SLIDE 1
Vortex Dynamos
Steve Tobias (University of Leeds) Stefan Llewellyn Smith (UCSD)
SLIDE 2 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
Obtain dynamo from self-consistent solution to equations
SLIDE 3 Vortices in Accretion Flow.
Bracco et al (1999) Li et al (2001) Vortices arise due to the interaction of shear and rotation due to the nonlinear saturation
May be important for planet formation? See e.g. Colgate & Hui (2002) Tagger (2001) Balmforth & Korycansky (2001) Godon & Livio (2000) …and many others.
SLIDE 4
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 These vortices are observed in the solar granulation (see Simon & Weiss 1997). Brummell et al (2002)
SLIDE 5 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.
SLIDE 6
dynamos which continue to amplify field in the limit of high Rm.
carefully prescribed flows with Lagrangian Chaos (e.g. ABC flows, Galloway-Proctor Flows).
- Need chaotic trajectories
(Vishik 1989; Kapper & Young 1995).
Fast Dynamos
SLIDE 7 Ponomarenko Dynamos
(generation of magnetic 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.
Ponomarenko (1973), Gilbert (1988), Ruzmaikin et al (1998), Bassom & Gilbert (1997).
Dobler et al (2002) SIMPLE FLOW OF THE FORM U=(0, rΩ, w)
SLIDE 8 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.
SLIDE 9
Hydrodynamics I
Velocity field is independent of z, but has all three components. Define Vorticity Equation We consider an incompressible flow and set
SLIDE 10
Hydrodynamics: II
Integrate up (no vertical pressure gradient) 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. Substitute into vorticity equation: (1) (2) (3)
SLIDE 11 Formation of Vortices
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. q w
SLIDE 12 Equilibration of vortex flows I: Kinetic Energy
saturates when there is a balance between forcing and diffusion.
is a turbulent time- dependent state.
- For this choice of forcing
parameters (G0 = 0.05, F0 = -0.25), the energy in the horizontal flows is larger than that in the vertical flows.
depends on the diffusivity ν.
SLIDE 13 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).
number (Re) can only be calculated a posteriori (once the saturation level
- f the flow is established).
Re = <u2>1/2 L / ν. ν = 0.004 Re = 505 ν = 0.002 Re = 1280 ν = 0.001 Re = 2640
SLIDE 14 Chaotic Lagrangian Flows
- 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.
ν=1/1000 ν=1/500 ν=1/250 Λ = JT J (see e.g. Soward 1994)
SLIDE 15
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
SLIDE 16 Dynamo solutions
amplify magnetic field.
fields are amplified.
function of k, Rm, Re.
q w b1 <B2>
SLIDE 17 Growth Rates I: σ versus k, Rm
function of k for various Rm.
prescribed flow type dynamos.
dynamo (more integrations needed).
Proctor than Ponomarenko.
rate appears to move to higher k as Rm increases (but for high enough Rm is independent of Rm). Solid η=0.01 Dot-dashed η=0.002 Dot-dot-dashed η=0.001 Dashed η=0.0005
SLIDE 18
Dynamo solutions
q b1 w <B2> w b1 ME η=1/100 η=1/1000
Lb scales as Rm-1/2 Lv
SLIDE 19 Dynamo solutions
q b1 w ME η=1/5000 Hence at very high Rm we have very thin structures. Need high resolution (but only for induction equation – not so bad)
SLIDE 20 Growth Rates II: σ versus k, Rm, Re
Similar behaviour is found for Re = 1280, 2640. Continues to be an effective dynamo even if Re > Rm
- Formation of coherent structures important.
SLIDE 21 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).
SLIDE 22 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.
SLIDE 23 Vortices in Boussinesq Convection
Boussinesq convection which has up-down symmetry and no net helicity, vortical motions appear to be important for small- scale field generation.
SLIDE 24 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 Re, Rm).
- We consider the incompressible N-S equations
with no z-dependence.
SLIDE 25
Evolution of inviscid invariants
(and some inviscid non-invariants!)
SLIDE 26 Choice of (helical) forcing
- In the absence of forcing
the turbulence will decay (see previous slide).
flow with a steady forcing at wavenumber 4 (cf. Ohkitani 1991).
Gz = G0 sin(4x) sin(4y). For simplicity we also set Fz = F0 sin(4x) sin(4y). So the forcing is. maximally helical. Gz Fz (other forcings at higher wavenumber and of non-helical nature need to be investigated.)
SLIDE 27 Straining motions in vortex flows
- Stretching properties of the flow can be calculated by
examining the strain matrix.
- strain matrix has 3 eigenvalues, λ1,
, λ2, , λ3.
λ1 = = −λ −λ3; ; λ2 = = 0.
q λ1
