Instability of Grass caused by Wave Overtopping

This paper presents a method to deal with the instability of turf in a grass cover that could be considered as an engineering approach, in which both turbulence and two strength models (root model and turf-element model) are discussed in a heuristic way. Turbulence is discussed because of its essential role in erosion. The root model detenllines the shear strength of rooted soil in the vertical direction. The turf-element model based on a force balance predicts not only the instability of the grass cover but also the initiation of motion of grass. The results of the turf-element model are compared with experimental data of some prototype experiments at Dutch sea dikes. Introduction Grass prevents erosion and is an effective control measure. This fonn of protection has long been used for agricultural drainage channels and on slopes of dikes. In addition, grass-lined emergency spillways are being used as an alternative to costly concrete lining. Following Samani and Kouwen (2002), five methods to evaluate the stability of grass-lined channels have been proposed: maximum pennissible velocity, maximum depth, equivalent stone size, pennissible tractive force and pennissible deflection. All these methods are based on hydraulics, and are probably good enough for steady, unifonn flow for the evaluation of Toughness coefficients (e.g. Temple 1999). De Baets et al. (2005) investigated the impact of root density and root length density of grass on the erodibility of root-penneated saturated top soi ls and added this aspect to the available methods based on hydraulics. As hydraulics (load) and geotechnical and vegetation (strength) aspects of the grass cover interact, this requires more insight. Dutch river dikes usually have clay layers covered with grass on the crest and on both the inner and outer slopes. Sea dikes with hard revetments in the wave impact zone of the seaward slope also have a grass cover on the crest and the inner slope. The grass cover includes grassland vegetation rooted in soil with two layers: the topsoil and the subsoil (Fig. 1, Muijs 1999). The porous turf layer consists of organic matter and sandy clay with a high root density and is elastic in moist conditions. The root structure connects the small clay aggregates and prevents them from being washed out, whereas the underlying clay substrate is stiffer or plastic and less permeable. On the inner slope, the sward contributes to the strength of the grass cover by covering the clay aggregates during overtopping flows, although its contribution to the actual grass strength may be modest. Near the surface, the grass strength is dominated by the root reinforcement, whereas deeper below the surface, where the number of roots decreases rapidly, the soil cohesion and the submerged weight of the soil dominate its strength. Turbulence For a grass-clay aggregate, relatively large forces are required to break up the aggregates within the bed, while relatively small forces may suffice to transport pure sand and small clay aggregates. Therefore, at the onset of dislocation, a grass cover or other types of vegetation on dike slopes will experience considerable turbulence, especially at


Introduction
Grass prevents erosion and is an effective control measure. This fonn of protection has long been used for agricultural drainage channels and on slopes of dikes. In addition, grass-lined emergency spillways are being used as an alternative to costly concrete lining. Following Samani and Kouwen (2002), five methods to evaluate the stability of grass-lined channels have been proposed: maximum pennissible velocity, maximum depth, equivalent stone size, pennissible tractive force and pennissible deflection. All these methods are based on hydraulics, and are probably good enough for steady, unifonn flow for the evaluation of Toughness coefficients (e.g. Temple 1999). De Baets et al. (2005) investigated the impact of root density and root length density of grass on the erodibility of root-penneated saturated top soi ls and added this aspect to the available methods based on hydraulics. As hydraulics (load) and geotechnical and vegetation (strength) aspects of the grass cover interact, this requires more insight.
Dutch river dikes usually have clay layers covered with grass on the crest and on both the inner and outer slopes. Sea dikes with hard revetments in the wave impact zone of the seaward slope also have a grass cover on the crest and the inner slope. The grass cover includes grassland vegetation rooted in soil with two layers: the topsoil and the subsoil (Fig. 1, Muijs 1999). The porous turf layer consists of organic matter and sandy clay with a high root density and is elastic in moist conditions. The root structure connects the small clay aggregates and prevents them from being washed out, whereas the underlying clay substrate is stiffer or plastic and less permeable. On the inner slope, the sward contributes to the strength of the grass cover by covering the clay aggregates during overtopping flows, although its contribution to the actual grass strength may be modest. Near the surface, the grass strength is dominated by the root reinforcement, whereas deeper below the surface, where the number of roots decreases rapidly, the soil cohesion and the submerged weight of the soil dominate its strength.

