Breaching of sea dikes initiated from the seaside by breaking wave impacts [Elektronische Ressource] / by Grzegorz Stanczak

Informations

• Publié par: technische_universitat_carolo-wilhelmina_zu_braunschweig
• Publié le: 01 janvier 2008
• Nombre de lectures: 19
• Langue: English
• Poids de l'ouvrage: 9 Mo

Extrait

Breaching of sea dikes initiated from the seaside by breaking wave impacts Dissertation submitted to and approved by the Faculty of Architecture, Civil Engineering and Environmental Sciences University of Braunschweig – Institute of Technology

and the

Faculty of Engineering University of Florence in candidacy for the degree of a DoktorIngenieur (Dr.Ing.) / *) Dottore di Ricerca in Risk Management on the Built Environment by Grzegorz Stanczak Born 24.03.1980 from Szczecin, Poland Submitted on 17 March 2008

Oral examination on

29 May 2008

Professorial advisors Prof. Hocine Oumeraci  Prof. Pierluigi Aminti 2008 *) Either the German or the Italian form of the title may be used.

The researches described in the following thesis were performed during my stay on two universities: Technische Universität Braunschweig (Germany) and Universita degli Studi di Firenze (Italy) in the frames of the International Graduate College GRK802.

First of all, I would like to thank my two tutors: prof. Hocine Oumeraci and prof. Pierluigi Aminti for their continuous support and inspiration. I am very grateful to my coworkers in the frames of the FloodSite project: Andreas Kortenhaus, Claudia D’Eliso and Peter Geisen hainer for the numerous discussions we had and for the time we spent together on conferences and meetings. I would also like to thank to the entire LWI staff for their continuous help: Gabi Fournier, Agnieszka Strusinska, Juan Recio, Rainer Kvapil and Markus Brühl as well as to the students who performed most of the laboratory experiments: Benjamin Rohloff, Christian Klein and Semeidi Husrin.

The support of the German Research Foundation in the frames of the GRK802 represented by prof. Udo Peil and prof. Claudio Borri as well as of the European Community in the frames of the FloodSite project is gratefully acknowledged.

Mojejz ˙ onieKamiliwpodzi˛ekowaniuzanieustannewsparcie,inspiracje˛orazmotywacj˛e dedykuje˛ niniejsza prace˛.

Contents 1 Introduction 1.1 Problem definition and motivations . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

2.1 2.2 2.3 2.4 2.5

Dike breaching processes and modeling: state of the art and study specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Causes of breach initiation and loading case considered . . . . . . . . . . 2.1.1 Morphological and hydraulic boundary conditions . . . . . . . . 2.1.2 Breaking wave impact and flow of wave runup and rundown . . 2.1.3 Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Implications for the present study . . . . . . . . . . . . . . . . . Erosion and mass instability processes and models . . . . . . . . . . . . 2.2.1 Breach location along the dike route . . . . . . . . . . . . . . . . 2.2.2 Erosion of the grass cover . . . . . . . . . . . . . . . . . . . . . 2.2.3 Erosion of the clay cover . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Erosion of the dike core . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Washout of the sand core . . . . . . . . . . . . . . . . . . . . . 2.2.6 Implications for the present study . . . . . . . . . . . . . . . . . Sediment transport processes and models . . . . . . . . . . . . . . . . . 2.3.1 Sediment transport processes . . . . . . . . . . . . . . . . . . . . 2.3.2 Sediment transport models . . . . . . . . . . . . . . . . . . . . . 2.3.3 Implications for the present study . . . . . . . . . . . . . . . . . Available breach models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Limitations of available breach models . . . . . . . . . . . . . . 2.4.2 Implications for the present study . . . . . . . . . . . . . . . . . Specification of objectives and methodology . . . . . . . . . . . . . . . . 2.5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Methodology and procedure . . . . . . . . . . . . . . . . . . . .

Preliminary model  development and implementation 3.1 Mathematical formulation . . . . . . . . . . . . . 3.2 Input module . . . . . . . . . . . . . . . . . . . . 3.2.1 Hydrodynamic boundary conditions . . . . 3.2.2 Dike geometry . . . . . . . . . . . . . . . 3.3 Hydrodynamic module . . . . . . . . . . . . . . . 3.3.1 Breaking wave impact on the outer slope . 3.3.2 Flow through the breach channel . . . . . . 3.3.3 Wave overtopping . . . . . . . . . . . . . 3.3.4 Combined wave overtopping and overflow . 3.3.5 Overflow . . . . . . . . . . . . . . . . . . 3.3.6 Additional simplifications and assumptions 3.4 Morphodynamic module . . . . . . . . . . . . . . 3.4.1 Erosion of the grass cover . . . . . . . . . 3.4.2 Erosion of the clay cover . . . . . . . . . . 3.4.3 Frontface erosion of the sand core . . . . . 3.4.4 Breach deepening and widening . . . . . .

