1 Introduction

1.1 General introduction of bedforms

Sand grains on Earth surfaces are carried from mountains to deep seas by flows, e.g., river, winds, tidal flows, and sediment-laden gravity currents. The transport of these sediments is not only one of the primary processes of the Earth's surface material circulation, including the carbon cycle (Schlünz and Schneider 2000; Galy et al. 2007), but also a cause of severe geohazards such as landslides or debris flows. Such past geohazards can be reconstructed from the characteristics of their deposits (e.g., Mitra et al. 2021), suggesting the disaster risk of each region.

Interaction between flows and solid particles during sediment transport results in various morphologies including bedforms that display wavy or flat topographies. Bedforms have been found ubiquitously on sediment beds; for instance, in open-channel flows (e.g., Collinson 1970; Ma et al. 2017; Galeazzi et al. 2018) and oscillatory flows (e.g., Miller and Komar 1980; Masselink et al. 2007; Wu and Parsons 2019).

In the past decades, numerous laboratory experiments (e.g., Gilbert 1914; Guy et al. 1966; Southard 1991; Dumas et al. 2005; Perillo et al. 2014; Bradley and Venditti 2019) and field observations (e.g., Colby and Hembree 1955; van den Berg 1987; Cisneros et al. 2020) were conducted to obtain empirical relationships between flows and bedforms. Bedform phase diagrams have been used to understand the formation of bedforms in response to flow behavior (e.g., Chabert and Chauvin 1963; Southard and Boguchwal 1990). Recently, Ohata et al. (2017) proposed bedform phase diagrams in three-dimensional parametric space, and their diagrams successfully characterized paleo-flow conditions from sedimentary structures.

1.2 Review of plane beds

Plane beds are bedforms that develop in various environments such as alluvial riverbeds (e.g., Hauer et al. 2019) and beach faces (e.g., Vaucher et al. 2018; Vaucher and Dashtgard 2021). The plane bed phase is characterized by nearly flat topography or low-relief bed waves. Flume experiments have explained that the migration of low-relief bed waves over plane beds results in planar lamination with a few millimeters in thick (Paola et al. 1989, Bridge and Best 1997). Planar lamination in sandstone beds (Fig. 1) is ubiquitously found in fluvial (e.g., Umazano et al. 2012; Mazumder and Van Kranendonk 2013), estuaries (e.g., Hori et al. 2001), tidal flat (e.g., Chakraborty et al. 2003), shoreface (e.g., Kikuchi 2018), and submarine fan environments (e.g., Eggenhuisen et al. 2011; Jobe et al. 2012). For example, deposits from turbidity currents (i.e., turbidites) in submarine environments exhibit succession of sedimentary structures called Bouma sequence, in which parallel lamination is observable in Tb and Td divisions (Bouma 1962).

Fig. 1
figure 1

Photographs showing planar lamination in sandy deposits. A Snapshot of a laboratory experiment conducted at Kyoto University. Flow is from right to left. Two types of sands (D50 = 0.12 mm and 0.53 mm) were used. B Field photograph showing a turbidite sandstone in the Aoshima Formation of the Miocene Miyazaki Group, Japan

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Plane beds were initially referred to as “the smooth phase of bedforms” (Owens 1908; Gilbert 1914). Later, Simons et al. (1961) used the term “plane” to avoid confusion with the hydraulically smooth phase of flows. Owens (1908) and Strahan et al. (1908) observed that the bed state changes from ripples to the smooth phase of bedform (i.e., plane bed) and then to antidunes with increasing flow velocity. In addition, the plane bed phase also appears under the condition when the bed shear stress is lower than that forming ripples or dunes (e.g., Simons et al. 1961; Southard 1971). Thus, previous studies have recognized that there are two conditions of plane bed formation and they are separated by formation conditions of ripples or dunes.

However, the formative process and conditions for phase transition of plane beds remain unclear. It is known that the plane bed is formed under two different hydraulic conditions, but the reason why the formation conditions are divided into two regions is not well understood. In previous studies, the two types of plane beds have been classified depending on the (1) sediment motion (Simons et al. 1961; Simons and Richardson 1962), (2) criticality of Froude number (Venditti 2013; Dey 2014), and (3) size of bed material and flow intensity (Allen 1968; Southard 1971).

