Wave competence and morphodynamics of boulder and gravel beaches

A reformatted Shields relation, {dollar}tausb{lcub}crit{rcub}=0.045(rhosb{lcub}s{rcub}-rhosb{lcub}w{rcub})gDsbsp{lcub}50{rcub}{lcub}0.6{rcub}Dsbsp{lcub}rm max{rcub}{lcub}0.4{rcub},{dollar} is used to estimate the shear stress required to remobilize the maximum-size particles from high-gradient river reaches having bed material ranging in size from gravel to boulders. The analysis uses a mobility ratio, expressed as the fluid shear stress during bank-full conditions, {dollar}tausb{lcub}b{rcub}=1/8 fsb{lcub}r{rcub}rhosb{lcub}w{rcub}Usp2,{dollar} over the reformatted Shields entrainment stress calculated from the size of the bed material. The mobility ratio, {dollar}tausb{lcub}b{rcub}/tausb{lcub}crit{rcub},{dollar} for bank-full discharge is much less than unity for most of the 33 sites analyzed. It is concluded that extension of the reformatted Shields equation to flood deposits and boulder-bed rivers to estimate past hydraulic conditions or to determine flushing flows for regulated rivers could result in significant error.; The concept of wave competence is developed here and applied to boulders and gravel on a beach. Two equations are derived, one to estimate critical threshold mass, {dollar}Msb{lcub}Ru{rcub},{dollar} where{dollar}{dollar}Msb{lcub}Rsb{lcub}u{rcub}{rcub}={lcub}rhosb{lcub}s{rcub}fsb{lcub}BF{rcub}Usb{lcub}max{rcub}Rsb{lcub}u{rcub}sp32{lcub}rm f{rcub}over Ksb{lcub}r{rcub}left({lcub}rhosb{lcub}s{rcub}-rhosb{lcub}w{rcub}over rhosb{lcub}w{rcub}{rcub}right)g {lcub}rm tan{rcub} theta{rcub}eqno {lcub}rm (III-41){rcub}cr{dollar}{dollar}and another to estimate minimum stable mass, {dollar}Msb{lcub}Hsb{rcub},{dollar} where{dollar}{dollar}Msb{lcub}Hsb{lcub}sb{rcub}{rcub}={lcub}rhosb{lcub}s{rcub} fsb{lcub}BF{rcub}Usb{lcub}max{rcub}Rsb{lcub}u{rcub}Hsbsp{lcub}sb{rcub}{lcub}2{rcub}2{lcub}rm f{rcub}over Ksb{lcub}r{rcub}left({lcub}rhosb{lcub}s{rcub}-rhosb{lcub}w{rcub}overrhosb{lcub}w{rcub}{rcub}right)g {lcub}rm tan{rcub} theta{rcub}eqno {lcub}rm (III-42){rcub}cr{dollar}{dollar}Estimates from III-41 accurately match field data giving the largest boulder transported on a beach during storm events. Equation III-42 predicts stable stone mass in the range defined by the Hudson formula, but has the advantage over the Hudson formula of incorporating the physically important parameters of wave period and swash velocity into a practical expression and thus avoids the need to guess a value for an empirical stability coefficient.; The beach crest is a common morphological feature formed by the deposition of sediment carried up-slope by wave swash. Two equations are derived that relate the height of the beach crest, {dollar}hsb{lcub}c{rcub},{dollar} to the wave forces and the beach material:{dollar}{dollar}hsb{lcub}c{rcub}= {lcub}1over2{rcub}left({lcub}rhosb{lcub}s{rcub}-rhosb{lcub}w{rcub}overrhosb{lcub}w{rcub}{rcub}right)left({lcub}gTD sb{lcub}i{rcub} {lcub}rm tan{rcub} thetaover Csb{lcub}d{rcub}Usbmax{rcub}right)eqno {lcub}rm (IV-11){rcub}cr{dollar}{dollar}and{dollar}{dollar}hsb{lcub}c{rcub}= {lcub}rhosb{lcub}w{rcub}Hsbsp{lcub}sb{rcub}{lcub}2{rcub}over 8m{rcub}eqno {lcub}rm (IV-15){rcub}cr{dollar}{dollar}Equation IV-11 compares the wave force acting to move a stone up the beach face with a weight force acting to hold the stone in place. Equation IV-15 relates the potential energy per unit area of the beach crest to the total wave energy that lifts and deposits the material above a given sea-level datum. The actual crest height of a natural gravel beach was accurately estimated by both equations.