Effect of bed clay on surface water-wave reconstruction from ripples

Abstract

Wave ripples can provide valuable information on their formative hydrodynamic conditions in past subaqueous environments by inverting dimension predictors. However, these inversions do not usually take the mixed non-cohesive/cohesive nature of sediment beds into account. Recent experiments involving sand–kaolinite mixtures have demonstrated that wave-ripple dimensions and the threshold of motion are affected by bed clay content. Here, a clean-sand method to determine wave climate from orbital ripple wavelength has been adapted to include the effect of clay and a consistent shear-stress threshold parameterisation. From present-day examples with known wave conditions, the results show that the largest clay effect occurs for coarse sand with median grain diameters over 0.45 mm. For a 7.4% volumetric clay concentration, the range of possible water-surface wavelengths and water depths can be reduced significantly, by factors of three and four compared to clean sand, indicating that neglecting clay when present will underestimate the wave climate.

Introduction

Bed-surface structures in sediments and sedimentary rocks of past subaqueous environments provide important information on flow hydraulics¹. These structures tend to be classified on the basis of the presence or absence of cohesion in equivalent modern environments: (a) cohesive structures, associated with physical cohesion by clay particles and biological cohesion by extracellular polymeric substances (EPS), e.g. biofilms², where the bed is stabilised by cohesion between grains and sudden catastrophic failure may occur under high bed shear stress, e.g. during storms; and (b) non-cohesive bedforms (e.g. wave and current ripples^3,4), where grains can move individually and tend to respond more rapidly and continuously to changes in flow forcing⁵. Examples of these types of bed-surface structure include roll-ups⁶ and non-cohesive wave ripples⁷. Davies et al.⁸ argued that the distinction between cohesive and non-cohesive sedimentary structures is artificial, as they represent end members of a spectrum, and thus predictions based on one classification may result in misinterpretations. This position is re-enforced by recent experiments that have shown how sandy bedforms can be affected by small amounts of biological and physical cohesion^9,10,11,12. The consequence for wave ripples of physical cohesion associated with kaolin clay in the bed has been detailed by Wu et al.¹³. Building on previous work⁷, Diem¹⁴ developed a clean-sand analytical method for the prediction of paleowave climate based on the dimensional measurement of wave ripples in the rock record^{15,16,17,18,19}. Here, the Diem¹⁴ approach is adapted for sand–clay mixtures, using the synthesis proposed by Wu et al.¹³.

The Diem¹⁴ approach starts by determining the wave orbital diameter from the ripple wavelength, without requiring a specific wavelength predictor. Here, the formulation starts as Diem¹⁴ did, with linear wave theory and additional constraints based on threshold of motion and wave breaking. It will then return to the effect of clay on ripple-wavelength prediction and the threshold of motion. Since wave conditions from the rock record are unknown, the importance of clay content is demonstrated with the use of present-day examples from the laboratory and field, where the wave conditions were known.

Results

The Diem¹⁴ approach

Wave conditions

Based on linear wave theory, the Diem¹⁴ approach uses expressions for the dispersion relation and the wave-velocity amplitude, U₀, together with conditions for the threshold of motion and wave breaking, to give

$$x<\tanh kh,$$

(1a)

$$x \geqslant A\cosh kh,$$

(1b)

(see “Methods”) where x = L/L_t∞, L is the water-surface wavelength, L_t∞ = πg(d₀/U_t)²/2 is the deep-water surface wavelength corresponding to the threshold of motion, k = 2π/L, h is the water depth, g is the acceleration due to gravity (= 9.81 m s²), d₀ is the orbital diameter (= H/sinhkh, H is the wave height), U_t is the critical wave-velocity amplitude associated with the threshold of motion and A = d₀/0.142L_t∞. As will be seen later, U_t is a function of d₀ and D₅₀, the median grain diameter, and A can be defined as (U_t/U_m)²/2, where U_m = (0.0355πgd₀)^1/2 is the maximum wave-velocity amplitude. Since wave-velocity amplitude and orbital diameter are related by U₀ = πd₀/T, where T is the wave period, U₀ and T are in the ranges U_t < U₀ ≤ U_m and πd₀/U_m ≤ T <  πd₀/U_t.

