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Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

Mahieddine Adim, Roberta Bianchini, Vincent Duchêne

Abstract

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.

Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

Abstract

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.
Paper Structure (15 sections, 15 theorems, 154 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of our results
  4. Related literature
  5. Outline
  6. Notations
  7. The bilayer shallow water system
  8. The system without thickness diffusivity
  9. The system with diffusivity
  10. The hydrostatic Euler equations
  11. Stability estimates
  12. Refined approximation
  13. Convergence
  14. Conclusion
  15. Acknowledgements

Key Result

Lemma 2.1

Let $0<\rho_s<\rho_b$ and $\bm{U}:=(H_s,H_b,U_s,U_b) \in\mathbb{R}^4$ be such that that $H_s,H_b>0$. There exist two values $0 < \mathop{\mathrm{Fr}}\nolimits_- < \mathop{\mathrm{Fr}}\nolimits_+$ such that the following holds: Moreover, $\mathop{\mathrm{Fr}}\nolimits_-$ and $\mathop{\mathrm{Fr}}\nolimits_+$ depend only and smoothly on $H_s/H_b \in (0,+\infty)$ and $\rho_s/\rho_b \in (0,1)$.

Figures (1)

  • Figure 1: Solutions to \ref{['eq.Hyperbolic-geometric']} with $H_s=1/3$, $H_b=2/3$, and differnet values for $\rho_s/\rho_b$. Solutions to the quartic equation are in black (plain). Solutions to the linear equation with $(U_b-U_s)/\sqrt{H_b}=1/2$ (green, plain), $(U_b-U_s)/\sqrt{H_b}=3/2$ (red, dashed) and $(U_b-U_s)/\sqrt{H_b}=5/2$ (blue, dot-dashed).

Theorems & Definitions (30)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: Well-posedness
  • Proposition 2.5: Small time well-posedness
  • proof
  • Lemma 2.6: Stability
  • proof
  • Proposition 2.7: Large-time well-posedness
  • ...and 20 more