Torsional Alfvén waves
in Jupiter‘s metallic hydrogen region

Kumiko Hori1,2, Rob Teed3, Chris Jones2
1) Graduate School of System Informatics, Kobe University.
2) Department of Applied Mathematics, University of Leeds, UK.
3) School of Mathematics and Statistics, University of Glasgow, UK.

Kobe, 19 July 2018

Jupiter, but the const
because it is inhibited by turbulent diffusionneon (16 ), and water] that is two to four times
weak because some of(24 ). Helium (and tentatively any late supply
larger than in the sun, which is in agreement
deep metallic envelopeof heavy elements to the outermost layers)
with interior models using the new helium
for as core mass, and vi
should be able to sink through the radiative
mixing ratio but not the one derived fromtant to note that heavyzone thanks to a salt-finger type of instability
Voyager data (Fig. 2). The lack of abundant
and Saturn’s molecular(25), thus ensuring an almost uniform chemwater in the Galileo measurements is thought
those
in their metallic reical composition of the radiative zone. (ii) In
to be due to jovian meteorology,
and its internalbulk
Predicted
structure
the region of varying helium concentration,(Guillot 1999)
in which convection may be suppressed. The
Fig. 1. Schematic repextent
of this inhomogeneous
poorlyknown
in data region (Fig. 1) resentation of the inis estimated from fully ionized phase separateriors of Jupiter, Saturn, Uranus, and Neption models (19). In reality, the inhomogetune. The hashed reneous region could be narrower or wider. It isgion indicates a possinot included in any interior models so far, but
metallic H-He
ble radiative zone [inthis seems justified because the gravitational
Jupiter, it corresponds
throughthe magnetic field:(conducting)
moments only provide constraints on quantiai
to P $ 0.15 to 0.6
ties that are averaged
relatively
extendedGPa, axial
T $ 1450 to• pre-Juno(n ~< 4):overstrong,predominantly
1900 K, and R $ 0.990
regions. More important, this region could be
dipole,secularvariation?
to 0.984 RJ; in Saturn,a relatively efficient barrier to the mixing of
it is located around P
minor species,(nbecausethey
would
have toclosestbe• post-Juno<
10
andmore?):
as everGPa, T to$ 1700~ diffusion processes. The $ 0.5
transported by slow
K, and R $ 0.965 RS,
asame
dynamoregion:would occur
at the localizedboundary ofpatchesa firstbut it is probably veryorder transition from the molecular to themarginal (23)]. The
range of temperatures
metallic phase (18).
throughanyoscillations/waves?
Br inferred at surface 0.96 RJ
for Jupiter and Saturn
Interior models of Jupiter and Saturn are
is for models neglect(JRM09: Connerney et al. 2018)calculated
by solving the standard quasi-hy• anelectrically-conducting,low-viscousing the fluid
presence ofdrostatic differential equations, including thethe inhomogeneous reina rapidly-rotatingrotational
potential calculated sphericalwithin the the-shellgion. Helium mass mixory of figures (26 ). In recent calculations,ing ratios Y are indipermeatedby themagneticconfield cated.
In the case of
only models that match
all observational
Saturn,straints
are considered. Uncertainties– Lorentz/Coriolis= O(1)? in the that Y/(Xit !isY)assumed" 0.16equation of state, surface temperature, opacin the molecular reities,
rotation, andobservational– internal
the rotating
MHD
hosts aerror
varietygion.
of waves
The size of thebars on the gravitational moments are taken
central rock and ice cores of Jupiter and Saturn is very uncertain (see tex
into account to determine the allowed range
models of Uranus and Neptune are shown, but their actual interior structur
different (34). The figure is adapted and updated from (19).
of internal compositions of these planets. The

Jupiter’s metallic hydrogen region• The origin of the magnetic field/dynamo action–• How to infer the deep interior dynamics?
–

Jupiter dyna–

-2.2mT

2.2mTRotating MHD waves• Waves in the presence of both magnetic field and rotation
have been studied for incompressible fluids and applied to
Earth’s liquid iron core
– torsional Alfvén waves (e.g. Braginsky 1967, Zatman & Bloxham 1997)• e.g. ~ 6 yrs variation à core internal field Bs >~ 2 mT (Gillet et al. 2010)• accounting for the interannual length-of-the-day variations?

