengal Fan [Schwenk et al., 2003]. However, Peakall et al. [2000] argue that submarine channel bends undergo cutoffs infrequently compared to fluvial channels of similar sinuosity. Additionally, the drainage density, bifurcation ratio, length ratio, and slope ratio of submarine channel drainage networks fall within the range of variability found in fluvial drainage networks [Pratson and Ryan, 1996; Straub et al., 2007]. The similarities and differences of the morphology and formational processes of these features have been discussed in detail [Peakall et al., 2000; Keevil et al., 2006; Kolla et al., 2007; Wynn et al., 2007]. In contrast, there are few systematic comparisons of the relationships between channel width, depth, and slope of fluvial and submarine channels [e.g., Straub et al., 2012], despite the availability of relevant data.

[4] The cross-sectional geometry of river channels has long been a subject of study in the field of geomorphology. In their classic work, Leopold and Maddock [1953] demonstrated that the hydraulic geometry of river channels shows relatively consistent behavior over a wide range of environments, despite variations in geology, vegetation, and climatic regime. Specifically, the width, depth, and velocity of river channels all increase as power functions of discharge [Leopold and Maddock, 1953]. This begs the question of whether or not such power law relationships extend to the submarine environment. Power law hydraulic geometry relationships have been determined for the Cascadia channel [Wilde, 1965], the Ascension-Monterey east channels [Wilde, 1965], and the Cocos Ridge deep sea channels [Wilde, 1966]. However, a hydraulic geometry analysis must be applied to a large number of submarine channels to determine if such power law behavior is a universal characteristic.

[5] In this study, we first assess the relationships between width, depth, and slope for 177 cross sections from 23 submarine channels and 231 cross sections from rivers [Parker et al., 2007; Latrubesse, 2008; Wilkerson and Parker, 2011]. We then infer values for volumetric sediment concentration of the turbidity currents by assuming that rivers and submarine channels follow a similar relationship between the net driving force of flow and channel size. These sediment concentration values, and a modified Chezy equation [e.g., Hurley, 1964; Wilde, 1965] are used to estimate mean flow velocities for the submarine channels. We assume values for the channel bed friction coefficients for submarine channels based on those found for large sand-bed rivers. Finally, the estimated mean flow velocities are used to establish hydraulic geometry relationships for a combined dataset of submarine channels and rivers.

[6] Thus, by using rivers as an analog, rough estimations of channelized turbidity current concentration and velocities may be made. These calculations rest on the assumption that a meaningful, semiquantitative analogy can indeed be drawn between rivers and submarine channels. We make this assumption in our choice of sediment concentration and friction coefficient, as well as in the use of a modified Chezy equation. This assumption is a working hypothesis grounded in the fact that both types of flows are subject to the same three forces: gravity, friction, and buoyancy. In the absence of accurate field measurements for channelized turbidity currents, the approach presented here provides a novel means of estimating velocity and concentration data necessary to constrain laboratory and numerical models.

2 Methods

[7] A database of submarine channel width, depth, and slope is obtained from an extensive review of bathymetric data reported in the literature (see Supplementary Information). Channel dimensions are extracted from plots of channel width, depth, and slope, typically plotted as a function of streamwise distance, and reported values published in extant literature (see Supplementary Information), as well as measured directly from published echosounder images. We define channel depth (H) as the vertical distance between the deepest point of the channel bed to the average height of the levee crests (Figure 1). We choose this measure rather than a cross-sectionally averaged depth due to some ambiguity in defining the latter and to maintain consistency with reported values. Channel width (B) is defined as the distance between the levee crests, or maximum width between the channel banks [Leopold and Maddock, 1953]. These parameters correspond to bankfull conditions, in that they describe the geometry at which a turbidity current just spills over its levees. We assume that these channels are depositional in origin and in equilibrium with formative flow events. The channel slope (S) is taken as reported, or calculated as the change of bed elevation between two cross sections, divided by the streamwise distance. In some cases, channel slope data were not available (see Supplementary Information). A total of 23 submarine channels are included in the analysis, although more cross sections are available for some channels than others (Table 1).

Figure 1.

Sketch of a typical submarine channel cross section showing the bankfull top width (B) and the bankfull depth to thalweg (H). Vertical exaggeration is approximately 30.