Simons et al. (1961) recognized two conditions of plane bed in their flume experiments using the medium sand—plane beds with and without sediment motion. On the basis of the flume experiments with the two types of medium sand (D = 0.28 mm and 0.45 mm), Simons et al. (1961) and Simons and Richardson (1961, 1962) classified alluvial flows based on bedforms into three regimes—a lower flow regime (plane beds without sediment movement, ripples, dunes with ripples superposed, and dunes), a transition zone (washed-out dunes), and an upper flow regime (plane beds with sediment movement, standing waves, antidunes, and chutes-and-pools). Simons et al. (1961) and Simons and Richardson (1962) recognized this sequence of flow regimes with the increase in flow intensity. The definition of Simons et al. (1961) has been employed in the analyses of bedforms (van Rijn 1984b; Julien and Raslan 1998). For example, the bedform stability diagram proposed by Colombini and Stocchino (2008) has described plane bed phases as plane beds without sediment motion and upper plane bed. Bradley and Venditti (2019) cited the phase diagram by Colombini and Stocchino (2008) and described the phase of plane beds without sediment motion as lower plane beds.

Although Simons et al. (1961) noted that flows change from the lower to upper flow regimes at a Froude number (Fr) less than unity, Venditti (2013) and Dey (2014) stated that lower and upper flow regimes are consistent with Froude-subcritical (Fr < 1) and Froude-supercritical flow regimes (Fr > 1), respectively. In other words, following the definition by Venditti (2013) and Dey (2014), plane bed in the lower regime is generally interpreted to exist at Fr < 1, and plane bed in the upper regime is at or above Fr = 1.

Allen (1968) and Southard (1971) described bedform phase diagrams based on a compilation of observational data (Fig. 2). In their phase diagrams, there is a stable plane bed region where coarse sediment (median diameter \({D}_{50}\) > 0.6–0.7 mm) is transported at low flow velocities (< 0.5 m/s). Allen (1968) termed this bed state as a lower-phase plane bed. That is, there are two types of plane beds with sediment movement, namely lower plane beds with low flow intensities on coarse sediment and upper plane beds with high flow intensities on fine sediment. Liu (1957) and Bogárdi (1961) also reported a region of flow conditions in which sediment motion occurred but ripples or other bedforms did not appear.

Fig. 2
figure 2

Bedform phase diagram for unidirectional currents digitized after Southard and Boguchwal (1990) for a flow depth of 0.25–0.40 m. The mean velocity U10, flow depth H10, and grain size D10 denote 10 °C equivalent values

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In summary, researchers have proposed various definitions and formation conditions of the two types of plane beds: (1) sediment motion (Simons et al. 1961; Simons and Richardson 1962), (2) criticality of Froude number (Venditti 2013; Dey 2014), and (3) size of bed material and flow intensity (Allen 1968; Southard 1971).

In this paper, we compiled a total of 935 sets of existing data from the literature. The dataset indicates that two separate regimes of plane bed exist and the boundary between the two plane bed regimes matches the threshold condition for the particle entrainment into suspension (Niño et al. 2003) or sheet flow (traction carpet; Gao 2008). We compiled data pertaining to plane beds in unidirectional open-channel flows and plotted them using dimensionless parameters as axes without any a priori assumption of types of plane beds. Based on this data analysis, we recognized that the upper plane beds are always accompanied by suspension or sheet flow.

2 Methods

2.1 Data sources

We compiled from the literature a total of 935 sets of data pertaining to plane beds to investigate the plane bed regimes. The dataset consisted of 890 sets of laboratory data and 45 sets of field data (Tables 1 and 2). A wide range of hydraulic conditions and sediment calibers are encompassed in the datasets. The median sediment diameter (\({D}_{50}\)) ranges from 1.1 × 10–2 to 44.3 mm, the flow depth (\(h\)) ranges from 1.2 × 10–3 to 2.74 m, and the flow velocity (\(U\)) ranges from 0.058 to 2.38 m/s. To compare the transport mode of plane bed formation with the threshold conditions of sediment motion and the initiation of suspension, we classified the data with sediment movement into three types based on the suspended sediment concentration (\({C}_{\mathrm{s}}\)), as follows: (a) The suspended sediment concentration was measured (hereafter referred to as data \({C}_{\mathrm{s}}\) > 0), (b) the suspended sediment concentration was recorded as zero (hereafter referred to as data \({C}_{\mathrm{s}}\) = 0), and (c) the suspended sediment concentration was not available (hereafter referred to as data no \({C}_{\mathrm{s}}\)).

