Wave runup on vegetated beaches: data-driven empirical equations for rigid and flexible vegetation types

Coastal zones are vital to economies and societies, supporting diverse ecosystems, human settlements, and critical infrastructure. However, these areas face increasing threats from storm surges and coastal flooding. Traditional engineering solutions like seawalls and groins often disrupt natural processes and encourage unsustainable development, leading to a growing interest in Nature-based Solutions (NbS) such as wetlands and coastal vegetation. An assessment of the protective function of these NbS requires improved understanding of their influence on coastal processes, including wave runup. Despite the availability of many empirical equations for predicting wave runup on bare beaches, comparable equations for vegetated beaches remain largely undeveloped. This study develops empirical equations for predicting short wave runup on vegetated beaches through integrated numerical modeling and expression programming techniques. The research investigates four key parameters: significant wave height ( H s ), peak wave period ( T p ), beach slope ( S ) and vegetation density ( N v ) across two vegetation types: rigid and flexible. Orthogonal sampling generates 768 simulation scenarios per vegetation type. The XBeach Non-Hydrostatic is then utilized to simulate wave runup under the generated wave and vegetation scenarios. To develop runup equations, dimensional analysis transforms parameters into three dimensionless variables: Iribarren number, wave steepness, and vegetation characteristics (number of stems). The Artificial Bee Colony Expression Programming ( ABCEP ) algorithm derives vegetation-specific runup empirical equations from the resulting dataset. Results demonstrate acceptable predictive performance with R² values exceeding 0.94 for both vegetation types. Validation against independent numerical data confirms superior performance compared to widely-used wave runup formulations established for non-vegetated beaches. The empirical equations developed in this research provide coastal engineers and practitioners with tools explicitly incorporating vegetation density for short wave runup prediction.