MODELING OF LEVEE EROSION UNDER IRREGULAR WAVES

A levee erosion model is developed to predict the temporal and cross-shore variations of vertical erosion depth under irregular wave action. The product of the erosion rate and the turf resistance force is related to the wave energy dissipation rates due to wave breaking and bottom friction. The turf resistance force is expressed using the turf thickness and the surface and underneath resistance parameters. The empirical parameters are calibrated using available data. The calibrated model has been shown to reproduce the relation between the limiting velocity and steady flow duration, the erosion rate on a seaward grassed slope, and the eroded profile evolution of a seaward clay slope. The levee erosion model is also compared with field tests for erosion on the landward slope caused by wave overtopping. It is found to be difficult to reproduce the observed erosion initiation and progression because of the wide variations of the grass cover and clay resistance. The turf resistance parameters will need to be calibrated for specific levees.


INTRODUCTION
Levees (dikes) have been constructed to protect some coastal areas against flooding by the combined action of storm surge and wind waves.A number of large-scale laboratory experiments were conducted to quantify the resilience of seaward levee slopes (e.g., Klein Breteler et al. 2012).Erosion of landward levee slopes has been investigated on actual levees using the Wave Overtopping Simulator (e.g., van der Meer et al. 2010) which mimics irregular wave overtopping events for the specified wave overtopping rate.The separate studies for the seaward and landward levee slopes have not been synthesized probably because of the different hydrodynamics involved on the seaward and landward slopes.Erosion on the seaward slope is caused by irregular breaking waves.Erosion on the landward slope is caused by intermittent wave overtopping of the levee whose crest elevation is normally designed to be higher than the design storm tide to avoid overflow.The similarity of the hydrodynamics involved in levee erosion and dune erosion is utilized herein to develop a numerical model for predicting the cross-shore and temporal variations of the erosion depth on the entire levee.
A levee erosion model is proposed by generalizing the work-based formula by Dean et al. (2010).The rate of work for levee erosion is expressed as the product of the vertical erosion rate and the resistance force of the grass and cohesive sediment.This rate of work is related to the energy dissipation rates due to wave breaking and bottom friction.The resistance force is assumed to decrease downward linearly in the turf zone with roots and be represented by the turf thickness and the surface and underneath resistance forces.The three parameters for the grass and soil characteristics are calibrated using available data.

LEVEE EROSION MODEL
The cross-shore model CSHORE for levee erosion (Kobayashi and Weitzner 2014) is explained for an emerged levee as depicted in Figure 1.Alongshore uniformity and normally incident waves are assumed.The cross-shore coordinate x is positive onshore with 0 x = at the toe of the levee.The vertical coordinate z is positive upward with 0 z = at the datum.The still water level (SWL) is located at the elevation of z S = with S = storm tide.For comparisons with laboratory data with constant S , the datum is taken at SWL and 0 S = .The hydrodynamic model in CSHORE predicts the mean (η and U ) and standard deviation ( η σ and U σ ) of the free surface elevation η and depth- averaged cross-shore velocity U where the overbar denotes time averaging.The mean water level The erosion model attempts to predict the temporal and cross-shore variations of the vertical erosion depth E defined as where the initial levee profile ( ) b z x at 0 t = is input to the numerical model.

R
ρ with ρ = fluid density and R = resistance force divided by ρ with its unit being m 2 /s 2 .The rate of erosion work is expressed as the product of the resistance force and the vertical erosion rate with 0 at 0 where D = energy dissipation rate per unit horizontal area corresponding to the rate of erosion work.Subsequently, D is related to the rate of fluid energy dissipation.The vertical distribution of the turf resistance depends on the detailed root and soil structures.The simple distribution of R shown in Figure 2 is adopted and R is expressed as ( ) where d = turf thickness; and 0 R and d R = turf surface and underneath resistance parameters, respectively.The substrate resistance is assumed to be invariant vertically and represented by d R .
The grass cover is characterized by the three parameters 0 , d R and d R which are allowed to vary spatially.Equation 2 can be integrated analytically for R given by Equations 3 and 4 for arbitrary D .
In the wet zone where water is present always, the dissipation rate D is assumed to be given by where the efficiency d e is chosen so that the values of D given by Equations 5 and 6 are the same at SWL x x = for the smooth transition between the two zones.The energy dissipation rate f D due to bottom friction is expressed analytically, assuming the exponential probability distribution of the instantaneous water depth (Kobayashi and Weitzner 2014).The computation procedure is as follows.The initial levee profile and the cross-shore variations of 0 , d R and d R in Equations 3 and 4 are specified at time 0 t = before the time-marching computation.
The time series of the still water level S and the spectral significant wave height mo H and the peak period p T at the seaward boundary 0 x = are also specified as input.The hydrodynamic model in CSHORE is used to compute the dissipation rate D given by Equations 5 and 6.The time step for the time-marching computation for large-scale and prototype levees is selected so that the computed erosion depth E does not increase more than 5 cm.The levee profile b z at the next time level is obtained using Equation 1.This time-marching computation is repeated until the end of a levee erosion test.The computation time is of the order of 10 -3 of the test duration.The computational efficiency is essential for the development of the levee erosion model which is empirical and requires the calibration of several parameters.

