A  brief  introduction  to  dynamos

Steve Tobias (Leeds)

5th Potsdam Thinkshop, 2007

Geophysical  and  Astrophysical  Magnetism:    The  Earth  

Earth has magnetic field!

Paleomagnetism: around for long time Largely Dipolar

Reverses every million or so years

Geophysical  and  Astrophysical  Magnetism:    

Solar  System  Planets   These tend to have magnetic fields

Geophysical  and  Astrophysical  Magnetism:    

The  Sun  and  Stars   These tend to have magnetic fields (more on this later)

Geophysical  and  Astrophysical  Magnetism:    Galaxies  

These tend to have magnetic fields M51 Spiral Galaxy

Geophysical  and  Astrophysical  Magnetism:  Experiments  

These tend not to have magnetic fields (only joking)

VKS dynamo experiment See Cary’s talk

A  brief  introduction  to  dynamos

“In mathematics the art of proposing a question must be held of higher value than

solving it” Georg Cantor

“There are more questions than answers…”

Johnny Nash

A  brief  introduction  to  dynamos Review of basics of dynamo theory:

C.A. Jones “Dynamo Theory” (Les Houches Lectures) http://www-lgit.obs.ujf-grenoble.fr/houches/

H.K. Moffatt “Magnetic Field Generation in Electrically

Conducting Fluids”, 1978, CUP.

Turbulent small-scale dynamos (and MHD turbulence): “MHD Dynamos and Turbulence”, Tobias, S.M., Cattaneo, F.

& Boldyrev, S. in “Ten Chapters on Turbulence” eds Davidson, Kaneda and Sreenivasan CUP (2013)

http://www1.maths.leeds.ac.uk/~smt/tcb_review13.pdf

The Induction Equation ⇥B

⇥t= r⇥ (u⇥B) + �r2B

�B

�t= r⇥ (u⇥B) +

1

Rmr2B

�B

�t+ (u.r)B = (B.r)u+

1

Rmr2B

For what choices of u does B remain for large times? Clearly if u=0 then the field will decay away on diffusive timescale.

Rm =Td

Tturn=

L2/�

L/U=

UL

�

stretching

Magnetic Reynolds Number Rm For geophysical and astrophysical flows Rm is large enough that stretching wins. In fact theoretical problems can arise because Rm is so large (see later) Earth’s Core Jupiter Solar Convection Zone Galaxy Experiments

102 � 103⇡ 105

⇡ 108

⇡ 10� 102

(though see Cary’s talk for plasma Rm!)

Stretching vs Diffusion �B

�t+ (u.r)B = (B.r)u+

1

Rmr2B

The induction equation is a linear problem in B. For a given u solutions for the field either grow or decay exponentially (on average) with (average) growth-rate Categorise flows into those for which (kinematic dynamos) Or... (non-dynamos) … sounds easy.

�� > 0� < 0

Antidynamo Theorems Cowling’s Theorem (1934) • An axisymmetric magnetic field can not be generated via dynamo action.

•  Field must be inherently 3D for

Zel’dovich Theorem (1958) •  A planar velocity field is not capable of sustaining dynamo action

All calculations and numerics will have to be 3D (or something clever will have to be done!).

� > 0

Bounds on Rm Backus (1958) •  Demonstrated that in a sphere that for dynamo action to occur

(Rm calculated on rate of strain) Childress (1969)

• Need enough stretching to overcome the diffusion

Rm =a2em�

> ⇥2

Rm =a2um

�> ⇥

Numerical Spherical Dynamos Bullard & Gellman (1954) •  Found numerical evidence of dynamos with a non-axisymmetric velocity

• Higher resolution calculations proved these not to be dynamos

• Under-resolving dynamos can seriously damage your health! •  Give dynamos where none exists.

Spherical Dynamos Dudley & James (1989) Kumar & Roberts

Cellular Dynamos G.O. Roberts(1970,1972) 2.5 dimensional integrable flow

Can solve for monochromatic solutions in z. Essentially a 2D-problem

Cellular Dynamos G.O. Roberts(1970) Good dynamo at low to moderate Rm.

