2016 GTP WORKSHOP, TURBULENCE AND WAVES IN FLOWS DOMINATED BY ROTATION:LESSONS FROM GEOPHYSICS AND PERSPECTIVES IN SPACE PHYSICS AND ASTROPHYSICS

The Turbulent MHD Geodynamo

John V. Shebalin, NASA/JSC

Geodynamo: Fundamental Questions*

(i) how does the field regenerate itself?

(ii) why is the external field dipole-like?

(iii) why is the dipole aligned (more or less) with

the rotation axis?

(iv) what triggers a reversal in the dipole field?*P.A. Davidson, Turbulence in Rotating and Electrically Conducting Fluids

(Cambridge U.P, Cambridge, UK, 2013) p. 532.

Earth has a Magnetofluid Outer Core

CrustLithosphere

Upper Mantle

Mantle

Outer Core: Liquid Fe, Ni, S, O, …

Inner Core: Solid Fe, Ni

RE = 6378 km

RCMB = 3480 km, = 10 gm/cm3

RI = 1220 km , = 12 gm/cm3

TroposphereStratosphere

MesosphereThermosphere

Exosphere

Magneto-fluid

Temperature too high (T > TC 1000 K) for ferromagnetism. Need a dynamo process (Larmor, 1919).

T = 4100 K

T = 1300 K

T = 5600 K

(from Wikipedia)

Outer Core Reynolds Numbers

• Reynolds number: Re =VARI / 5108

is the kinematic viscosity

• Magnetic Reynolds number: Rm =VARI / 3103

is the magnetic diffusivity (~ resistivity)

• Large Re and Rm MHD turbulence

Basic EquationsMathematical model based on the magnetohydrodynamic (MHD) equations with buoyancy the Boussinesq approximation; compositional variation is not included because of strong mixing.

ωgBjωuω 21o

Tt

ΒΒuΒ 2

t

sources)heat :(,2 hhTΤt

T

u

uωuu :Vorticity ;0, :Velocity

bjbb :Current ;0 , :field Magnetic

Navier-Stokes eq.:

Magnetic induction:

Temperature variation:

Geodynamo Simulations

• Codes use Chebyshev-spherical harmonics, other methods

• Computation is limited by computer memory & speed

Simulations based on approximate models (eqs. & b.c.s)

Turbulence not resolved, but modeled (hyperdiffusion, etc.)

• Important simulation by Glatzmaier & Roberts Nature 377, 203-209, 1995; Phys. Earth Plant. Int. 91, 63-75, 1995.

Demonstrated a magnetic dipole field reversal

Substantiated the geodynamo as an MHD process

Traditional Approaches to Turbulence• Homogeneous and incompressible: = o,

but some compressible research.

• Fourier analysis and numerical simulation:

waves in an infinite domain or periodic box.

• Energy: E = E(k)dk, energy spectrum: E(k).

• No dissipation in inertial range ko << k << kD,

ko: large scales; kD: dissipation scale; : energy input rate;

Kolmogorov inertial range law: E(k) = cK k .

MHD Turbulence•Turbulence usually simulated in periodic box:

Fourier expansions: u(x,t) = kũ(k,t)exp(ikx), etc.

•Ideal MHD turbulence: : Inverse energy cascade; Large-scale coherent structure; Statistical mechanics with broken ergodicity.

•Real MHD turbulence: Forced, dissipative case; large scales ~ ideal case?

•Outer core ~ spherical shell: Fourier method spherical Galerkin method.

MHD Turbulence in a Spherical Shell

http://www.es.ucsc.edu/~glatz/geodynamo.html

Need to move beyond a periodic box to a spherical shell.

One way: Define simulation boundaries inside of physical boundaries.

Homogeneous boundaries at ri, ro; b.c.s are ru = rb = r = rj = 0.

Core-mantle boundary:Gray circle

Computational boundary:Red circle

Galerkin Method for Spherical Shells

Use b(x,t) = lmn [blmn(t)Tlmn(x) + almn(t)Plmn(x)], etc.,

where Tlmn = Plmn, Plmn = kln2 Tlmn .