Turbulence
For a grass-clay aggregate, relatively large forces are required to break up the aggregates within the bed, while relatively small forces may suffice to transport pure sand and small clay aggregates. Therefore, at the onset of dislocation, a grass cover or other types of vegetation on dike slopes will experience considerable turbulence, especially at  (Muijs 1999) steep slopes. Usually the bed roughness is characterised by the Chezy coefficient (C) as In Uo = C R " S b (I) in which R" is the hydraulic radius that may be taken equal to the flow depth (h) for overtopping flow at dikes, Sb is the dike slope and U o is the depth-averaged flow velocity. The depth-averaged relative turbulence intensity (ro) is defined as (Hoffmans 2010) Fa in which ko is the depth-averaged turbulence energy and 0;" 0;, and 0"" . are standard deviations of the fluctuating velocities in the streamwise, transverse and normal (z') directions. For uniform flow and hydraulic smooth conditions, Nezu (1977) found 0;,( z') = a"u, exp (-z'l h) , o;.(z') = y"o;,(z') and o;.,(z') = y" .o;,(z') where u. is the bed shear velocity, a" = 1.92, Yv = 0.71 and y" , = 0.55. For channels with smooth and rough bed, the turbulence distributions are reasonable the same (Graf 1998 Usually in k-s models the near bed turbulent energy is defined as k" = (U.)2/(C Il )0.5 in which c ll = 0.09 (e.g. Launder and Spalding 1972). Assuming that f,:"(z') is linear distributed ko = 1. 65(u.)2 Since k(z') is an exponential distribution the value of 1.65 is less, so an appropriate value of £lo for all types of uniform flow is £lo = 1.2. Using the Chezy equation, ro is for uniform flow u. .Ji where g is the acceleration due to gravity. Prototype experiments on different sea dikes using a wave-overtopping simulator [typical wave characteristics are Hs = 2 m (significant wave height) and T... = 5.7 s (peak wave period) for a varying dike crest freeboard] showed that at the inner slope of sea dikes, the steepness of which is about lV:3H, the maximum depth-averaged flow velocities (Um ) per overtopping event reach values up to 8 mls (corresponding wave volume per unit width is 5,500 tim) (Fig. 2). The flow depths vary from 2 em to 40 em resulting in high turbulence intensities (0.1 < ro < 0.3). When grass-clay aggregates are uplifted, pm represents the maximum lowering of the local pressure caused by eddies. According to Nezu (1977) O"p. J To depends on the Froude number (Fr). If the turbulent structure is dominated by the Reynolds shear stresses, O"p.bl TO is about 3 for Fr < 0.5 and Fr > 2. At the transition from sub to supercritical flow, O"p.bl TO reaches an evident maximum, thus a, has no universal value. The mean bed shear stress, defined as TO = p(u. )", using Eq. 3 reads