Equation (1a, b) represent the range of possible conditions between threshold and wave breaking for the wave climate, such that x is in the range Acoshkh ≤ x < tanhkh, and A < 1/2 for there to be any allowable conditions (see “Methods”). Figure 1 shows kh versus x for the limiting single-valued A = 1/2 case (U_m² = U_t²) from shallow water (kh ≪ 1) to deep water (kh ≫ 1). Figure 1 also shows the A = 1/4 case (U_m² = 2U_t²), corresponding to typical above-threshold wave conditions. In the latter case, the shaded region in Fig. 1 shows the allowable values of x [the “Methods” section explains how intersection points, (x_min, kh_min) and (x_max, kh_max), are determined]. The breaking-wave curves (x = Acoshkh) are concave downward and the threshold curve (x = tanhkh) is concave upward. Notice for the breaking-wave curves, x → A for kh ≪ 1 and A also controls the slope for larger kh. In dimensional terms, L is therefore limited by the threshold scale (L_t∞) and breaking-wave scale (AL_t∞) according to AL_t∞ < L ≤  L_t∞. Then, from equations (1a, b), the range of h can be expressed as a function of L as.

Fig. 1

kh versus x (L/L_t∞) for the limiting case of A = 1/2 and also A = 1/4. The dots correspond to x = 2^–1/2 and kh = arctanh2^–1/2, for A = 1/2; x_{max, min} = (2 ± 3^1/2)^1/2/2 and kh_{max, min} = arctanh[x_{max, min}], for A = 1/4 (see “Methods”, Eq. (9)), and shading represents allowable values of x and kh for A = 1/4 (Acoshkh ≤ x < tanhkh).

$$(L/2\pi ){\text{arctanh}}(L/{L_{{\text{t}}\infty }})<h \leqslant (L/2\pi ){\text {arccosh}}(L/A{L_{{\text{t}}\infty }})$$

(2)

The ripple predictor

Diem’s¹⁴ central assumption is that the orbital diameter can be expressed in terms of a ripple wavelength, which is assumed to be in equilibrium, λ_e, as

$${\uplambda _{\text{e}}}={\upalpha _0}{{\text{d}}_0},$$

(3)

where α₀ = 0.65, based on the experiments²⁰, provided that λ_e < 200 mm (d₀ = 308 mm). Above this limit, Diem¹⁴ used the Sleath²¹ predictor. This arbitrary 200-mm limit represents the lower boundary of the suborbital and anorbital ranges, where the wavelength is dependent on both d₀ and D₅₀ for suborbital ripples and only dependent on D₅₀ for anorbital ripples. However, while a value of α₀ in the range 0.5 ≤ α₀ ≤ 0.75 in Eq. (3) is widely accepted, there is little agreement in the literature on the precise nature of the orbital, suborbital, and anorbital limits²². Wiberg and Harris³ defined orbital, suborbital and anorbital ripples by d₀/D₅₀ ≤ 1754, 1754 < d₀/D₅₀ ≤ 5587 and d₀/D₅₀ > 5587, respectively²³, whereas other researchers have argued the anorbital limit should have wave-period dependence^24,25. Provided that Eq. (3) does hold, which is what will be assumed here, all quantities involving d₀ can now be expressed in terms of λ_e. Wu et al.¹³, and other researchers previously^26,27, have demonstrated that d₀ can be modified by the presence of a current. However, as the method relies on there being a one-to-one correspondence between d₀ and λ_e, this effect cannot be included and so conditions must be restricted to waves in isolation.

Adaptions to the Diem¹⁴ approach

Threshold of motion parameterisation

Based on the Soulsby²⁸ critical threshold of motion for clean sand, the “Methods” section derives an expression for U_t², Eq. (11), such that U_t², L_t∞ and AL_t∞ can be written

$$U_{\text{t}}^{{~2}}=~B(gd_{0}^{{~0.52}}D_{{50}}^{{~0.48}}),$$

(4a)

$${L_{\text{t}\infty }}~=\frac{{\uppi {d_0}}}{{2B}}{\left( {\frac{{{d_0}}}{{{D_{50}}}}} \right)^{0.48}},$$

(4b)

$$A{L_{\text{t}\infty }}~=\frac{{{d_0}}}{{0.142}},$$

(4c)

where B = 3.653(s–1)θ₀, s is the relative density of sediment in water, θ₀ is the critical skin friction Shields parameter, Eq. (10), and d₀ = λ_e/α₀ from Eq. (3). Diem¹⁴ used the Komar and Miller²⁹ mobility threshold prescription. Here, the Soulsby²⁸ expression has been used, as it allows U_t² to be directly related to θ₀ and it avoids the need for two different functional forms, for D₅₀ < 0.5 mm and D₅₀ ≥ 0.5 mm (Eqs. (12) and (13)).

