Magnetic Rossby waves in the Earth’s core

Kumiko Hori1,2, Chris Jones1, Rob Teed3, Steve Tobias1
1) Department of Applied Mathematics, University of Leeds.
2) Graduate School of System Informatics, Kobe University, Japan.
3) School of Mathematics and Statistics, University of Glasgow.

Waves in the Earth’s fluid core
Waves provide us with information about the ‘invisible’ system• torsional Alfven waves (e.g. Braginsky 1967, Zatman & Bloxham 1997)– axisymmetric, travelling in radius s– ~ 6 yrs traveltime: Bs >~ 2 mT (Gillet et al. 2010, 2015)• axisymmetric MAC oscillations (e.g. Braginsky 1993)– in a thin, stably stratified layer at the top of the core?
– ~ 60 yrs geomagnetic variation: H ~ 140 km? (Buffett 2014)• slow magnetic Rossby waves (e.g. Hide 1966, Acheson 1978)– nonaxisymmetric, travelling in azimuth f– ~ 300 yrs westward drift: Bf ~ 10 mT? (Hori et al. 2015)• (fast magnetic) Rossby waves in a thin stable layer (e.g. Braginsky 1984)– ~ 6 yrs westward drift? (Chulliat et al. 2015)– in the solar tachocline also?: ~ 2 yrs westward? (McIntosh et al. 2017)

An axisymmetric mode: torsional Alfven waves• A special class of Alfven waves(Braginsky 1970; also Roberts & Aurnou 2012)– the azimuthal momentum eq on cylindricalver.2.2 September 4, 2017 KH
surfaces in the magnetostrophic balanceff2 | gives
ff
a2steady state (Taylor 1963)
art (such as |Bs and |B |) are greater than those of distinct ones (|Bs B |), the equation can berastically simplified
to only leave
the homogeneous– cylindricalperturbations
onpartthe state0@ 2 hu i

@t2 s

»

=

1
@
s3 hh⇢i @s

hu0 i
2 @

s3 hh⇢iUA

@s s

!

⌘

@ FLR
@t sh⇢i(9)

travel in radius s with the the z-mean

174
P. H. Roberts
J. M. Aurnou
0 i/sandf2 i/µ0 h⇢i, implying a wave
1/2
here UA2 = hB
for angularvelocityhu
inthe anelastic case (Jault
Alfven speed UA =equation(<Bs2>/<r>µ)s0& Finlay, 2015). Here the restoring force is represented by FLR , while the remaining terms can be summed
p to a forcing
term FLD = FL0 FLR , as well as FR0 . A perturbation of angular velocity can propagate in• Data:ylindrical radius s with an Alfvén speed UA . The speed depends on the magnitude of the backgroundprobably
responsible
variations
oloidal field B–s2 andthe backgrounddensityfor⇢, 6-7bothyearof which
may vary with s. As expected for the» modecan accountfor the 6 year LODchangeMHD wave, this special
is also nondispersive,i.e. thespeed independent of wavenumbers. Since thequation allows–both
inwardand outward
a superposition
theobserved
wavepropagation,
speed is used
to infer of those modes, provided a similar
mplitude and proper excitation, could yield its standing waves and enable normal mode solutions. In
the field strength within the corearth, neverthelss, data as well as numerical simulations indicate its propagating nature, preferable to2 1/2 ≥ 2 mT
» 1).<Bs
tanding ones (see sec.
We> shall
discuss below that standing TWs could be chosen in Jupiter.

»

better fits with the scaling law
ber 2012

Numerical simulations

4. Schematics(a) threea geostrophic
flow
in the core,Vg, and (b)a planwereview of an initiallyo explore potential excitation of TW inFigure
the gasgiant, weshowing
chooseJoviandynamomodels,which

Filtered LOD cha

Cohe

0.4

oscillation. Time–cylindrical radius map of the bandpass-filtered angula0
The colour scale ranges between 20.4 km yr21 (blue) an
velocity ũg(s, t).
10.4 km yr21 (yellow) with contours every 0.02 km yr21. a, Filtered–0.1
ensemble average.
As inside the black box in Fig. 2b (outside the tangencylinder for 1960–1982) but truncated at spherical harmonic degree n 5This corresponds~obs–0.2 to the observation ug ðs, t Þ used for the data assimilatio
U
in
a
core
flow
model
inverted from
Assimilationf (n ( 9). Predictionsb, Assimilation output~predg ðs, t Þ resulting from the$ obs % u
theofgeomagnetic
variation
gufm1 paramet
for damping
torsional wave assimilation
the ~tn0 ðt Þ n~1,3,...,901
27b(aG, aB) 5 (10 , 3 3 10 ), with a normalized misfit of 0.87 (seeSupplementary Information for details).

An axisymmetric mode: torsional Alfven waves
0.2
0

b 180

• A special class of Alfven waves
120

Phase (degree)(Braginsky 1970; also60Roberts & Aurnou 2012) :!

Cylindrical radius0– the azimuthal
momentum eq on cylindricalver.2.2 0.8 September 4, 2017 KH−60
surfaces in themagnetostrophic balancechanges of the core angular momentum. We are in the presence
0.6−120
forced torsional
waves24, the triggering mechanism of which remaif
2steadyf2 | gives
f
f
a
state
(Taylor1963)
rt (such as |B
and
|
B
|)
are
greater
than
those
of
distinct
ones
(|
B
B
|),
the equation can be
s
s
to
be
understood.
Assuming that the geostrophic velocity at s 5 0.3−180
astically simplified
to only leavethehomogeneous
30
20 15
10 on
8 part7the
6 5 state
430.4– cylindricalperturbations
does not differ much from the angular rotation of the solid inn
Period (years)