Table 1. Inventory of Submarine Channels Analyzed
ChannelLocation# Cross Sections
  • * Indicates no slope data found.
NAMOCLabrador Sea35
VidalNorthern Guiana Basin23
Cascadiaoffshore Oregon/Washington23
AmazonE. coast S. America14
Montereyoffshore California12
TanzaniaEast African Margin10
ToyamaYamato Basin10
Bountyeast of S. Island New Zealand10
SurveyorAlaskan Slope11
BengalIndian Ocean7
ValenciaW. Mediterranean Sea4
La Jolla*off S. California3
Lofoten*Norwegian Margin3
MagdalenaCaribbean Margin of Columbia3
Falkland*Atlantic1
Faroe-Shetland*West Shetland Slope (off Scotland)1
Gollum*European NE Atlantic margin1
Green*Offshore Angola1
Iberia-Biscay Interplain ChannelAtlantic1
Moresby*Pacific1
Scott*Pacific1
Seamap*Pacific1
Vema*Atlantic1
 Total177

[8] Parker et al. [2007], Latrubesse [personal communication], and Wilkerson and Parker [2011] compiled the data for bankfull top-width, cross-sectionally averaged bankfull depth, bed slope, bankfull discharge, and mean bankfull velocity for the 231 river cross sections used in this study. It is important to note that the bankfull discharges and mean bankfull velocities collected for rivers are measured values, rather than calculated. The widths of these channels range from 2 m to 3,400 m and depths range from 0.22 m to 48 m (Figure 2). Additionally, the characteristic bed material size ranges from silt-size particles (D50 < 0.062 mm) to gravel (the latter including cobble sizes, up to 167.5 mm). The data are stratified into five size ranges in Figure 2. The data for all ranges show a common trend. This result suggests that a similar trend may hold for the case of submarine channels, for which characteristic channel bed material sizes are rarely available.

Figure 2.

Bankfull width plotted against bankfull depth for rivers, sorted by median bed material grain size (D50 given in mm).

[9] We also apply the classic hydraulic geometry analysis of Leopold and Maddock [1953] to bankfull characteristics of both the rivers and the submarine channels. In a hydraulic geometry analysis, power law relationships are determined between discharge (Q) and either the width (B), depth (H) or velocity (U) of the flow (equations (1)-(3) [Leopold and Maddock, 1953]. A hydraulic geometry analysis can be applied at-a-station, using the changing cross-sectional geometry of the fluid in the channel with discharge at a single cross section, or downstream, which relates the channel geometry at different cross-sections to a single discharge of given flow frequency. At-a-station hydraulic geometry is constrained by the continuity requirement, meaning the exponents in equations (1)-(3) below (b, f, and m) should add to one, and the coefficients (a, c, and k) should multiply to one. In contrast, the downstream hydraulic geometry relationships are not constrained by continuity for any discharge that is not a bankfull flow everywhere along the channel. We assume that all error associated with the relaxation of the continuity requirement is encompassed by the relationship between velocity and discharge.

display math(1)
display math(2)
display math(3)

[10] The downstream hydraulic geometry analysis for submarine channels is based on measurements of channel cross-sectional geometry and estimates of bankfull mean velocity and discharge at each cross section. We therefore assume that: (a) the bankfull discharge has the same flow frequency at every cross section used in our analysis and (b) the continuity requirement should be satisfied. Although Khripounoff et al. [2003] report on direct measurements of a turbidity current in the Zaire submarine channel 150 meters above the channel bed, there appear to be no actual measurements of mean (cross-sectionally averaged) flow velocity or discharge of turbidity currents in submarine fan channels. In order to fill this gap, we estimate bankfull mean velocity and discharge using a modification of the Chezy formula incorporating flow resistance from both the channel bed and the mixing interface between the turbidity current and the ambient fluid above.

3 Geometric Properties of Submarine Channels and Estimation of Characteristic Suspended Sediment Concentration

3.1 Width, Depth, and Slope of Submarine Channels and Rivers

[11] The power law relationship between river channel width and depth is well documented. For example, the power relations of Leeder [1973], R. C. Crane (unpublished PhD dissertation, 1982), and Bridge and Mackey [1993] are given in Table 2. A power law relationship between width and depth is also consistent with the data collected for rivers and submarine channels in this study (Figure 3). The results of power law regression for rivers only, submarine channels only, and the combined set of rivers and submarine channels are shown in Figure 3 and listed in Table 2. The submarine channel data show more scatter than the river data, but nevertheless follow a trend that can be fitted to a power law. A combination of the river and submarine channel data is well represented by a power law which is similar to that obtained from just the river data. Although the two data sets show overlap, it is notable that submarine channels can be up to an order of magnitude wider and deeper than the largest rivers (Figure 3, Table 3). Table 3 also shows that submarine channels have a greater average width/depth ratio than rivers. This result is in contrast to a previous report (Table 3) [Peakall et al., 2000]. As explained above, a bankfull depth corresponding to a cross-sectional average was used in the case of the rivers, whereas the maximum depth of the cross-section was used in the case of the submarine channels. Were a precisely common definition used for both, the difference in average width/depth ratio between submarine channels and rivers would only become more pronounced.

Table 2. Power Law Regressions for Channel Width vs. Depth
Regression EquationSource
  1. *This study.

B = 22.3(Hmax)1.13Combined river/submarine channel data*R2 = 0.86
B = 47.4(Hmax)0.94Submarine channels only*R2 = 0.39
B = 18.8(Hmax)1.41Rivers only*R2 = 0.84
B = 6.8(Hmax)1.54

Leeder (1973)

B = 12.82(Hmax)1.59Crane (1982)
B = 8.88(Hmax)1.82

Bridge and Mackey (1993)

Figure 3.