Inclusion of the effect of clay

Wu et al.¹³ showed that the ratio of wavelength to orbital diameter, α, which replaces α₀ in Eq. (3), can be expressed as

$${\upalpha}\,=\,{\upalpha_0}\left\{ {\begin{array}{*{20}{l}} {1,}\quad\quad\quad\quad\quad\quad\quad\quad\quad{{C_0}}\leqslant{C_{0{\text{m}}},} \\ {1-5.5}({C_0}-{C_{0{\text{m}}}),}\quad{C_0} >{C_{0{\text{m}}},} \end{array}} \right.$$

(5)

where α₀ is the clean-sand constant of proportionality (= 0.61), C₀ is the clay content in the bed, C_0m = 7.4% is the minimum value of C₀ where α can change from α₀ and α = α₀/2 for C₀ = 16.3%. Whitehouse et al.³⁰ showed that the threshold of motion is enhanced by the clay content according to

$${\uptheta_{0\text{E}}}={\uptheta _0}{B_\theta },$$

(6)

where θ₀ is the clean-sand threshold, B_θ = 1 + P_θC₀ and P_θ is a constant that depends on the sediment properties. Based on their experiments, Wu et al.¹³ determined that P_θ = 6.3 for D₅₀ = 0.143 mm and P_θ = 23 for 0.45 ≤ D₅₀ ≤ 0.5 mm (between these two ranges, it will be assumed that P_θ can be linearly interpolated). Notice in Eq. (6) that even small amounts of clay produce an enhancement which is strongly dependent on grain size.

Thus, the two main effects of including clay are that α is reduced and θ_0E is increased. Substituting Eqs. (5) and (6), into Eqs. (4a, b, c) gives U_t², L_t∞ and AL_t∞ as

$$U_{\text{t}}^{{~2}}=~B\left[ {\frac{{{B_\theta }}}{{{\upalpha ^{0.52}}}}} \right](g\uplambda _{\text{e}}^{{0.52}}D_{{50}}^{{~0.48}}),$$

(7a)

$${L_{\text{t}\infty }}~=\frac{{\pi {\uplambda _{\text{e}}}}}{{2B[{B_\theta }{\upalpha ^{1.48}}]}}{\left( {\frac{{{\uplambda _{\text{e}}}}}{{{D_{50}}}}} \right)^{0.48}},$$

(7b)

$$A{L_{\text{t}\infty }}~=\frac{{{\uplambda _{\text{e}}}}}{{0.142[\upalpha ]}},$$

(7c)

where λ_e is the mixed clay–sand ripple wavelength and only the square-bracketed quantity in each expression depends on C₀.

The adapted procedure

The procedure begins with the determination of the ripple wavelength, λ_e, and bed-clay content, C₀. Once these have been determined, the following calculations are undertaken:

• (i) Use λ_e and C₀ in equations (7a, b,c), with α and B_θ given by Eqs. (5) and (6), to determine U_t, L_t∞ and AL_t∞

• (ii) Use A in Eq. (9) to determine x_{max, min}, so that x_min ≤ x ≤ x_max

• (iii) Use L = L_t∞x to determine the range of h based on Eq. (2) and U_t and U_m = (0.0355πgλ_e/α)^1/2 to determine the ranges of U₀ and T: U_t < U₀ < U_m and πλ_e/αU_m < T  <  πλ_e/αU_t

Example cases

With specific examples from the rock record, Diem¹⁴ was able to show how local considerations and context could be used to further limit the theoretical ranges described in the previous section. Here, modern-day examples, where the wave properties are known, are used, so that attention can be focussed on the effect of clay on the theoretical ranges alone. The example cases correspond to clean, coarse-, medium- and fine-grained sand from the laboratory and field, and involve determining how wave conditions based on the measured ripples change if the clay content is varied in the range 0 ≤ C₀ ≤ 16.3%.