TURF FAILURE UNDER STEADY FLOW
The present levee erosion model is applicable to predict turf failure under steady flow on a gentle slope.The failure of a turf may be assumed to occur when the erosion depth E becomes equal to the turf thickness d which is taken as d = 0.1 m on the basis of the vertical decrease of the number of roots per unit surface area plotted by Hoffmans et al. (2008).Hewlett et al. (1987) plotted the limiting steady flow velocity as a function of its duration for turf failure.Their curves for the good, average, and poor grass covers are used to obtain ( )  Smith et al. (1994) excavated 16 blocks from a section of an existing sea levee in the Netherlands.The length, width, and thickness of each block were 2.5 m, 2.5 m, and 1.0 m, respectively.The blocks were transported to reconstruct the levee section in a large wave flume.The grass cover and clay substrate layer of 1-m thickness was placed on the seaward and landward slopes of 1/4 and 1/2.5, respectively.The grass cover was inspected in detail and found to be good.The width of the concrete crest was 2 m and the crest height was 7 m above the flume horizontal bottom.The levee erosion model is compared with the erosion tests 6 and 7.The water depth at the levee toe was 4.8 and 3.5 m for tests 6 and 7, respectively.The significant wave height  For test 7 with s H = 0.75 m, the grass cover inspection and profile survey were performed every 4 hours.The computed maximum erosion depth at t = 20 h is less than 3 cm and occurs slightly below SWL ( SWL X = 14 m) as shown in Figure 4.The measured maximum erosion depth was about 2 cm at t = 20 h in the zone of x = 12.2 -13.1 m.The landward limit of computed wave uprush is near x = 21 m well below the crest ( x = 28 -30 m).occurred 1 m below SWL.The diameter and depth of the hole were 0.75 m and 0.12 m at t = 9 h, respectively, and increased to 1.0 m and 0.15 m at t = 11 hr.No hole was observed in test 7.The present levee erosion model based on the assumption of alongshore uniformity cannot predict the development of a three-dimensional hole.Erosion of a strip of a poor grass cover is analyzed instead.The strip of cross-shore width of 1.2 m is assumed to be located in the zone of 1.0 to 1.3 m below SWL.The surface resistance parameter 0 R is reduced to 0 R = 200 m 2 /s 2 for the poor grass cover from 0 R = 1,000 m 2 /s 2 for the good grass cover.Figure 5 shows the computed cross-shore variations of the erosion depth at t = 1, 2, … , 11 h for test 6 H where the letter H indicates a two-dimensional hole.
The strip of the poor grass cover on the levee section in Figure 5 corresponds to the 1.2-m wide zone between the good grass cover indicated by short vertical lines.The erosion depth E for the good grass cover increases slowly with time t and is about 4 cm at t = 11 hr.The computed value of E for the poor grass cover becomes larger than the grass cover thickness of d = 10 cm at t = 9 hr and increases rapidly during t = 9 -11 h.The computed erosion depth of E = 40 cm at t = 11 h is too large partly because the computed depth-averaged velocity U does not represent the reduced velocity inside the deep hole.The present levee erosion model does not predict the increase of the hole width after the hole development, probably because lateral erosion underneath the grass cover is not included in the model. .The wave conditions starting from 0 t = were varied in six steps at t = 0.5, 1.31, 2.31, 3.55, 5.23, and 5.98 h (end of the last step).The spectral significant wave height mo H was 1.12, 1.17, 1.51, 1.56, 1.58, and 1.57 m in the first to sixth steps.The corresponding peak period p T was 4. 97, 5.01, 5.73, 5.77, 5.72, and 5.98 s, respectively.The eroded clay profile was measured at the end of each step.  .31,2.31, 3.55, 5.23, and 5.98 h.The downward erosion is predicted well at t = 0.5 and 1.31 h but overpredicted near x = 8.7 m at t = 2.31 to 5.98 h where the boundary between the boulder clay and compacted clay was located at x = 8.7 m.The overprediction may be related partly to the deposition of the eroded clay in the zone of x < 8.7 m where the eroded clay is assumed to be transported seaward of the levee toe at 0 x = in this levee erosion model.The computed erosion depth E is proportional to 1 d R − and can be decreased by reducing the breaking wave efficiency B e = 0.0002 in Equation 5.The adopted values of d R and B e are based on a number of computations made using the different combinations of d R and B e .The horizontal erosion of the eroded profile above SWL is predicted well at t = 0.5 to 3.55 h but underpredicted at t = 5.23 and 5.98 h.The boundary between the boulder clay and compacted clay was located at x = 20.7 m and the measured profile close to this boundary was not presented by Wolters et al. (2008).Steendam et al. (2010) performed a number of tests on real levees to examine the behavior of the grass cover on the landward slope.These tests were conducted on the good grass cover.The numerical model is compared with test V1 because the cross section of the landward slope and crest of the Vecht levee for test V1 was presented by Steendam et al. (2010).The Wave Overtopping Simulator (van der Meer et al. 2010) simulates the overtopping wave volumes at the crest of an actual levee.The overtopping volumes of irregular waves were calculated for the specified significant wave height s H and peak period p T at the toe of a seaward slope of 1/4.The cross section of the levee in Figure 1 is based on that of test V1.The crest height of the levee above the datum 0 z = was 3.7 m and the crest width was 3m.The toe of the seaward 1/4 slope is assumed to be located at  For s H = 1 m, o q = 1 -90 ℓ /s/m for S = 1.8 -3.4 m where S = 3.7 m corresponds to the levee crest elevation.The rate o q increases more rapidly with the increase of S for s H = The landward slope of test V1 is depicted in the bottom panel of Figure 9.A maintenance road was located in the middle of the slope.The grassed zones above and below the road are indicated by short vertical lines in Figure 9.The road consisted of bricks that allowed grass growth.The landward slope was subjected to the sequence of o q = 1, 10, 30 and 50 ℓ /s/m based on s H =2 m.The duration of each value of o q was 6 hours.Initial damage developed at the seaward edge of the maintenance road during o q = 1 ℓ /s/m.The initial damage progressed slowly during o q = 10 ℓ /s/m and evolved into two substantial holes undermining the road during o q = 30 ℓ /s/m.At the beginning of o q = 50 ℓ /s/m, some bricks were dislodged and some of the grass cover was eroded from the toe and berm in the zone of x = 48 -50 m in Figure 9.The numerical model cannot simulate the observed damage progression in test V1 partly because local flow disturbance and damage initiation are not taken into account.Figure 9 shows the computed erosion depth E on the landward slope at t = 1, 2, … , 6 h for o q = 50 ℓ /s/m where 0 E = at 0 t = .The maintenance road is crudely represented as a strip of a poor grass cover with 0 R = 200 m 2 /s 2 where 0 R = 1,000 m 2 /s 2 for the good grass cover.The computed E of the poor grass cover at t = 1 h is almost continuous with that of the good grass cover at t = 5 h because E is approximately proportional to 1 0 R − and t for the small erosion of 2 mm or less.The numerical model predicts the downward increase of E on the inclined slope but the predicted value of E is too small.The sensitivity of E to o q in the range of o q = 1 -90 ℓ /s/m is examined by computing the cross-shore variation of E on the landward slope of the same profile but covered with only the good grass cover.The maximum value of E occurs near the toe of the landward slope.The maximum erosion depth max E increases with the increase of t and o q but max E is about 1 mm at t = 6 h for o q = 90 ℓ /s/m.

