Reconsidering the seawater-density parameter in hydrodynamic flow transport equations for coastal boulders

ABSTRACT

Existing hydrodynamic flow transport equations for coastal boulder transport are useful for estimating post-event the characteristics of extreme storm waves and tsunamis. However, the effect of suspended sediment concentration (SSC) on seawater density is normally ignored. This is unrealistic given that turbulent runup flows easily entrain available fine sediment. Proper consideration of SSC can be encouraged by including a mixed-fluid density coefficient (C_ρ) as a multiplier for clear-seawater density, where elevated sediment content can be assumed and estimated. Minimum flow velocities required for boulder transport are shown to reduce as sediment concentrations increase.

KEYWORDS:

• Coastal boulders
• flow transport equations
• sediment concentration
• seawater density
• tsunamis
• storm waves

Introduction

Coastal boulder proxies for extreme wave events

Geological evidence is increasingly being utilised by coastal scientists to identify and characterise the nature of palaeo-high energy wave events on coasts such as tsunamis and storms. Such evidence enables comparisons in magnitude between events and, through cautious interpretation of age-datable material, provides insight into the temporal frequencies of the strongest wave events experienced on exposed coastlines over timescales much longer than written records permit (Yu et al. 2009; Terry et al. 2015, 2016; Lau et al. 2016). Coastal boulder deposits represent one kind of geological evidence that may be investigated (Figure 1; Hearty 1997; Etienne 2012; Yu et al. 2012). First described as a potential proxy for big storms in the early twentieth century (Hedley and Taylor 1907), boulder analysis is now a well-established method in coastal geomorphological research, and has been applied across a range of tropical and extra-tropical regimes (Etienne and Paris 2010; Goto et al. 2010; Engel and May 2012; Salzmann and Green 2012; Lau et al. 2015; Kennedy et al. 2017). Measuring the size, position and orientation of transported coastal boulders reveals clues concerning the power and direction of the onshore flows that deposited them, as generated by extreme waves striking the coast (Scicchitano et al. 2007; Goto et al. 2009; Terry et al. 2013; May et al. 2015; Shah-Hosseini et al. 2016).

Figure 1. Sampling and measurement of coastal boulder fields: reef-derived carbonate clasts near Chaweng, Ko Samui island, southern Thailand (above), and large clasts of volcanic breccia and limestone on Ludao island, SE Taiwan (below).

Hydrodynamic flow transport equations

Hydrodynamic flow transport (HFT) equations quantifying relationships between coastal boulder size and minimum wave height required for transport were originally developed by Nott (1997, 2003). Nott’s HFT equations were rearranged by Nandasena et al. (2011) to enable calculation of the minimum flow velocity (MFV) necessary to initiate boulder movement, according to various possible modes of transport: sliding, rolling or lifting. In the HFT equations below (Nandasena et al. 2011), u is the calculated MFV in m/s and other parameters are as listed in Table 1.

Table 1. Parameters in the widely-used hydrodynamic flow transport equations for transport initiation of coastal boulders.

Equation elements	Corresponding parameter	Units
	Variables
ρs	Sediment density	g/cm^3 (t/m^3)
a	Boulder a-axis (long axis)	m
b	Boulder b-axis (intermediate axis)	m
c	Boulder c-axis (short axis)	m
θ	Bed slope angle	°
ρw	Seawater density	g/cm^3 (t/m^3)
	Coefficients
Cd	Coefficient of drag	—
Cl	Coefficient of lift	—
µs	Coefficient of static friction	—
	Constants
g	Gravitational acceleration	m/s^2