Root model
The Mohr-Coulomb equation describes the shear failure of soil in terms of shear stresses as well as normal stresses and can be written as (e.g. Lambe and Whitman 1969) where Ce is the effective soil cohesion, pw is the soil pore water pressure, er is the soil normal stress, T: , is the soil shear strength and fPe is the effective internal friction angle. The soil cohesion (c) is the result of cementation, weak electrical bonding of clays and organic colloids and capillary tension, whereas ¢e represents the frictional interaction of individual particles and the interlocking of particles. The magnitude of er depends on the weight of the soil and the soil moisture, whereas the buoyancy generated by pw reduces the normal stress. For dry soil when p". = 0 both C e = C and ¢e equals the internal friction angle (¢).
Typically the strength of roots is modelled by an artificial additional cohesion (C,. ). In a root permeated soil Eq. 6 can be modified to include c,.
Most attempts to determine the effects of root reinforcement by grassland vegetation have used root-cohesion estimates according to the root equation of Wu et al. (1979), which requires the root tensile strength (0;.) and the mean root diameter (d,.). Where a root crosses a shear zone, 0;. can be resolved into components parallel (Hr) and perpendicular (V,.) to the shear zone (Fig. 3). Thus, c,. is where AriA \ is the root area ratio also known as RAR and B is the angle of shear rotation. Though little is known about B, from field observations of conifers, Wu et al. (1979) suggested a range of 45° to 70"' Since Eq. 8 is insensitive to changes in B (it is close to 1.2 for a large range of (/), c,. may be rewritten as c,. '" 1.20;.ArIA\.
Although horizontal roots may have some impacts on the threshold condition for vertical motion, here the vertical grass strength (Vg,.ass ) and the critical vertical grass strength (Vgrass.c ) are approximated by where erg,.ass and ergrass.c represent the normal grass strength and the critical normal grass strength. Hence, Vgmss.c does not include the critical friction strength of roots on clay (cgm.".c ). If cgrass.c « Vgmss. c all roots are well-anchored and they will all break simultaneously, so they will not be pulled out before breaking owing to a lack of anchoring. However, prototype and laboratory experiments have demonstrated that the roots do not all break simultaneously (e.g. Pollen and Simon 2003) so if deformations are not included Eqs. 8 and 9 overestimate the critical values of c,. and v"rass .
The critical tensile strength depends strongly on the type and the quality of the grass. Sprangers ( 1999) measured grass parameters such as the root length and the dry root mass densities from 24 Dutch dike sites. He found that A,.IAI of dike grassland decreases exponentially with depth and that about 50% of the roots can be found in the top 6 em of the turf, while about 75% of the roots were located within the top 20 cm. Above Arel (= 10 em) about 73 of all roots are to be found. The critical vertical grass strength, which is at maximum near the surface and decreases with depth, is here estimated by Based on Spranger's work, the Dutch guidelines for assessing primary dikes (VTV 2006) distinguish 4 different qualities for the grass cover in which the quality of the grass is strongly correlated to the number (No) of roots. The mean root diameter of Dutch grasses is 0.113 mm (Paulissen 2009), which is significantly finer than the range of d r tested by the Chinese researchers Chengh et al. (2003). If the following assumptions are made: dr = 0.113 mm, O;·.c = 20'10 6 N/m2, /1= 45°, ¢= 30° and Cgrass.c is neglected with respect to Vgrass.c then the root properties can be determined (Table I).  Turf-element model If a grass-clay aggregate with the dimensions of a cube is considered, the following forces acting on this cube can be distinguished: the load due to the lift force caused by pressure fluctuations perpendicular to the grass cover, and the strength, i.e., the submerged weight of the soil, and the forces caused by shear, cohesion and the roots. Figure 4 shows a cube f//:= = t at a horizontal plane where -ex, f v and f= are length scales in the x, y and z direction respectively and -e is the representative aggregate scale. A grass-clay aggregate is unstable if the load is larger than the strength, thus z SCOUR AND EROSION which act on the four sidewalls, depends on the critical rupture strength of clay (Celay.c) and the critical friction strength of roots (Cgrass.c ) averaged over f=. The sum of the cohesion forces (Fc) is approximated by Fc = (1n)(Cc'ay.c + Cgrass.c)2(fx + f,,)f= and the total force at the bottom-element is The assumption of Eq. II is applicable if pm at the top of the grass-clay aggregate significantly decreases with depth or if the penetration length (fp) is f"lf « I. According to De Groot et al. (1996) Klar (2005) found that the turbulence energy at 2 em below the bed level is about 10% of k" . Using C". = fv = ( = -z, Eq. II can be rewritten as

