Abstract

We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ (t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e3). Here r is a remainder of order Ro1/2 α where Roα = (H0U0)/(Ω0L2 0) is the anisotropic Rossby number (H0 - height, L0 - horizontal length scale, Ω0 - angular velocity of background rotation, U0 - horizontal velocity scale); Roα = (H0/L0)Ro where H0/L0 is the aspect ratio and Ro = U0/(Ω0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roα is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* → +∞, as Roα → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case.

Original languageEnglish (US)
Pages (from-to)35-42
Number of pages8
JournalComputers and Mathematics with Applications
Volume31
Issue number9
DOIs
StatePublished - May 1996

Fingerprint

Rotating Fluid
Integrability
Euler
Turbulence
Regularity
2D Turbulence
Navier Stokes equations
Navier-Stokes Equations
Horizontal
Fluids
Regular Solution
Euler Equations
Length Scale
Vertical
Euler System
Wave Operator
Navier-Stokes System
Curl
Euler equations
Angular velocity

Keywords

  • Euler equations
  • Navier-Stokes equations
  • Phase turbulence
  • Rotating fluids

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Modeling and Simulation

Cite this

Integrability and regularity of 3D Euler and equations for uniformly rotating fluids. / Babin, A.; Mahalov, Alex; Nicolaenko, B.

In: Computers and Mathematics with Applications, Vol. 31, No. 9, 05.1996, p. 35-42.

Research output: Contribution to journalArticle

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N2 - We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ (t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e3). Here r is a remainder of order Ro1/2 α where Roα = (H0U0)/(Ω0L2 0) is the anisotropic Rossby number (H0 - height, L0 - horizontal length scale, Ω0 - angular velocity of background rotation, U0 - horizontal velocity scale); Roα = (H0/L0)Ro where H0/L0 is the aspect ratio and Ro = U0/(Ω0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roα is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* → +∞, as Roα → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case.

AB - We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x1, x2, x3) = Ũ (t, x1, x2) + V(t, x1, x2, x3) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e3). Here r is a remainder of order Ro1/2 α where Roα = (H0U0)/(Ω0L2 0) is the anisotropic Rossby number (H0 - height, L0 - horizontal length scale, Ω0 - angular velocity of background rotation, U0 - horizontal velocity scale); Roα = (H0/L0)Ro where H0/L0 is the aspect ratio and Ro = U0/(Ω0L0) is a Rossby number based on the horizontal length scale L0. The vector field V(t, x1, x2, x3) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω0t, τ1(t), and τ2(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ1(t) and τ2(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e3 and velocity Ū3(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x1, x2, x3) and 2D turbulence for vertically averaged fields Ū(t, x1, x2) if the anisotropic Rossby number Roα is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* → +∞, as Roα → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case.

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