Table 1 Summary of flume data used for the analysis
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Table 2 Summary of field data used for the analysis
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2.2 Dimensionless parameters for morphodynamic conditions

First, we focused on the grain size and the sediment transport mechanism (i.e., no sediment movement, bed load, and suspended load) as controls on plane bed regimes. Bedform phases were expressed in a space of dimensionless parameters that reflected the properties of flows and sediment particles (e.g., van den Berg and van Gelder 1993; Ohata et al. 2017). We employed the following dimensionless parameters to represent hydraulic conditions and sediment properties: the particle Reynolds number (\({\mathrm{Re}}_{\mathrm{p}}\)), Shields number (\({\tau }_{*}\)), and the suspension index (\({u}_{*}/{w}_{\mathrm{s}}\)).

The particle Reynolds number (\({\text{Re}}_{{\text{p}}}\)) is defined as (Garcia 2000):

$$\begin{array}{*{20}c} {{\text{Re}}_{{\text{p}}} = \frac{{\sqrt {RgD_{50} } D_{50} }}{\nu }} \\ \end{array}$$
(1)

where \(R\) represents the submerged specific density of the sediment, \(g\) denotes the gravitational acceleration, and \(\nu\) denotes the kinematic viscosity of the fluid. The submerged specific density is defined as \(R=({\rho }_{\mathrm{s}}-{\rho }_{\mathrm{f}})/{\rho }_{\mathrm{f}}\), where \({\rho }_{\mathrm{s}}\) and \({\rho }_{\mathrm{f}}\) represent the sediment and fluid densities, respectively. The kinematic viscosity (\(\nu\)) was assumed to be a function of temperature according to the relationship for water (van den Berg and van Gelder 1993):

$$\begin{array}{*{20}c} {\nu = \left[ {1.14 - 0.031\left( {T - 15} \right) + 0.00068\left( {T - 15} \right)^{2} } \right]10^{ - 6} } \\ \end{array}$$
(2)

where \(T\) represents the water temperature in degree Celsius. Following van den Berg and van Gelder (1993), we assumed a value of 20 °C for data where \(T\) was not reported. Shields number (\(\tau_{*}\)) is defined as

$$\begin{array}{*{20}c} {\tau_{* } = \frac{{u_{*}^{2} }}{{RgD_{50} }}} \\ \end{array}$$
(3)

Here, the shear velocity (\({u}_{*}\)) for the field data was computed as \({u}_{*}=\sqrt{ghS}\), where \(S\) represents the bed slope. For laboratory data, we removed the sidewall effect and calculated the bed component of the shear velocity using the method proposed by Chiew and Parker (1994) (see Ohata et al. (2017) for details). We focused on the analysis of plane bed data in this study; therefore, we assumed the flow resistance induced by the bedforms is negligible.

The suspension index is expressed as the ratio of the shear velocity (\({u}_{*}\)) to the settling velocity of sediment (\({w}_{\mathrm{s}}\)). The settling velocity was estimated using the relationship formulated by Ferguson and Church (2004):

$$\begin{array}{*{20}c} {w_{{\text{s}}} = \frac{{RgD_{50}^{2} }}{{C_{1} \nu + \sqrt {0.75C_{2} RgD_{50}^{3} } }}} \\ \end{array}$$
(4)

where constants \({C}_{1}\) and \({C}_{2}\) were set to 18 and 1, respectively, which were the values for natural sands (Ferguson and Church 2004).

The sediment transport regimes are classified based on the transport mechanism—no sediment movement, bed-load-dominated, mixed-load, and suspended-load-dominated regimes (Church 2006). The boundary between no sediment movement and other regimes is defined by the Shields curve, which is the critical condition for the initiation of particle motion (e.g., Shields 1936). Based on Shields’ experimental data, Brownlie (1981) proposed a function describing the threshold condition of particle motion as:

$$\begin{array}{*{20}c} {\tau_{{*{\text{c}}}} = 0.22{\text{Re}}_{{\text{p}}}^{ - 0.6} + 0.06\exp \left( { - 17.77{\text{Re}}_{{\text{p}}}^{ - 0.6} } \right)} \\ \end{array}$$
(5)