– magnetic Rossby waves (Hide 1966)• e.g. ~ 300 yrs westward drift à Bf ~ 1-10 mT? (Hori et al. 2015)– MAC waves in a thin stably-stratified layer, at the top of the core?
• axisymmetric (e.g. Braginsky 1993; Buffett 2014), fast magnetic Rossby (Chulliat et al. 2015)• What about in Jupiter’s interior?
– density significantly varies with radius: r(rcore)/r(rmetallic) ~< 20* anelastic approximation for compressible fluids adopted

Torsional Alfvén waves
ver.2.2
September
6, 2017KH• A special class of Alfvén waves (Braginsky 1970; also
Jault &
Finlay 2015):
DE
D overE– The azimuthal
momentumequation
integrated@h⇢ u i =ê · (r · ⇢uu)
2⌦h⇢ us i + ê · (J ⇥ B)@t surfaces C = 2ps h(s) about the rotation axis:
cylindrical@@tZ⇢u dS +CZ

Cê · (r · ⇢uu)dS + 2⌦Z

⇢us dS =
C

Z

C

(10)

ê · (J ⇥ B)dS = 0(11)– For anelastic/incompressible fluids, theCoriolis
term vanishes↵
1 1 @ 2 ⌦s h Bs B(12)
L =– The magnetostrophicFbalanceµ0 s2(Ro,h @sE<<1 & L=O(1) ) yields a
steady state (Taylor 1963)
ver.2.2 September 4, 2017 KH
1 @ 2
1
s
h
h⇢
u
u
i
[⇢ uz u--]zz+s-2
s h @sf2
Cylindricalperturbations
on the state, <uh ’> = <uf’>(s,t),
f
FR =

(13)

f2–| and |B |) are greater than those of distinct fones (|Bfs B |), the equation can be(such as |Bs
174
P. H. Roberts and J. M. Aurnou
canonlybe leavegovernedby a homogeneous
equation:stically simplified tothe homogeneouspart!
0Li FLR
0 i
FLD
=
F
(14)
2
hu
hu
@
1
@
@ FLR
3
2" @
!#
=
s hh⇢iU
⌘
(9)
Z ⌧A
0 i
hu
⌦
↵
@t2 s1 1 s@3 hh⇢i
@s
@s
s
@t
sh⇢i
1
@
@
2
3
2
=

µ0 s2 h @ss h Bs B

s3 hh⇢i @ss hh⇢iUA

@s s

dt

(15)

ber 2012• implying
propagation
radius sforwith
Alfvénvelocity
speed Uf2 i/µ0 h⇢i,re UA2 = hB
a waveinequation
angularin the anelastic case (Jault-- 2 -- huA0 i/ss
1/2
given byforce
z-meanquantities:s >/<r>µ0)
inlay, 2015). Here the restoringis represented
byUFA=(<Bthe remaining
terms can be summed
LR , whileo a forcing term FLD
FL0 outward
FLR , as well
FR0 inward. A perturbationof angular velocity can propagate in• =both(+s)asand(-s) propagation,
ndrical radius s with anorAlfvén
speed
UA . possible
The speed depends on the magnitude of the backgroundstanding
waves,
2
oidal field Bs and the background density ⇢, both of which may vary with s. As expected for the
D wave, this special mode is also nondispersive, i.e. the speed independent(Robertsof&wavenumbers.
Aurnou 2012) Since theation allows both inward and outward propagation, a superposition
ofshowing
those(a)modes,
providedsimilarFigure 4. Schematics
a geostrophic
flow in the acore,
Vg, and (b) a plan view of an initiallyfield model at recent epochs may have increased the sensitivity in theshipreequatorial area. We confirm the slower propagation inferred by Gillet etTaya
the equator and find no evidence for reflection at the equator.

Torsional waves in Earth’s core
– Alfvén speed UA for constant r• Early studies sought its standing form
(e.g. Braginsky 1970; Zatman & Bloxham 1997)• More likely travelling to the equator– data: 4-9 year periods (Gillet et al. 2010, 2015)• the internal field strength of <Bs2>1/2 ≥ 2 mT

– geodynamo simulation (Wicht & Christensen
2010; Teed et al. 2014; Schaeffer et al. 2017)• no obvious reflection, no standing‘oscillations’• due to strong dissipation around CMB?