Fast Dynamos Vainshtein & Zel’dovich (1978) Childress & Gilbert (1995)

•  Easy to get dynamos now •  The “Fast Dynamo problem” is can flows be found for which

•  Growth-rate independent of diffusivity as the diffusivity becomes small. •  Clearly important for some astrophysical flows •  In order for this to occur the flow must have chaotic particle paths

� > 0 as Rm ! 1

Numerical Fast Dynamos Otani (1993) Galloway & Proctor (1992)

•  Can get chaos by making flows three-dimensional or time-dependent

Numerical Fast Dynamos Galloway & Proctor (1992)

•  Can get chaos by making flows three-dimensional or time-dependent

Here stretching beats the small diffusion and field is generated on a small scale

/ Rm�1/2

Bz⇤1

Numerical Fast Dynamos Galloway & Proctor (1992)

•  Growth rate rapidly becomes independent of Rm

•  and appears to stay independent of Rm as Rm gets large.

Numerical/Experimental Fast Dynamos Khalzov et al

Turbulent Dynamos

•  In astrophysics, geophysics and experiments the flow is not usually at one scale! •  Usually high Re and turbulent. Many scales in the flow and the field. •  Two important cases: high and low Pm

Pm = ⇥/�

High Pm

Low Pm

Low Pm Dynamos

•  For high Pm dynamos there is no problem (except field is generated on very small scales) •  For low Pm dynamos the magnetic field dissipates in the inertial range of the turbulence. •  This may (and does) lead to an inhibition of dynamo action so the critical Rm goes up as Pm goes down. •  Eventually (though possibly not soon enough for the experiments) the critical Rm becomes independent of Pm.

•  Short correlation time (Kazantsev) flows (Boldyrev & Cattaneo 2004)

Low Pm Dynamos

•  Numerical Demonstration

• Schekochihin et al. (2007) • Also shown for turbulent dynamos with coherent structures by Tobias & Cattaneo (2008)

Dynamo Saturation

•  Eventually the magnetic field will grow to a level where it is large enough to modify the flow that is generating it.

•  Good question is when this happens •  At this point the dynamo will saturate and the exponential growth will stop. •  How this is achieved is an open question and may be different in different situations

• Schekochihin et al. (2007) • Also shown for dynamos with coherent structures by Tobias & Cattaneo (2008)

Dynamo Saturation

•  Possible Mechanisms include •  Scale-by scale suppression of the flows •  Renormalisation of the flow to its marginal state •  Modification of the chaotic properties of the flow •  Back-reaction on the driving mechanisms

•  None of these seem to work in all cases (see e.g. Cattaneo & Tobias 2009 who showed that the exponential growth only stops for a set of vector fields of measure zero.)

Systematic “Large-Scale Dynamos”

•  So far… •  can we get dynamos, growth of magnetic energy

for a given flow.

• Astrophysics/Geophysics •  We want to get a large-scale systematic dynamo,

with a significant fraction of the energy on scales comparable with the size of the astrophysical object.

• Use as a paradigm the solar dynamo…

l  Question: How can we understand the dynamo properties of turbulent flows with a large range of scales l  Random fluctuations l  Coherent structures

l  Question: How can an astrophysical object such as a star or galaxy generate a systematic (large-scale) magnetic field at high Rm? How can it overcome its tendency to be dominated by fluctuations at the small scales?

Courtesy D. Hathaway

SUNSPOT NUMBER: last 400 years

Modulation of basic cycle amplitude (some modulation of frequency) Gleissberg Cycle: ~80 year modulation MAUNDER MINIMUM: Very Few Spots , Lasted a few cycles Coincided with little Ice Age on Earth

Abraham Hondius (1684)

Maunder Minimum

Observations: Stellar (Solar-Type Stars)

Stellar Magnetic Activity can be inferred by amount of Chromospheric Ca H and K emission Mount Wilson Survey (see e.g. Baliunas ) Solar-Type Stars show a variety of activity.

Cyclic, Aperiodic, Modulated, Grand Minima

Activity is a function of spectral type/rotation rate of star As rotation increases: activity increases modulation increases Activity measured by the relative Ca II HK flux density (Noyes et al 1994) But filling factor of magnetic fields also changes (Montesinos & Jordan 1993)

Cycle period

–  Detected in old slowly-rotating G-K stars. –  2 branches (I and A) (Brandenburg et al 1998)

  ΩI ~ 6 ΩA (including Sun) Ωcyc/Ωrot ~ Ro-0.5 (Saar & Brandenburg 1999)

HKRʹ′1−∝ʹ′ RoRHK

9.0−∝ RoF

Observations: Stellar (Solar-Type Stars)

SmaSSmall Scale Dynamos

The Large-Scale Solar Dynamo Helioseismology shows the

internal structure of the Sun.

