Using a Quadratic Model to Theoretically Describe the Nature of Equilibrium Shorerise Profiles

Abstract

Given that the assumptions of a quadratic model are true, the results of this study reveal new insights about the nature of equilibrium shorerise profiles. When an equilibrium state has been achieved between wave forcings and the shape of a shorerise profile, the quadratic model assumes that along a traverse of shoaling waves the acceleration rate of onshore wave energy expenditure on slope bottom areas per unit of horizontal and longshore distance remains constant. Given this constant onshore energy condition, it is mathematically shown that a quadratic function predicts the shape of a shorerise profile in the form of z₁ = ax₁² + bx₁ + c, where z₁ is the relief of a shorerise point above the origin located at the shorerise-ramp juncture, x₁ is the horizontal shoreward distance from the origin, and a,b,c are empirical coefficients. Using this quadratic function, observed z₁ values were correlated with corresponding x₁ values for 37 shorerise profiles stemming from Long Island, New York, and 37 profiles from Mustang-Padre Island, Texas. For the 74 profiles originating from two regions with diverse geomorphic histories, hence diverse degrees of profile curvature, the coefficients of determination (r²) ranged between 0.95 and 0.99. For each nine Duck, North Carolina, shorerise profiles taken at the same line from July, 1994 through December, 1995, and with varying degrees of curvature caused by varying wave conditions, r² was 0.99. No r² value was recorded at unity; therefore, no profile was interpreted as being at equilibrium. Profiles were regarded as moving closer to an equilibrium state as r² increased. Because the results show that a quadratic function effectively predicts the shape of shorerise profiles that varied over space and time, it follows that the geometric property of a bottom slope increasing onshore at a constant rate of 2a (d²z₁/dx₁² = 2a) is inva riant over space and time. Therefore, when r² is equal to unity, the geometric property of a bottom slope increasing onshore at a constant rate of 2a may be the signature of all shorerise profiles at equilibrium with waves that deliver a constant acceleration rate of onshore wave energy expenditure over slope bottom areas per unit of horizontal and longshore distance. Because wave conditions frequently vary, profiles should rarely achieve equilibrium. Using Airy wave theory and assuming that an open system operates across a shorerise, a discussion is presented in an attempt to explain why shoaling waves should maintain a profile with a slope that increases onshore at a constant rate.