Vortex Dynamos Steve Tobias (University of Leeds) Stefan Llewellyn Smith (UCSD)
An introduction to vortices • Vortices are ubiquitous in geophysical and astrophysical fluid mechanics (stratification & rotation). • Coherent structures that contain the local helicity/enstrophy of the turbulence. Here we examine the (fast?) dynamo properties of interacting vortices. Obtain dynamo from self-consistent solution to equations of motion (NS).
Vortices in Accretion Flow. Bracco et al Li et al (2001) (1999) Vortices arise due to the interaction of shear and rotation due to the nonlinear saturation of Rossby waves. May be important for planet formation? See e.g. Colgate & Hui (2002) Tagger (2001) Balmforth & Korycansky (2001) Godon & Livio (2000) …and many others.
Vortices in Compressible Convection In compressible convection there is a strong asymmetry between upflows and downflows . Dynamics dominated by strong downward sinking plumes from vertices of downflow network. VORTEX TUBES Brummell et al (2002) These vortices are observed in the solar granulation (see Simon & Weiss 1997).
What is a dynamo? • A dynamo is a system that maintains magnetic field against dissipation. • Kinematic dynamo problem: does a given flow amplify the magnetic field? Ignore Lorentz force: linear problem (in B). • Dynamic problem: add Lorentz force to get a nonlinear problem. • Use MHD (magnetohydrodynamic equations). • Cowling’s theorem: an axisymmetric flow field cannot act as a dynamo. • Governing parameters: Reynolds number and magnetic Reynolds number.
Fast Dynamos • Fast Dynamos are dynamos which continue to amplify field in the limit of high R m. • Usually investigated for carefully prescribed flows with Lagrangian Chaos (e.g. ABC flows, Galloway-Proctor Flows). • Need chaotic trajectories (Vishik 1989; Kapper & Young 1995).
Ponomarenko Dynamos • The dynamo effect SIMPLE FLOW OF THE FORM (generation of magnetic U=(0, r Ω , w) field) of a vortex in isolation has been modelled by prescribing the simple Ponomarenko flow and solving the induction equation in both the kinematic and nonlinear regimes. • See e.g. Lortz (1968), Ponomarenko (1973), Gilbert (1988), Ruzmaikin et al (1998), Bassom & Gilbert (1997). Dobler et al (2002)
Interacting vortices • The Ponomarenko dynamo relies on diffusion to work: the field is generated by a single vortex. • A flow field will generally have many interacting vortices, leading to a velocity field with chaotic Lagrangian properties. • Hence it has the potential to be a fast dynamo.
Hydrodynamics I Velocity field is independent of z, but has all three components. Vorticity Equation Define We consider an incompressible flow and set
Hydrodynamics: II Substitute into vorticity equation: (1) Integrate up (2) (no vertical pressure gradient) (3) The evolution equation for w is therefore the same as for a passive scalar in 2D-turbulence. 2 differences: (a) Forcing (source term) (b) w can take either sign .
Formation of Vortices q w Choose random random spectrum of initial conditions for both q and w. Competition between vortex formation and vortex merging. • q evolves via an inverse cascade to form vortex patches. (cf. Babiano et al 1987, McWilliams 1990, Arroyo et al 1995, Provenzale 1999). • w evolves as a passive scalar and becomes aligned with q (gets trapped in vortices and cancels outside of them)! • Hence we have the natural formation of interacting vortex tubes.
Equilibration of vortex flows I: Kinetic Energy • The velocity field saturates when there is a balance between forcing and diffusion. • For small enough ν this is a turbulent time- dependent state. • For this choice of forcing parameters (G 0 = 0.05, F 0 = -0.25), the energy in the horizontal flows is larger than that in the vertical flows. • The level of saturation depends on the diffusivity ν .