Each Tlnm(x) and Plnm(x) satisfies homogeneous b.c.s.

Basis functions Tlmn(x) & Plmn(x) are products of

spherical Bessel & Neumann functions

and vector spherical harmonics.

Mininni and Montgomery, Phys. Fluids 18, 116602, 2006; Shebalin, Geophys. Astrophys. Fluid Dyn. 107, 353–375, 2013.

+/ Helicity Expansions

.,0)(

,),()()(

:0:0,||1,

*

||||

,snqmpl

Vlmnpqsrlmn

lmnlnlmnmnlmnllnlmn

lnln

dV

kk

helicity,lhelicity,lLlkk

oi

JJxJ

JJxPxTxJ

(Mininni and Montgomery, Phys. Fluids 18, 116602, 2006)

“Chandrasekhar-Kendall functions”: Jlmn

.),(,),(,,,,

nmllmnlmn

nmllmnlmn tt JxbJxu

., ||||1

21

||||21

mnlmnllnlmnmnlmnllnlmn abkuwk

l,m,n-Space Dynamical System

^

.,

,2,

*2

*2

dVGkGk

dVFkFk

Vlmnlmnlmnlnlmnlnlmn

Vlmnolmnlmnlnlmnlnlmn

Jbu

JbjΩωu

= u = l,m,n klnlmnJlmn, j = b = l,m,n klnlmnJlmn.

If ideal MHD turbulence; can apply statistical mechanics.

Creating a spherical Galerkin transform method simulation is challenging.

.0, 22

if Theorem Liouvillekk ln

lmn

lmnln

lmn

lmn

Periodic Box as a Surrogate Volume‘Periodic box’ is a 3-torus, i.e., a compact manifold without boundary.

Topologically not a spherical shell, but both have a largest length scale.

Same ideal MHD statistical mechanics in sphere and periodic box.

Thus, we can use the periodic box as surrogate for a spherical shell.

Homogenous b.c.s Periodic b.c.s

2004 Kageyama

• 2 invariants if o 0 (3 invariants if o = 0):

Energy: E = (2)½(u2 + b2)dx3, (Cross Helicity: HC = (2)½ubdx3)

Magnetic Helicity: HM = (2)½abdx3, (b = a).

• Phase space ũn(k), bn(k) canonical ensemble theory.T. D. Lee, Q. Appl. Math. 10, 69-74, 1952; R. H. Kraichnan, J. Fluid Mech. 59, 745–752, 1973.

• Probability density function: D = Z1exp(E HM).

• MHD inverse cascade: magnetic energy smallest k.Frisch, et al, J. Fluid Mech. 68, 769-778, 1975. Fyfe & Montgomery J. Plasma Phys. 16,181-191, 1976.

~

Stat Mech of Ideal MHD Turbulence

‘Absolute Equilibrium Ensemble Theory’

.)(~)(~)(~)(~)(

,|)(~||)(~||)(~||)(~|)(

,1,2,2,112

22

22

21

21

1

3

3

kkkkk

kkkkk

IRIRNM

N

bbbbkH

bubuE

• Probability density: D = k Dk, Dk = ZkeE(k) –HM (k)

• Partition Function: k eE(k) –HM (k)dk

• Expectation values: A(k) = A(k)Dk dk

• Each k denotes a mode and the modal E & HM are:

• Ergodicity: A(k) =Ᾱ T 0TA(k,t)dt

?

Modal Probability Density for o 0

.

