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On Three-dimensional Rotating Turbulence. Shiyi Chen Collaborator: Q. Chen, G. Eyink, D. Holm. z. y. x. For solid-body rotating flow*,. Governing equations. In general, N-S :. Coriolis force. Centrifugal force.

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## On Three-dimensional Rotating Turbulence

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**On Three-dimensional Rotating**Turbulence Shiyi Chen Collaborator: Q. Chen, G. Eyink, D. Holm z y x**For solid-body rotating flow*,**Governing equations In general, N-S : Coriolis force Centrifugal force * All quantities here are relative to the rotating frame. * Centrifugal force * Coriolis force (angular momentum) * Nonlinear term**Governing equations**Non-dimensionalized N-S equation: Rossby # Ekman #**Climate prediction**Engineering Applications Turbomachinery (i.e. centrifugal Pumps) Geophysical Fluid Dynamics Solid boundary is present**Classic Taylor-Proudman theorem**Dimensionless form: If and (geostrophic flow) 3D rotating flow is two-dimensionalized. Taking a curl , we have**Which one dominates?**Dynamic Taylor-Proudman Theorem • For turbulence, • Coriolis force and nonlinear term?**Helical decomposition**In k-space:**q**k p • Nonrotating: • Rotating: Helical Representation of N-S Eq. 3D mode/fast mode: 2D mode/slow mode:**Helical waves and Inertial waves**Or In k-space, Eigenmodes (Inertial waves) Greenspan (1969)**O(1)**0 3D nonrotating turbulence Our interest: 1)How does 3D flow become two-dimensional ? 2) Resonant triadic interactions’ role in the two-dimensionalization when . Research Scope: 3D homogeneous turbulence ( no boundary effects, small Rossby number)**3D turbulence under rapid rotation**Solution:**3D turbulence under rapid rotation**Resonant condition: “averaged equation”:**Helical Representation of N-S Eq.**Conservation of energy and helicity in each triad gives: And triadic energy transfer function: (general)**3D turbulence under rapid rotation**From resonant condition and triad condition : Combined with How energy is transferred among different modes? Note:**Three resonant triadic interactions**k q p k q p k p 1. “fast-fast-fast” interactions 2. “fast-slow-fast” interactions 3. “slow-slow-slow” interactions q 4. Slow-Fast-Fast**“Fast-fast-fast” resonant triadic interactions**Instability assumption (Waleffe92): energy transfer is from the mode whose coefficient is opposite to the other two. One transfer function is negative and the other two are positive. If we normalize three wave numbers by the middle one,**1**w v “Fast-fast-fast” resonant triadic interactions**k**q p k “Fast-fast-fast” resonant triadic interactions “Fast-fast-fast” resonant triadic interactions tends to drive flow quasi-2D.**q**k p “Fast-slow-fast” resonant triadic interactions e * Energy exchange only happens between two 3D modes!**k**p “Slow-slow-slow” resonant triadic interactions q Since Using “averaged equation”**Dynamic Taylor-Proudman Theorem(2D-3C)**Let “slow-slow-slow” resonant interactions split into two parts: 1. (2D N-S) 2. (2D passive scalar) Note: Emid & Majda(1996); Mahalov & Zhou(1996) • “slow-slow-slow” resonant triadic interactions can split into • 2D turbulence and 2D passive scalar as .**Passive scalar T(x,y)**Dynamic Taylor-Proudman Theorem(2D-3C)**Numerical plans**Whether resonant interaction is responsible for flow two dimensionalization as ? 1) “slow-slow-slow” interactions. 3D simulation under rotation 2D nonrotating turbulence (2D-3C) Passive scalar T(x,y) 2) “slow-fast-fast” non-resonant interactions disappear? 3) “fast-fast-fast” resonant interactions.**Numerical schemes**DNS with hyperviscousity Forcing: Energy injected at • 3D rotating flow • 2. 2D turbulence • 3. 2D passive scalar**Inverse energy cascade**Energy injection scales t**“Fast-fast-fast” triadic interactions*** Fast-mode energy from “fast-fast-fast” triadic interactions tends to accumulates at small**Passive scalar T(x,y)**3D averaged field compared with the solution of 2D-3C equations**Dynamics of**t 2D * Inverse cascade for the vertically averaged horizontal velocity**Dynamics of :**2D • behaves like 2D passive scalar when Rossby number decreases.**3D averaged field compared with the solution of 2D-3C**equations • The difference decreases • as Rossby number decreases.**“Slow-fast-fast” triadic interactions*** Energy from non-resonant triads into small wavenumbers decreases with Rossby number.**Conclusions**• The rate of two-dimensionalization of 3D rotating flow decreases when Rossby number decreases. • Slow-mode energy spectrum approaches and its energy flux is closer to a constant. • The vertically averaged velocity and the solution of 2D-3C equations converge as • The energy flux from non-resonant triads into small in the 2D plane decreases as • The fast-mode energy is transferred toward the 2D plane, consistent with the consequence of “fast-fast-fast” resonant triadic interactions.

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