Wu et al.¹¹, coarse-sand laboratory data

Wu et al.¹¹ conducted a series of experiments involving a single-wave condition over a bed composed of well-sorted coarse sand, D₅₀ = 0.496 mm (θ₀ = 0.032), and varying clay content, 0 ≤ C₀ ≤  7.4%. For the clean sand experiment (C₀ = 0%), the wave conditions were given by h = 0.6 m, H = 0.16 m and T = 2.49 s (L = 5.62 m), corresponding to d₀ = 0.223 m. Despite the coarse grain size, which allowed clay and sand particles to be distinguished more readily in the experiments, Wu et al.¹² demonstrated that these hydrodynamic conditions were comparable to an intertidal site in the macrotidal Dee Estuary with a medium grain size³¹. This experiment produced ripples with a wavelength λ_e = 278D₅₀¹³. Figure 2a shows the threshold and wave-breaking scales, L_t∞ and AL_t∞, versus C₀. L_t∞, which is smallest at C₀ = 7.4%, has a much larger range than AL_t∞, which is constant for C₀ ≤  7.4% and then doubles up to C₀ = 16.3%. Figure 2d shows the corresponding L-h phase space, based on Eq. (2), for the clay contents depicted in Fig. 2a (C₀ = 0, 7.4 and 16.3%). Compared to the dimensionless x–kh plot (Fig. 1), the threshold curves are still concave downwards, but more exaggerated, and the breaking-wave curves are close to straight lines. The change in ranges is largely due to changes in L_t∞. The reduction in range between the largest and smallest (corresponding to C₀ = 0% and 7.4%) is by a factor of 3 and 4 for the water-surface wavelength and water depth, respectively (Fig. 2d). Notice that the actual surface wavelength and water depth (L = 5.62 m, h = 0.6 m) are within all three ranges. Figure 2a, d can be compared with Fig. 4a, b to see the effect of using the Komar and Miller²⁹ clean-sand mobility description for the threshold. This shows L_t∞ to be about 63% of its value in Fig. 2a, because B = 0.21(s–1) = 0.34 as opposed to 3.653(s–1)θ₀ = 0.19. Thus, using the Diem¹⁴ clean-sand mobility description underpredicts the range of water-surface wavelengths and heights in an absolute sense. In a relative sense, the change in the ranges with clay content is similar, because the powers of d₀ and D₅₀ are similar (Eq. (13), for D₅₀ < 0.5 mm, and Eq. (7a)), but this will not be the case for D₅₀ > 0.5 mm. Also, the measured L and h are not within the C₀ = 7.4% range (Fig. 4b). As L and h are below the threshold curve, this would imply that ripples of this size are relict for this clay content. This is inconsistent with the experimental results, since Wu et al.¹³ showed no reduction in λ_e for C₀ ≤  7.4%.

Fig. 2

(a–c) Threshold, L_t∞, and wave-breaking, AL_t∞, scales, equations (7b, c), versus C₀ and (d–f) L–h phase space from Eq. (2), showing the different ranges for C₀ = 0, 7.4 and 16.3% and the measured L and h. For (a,d) Wu et al.¹¹, λ_e = 278D₅₀, D₅₀ = 0.496 mm, θ₀ = 0.032, L = 5.62 m and h = 0.6 m; for (b,e) Doucette³², λ_e = 250 mm, D₅₀ = 0.22 mm, θ₀ = 0.045, L = 11.9 m and h = 0.47 m, and for (c,f) Boyd et al.³³, λ_e = 180 mm, D₅₀ = 0.11 mm, θ₀ = 0.076, L = 50.7 m and h = 10 m. Legend applies to (d–f); colours in (a–c) are consistent with the legend.