0000

000000

,

)(~)(~)(~)(~

1

ldimensiona8,)exp(

P.D.F. modal theis)exp(

2

1

2

1

†

†1

23

ki

kikM

bbuu

NX

, dΓdΓXMXZ

XMXZD

kkkk

k

kkkkkk

kkkkk

Expectation values: ũn(k) = 0, etc. Shebalin, Phys. Plasmas 15, 022305, 2008

Rotational Helical Eigenmodes

.0

,||,||

,,

,

0000

000000

)4(

)3(

)2(

)1(

kk

kiki

M

k

k

k

k

k

Helicity.,)(~)(~)(~

Helicity;,)(~)(~)(~Helicity;),ˆ2exp(~)(~)(~)(~Helicity;),ˆ2exp(~)(~)(~)(~

1221

4

1223

o1221

2

o1221

kkk

kkk

ΩkkkkΩkkkk

bibv

bibv

tiuiuvtiuiuv

i

i

• Then, the eigenvariables ṽn(k) may be written as helical waves:

• If HM > 0 and o 0, then Mk and its eigenvalues k(n) are

• ṽn(k), n = 1,2 are linear inertial and 3,4 are nonlinear eigenmodes.

Shebalin, Phys. Plasmas 16, 072301, 2009

Entropy Functional: () = lnD = so k,n ln k(n)()

• k2 = 1 terms of G must be very large in magnitude.

• Then, E+|HM| 2 ~ 0 and thus 1(4)k

(n) << 1.

.2/,,)2(||

||)(

;0)(,0)]()()[(2)(3

EHEEH

H

MM

M

kkG

FGGFNd

d

k

).1O(~,,2,2 3

rErNM

M

HE

E(E and HM are initial values; HM > 0 < 0)

Shebalin, Phys. Plasmas 16, 072301, 2009.

Eigenvariable Energies• Normalization: energy/unit-volume is E = 1.

• Since 1(4)/k

(n) ~ O(N) and En(k) = 1/k(n,

E4(κ) ~ O(1), | = 1; other En(k) ~ O(N ).

• Positive helicity e.v. ṽ4() is large for HM > 0.

• b(x,t) has a large-scale, helical component,

but is it quasi-stationary?

Fourier Method Computation

• Galerkin expansion: Each term satisfies the b.c.s.

• Fourier spectral or pseudo-spectral transform method.

• 3rd-order, Adams Bashforth/Moulton time integration.

• Isotropic truncation in k-space for ideal & real runs.

• Patterson-Orszag de-aliasing for ideal runs.

• Millions of time-steps for statistical stationarity.

Fourier Modes per k2

Spherical Galerkin expansions: Number of modes increases smoothly, lmn ~ n(2l+1).

Ideal Spectra

(Shebalin, Phys. Plasmas 20, 102305, 2013)

• Predicted and simulation spectra from a 323 ideal run are given below.

• Match is close, but there are some differences that don’t disappear with

run time, which indicates possible structure.

Ideal Runs: Phase Plots, k2 = 2

• Phase plots of (a) velocity and (b) magnetic components, starting at t = 1000(black dots) and running until t = 2000 showing statistical stationarity.

• These components appear to be zero-mean random variables; black circles represent predicted standard deviations.

Shebalin, Phys. Plasmas 20, 102305, 2013

Ideal MHD: Broken Ergodicity

Shebalin, Geophys. Astrophys. Fluid Dyn. 107, 411-466, 2013

ῦ4(ẑ), o = 1ẑ; t = 0 to 103

• Again, since |ῦ4()|2N ~ O(1), = 1, then for the

un-normalized eigenvariable, |ῦ4()| ~ O(N ).

• Broken ergodicity! ῦ4() ~ O(N3/2) becomes quasi-stationary.

323 Ideal Runs: Rotating and Non-Rotating Cases

ṽ4(k)

(Shebalin, Phys. Plasmas 20, 102305, 2013)

Origin of Broken Ergodicity for o 0(|| = 1, i.e., longest length-scale)

.)(~)(~)(~)4,)(~)(~0)3

,)(~)(~0)2,)(~)(~0)1

21242121

2122121

κκκκκ

κκκκ

bibvbib

uiuuiu

i

i

.0)(~)(~and0)(~,0)(~2121 κκκκ bibuu

• The 1st, 2nd and 3rd equations tell us that, for HM > 0,

• Ideal statistics tells us that (4)(1,2,3)

• N|ṽn(k)|2 = 1/k(n), so if (4) (1,2,3) then

Large-Scale Mode (|| = 1) ~ Quasi-Stationary

.)O(~)(lnothers allfor but ),O(~)(ln

then,)O( ~)(~)(~and)O(~)(~~)(~If

2/32/34

2/312

2/312

Nvdtd,Nv

dtd

NbibNuiu

n kκ

κκκκ

• Once ṽ4() becomes big, then ṽ4() ~ quasi-stationary.