00 i core, only a tiny inner core oscillation is predicted from our study,hu@ 2 1hu@ not only@
Figure| Highi coherence1is foundonlongtimescales,
but also on @ FLR0.2 22
32
theareorder of 10 degrees per year. Although
it is difficult to dete
=
s
hh⇢iU
(9)
A
an approximately
six-yearperiod. Coherence (a) and phase (b) spectra⌘
2 s
3 hh⇢i
26
@t
s
@s
@s
s
@t
sh⇢i
calculated over the time span 1925–1990 for the DLOD time series from seismological studies , this quasi-harmonic signal could co0
LUNAR97 (ref. 13) and the predictions from the average of an ensembleof a target
stituteto1965
focus.1970 1975
1955on which
1960
1980
1985
» travelinradius
s
with
the
the
z-meanquasi-geostrophic
core
flow
models(see
Supplementary
Information).
2
0
f
2
Time
(years)2
1/2
ere UA = hBs i/µ0 h⇢i, Alfven
implying
acorrespondwave
for angularvelocity hu i/s in the anelastic case (Jault
speed
UA =equation
Green
segmentsto(<Bfrequency
rangesovers >/<r>µ0 )which the phase shift is
Figure 2 | 6The terms
six-year DLOD
is carried by geostrophic wave-likethan 30u.force is represented by FLR , while the remaining
Finlay, 2015). Here thesmaller
restoringcansignal
be summed
patterns travelling from the inner core to the outer core Equator.
to a forcing term FLD = FL0 FLR , as well as FR0 . A perturbation of angularvelocity can propagate ina, Comparison of DLOD time series, bandpass-filtered between five and
5
from
the
tangent
cylindertotheEquator.
Such
fastpropagationis13
years, of the LUNAR97(green), the predictions from theindrical radius s with an Alfvén speed UA . The speed depends on theeight
magnitudeof the data
backgroundmadepossiblebyalarge
magnetic
field
insidethecore,ofamplitude
average of the quasi-geostrophic core flow models (black), and theloidal field Bs2 and theseveral
backgrounddensity⇢, bothvaryensemble
withs. As expected for the
millitesla. At large
cylindricalradii, inof
thewhich
Equatorialmay
region,result(red)$ obs % 4 of the torsional wave assimilation of the flow coefficients~tn0 ðtof
» modecanaccount
fordown.
the 6In year
LODchangethe propagationa torsional
scenario,
obserHD wave, this special
is also slows
nondispersive,i.e.wavethe
speedthatindependent
wavenumbers.
Since
the Information). b, TimeÞ n~1,3,...,9for 1960–1982 (seeSupplementary
versus cylindricalradiusmap of theaband-passfiltered angular velocity
vationandis consistenta weaker field aclose
to the Equator.of those3
uation allows both inward
outwardwith
propagation,
superposition
modes,
providedsimilar
(s,
t)
for
the
ensembleaverageofthequasi-geostrophic
core flow model.
ũ
Furthermore, the absence of a reflected wave suggests the presencegmplitude and proper excitation,
could
yield
its
standingwaves
andenablenormalmode
solutions.
Inin outer core radius units. The colour scale ranges betweenof significant Ohmic dissipation. This is due either to large gradients Distance is 21
2 (blue) and 10.4 km yr21 (yellow) with contours every
20.4
km
yr
rth, neverthelss, data ofasthewellas magnetic
numerical
simulations
indicate its
nature, preferable toinducedfield, resulting
from inhomogeneities
in thepropagating
2> 1/2 ≥ 2 mT
0.02 km yr21. The horizontal dashed line at s 5 0.35 corresponds to the»<Bsvelocity,or to thepresenceof a conducting
at thecould be chosen in Jupiter.
nding ones (see sec. 1).Alfvén
Wewaveshall
discuss
below
thatstandinglayer
TWsposition of1the tangent cylinder. The black box corresponds to the space19
base of thefitslowermantlehave explored» betterwiththe. We
scalinglaw this second hypothesis. time domain used for the assimilation of torsional waves (Fig. 3).

– probably responsible for 6-7 year variations– the observed wave speed is used to infer
the field strength within the core

Bs r.m.s. (mT)

• Data:The ensemble average of the filtered zonal flow coefficients are now
0considered as observations for a second inversion. To obtain not only
0.4 only0.5
0.9
Numerical simulations
we
thus
providea lower0.6
bound 0.7
for the 0.8
field intensity
in 1that
the strength but also the profile of B~s ðsÞ, we use a variational data
(Gilletet al. region.
2010)Our findings are inCylindricalline with radiusestimates (1) inferred fromassimilation framework20,21 that rests on the torsional wave
equation
quasi-geostrophic
core-flow
inversions
assuminga magnetostrophic~
19
Figure4 | Themagnetic
field B whichðsÞ exceeds
in most of the fluid
explore potential excitation of TW in the gas
giant, we choose three
Joviandynamomodels,were2–3 mTwere previously obscured.Ratherthan a statnonaxisymmetricradial
ic picture with small-amplitude features
magnetic field, highridingon top, the processed data of the residpass filteredwith a cutual field reveal a dynamic f1eld morphology
off period of 400 years
studyears
400
the
over
rapidly
evolvesthat(referredto in the text
as the residual field),ied (Fig. 1) (movie S3). In the equatorial
shown at the core surregion we observe a series of high-amplitude
face in 1850.
flux foci moving westward.Field changes aremost obvious under the Atlantic hemisphere
while less activity occurs under the Pacif1chemisphere, suggesting some longitudinal• significant in the Atlantic hemisphere: period ~ 3*102 yrs
modulationof the field or of the mechanism
causing its motion (12, 13).
(Bullard+ 1950)(Hide 1966)
We constructedtime-longitude diagrams
01
0.05
0
-0.05
-0.1
of the residual f1eld every 2° of(14-1
Residual radial magneff¢ ffeZdI mT
latitude in order to view zonal motions,which are importantin rapidlyrotatingfluids
lo*1
such as Earth's liquid outer core because of
the influence of strong Coriolis forces. WestI oxQe
ward motion of a succession of flux foci was
I o.06lessand
2A)
observed at the equator (Fig.
ooooclearly at mid-latitudes (e.g., Fig. 2B atl oloS
40°S). Two-dimensional frequency-wavel oso
4>numberpower spectra were calculated fromE
4.02
i
the time-longitude diagrams. Peaks in these
4.04
spectra pinpoint the preferred zonal waveI 4.06
numbersm (where m = 360°/A and A is theangularwavelength in degrees) and frequen14::>V 1WV-IW -:tU
I 0.1
ciest(wheref = 1/T and T is the period in
Longitude J degrees east
years) of the zonal motion of the residual
a
_
10
at
f1eld at each latitude. At the equator, the
:
;
O.01
I
D
0-01
*
g
s
c
*
8
dominantwavenumberwas m = 5 (i.e., A =*e72°) andf = 3.75 x 10-3 year-l (i.e., T=*s--;--^@*O-QQ6'- - - - - ' " - - - - - - '-- - - - - - i
0.006 -- @--bt
270 years), whereas at 40°S, the f1eldchange
powerwith morepartwas less monochromatic
Nonaxisymmetric
of at
Br
^ j
e
--v-^0,004
gg2:::::::J@::::::,::::::::}
0002- - - - - - - - * - w- " - - - - - - - - t
04002' - - - a
found
we
20°N,
At
lower wavenumbers.
at the surface of the core L
&
_O
.
.
o
with highnconsistentstrong m = 8 signal
.
O
10
5
0
at theequator/ 40°S
4
-10
10
5
0
4
-10
E
W
resolutionmaps of the radialmagneticfield at
EW(gufm1: Finlay & Jackson 2003)
wavenumber
spaffalZonal
the core surface, recently obtained from satZonal spatial wavenumber
2. Time-longitudediagramsof the residualfield at specificlatitudes(A) 0° (the equator)and

Nonaxisymmetric waves in the core?

