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Understanding Stokes drift mechanism via crest and trough phase estimates

Anirban Guha, Akanksha Gupta

Abstract

By providing mathematical estimates, this paper answers a fundamental question -- "what leads to Stokes drift"? Although overwhelmingly understood for water waves, Stokes drift is a generic mechanism that stems from kinematics and occurs in any non-transverse wave in fluids. To showcase its generality, we undertake a comparative study of the pathline equation of sound (1D) and intermediate-depth water (2D) waves. Although we obtain a closed-form solution $\mathbf{x}(t)$ for the specific case of linear sound waves, a more generic and meaningful approach involves the application of asymptotic methods and expressing variables in terms of the Lagrangian phase $θ$. We show that the latter reduces the 2D pathline equation of water waves to 1D. Using asymptotic methods, we solve the respective pathline equation for sound and water waves, and for each case, we obtain a parametric representation of particle position $\mathbf{x}(θ)$ and elapsed time $t(θ)$. Such a parametric description has allowed us to obtain second-order-accurate expressions for the time duration, horizontal displacement, and average horizontal velocity of a particle in the crest and trough phases. All these quantities are of higher magnitude in the crest phase in comparison to the trough, leading to a forward drift, i.e. Stokes drift. We also explore particle trajectory due to second-order Stokes waves and compare it with linear waves. While finite amplitude waves modify the estimates obtained from linear waves, the understanding acquired from linear waves is generally found to be valid.

Understanding Stokes drift mechanism via crest and trough phase estimates

Abstract

By providing mathematical estimates, this paper answers a fundamental question -- "what leads to Stokes drift"? Although overwhelmingly understood for water waves, Stokes drift is a generic mechanism that stems from kinematics and occurs in any non-transverse wave in fluids. To showcase its generality, we undertake a comparative study of the pathline equation of sound (1D) and intermediate-depth water (2D) waves. Although we obtain a closed-form solution for the specific case of linear sound waves, a more generic and meaningful approach involves the application of asymptotic methods and expressing variables in terms of the Lagrangian phase . We show that the latter reduces the 2D pathline equation of water waves to 1D. Using asymptotic methods, we solve the respective pathline equation for sound and water waves, and for each case, we obtain a parametric representation of particle position and elapsed time . Such a parametric description has allowed us to obtain second-order-accurate expressions for the time duration, horizontal displacement, and average horizontal velocity of a particle in the crest and trough phases. All these quantities are of higher magnitude in the crest phase in comparison to the trough, leading to a forward drift, i.e. Stokes drift. We also explore particle trajectory due to second-order Stokes waves and compare it with linear waves. While finite amplitude waves modify the estimates obtained from linear waves, the understanding acquired from linear waves is generally found to be valid.
Paper Structure (7 sections, 58 equations, 1 figure, 1 table)

This paper contains 7 sections, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The 1D problem: longitudinal waves
  3. The 2D problem: intermediate depth water waves
  4. Linear Waves:
  5. Second-order Stokes waves
  6. Calculation of Stokes drift
  7. Summary and Conclusion

Figures (1)

  • Figure 1: A rightward propagating linear water wave. A particle initially located at 'a' traverses an open trajectory a--b--c--d--$\Tilde{\mathrm{a}}$ in the clockwise direction. The endpoint $\Tilde{\mathrm{a}}$ signifies completion of $2\pi$ rotation. Each point marked on the particle trajectory corresponds to that on the wave. The linear distance between 'a' and '$\Tilde{\mathrm{a}}$' is the Stokes correction $x^{\textnormal{L,net}}$.