Doucette³², medium-sand field data

The Doucette³² field measurements were taken on a microtidal beach of Wambro Sound, Western Australia, near Perth (run 1) where h = 0.47 m, H = 0.2 m and T = 5.6 s (L = 11.9 m), corresponding to d₀ = 0.79 m. The bed was composed of medium sand, with D₅₀ = 0.22 mm (θ₀ = 0.045), and the measured ripples had a wavelength of λ_e = 250 mm. This example case builds on the Wu et al.¹¹ example case by corresponding to both the hydrodynamics and medium grain size of the Dee Estuary UK³¹. Since d₀/D₅₀ = 3591, the ripples were in the suborbital range, where the wavelength is dependent on both the orbital and grain diameters. Notice this wavelength is above Diem’s¹⁴ 200-mm limit. The methods section demonstrates that, whilst using the Sleath²¹ predictor for C₀ = 0 produces a difference, it is similar to the other two example cases, which are below the Diem¹⁴ limit, and so is not considered significant. From interpolation, P_θ in equation (6) is determined to be 10. Figure 2b shows L_t∞ and AL_t∞ versus C₀ and Fig. 2e shows the L–h phase space, for C₀ = 0, 7.4 and 16.3%. These reveal similar behaviour to that of the coarse-grained sand case, but less extreme: in Fig. 2b, L_t∞ is still at its minimum at C₀ = 7.4% and AL_t∞ shows the same enhancement as in Fig. 2a. The reduction in range between the largest and smallest (C₀ = 0% and 7.4%) is by a factor of 2 for both the water-surface wavelength and water depth (Fig. 2e). Again, the actual surface wavelength and water depth (L = 11.9 m, h = 0.47 m) are within all three ranges.

Boyd et al.³³, fine-sand field data

The Boyd et al.³³ field measurements were undertaken about 1 km from Martinique Beach on the Atlantic coast of Nova Scotia during a period of relative calm (day 167, hour 9) where h = 10 m, H = 0.5 m and T = 6.2 s (L = 50.7 m), corresponding to d₀ = 0.32 m. The bed was composed of well-sorted fine sand, with D₅₀ = 0.11 mm (θ₀ = 0.076), and the measured ripples had a wavelength of λ_e = 180 mm. Fine sands are common in many estuaries around the world, for example³⁴. d₀/D₅₀ = 2873 puts the ripples into the suborbital range. Assuming that Pθ in equation (6) is the same as for 0.143 mm (P_θ = 6.3), Fig. 2c shows L_t∞ and AL_t∞ versus C₀ and Fig. 2f shows the L–h phase space, for C₀ = 0, 7.4 and 16.3%. L_t∞ in Fig. 2c is still at its minimum at C₀ = 7.4%, but, because of far weaker clay enhancement of the threshold for fine sands in equation (6), L_t∞ is largest for C₀ = 16.3%. In Fig. 2f, the measured water-surface wavelength and water depth (L = 50.7 m, h = 10 m) are below the threshold curve and outside the range for the C₀ = 0 and 7.4% clay contents, and just above the threshold curve and within range for C₀ = 16.3%, because, unlike the previous cases, C₀ = 16.3% produces the largest L_t∞. Since there was little clay at the field site, the wave conditions were probably below threshold, implying that the observed ripples were relict. This is supported by the fact that Boyd et al.’s³³ previous observation at day 167, hour 3, showed the same wavelength and no ripple migration. The reduction in range between the largest and smallest (C₀ = 16.3% and 7.4%) is again by a factor of 2 for both the water-surface wavelength and water depth (Fig. 2f).

Discussion

The range of L shown in Fig. 2 is largely controlled by L_t∞, so it is of interest to determine how the change in clay content affects L_t∞, Eq. (7b), compared to the original clean-sand Diem¹⁴ method using Komar and Miller²⁹, L_t∞KM, Eq. (13) with C₀ = 0%. The net effect is shown as a ratio in Fig. 3 for C₀ = 0, 7.4 and 16.3% and 0.1 ≤ D₅₀ ≤ 0.8 mm, for the approximate limits in the range of λ_e/D₅₀ of 250 and 1,000. There are two competing effects: the reduction because of clay content (Fig. 2) and the increase because of using the Soulsby²⁸ threshold condition rather than the Komar and Miller²⁹. Figure 3 shows a discontinuity for clean sand at D₅₀ = 0.5 mm as a result of Eq. (13), leading to the largest difference (L_t∞ is increased by up to 161% for λ_e/D₅₀ = 250), which decreases with increasing λ_e/D₅₀ (although the Diem¹⁴ method has rarely been applied for D₅₀ > 0.5 mm). Otherwise for D₅₀ ≤ 0.19 mm, L_t∞ is reduced by up to 36%, and for 0.19 < D₅₀ ≤ 0.5 mm, L_t∞ is increased by up to 64%. For C₀ = 7.4%, L_t∞ is consistently decreased by between 35 and 56%, and for C₀ = 16.3%, L_t∞ varies only slightly (increased by up to 14%, for 0.12 ≤ D₅₀ ≤ 0.37 mm, and otherwise reduced by up to 15%). The absence of a discontinuity in the present formulation, compared to Diem’s¹⁴ original formulation, is clearly preferable. Also, the net effect of the clay on L_t∞ will be stronger for smaller than for larger clay contents. It should be remembered that the present method, like the original Diem¹⁴ method, is only applicable to waves alone.