• Possibly related to the geomagnetic westward drift– the nonaxisymmetric part of the field moving in azimuth– probably a mixture of flow advection
and wave propagation• à How can we separate the signal due to waves?
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229
using
our
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0
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theoryf
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djz
bstitution
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into this andzi
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+
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+
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(r
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(17)
0xzE 1978;
2ingredients2 ) dt s
2 also2 Hori et al. 2015):
z uy =
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(r
s
s
@
µ
=
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L
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o
*equation+can be solved analytically (see (18)230
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this0xCanet et2′ 2 0
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′
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foruequation
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s
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2 @analytically2
r 2some simple
fieldssolveddt
E (ro2 232
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00(see Canet et∂t∂x(L=O(1);Ro,E<<1)geostrophic, so that us (H) + us ( H) ⇡0 2hus i, and that the radial gradient(2014) for detailed
analysis).
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us istheapproximately′′ ⇠ 0 ,suppose
233
of
the
axial
vorticity,issmallerthanazimuthal gradient, consistent
∂ξ
∂u
z
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′ the
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fieldsthisequationbesolvedanalytically
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#
ostrophic,
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u
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and
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234
with
our
previous⇡
hu0s i. This is valid (19)only for
z
′
′
2014)
for
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analysis).
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insteadsupposethatuisapproximately
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∂t
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s
∂ξ
∂u
the axial vorticity,⇠,issmallerthantheazimuthal
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consistent′ it #zz 0ρ ′ z large
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m0s0 i,components,
but
considerably simplifies the (19)
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·
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× gradientB)
trophic, so that235ucoupled
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and
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th our previous
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is valid only for
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=than
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· ∇u
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s i. This236 ⇠ 0leaving
he axial vorticity,theazimuthal
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itsimplifies
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i.e.=⇠z0 B⇡· ∇u
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#but
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aving
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f
onably large m components,∂tbut
simplifies
theproblem
2 it considerably00
d 1 @hus i P m
2sdhus i P m B @ us
caling
1
=0.
(19)
ing
– spherical geometry
2 s @ (topographic
2 * b-effect)2 ) dt +
3 @ 3
dt
E
(r
s
E
s
f2 @ 3 u00
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11 @hud2thePmdhuof0s iz*(quasi-geostrophic)
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nsionalise
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f2 @ 3 u0=0.
(19)
d2 1 2@hu0s i P m
2s 2 dhu0s2i P m B
s s3 @ 3 0
ı(m
!t)
dt
s
@
E
(r
s
)
dt
E
=
0
.
(19)
ensionalise
with
Here
solutionswitha formofhus i ones⇠e
at given s and obtain
2wetsseek2o) dt3 @3than–y237equations
azimuthallengthscalesshorterradialxdt2 s @the
v
g
ψ
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o
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and
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as
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d
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c
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andmodesslow
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+B!
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(20)
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for Bf = B0 s ef
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(20)
adv!
c!
(22)(after Malkus 1967)
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R 2 Alfvén,
!
ˆ
2
239
where
the
Rossby,andadvection
22!
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should rule them. This
where Rossby and Alfven frequencies
equationseredimensionlessvelocity
potentialψchosen sofrequencies
that
the areinduction equation
should rule them. This
the Rossby,
Alfvén,andis advection
2 f2
2here the Rossby, Alfvén,andadvection
frequencies
are
fi
m
h
B
i
Pm
Pm
2s
mhU
2!
he dimensionless equations"!
ˆ R∆= ψ) c222 f, c!
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and !adv =, (21)
2∂(g, ∆2 g)
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∂
∂(ψ,
∂ψ
2
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f
E
(r
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s
s
R
R
m
h
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i
P m ∆2s"ϵ, !ˆ 2 = P m− o2 2 m2 hand=
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2ψ + 2
B
R =
M
P
m
P
m
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i
∂
∂(ψ,
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c
∂ψ
c
∂
∂(g,
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2
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2
∂t
∂(y,
x)
c
∂x
c
∂x
∂(y,
x)
2
2
2
sR
A
ARand ∆!
ˆ R =E (ro ∆22s240
ˆ ME =−We!sadv=ϵ, (21)ψ)m+2respectively.
ϵ , !
(23)
2g +
is thesum of the dynamical
2= the2 see that
2 wave frequency

Magnetic Rossby waves

1

0.8

0.8

0.6

0.6
0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4
−0.6

−0.4

−0.8

−0.6

−1

−1

−0.5

0

0.5

1

Magnetic Rossby waves (cont’d)• Fast modes:– w+ à + wR (1+ wM2/wR2) in the limit wM2/wR2 << 1
The fast modes !
ˆ + essentially recall the hydrodynamic Rossby waves,–essentially(nonmag) Rossby
waves (Busse 1986)
hich travel prograde with the frequency !
ˆ R about the advection part. They
– travelling
progradely
timescales
rise from a balance
betweenthe first(eastward)
two terms withofver.2.0(19),dh⇠⌅C . KHz i/dt andJanuary
9, 2018
O(months)
incore solution of rotating MHD,y contrast, theofslow
modes !
ˆ thearefluida uniqueometimes called MR waves or MC-Rossby waves. Their properties shallow
become B B’!
"2
2
shallow
videntwhen
takingthe4Ωχlimit
ˆ M /ˆ
!RB⌧
1 onˆg) , toobtain• ∂(ψ,
Slow
modes:∂∆2 ψ)∂ψ!
∂ the slow
1 mode,∂(g, ∆2!
y
0x
B B’
∆2the
ψ +binomial approximation)−=∆ver.2.0
January 9, 2018 KH (14)
2g +
using
∂t
∂(y, x)L2 ∂xρµ0 ∂x 2
2 B0x ∂(y, x)– w- à - wM /wR in the limit wM /wR << 1