(A) For boulder sliding:(1) $u^2\ge \frac{2\left({\rho }_s/{\rho }_w-1\right)gc\left({\mu }_s\cos \theta +\sin \theta \right)}{C_d\left(c/b\right)+{\mu }_s{C}_l}$(1) (B) For boulder rolling:(2) $u^2\ge \frac{2\left({\rho }_s/{\rho }_w-1\right)gc\left(\cos \theta +\left(c/b\right)\sin \theta \right)}{C_d\left(c^2/b^2\right)+C_l}$(2) (C) For boulder lifting:(3) $u^2\ge \frac{2\left({\rho }_s/{\rho }_w-1\right)gc\cos \theta }{C_l}$(3)

The seawater density problem

Modelling fluid mechanics at both small and large scales necessarily requires approximation of the processes involved. Models depicting the objects of study are therefore simplifications, in part to avoid superfluous detail and to reduce expensive computational effort in their application (Munson et al. 2013; Çengel and Cimbala 2015). Broadly speaking, the current HFT equations represent a one-dimensional depiction of boulder transport, which is appropriate for pragmatic approximations.

Regarding the application of HFT equations, however, a persistent issue can be identified with the seawater density parameter (ρ_w). Although ρ_w is a variable, most authors treat it as a de facto constant by simply substituting a density value of 1.025 g/cm³ or similar. This value corresponds to the density of clear seawater and ignores any contribution of suspended sediments in the turbulent flow. Yet, sediment and seawater mixtures are more viscous fluids than clear seawater, potentially altering the fluid dynamics between the flow and boulders interacting with the flow stream.

González et al. (2007) emphasised that sediment content must be taken into account when computing tsunami hydrodynamic forces. For modelling tsunami impacts on coastal structures, Jiffry et al. (2015) and Attary et al. (2017) are among the few studies using fluid density values greater than for clear seawater (1200 kg/m³ and 1080–1320 kg/m³, respectively). Nonetheless, the effect of elevated sediment concentrations on hydrodynamics is recognised as a continuing source of epistemic uncertainty in tsunami sediment transport modelling, and sediment-enhanced fluid density is not yet addressed in models for coastal boulder transport (Jaffe et al. 2016). Kain et al. (2012) believe that water alone is incapable of moving large clasts, and that movement is only possible by debris-rich flows. They also advocated including the effects of increased fluid density from fine sediment in the water.

We agree with these views. In response, our aim here is to draw closer attention to the contribution of suspended sediment on fluid density. One way to avoid suspended sediment being overlooked or ignored in the application of existing HFT equations for coastal boulder transport is to introduce a simple coefficient that adjusts the clearwater fluid density according to the suspended sediment concentration (SSC), where this is known, assumed or can be estimated.

Anticipating elevated SSC in tsunamis and breaking storm waves

Although direct empirical evidence is limited, it is intuitive that sediment fractions will be non-zero values in high-energy waves and fast-moving runup (and backwash) on affected coastlines. Beaches, reefs and deltas provide sources of unlithified sediment for erosion and entrainment near the shoreline. The most intense sediment transport in the coastal zone is often beneath breaking waves in the surf on a beach. Neglecting wave breaking processes might therefore lead to an underestimate of SSC in the turbulent surf zone (Soulsby 1997). Accounts of black or brown tsunami waves (Houtz 1962; González et al. 2007) and deposition of sand and mud layers inland (Etienne and Terry 2012; Yamada et al. 2014) likewise indicate elevated SSC. From investigation of the 2018 Indonesian (Sulawesi) tsunami, Sassa and Takagawa (2019) suggest that significant earthquake-generated coastal sediment liquefaction can result in a liquefied gravity flow-induced tsunami. Yoshii et al. (2018) similarly conceive that earthquake shock can liquefy sediment prior to tsunami arrival. For localised tsunamis caused by subaerial or submarine landslides (Tappin et al. 2008; Goff and Terry 2016; Lau et al. 2018), it is plausible that the landslide mass would contribute to elevated suspended sediment concentrations.