– lab experiments also? (Nataf et al.)Cylindrical radius s /rcore• Suppose the incompressible case

Now the theory of “magnetostrophic dynamos” [see, e.g., Roberts andoped to account for the Earth’s magnetic field, gives us a tool to interp
Axisymmetric zonal velocity usionof frequency. We note above that this ratio remains small andf does not
in
a
core
flow
model
(Gillet
et
al.
2015)
and
8 years (see Figure 9). Taylor [1963] demonstrated that in the absence
1.0 CMB
lor’s
Finl
∀s,
Rob
0.8
∫Σ
with
with Σ
0.6
Robert∀sFinlay0.4
lor’s th
ICB
su
1940
1960
2000
8 years(see Figure9). Taylor 1980
[1963] demonstrated
that in theandabse
froYearthat this ratio remains small and sionof frequency. We note above
does n𝜕
oped to account for the Earth’s magnetic field, gives us a tool to inte
𝜕
Now the theory of “magnetostrophic dynamos” [see, e.g., Roberts anin adynamosimulation(Schaeffertheequatorand findno evidence
for reflectionetatal.
the2017)equator. Taylor
equatorial area. We confirm the slower propagation inferred by ship
Gilletbee
field model at recent epochs may have increased the sensitivityuin
anG the
icantly larger than the spread in the flow ensemble (even at earlier
ep
which
UA
insidewaveform at about 10∘ latitude (see Figure 10, bottom). The amplitude
thelatitmaFigure 10. Ensemble mean of the geostrophic flow (in km/yr),band-pass
ICBfiltered between∘4 and 9.5 years, as a function of time. The
sion
1
deviation).
Flows aretotruncatedat degree𝓁 =grey
14. lines correspond to(top) The
black
line correspond
10 latitude.
oves3 H
averageratio overthe ensemble
grey line:
standard
C basedon a r.m.s.members
cylindrical(thinmagnetic
field±of1 1.9
Alfvénvelocities
Geo
the0.6ratio
the ensemble
averageof toflow
boldclosegreytotheandmTfor
in regions
respectively
close
thesolutions.
inner coreInand=−Y axisproportional
the line
the
Bottompannel:theequator.
zonal andnonzonalflows
as aincrementsfunction ofareperiod.
In boldto
blackdow
Z ∕sNZ+ between𝛿s (dY ∝ sin⇒ Yspectral∝ 1 − cos𝜃 ).
surface
Figurebetween
9. Ratio sandthe𝜃d𝜃
power
densities(PSD) forin tandJovian dynamo models
The reference state used in the model• Setup (Jones 2014; also Gastine et al. 2014):10

ρ/ρ(rm), σ/σ(rm)– model a metallic region & a transition to
the molecular region: 0.09RJ ~< r < 0.96RJ~– dynamos driven by rotating, anelastic
convection (Lantz & Fan 1999; Braginsky & Roberts 1995)– a reference state (French et al. 2012):• density contrast, r(rcore)/r(rcutoff) ~ 18
• electrical conductivity s drops at r ~ 0.85RJ
by more than five orders

density r/rm

1

0.1

electrical conductivity
s/sm

0.01

0.001
0.1

rcore

metallic regiontransition

0.2

0.7

0.3

0.4 0.5 0.6
radius r/RJ

Radius r/RJ

0.8

0.9

rcutoff

Jupiterdynamomodel
snapshots• Some features:– jupiter-like magnetic fields reproduced

Br at the cutoff radius rcutoff ~ 0.96 RJ
truncated up to n=10 (after Jones 2014)

2.1
5

0.1

8.0

6.0

4.0

2.0

xulfpot-H

Jovian dynamo models
4

ek-H

ek-H

0.0
5

)b(

4

3

3

2

2

2.1
6.0

0.1

4.0

8.0

6.0

4.0

em-H

2.0
0.0

1
1
The
reference state usedin the model• Setup (Jones 2014; also Gastine et al. 2014):

