and for irregular waves:
1.11 3 ,1/3 2.22 10 ,1/3/ 50 b b A A D (2.41)Solving these equations for ripple wavelength results in:
0.68 0.32 ,1/3 50 6.76 Ab D
(2.42) 3 0.11 1.11 ,1/3 50 2.22 10 Ab D
(2.43)for ripples under regular and irregular wave forcing, respectively. Equation (2.42) clearly shows that for regular waves, orbital semi-excursion is the dominant parameter
controlling ripple wavelength while for irregular waves (equation (2.43)) the median grain diameter is the dominant scaling factor. Based on the range of Ab,1/3/D50 values
found in these data sets (10 to 104 for regular waves, 102 to 5×105 for irregular waves) the
monochromatic wave conditions extend to typically orbital scale ripples (Ab,1/3/D50<877)
with some suborbital (877<Ab,1/3/D50<2,794) which partially explains the greater
dependence on semi-excursion. On the other hand, ripples formed by irregular waves fall in the suborbital and anorbital (Ab,1/3/D50>2,794) regimes where grain size becomes more
important. However, at the overlapping suborbital region, the data from the
monochromatic wave conditions (Figure 2.13a) trend in agreement with equation (2.42) well into the suborbital regime, while irregular wave formed ripples continue on an opposing trend into the orbital regime (see Figure 2.13b). One explanation for the scaling differences between these two types of waves is the consistency of the forcing present. Under monochromatic waves, the bed is subjected to the same orbital excursion with each passing wave, while under irregular waves the orbital excursion varies for each wave. Thus, the grain size acts as a filter and a limiting factor preventing or delaying the transition from suborbital to orbital scaling.
Figure 2.13. Top row: scatter plots of (a) ripple wavelength (λ) normalized by particle size (D50) for regular wave conditions (equation (2.42)); (b) ripple wavelength (λ)
normalized by wave semi-orbital excursion (Ab,1/3) for irregular wave conditions
(equation (2.43)); and (c) the same as (b) but with all data combined (equation (2.44)). Bottom row: scatter plots of ripple height (η) vs. wavelength (λ) for (d) regular wave (equation (2.45)), (e) irregular wave (equation (2.46)), and (f) both regular and irregular wave conditions (equation (2.49)).
The best attempt to collapse both regular and irregular data using a common parameterization is achieved when λ/Ab,1/3 is plotted against Ab,1/3/D50 (see Figure 2.13c)
in a similar manner as in Soulsby and Whitehouse [2005]. Comparing the enriched data set to that used by Soulsby and Whitehouse [2005], the new data set includes significantly more field observations with smaller Ab,1/3/D50 values that allow to define the trend of
ripple characteristics over a larger range. Least squares fit, on a log-log scale, of an equation similar to that of SW (equation (2.27)) through the expanded data set leads to:
1 1.15 3 4 ,1/3 0.72 2.0 10 ,1/3/ 50 1 exp 1.57 10 ,1/3/ 50 b b b A A D A D (2.44)with a correlation coefficient (r2) of 0.73. This equation captures the data trend for values
of Ab,1/3/D50 < 7,000 but may over predict irregular wave ripples at larger Ab,1/3/D50
values.
2.5.2. Ripple Height and Steepness
During the comparison of the data with existing predictors (see section 2.4), there were a number of occasions where ripple steepness appeared to be constant for a large range of values of the parameter used in the x-axis. Similar to wavelength, ripple
steepness data (note that the new data set described in section 2.3.2 did not include ripple heights) are examined for monochromatic and irregular wave conditions, separately. A log-log scatter plot of ripple height vs. wavelength shows a linear trend. Least squares analysis on the data reveals the following best fit relationships for regular (Figure 2.13d) and irregular (Figure 2.13e) wave conditions:
0.89
0.115
(2.45) 1.050.126
(2.46)with correlation coefficients (r2), on a log-log scale, of 0.78 and 0.81, respectively. It
should be noted that in the above equations both η and λ are in meters. Dividing both parts by wavelength, the following relationships for wave steepness are established
0.11 0.075 0.035 ,1/3 50 0.115 0.093 Ab D
(2.47) 0.05 2 0.006 ,1/ 0 05 5 6 3 0 . 0.126 0.185 10 Ab D
(2.48)for regular and irregular wave conditions, respectively. The data and equations (2.47) and (2.48) suggest that ripples formed under irregular wave conditions are slightly steeper (0.126) than those formed under regular wave conditions (0.115) with both having an almost constant steepness. The scatter of data points under the predicted line in Figure 2.13e are attributed to large errors associated with the measurements of very small ripple height (η < 1 mm) in the field and are considered to represent the experimental error.
Unlike for wavelength, when the two data sets are overlaid (see Figure 2.13f), they collapse on a single trend and least squares fitting produces the following steepness relationship:
0.056
0.120
(2.49)
with a correlation coefficient r2 of 0.79. The small correction for very long wavelengths
is attributed to the enhanced turbulence at the crests due to flow contraction over the ripples [Du Toit and Sleath, 1981; Nielsen, 1992]. Since ripple height increases with increasing wavelength, the enhancement will be greater for larger ripples.
2.5.3. Model Errors
The performance of the new models described in equations (2.40) to (2.49) is compared to that of all previously presented models (see section 2.2). For each model, the predicted ripple wavelength and height values are compared against the measured ones from the database; the scatter plots together with the 1:1 line are shown in Figures 2.14 and 2.15 for ripple wavelength and height, respectively. The least squares error of each predictor (in a log-log scale) was estimated using the root mean square error (RMS), normalized by the range of the observed values for each parameter predicted:
2
10 10 10 10
1 log log log max
p m m m
X X X log min X
N