Fig. 3

Relative size of L_t∞(C₀) from Eq. (7b) normalised by the clean-sand L_t∞ from Eq. (13), L_t ∞ KM, for C₀ = 0, 7.4 and 16.3%. λ_e/D₅₀ = 250 and 1000, and the dots correspond to the Wu et al.¹¹ clean-sand experiment in Fig. 2a.

It is important to clarify how a representative clay content, C₀, for the ripples should be determined. In the modern environment this usually involves measuring C₀ below the active layer (below trough level), as efficient winnowing often removes clay from the body of the ripples during development¹². In the geological record, where the method still requires testing, clay content in deposits should be based on primary clay minerals and diagenetic alterations for which it can be established that the original mineral was part of the primary clay fraction. The laboratory experiments used in this study¹³ are based only on the physical cohesion associated with kaolin, and not the biological cohesion associated with EPS that often occur together with clay in the field³⁵. Since, the effect of EPS can be quite substantial^2,9, the present method should be interpreted as a conservative estimate in the field. For instance, Baas et al.³⁵ concluded that ripple development in the field compared to the laboratory was significantly delayed due to the additional presence of EPS leading to a greater enhancement in the threshold of motion, see Eq. (6). EPS delaying development may also challenge the equilibrium wavelength assumption, see Eq. (3). Further study of the effects of combined clay and EPS on wave ripples is thus still required.

Conclusions

Preserved sedimentary bedforms provide important information for reconstructing past hydraulics in subaqueous environments by inverting bedform predictors, but this is usually based exclusively on non-cohesive sand. The present work incorporates the effects of sand-clay mixtures on bedforms, using the experimental results of Wu et al.¹³ in the non-cohesive inversion method of Diem¹⁴ for waves alone. Based on wave breaking and threshold of motion limitations, the Diem¹⁴ approach results in ranges for wave conditions. Here we have shown that the inclusion of as little as 7.4% clay in the most extreme case of coarse sand, D₅₀ ≥ 0.45 mm, reduces the possible ranges of water-surface wavelengths and water depths by factors of 3 and 4, respectively. For fine sand, the ranges are reduced by a factor of two. In short, not accounting for the modifying effect of clay in ripple growth and equilibrium geometries, may lead to underestimating the prevailing wave conditions if clay is present.

Methods

Derivation of the Diem¹⁴ approach

For linear wave theory, the dispersion relation and the wave-velocity amplitude, U₀, are

$${\upsigma ^2}=gk\tanh kh,$$

(8a)

$${U_0}=\uppi {d_0}/T,$$

(8b)

where σ = 2π/T, d₀ = H/sinhkh and k = 2π/L. The wave properties, characterised by equations (8a, b), are subject to two constraints: threshold of motion and wave breaking. For sediment movement U₀² > U_t², where U_t is the threshold wave-velocity amplitude to be determined below, which when combined with equations (8a, b) gives x < tanhkh, Eq. (1a), where x = L/L_t∞ and L_t∞ = πg(d₀/U_t)²/2. The wave-breaking criterion³⁶ defines the maximum possible wave steepness as, H/L ≤ 0.142tanhkh, which when combined with d₀ and L_t∞ gives x ≥ Acoshkh, Eq. (1b), where A = d₀/0.142L_t∞. It will be shown below, that A < 1/2, so if A = (U_t/U_m)²/2, where U_m = (0.0355πgd₀)^1/2 is the maximum wave-velocity amplitude (U_m > U_t), then U_t < U₀ ≤ U_m and from Eq. (8b) πd₀/U_m ≤ T <  πd₀/U_t.

The limits of the possible values of x can be found by combining Eqs. (1a, b) using the identity 1–tanh²kh = sech²kh, such that the maximum and minimum in x satisfy the equation x⁴–x² + A² = 0, so that x_{max, min} are

$${x_{\hbox{max} ,\hbox{min} }}={[1/2 \pm {(1 - 4{A^2})^{1/2}/2}]^{1/2}},$$

(9)

where A < 1/2, for there to be two distinct values. Here, x and kh are in the ranges x_min ≤ x < x_max and arctanhx < kh ≤ arccosh(x/A), respectively. In Fig. 1, for the limiting A = 1/2 (U_m² = U_t²) single-valued case the dot corresponds to x = 2^–1/2, from Eq. (9), and kh = arctanh2^–1/2 ~ 0.88. Likewise, in the A = 1/4 (U_m² = 2U_t²) case the dots mark x_{max, min} = (2 ± 3^1/2)^1/2/2~ 0.97, 0.26 and kh_{max, min} = arctanh(x_{max, min}).