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34 35 35 35 35 36 37 41 42 42 43 44 44 46 47 48 50

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6

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3.5

3.6

3.4.5 Limitations of the mathematical formulation . . . . . . . . . . . . . Model implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Computational aspects, results and discussion . . . . . . . . . . . . . 3.5.2 Tentative validation of the preliminary model . . . . . . . . . . . . . Capabilities and limitations of the preliminary model . . . . . . . . . . . . .

Laboratory experiments on the erosion and breaching processes 4.1 Experiments on the surface erosion of clay cover . . . . . . . . . . 4.2 Experiments on the shear failure in cracks . . . . . . . . . . . . . . 4.2.1 Experimental verification of the model by Führböter (1966) 4.2.2 Improvement of the Führböter (1966) model . . . . . . . . 4.3 Reinforcement properties of the grass roots . . . . . . . . . . . . . 4.3.1 Laboratory experiments on the root volume ratio . . . . . . 4.3.2 Tests with grass cover  shear strength . . . . . . . . . . . . 4.3.3 Surface erosion of grassreinforced clay cover . . . . . . . . 4.4 Dike breaching tests in a small wave flume . . . . . . . . . . . . . .

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Development of the detailed model 5.1 Improvements of the hydrodynamic module . . . . . . . . . . . . . . . . . . 5.1.1 Free surface flow  simulation of breaking wave impact and flow field on the outer slope . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Simulation of the erosion of the front face of the sand core . . . . . . 5.1.3 Wave overtopping, overflow and combined overtopping and overflow 5.1.4 Water infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Improvements of the morphodynamic module . . . . . . . . . . . . . . . . . 5.2.1 Breach initiation  cover erosion . . . . . . . . . . . . . . . . . . . . 5.2.2 Undermining and mass instability of the clay cover . . . . . . . . . . 5.2.3 Front face erosion of the sand core  beach profile formation . . . . . 5.2.4 Transition phase for the sand core erosion . . . . . . . . . . . . . . . 5.3 Overall implementation of the model . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Simulation of the dike cover erosion . . . . . . . . . . . . . . . . . . 5.3.2 Sand core erosion  beach profile formation and transition phase . . . 5.3.3 Example application, results and discussion . . . . . . . . . . . . . .

Uncertainties, sensitivity and reliability analyses 6.1 Uncertainties . . . . . . . . . . . . . . . . . . . . . 6.1.1 Main uncertainties in the preliminary model . 6.1.2 Main uncertainties in the detailed model . . . 6.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . 6.2.1 Grass erosion time . . . . . . . . . . . . . . 6.2.2 Clay erosion time . . . . . . . . . . . . . . . 6.2.3 Core erosion and washout time . . . . . . . 6.3 Reliability analysis . . . . . . . . . . . . . . . . . . 6.3.1 Results of the analysis . . . . . . . . . . . .

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Summary, conclusions and recommendations 7.1 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Applicability of the results and future research . . . . . . . . . . . . . . . . .

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57 58 59 62 64

66 66 70 70 74 76 76 78 79 81

84 85

86 90 92 95 100 100 104 106 108 108 111 112 113

118 118 118 119 119 120 124 127 129 129

134 134 136

List of Figures 1.1 Riskbased approach for the design of flood defence structures (Modified from Oumeraci, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overall methodology of the present study . . . . . . . . . . . . . . . . . . .

