SQG Vortex Dynamics and Passive Scalar Transport

Transcription

1 SQG Vortex Dynamics and Passive Scalar Transport Cecily Taylor Stefan Llewellyn Smith Mechanical and Aerospace Engineering University of California, San Diego ONR, Ocean 3D+1, MURI September 28, 2015

2 Ocean Dynamics The ocean is a thin fluid envelope spread on a rapidly rotating Earth. Dynamics are largely two-dimensional with weak vertical flow. Vortices are common in the ocean. Goal: understand effect of weak 3D variation using dynamically consistent models with weak vertical velocity Use idealized vortices as model problems.

3 Ocean Dynamics The ocean is a thin fluid envelope spread on a rapidly rotating Earth. Dynamics are largely two-dimensional with weak vertical flow. Vortices are common in the ocean. Goal: understand effect of weak 3D variation using dynamically consistent models with weak vertical velocity Use idealized vortices as model problems.

4 Ocean Dynamics The ocean is a thin fluid envelope spread on a rapidly rotating Earth. Dynamics are largely two-dimensional with weak vertical flow. Vortices are common in the ocean. Goal: understand effect of weak 3D variation using dynamically consistent models with weak vertical velocity Use idealized vortices as model problems.

5 Ocean Dynamics The ocean is a thin fluid envelope spread on a rapidly rotating Earth. Dynamics are largely two-dimensional with weak vertical flow. Vortices are common in the ocean. Goal: understand effect of weak 3D variation using dynamically consistent models with weak vertical velocity Use idealized vortices as model problems.

6 Quasigeostrophic (QG) Equations Boussinesq, hydrostatic. Constant f, N. Reduced equation of motion for Ro 1. ζ θ = f z ψ ψ Vorticity Buoyancy ( density) Streamfunction bulk t ζ = J(ψ, ζ) + f z w surface t θ = J(ψ, θ) N 2 w [ ( ) f 2 bulk q = xx + yy + zz] ψ N Pedlosky 1982

7 Surface Quasigeostrophic Equations Surface QG (SQG) assumes potential vorticity q = 0, so the inherent dynamics are on the boundary θ =! f ψ!!!!!evolves! z on!the!surface! w=o(ro)) Mo0on!induced! below!where!q=0! Possible!second! layer?!

8 Interior Flow Field We wish to determine the consistent velocity solution to O(Ro). In the asymptotic procedure to obtain QG equations, begin by defining and expanding potentials ψ, F, G. v u θ = ψ + F G 0 ɛ = Ro ψ ψ 0 + ɛψ 1 + ɛ 2 ψ 2 + F ɛf 1 + ɛ 2 F 2 + G ɛg 1 + ɛ 2 G 2 + Muraki et al. 1999

9 Point Vortices We want the simplest possible dynamically consistent flow field: Three point vortices have regular motion but induce chaotic flow. Followed analysis by Kuznetsov & Zaslavsky (1998) for interaction of three vortices of equal strength.

10 SQG Point Vortex at O(1) Singular vorticity distribution at (x n, y n, 0) θ s 0 = n κ n δ(x x n )δ(y y n ) 2 ψ 0 = 0 ψ 0 = n ψ s 0 z = θs 0 κ n 2π x x n Dynamically (asymptotically) consistent

11 Poincaré Maps at Various Heights z = 0 z =.25 z =.5

12 Arbitrary Strength Following Aref (1979) use momentum conservation to define a constant C and trilinear coordinates κ 1 κ 2 l κ 2κ 3 l κ 3κ 1 l 2 31 = 3κ 1κ 2 κ 3 C From b 1 = l2 23 κ 1 C, b 2 = l2 13 κ 2 C, b 3 = l2 12 κ 3 C = κ ( 1κ 2 κ 3 2π C 1/2 H = 1 κ α κ β 4π l αβ α,β 1 b 1 κ 1 1/2 κ b 2 κ 2 1/2 κ b 3 κ 3 1/2 κ 3 )

13 Trajectory Curves

14 κ = (2, 1, 3)

15 O(Ro) Corrections The point vortices prove problematic when we attempt to find the O(Ro) velocities. Consider 2 F 1 = 2J ψ 0 = n ( ψ0 z, ψ ) 0 x κ n 2π x x n So the equation of motion of F 1 involves multiplying derivatives of a singular function! Can only calculate w = D 0θ 0 Dt

16 Topological Entropy

17 Topological Entropy Literature is dense: based on mapping analysis A single number for a given flow that doesn t involve a tracer advection equation Braiding Entropy - code from Jean-Luc Thiffeault and Marko Budisic Built for 2D transport, but works for 3D trajectories (where z component is ignored) - is info lost? Well commented and documented, no additional parameters

18 Initial Findings: FTBE vs. H κ = (1, 1, 1) κ = (2, 1, 3)

19 Initial Findings: FTBE vs. depth H =.53 H =.57

20 FTBE Variance

21 Dritschel Vortex Dritschel (2011) described vortices that maintain their shape in SQG. Start with an ellipsoid oriented along Cartesian axes that contains a region of constant potential vorticity Q. In QG flow, this ellipsoid will rotate as a rigid body. Dritschel et al. 2004

22 Limit to SQG Take the limit as the vertical axis of the ellipsoid c 0. We find a buoyancy distribution θ s 0 = κ 1 x 2 /a 2 y 2 /b 2 where κ is related to vortex strength. This is continuous so the flow field is now finite and regular, even at the edge.

