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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