• Define RMS dipole moments and angles w.r.t. z-axis.

• Alignment is seen numerically, but needs explanation.

• Order of magnitude analysis gives:

Dissipation, Forcing & Rotation• Undertook some 643 runs with & adjusted so that kD kmax.

• Compared runs with o = 0 and o = 10.

• Various ratios of kinetic to magnetic helicity injection energy.

• ‘Quasi-stationary forcing’ at kf = 9, kinetic energy fraction: c2.

Shebalin, Phys. Plasmas 23, 062318, 2016.

• Newer runs have helically symmetric forcing at same kf = 9.

Dissipative, Forced RunsShebalin, Phys. Plasmas 23, 062318, 2016

• All 643 runs had same initial conditions.

• Quasi-stationary forcing with c2 = 0.9, 0.5 and 0.1.

Input E: 90% EK, 10% EM Input E: 10% EK, 90% EM Input E: 10% EK, 90% EM

x-Space RepresentationShebalin, Phys. Plasmas 23, 062318, 2016

Shebalin, Phys. Plasmas 23, 062318, 2016.

Contours of as, (a) s = êx, (b) s = êy, (c) s = êz, at t = 1100; run KM07: o = 10, c2 = 0.1.

Shape, strength, direction are time-dependent and contours give only partial information.

Finding optimal computer graphic representation is a challenging task, perhaps more

revealing for spherical shell rather than periodic box simulations.

New 643 grid-size simulations, forced with + helicity & varying amounts of helicity symmetrically injected at k = (9,0,0), (0,9,0) and (0,0,9).

Recent Forced, Dissipative Run N06

(1) EM , HM and other global quantities.

(2) Energy spectrum with forcing & dissipation

(3) Homogeneous dipole angle z.

3-D Spectra are Asymmetric

• Forcing is helically symmetric; injected EK/EM can vary.• Asymmetry persists in dissipative, forced runs.• However, ideal & dissipative spectra are isotropic.

• ‘Inertia tensor’ of 3-D spectra can be determined.• Principal moments related to homogeneous ellipsoid.

Homogeneous Dipole Moment

RMS dipole moment:

Homogeneous dipole angle:

New 643 grid-size simulations, forced with + helicity & equal amounts

of kinetic & magnetic helicity injected at k = (9,0,0), (0,9,0) and

(0,0,9).

(a) t = 0 to 567, (b) t = 567 to 1700.

Phase Plots for ῦ3,4(), = 1

Quasi-stationary, large-scale, coherent structure appears.

Wave associated with ῦ4(ẑ) undergoes ‘secular variation’.

N-KM06o = 10

New 643 grid-size simulations, forced with + helicity & equal amounts

of kinetic & magnetic helicity injected at k = (9,0,0), (0,9,0) and (0,0,9).

Phase Plots of o = 0 vs o = 10 Runs

Large-scale coherent structure dipole; angle depends on o.

Is MHD turbulence an essential ingredient of the geodynamo?

Lessons Learned?Geodynamo: Fundamental Answers? Perhaps some hints.

(i) how does the field regenerate itself?

MHD turbulence

(ii) why is the external field dipole-like?

Broken ergodicity

(iii) why is the dipole aligned (more or less) with the rotation axis?

Initial magnetic helical structure rolls over?

(iv) what triggers a reversal in the dipole field?

Disruption of the energy injection process?

Summary

The Turbulent MHD Geodynamo

• Analysis and computation based on wave expansions.

• Ideal case seems pertinent to forced, dissipative case.

• Large-scale coherent structure (~ dipole) emerges.

• Forced, dissipative spectra is seen to have structure.

• Fourier results suggest spherical Galerkin runs be done.

• Magnetic Prandtl number effects? Influence of forcing?

The End