10 0

ρ/ρ(rm), σ/σ(rm)– model a metallic region & a transition to
the molecular region: 0.09RJ ~< r < 0.96RJ~– dynamos driven by rotating, anelastic
convection (Lantz & Fan 1999; Braginsky & Roberts 1995)– a reference state (French et al. 2012):

lanoz-H

12.1

0.1

xulferoc-H

8.0

6.0

4.0

0

2.0-

10.0

4.02.1

2.0

t

1

pid-H

lanoz-H

0.1

8.0

density r/rm

6.0

4.0

t

)a( htob ni stinu noisuffid citengam ni t htiw H nur rof 02 g dna 01 g stneicfifeoc elopurdauq dna elopid sselnoisnemid eht fo s
eht si xufleroc-H ,yradnuob retuo eht morf xufl taeh sselnoisnemid eht si xuflpot-H )b( .egareva evitalumuc sti dna ygrene.egareva evitalumuc etoned srabrevO . /u fo trap cirtemmysixa eht fo ygrene citenik si lanoz-H " 601 dna y• density contrast, r(rcore)/r(rcutoff) ~ 18
• electrical conductivity s drops at r ~ 0.85RJ
by more than five orders• Some features:

0.1

electrical conductivity
s/sm

0.01

0.001
0.1

– a magnetic tangent cylinder formed• attaching to a top of the metallic region
at the equator• one strong jet outside the MTC; weakmultiple zonal flows inside– fluctuating: to be analyzed

transition

0.2

0.7

rcore
009.0

– jupiter-like magnetic fields reproduced

metallic region
0.3

0.4 0.5 0.6
radius r/RJ

Radius r/RJ )b(

551

Axisymmertric Bf&
0.00345.4
5.3

0.0073MTC

5.2

0.00135.1

009.0-

Uf

0.8

0.9

rcutoff
Tm2.1

951–840.00525.0

0.0091
5.05.15.25.35.4-

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0.005-

omanyd eht morf dlefi citengam eht fo tnenopmoc laidar eht fo tohspanS )b( .)3991 ,yenrennoC( suidar naem eht ta dlefi citAnelastic Alfvén speed in simulations• Predicted Alfvén speeds--- )1/2 :UA = (<Bs2>/µ0<r>– independent of wavenumbers, i.e. nondispersive– higher for low r, i.e. increasing with s– drops to the MTC

Alfvén speed UA

for the variable r

at E = 1.5*10-5, Pm=3, Pr = 0.1, H=1.4& fixed entropy-flux outer boundary

for constant r0

Cylindrical radius s/rcutoff

MTC

Torsional waves in Jovian simulations

Axisymmetric azimuthal velocity, -uf• Identified with the predicted speeds-- 2-- 1/2of UA = (<Bs >/µ0<r>)– travelling in s, outwardly or inwardly,
from an outer radius (0.6 < s/rcutoff < 0.8)s

• Reflected from the MTC

Cylindrical radius s/rcutoff– which acts as an interface to a
resistive zone
• 1D models helpful MTC

at E = 1.5*10-5, Pm=3, Pr = 0.1, H=1.4& fixed entropy-flux outer boundary--

Fluctuating, z-mean azimuthal velocity <u’f>UATime / magnetic diffusion time

Evolution of torsional waves• Waveforms can become sharp• Reflection from the MTC– as well as transmission to theoutside– reflected waves not identicalto incident waves--• typical for inviscid nonlinear
waves• e.g. water waves, shock waves• cf. dispersive, cnoidal/solitaty
Rossby ones (Hori et al. 2017)

Amplitude <uf’>– steepening; weak, unstable• due to its spherical geometries,
variable background fields,
nonlinearities, etc.

Cylindrical radius sη oneη(forx > x).
0 They are
B=B
, u=u
(x)ê
0 y forx +abweakly
y (x)êyconductingy=fluid (for negativex)0 êandpositive
Suppose that there is a basic magnetic field.
a uniform
magnetic field B0 in the x direction.
g equationsareBpermeated= B0 êx +byby (x)ê
ubackground= uy∂u(x)ê(1) Fory ,y y B0 ∂by(1)

Alfvén waves approaching a resistive layerning equations
Let

=

(3)