z

x
B0
B0

¤Ω

cold

cold

j

g
g

u

deep
∂
∂(ψ, ∆2 g)∂∆2 g +∆(15)2
2 2ψ 2f
2 = Bm0x3 hBju
ˆ M ! ∂x i(ro s )hot"∂t∂(y, x)!
deep
!
ˆ M∂ψ
=
=
,
(22)
∂(ψ, ∆22ψ) 4Ωχ
B
∂
1
∂(g,
∆
g)
R
2
ˆ R 0x2s4
ψ + ∂ψ ∂ ψ −=!
∆2 g +
(14)
hot
cold
= x) 2 = ∆2Lg =∂xg = 0 ρµat0 ∂xy = 0, D B0x ∂(y, x)
(16)
∂(y,
∂x
∂y– travelling retrogradely (westward) along thend the∂ observed
frequency
ˆ M R andthe advection
freshallow∂(ψ,∆2 g) will be ∂the sum of !
coldB’
2 years)
toroidalfield
Bontimescales
of
O(10
∆
g
+
=
B
∆
ψ
(15)
2
0x
2
f
uency∂t!adv . This∂(y,
implieslower frequency and a retrogradeshallow
propax) a much ∂xar theoryB’u’• cf. torsional
Alfven
waves
along
Bs
ation
flow
is
large
and
eastward.
The
corresponding
2 unless the advective∂ ψ= speed= ∆is2–ggiven=highlyg=y =and0, Dsimilarly for the Rossby and Alfvén
(16)
hase
VM0dispersive!
ˆ M R /m,
R = atjʼ j’ u’′∂y 24ρΩχ ′∂jz
j’ x Bdeep
hase speeds.–Themagnetic
Rossby
speed
goes
up
as
the
wavenumber
m
in−
u
=
B
(17)
j’
0x
y
the governingLequations(Cartesian)jʼ∂xbetween the last two terms, ⌅hotj’ x Beases or the radius s decreases.
Abalance
deep
C
′
′
eory
∂j
B0x ∂ξzwave propagationnd ⌅L , is vital for this mode,z indicatingthat the time variations arise from (18)
=
hot
∂t
µ0 ∂x
he induction equation while the momentum
equation is almost in balance.
wave propagation′
wave propagation
4ρΩχ
∂j
z
′
hese slow waves willfrom Alfvén or Rossby (fast MR)− be distinguisheduy = B0x(17)
L∂x
odes in terms of dispersion
relations! = !(m),phase velocity !/m, and∂ξz′′∂u′z ′(Hori, Takehiro & Shimizu, 2014)
′
#
∂j
B
∂ξ
ρ
−
2ρΩ
=
ê
·
∇
×
(j
×
B)
(19)
0x
z
z
z
orticity balances. ∂t =
(18)
∂z
0

0

Waves hint at strong-field dynamos?
• Linear, rotating magnetoconvection(e.g. Chandrasekahr 1961, Fearn 1979; also Zhang & Schubert 2000):– as magnetic field is strengthened to L=O(1), thethermal stability Racrit , the preferred wavenumber
kcrit , and wave frequency wcrit drop
– dynamos hypothesized in the regime:‘strong-field’ dynamos (e.g. Roberts 1978)– Note: all three effects not necessarily• depend on the background magnetic
field, boundary conditions, etc.

Flow vigor
U or Rm
Rmd
RmdmagField strength
B2 or LRacmag= O(E-1)

Ra
Rac= O(E-4/3)
Strong field dynamo

L=O(1)

Weak field dynamo

L=O(E)
RadmagRad

Ra

Waves hint at strong-field dynamos?
• Linear, rotating magnetoconvection(e.g. Chandrasekahr 1961, Fearn 1979; also Zhang & Schubert 2000):– as magnetic field is strengthened to L=O(1), thethermal stability Racrit , the preferred wavenumber
kcrit , and wave frequency wcrit drop
– dynamos hypothesized in the regime:‘strong-field’ dynamos (e.g. Roberts 1978)– Note: all three effects not necessarilyhydro• depend on the background magnetic
field, boundary conditions, etc.

• Convection-driven spherical dynamos likely
approaching the regime (e.g. Yadav et al. 2016; Dormy 2016)

dynamo– force balances– flow properties? (vigor/heat transfer/subcriticality,
azimuthal length scales, and wave time scales)• cf. plane layer models

Fig. 2. (A−D) Radial velocity, given in terms of the Reynolds number (u D=ν, where u is the local velocity), in the equatorial plane of the HD simulations.
Radial velocity in the equatorial plane(E−H) The same for the corresponding magnetohydrodynamic cases. (A and E) E = 10−3, Ra = 6 × 105; (B and F) E = 10−4, Ra = 7 × 106; (C and G) E = 10−5, Ra =-6, Ra/Ra = 10, Pm/Pr = 0.58−69Ecolor= 10mapsThe
are saturated
at values lower
10 ; and (D and H) E = 10 , Ra = 2 × 10 . The Rayleigh number of all of the cases shown is about 10 times Rac .at
cthan the extrema to highlight fainter structures.
(Yadav et al. 2016)

ver.1.4

March 4, 2015 KH

March 4, 2015 KH
Convection-driven,
spherical
dynamo ver.1.4
simulations
Formulation/scaling
for numericalsimulations

ased onnote in September 2014; Teedet al. (2014,b);
3 RT’sFormulation/scaling
for numericalsimulations• Greatly studied for the past decades (e.g. Glatzmaier & Roberts 1995; Kageyama & Sato
1995; also reviews by Christensen & Wicht 2007; Jones 2011)

Based
RT’s note in September 2014; Teed et al. (2014,b);Scalingtheonvariables:– successful for reproducing
observed featuresof planetary magnetic fields
2
2
!
D
ϵD
Scalingthevariables:– a tool forthewithfieldsx =understanding
Dx∗ , t =t∗ , dynamics
T =T∗ , self-generated
B = ρµ0 ΩηBmagnetic∗
x = Dx ,

η D2

t=

t ,

η ϵD2

T =

T ,

B=

!