Methods

Incorporating suspended sediments

Consider the three situations illustrated in Figure 2. Figure 2a shows an evenly distributed flow stream where the sediments are equally mixed. Such a scenario is possible in highly turbulent conditions. Figure 2b represents another situation where a depth-profile exists. Sediment-induced density stratification occurs where large amounts of sediment are suspended in the water column (Apotsos et al. 2011), although stratification is largely absent in published tsunami sand transport models (Jaffe et al. 2016). If the entrained sediments are a mix of fines and coarse material, transported in suspension and in traction, then the flow will potentially have this type of gradient. Figure 2c represents a heterogeneous mixture with an instantaneous profile showing high levels of concentration in the upper layer. Such situations will temporarily create an uneven momentum profile. The purpose of this exercise is to demonstrate that there are various possible within-flow sediment distributions during flow contact with any coastal boulders near the shoreline.

Figure 2. Three simplified variations of an incoming flow stream with different distributions of sediments, as represented by black dots. Side views are shown. (a) A homogenously mixed flow stream where the sediment content is evenly distributed. (b) A depth-gradient where gravitational effects give higher sediment concentrations in the lower layers of the flow than the uppermost layers. (c) An unevenly mixed flow stream with a densely concentrated area at the top left ‘corner.’

Figure 2b and 2c are two-dimensional problems that would introduce extra complexity if we were to model them. An alternative, however, is to continue using the current HFT equations in their one-dimensional form, but to multiply clear seawater density (ρ_w) by a ‘mixed-fluid density coefficient’ (C_ρ), where appropriate. This avoids a de facto assumption of a zero-sediment concentration. A mixed-fluid density value (ρ_m) can be calculated as follows, based on volume ratios of sediment and seawater:(4) ${\rho }_m=\left(f_w\times {\rho }_w\right)+\left(f_s\times {\rho }_s\right)g/c{m}^3$(4) where, ρ_m is the average density of the seawater and sediment mix, f_w is the seawater fraction by volume between 0 and 1, f_s is the sediment fraction by volume between 0 and 1, ρ_w is the density of clear seawater (1.025 g/cm³), ρ_s is the sediment density, and f_w + f_s = 1

The mixed-fluid density coefficient (C_ρ) is then calculated:(5) $C_{\rho }={\rho }_m/{\rho }_w$(5)

The C_ρ coefficient can now be incorporated into Equations (1)–(3) as a multiplier for clear seawater density (ρ_w). The revised HFT equations for boulder transport are written thus:

(A) For boulder sliding:(6) $u^2\ge \frac{2\left({\rho }_s/\left(C\rho .{\rho }_w\right)-1\right)gc\left({\mu }_s\cos \theta +\sin \theta \right)}{C_d\left(c/b\right)+{\mu }_s{C}_l}$(6)

(B) For boulder rolling:(7) $u^2\ge \frac{2\left({\rho }_s/\left(C\rho .{\rho }_w\right)-1\right)gc\left(\cos \theta +\left(c/b\right)\sin \theta \right)}{C_d\left(c^2/b^2\right)+C_l}$(7)

(C) For boulder lifting:(8) $u^2\ge \frac{2\left({\rho }_s/\left(C\rho .{\rho }_w\right)-1\right)gc\cos \theta }{C_l}$(8)

Results and Discussion

Effects of increasing sediment concentrations on fluid density

Table 2 shows the increase in mixed-fluid density (ρ_m) with increasing seawater sediment concentrations. The density coefficient (C_ρ) increases correspondingly (Figure 3). Appropriate ranges of C_ρ values can be selected from Figure 3, according to the sediment type, density (ρ_s) and estimations of SSC (see below).

Figure 3. Values of the mixed-fluid density coefficient (C_ρ) for a flow stream of seawater containing suspended sediment, according to sediment concentrations and sediment (grain) density, based on equation (5).

Table 2. Relationships between the concentration of example types of sediments in seawater and the resulting density of the seawater and sediment mix (ρ_m), based on equation (4).