Surface Differential Rotation is maintained throughout the Convection zone

Solid body rotation in the radiative interior

Thin matching zone of shear known as the tachocline at the base of the solar convection zone (just in the stable region). Shear is very important

Tachocline instabilities: Knobloch & Spruit (1982) (GSF et al) Gilman & Fox (1997) (Joint diff rotn mag field)

Large-Scale Computation

cf Toomre, Miesch, Brun, Browning, Brown (ASH code)

Basics for the Sun

.

terms,loss)(

,0).(

,.

),0.()( 2

TRpDtpDt

pt

t

otherviscous

ρ

ρ

ρρ

ρρ

η

γ

=

=

=∇+∂∂

+++×+−∇=⎟⎠

⎞⎜⎝

⎛ ∇+∂∂

=∇∇+××∇=∂∂

−

u

FFgBjuuu

BBBuB

Dynamics in the solar interior is governed by the following equations of MHD

INDUCTION

MOMENTUM

CONTINUITY

ENERGY

GAS LAW

Basics for the Sun

Ra ≡ gΔ∇d4

νχHP

νUL≡Re

ηULRm ≡

χν=Pr

ην=Pm

LURo Ω= 2

scUM =

202Bpµβ =

1020

1013

1010

10-7

105

10-3

10-4

0.1-1

1016

1012

106

10-7

10-6

1

1

10-3-0.4

BASE OF CZ PHOTOSPHERE

Ossendrijver (2003)

The Puzzle… •  Dynamo models at moderate rotation rates…

•  At low Rm, or for short correlation time turbulence can get large-scale systematic magnetic fields.

•  As Rm is increased (still not nearly close to astrophysical values) systematic magnetic field breaks down and small-scale dynamos emerge.

•  Can we fix this?

Basics for the Sun

.

terms,loss)(

,0).(

,.

),0.()( 2

TRpDtpDt

pt

t

otherviscous

ρ

ρ

ρρ

ρρ

η

γ

=

=

=∇+∂∂

+++×+−∇=⎟⎠

⎞⎜⎝

⎛ ∇+∂∂

=∇∇+××∇=∂∂

−

u

FFgBjuuu

BBBuB

Dynamics in the solar interior is governed by the following equations of MHD

INDUCTION

MOMENTUM

CONTINUITY

ENERGY

GAS LAW

Small-Scale Dynamos •  Small-scale dynamos rely on chaotic stretching and

reinforcement of the field (see e.g. Childress & Gilbert 1995) •  More coherent (in time) the velocity the better the

stretching (usually)

•  Any sufficiently chaotic flow will tend to generate magnetic field on the resistive scale.

•  Interesting questions do remain… •  e.g. Low Pm problem.

•  What happens when magnetic field dissipates in inertial range of the turbulence?

•  Coherent structures versus random flows

Starting point is the magnetic induction equation of MHD:

,)( 2BBuB∇+××∇=

∂∂

ηt

where B is the magnetic field, u is the fluid velocity and η is the magnetic diffusivity (assumed constant for simplicity).

Assume scale separation between large- and small-scale field and flow:

,, 00 uUUbBB +=+=

where B and U vary on some large length scale L, and u and b vary on a much smaller scale l.

,, 00 UUBB =〉〈=〉〈

where averages are taken over some intermediate scale l « a « L.

Large-Scale Dynamos

For simplicity, ignore large-scale flow, for the moment. Induction equation for mean field:

where mean emf is

This equation is exact, but is only useful if we can relate to

,)( 20 bGBub

∇+×∇+××∇=∂∂

ηt

.〉×〈−×= bubuGWhere “pain in the neck term”

Consider the induction equation for the fluctuating field:

⇥B0

⇥t= ⇥� E + �⇥2B0

E = ⇥u� b⇤

Traditional approach is to assume that the fluctuating field is driven solely by the large-scale magnetic field.

i.e. in the absence of B0 the fluctuating field decays. i.e. No small-scale dynamo (not really appropriate for high Rm turbulent fluids)

Under this assumption, the relation between b and 0B (and hence between

and 0B ) is linear and homogeneous. E

Postulate an expansion of the form:

where αij and βijk are pseudo-tensors, determined by the statistics of the turbulence. Simplest case is that of isotropic turbulence, for which αij = αδij and βijk = βεijk. Then mean induction equation becomes:

.)() 02

00 BB(B

∇++×∇=∂

∂βηα

t

α: regenerative term, responsible for large-scale dynamo action. Since is a polar vector whereas is an axial vector then α can be non-zero only for turbulence lacking reflexional symmetry (i.e. possessing handedness). β: turbulent diffusivity.