Equilibration of vortex flows II: Helicity and Reynolds Numbers. • The helical forcing leads to a turbulent state with net (negative) helicity. • The vertical velocity (w) is anti-correlated with the vertical vorticity (q). • The fluid Reynolds number (Re) can only be calculated a posteriori (once the saturation level of the flow is established). Re = <u 2 > 1/2 L / ν . ν = 0.004 Re = 505 ν = 0.002 Re = 1280 ν = 0.001 Re = 2640
Chaotic Lagrangian Flows Λ = J T J ν =1/250 ν =1/500 ν =1/1000 (see e.g. Soward 1994) • The velocity field driven by the vortex interactions is very chaotic. • Positive finite time Lyapunov exponents in all the flow, no regions of integrability. • Average Lyapunov exponent O(1) (lots of stretching). • Chaos increases slightly as Re increases .
Dynamo equations Use these vortex solutions as velocity fields in a kinematic dynamo calculation. i.e. solve with u as before. The dynamo problem is separable in z, so set and hence solve
Dynamo solutions q b 1 • Vortices advect and amplify magnetic field. • Small-scale magnetic fields are amplified. <B 2 > w • Growth-rate is a function of k, R m , R e.
Growth Rates I: σ versus k, R m • Look at growth-rate as a function of k for various R m . • Not as good a dynamo as prescribed flow type dynamos. • Good candidate for fast dynamo (more integrations needed). • More like Galloway- Proctor than Ponomarenko. • Mode of maximum growth- Solid η =0.01 rate appears to move to Dot-dashed η =0.002 higher k as R m increases Dot-dot-dashed η =0.001 (but for high enough R m is Dashed η =0.0005 independent of R m ).
Dynamo solutions q η =1/100 η =1/1000 b 1 b1 <B 2 > w ME w L b scales as Rm -1/2 L v
Dynamo solutions q η =1/5000 Hence at very high Rm we have very b 1 thin structures. Need high resolution (but only for induction equation – not so bad) ME -good to be in 2D! w
Growth Rates II: σ versus k, R m , R e Similar behaviour is found for Re = 1280, 2640. Continues to be an effective dynamo even if R e > R m • Formation of coherent structures important.
Conclusions • Interacting vortices are important geo/astrophysically and have chaotic Lagrangian properties. • Good dynamos – candidates for fast dynamo action. • Formation of coherent structures are important. Astrophysical and geophysical flows may continue to be dynamos if Re >> Rm (Pm <<1). If coherent structures are important then spectra may have a limited role in determining dynamo action (phase information may be important).
Future Work Nonlinear saturation – project onto one k-mode to keep two- dimensional – not so bad if modelling rapidly rotating flows? • Consider dynamo properties of vortices arising from instability of shear flow (shear contributes to formation of vortices and generation of magnetic field). • Inclusion of rotation via β -plane approx (QG). • Look at flows with similar spectra but different phases (one with coherent structures, one without) to determine the importance of coherent structures for dynamos where Re >> Rm. • Three-dimensional instabilities of these vortices.
Vortices in Boussinesq Convection • Even in non-rotating Boussinesq convection which has up-down symmetry and no net helicity, vortical motions appear to be important for small- scale field generation .
The velocity field • The formation of vortex patches in forced or decaying turbulence is well known from geophysical (2D) studies (e.g. McWilliams 1984, 1990; also MHD in applied field Kinney & McWilliams 1998). • For a dynamo we need a velocity field with all three velocity components. • But for investigation of fast dynamo properties it is helpful to have a 2D computation (can get to higher R e , R m ). • We consider the incompressible N-S equations with no z-dependence .
Evolution of inviscid invariants (and some inviscid non-invariants!)
Choice of (helical) forcing • In the absence of forcing the turbulence will decay (see previous slide). G z • We choose to force the flow with a steady forcing at wavenumber 4 (cf. Ohkitani 1991). • We set G z = G 0 sin(4x) sin(4y). For simplicity we also set F z F z = F 0 sin(4x) sin(4y). So the forcing is. maximally helical. (other forcings at higher wavenumber and of non-helical nature need to be investigated.)
Straining motions in vortex flows • Stretching properties of the flow can be calculated by examining the strain matrix. • strain matrix has 3 eigenvalues, λ 1 , , λ 3 . , λ 2 , • As d/dz = 0, λ 1 = = 0 . = −λ −λ 3 ; ; λ 2 = q λ 1
Recommend
More recommend