2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8 2.9 2.10

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25

4.1 4.2 4.3 4.4

Causes of breach initiation in sea dikes . . . . . . . . . . . . . . . . . . . . . Simplified example of typical sea dike in Germany . . . . . . . . . . . . . . Structure of a grass cover  principle sketch and a picture of a crosssection . Dike breaching processes and phases . . . . . . . . . . . . . . . . . . . . . . Phases of a dike breaching to be considered in simulation . . . . . . . . . . . Impact of a plunging breaker on a crack in a dike  photoelastic investigation (Berkenbrink et al, 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wave impact approach  principle sketch . . . . . . . . . . . . . . . . . . . . Equilibrium profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selected breaching models . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structure and main components of the preliminary model . . . . . . . . . . . Flow conditions dike breaching simulation . . . . . . . . . . . . . . . . . . . Plunging breaker on a dike slope . . . . . . . . . . . . . . . . . . . . . . . . Location of the impact on a slope  definition sketch . . . . . . . . . . . . . . Impact pressure distribution on the dike slope . . . . . . . . . . . . . . . . . Incidence angle of plunging breaker on a dike slope (after Führböter, 1966) . Overflow  breach channel geometry . . . . . . . . . . . . . . . . . . . . . . Flow through dike breach  loading cases . . . . . . . . . . . . . . . . . . . Phases of dike breach simulation . . . . . . . . . . . . . . . . . . . . . . . . Laboratory tests on the erosion resistance of the grass cover (Smith et al, 1994) Wave impact approach for the sand core . . . . . . . . . . . . . . . . . . . . Definition of the end of Phase 3 (frontface erosion of the sand core) . . . . . Shape of the partial breach at the outer slope of sea dikes . . . . . . . . . . . Initial conditions for the 3D modeling y−zplane . . . . . . . . . . . . . . Finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . Breach channel evolution (modified from D’Eliso, 2007) . . . . . . . . . . .

Definition sketch for the volumeaveraged approach (modified from Tuan, 2007)

Flow chart of the preliminary model . . . . . . . . . . . . . . . . . . . . . . Geometry of the prototype dike . . . . . . . . . . . . . . . . . . . . . . . . . Progress of erosion in xdirection . . . . . . . . . . . . . . . . . . . . . . . . Outflow hydrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shape of the full breach channel x−yplane . . . . . . . . . . . . . . . . . Damage of the sea dike in Neuwerk observed in 1962 (Führböter et al, 1976) Comparison of calculated and measured damage of the dike . . . . . . . . . . Dike breach in the Ülversbüller Koog . . . . . . . . . . . . . . . . . . . . .

Wave impact simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of eroded clay vs. water layer thickness . . . . . . . . . . . . . . . . Measured volume of eroded soil vs volume calculated using Eq. 2.1 . . . . . Measured volume of eroded soil vs volume calculated using Eq. 4.1 . . . . .

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4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17

6.1 6.2 6.3 6.4 6.5

6.6 6.7 6.8

Forces inside a crack subject to an impact pressure (after Führböter,1966) . . Artificial crack in the soil sample  side view . . . . . . . . . . . . . . . . . . Artificial crack in the soil sample  top view . . . . . . . . . . . . . . . . . . Mass of water impacting the sample . . . . . . . . . . . . . . . . . . . . . . Crack development recorded after an impact  angle of shear failureα. . . . Predicted and observed shear failure  principle sketch . . . . . . . . . . . . Calculated and measured shear failure angleα. . . . . . . . . . . . . . . . . Forces acting on the block of soil . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the original and modified approach of Führböter (1966) . . . . Flexible elastic perpendicular root reinforcement (Wu et al, 1979) . . . . . . Measured root volume ratio for the samples taken from the dike . . . . . . . Planes of shear strength measurements . . . . . . . . . . . . . . . . . . . . . Measured and calculated increase of shear strength . . . . . . . . . . . . . . Measured and calculated values of the detachability parameterk d,g,p with re spect to the depth under the surface . . . . . . . . . . . . . . . . . . . . . . . Experimental setup in the small LWI wave flume . . . . . . . . . . . . . . . Progress of dike erosion and breaching observed in the small LWI flume . . .

71 72 72 72 73 73 74 75 75 76 78 78 79

80 81 83

Strategy for the detailed computational model . . . . . . . . . . . . . . . . . 85 Wave breaking simulated by COBRAS  free surface elevation . . . . . . . . 89 Distribution of impact pressures, velocity and layer thickness on the dike slope 89 Definition sketch for the calculation of the overtopping parameters (D’Eliso, 2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Flow properties at the dike within a wave period (D’Eliso, 2007) . . . . . . . 94 Combined flow  definition sketch (D’Eliso, 2007) . . . . . . . . . . . . . . . 95 Comparison of the saturated fronts calculated using Wang Q and Wang Z models 99 Erodibility of the clay as a function of water content . . . . . . . . . . . . . . 101 Transition phase between cover and core erosion . . . . . . . . . . . . . . . 105 Undermining of the clay layer during the transition phase . . . . . . . . . . . 105 Bending of the undermined clay cantilever  definition sketch . . . . . . . . . 106 Formation of beach profile and crossshore transport zones . . . . . . . . . . 106 Detailed model  complete flow chart . . . . . . . . . . . . . . . . . . . . . . 110 Detailed model  complete flow chart (continued) . . . . . . . . . . . . . . . 111 Progress of the cover erosion  comparison of the preliminary and detailed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Final breach shape  comparison of the preliminary and detailed model . . . . 116 Damage of the dike (after Forschungs und Vorarbeitenstelle Neuwerk) and comparison with the results from preliminary and detailed model . . . . . . . 117