ρµ ΩηB

(16)
(16)

∗
∗
∗
0
∗
he dimensionlessηη• MHD governing
dynamosequations:driven by Boussinesq
convection
in rotating spherical shells:

The dimensionlessgoverning– Governing
equations(dimensionless)∂u
P mequations:

P m2 Ra2+ u · ∇u =[2êz × u − ∇p + (∇ × B) × B] +
T
ê
+
P
m∇
u
r
2
∂t
E Pm
r Ra
∂u
PPm
+
u
·
∇u
=
[2ê
×
u
−
∇p
+
(∇
×
B)
×
B]
+
T êr + P m∇2 u
z
∂T
P
m
2
Pr
+ u∂t· ∇T =
∇E
T −1
∂T
∂t
Pr Pm 2+ u · ∇T =∇ T −1
∂t ∂B
Pr
2
=
∇
×
(u
×
B)
+
∇
B
∂t ∂B = ∇ × (u × B) + ∇2 B
∇ · u =∂t 0 , ∇ · B = 0∇·u = 0,∇·B =0

(17)
(17)

(18)
(18)

(19)
(19)

(20)

(20)

here the non-dimensional parameters are

modified
Rayleigh,where –
theParameters:
non-dimensionalparameters
are Ekman, kinetic/magnetic Prandtl numbers
gα|ϵ|D5νννgα|ϵ|DνννRa =, 5E =
,
P
r
=
,
P
m
=
Ra νκη
=, EΩD= 2 2 , Pr =
,
P
m
=
κ
η
νκη

ΩD

= 10-4 - 10-6

κ

η

(21)

(21)

~ 16 Racrit electrically insulating =(except
1= 1-5 dipole of magnitude B0 ),• magnetoconvection:
rigid/no-slip,the• magnetoconvection:
rigid/no-slip, electrically insulating (except
theaxial
axial dipole of magnitude B0 ),fixed temperatureb.c.s, b.c.s,
anddiﬀerential
fixed
temperatureand diﬀerential
heating– Leeds
spherical
dynamocode: heating
based
on pseudo spectral method (e.g. Jones et al. 2011)• dynamos:
rigid/no-slip,
electricallyinsulating,fixedheat-flux
b.c.s,• dynamos:
rigid/no-slip,
electricallyinsulating,fixedheat-flux
b.c.s,andandvolumetricvolumetric sinksink

Slow MR waves in dynamo simulations
z-mean radial velocity <us>in the equatorial plane• Slow modes identified:

The fast modes !
ˆ + essentially recall the hydrodynamic Rossby waves,–
retrograde
commonlyin
which travel prograde with thedrifts
frequency!
ˆ R about seen
the advection
part. Theyarise from a balance
betweenthe first two terms of (19), dh⇠z i/dt and ⌅C .
dynamosimulations
By contrast, the slow modes !
ˆ are a unique solution of rotating MHD,–
their
speedsfor byTheir
total
sometimes called MR waves or accounted
MC-Rossby waves.
properties become22evident when taking
limit !
ˆM/ˆ
!wave
themeanslow mode,
phasethespeedsofandflow!ˆ , to obtainR ⌧ 1 on(using the binomial
approximation)
advection,(w + w )/m, whereMR!
ˆM R =

2
!
ˆM
=
!
ˆR

adv

f2 i(r2
m3 h B
o
2s4

s2 )

,

(22)
<us’> at s=0.5ro(a)Time / mag. diff.time

and the observed frequency will be the sum of !
ˆ M R and the advection frequency !adv . This implies a much lower frequency and a retrograde propa(wMR (m=8)+wadv)/m
gation unlessis largeisand
eastward.
– the
2Dadvective
spectralflow
analysis
crucialto The corresponding
phase speed is given VM R = !
ˆ M R /m, and similarly for the Rossby and Alfvéndistinguisheach component
adv/mphase speeds. The magnetic Rossby
speed goes up as the wavenumber mwincreases or the radius s decreases. A balance between the last two terms, ⌅Cand ⌅L , is vital for this mode, indicating that the time variations arise from• Note: wave contribution
the induction equation while the momentum equation is almost in balance.
(wMR(m=5)+wadv)/m
These slowdepends
waves will on
be distinguishedthe radiusfrom
s Alfvén or Rossby (fast MR)
modes in terms of dispersion relations ! = !(m), phase velocity !/m, and– wave ~< advection at larger s
Azimuth / radianvorticity balances.
f2 i, all dispersion relations (20) are comprised of-5
At fixed s and hence hB

at E = 10 , Pm/Pr = 5, Ra/Rac = 8 & L ~ 22
MR branches at lower wavenumber m and Alfvén branches at higher m. The(Hori, Jones & Teed, 2015)
2

2

4

6

2

2 2

f2

Slow MR waves in dynamo simulations
z-mean radial velocity <us>
d_E5_Pm5lf_seg2-4
in thet=0.00000
equatorial plane• Slow modes identified:

The fast modes !
ˆ + essentially recall the hydrodynamic Rossby waves,–
retrograde
commonlyseen
in part. They
which travel prograde with thedrifts
frequency!
ˆ R about theadvectionarise from a balance
betweenthe first two terms of (19), dh⇠z i/dt and ⌅C .
dynamosimulations
By contrast, the slow modes !
ˆ are a unique solution of rotating MHD,–
their
speedsaccountedwaves.
for bytotal
sometimes called MR waves or MC-Rossby
Theirproperties become2
evident when taking
thespeeds
limit !
ˆM
/ˆ
!R2wave⌧ 1 onthemeanslow mode,
phase
of
and
flow!ˆ , to obtain(using the binomial
approximation)
advection,(w + w )/m, whereMR!
ˆM R =

2
!
ˆM
=
!
ˆR

adv

f2 i(r2
m3 h B
o
2s4

s2 )

,

(22)

717.9
430.7
143.6
-143.6-430.7-717.9<us’> at s=0.5ro

Frequency f = w/2p

wM+wadv
Eastward
and the observed frequency will be the sum of !
ˆ M R and the advection
frequency !adv . This implies a much lower frequency and a retrograde propagation unless– theflowis large isandeastward.
2Dadvective
spectralanalysis
crucialto The corresponding
phase speed is given VM R = !
ˆ M R /m, and similarly for the Rossby and Alfvéndistinguisheach speed
component
phase speeds. The magnetic Rossby
goes up as the wavenumber m inwadv
creases or the radius s decreases. A balance between the last two terms, ⌅Cand ⌅L , is vital for this mode, indicating that the time variations arise from• Note: wave contribution
the induction equation while the momentum equation is almost in balance.
These slowdepends
waves will on
be distinguishedMR)
the radiusfrom
s Alfvén or Rossby (fastWestward
wMR+wadv
modes in terms of dispersion relations ! = !(m), phase velocity !/m, and– wave ~< advection at larger s
vorticity balances.
Azimuthal wavenumber m
f2 i, all dispersion relations (20) are comprised of
At fixed s and hence hB