Ei = �ijB0j + ⇥ijk⇤B0j

⇤xk+ . . .

Growth-rates for dynamos at a single scale

• A fast dynamo is has an asymptotic growth rate as Rm gets large

• A quick dynamo is one which reaches this maximum growth rate close to (Tobias & Cattaneo, JFM, 2008)

• We define high Rm to be well into “the green zone” • Certainly the case for astrophysical flows

• Not usually true for numerical simulations (if we are talking about small-scale flows)

Rmcrit

�

�⇤

Rmcrit

Dynamos at a single scale l  For a velocity field

imposed at a finite scale l  Competition between

stretching and diffusion. l  If stretching strong

enough and coherent enough get exponential growth of field.

l  Field is usually amplified at small scales

-  Resistive scale

lB ⇠ Rm�1/2

Note:This flow lacks reflexional symmetry (helical) And should be a good large-scale dynamo

Gal

low

ay &

Pro

ctor

, (19

92) N

atur

e

Dynamos at a single scale l  Field is amplified on

local turnover time of the flow l  Independent of diffusion

as Rm gets large (fast)

l  Relies on l  Chaotic stretching of

fieldlines by velocity l  Measured by the finite time

Lyapunov exponent l  Not too much cancellation

l  measured by the cancellation exponent

Two important Ratios

1

1 ~10

= σ ⁄ σ★

χ Rm = ⁄

σ χ

χ Rm

σ χ

Dynamos at two scales l  Can get very interesting

dynamics at high Rm l  Mode crossing between

modes driven by large-scale flows and small-scale flows

l  Suppression of growth of small-scale field by large-scale flow at high Rm l  Shear enhanced

dissipation l  Reduced stretching

l  Decrease of Lyapunov exponents

l  Enhanced cancellation

Cattaneo & Tobias (2005) Physics of Fluids, 17, 127105

The effect of a large-scale flow at high enough Rm is to decrease the efficiency of a small-scale dynamo.

k=64

Dynamos at two scales C

atta

neo

& T

obia

s, P

hysi

cs o

f Flu

ids

2005

Large-scale versus small-scale l  All the dynamos described above have the ingredients

required to be a large-scale dynamo l  Lack of reflexional symmetry in the flow l  Leads to the generation of a mean EMF

l  However at high RRm (in the green zone - and even in the amber zone) the large-scale magnetic field generated by this EMF is completely dominated by the small-scale fluctuations provided by the small-scale dynamo (cf Cattaneo & Hughes 2006)

l  One idea is to use a shear flow to “boost” the EMF (and indeed the dynamo growth) via one of many effects (shear-current effect etc) (see e.g. Yousef et al 2008, Käpylä & Brandenburg 2009, Sridhar & Singh 2010, Hughes & Proctor 2013)

l  Though see Courvoisier & Kim (2009)

l  An alternative is to use the shear to control the fluctuations

E = ⇥u0 � b0⇤

High Rm effects of shear

l  Need to get to very high Rm (so growth-rate is asymptotic for small-scale flow.) l  Very hard to do in 3D flow.

l  Resolutions up to 40962 l  Use multi-scale generalisation of the GP/CT2005 flow (2.5 D)

l  Velocity amplitude decreases with scale; shear rate and turnover frequency increase with scale

l  Scale dependent renewal time l  comparable with local turnover time (in asymptotic regime) l  much shorter than local turnover time (poor dynamo)

+ +…

Rmk =Uk

k�⇡ 2500Rms ⇡ 0 ! 105

Tobias & Cattaneo (2013, Nature) Cattaneo & Tobias (2014 ApJ)

No shear: long correlation time

l  Great small-scale dynamo (no real surprise) l  Filamentary field with

l  Length comparable to scale of velocity l  Width controlled by diffusion

l  Overall pattern changes on the turnover time l  Comparable with correlation time