Channel width vs. depth. The black dotted line shows the power law regression of the combined dataset of river and submarine channel cross sections, the red line shows the power law regression of just the submarine channel data, and the blue line shows the power law regression of just the river data.

Table 3. Comparison of Width, Depth, Slope, and B/H
 Width (m)Depth (m)SlopeB/H
 Submarine ChannelsRiversSubmarine ChannelsRiversSubmarine ChannelsRiversSubmarine ChannelsRivers
Min600.02.33.40.20.000120.000016.14.1
Max29000.03400.0450.048.10.020900.05200638.0230.0
Mean7820.3349.2135.75.10.002230.0034371.1141.0
Standard Deviation7507.8687.778.57.90.002690.0068180.4343.5
Median4821.140.0128.61.90.001270.0011043.122.9

[12] Plotting channel slope against width, depth, and area serves to underline the differences between submarine channels and rivers (Figures 4a–4c). A clearly defined relationship exists between channel dimension and slope for rivers, in which slope tends to decrease with increasing channel size. In contrast, the same trend is quite muted in the case of the submarine channels. But above all, these plots show that for a given channel width or depth, submarine channels are 1–2 orders of magnitude steeper than rivers.

Figure 4.

(a) The slope of the channel cross sections plotted against bankfull depth to thalweg (m). (b) The slope of the channel cross sections plotted against bankfull top width (m). (c) The slope of the channel cross sections plotted against bankfull cross sectional area (m2). The river cross sections are plotted as squares and the submarine channel cross sections are plotted as circles.

3.2 Unit Driving Force and Turbidity Current Concentration

[13] As stated above, gravity, friction, and buoyancy are the three main forces that act on channelized flows. The driving force of flow on a control volume of fluid is the downslope component of the immersed weight of the control volume, while friction acts as the resisting force (Figure 5). The force of buoyancy thus reduces the effective driving force of the flow. This effect is particularly strong when the density of the flowing fluid does not differ significantly from the ambient fluid.

Figure 5.

Force balance diagram for a control volume of fluid on a slope, where A is the cross sectional area of the control volume, L is the length of the control volume, ρamb is the density of the ambient fluid, ρw,t is the density of water or the density of the turbidity current, θ is the angle of incline of the slope, Cfi is the coefficient of friction between the flow and the ambient fluid, and Cfb is the coefficient of friction at the bed.

[14] Consider the control volume illustrated in Figure 5, in which a portion of channel with slope S, area A, and length L is depicted. Consider the case of a river, where water of density ρw flows under (i.e., is immersed in) air with density ρair and g is the gravitational acceleration. The driving force Fdrive, river for this case is the down-channel component of the immersed weight of the control volume of water, given as equation (4).

display math(4)

[15] The approximation is possible because the density of air is negligible compared to that of water. (This result is not changed by the presence of suspended sediment, as long as the concentration is sufficiently dilute that the density of sediment-water mixture differs little from the density of the water alone.)

[16] We next consider the case of a turbidity current: a mixture of water and suspended sediment with density ρt that flows under clear (sediment-free) water with density ρw. Where C denotes the volume concentration of suspended sediment, ρsed denotes the density of the sediment itself and R = (ρsed - ρw)/ρw (~ 1.65 for quartz). ρt is given as equation (5).

display math(5)

[17] The driving force Fdrive,turb acting on the control volume is then given by equation (6).

display math(6)

[18] The driving forces per unit distance downstream (i.e. unit driving forces fdrive,river and fdrive,turb) for a river and turbidity current, respectively, are given as equations (7a) and (7b). Therefore, the ratio between fdrive,river and fdrive,turb, denoted rdrive, is given as equation (8).

display math(7a)
display math(7b)
display math(8)

[19] Turbidity currents differ from submarine debris flows in that they are dilute, such that C ≪ 1 [e.g., Marr et al., 2001]. Recalling that R ≈ 1.65, the above equation illustrates that for the same channel slope, turbidity currents have a greatly reduced unit driving force as compared to rivers. Consequently, for a turbidity current in a submarine channel the net unit driving force is reduced by the factor RC compared to a corresponding river flow.

[20] As shown above, the unit driving force (~ S) of rivers clearly decreases as channel width, depth, and area increase (Figures 4a–4c). We are unable to calculate the unit driving force (~ RCS) for submarine channels without a value for volumetric sediment concentration. The clear relationship between slope and channel size for rivers, as well as the analogous physics of rivers and turbidity currents, suggests that the latter have concentrations such that they obey the same trend in unit driving force vs. channel size as rivers. In considering this, it is useful to recall that both types of flows are driven by the same force (gravity), and the driving force differs only in the degree to which gravity is reduced by buoyancy.