MR branches at lower wavenumber m and Alfvén branches at higher m. The
2
2
4
6
2
2 2 f2

Exploring more cases
E=5*10-6, Pm=2, Ra/Rac = 16
&L~6

• MR waves were found in models
when torsional waves were
found

(wMR (m=9)+wadv)/m
Azimuth f
E=10-4, Pm=5, Ra/Rac = 8& L ~ 22

• Note: excited azimuthal wavenumbers m vary(wMR (m=5)+wadv)/m
Time t– chosen by the convective
instability

wadv/mTime t– generated magnetic fields of
non-reversing dipole– for strong-field solutions (L >~ 2;
Pm >= 5 or E =< 10-5) , good
Taylorization (< 0.2), good
geostrophy (U’C > 0.4)

• dependent on E, Ra, L, etc

wadv/mAzimuth fNonlinearity on waveforms?
t = 0.006

The observed waves illustrate• no wave packets• isolated, sharp waveforms– steepening– shifted to positivet = 0.005•

reminiscent of cnoidal/solitary
waves in weakly nonlinear,dispersive waves (e.g. Whitham 1974)– cf. (nonmag) solitary Rossby(e.g. Redekkop 1977, Yamagata 1982)

t = 0.004

Evolution of amplitude <us’> at s=0.5ro(Hori, Teed & Jones 2017)

Azimuth / radiann the background state given by the axisymmetric component, we further
eparate the mean parts into axisymmetric and non-axisymmetric parts, suchhat
' (s, φ, z) + u′ (s, φ, z, t)
u=U
(9)
n
′
′a
'
'
= U (s, z) + U (s, φ, z) + ⟨u ⟩(s, φ, t) + u (s, φ, z, t)

The role of nonlinear Lorentz force• Coriolis and Lorentz terms are
dominant in the axial vorticity eq.

' φ, z) + b′ (s, φ, z, t) = B(s,' z) + B' n (s, φ, z) + b′ (s, φ, z, t) . (10)B = B(s,VMC(m=5)+Vadv

Time t

The averaging operators and fluctuating parts appearing here are defined by– Reynolds( term
remains minorτ1' φ, z) =
',
A(s,
A dt,
A′ (t, s, φ, z) = A − A
(11)
τ 0
( 2π
• The Lorentz
X can nbe
1 termA(t, s, z) =A dφ,LA (t, s, φ, z) = A − A(12)
Vadv=z2πexpanded, in0 terms of the mean

Coriolis term XC
at s=0.5ro

ubstituting (10)the Lorentzparts,
term, ΞasL , we find its individual terms:
andintofluctuatingP m) '
ΞL =⟨B · ∇jz′ ⟩ + ⟨b′ · ∇jz′ ⟩′′om ΞL can 646
modelof Ξthe⟨u⟩.
In
Figure11
test
this11 we testfrom
ΞLpattern
can model
pattern
of
Ξ
,
or
⟨u
In
FigureE the
C , orC(13)ss ⟩. weSelected
Lorentz
terms
*
n
′
′
' · ∇j ′ ⟩++(other!φ ∂j ′′!φ ∂j ′bb
+ ⟨B
⟨B · ∇terms)
J'z ⟩ − ⟨J] · ∇Bz ⟩ .
B
B
∂j
z
φ
z
647
by thetakinga sum ofthe dominant⟨ s ∂φz ⟩,+ andnonlinear,⟨ sφtaking a sumofdominant
restoring,⟨ s ∂φzrestoring,⟩, and nonlinear,⟨ s ∂φ⟩,– first term for the restoring force
Time t648
terms.
The
selected terms
reproducesome features
includingrms. The selected
terms
reproducesome features
includingsteepenedcrestssteepened cr
7– second term for the leading649
andtroughs.
We that,hencealthough
speculatethethat,
althoughtheory is essend troughs. We
hence
speculatelineartheorythe
is linearessential
nonlinearpart
explaining
speeds,Lorentz
the nonlinear
termforis importantr explaining650itsfor
wave
speeds, its
thewavenonlinearterm isLorentz
important651
the observed
will help
us to study the fundamen
sumcreatingof thedominant
eating•theTheobserved
waveforms.
Thiswaveforms.
will help usThis
to study
the fundamentals
652
of andthe nonlinear
nonlinear
dynamics,
example,by adopting
restoringterms
the nonlinear
dynamics,
for example,
by foradopting
reducedmodels.reduced models.

reproduces steepened shapes
653
3.6. Space-time
analysisof surface
6. Space-time
analysisof surface
magnetic
field magnetic field(Hori,
Teed & Jones
2017)
Azimuthf waves could be detectab
654
We
now
addressthequestionwhetherMR
We now address the question whether MR waves could be detectable inToroidal field strength within the Earth’s core
• The dispersion relation tells us about
waves riding on mean flow advection
1.4

magnetic Rossby: m=5: m=8
gufm1 at 40S
gufm1 - zonal flow

– a geomagnetic drift speed of
0.56 º/yr at 40º S (Finlay & Jackson 2003)– suppose a mean flow of 0.24 º/yr
(Pais et al. 2015)– Given m=5, this implies a z-mean
toroidal field Bf ~ 12 mT at s ~ 0.8ro

Phase speed [ deg/yrs ]1.2• equivalent to, or stronger than, thepoloidal field Bs ≥ 3 mT (Gillet et al. 2010)– constrains the dynamo mechamism?
• e.g. a2-type or aw -type• stronger poloidal fields in dynamo
simulations

1
0.8
0.6
0.4
0.2
0

0

5

10

15

Toroidal field strength / mT (=10 G)

20(Hori, Jones & Teed, 2015)

Latitude (de0In thin, stably stratified layers

–30
–60
–90

• A stableAIA/EUVI
layer
at the top of the Earth’s coreBP (density 28-day aggregate (centre: 20120111)
1984, 1999)

30• Solar tachocline at the bottom of the0convection zone

LETTERS
20
25

30

–30
–60
–90
–180

15 a

– SW models applied by toroidal field

10(Gilman 2000; Zaqarashvili et al. 2007)5– ~ 3 m/s westward drifts and eastward–120−60 (McIntosh
0 et al. 2017)60
120
wavetrains?