f

∂t
µρ!
∂x with ρ0 being constant density.
simplicity we assume an incompressible
fluid
s are
2
0 , foru =xuy<(x)ê0 y= B0 êx +ηby=
(x)ê
(1)
∂bmagnetic∂uyyb and∂ uby, orB
there
is a basic
field.
he continuity
for∂by /∂x,
across x =y 0!yng
equations
Wecondition
thenthevariablesy+ η= suppose
where
B
(2)
0
∂xη2 = 0 for x <η 0= η0 for x > 0∂by the ∂u∂ 2 bequationsy∂ty ∂x
governing
are
B0
= B•
+ η 2 where(2)
Consider1d models:
0
−
+
∂b
∂b
f(x)
∂x field. B = B
re ∂tis a basic∂xmagneticη 0=xy > 0 y
0+byforê=xηb+(C.1)
b−, (x)êy2, =u = uy (x)ê!y 6(4)yyη = 0 for x < 0 (1)∂yy(x)êby y∂x∂uy , ∂uyu
B0=∂b
B = B0 êx +∂bbyy(x)êu∂x= B0= + η 2 where
(2)
(3)
4
∂uy
B0 ∂b
y ∂t
∂t
∂x
∂x
η
=
η
for
x
>
0
µρand
∂xmomentum as0
to rewrite
theof induction=governing(3)
arekequations
for a solutioninform
ofequationsthentheequations
2
∂t
µρ
∂x
B=
by (x)êuuy (x)ê(1)0 êx +
y= 0
" yy,/∂x,#!=
nuity condition for
byyBandu∂u∂bacrossx
y ,y or
∂u
∂b
B
∂b
∂
∂b
∂u
B
∂b
0
y
y
0
iωt
−ikx
+ikx
2
ition for by ∂b
andacross∂u∂b
bye + xe η= 0 y+ ηR=ande 0 for=foryx=<x 0<00 y ,(5)(3)
Bby0∂ =
(C.2)
y uy , ory y=/∂x,
uations are= B0 ∂t + η ∂x2 where(2)∂t + ∂tµρ ∂x∂x ∂x −
µ0 ρ-20 ∂x
perfect conductor
resistive layer
∂t
∂x
∂x−
ηy=iωt
η0 ∂b
x>0
∂b
λxfor
+
y
−
+
!
for acrossx >-40x = 0∂bbyh =∂by =T(zero h)(6)(nonzero h)yby,bandthecontinuity
conditionforuy ,e =orhe∂b/∂x,
byconstants;
(4)
−
+
where
B
and
are
=
for
x
<
0
y0
y
2
respectively.
Herethemagnetic
diffusivity,
η(x)
=
1/µ
σ(x),
varies
in
x:
it
is
0
b
=
b
,
=
(4)
0
η = 0 ∂xfor x < 0y∂uy y
∂by
∂ ∂x
by
for
for x > 0
∂xB0 ∂by ∂x
-6
∂u
=
+
where
B
η
(2)x < 0
y
2
0
−
+
2
n VA ! ωη(3) -5for x < 0∂x
and to a=constant∂t0 set zero ∂x
∂b
0xfor
-150.
-10
0
5
η −= η0+valuefor η∂b$ nonzeroy> x0 >y At the interface∂tµρ ∂xolutionoform of in •form
b
=
b
,
=
(4)
ωy
y
solutions
form
of
∂x
λ
=
(−1
i) ∂x condition(7)
theSeekfield
andvelocity"are#in
continuous,i.e.
the+continuity
across x = 0"#
uity condition for
b
and
u
,
or
∂b
/∂x,
across
x
=
0
2η
y
y
y
∂u
iωt
+ikx
iωt
y −ikxB
0 ∂bxy <+ikx
by =erequired
e−ikx
+b R
eform
for
00
(5)
bfor
e
e
+
R
e
for
(5)
for xx<<0 0 6
=
(3)
Look
a solution
in
of
y =
is for
and
∂b
/∂x.
y
y
∂t
µρ
∂x
So at the interface it should beiωtsatisfied:
"+
# 4
∂b−
∂b
λx
yx>
yλx
iωt
−
+
b
=
T
e
e
for
0
(6) l) (4)
for
x
>
0
complexyiωt−ikx for+ikx
Eq. (C.2) maybe
reduced
to
b
=
b
,
=
b
=
T
e
e
x
>
0(with(6)y bacrosse 0 +Re
for
x
<
0
(5)
y =e x =y condition for by and uy ,y or ∂by y /∂x,2