EROSION OF LANDWARD GRASSED SLOPE
For the poor grass cover, max E is about 5 mm at t = 6 h for o q = 90 ℓ /s/m as shown in Figure 10.
The computed max E at t = 1, 2, … , 6 h as a function of o q for the landward clay slope with 0 10 d R R = = m 2 /s 2 is about 10 cm at t = 6 h for o q = 90 ℓ /s/m.The surface resistance parameter 0 R determines the degree of erosion on the landward slope.The wave overtopping rate o q and duration t are also important but more predictable than the uncertain parameter 0 R .The computed difference between s H = 2 m and 1 m is found to be small and o q may be used to represent the erosion work by overtopping waves.The computed results presented above are discussed in light of other available data.Thornton et al. (2011) examined resiliency of different landward slope surfaces using a large-scale wave overtopping test facility.The levee geometry consisted of a downward 1/3 slope with a height of 2.7 m that transitioned to a 3.6-m long berm on a 1/25 slope.This geometry is similar to that shown in Figure 9. Bermuda grass in planter trays was cultivated and well maintained for 6 months prior to testing.This grass cover with dense roots, ample thatching, and few imperfections suffered little damage under the 24-hour incremental increase of o q up to 370 ℓ /s/m.The Bahia grass cover suffered little damage under the 17-hour incremental increase of o q up to 279 ℓ /s/m.The grass cover model in Figure 2 with 0 R = 200 -1,000 m 2 /s 2 , d = 0.1 m, and d R = 10 m 2 /s 2 appears to correspond to the nearly perfect grass cover with few imperfections.The previously tested Bermuda grass went dormant during winter.The dormant Bermuda grass was tested in sequence of o q = 186,232 and 186 ℓ /s/m with a 1-hour interval.Minor erosion evolved into a 2.7-m long trench with a cross-flume width of 15-30 cm and a depth of 5-15 cm.This trench development is somewhat similar to the hole development discussed in relation to Figures 5 and 6.Thornton et al. (2011) also tested a bare clay surface.Significant soil loss occurred in one hour under o q = 9.3 ℓ /s/m.The bare clay test was continued under o q = 18.6 ℓ /s/m but suspended after 20 min because the clay slope failed catastrophically.Steendam et al. (2012) conducted wave overtopping tests on a real levee with a poorly-maintained grass cover in contrast to the good grass cover tests by Steendam et al. (2010).The bad grass cover with holes made by moles had no resistance against overtopping waves.They concluded that a bad grass cover should be considered as an unprotected clay layer.The wave overtopping tests on real levees indicate the wide variations of grass covers.The simple grass cover model in Figure 2 may become more realistic if the spatial variations of 0 , d R R and d are known because damage tends to initiate from weak spots.