Determination of U _t ² based on Soulsby²⁸

According to Soulsby²⁸, the Shields parameter for the critical threshold of motion of clean sand is

$${\uptheta _0}=\frac{{0.3}}{{1+1.2{D_*}}}+0.055(1-{\text{e}^{-0.02D*}}),$$

(10)

where D_* = [(s–1)g/ν²]^1/3D₅₀, s = ρ_s/ρ, ρ_s and ρ are the sediment and water densities and ν is the kinematic viscosity (~ 1 mm² s^-1). For waves, θ₀ = f_wU_t²/2(s–1)gD₅₀, where f_w = 1.39(6d₀/D₅₀)^–0.52 is the skin friction factor³⁷. Rearranging the θ₀ wave expression gives U_t² as

$$U_{\text{t}}^{{~2}}=~B(gd_{0}^{{~0.52}}D_{{50}}^{{~0.48}}),$$

(11)

where B = 6^0.52(s–1)θ₀/0.695 = 3.653(s–1)θ₀. Equation (11) can be compared with Eq. (12).

Diem¹⁴ threshold of motion constraint

Diem¹⁴ used the Komar & Miller²⁹ expression for U_t², namely

$$U_{\text{t}}^{{~2}}=(s - 1)g{d_0}~ \times \left\{ {\begin{array}{*{20}{l}} {0.21{{({d_0}/{D_{50}})}^{ - 0.5}},}&{{D_{50}}<0.5~\;{\text{mm}},} \\ {0.46\uppi {{({d_0}/{D_{50}})}^{ - 0.75}},}&{{D_{50}} \geqslant 0.5~\;{\text{mm}},} \end{array}} \right.$$

(12)

such that with the inclusion of clay L_t∞ = πgd₀²/2B_θU_t², d₀ = λ_e/α and B_θ and α are given by Eqs. (5) and (6), giving L_t∞ as

$${L_{\text{t}\infty }}=\frac{{\uppi {\uplambda _{\text{e}}}}}{{2(s - 1)}} \times \left\{ {\begin{array}{*{20}{l}} {\frac{1}{{0.21[{B_\theta }{\alpha ^{1.5}}]}}{{\left( {\frac{{{\uplambda _{\text{e}}}}}{{{D_{50}}}}} \right)}^{0.5}},}&{{D_{50}}<0.5\;~{\text{mm}},} \\ {\frac{1}{{0.46\uppi [{B_\theta }{\alpha ^{1.75}}]}}{{\left( {\frac{{{\uplambda _{\text{e}}}}}{{{D_{50}}}}} \right)}^{0.75}},}&{{D_{50}} \geqslant 0.5\;~{\text{mm}},} \end{array}} \right.$$

(13)

and AL_t∞ as λ_e/0.142[α] remains the same. Figure 4 shows the effect of this parameterisation of the threshold of motion for the first example case of Wu et al.¹¹ depicted in Fig. 2a, d. Unlike Fig. 2d, the measured values of h and L are outside the range predicted for C₀ = 7.4%.

Fig. 4

(a) Threshold, L_t∞, and wave-breaking scales, AL_t∞, Eqs. (13) and (7c), versus C₀ and (b) L–h phase space from Eq. (2) showing the different ranges for C₀ = 0, 7.4 and 16.3% and the measured h and L for Wu et al.¹¹, λ_e = 278D₅₀, D₅₀ = 0.496 mm, h = 0.6 m and L = 5.62 m.