Influence of the grass erosion coefficientE g 121on the time of grass erosion . . . Influence of the root volume ratio RVR on the grass erosion time . . . . . . . 121 Influence of the damping coefficientwon the grass erosion time . . . . . . . 121 Influence of the grass cover factorC f . . . . . . . . on the grass failure time 122 Influence of the clay percentage in the substrate soilc%on the grass erosion time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Influence of the initial infiltration frontz inf 123. . . . . on the grass erosion time Influence of the initial volumetric water contentθ i on the grass erosion time . 123 Influence of the saturated volumetric water contentθ s on the grass erosion time 123

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6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18

6.19 6.20 6.21 6.22 6.23 6.24 6.25

7.1

Influence of the clay erosion coefficientE c . . . . .on the clay erosion time

124 Influence of the clay percentage in the substrate soilc%on the clay erosion time124 125 125 126 126 126 127 127 128 128 128 130 131 131 132 132

Influence of the initial infiltration frontz inf on the clay erosion time . . . . . Influence of the initial volumetric water contentθ i on the clay erosion time . . Influence of the saturated volumetric water contentθ s on the clay erosion time Influence of the minimal depth of cracks on the clay erosion time . . . . . . . Influence of the maximal depth of crack on the clay erosion time . . . . . . . Influence of the grain size on the sand core erosion time . . . . . . . . . . . . Influence of the grain size on the peak outflow discharge . . . . . . . . . . . Influence of the breach growth coefficient on the core erosion and washout time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the breach growth coefficient on the peak outflow discharge . . . Influence of the breach growth coefficient on the final breach width . . . . . . Monte Carlo simulation results for the time of grass erosion . . . . . . . . . . Monte Carlo simulation results for the time of clay erosion . . . . . . . . . . Monte Carlo simulation results for the time of the full breaching . . . . . . . Monte Carlo simulation results for the final breach width . . . . . . . . . . . Monte Carlo simulation results for the peak outflow discharge . . . . . . . .

Overview of the undertaken work . . . . . . . . . . . . . . . . . . . . . . . .

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List of Tables 2.1 Selection of the available sediment transport models . . . . . . . . . . . . . . 2.2 Selection of the available breaching and erosion models and indications of their applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6

4.1 4.2 4.3 4.4

4.5 4.6

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

5.9 5.10 5.11 5.12 5.13

6.1

6.2 6.3

6.4

7.1 7.2

Classificaton of breaking types on sea dikes (Schüttrumpf, 2001) . . . . . . . Summary of the models implemented in the preliminary model  c.f Fig.2.9 . Input parameters for the simulation of breaching . . . . . . . . . . . . . . . . Main outcomes from the model . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the measured and calculated dike damage . . . . . . . . . . . Comparison of the observed and calculated dike breach parameters . . . . . .

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38 58 60 61 63 64

Properties of the three clay types used in the laboratory tests (IGBFT,2001) . 67 Classification of clay for dikes according to the Dutch requirements (TAW,1996) 67 Detachability coefficients for the tested types of clay . . . . . . . . . . . . . 68 Comparison of the mean and standard deviation ofα meas /α calc for the original and modified approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Measured root volume ratio for the ten samples taken from the dike. . . . . . 77 Results of the direct shear tests. . . . . . . . . . . . . . . . . . . . . . . . . . 79

Main inputs for the calculation of the wave loading on the prototype dike . . 88 Typical values of the hydraulic conductivityk s . . . . . 96(D’Eliso et al, 2007) Typical values of the pore size distribution indexN. . . (D’Eliso et al, 2007) 97 Typical values of the air entry valueh b 97( D’Eliso et al, 2007) . . . . . . . . . Typical values of the saturated volumetric water contentθ s 98(D’Eliso et al, 2007) Typical values of the initial volumetric water contentθ i 98(D’Eliso et al., 2007) Typical values of the residual volumetric water contentθ r (D’Eliso et al, 2007) 98 Coefficients describing the grass roots distribution and their effect on soil erodibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Grass cover factor and curve retardance factor for idealized scenarios . . . . . 104 Input parameters for the simulation of breaching . . . . . . . . . . . . . . . . 114 Main outputs of the simulation and comparison with the preliminary model . 114 Main outputs of the simulation and comparison with the preliminary model . 117 Main outputs of the simulation and comparison with the preliminary model . 117

