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# On Three-dimensional Rotating Turbulence

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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1. On Three-dimensional Rotating Turbulence Shiyi Chen Collaborator: Q. Chen, G. Eyink, D. Holm z y x

2. 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

3. Governing equations Non-dimensionalized N-S equation: Rossby # Ekman #

4. Climate prediction Engineering Applications Turbomachinery (i.e. centrifugal Pumps) Geophysical Fluid Dynamics Solid boundary is present

5. Classic Taylor-Proudman theorem Dimensionless form: If and (geostrophic flow) 3D rotating flow is two-dimensionalized. Taking a curl , we have

6. Which one dominates? Dynamic Taylor-Proudman Theorem • For turbulence, • Coriolis force and nonlinear term?

7. Helical decomposition In k-space:

8. q k p • Nonrotating: • Rotating: Helical Representation of N-S Eq. 3D mode/fast mode: 2D mode/slow mode:

9. Helical waves and Inertial waves Or In k-space, Eigenmodes (Inertial waves) Greenspan (1969)

10. 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)

11. 3D turbulence under rapid rotation Solution:

12. 3D turbulence under rapid rotation Resonant condition: “averaged equation”:

13. Helical Representation of N-S Eq. Conservation of energy and helicity in each triad gives: And triadic energy transfer function: (general)

14. 3D turbulence under rapid rotation From resonant condition and triad condition : Combined with How energy is transferred among different modes? Note:

15. 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

16. “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,

17. 1 w v “Fast-fast-fast” resonant triadic interactions

18. k q p k “Fast-fast-fast” resonant triadic interactions “Fast-fast-fast” resonant triadic interactions tends to drive flow quasi-2D.

19. q k p “Fast-slow-fast” resonant triadic interactions e * Energy exchange only happens between two 3D modes!

20. k p “Slow-slow-slow” resonant triadic interactions q Since Using “averaged equation”

21. 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 .

22. Passive scalar T(x,y) Dynamic Taylor-Proudman Theorem(2D-3C)

23. 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.

24. Numerical schemes DNS with hyperviscousity Forcing: Energy injected at • 3D rotating flow • 2. 2D turbulence • 3. 2D passive scalar

25. Flow structures in 3D rotating turbulence

26. Inverse energy cascade Energy injection scales t

27. Flow two-dimensionalization

28. Flow two-dimensionalization

29. Flow two-dimensionalization

30. Flow two-dimensionalization

31. “Fast-fast-fast” triadic interactions * Fast-mode energy from “fast-fast-fast” triadic interactions tends to accumulates at small

32. Passive scalar T(x,y) 3D averaged field compared with the solution of 2D-3C equations

33. Dynamics of t 2D * Inverse cascade for the vertically averaged horizontal velocity

34. Dynamics of : 2D • behaves like 2D passive scalar when Rossby number decreases.