23 Streamfunction The streamfunction is then calculated from 2 ψ = 0 and ψz s = θ0 s (Dritschel et al. 2004) ψ = 3κ du (1 x 2 4 (u + a 2 )(u + b 2 )u u + a 2 y 2 ) u + b 2 z2 u λ λ is the largest root of x 2 λ + a 2 + y 2 λ + b 2 + z2 λ = 1

24 Vortex Interactions Each vortex is represented by a set of point vortices, the strengths and positions of which are chosen to match the spatial moments of the initial elliptical vortex up to the desired order (Dritschel et al. 2004)

25 Simulation a 1 /b 1 t=0 = 2, a 2 /b 2 t=0 = 1.5, R t=0 = 10, κ 1 = κ 2 = 1

26 Velocities: X-Y & X-Z slices ɛ = 0.1

27 Conclusions SQG vortices constitute an interesting model system: 2D dynamics and 3D transport. Vortex motion is regular but transport can be chaotic. Arbitrary vortex strengths introduce additional free parameters. Is it possible to reconcile singularities of point vortices to get O(Ro) solution?

28 Conclusions SQG vortices constitute an interesting model system: 2D dynamics and 3D transport. Vortex motion is regular but transport can be chaotic. Arbitrary vortex strengths introduce additional free parameters. Is it possible to reconcile singularities of point vortices to get O(Ro) solution?

29 Conclusions SQG vortices constitute an interesting model system: 2D dynamics and 3D transport. Vortex motion is regular but transport can be chaotic. Arbitrary vortex strengths introduce additional free parameters. Is it possible to reconcile singularities of point vortices to get O(Ro) solution?

30 Next Steps Run diagnostic for Dritschel vortex solution & interactions use steady state solutions from Dritschel & Poje. Include O(Ro) corrections and examine mixing dependence on Ro. At what time do the transport properties of the 3D model deviate from 2D? O(1/Ro)? What about a periodic domain? What if we add a second boundary, providing a vertical length scale?

31 Next Steps Run diagnostic for Dritschel vortex solution & interactions use steady state solutions from Dritschel & Poje. Include O(Ro) corrections and examine mixing dependence on Ro. At what time do the transport properties of the 3D model deviate from 2D? O(1/Ro)? What about a periodic domain? What if we add a second boundary, providing a vertical length scale?

32 Next Steps Run diagnostic for Dritschel vortex solution & interactions use steady state solutions from Dritschel & Poje. Include O(Ro) corrections and examine mixing dependence on Ro. At what time do the transport properties of the 3D model deviate from 2D? O(1/Ro)? What about a periodic domain? What if we add a second boundary, providing a vertical length scale?

33 Next Steps Run diagnostic for Dritschel vortex solution & interactions use steady state solutions from Dritschel & Poje. Include O(Ro) corrections and examine mixing dependence on Ro. At what time do the transport properties of the 3D model deviate from 2D? O(1/Ro)? What about a periodic domain? What if we add a second boundary, providing a vertical length scale?

34 Interior Flow Field Our system is governed by 2 F 1 = 2J 2 ψ 0 = 0 ( ) ψ0 z, v 0 ψ s 0 z = θs 0 2 G 1 = 2J ( ) ψ0 z, u 0 (F 1 ) s = (G 1 ) s = 0 2 ψ 1 = q 1 + ψ 0 2 ( ) s ψ1 z = θ1 s z = 0 where s indicates at the surface, z = 0. Muraki et al. 1999

35 Solving Poisson Equations In 2002, Hakim et al. determined the particular solutions to the QG +1 potentials. The resulting Laplace equation can be solved in horizontal 2D Fourier Space. For example: particular solution to Poisson F 1 = ψ 0 y ψ 0 z + F 1, new gov ing equations 2 F1 = 0, F1 s = [ ψ0 y ] ψ s 0, z solution to Laplace ˆ F 1 = ˆ F s 1 e κz And similarly for G 1, ψ 1

36 Procedure 1 Specify θ s 0 with θs 1 = 0 2 Solve 2 ψ 0 = 0 with ψ 0 z = θ 0 at z = 0 3 Solve Laplace equations for F 1, G 1, ψ 1. 4 Obtain velocities from derivatives of potentials. 5 Advect one time step and iterate.

37 Velocities Now we have u ψ ( 0 y ɛ ψ1 y + F ) 1 z v ψ ( 0 x + ɛ ψ1 x G ) 1 z ( F1 w ɛ x + G ) 1 = D 0θ 0 y Dt Dθ 0 Dt known from θ 0 = ψ 0 z Muraki et al. 1999