Next, the threshold condition for the initiation of suspension is expressed using the suspension index (\({u}_{*}/{w}_{\mathrm{s}}\)) (Bagnold 1966; van Rijn 1984a; Niño et al. 2003). The threshold condition of the particle entrainment into suspension represents the boundary between the bed-load-dominated regime and a regime where the bed materials are transported with suspension. The threshold condition for the particle entrainment into suspension, obtained by Niño et al. (2003), is:

$$\begin{array}{*{20}c} {\left( {\frac{{u_{*} }}{{w_{{\text{s}}} }}} \right)_{{\text{c}}} = \left\{ {\begin{array}{*{20}l} {21.2{\text{Re}}_{{\text{p}}}^{ - 1.2} ,} \hfill & {1 < {\text{Re}}_{{\text{p}}} < 27.3} \hfill \\ {0.4,} \hfill & {27.3 \le {\text{Re}}_{{\text{p}}} } \hfill \\ \end{array} } \right.} \\ \end{array}$$
(6)

We used Eq. (5) (dashed line) and 6 (dotted line) to define the boundaries of the sediment transport regimes (Figs. 3, 4). The region on phase diagrams below the dashed line denotes the no sediment movement regime, the region between dashed and dotted lines denotes a bed-load-dominated regime, and the region above the dotted line denotes a mixed-load or suspended-load-dominated regime where bed materials are moved in bed load and suspension. To describe the threshold condition of sediment motion (Eq. 5) on the \({\text{Re}}_{{\text{p}}} {-}u_{*} /w_{{\text{s}}}\) diagram (Fig. 4) and the initiation of suspension (Eq. 6) on the \({\text{Re}}_{{\text{p}}} {-}\tau_{*}\) diagram (Fig. 3), Eqs. (5 and 6) are rearranged using Eqs. (1, 3, and 4), where \(R\) = 1.65 (the value for quartz) and ν = 1.0 × 10–6 (the value for water with 20 °C).

Fig. 3
figure 3

Plane bed regime in a space of particle Reynolds number \({\mathrm{Re}}_{\mathrm{p}}\) versus Shields number \({\tau }_{*}\). The dotted line denotes the threshold condition for particle motion (Eq. 5). The solid line denotes the threshold condition for sediment suspension (Eq. 6). Both lines were extended to \({\mathrm{Re}}_{\mathrm{p}}\) = 10–1 and 105. The plane bed data are divided into two separate regions: the region around the threshold condition of particle motion and the region above the threshold condition of suspension

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Fig. 4
figure 4

Plane bed regime in a space of particle Reynolds number \({\mathrm{Re}}_{\mathrm{p}}\) versus suspension index \({u}_{*}/{w}_{\mathrm{s}}\). The dotted line denotes the threshold condition for particle motion (Eq. 5). The solid line denotes the threshold condition for sediment suspension (Eq. 6). Both lines were extended to \({\mathrm{Re}}_{\mathrm{p}}\) = 10–1 and 105. The plane bed data are divided into two separate regions: the region around the threshold condition of particle motion and the region above the threshold condition of suspension

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Second, the plane bed conditions were investigated using Froude number. Phase diagrams were generated using \({\text{Re}}_{{\text{p}}}\), \(u_{*} /w_{{\text{s}}}\), and Froude number as the axes. Froude number is the ratio of the inertial force to the gravitational force, defined as:

$$\begin{array}{*{20}c} {\mathrm{Fr} = \frac{U}{{\sqrt {gh} }} } \\ \end{array}$$
(7)

3 Results

3.1 Influence of grain size and sediment transport mode

First, the plane bed data were plotted in \({\text{Re}}_{{\text{p}}} {-}\tau_{*}\) space (Fig. 3) to investigate the relationships among the plane bed regimes, the particle diameter, and the criteria for sediment movement. The threshold conditions of particle motion (Eq. 5) and suspension (Eq. 6) are shown in Fig. 3 as the dashed and dotted lines, respectively. The median diameter \({D}_{50}\) is recast from \({\mathrm{Re}}_{\mathrm{p}}\) with \(R\) = 1.65 and ν = 1.0 × 10–6, which is represented on the top axis of Fig. 3.