Seawater sediment concentration by volume (%)	Sediment fraction fs	Seawater fraction fw	Mixed fluid density for seawater containing carbonate sediments^a ρm (g/cm^3)	Mixed fluid density for seawater containing basalt sediments^b ρm (g/cm^3)
0	0	1	1.025^c	1.025^c
1	0.01	0.99	1.031	1.043
2	0.02	0.98	1.037	1.061
3	0.03	0.97	1.042	1.078
4	0.04	0.96	1.048	1.096
5	0.05	0.95	1.054	1.114
6	0.06	0.94	1.060	1.132
7	0.07	0.93	1.065	1.149
8	0.08	0.92	1.071	1.167
10	0.10	0.90	1.083	1.203
12	0.12	0.88	1.094	1.238
15	0.15	0.85	1.111	1.291
20	0.20	0.80	1.140	1.380
25	0.25	0.75	1.169	1.469
30	0.30	0.70	1.198	1.558

^asediment density 1.6 g/cm³.

^bsediment density 2.8 g/cm³.

^cequivalent to density of clear seawater, p_w.

It is seen that incorporating increasing C_ρ values into the HFT equations has an inverse influence on the calculated minimum flow velocity (MFV) required to transport coastal boulders. This means that lower flow speeds are sufficient for boulder transport in seawater containing greater proportions of sediment (Figure 4). For example, for a reef limestone boulder subjected to flow in seawater containing 5% or 20% sediment concentration of similar carbonate lithology (i.e. coral sands), the MFV required for boulder movement decreases to 0.96 and 0.85 of the equivalent MFV in clear seawater, respectively. At the same sediment concentrations, for a basalt boulder in a flow stream containing basaltic sands, the corresponding reductions in MFV required for boulder transport are 0.94 and 0.77, as compared to flow transport in clear seawater. Figure 4 has an important implication for MFV results reported in the literature from coastal boulder analysis: MFVs are likely to be overestimations because boulder transport in clear seawater is normally assumed.

Figure 4. Reduction in the minimum flow velocity (MFV) required for coastal boulder transport in a fluid mixture of seawater and sediment. The MFV is expressed as a fraction of the MFV calculated for clear seawater without suspended sediment. Two examples are given for coastal boulders of different lithologies: limestone and basalt. The sediments entrained in seawater are assumed to have the same lithology as the coastal boulders. The graphs apply to any mode of boulder transport – sliding, rolling or lifting.

Inferring realistic sediment concentrations

Inherent challenges exist in collecting empirical measurements of sediment concentrations during HEW events. Defining sediment concentrations that are representative of all natural situations is therefore difficult. Goto et al. (2014) suggested 0–10% (average 2%) sediment concentrations within the 2011 Tohoku-oki (Japan) tsunami runup as it crossed the flat topography of the Sendai Plain. Greater concentrations might be anticipated at the shoreward edge at initial wave impact. Battisto et al. (1998) reported SSC (sand and mud) values <1% in the surf zone at the Duck research station in North Carolina, measured in a sampling campaign during a storm in April 1997, but believed their concentrations were significant underestimates. For tsunami transport of sediment released by underwater landslides, a sediment concentration of 55 ± 15 kg/m³ (∼3% concentration, assuming ρ_s = 2.0 g/cm³) was given by Brizuela et al. (2019) as their most conservative estimate. In the ‘Tsunami Sand Transport Laboratory Experiments’ of Yoshii et al. (2018), sediment concentrations were mostly <5%, but considerable differences existed between experimental cases, with some situations producing very high concentrations up to 14% and 30%. A guideline value of 7% was assumed for vertically-averaged tsunami sediment concentration (Applied Technology Council 2019), but such averages mask the possibility of much higher concentrations at the base of flows produced by suspended sediment-induced density stratification and hindered particle settling (Li and Huang 2013).