[21] In all cases, the slopes (S) of the submarine channels plot substantially higher than the rivers (Figures 4a–4c). Equations (7a) and (7b) indicate that for the same slope, the unit driving force ρwRCgS of the turbidity currents that sculpted the submarine channels can be brought into accord with the unit driving force of the rivers ρwgS by choosing an appropriately low value of concentration C. Here we use the hypothesis of approximate similarity in driving force to back-calculate estimates for C for turbidity currents. This hypothesis is motivated by the idea that rivers and turbidity currents represent end-member cases of a continuum of sediment-laden flows that are governed by a common set of relations.

[22] The results of such a back-calculation are shown in Figures 6a–6c. The value of C that provides the best fit between the river and submarine channel data is 0.2%. This value is low compared to the values given by Pirmez and Imran [2003], which are quoted extensively in the literature. Nevertheless, even the lowest value reported by Pirmez and Imran [2003], C = 0.6%, provides a reasonably good correspondence between the two data sets. The method described above does not provide a means to estimate the suspended sediment concentration C of the formative turbidity current of each individual submarine channel. The best it can do is provide a single estimation of the range of realistic concentrations for the dataset of this present study. The analysis here suggests that the range 0.2% ~ 0.6% is reasonable. Indeed, the tentative and approximate nature of the similarity hypothesis notwithstanding, it provides the first quantitative means to estimate a parameter that heretofore has been constrained mostly by informed speculation.

Figure 6.

(a) Unit driving force (fdrive) plotted against bankfull depth to thalweg (m). (b) Unit driving force (fdrive) plotted against bankfull top width (m). (c) Unit driving force (fdrive) plotted against bankfull cross sectional area (m2). Squares show measured data for rivers, circles show calculated values for submarine channels with C = 0.2% and Cfb = 0.005, and triangles show calculated values for submarine channels with C = 0.6% and Cfb = 0.002.

4 Comparison of River and Submarine Channel Geometry

[23] In the case of turbidity currents, the non-negligible buoyant force effectively reduces the gravitational driving force of the channelized flow by the factor RC. Yet, in spite of the reduced gravity environment, turbidity currents sculpt some of the largest channel features anywhere on earth. Our data analysis shows that for a given channel dimension (width, depth, or area) submarine channels tend to be steeper than rivers by more than one order of magnitude. This trend can be readily explained by examining the equation for the driving force on a control volume of fluid (equations (7a), (7b)). For a given channel size, this relationship shows that a turbidity current requires a steeper slope to have the same driving force as a river. Similarly, for a given channel slope, submarine channels must have larger dimensions than rivers to have the same driving force. In fact, most submarine channels tend to be wider and deeper than rivers, with some overlap between the largest rivers and the smallest submarine channels.

[24] As noted in Table 2, rivers and submarine channels follow a similar power law trend in width vs. depth, where B = 22.2Hmax1.13 (R2 = 0.86) for the combined dataset of rivers and submarine channels (Figure 3). However, the power law regression for just the rivers (B = 18.7Hmax1.42, R2 = 0.84) also provides a good fit for a large fraction of the submarine channels. Crane [1982] reported B = 12.82Hmax1.59 for 57 rivers with sinuosity greater than 1.7, which is in general accordance with the results presented here. It should be noted that a key difference between rivers and submarine channels is that width and depth tend to increase in the downstream direction for rivers, while the opposite holds true for submarine channels, particularly within the submarine fan systems [Flood and Damuth, 1987; Babonneau et al., 2002; Pirmez and Imran, 2003]. This is at least due in part to the fact that most submarine channel networks are distributary systems. This is not always the case, as collective submarine networks have also been identified [Straub et al., 2007, 2012]. In addition, turbidity currents can wane downstream due to entrainment of ambient water and deposition of suspended sediment, whereas rivers do not lose their driving power as sediment settles out.

[25] It is interesting to note the conspicuous lack of submarine channels in this dataset that are smaller than ∼10 m deep or ∼1000 m wide. The cause of the relative scarcity of small submarine channels in the literature is still a relatively open question. The bias toward large submarine channels may arise from the limited resolution of bathymetric surveying instruments. Previous studies of submarine channel morphology may tend toward large channels as they are easier to identify and image. Recent advances in bathymetric surveying technology [e.g., Maier et al., 2011, 2012] have enabled the imaging of a submarine channel system with vertical relief < 10 meters. Future applications of this technology will help identify the cause of this apparent bias toward large channels in this dataset.

5 Estimation of Bankfull Velocity and Discharge in Submarine Channels

[26] In the classic hydraulic geometry analysis of Leopold and Maddock [1953], the width, depth, and velocity of river flows are all shown to increase as power functions of discharge. While these relations are well established for rivers, the difficulties of obtaining measurements of velocity in channelized turbidity currents preclude a similar hydraulic geometry analysis based on directly-measured field data. A simple balance of driving and resisting forces, based on governing equations developed for rivers but adapted to turbidity currents can, however, be used to estimate the mean bankfull flow velocities in submarine channels. Using the flow velocities and channel dimensions to estimate bankfull discharge, it is then possible to perform a hydraulic geometry analysis on the submarine channels.