Shields numbers associated with plane bed conditions range from 0.05 to 10, and the plane bed data are divided into two separate regions at \({\tau }_{*}\) = 0.1–0.2 (Fig. 3). The plane bed data with low \({\tau }_{*}\) plot around the threshold condition of particle motion, and the region with high \({\tau }_{*}\) plots above the threshold condition of suspension. The threshold condition of particle motion does not divide the plane bed regime; rather, the threshold condition of suspension runs between the two regions of data. The particle Reynolds numbers of the data with low \({\tau }_{*}\) range 9 < \({\mathrm{Re}}_{\mathrm{p}}\)  < 4.0 × 104 (0.2 mm < \({D}_{50}\)  < 40 mm). In the case of the data with high \({\tau }_{*}\), the particle Reynolds number ranges from 0.1 to 300 (0.01 mm < \({D}_{50}\)  < 2 mm). The data \({C}_{\mathrm{s}}\) > 0 and field observation data are included in the region of high \({\tau }_{*}\). The data \({C}_{\mathrm{s}}\) = 0 plot around the threshold condition for particle motion and \({\tau }_{*}\) = 0.4.

Second, the relationships among the plane bed regimes, the sediment diameter, and the criteria for suspension were examined by plotting \({u}_{*}/{w}_{\mathrm{s}}\) versus \({\mathrm{Re}}_{\mathrm{p}}\) (Fig. 4). Median diameter \({D}_{50}\) is shown on the top axis of Fig. 4.

In \({\mathrm{Re}}_{\mathrm{p}}\)–\({u}_{*}/{w}_{\mathrm{s}}\) space, the plane bed data plot in two regions separated by the threshold condition of suspension (Fig. 4), similar to our results in \({\mathrm{Re}}_{\mathrm{p}}\)–\({\tau }_{*}\) space (Fig. 3). The threshold condition of particle motion runs through the middle of the data with a low suspension index. With respect to the particle Reynolds number, the data with a high suspension index plot in the region 0.1 < \({\mathrm{Re}}_{\mathrm{p}}\)  < 3.2 × 103 (0.01 mm < \({D}_{50}\)  < 2 mm) and the data with low suspension index plot in the region 9 < \({\mathrm{Re}}_{\mathrm{p}}\)  < 4.0 × 104 (0.2 mm < \({D}_{50}\)  < 40 mm). The data \({C}_{\mathrm{s}}\) > 0 and field data plot above the threshold condition of suspension. The data \({C}_{\mathrm{s}}\) = 0 plot below and above the suspension criteria.

3.2 Effect of Froude number

We investigated the consistency of the flow regime concept based on Froude number (Venditti 2013; Dey 2014) to plane bed regimes, plotting data in \({\mathrm{Re}}_{\mathrm{p}}\)–\(\mathrm{Fr}\) space (Fig. 5). The data include Froude numbers ranging from 0.1 to 3.5. The data are not separated with regard to Froude number values. The field data have lower Froude numbers (\(\mathrm{Fr}\) < 0.5) and higher Shields numbers and exceed the threshold condition of suspension (Figs. 3 and 4). For \({C}_{\mathrm{s}}\) > 0, the data fall in the domain 0.1 < \(\mathrm{Fr}\)  < 1.3 and the data \({C}_{\mathrm{s}}\) = 0 distribute in the domain 0.2 < \(\mathrm{Fr}\)  < 3.5.

Fig. 5
figure 5

Plane bed regime in a space of particle Reynolds number \({\mathrm{Re}}_{\mathrm{p}}\) versus Froude number \(\mathrm{Fr}\). The data points of plane beds are not discriminated by Froude number

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4 Discussion

4.1 A new definition for plane beds

Data from the literature including field observations and flume experiments for plane beds were used to create phase diagrams for plane beds (Figs. 3, 4, 5). Traditionally, the particle diameters or the critical values of sediment motion or Froude number were used to discriminate lower and upper plane beds. However, our phase diagrams indicate that this is unlikely (Figs. 3, 4, 5). Indeed, the phase regions of the plane beds are not separated by the traditional indicators such as particle diameters. In contrast, observational data of the plane beds plot in two separate regions in the \({\mathrm{Re}}_{\mathrm{p}}\)–\({\tau }_{*}\) and \({\mathrm{Re}}_{\mathrm{p}}\)–\({u}_{*}/{w}_{\mathrm{s}}\) diagrams (Figs. 3, 4), and the boundary between the two regions corresponds to the threshold condition for sediment suspension. Therefore, we suggest that the upper and lower plane bed phases separated by the ripple and dune phases (Fig. 2) should be redefined—upper plane beds are plane beds in which the lower boundary is the threshold condition for sediment suspension, and lower plane beds are plane beds dominated by bed load. That is, there is a necessity to modify the classic bedform phase diagram.