Tobi

as &

Cat

tane

o (2

013)

x

y Computations using UK MHD Consortium Machine at University of Leeds

B

x

(x, y, 0)

No shear: Long correlation time

l  No systematic large-scale behaviour l  E.g. average Bx over x and plot as a function of y and t

l  Can also construct a velocity field with no net helicity when averaged over time l  This has comparable growth-rate as a small-scale dynamo

l  Similar stretching, similar cancellation, similar pictures…

Tobi

as &

Cat

tane

o (2

013)

time

y

hBx

i(y, t)

Exponential growth removed…

Add some shear; long correlation time

No net helicity Helicity

With shear… To

bias

& C

atta

neo

(201

3)

time

y

hBx

i(y, t)

y

hBx

i(y, t)

HEL

ICA

L N

ON

-HEL

ICA

L

Exponential growth removed…moved to middle of the domain for clarity…

time

Mechanism? Long correlation time To

bias

& C

atta

neo

(201

3)

hBx

i(y, t)

l  Is it boosting the EMF?

l  Nope it is suppressing the small-scale dynamo (CT05)

l  This only happens for high enough Rm l  Flows in the green zone! l  Really for efficient enough small-scale dynamos

Ex

� � #

Are these really dynamo waves? To

bias

& C

atta

neo

(201

3)

l  Yes, they have all the right properties l  Propagate in correct direction l  Swap direction with direction of shear l  Period Decreases with increasing shear.

Short correlation time Of course reducing the correlation time reduces the efficiency of the small-scale dynamo (at fixed Rm) so χRm is decreased (even though Rm is still very high)

l  This flow is a good mimic of current 3D numerical dynamo calculations l  These small-scale flows are not usually optimised for small-

scale dynamo action and are usually run at Rm close to onset - χRm order unity

l  Not usually in the asymptotic regime (green zone) l  Growth-time is again on local turnover time but gets

longer as correlation time decreases.

Cattaneo & Tobias (ApJ 2014)

Short correlation time

Cattaneo & Tobias (ApJ 2014)

x

0 5 10 15 20V0

0

1

2

3s

Short correlation time

Cattaneo & Tobias (ApJ 2014)

x

Short correlation time

Cattaneo & Tobias (ApJ 2014)

x

0 5 10 15 20V0

0.00

0.02

0.04

0.06

0.08

0.10

-Ex

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-Ex

1

10

100

1000

Variance of EMF is a simple function of shear rate

(Tobias & Cattaneo 2014)

•  Clearly there are scales in a turbulent cascade that are too small to •  Feel the effects of a shear •  Feel the effects of rotation

•  These have very fast turnover times and low amplitudes and so will grow and saturate quickly?

•  These will provide a background noise to the large-scale dynamo (cf the magnetic carpet and the solar cycle)

•  Can only tell by performing a nonlinear calculation •  Currently underway…

Further Thoughts and Further Work

•  It is all very well to try and boost the EMF to sustain large-scale dynamo action, but… •  The small scale dynamo will win unless some agent acts

to suppress it. •  Shear •  Nonlinear suppression (faster eddies have less energy) •  Roughness of the turbulence (Subramanian & Brandenburg 2014, ArXiv)

The Suppression Principle (Cattaneo & Tobias ApJ 2014)

•  At high Rm •  Shear may suppress the small-scale dynamo

•  If small-scale dynamo is “as good as it gets” ratio χRm large

•  If small-scale dynamo is “weak” then shear may help ratio χRm small (2-3)

•  Shear may suppress the average EMF •  But narrows distribution (reduced intermittency) potentially •  making statistical approaches more accurate.

•  So what do we mean by a high Rm small-scale dynamo? •  One where adding a systematic large-scale flow decreases the

efficiency of dynamo action by making the flow more integrable…

Conclusions

•  So when do we get a large-scale dynamo and when a small-scale?

•  For a turbulent cascade… 1.  (a) Can calculate which scales are active in creating small-scale dynamos 2.  (T&C JFM, 2007). 3.  (b) Can calculate which scales contribute to mean and variance of EMF 4.  (T&C 2014) 5.  If timescale of (a) << growth-time of large-scales in (b) then SSD wins

6.  Note large-scale shear can modify both (b) and more importantly (a)!

Large vs Small