[27] Turbidity current velocities were estimated using a Chezy equation that has been modified to include the effect of the interface between the turbidity current and the ambient water above [e.g. Komar, 1977]. The original Chezy equation (equation (9a)) (for a wide channel) is a simplified force balance between the driving force of gravity and the resisting force of friction at the bed, which is encompassed by the Chezy coefficient (Cz) (related to the bed friction coefficient Cfb in equation (9b)).

display math(9a)
display math(9b)

[28] In the submarine environment, we must account for the additional complications of the non-negligible buoyant force and resistance at the mixing interface between the flow and the ambient fluid. The Chezy equation can then be rewritten for turbidity currents as shown in equation (10):

display math(10)

[29] The force of buoyancy is accounted for by multiplying the driving force by a factor of RC (equation (10)). Friction at the interface between the flow and the ambient fluid is included in the conversion of the Chezy coefficient to a friction coefficient (Cf) that is composed of the sum of the friction coefficient at the bed (Cfb) and the friction coefficient at the interface (Cfi) (equation (11)).

display math(11)

[30] Equation (10) can then be rewritten in terms of the bed friction coefficient and r, the ratio of the interfacial friction coefficient to the bed friction coefficient (equation (12)), to give equation (13).

display math(12)
display math(13)

[31] To solve equation (13) for velocity, assumptions must be made about the sediment concentration and the friction coefficient at the bed. For sediment concentration, we use the results from above (C = 0.2%) and a value reported by Pirmez and Imran [2003] (C = 0.6%). The bed friction coefficient is influenced by a number of parameters, such as grain size, presence or absence of bedforms, channel curvature, and large-scale channel roughness elements. Since measurements of these parameters are not readily available for flows in large submarine channels, we use the bed friction coefficients of large river channels (defined as Qbankfull ≥ 20,000 m3/s) as an analog. Out of the 19 large rivers in the collection used here, 12 have bed friction coefficients between 0.002 and 0.005 (Figure 7). There is reasonable agreement between river discharges calculated using equation (9a) with Cfb = 0.002 and 0.005 and the measured discharge values (Figure 8). We will therefore use both these values of bed friction coefficient to bracket our calculations of velocity for the submarine channels.

Figure 7.

Streamwise channel slope plotted against bed friction coefficient (Cfb) for large rivers (black circles), and the minimum and maximum values of bed friction selected for the hydraulic geometry analysis (solid line indicates Cfb = 0.002 and dashed line indicates Cfb = 0.005).

Figure 8.

Discharge predicted using the Chezy equation (Eq. (9a) plotted against measured discharge values for large rivers (diamonds show predictions using Cfb = 0.002, squares show predictions using Cfb = 0.005, and the solid black line shows Qpredicted = Qmeasured). Discharge is in units of m3s-1.

[32] A third component necessary to estimate turbidity current velocity is the friction at the interface between the flow and ambient fluid, which is caused by entrainment of the ambient fluid into the turbidity current. First we define the bulk dimensionless Richardson number Ri (equation (14)).

display math(14)

[33] Parker et al. [1986] derived a set of layer-averaged equations governing turbidity currents that can be used to characterize interfacial friction. An application of the layer-averaged momentum equation [equation (17)c) from Parker et al., 1986] to the case of normal flow, for which layer-averaged velocity U remains constant and net entrainment/deposition of sediment is negligible, leads to equation (15). The friction coefficient at the interface between the flow and ambient fluid is given by equation (16) and ew is a dimensionless coefficient of entrainment of ambient water into the current, given by equation (17). Parker et al. [1987] derived an empirical relation for ew as a function of bulk Richardson number (equation (18)).

display math(15)
display math(16)
display math(17)
display math(18)

[34] Setting equation (13) equal to equation (15) gives equation (19), which can be iteratively solved for Ri, given slope S, bed friction Cfb, and equations (12), (16), and (18). The mean flow velocity can then be calculated from equation (14), given values for R, C, and H.

display math(19)

[35] In the present study, velocities were estimated by solving equation (19) with four different combinations of sediment concentration and bed friction coefficient (C = 0.2% and 0.6%; Cfb = 0.002 and 0.005). Examining the modified Chezy equation (equation (13)) shows that sediment concentration is directly related to velocity, whereas the friction coefficient is inversely related. Therefore, to simplify the presentation of the results, only the two end member values of C and Cfb (C = 0.2% and Cfb = 0.005; C = 0.6% and Cfb = 0.002) are shown here. Once velocity was computed, bankfull discharge values for the submarine channels were computed using the continuity equation (equation (20)), assuming a rectangular channel cross section.

display math(20)