Simons et al. (1961) defined lower plane beds as plane beds without sediment motion. For the data with a low Shields number (\({\tau }_{*}\) < 0.1–0.2), there are overlaps of data points pertaining to plane beds with sediment movement and without sediment movement (Fig. 3). There is a range of shear stresses in which bed particles could move because, for example, the sediment bed is composed of natural sands that are somewhat angular and of various sizes. Hence, some data points of plane beds with sediment motion plotted below the line are defined by the threshold of sediment motion. Also, plane beds with sediment motion have been observed on a coarse-grained bed (Guy et al. 1966; Southard and Boguchwal 1973; Costello and Southard 1981; Cao 1985). These results of flume experiments demonstrate that the classical definition of plane beds by Simons et al. (1961) is not appropriate.

Similarly, Froude number cannot be used to discriminate the two types of plane beds (Fig. 5). Venditti (2013) and Dey (2014) stated that the upper and lower flow regimes are defined by Froude number. However, both plane beds with and without suspension can develop under various Froude numbers (Fig. 4). For example, plane beds appear with suspended loads in natural rivers where \({\tau }_{*}\) is high and Fr is less than unity due to the great depth (e.g., Ma et al. 2017). Therefore, we propose that lower and upper plane beds are not classified by the criticality of Froude number.

It is noteworthy that our analyses indicated that the regime of the lower plane bed extends to the region of fine-grained sediment (Figs. 3, 4). Southard (1971) defined lower plane beds as plane beds just above the threshold of sediment motion under subcritical flow with relatively coarse grains (\({D}_{50}\) > 0.7 mm). Later, Southard and Boguchwal (1990) proposed bedform phase diagrams using a compilation of large datasets, although they omitted the data of relatively fine-grained plane beds (10 °C-equivalent sediment diameter is smaller than 0.7 mm) under low-velocity flows in their phase diagrams. Southard and Boguchwal (1990) did not include such data in their diagram because they suspected that equilibrium had not been attained. However, the duration of experiments performed by Guy et al. (1966) and Taylor (1971), which observed low-velocity plane beds in fine sediments, was from several hours to more than 24 h. The sediment discharge rates for low-velocity plane beds in the experiments of Guy et al. (1966) and Taylor (1971) are comparable to those of ripples in their experiments. Further, the data points of low-flow velocity plane beds in fine sediment are continuous from those of low-flow velocity plane beds in coarse sediment (Figs. 3, 4). Therefore, the stable field of the plane bed without suspended load can be identified from fine sand to gravel.

The origin of the lower plane bed phase in the new definition is puzzling. In previous studies, one of the controlling factors in the lower and upper plane bed regimes was assumed to be the grain size and flow intensity (Allen 1968; Southard 1971). Leeder (1980) interpreted that coarse-grained plane beds are formed because flow separation is prevented by the bed roughness. Recently, Blois et al. (2014) proposed that bed permeability may be another explanation for the formation of coarse-grained plane beds. However, this study established that the region of coarse-grained plane beds extends continuously to the fine-grained region, and the conditions for fine-grained plane beds were distributed separately in the two regions: the high- and low-bed shear stresses. Thus, the formation of the lower plane bed cannot be attributed to bed roughness or permeability owing to the grain size distribution.

It should be noted here that bed particles are also transported in the sheet flow (traction carpet) regime (Sumer et al. 1996), which can be related to the plane bed. Sheet flows consist of a shear layer of bed load that moves under high shear stress (\({\tau }_{*}\) > ~ 0.5) where ripples and dunes are washed out (Sumer et al 1996; Pugh and Wilson 1999). Williams (1970) observed that plane bed developed within sheet flows, and the data from Williams (1970) plotted above the threshold condition of suspension, yet no active suspension was observed in the experiments of Williams (1970) (Figs. 3, 4). Recently, Hernandez-Moreira et al. (2020) suggested that plane beds can develop under sheet flows in the intermediate condition between upstream and downstream migrating antidunes. However, experiments of sheet flow in open channels are scarce because laboratory experiments on sheet flows have been conducted using closed conduits in order to achieve high shear stresses (Nnadi and Wilson 1992; Sumer et al. 1996) or using oscillatory flow tunnels to simulate storm conditions (Dick and Sleath 1992; O'Donoghue and Wright 2004). Currently, it is difficult to determine whether sheet flow is another possible mechanism that generates plane beds. Further laboratory experiments, numerical experiments, and theoretical considerations are required to provide explanations for the plane bed regimes.