In modelling experiments by Tehranirad et al. (2016), tsunami SSCs reached maximum values of 100 kg/m³ (∼4% concentration, assuming ρ_s = 2.65 g/cm³). According to process-based modelling studies by Apotsos et al. (2011), erosive capabilities of tsunami flows may produce SSCs >200 kg/m³ (>8% concentration, assuming ρ_s = 2.65 g/cm³) in runup and up to 600 kg/m³ (∼23% concentration, ρ_s = 2.65 g/cm³) in high-velocity backwash in the offshore zone. Li and Huang (2013) also modelled SSC values up to >750 kg/m³ (>28% concentration) at the base of backwash flows, when allowing for density stratification and hindered settling. In their numerical simulations of sediment transport in the 2011 Tohoku tsunami, Yamashita et al. (2016) set maximum permissible SSC to 38%, while Larsen et al. (2017) imposed a maximum boundary value of 60% concentration.

Whether or not instantaneous maximum sediment concentrations above 20% in breaking storm waves or tsunamis are realistic remains to be confirmed. Until such time as further empirical measurements are available for real HEW events or from wave tank experiments, conservative maximum SSC values within the range 2–10% are suggested. However, recall that other types of hyperconcentrated flows do exist in nature, such as sea bottom fluid-mud layers (Sheremet et al. 2005a, 2005b), erosion gully flow (Jiongxin 2004), and lahars that are able to maintain sediment concentrations >50% (Pierson and Scott 1985; Cronin et al. 1997; Lavigne and Thouret 2002).

Caveats to findings

Several caveats need to be mentioned. (1) Any change in fluid properties will have implications for the contact dynamics with obstacles positioned within the flow. Chosen values for the coefficient of drag (C_d) and the coefficient of lift (C_l) to be substituted in HFT equations would also need to incorporate the increased density of the mixed fluid. (2) It could be argued that temperature and salinity also affect seawater density. However, reasonable estimations indicate that such effects are minimal. For seawater temperatures of 1–35°C, representing environments ranging from high-latitude winters to the hot tropics, seawater density (ρ_w) only changes by 0.77% (Haynes 2017). Similarly, 0–4% salt percentage (by mass) for freshwater or seawater causes ρ_w to vary by just 2.72%. Accordingly, salinity and temperature considerations can largely be discounted. (3) The contribution to boulder movement by the transfer of momentum from collisions of pebbles or cobbles entrained as traction load is unknown. (4) A fluidised bedload of mobile sands and gravels at the base of the flow stream might reduce friction, thereby increasing the efficiency of boulder transport.

Addressing the issues above is beyond the present scope, but the purpose of mentioning them is to recognise the implications and limitations of including a mixed-fluid density coefficient to represent elevated sediment concentrations in seawater. Such additional effects of sediment content on boulder transport processes could be investigated and incorporated into the HFT equations in future refinements. Field measurements during extreme storm waves and quasi-empirical approaches in wave-tank experiments offer good possibilities for further work.

Conclusions

We recognise the popularity, value and simplicity of existing hydrodynamic flow transport (HFT) equations for coastal boulder transport to estimate characteristics of tsunamis or storm waves as they strike coastlines. One-dimensional HFT equations are easily applied and are widely used. Yet, many workers overlook or ignore the influence of sediment content on fluid density. This is an omission, given the probability of significant sediment entrainment in the high-velocity turbulent flows generated by tsunami runup or breaking storm waves. Incorporating a simple ‘mixed-fluid density coefficient’ (C_ρ) as a multiplier for clear seawater density (ρ_w) can easily adjust for elevated sediment concentrations, where these are known, assumed or can be estimated. The additional coefficient introduces no greater complexity when applying the existing HFT equations, but encourages proper consideration of sediment concentrations in the flow responsible for coastal boulder transport.

Acknowledgements

Miss Maitha Eisa Almeer is thanked for her efforts with literaturesearching. Two anonymous reviewers provided helpful advice for improving the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Zayed University [Grant Number R19088].