V
+ (Cday .c + Cgm".c ) -( ( 'Ia,'.,. + g'a".,' (z)) (12) where Vsoil(Z) is the vertical soil strength as function of z. When describing incipient motion, horizontal forces are usually considered. The critical condition for moving grass-clay aggregates is reached if TO equals the critical mean bed shear stress (Tc) . If there is neither clay nor grass and neglecting the shear forces, Tc of loosely packed materials is (z = -d) In turbulent flow, the critical Shields parameter ('I' c) varies from 0.03 to 0.06 for coarse sand and gravel, whereas for small Reynolds numbers up to fully laminar flow 'l'c increases from 0.03 to 0.2 . Hence, the first term in Eq. 12 confirms the earlier research of Shields. Substituting Z = -Are;; Eq. 12 can be rewritten as (14) Compacted clay has a high resistance against erosion and a low hydraulic conductivity (k) provided the clay is kept in sufficiently moist condition. However, k of a grass-clay aggregate is s ĩ n i f i c a n t l y higher owing to atmosphere, flora and fauna and varies from 10-5 rnls to 10-rnls owing to cracks and other disturbances (Kruse 1998). Hence, near the surface the submerged weight and the rupture strength of clay are neglible with respect to the friction strength of the roots. If (Ps -p)( I + 2tan¢)gA re ! « C elay.c« C g,ass.c then Eq. 14 reduces to ( 15) Per sidewall the strength of roots averaged over Are! is where ro.c is the critical depth-averaged relative turbulence intensity. When the slope steepness increases or the bed becomes rougher the bed turbulence increases.

Examples of some prototype experiments
At several test locations in the Netherlands (Delfzijl, Boonweg, St. Philipsland, Kattendijke and Afsluitdijk) both the grass and dike core were inspected to determine the grass cover stability and the resistance of the grass to erosion. Different strength parameters were measured, such as the number of roots, soil cohesion, internal friction angle and liquidity index (Van Hoven et al. 2010). The wave-overtopping simulator (Van der Meer, 2007) was used to test the erosion resistance of the inner slope. Cis per m after testing no significant damage to the grass cover was observed.
The number of roots near the surface varied from 20 to 50 according to the Dutch standard area (VTV 2006). For turbulent flow the boundaries between occasional particle movement at some locations and general transport are 0.03 < 'Pc < 0.06 .

Conclusions and recommendations
The critical normal grass strength as given in Table I is predicted to vary from 5 kN/m 2 to IS kN/m 2 and represents the range of different qualities of grass. The critical vertical grass strength is influenced by No, d r (= 0.113 mm), O;·.c (= 20'10 6 N /m2) , ,1., e! (= 10 cm) and B (= 45°). Since these parameters can easily be determined to a reasonable degree of precision, the computed value of " r a s s . c should al so be accurate in this respect. However, Vgrass.c does not include the heterogeneity of the grass cover, e.g., the decrease of d r with depth and the standard deviations of No, dr , O;·. c, ,1.re! and B respectively.
Although the modelling is based on physical considerations and should be representative for the strength of the grass cover, the model is based on static equilibrium conditions (Cgrass.c « Vgra.,s.c)' Therefore, it is necessary to examine the deformation of grass as function of the dynamic load, i.e. the relation between cgras.,.c, Vgrass.c and the elastic modulus for grass, in greater detail and to validate the incipient motion of grass by carrying out sufficient experiments. Moreover, it is recommended to analyse the penetration of the pressure fluctuations close to bed.
The turf-element model should be considered as a conceptual approach, which incorporates a good description of the physics. The model predicts the initiation of motion of turf as an element with a length scale of 10 cm. To update the VTV in 2011 it is necessary to examine the deformation of grass as function of the load and to understand the physical significance of the parameters used in the turf model for different types of grass and clay.