4.2 Theoretical explanation of plane bed formation

The transition from dunes to plane beds has been theoretically explained by the spatial lag between the dune crest and the location of maximum sediment transport rate (Kennedy 1963). Dune height decreases when the maximum sediment transport rate occurs at the downstream of the dune crest. Recently, the spatial lag has been quantitatively observed under suspended-load-dominated flows (Naqshband et al. 2017), whereas there was no spatial lag in mixed-load flows (Naqshband et al. 2014). Therefore, the flume experiments by Naqshband et al. (2014, 2017) indicated that the increase in suspended load flux could lead to the spatial lag between the dune crest and the location of the maximum sediment transport rate. Figures 3 and 4 show the data of plane beds with suspension plotted at a region separated from the data of plane beds without suspension. That is, the compilation of the dataset supports that the spatial lag is caused by the existence of the suspended load.

Further, the influence of suspended load on the formation condition of bedforms has been demonstrated in the linear stability analyses by Nakasato and Izumi (2008). Nakasato and Izumi (2008) showed that the maximum Froude number for dune formation decreased by including the effect of suspended load; that is, the dune formation is partly suppressed. Although the results of linear analyses by Nakasato and Izumi (2008) were verified using the experimental data of dune and antidune of Guy et al. (1966), the hydraulic conditions of analyses were limited, and they did not make a comparison with the data of plane beds. In future theoretical research, it is required to analyze various conditions (e.g., sediment diameter and flow depth) and compare the results with the plane bed data.

4.3 Implication for rock record

Planar parallel lamination is often observed in ancient sedimentary successions and utilized to interpret the depositional environments (e.g., Clifton 1976; Plink-Björklund 2005). For example, planar-laminated sandstones from shallow marine conditions are interpreted to be formed in nearshore environments where wave collapse generates back-and-forth currents on the sea floor (Clifton 1976; Dumas and Arnott 2005; Vaucher et al. 2018).

In deep marine environments, turbidites may form planar laminations as the Tb and Td divisions (Bouma 1962), although both may be absent (e.g., Sumner et al. 2012). The origin of the Tb division is attributed to plane bed deposition, whereas that of the Td division is enigmatic (Middleton 1993; Talling et al. 2012). Following the definitions of Southard and Boguchwal (1990), Talling et al. (2012) argued that the origin of Td division in turbidites is not the lower plane beds because the lower plane beds occur only on the coarse-grained beds. However, our phase diagrams imply that the origin of planar laminae in sandstone cannot be presumed depending only on its particle size. Although Southard (1971) and Southard and Boguchwal (1990) suggested that lower plane beds cannot be formed with sediment finer than 0.7 mm in diameter, this study shows that fine-grained plane beds can develop at low-flow velocity (Figs. 3 and 4). Also, Hesse and Chough (1980) pointed out the possibility of the existence of fine-grained plane beds at low shear stress based on the flume experiments using silt-sized sediment by Rees (1966). Fine-grained plane beds can be stable at low shear stress because the critical shear stress to maintain the sediment movement under supersaturated flows where the sediment is already suspended is different from the critical shear stress to erode the sediment from the bed (Rees 1966). Also, Td division is composed of silt- and clay-sized particles (Bouma 1962), and the cohesion is one of the possible mechanisms for the formation of plane beds in open-channel flows (Schindler 2015). Therefore, gaps in the existing data to explain the origin of Td division should be filled by flume experiments of sediment-laden gravity currents and open-channel flows under controlled conditions.

5 Conclusions

In this study, we compiled 935 flume and field datasets pertaining to plane beds and analyzed the dataset in the nondimensional parametric space. The results of our analysis indicate that the formation conditions of plane beds do not show a clear boundary at the threshold condition of sediment motion, a unique value of particle diameter, or a Froude number of unity, which have been traditionally used to define lower and upper plane beds. Conversely, the formation conditions of plane beds are distributed in two separate regions, a lower plane bed dominated by bed load and an upper plane bed in which the lower boundary is the threshold condition for sediment suspension. This study demonstrates that suspended load significantly contributes to the formation of plane beds. Further experiments are needed to understand the mechanisms that can produce fine-grained plane beds in flows with low-bed shear stress.