6 Relations for Hydraulic Geometry as Functions of Bankfull Discharge

[36] Results from the hydraulic geometry analysis show that rivers and submarine channels follow similar power law relations for both width and depth plotted as a function of bankfull discharge, with relatively modest variation among the different cases of C and Cfb (Figures 9 and 10). The plot of velocity vs. discharge shows the weakest power law trend for both rivers and submarine channels (Figure 11). This is expected based on the assumptions inherent in a downstream hydraulic geometry analysis (see Methods section). The combination of C = 0.6% and Cfb = 0.002 gives the best power law regression for the combination of the river and submarine channel data (R2 values are given in Table 4). The power law regressions given in equations (1) to (3) take the forms shown in equations (21)(23) (these are the power law regressions found for the combined dataset of the rivers with the two end member cases of the submarine channels), where the units are meters and seconds.

display math(21)
display math(22)
display math(23)
Figure 9.

Bankfull top width (m) plotted against discharge (m3s-1). Squares show measured data for rivers, circles show calculated values for submarine channels with C = 0.2% and Cfb = 0.005, and triangles show calculated values for submarine channels with C = 0.6% and Cfb = 0.002.

Figure 10.

Bankfull depth to thalweg (m) plotted against discharge (m3s-1). Squares show measured data for rivers, circles show calculated values for submarine channels with C = 0.2%, and Cfb = 0.005 and triangles show calculated values for submarine channels with C = 0.6% and Cfb = 0.002.

Figure 11.

Bankfull mean velocity (m) plotted against discharge (m3s-1). Squares show measured data for rivers, circles show calculated values for submarine channels with C = 0.2% and Cfb = 0.005, and triangles show calculated values for submarine channels with C = 0.6% and Cfb = 0.002.

Table 4. Continuity Analysis of Best Fit Power Law Functions for Hydraulic Geometry Analysis
 Cfb = 0.005 C = 0.2%Cfb = 0.002 C = 0.6%
 CoefficientExponentR2CoefficientExponentR2
B = aQb3.3600.5430.9633.7730.5170.956
H = cQf0.2500.4380.9340.2660.4210.947
U = kQm1.1910.0190.0250.9950.0620.264
a∙c∙k1.000051.0001
b + f + m1.0000.99990

[37] These equations are also shown on Figures 9-11. The coefficients from those regressions very nearly multiply to unity, and the exponents very nearly sum to unity, satisfying the constraints required for continuity (Table 4). The power law exponents reported here are consistent with previous results for rivers, which tend to be near 0.5, 0.4, and 0.1 for width, depth, and velocity, respectively [Table 5.4 in Knighton, 1998; Bridge, 2003].

[38] The densimetric Froude number of the flow Frd is given as equation (24). The issue as to whether formative flows in submarine channels are Froude-supercritical (Frd > 1) or Froude-subcritical (Frd < 1) has been the subject of some discussion [e.g., Pirmez and Imran, 2003], and has relevance in regard to secondary flow in meandering submarine channels [e.g., Abad et al., 2011]. A manipulation of equations (21)-(23), along with the assumption of a concentration of 0.6% and a value of R of 1.65, corresponding to quartz, yields equation (25), where discharge is given in m3s-1.

display math(24)
display math(25)

[39] In Figure 12, computed values of Frd for each submarine channel, along with equation (25), are plotted against Q. The present analysis indicates that formative flows of submarine channels tend to be Froude-subcritical for discharges in excess of about 80,000 m3/s. Also shown for reference are (a) the corresponding values of Froude number Frd = inline image for each river, (b) a line corresponding to the average value of densimetric Froude number for the submarine channels (0.63), and (c) a line corresponding to the average value of the Froude number for large rivers (0.13). The upward bias in the case of submarine channels is apparent from Figure 12.

Figure 12.

Densimetric Froude number (Frd) plotted against calculated bankfull discharge for submarine channels and measured bankfull discharge for rivers (discharge in units of m3s-1). For submarine channels Frd is given by equation (24), while in the case of rivers Frd is the standard Froude number for open channel flow (RC ~ 1). The solid black line shows equation (25), Frd = 0.646Q− 0.164, the thin dotted line shows the average densimetric Froude number of the submarine channels, and the thick dashed line shows the average Froude number of the set of large rivers.

[40] As noted above, the reference depth used in our calculations of submarine channel velocity and discharge is the maximum channel depth rather than the cross-sectionally averaged depth, which would be smaller. This differs from the reference depth used for rivers, which is a cross-sectional average. The hydraulic geometry relationships and densimetric Froude numbers shown in Figures 9-12 reflect this discrepancy. To assess the effect of reducing the reference depth used for submarine channels, we also performed calculations of flow velocity (equation (19)) and discharge (equation (20)) using 50% of the maximum depth. Half of the maximum depth was chosen as it roughly corresponds to the average depth of a triangular channel, and we view this as a lower bound for the cross-sectional average of bankfull depth. The average densimetric Froude number of the submarine dataset decreased from 0.63 to 0.62 as reference depth was reduced by half. A reduction of the reference depth by half also results in approximately 30% reduction of estimated velocity (equation (19)), and approximately 65% reduction in estimated bankfull discharge (equation (20)). Given that these differences in estimated velocities and discharges represent a lower bound for mean channel depth, the bankfull mean velocity and discharge would likely have a value between these lower bound estimates and those described in this paper representing an upper bound. Moreover, the power law regressions calculated using the 50% reduction in depth do not differ significantly from those reported in equations (21)-(23).

