### 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, x_{1}, x_{2}, x_{3}) = Ũ (t, x_{1}, x_{2}) + V(t, x_{1}, x_{2}, x_{3}) + 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 e_{3}). Here r is a remainder of order Ro^{1/2}
_{α} where Ro_{α} = (H_{0}U_{0})/(Ω_{0}L^{2}
_{0}) is the anisotropic Rossby number (H_{0} - height, L_{0} - horizontal length scale, Ω_{0} - angular velocity of background rotation, U_{0} - horizontal velocity scale); Ro_{α} = (H_{0}/L_{0})Ro where H_{0}/L_{0} is the aspect ratio and Ro = U_{0}/(Ω_{0}L_{0}) is a Rossby number based on the horizontal length scale L_{0}. The vector field V(t, x_{1}, x_{2}, x_{3}) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω_{0}t, τ_{1}(t), and τ_{2}(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ_{1}(t) and τ_{2}(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e_{3} and velocity Ū^{3}(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, x_{1}, x_{2}, x_{3}) and 2D turbulence for vertically averaged fields Ū(t, x_{1}, x_{2}) if the anisotropic Rossby number Ro_{α} 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 language | English (US) |
---|---|

Pages (from-to) | 35-42 |

Number of pages | 8 |

Journal | Computers and Mathematics with Applications |

Volume | 31 |

Issue number | 9 |

DOIs | |

State | Published - May 1996 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Computers and Mathematics with Applications*,

*31*(9), 35-42. https://doi.org/10.1016/0898-1221(96)00040-5

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

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 31, no. 9, pp. 35-42. https://doi.org/10.1016/0898-1221(96)00040-5

}

TY - JOUR

T1 - Integrability and regularity of 3D Euler and equations for uniformly rotating fluids

AU - Babin, A.

AU - Mahalov, Alex

AU - Nicolaenko, B.

PY - 1996/5

Y1 - 1996/5

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.

KW - Euler equations

KW - Navier-Stokes equations

KW - Phase turbulence

KW - Rotating fluids

UR - http://www.scopus.com/inward/record.url?scp=0030150311&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030150311&partnerID=8YFLogxK

U2 - 10.1016/0898-1221(96)00040-5

DO - 10.1016/0898-1221(96)00040-5

M3 - Article

AN - SCOPUS:0030150311

VL - 31

SP - 35

EP - 42

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 9

ER -