∂x 1 + R = T
(8)
iωt
λx
with continuous
conditions
acrosstheinterface
x
=
0:
∂ by
∂
b
$ by = T e e 0 for x > 0 (C.3)
y
2
(6)
$
= V∂b
ution
ωη
A − 2 ∂b+for ω x < 0
0 in form of2y∂x = T y−ω∂t +−ik +$(−1 + i) -2(9)by +, i) ikR=#(4)
λ = by "=
(7)
2 (−1
2ω
3 0
2
2η
∂
b
∂
b
∂
b
when VA !
∂x + i)y λ = ∂x(7)by ωη
= 0eiωt2ηe0 −ikx
+ RVAe2+ikxy (−1xy$< 0 for x -4> 0 ,=+ ηfor(C.4) (5)
0
2
2
2 ω
2η
2
0
∂t
∂x
∂t∂x
to yield the reflection
coefficientsfor>> VA /h0 :itngshouldbeofsatisfied:
λ=
(−1 w
+ -6
i)
(7)
on
in formiωt λx
%
b
=
T
e
e
for
x
>
0
(6)
2η
y
√
0
-15
-10
-5
0
interface itwhereshouldbe satisfied:ik V+A =ω/2η
i) we seek solutions of the form
" speedthe Alfvén
B0 /#0 (−1ρ0 µ 0 +
. Now
%
R
=
, T =1+R
(10)
iωt
−ikx
+ikx
1++RRe=Tsatisfied:(8)
So atbythe
it ikshouldbe= interface
e
e $
for
x
<
0
(5)
−
ω/2η
(−1
+
i)
η0
0
$ iωt ! −ikx
1Re+ikxR" = Tfor x < 0
(8)
ω
b
=
e
e
+
(C.5)
ω
y
iωt
λx
−ik
+
ikR
=
T
(−1
+
i)
(9)
2
1+R
=T
(8)
=−1
T and
e (−1
e +$
x>
n ω # k η0 , this ends up to Rhence
T ∼ 0,
i.e.0 perfectreflection.
λb=i) for(7)(6)y∼2η
0 2ηk2h , thenω
– for large
w
>>
R
~
-1
&
T
~
0:
perfect
reflection
$
iωt 0
λx
− ,0thee this all about by : since u−
negativereflection
reflection in uy(9).
b∼y ±ikb= ikR
for(−1x >+0i) in(C.6)+
ω by yields a positive
y−ik
yT e =eT
√
−ik
+
ikR
=
T
(−1
+
i)
(9)
2η
0
+ ∼/dx:+ ∼+ , suggesting+ , or same
it should
be–wavesuysatisfied:~udbanegativereflection
inbyieldsapositivereflectioninu,nterface
for transmitted
λb
(−1
+
i)/
2b
a
3π/4-shift
in
u
phase
as
%
2η
y
y
y
$
0
y
y
y
y
ω
2
2
2
ik
+
ω/2η
(−1
+
i)
where
λ, being
R andT
are0complex
k = ω /VA . For x > 0, substituting
reflected wave
while
%absorbed.
R=(10) (C.6) (8)(7)λ=(−1and++i)
1
R=T
leaving%2η0+ i)ik − ω/2η (−1
%
2

df

∂x

2

5

@
h⇢ u i = FR +"FL + FV =
@t
Z ⌧
0

h⇢i

@hu i

h⇢i

hu0 i

@t0

@hu i

@
@t@ hu
s0 i

@t

1 @

Z ⌧"

Z ⌧"

s2 h @s1 @s2 h @sê · (r · ⇢uu) +!#ê · (J ⇥ B)

s3 hh⇢iUA2

hu0 i

@
!#s
@s
0

(11)

Excitation mechanism

s3 hh⇢iUA2@ hu i@s s

dt = FR + FLD + FV

hu0 i

1
@
3
2 @ !#
s
hh⇢iU
A
0 i
hu@s
s31hh⇢i@ @s 3s2 @dt = FR + FLD + FV

!#

(12)

1
(FR + FLD + FV )sh⇢i

• The momentum equation canbe split1s hh⇢iUdt =(F + F + F )@t sinto thes hh⇢i@s@s s forcing
sh⇢iparts:
restoring
and
Z ⌧"

3

A

dt =

R

LD

V

MTC
1 @ 2
1
z+
FR = 12 @ 2s h h⇢ us u i1[⇢ uz u ]z
FR = s2 h @ss h h⇢ us u i[⇢ huz u ]zz+s h @sh
FFLD =F
F
L
LR
FLR
LD = FL"Z ⌧ "!# 0 !#
Z0hu i⌧
11 @ @ 2 2⌦ ⌦ ↵ ↵1 @ 1 3@32 @ hu 2i @
=
s
h
B
B
s
hh⇢iU
dt
s h Bs s
s hh⇢iUA
dtµµ00ss22hh@s@s
s2 h @s@s s A @s s
s2 h @s
11 11 @ @ 2 ⌦ ⌦ ↵ ↵F=
s sh2 hBsB
B B
L
FL–= TWs the Reynolds force at anµ0 s22hinitiatedbyµ0 s h@s@s

outer radius, 0.6 < s/rcutoff < 0.8– at which convection is beating ontimescales of (hydro) Rossby waves