Using the Sleath²¹ expression to predict d ₀/λ_e

This section considers the effect of using the Sleath²¹ expression for d₀/λ_e when λ_e ≥ 200 mm and applies it to the three examples considered in the “Results” section for the clean-sand case. In the clean-sand case, according to Diem¹⁴, d₀/λ_e can be expressed as

$$\frac{{{d_0}}}{{{\uplambda _{\text{e}}}}}=\left\{ {\begin{array}{*{20}{l}} {\upalpha _{0}^{{ - 1}},}&{{\uplambda _{\text{e}}}<200\;~{\text{mm}},} \\ {0.778R_{\text{s}}^{{~0.151}},}&{{\uplambda _{\text{e}}} \geqslant 200~\;{\text{mm}},} \end{array}} \right.$$

(14)

where α₀ was taken to be 0.65, but here is 0.61, see Eq. (6), and R_s = (U₀d₀/2ν)^1/2. If λ_e < 200 mm, then α₀^–1λ_e can be substituted for d₀, as explained in the main paper. However, for λ_e ≥  200 mm, since the d₀/λ_e ratio can change, Diem¹⁴ showed that an additional step in the calculation was required. The wave-velocity amplitude must still be in the range U_t < U₀ ≤ U_m, so that R_s is in the range (U_td₀/2ν)^1/2 < R_s ≤ (U_md₀/2ν)^1/2, and therefore from Eq. (14), for λ_e ≥  200 mm, d₀/λ_e must be in the range 0.778(U_td₀/2ν)^0.0755 < d₀/λ_e ≤  0.778(U_md₀/2ν)^0.0755. From Eq. (11), U_t = (Bg)^0.5d₀^0.26D₅₀^0.24, where B = 3.653(s–1)θ₀, and U_m = (0.0355πgd₀)^0.5, so that the d₀/λ_e range is

$${P_1}{\left( {\frac{{{g^{0.5}}d_{0}^{{~1.26}}D_{{50}}^{{~0.24}}}}{\upnu }} \right)^{0.0755}}<~\frac{{{d_0}}}{{{\uplambda _{\text{e}}}}}~ \leqslant ~{P_2}{\left( {\frac{{gd_{0}^{{~~3}}}}{{{\upnu ^2}}}} \right)^{0.03775}},$$

(15)

where P₁ = 0.778(B/4)^0.03775, P₂ = 0.778(0.0355π/4)^0.03775, and the minimum and maximum in the d₀/λ_e range correspond to the threshold of motion and wave breaking, respectively. For a given measured λ_e, the solution to Eq. (15) requires an iteration starting from d₀ = α₀^–1λ_e = 1.64λ_e. Substituting λ_e(d₀/λ_e)_min and λ_e(d₀/λ_e)_max, from Eq. (15), into equations (4a, b,c) allows the threshold of motion and wave breaking scales, L_t∞ and AL_t∞, to be expressed as

$${L_{\text{t}\infty }}~=\frac{{\uppi {\uplambda _{\text{e}}}({d_0}/{\uplambda _{\text{e}}})_{{\hbox{min} }}^{{1.48}}}}{{2B}}{\left( {\frac{{{\uplambda _{\text{e}}}}}{{{D_{50}}}}} \right)^{0.48}}$$

(16a)

$$A{L_{\text{t}\infty }}~=\frac{{{\uplambda _{\text{e}}}{{({d_0}/{\uplambda _{\text{e}}})}_{\hbox{max} }}}}{{0.142}}.$$

(16b)

For each of the three example cases considered in the “Results” section, (d₀/λ_e)_min and (d₀/λ_e)_max are listed in Table 1, even though the λ_e ≥ 200 mm condition is only met in the Doucette³² case. In all three example cases, d₀/λ_e = 1.64 lies between (d₀/λ_e)_min and (d₀/λ_e)_max. The observed and predicted d₀/λ_e are in close agreement, apart from the Doucette³² case, where both the d₀/λ_e range from Eq. (15) and d₀/λ_e = 1.64 underpredict by approximately a factor of two. The values of L_t∞R and AL_t∞R from Eqs. (16a, b) and (4b, c) are also given in Table 1. Since these values for L_t∞R and AL_t∞R are largely similar in all three cases, this suggests that using the orbital approximation d₀/λ_e = 1.64 for the Doucette³² case is reasonable, even though λ_e ≥  200 mm.

Table 1 Measured D₅₀, λ_e and d₀/λ_e and predicted θ₀ from Eq. (10), (d₀/λ_e)_min and (d₀/λ_e)_max from Eq. (15), L_t∞R = [0.61(d₀/λ_e)_min]^1.48 and AL_t∞R = 0.61(d₀/λ_e)_max, based on Eqs. (16a, b) and (4b, c) for the three example cases.