[41] Our estimates of flow velocity for the case C = 0.6% and Cfb = 0.002 range from 0.33 m/s to 6.55 m/s. For C = 0.2% and Cfb = 0.005, our velocity estimates range from 0.14 m/s to 3.34 m/s. The minimum velocities come from the Cascadia channel (cross section located 357 km from the head of Willapa Canyon), while the maximum velocities were found for the Amazon channel (near the transition from canyon to channel). The average velocity for all the channels was 1.02 m/s for C = 0.2% and Cfb = 0.005, and 2.21 m/s for C = 0.6% and Cfb = 0.002.

[42] It is useful to compare our estimates for velocities with previous estimates. Pirmez and Imran [2003] use a variety of methods to estimate velocities in the Amazon fan channels ranging from 1.5 to 3 m/s in the upper fan channel and 0.5 to 1.5 m/s in the middle to lower fan [Figure 7 of Pirmez and Imran, 2003]. These values are in close agreement with our velocity estimates for the case C = 0.2% and Cfb = 0.005, which range from 1.34-3.34 m/s in the upper fan channel and 0.45-1.09 m/s in the middle fan channel. For C = 0.6% and Cfb = 0.002 our velocity estimates range from 2.67-6.55 m/s in the upper fan channel and 0.99-2.23 m/s in the middle fan channel. Komar [1969] used the Bagnold equation to find velocities in the Monterey fan channels. Our velocity estimates range from 0.47-1.84 m/s using C = 0.2% and Cfb = 0.005, and 1.12-3.61 m/s using C = 0.6% and Cfb = 0.002. These values are lower than Komar's [1969] estimates for the same channel cross sections, which ranged from 6 m/s to 24 m/s.

[43] A major assumption made in the present analysis is the use of the same sediment concentration and bed friction coefficient for all the channels. This partially accounts for the differences between our estimated velocities and previously reported values. Although our velocities are first order approximations based on very simple calculations, they are the only ones based on an objective estimate of sediment concentration.

[44] As noted above, turbidity currents are subject to an additional source of friction (beyond friction at the channel bed) caused by entrainment of fluid at the interface between the flow and the ambient fluid. A simple shift downward in the unit driving force (as per section 3.2) does not account for this difference in frictional resistance between the rivers and submarine channels. Thus, the unit driving force analysis is expected to underestimate suspended sediment concentration, and this value should therefore be treated as a lower limit. In contrast, the balance of forces in the modified Chezy equation (equation (13)) includes interfacial friction in the calculation of turbidity current velocity. Consequently, the best agreement in hydraulic geometry trends (Figures 9-11) between rivers and submarine channels is given by a somewhat higher concentration (C = 0.6%) than that predicted in section 3.2 (C = 0.2%).

[45] There are numerous simplifications involved in the estimates of turbidity current velocity presented here. The Chezy equation was developed for river flow under the conditions of steady, uniform flow. While this is a good assumption for many river flows, turbidity currents are more event-based, and are subject to complex effects like self-acceleration [e.g., Parker et al., 1986; Sequeiros et al., 2009], density stratification [e.g., Sequeiros et al., 2010], and a diffuse upper interface that may extend well above the levees, even when the body of flow is more or less confined to the channel [e.g., Peakall et al., 2000; Straub et al., 2008, 2011]. Although channel-forming turbidity currents likely occur as events that are more intermittent than those in rivers, we assume the case of relatively sustained turbidity currents to satisfy the assumption of steady, uniform flow. Lamb et al. [2004, 2005] have illustrated experimentally the role of both pulse-like and sustained flows. The occurrence of sustained flows in the field has been offered as an explanation for thick, massive sand deposits [Kneller and Branney, 1995] and their occurrence has been linked to the process of breaching [Mastbergen and Van Den Berg, 2003; Eke et al., 2011] as well as hyperpycnal flows [Mulder et al., 2003; Best et al., 2005]. Additionally, Peakall et al. [2000] cite spatially extensive sediment layers in levee deposits as evidence of sustained deposition by a single turbidity current, and Spinewine et al. [2011] argue for the role of sustained currents in the co-evolution of a channel and its bounding levees over hundreds to thousands of kilometers.