Uncertainties related to the input parameters for the hydrodynamic module (partially taken from D’Eliso, 2007 and Kortenhaus, 2003) . . . . . . . . . . Input parameters for the Level I analysis . . . . . . . . . . . . . . . . . . . . Uncertainties related to the input parameters for the morphodynamic module (partially after D’Eliso, 2007 and Kortenhaus, 2003) . . . . . . . . . . . . . Main outcomes from the Monte Carlo simulation . . . . . . . . . . . . . . .

Summary of processes included in the model system Summary of processes included in the model system

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List of symbols

1 :n 1 :m a A B B B B B,ini B D c c % c r C 0 C g C f db d b d c d c,max d g d g,max d max dz D D 50 D eq e b e s E E c E g E k E s F F crack F s g h h c h b h l h s h of h p H

                                           

tangent of the outer slope [−] tangent of the inner slope [−] depth of the crack[m] 2 Breach channel crosssection [m] Breach width [m] Breach width at the location of overflow [m] Width of the initial breach channel [m] width of the dike crest [m] 2 undrained soil cohesion [N/m] clay percentage [−] 2 apparent root cohesion [N/m ] wave phase speed in deep water [m/s] wave group celerity[m/s] grass cover factor [−] Change in the breach width [m] depth at the wave breaking [m] thickness of the clay cover [m] Maximal depth of clay erosion for a single impact [m] thickness of the grass cover [m] Maximal depth of grass erosion for a single impact [m] Maximal allowable distance between point of impact andith point on the dike [m] Change in the breach depth [m] 3 wave energy dissipation per unit water volume [N m/m /sec] median grain size [mm] 3 equilibrium energy dissipation per unit water volume [/secN m/m ] Bed load efficiency coefficient [m] Suspended load efficiency coefficient [m] 2 wave energy density[N m/m] −1−1 Clay erodibility coefficient [m∙s] −1−1 Grass erodibility coefficient [m∙s] kinetic energy of impact [J] 2 stable wave energy density [N∙m/m] wave energy flux [N∙m/m∙s] force acting on the wall of the crack[N] stable wave energy flux [N∙m/m∙s] 2 the gravity acceleration [m/s] water depth or water layer thickness [m] layer thickness on the crest [m] air entry value[cm] layer thickness on the inner slope [m] layer thickness on the outer slope [m] Overflow head[m] Backwater level [m] wave height at the dike toe [m]

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H b H d H s J k d k s l crit L 0 L c L s n n tot N p p max,i P q sb q overtopping Q Q single Q st r h R R d R c S S p S Q S xx t c t cf t d t f t gf t i t sf t t T p T R T w v v c v l v s V V R

                                             

breaker height [m] dike height [m] significant wave height [m] Energy slope [−] 3 empirical detachability coefficient [m /P a] saturated hydraulic conductivity [m/s] critical length of the undermined clay cover [m] deep water wave length [m] length of the crack[m] stem length [m] −1/3 Manning roughness [m∙s] −1/3 total Manning roughness [s∙m] pore size distribution index [−] Porosity of the soil [−] Impact pressure that is not exceeded fori%of the waves [P a] Wetted perimeter of the breach channel [m] 3−1 Volumetric bed load sediment transport [m∙s∙m] −1−1 Averaged volume of overtopping [l∙m∙s] 3−1 flow discharge[m∙s] 3 Volume of sand eroded for a single impact [m] Total volumetric sediment transport [m] length of the initial breach channel [m] Hydraulic radius [m] 3 volume of soil eroded after a single impact event [m] Freeboard [m] 2 shear stress on the shear failure plane[N/m] sand percentage [−] backwater level coefficient [−] radiation stress component directed onshore [m/s] core wash out time [h] clay erosion time [h] breach development time [h] breach formation time [h] grass erosion time [h] breach initiation time [h] sand erosion time [h] total breaching time [h] peak period [s] 2 root tensile strength [N/m ] 0 water temperature [C] −1 depth averaged flow velocity [m∙s] flow velocity on the crest [m/s] flow velocity on the inner slope [m/s] flow velocity on the outer slope [m/s] 3 total volume of soil including the roots [m] 3 volume of roots in soil [m]

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