Figure 2 ..
Figure 2. Resistance force parameter R as a function of vertical erosion depth E.

sH
and the peak period p T were 1.4 m and 4.7 s for test 6, respectively.For test 7, test duration was 11 h for test 6 and 20 h for test 7.For the following computations, the spectral significant wave height mo H is assumed to be the same as s H and given by 4 for the concrete sections of the constructed levee.These values should be regarded as order-of-magnitude estimates.For test 6, the grass cover inspection and profile survey were performed every hour.The erosion depth E is computed and plotted in Figure3as a function of x at t = 1, 2, …, 11 h.The concrete sections correspond to the zones of x = 0 -8 m and x = 28 -30 m.The measured maximum erosion depth for test 6 was about 4 cm at t = 11 h in the zone of x = 15.8 -17.5 m.The computed maximum erosion depth occurs closer to the still water shoreline located at SWL x = 19.2 m.The computed erosion depth increases almost linearly with time t and is negligible on the landward slope ( x > 30 m).

Figure 3 .H
Figure 3. Computed erosion depth increase with time t for test 6 with s H = 1.4 m

Figure 4 .
Figure 4. Computed erosion depth increase with time t for test 7 with s H = 0.75 m

Figure 5 .Figure 6 .
Figure 5. Computed erosion depth increase with time t for test 6 with poor grass patch and concrete sections ( x = 0 -8 m and x = 28 -30 m)Figure6shows the computed cross-shore variations of the erosion depth at t = 4, 8, … , 20 h for test 7H with the 1.2-m wide strip of the poor grass cover.The erosion depth E increases with time but the hole development defined as E d > = 10 cm through the poor grass cover does not occur because 3 E < cm at t = 20 h.Figures5 and 6indicate that the turf resistance is as important as the wave forcing.

Figure 7 .
Figure 7. Measured and computed profile evolutions of seaward clay slope Figure 7 compares the measured and computed profiles at t = 0.5, 1.31, 2.31, 3.55, 5.23, and 5.98 h.The downward erosion is predicted well at t = 0.5 and 1.31 h but overpredicted near x = 8.7 m at t = 2.31 to 5.98 h where the boundary between the boulder clay and compacted clay was located at x = 8.7 m.The overprediction may be related partly to the deposition of the eroded clay in the zone of x < 8.7 m where the eroded clay is assumed to be transported seaward of the levee toe at 0 x = in this levee erosion model.The computed erosion depth E is proportional to 1 break seaward of the levee toe located at 0 x = .The still water level S is chosen to produce the wave overtopping rate o q on the crest specified for test V1.The hydrodynamic model in CSHORE for the fixed bottom is used to obtain the relation between S and o q .

2 sH
= m, o q = 1 -90 ℓ /s/m for S = (−1.2) -2.0 m.The computed relation for s H = 1 m and p T = 4.0 s is also presented because van der Meer et al. (2010) examined the difference of individual wave overtopping volumes caused by the wave height difference for the same value of o q .

Figure 9 .
Figure 9. Computed erosion depth increase with time t for test VI with poor grass patch

Figure 10 .
Figure 10.Computed maximum erosion depth E max at slope transition of poor grass cover as a function of q o and t for a poor grass cover.These estimates may not be accurate but indicate the large variation of the turf and substrate resistance against erosion.
to 1200 m 2 /s 2 .The turf surface resistance parameter 0 R is expected to be much larger than the underneath resistance parameter d R .The erosion experiment of a seaward clay slope by Wolters et al. (2008) is used to calibrate