Scaling the shape of shoaling and overturning solitary waves

The shape of depth-limited overturning waves strongly influences turbulence and sediment transport in the surf zone. Bathymetric slope s <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>s</mml:mi> </mml:math> $s$ and initial wave height upper H 0 divided by h 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> $H_0/h_0$ are key parameters that govern wave overturning shape, yet their effect has not been quantified. We use a two-dimensional fully nonlinear potential flow model to study shoaling and overturning solitary waves across a range of s <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>s</mml:mi> </mml:math> $s$ and upper H 0 divided by h 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> $H_0/h_0$ representing spilling and plunging waves. Near-constant domain volume and energy demonstrate the model’s accuracy. Gentler slopes yield smaller, more inclined overturns with thinner jets, whereas steeper slopes produce larger, more horizontally inclined overturns with thicker jets. Wave breaking and overturning parameters such as breaker depth index, overturn area, jet area, aspect ratio and overturn angle are examined. Non-dimensional overturn and jet areas increase with steeper slopes and smaller upper H 0 divided by h 0 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> $H_0/h_0$ . Aspect ratios increase similarly but more slowly, while overturn angles shift from inclined to horizontal. A scaling parameter s divided by left parenthesis upper H 0 divided by h 0 right parenthesis Superscript 1 divided by 4 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>s</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> $s/(H_0/h_0)^{1/4}$ scales these wave breaking and overturning parameters well. During shoaling, front-face wave steepening occurs more rapidly for larger slopes and smaller <jats:alt-text content-typ