BP density (BPs per degree)

Latitude (degrees)60– SW models applied by poloidal field (Braginsky

AIA/EUVI BP density
Hovmöller(northern hemisphere)b

0 2013
20

180

60
60°

–45°
−45
0

45°

–30°
−30

30°
30

(McIntosh et al. 2017)

45°
45

0

2011

Carrington longitude (degrees)

ustrating the combined coverage of BPs in the solar atmosphere from three distinct vantage points. The figure shows a longitude versus

0
18
0

60

12

0

15°
15
0°
0

4

b

0

10
8
6
4
2

0
–1

0

–15°
−15
Coronal brightpoints
in Jan 2012
& at around0°15°N / 22°S
0

3

90°
90

60
12
0
18
0
–1
80
–1
20
–6
0

–45°
−45

90°
90
75°
75

0

45°
45

–90°
−90

–6
0

−60
–60°

90°
90

6

135
135°

20

–45°
−45

−90
−90°
−75
–75°

−135
–135°

10

–1

–90°
−90

135
135°

2012

–1
8

−135
–135°

Time (year)

13

180°
180

6

0

16

d AIA/EUVI BP density (northern hemisphere)

180°
180

8

2

BP density (BPs per degree)

AIA/EUVI BP density (southern hemisphere)

14
12
10

Carrington longitude (degrees)ca

AIA/EUVI BP density
Hovmöller(southern hemisphere)

Frequency

90

Frequencyb

Carrington longitude (degrees)

Figure 3 | Illustrating the phase and group velocities of solar Rossby

Figure 4visible inlifetimes
in Fig. 3
distributωx
0 the rotating
0 frame0 as
We consider a local Cartesian coordinate(x,theiry, z)components
in which are written in
Eqs.frame(1)–(3)the x axis is directed towards the rotation, the y axis is directed∂h
B xis ∂bx
towards the north pole of the sphere ∂uandx −thef uz axis
directed
−
g
,
(5)
=
y
Thi
∂t
4πρ ∂x∂x Thus, we can now perform a Fouri
vertically.
exp
(ik
y),
which
gives
the
dispersionybrar
Let us next consider that the unperturbed magnetic field,∂uyThen, afterBlinearizing∂hx ∂by(B x , 0, 0), is directed along the x axis.
2 wav+ f ux =− g , ω4 − [2k2x v2 + f 2 + C 2 (k2x + ky2(6)
)]ω −
C
disp
• b-planewater
modelsappliedan azimuthal
field A 0 0∂t in the by4πρ
∂x
Eqs.shallow(1)–(3) theircomponents
are written
rotating
frame
as∂y
ky2
+k2x v2A [k2x v2A + C02 (k2x +low∂u x∂h
B x ∂b x∂uy∂b x∂u x ∂by− f uy =
−g ,
= Bx
,
=ver.1.1
B(5), April 6, 2016 KH(7)x
dispersion relation containsvAh∂t
4πρ ∂x∂x∂t∂x∂t∂x Thisstar
branches, which respectively corresp"!
∂u x ∂uy
∂h
∂uy
Poi
∂h
B x ∂blow-latitudey
2.3 nonaxisymmetric,
+
H
+
=(6)
0, waves and to Alfvén and Rossby(8) wave0+ f ux =
−g ,
∂t
∂x
∂y
dispersion relation transforms into tha∂t4πρ ∂x∂y2ωWhen βy ≫ f0 ,low water” waves (Schecter et al. 2001!
"#
$ and magnetic field perturwhere
u
,
u
,
b
and
b
are
the
velocity
x
y
x
y
2 2
∂uyv 2∂b x d2 uy ∂u x ω 2∂byWe
next concentrate in the casewhi
of
ω
β
k
ωβ
x
A
2 Bx
2 is the
=
B
,
=
,
(7)
bations,h=H−H
perturbation
of
the
layer
thickness
and
x
0
+
−k 1+
− 2 2
y −
0 (37)• when ∂tf ~ dyby,, suchas corresponds to th
2A2 ) ≪uyC0=
2 ∂x C 2 ∂t x
2 v2 )
2 − k 2v
∂x
C
C
(ω
−
k
(ω
v
f
(with
Θ
the
latitude).
For bran00
0= 2Ω0 sin
x Θ
x A
A is the Coriolis parameterstars.
frequency! !2" " 2#$ Then, the high22ω2transforms
into
the
HD
shallowd ∂huyω ∂u x 2 ∂uy vAkzero
β systemPoincaré
x β magnetic field this2
whose dispersion rela
or
+
− k+
− 2 (Pedlosky
y uy waves,= 0 (38)x 1 + = 20, −
+
H
(8)
water
2
2 equations
2 1987).
2
2
2
2
2
0
dy∂t
C0 ∂x
ω(1 − kx vA /ω ) C0 (1 − kx vA /ω )∂y C0
2 time2 2(5)–(6)2 and using
Diﬀerentiation with respectωtoEqs.
= f02 of+CNot
0 (k x + ky ),This is a Schrödinger equation (nonlinear, second-order
ODE, gives
homogeneous): oscillatory solutions
Eqs. (7)–(8)(19where u x , uy , b x and by are the velocity and magnetic field pertur–cf.
nonmagnetic
case
(e.g.
Matsuno1966):
when |y| < |yc | and evanescent solutions when |y| >2 |yc |, where the criticallatitudes
yc 2low
is givenasthebranch yields!while" frequency
197
2 and
2
bations, h = H − H0 is the perturbation
of
the
layer
thickness
∂u
∂
u
∂
∂
u
∂
u
u
y
y
!
"
#
$
x
x
x
2
2
2 Coriolis
2f= vA kx ωβ−vΘ
+ C0+k β ,(9) shosinCΘ02 is(withlatitude).
For•f =a 2Ω
Schroedinger
(ωthe
−eq.
kx2 vA2 ) parameter
ω2
2
2
A the
2 0
2 ∂t2
x
∂t
∂x∂y
2
2
2
∂x
∂x
=
− kx 1 +into−inatω −(39)k x vA = 0.
2 transforms
2 the HD2 shallow
zeroycmagnetic
fieldsystem2 βthis2
2 v2 ) ω + 2
2
ω
C
C
(ω
−
k
k
+
k
•wateroscillatory
for
|y|
<
y
,
i.e.
equatoriallytrapped
waves
0
0
x
A
x
y
! $2
! c1987).
"2#equations22 "
2 (Pedlosky
2 2
2
2
2
2
∂
u
∂
u
u
∂
ω=
∂
u
∂u
ykx ωβ 2y(ω − kwith
ωC0timevA(5)–(6)x C02xx vA )respect
2 of yEqs.
Diﬀerentiation
to
and
using
=
v
+
+
f
+
C
,
(10)
– In the presenceof magnetic=− 2fieldk 1+− A 22(40)
that the dispersion
relation (17) w
2Note∂x − kx2 0vAω2β2β x ∂t2 C02 ∂t β 2 (ω)∂x∂y ∂y2
Eqs. (7)–(8) gives(1966) in the two dimensional caseThi(s• nonzero VA increases yc,! i.e. releasing" the trapped
waves
#
√
fect
This determines2 how the waves areequator.
1973).
Thistheformula
revealssome in2
2 trapped 2to thewhere
v
=
B
/
4πρ
and
C
=
gH
are
Alfvénandsurface
∂u
∂
u
A
x
0
0
∂
∂
u
∂
u
u
y
y
x
x
x
• large
VA gives
rise toa 02Besseleq.
la= v2Asmall+=C,(9) short wavelengths, i.e. large k x , thetoro
2 2 +gravity
speeds,respectively.
• for large ∂tC022/β−2 ,f i.e.,
for
ϵ
4Ω
R
/(gH
):
2
2
0
0∂x0
∂t
∂x∂y
∂x
over
one,
which
analysisofthe
thesecond
form exp(−iωt+ lead! " We now#perform a Fourier$inates
2
2 2
2
2
lon
(ω
− kx vA!) C0 ik
x ωβ" vAafter somekalgebra
2
2 x2x) andobtain222kx ∂ 1uy+ 2 + 2
(41)
mag
∂ uy
∂uyxc = −2x
2 v 2 ) ω = ±k x vA .
2 ∂ uy ω 2 2 ∂βu
C
(ω
−
k
⎤
⎡
=
v
+
+
f
+
C
,
(10)
0
x
A
two
A
0
k2x v2Auy2 ⎢⎢ ω2∂ f ⎥⎥
ω2 f 2kx ω∂t∂x∂y ∂2∂y
∂t2
∂x2
2

