Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

• Main Authors: James N. Steer, Mark L. McAllister, Alistair G. L. Borthwick, Ton S. van den Bremer
• Format: Article
• Language: English
• Published: MDPI AG 2019-06-01
• Series: Fluids
• Subjects: surface waves; crossing seas; modulational/Benjamin-Feir instability; coupled nonlinear Schrödinger equation (CNLSE); experiments
• Online Access: https://www.mdpi.com/2311-5521/4/2/105

The coupled nonlinear Schr&#246;dinger equation (CNLSE) is a wave envelope evolution equation applicable to two crossing, narrow-banded wave systems. Modulational instability (MI), a feature of the nonlinear Schr&#246;dinger wave equation, is characterized (to first order) by an exponential growth of sideband components and the formation of distinct wave pulses, often containing extreme waves. Linear stability analysis of the CNLSE shows the effect of crossing angle, <inline-formula> <math display="inline"> <semantics> <mi>&#952;</mi> </semantics> </math> </inline-formula>, on MI, and reveals instabilities between <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mn>0</mn> <mo>∘</mo> </msup> <mo>&lt;</mo> <mi>&#952;</mi> <mo>&lt;</mo> <msup> <mn>35</mn> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mn>46</mn> <mo>∘</mo> </msup> <mo>&lt;</mo> <mi>&#952;</mi> <mo>&lt;</mo> <msup> <mn>143</mn> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mn>145</mn> <mo>∘</mo> </msup> <mo>&lt;</mo> <mi>&#952;</mi> <mo>&lt;</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula>. Herein, the modulational stability of crossing wavetrains seeded with symmetrical sidebands is determined experimentally from tests in a circular wave basin. Experiments were carried out at 12 crossing angles between <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mn>0</mn> <mo>∘</mo> </msup> <mo>&#8804;</mo> <mi>&#952;</mi> <mo>&#8804;</mo> <msup> <mn>88</mn> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula>, and strong unidirectional sideband growth was observed. This growth reduced significantly at angles beyond <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#952;</mi> <mo>&#8776;</mo> <msup> <mn>20</mn> <mo>∘</mo> </msup> </mrow> </semantics> </math> </inline-formula>, reaching complete stability at <inline-formula> <math display="inline"> <semantics> <mi>&#952;</mi> </semantics> </math> </inline-formula> = 30&#8722;40<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>∘</mo> </msup> </semantics> </math> </inline-formula>. We find satisfactory agreement between numerical predictions (using a time-marching CNLSE solver) and experimental measurements for all crossing angles.