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The three limits of the hydrostatic approximation

Ken Furukawa, Yoshikazu Giga, Matthias Hieber, Amru Hussein, Takahito Kashiwabara, Marc Wrona

TL;DR

This work analyzes the hydrostatic approximation by studying the three formal limits arising from anisotropic scaling ν_z=ε^2δ in a vertically thin domain for the 3D Navier–Stokes equations. Using maximal L^2-regularity and a quadratic-inequality framework, it proves norm-convergence to (i) the primitive equations with full viscosity (δ finite, ε→0), (ii) the primitive equations with only horizontal viscosity (δ→0), and (iii) the 2D-Navier–Stokes equations (δ→∞) as ε→0, including convergence for ε→0 with δ fixed and, in some regimes, explicit rates like O(ε+δ) or δ^{-1/4}. The analysis relies on uniform linear estimates for the scaled Stokes problem, preservation of vertical regularity, and detailed nonlinear estimates in anisotropic function spaces, enabling a flexible treatment of multiple coupled limits. The results substantiate the stability of reduced models under parameter variations and provide a method applicable to related geophysical models with anisotropic diffusion and transport. The work also emphasizes the versatility of the maximal-regularity approach over purely energy-based methods in capturing sharp convergence behavior across regimes.

Abstract

The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $ν_z=\mathcal{O}(\varepsilon^γ)$ where $γ=2$, one obtains the primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take two more limit equations into consideration: For $γ<2$ the $2D$-Navier-Stokes equations are obtained. For $γ>2$ the primitive equations with only horizontal viscosity $-Δ_H$ as $\varepsilon\to 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $ν_z=\varepsilon^2 δ$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $\varepsilon,δ\to 0$. The flexibility of our methods is also illustrated by the convergence for $δ\to \infty$ and $\varepsilon\to 0$ to the $2D$-Navier-Stokes equations.

The three limits of the hydrostatic approximation

TL;DR

This work analyzes the hydrostatic approximation by studying the three formal limits arising from anisotropic scaling ν_z=ε^2δ in a vertically thin domain for the 3D Navier–Stokes equations. Using maximal L^2-regularity and a quadratic-inequality framework, it proves norm-convergence to (i) the primitive equations with full viscosity (δ finite, ε→0), (ii) the primitive equations with only horizontal viscosity (δ→0), and (iii) the 2D-Navier–Stokes equations (δ→∞) as ε→0, including convergence for ε→0 with δ fixed and, in some regimes, explicit rates like O(ε+δ) or δ^{-1/4}. The analysis relies on uniform linear estimates for the scaled Stokes problem, preservation of vertical regularity, and detailed nonlinear estimates in anisotropic function spaces, enabling a flexible treatment of multiple coupled limits. The results substantiate the stability of reduced models under parameter variations and provide a method applicable to related geophysical models with anisotropic diffusion and transport. The work also emphasizes the versatility of the maximal-regularity approach over purely energy-based methods in capturing sharp convergence behavior across regimes.

Abstract

The primitive equations are derived from the -Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an -thin domain and anisotropic viscosities with vertical viscosity where , one obtains the primitive equations with full viscosity as . Here, we take two more limit equations into consideration: For the -Navier-Stokes equations are obtained. For the primitive equations with only horizontal viscosity as . Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for . The flexibility of our methods is also illustrated by the convergence for and to the -Navier-Stokes equations.
Paper Structure (31 sections, 24 theorems, 195 equations, 2 figures)

This paper contains 31 sections, 24 theorems, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature on the limit equations
  3. Literature on the hydrostatic approximation and our approach to it
  4. Outline
  5. Rescaling procedure and the formal limit equations
  6. Rescaled $3D$ Navier-Stokes equations
  7. The anisotropic primitive equations
  8. The primitive equations with only horizontal viscosity
  9. Difference equations for $\varepsilon\to 0$ and $\delta\to 0$
  10. Rescaled $3D$ Navier-Stokes equations with $2D$ Navier-Stokes equations as subsystem
  11. Anisotropic $3D$ Stokes and anisotropic $2D$ Navier-Stokes equations
  12. Difference equation for $\varepsilon\to 0$ and $\delta\to \infty$
  13. Function spaces
  14. Isotropic function spaces on the torus
  15. Maximal $L^2_t$-$L^2_x$-regularity spaces
  16. ...and 16 more sections

Key Result

Lemma 3.1

Let $p,q\in (1,\infty )$ and $\varepsilon>0$. Then

Figures (2)

  • Figure 1: Convergences schematically
  • Figure 2: Convergences schematically for $\nu_z=\varepsilon^\gamma$ as $\varepsilon\to 0$

Theorems & Definitions (50)

  • Lemma 3.1: Characterization of the initial value spaces
  • Proposition 4.1: Local existence for \ref{['eq:NS_eps']}
  • Proposition 4.2: Global existence for \ref{['eq_PEH']} and higher regularity
  • Theorem 4.3: Convergence for $\varepsilon\to 0$ and $\delta \to 0$
  • Remark 4.4: Comparison to the results in LiTitiGuozhi_2022 by Li, Titi and Yuan
  • Proposition 4.5: Global existence for the $2D$ Navier-Stokes equations
  • Theorem 4.6: Convergence for $0<\varepsilon\leq 1$ and $\delta \to \infty$
  • Remark 4.7: Further convergences
  • Remark 4.8: Boundary conditions
  • Proposition 5.1
  • ...and 40 more