e.g. equatorial waves (cartesian)caled height η (Waves travelling Eastward).

In spherical shells• Nonaxisymmetric MAC waves classified:– inertio-gravity– Rossby– Kelvin

Fast Magnetic Rossby waves

Slow Magnetic Rossby waves
At small ✏ we have Rossby waves as well as MIG waves. These• Rossby:haveforfrequency
eq.symmetricBfspherical= B0 sinqO(⌦0 ), andharmonic
solutions. The term

Theseslow waves are in hydromagnetic equilibrium, so inertial(Marquez-Artaviaal., 2017)@⌘/@tetdrops
out of the mass conservation equation.

term in the equation of motion is negligible. Time-dependence– fast Theymodes
travelin westwards,
and at small ↵only
induction equation.

• goes westward
Frequency goes to zero as ↵2⌦
goesm to zero, so this branch doesn’t
02
2• in theexist
limit in
VMnon-magnetic/Vc << 1, ! case.
=.
n(n + 1)– slow modes
When ↵ and ✏ are small, eigenfunction are spherical harmonics and• goeseastward
At large✏ they turn into equatorially
2 trapped waves, described bymva(n(n + 1) 2),
parabolic• inthe limitcylinder
VM2/Vc2functions.
<< 1, ! =2⌦0 R02• slowly
westwardat
for↵n=m=1
As ↵ increases,= 0.5, = 0.5m. For ↵ > 0.5, they becomea Márquez-Artaviasuper-Alfvénic.
so they travel eastwards. The– even polartrapped at large VM2/Vnc2= 1 mode is anomalous, andtravels
verybecome
slowlyAt Waterlarge↵ theypolar
trapped, described by Whittaker’s
g Magnetic Shallow
Waves
in westwards.
a Sphere2– become
unstableat largeRossby
VM2/Vwaves
c function
Notethatassociatedmagnetic
in a fullsphere travel westwards.
equation,with
Laguerre
solutions.

At large ↵, slow magnetic Rossby waves become polar trapped,andatEigenfunctions
nearly the same
speed
as MR
the wavespolar trapped fast 64/69(4.5) Shallow water
MHDtravel
modelof fast/ slow
2
2 going east-12
2 than west.
magnetic
Rossbyratherfor m=1,
a (~Vwaves,
M /Vc ) = 0.1, e (~ VA /Vc ) = 0.01

Summary• Geo-/Jovian dynamo simulations are supporting theexcitation of magnetic Rossby waves for incompressible/anelastic fluids– crests/troughs going retrogradely on timescales of O(101-2 yrs)
in the Earth’s core, about mean zonal flows– excited when torsional Alfven waves were excited• for strong-field dynamos (Pm >=5 or E =< 10-4; L >~ 2)– the speeds accounted for by the linear theory, but theirwaveforms steepened, likely due to nonlinear Lorentz terms– their speeds potentially revealing the strength of the ‘hidden’
toroidal field– induced by topography but also by compressibility

Thank you

QG vs. non-QG modes• In spheres(n,m)=(18,8)

1

0.8

0.8

– e.g. for Malkus field (1967)
B f = B0 s e f

0.6
0.6
0.4

0.4

0.2

0.2

0
0

the solution, P=Pnm(µ) Pnm(µ)–– equatorially trapped for for small n– even (n-m): eq.symmetric (QG) modes
•

−0.2
−0.2
−0.4
−0.6

−0.4

−0.8

−0.6

−1

goes retrograde & faster (≈ −wM2/wb)– odd (n-m): eq.anti-symmetric modes

−1

−0.5

0

1

(n,m)=(19,8)

1• goes prograde & slower (≈ +wM2/wI)

0.5

0.8

0.8

0.6

0.6
0.4
0.4
0.2

– cf. MC waves in simple plane layers

0.2
0• slow modes has no preference in propagation
direction (≈ ±wM2/wC)• The geometrical effect splits the modes into afaster & slower ones

0
−0.2
−0.2
−0.4

−0.4

−0.6

−0.6

−0.8

−0.8
−1

−1

−0.5

0

0.5

1

Eigenfunctions for Malkus field (after Malkus 1967)