[46] Convincing evidence for frequent overbank flow through flow stripping and overspill has been summarized by Peakall et al. [2000]. Straub and Mohrig [2008] and Straub et al. [2008, 2011] have experimentally modeled the process by which these overbank flows build levees. Here, the term “flow stripping” is used to describe overbank flow on the outer bank of submarine channel bends, while “overspill” is used to describe spatially extensive overbank flow [after Peakall et al., 2000]. Additionally, stratification of density underflows has been well documented in the laboratory. Although flow stratification models differ in their details [e.g., Peakall et al., 2000, Figure 6], there is general agreement that turbidity current concentration decreases gradually with distance above the channel bed from a local maximum located near the bed [e.g., Parker et al., 1987; García, 1994; Straub et al., 2008, 2011]. Continuous or nearly continuous stratification models are characteristic of equilibrium subcritical flows [Peakall et al., 2000; Sequeiros et al., 2010]. Density stratification of turbidity currents has implications for the dynamics of overbank flow. Peakall et al. [2000] use a variety of continuous concentration profiles to show that flow stripping has only a minimal effect on the depth-averaged velocity of well-stratified, subcritical flows. Similarly, they argue that in the case of general overspill at both levees, when the initial flow depth is only slightly greater than the channel depth, the overbank flow will be relatively low in concentration compared to the in-channel fluid.

[47] We therefore argue that the Chezy equation is a good approximation for the depth averaged velocity of in-channel fluid of subcritical, stratified turbidity currents, even if some overbank flow occurs. We furthermore argue that the mechanisms described above are responsible for the self-formation and maintenance of channelized turbidity currents [Imran et al., 1998; Straub et al., 2008, 2011]. In other words, there is a balance between levee deposition by overbank flow and channel aggradation and erosion caused by the turbidity current. This suggests that the formative events for submarine channels are associated with turbidity currents with a main body that is modestly overbank [e.g., Spinewine et al., 2011] but which display a diffuse, low-concentration layer above the levees [Straub et al., 2008, 2011]. These observations support our selection of bankfull discharge for the hydraulic geometry analysis of channelized turbidity currents.

7 Conclusions

[48] The results of this study show that the submarine channels from this study can be up to one order of magnitude wider and deeper than rivers, yet follow a similar power law trend in the relation for width vs. depth. Furthermore, for a similar channel size between submarine channels and rivers, submarine channels tend to be between one and two orders of magnitude steeper than rivers. This increased slope is a consequence of the reduction of gravity due to buoyancy (such that the turbidity current sculpting the channel is only slightly heavier than the ambient water) and the added friction at the interface between the sediment laden flow and the ambient water. Since direct measurements of sediment concentration for turbidity currents are difficult to obtain in the submarine environment, we compared the unit driving force of rivers and submarine channels and obtained a best-fit, depth-averaged volumetric sediment concentration for the submarine case to be near 0.2%. Reasonable agreement was also obtained, however, with a concentration of 0.6%.

[49] To constrain the values of bed friction coefficients for submarine channels, we used large rivers as an analog, giving a range of values for bed friction coefficient Cfb ranging from 0.002 to 0.005. The estimated values of sediment concentration and bed friction were then applied to our modified Chezy formula to predict bankfull velocities and discharges for the submarine channel cross-sections. We used the estimated discharges to perform a hydraulic geometry analysis on the combined dataset of rivers and submarine channels. The results of the hydraulic geometry analysis show the best agreement between rivers and submarine channels for the case of Cfb = 0.002 and C = 0.6%. The additional friction at the flow–ambient fluid interface likely resulted in the need for a higher concentration than originally estimated in the unit driving force analysis. The results of this study suggest that C = 0.6% (a value previously reported by Pirmez and Imran [2003] for the Amazon submarine fan) is a good first order estimate of the volumetric sediment concentration of channel-formative turbidity currents. The present analysis, however, provides only a characteristic value of concentration for submarine channels in general, and does not allow estimation on a channel-by-channel basis.

[50] The estimations of sediment concentration and bed friction coefficient rest on the assumption that an analogy may be drawn between the simplified hydraulics of flow in rivers and in submarine channels. This analogy is based on the similarity in the balance of forces acting on these flows, which we suggest manifests in the relationships between channel width, depth, and slope. Specifically, the shared trend in width vs. depth suggests a continuum in scales between rivers and submarine channels. The broad morphologic differences between river and submarine channels, namely the comparatively large size and steep slopes of submarine channels, reflect the increased driving force needed to overcome buoyancy and interfacial friction in the submarine environment.

[51] The concentrations and velocities estimated here provide first order constraints on these parameters, which may be refined when better field measurements are available. The results of this study should serve as a basis for future studies and comparisons of channelized flows over a variety of scales and environments.

Acknowledgments

[52] The participation of the third author in this research was funded by Shell International Exploration and Production and the National Center for Earth-surface Dynamics, a Science and Technology Center of the US National Science Foundation. The authors would like to acknowledge and thank J. Hernandez for assistance during preliminary data acquisition and analysis. We also thank A. Densmore, A. Ashton, K. Straub, and two anonymous reviewers for their constructive reviews, which greatly improved this manuscript.

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