2. Quantifying Irreversible Transport Using a Tracer-Based Coordinate
Noboru Nakamura
University of Chicago
Douglas Allen and Jun Ma
Naval Research Laboratory

3. 1. Inhomogeneity in mixing (barriers, etc.)
2. Direction of transport (gross fluxes)
3. Eulerian eddy diffusivity revisited

4. 1. Mixing is inhomogeneous

5. 1. Mixing is inhomogeneous
Polar Vortex

6. 1. Mixing is inhomogeneous
Polar Vortex
“Surf Zone”

7. 1. Mixing is inhomogeneous
Polar Vortex
“Surf Zone”
Tropical Reservoir

8. Well-mixed regions are separated by semi-permeable barriers

9. Well-mixed regions are separated by semi-permeable barriers
But why are gradients maximal at the barriers?

10. Well-mixed regions are separated by semi-permeable barriers
But why are gradients maximal at the barriers?
The barriers are highly variable

11. Well-mixed regions are separated by semi-permeable barriers
But why are gradients maximal at the barriers?
The barriers are highly variable
The fluxes must be measured with respect to
a surface that is unsteady and irregular.

12. Well-mixed regions are separated by semi-permeable barriers
But why are gradients maximal at the barriers?
The barriers are highly variable
The fluxes must be measured with respect to
a surface that is unsteady and irregular.
Difficult to separate reversible and irreversible
transport.

13. Well-mixed regions are separated by semi-permeable barriers
But why are gradients maximal at the barriers?
The barriers are highly variable
The fluxes must be measured with respect to
a surface that is unsteady and irregular.
Difficult to separate reversible and irreversible
transport.
Both Eulerian-mean and finite-time (or -scale)
Lyapunov exponents are problematic.

14. 2D Advection-Diffusion Problem

15. 2D Advection-Diffusion Problem
A(q, t) : area in which at time t
area of the
domain A0 A q*
qmin q
qmax

16. Change of area is solely due to diffusion
Change of area is solely due to diffusion. (Advection does not affect the area.)
The rhs is the convergence of diffusive mass flux into the contour q*= q.

17. Exercise
Prove the above Hint: *

18. divergence theorem etc.
where
is an area-weighted average around the q - contour.
(No need of line integral!)

19. NB. Tracer PDF
obeys
Negative diffusion!

20. Let’s invert the relationship A(q,t) to q(A,t).q*
qmin q
qmax

21. Area coordinate is a material coordinate
and unaffected by the shape or position of
the tracer contour

22. Area coordinate is a material coordinate
and unaffected by the shape or position of
the tracer contour
Flux is solely due to small-
scale diffusion but enhanced by Le2

23. Area coordinate is a material coordinate
and unaffected by the shape or position of
the tracer contour
Flux is solely due to small-
scale diffusion but enhanced by Le2
Large Le2 means diffusion is enhanced,
whereas small Le2 means barrier to mixing

24. is related to the length
of the tracer contour and hence an instantaneous
measure of geometrical complexity
(Haynes and Shuckburgh 2000,
Shuckburgh and Haynes 2003)

26. A A z* L0 L0

27. : effective diffusivityA A z* L0 L0
: effective diffusivity
: normalized effective diffusivity
(“roughness” or “stretchedness”)

28. Le2/Lo2

29. Le2/Lo2

30. Le2/Lo2

31. Le2/Lo2

32. Le2/Lo2

37. Effective Diffusivity in the Stratosphere

38. log(Le2/L02) (color) and zonal mean wind (contours)

39. 2. Which way does a wave break?

40. Direction of transport does matter.

41. Problems with particle advection
Difficult to determine the surface of transport accurately
Gross flux dominated by small-scale, random motion. The flux depends sensitively on the residence time of the particle on one side of the surface, and it fails to converge at vanishing residence time (Hall and Holtzer 2003)

42. Diagnostic Strategy

43. Diagnostic Strategy

44. Outward vs. Inward Breaking

50. December K (~ 32 km)

52. 3. Is there an Eulerian analog of effective diffusivity?
K is nonunique.

59. Conclusions
A diagnostic framework has been developed to quantify irreversible transport from instantaneous tracer geometry. This includes barrier locations and strength (both contour-based and Eulerian) and direction of transport.
The technique is most useful (and robust) when the tracer micro-structure is governed by large scale flows (e.g., stratosphere), but it is in principle applicable to more general cases, including 3D advection-diffusion.