4072.5z-mean radial
velocity 2443.5
<us>
814.5
-814.5-2443.5-4072.5

(12)

(13)

Reynolds term FR

(13)

(14)

(14)

(15)

(15)

(16)

(16)

(17)

(17)

forcing Lorentz term FLD

Torsional ‘oscillations’ possible• Zonal flow fluctuations in another case– standing inside the MTC• travelling from an outer radius both inwardly and outwardly• superposition with reflected waves enables standing waves– only transmitted outside the MTC• while being absorbed

à The nature signifying the depth?

--

Fluctuating, z-mean azimuthal velocity <u’f>• a bound of the core fluid(e.g. Schaeffer & Jault 2016)

Cylindrical radius s/rcutoff– cf. Earth’s CMB

MTC

UA

0
at E = 1.5*10-5, Pm=3, Pr = 0.1, H=1.4& fixed entropy outer boundary

8.8

17.6

26.4

35.2

Dimensional time tJ [years]

44.1

Detectable on Jupiter?
• Typical timescales– Given a field of Bs ~ 3 mT & r ~ 853 kg/m3 at the equator at a topof the metallic region (~ 0.85 RJ), then Alfvén speed ~ 9.2*10-2 m/s– TW traveltimes across the metallic region can be 9-13 years• Note: the internal field uncertain– amplitude < 1/10 of ourzonal jet outside MTC– cf. changes in the zonal
wind at the cloud level?
(Tollefson et al. 2017)– cf. global upheavals??
(e.g. Rogers 1995)

Cylindrical radius s/rcutoff• TW seen on a spherical surface
above the metallic region----

Filtered, zonal velocity fluctuation u’f /max(Uf) at thecutoff boundary (~0.96 RJ) in the southern hemisphere
MTC
~28ºS

60ºS

0

UA

8.8

17.6

26.4

35.2

Dimensional time tJ [years]

44.1

• Zonal wind speed– In-situ (Cassini vs. Voyager 2) reported (Porco et al. 2003)– ground/HST campaigns (2009-2016) identified relevant variability
near 24 ○N & 5-7 year periods at lower latitudes (Tollefson et al. 2017)

1989JBAA...99..135R

Long-term changes at the cloud deck?

• Coloration, brightening, outbreak events, etc.
– sketched for > 100 years: ‘global upheavals’ (Rogers 1995; Fletcher 2017)– irregularly, but periodic at some epoch at NTB?
174

J. Tollefson et al. / Icarus 296 (2017) 163–178(Tollefson et al. 2017)(Rogers 1989, 1995)
Fig. 10. Left: Time series of low-latitude zonal winds from WFC3 (2009–2016), combined with zonal winds measured from Voyager (not shown) and HST/WFPC2 data

350

through MHD waves ([10]; also sec. 1). One may envisage an analogous cou-

Length-of-day variations

pling in Jupiter between the deeper conducting metallic region and the overlying
insulating molecular envelope and hence a Jovian LOD fluctuation.

•WeTW
transport
thebyangular
evaluate
the influencecalculatingmomentum
the axial angular momentum change– almost-perfectly
exchanging
the angularmomentumds
that is deduced
from the axisymmetricdisturbancesin ourmetallic hydrogen355

region,with the overlying molecular region, whereδσ = 2π! smtc ! z+stcz−h"ρeq #s2 "u"φ #dzds ,(9)and those outside the region,• This may fluctuate! therotation rate (LOD)!planet’s
r
z
cut

+

2 "
δσomtcds= 2π#dzds. variation dP(10)– the change= -2p I dP/P2h"ρ
implying
LOD
eq #s "uφan
smtc

z−

s) changes
the display the time evolutions of δσ andIn figure– 6 cf.
theO(10solid-2 anddotted ofcurvesemission (Higgins et al. 1996)

Axial angular momentum
change ds

System?: decametricinsideMTC
δσomtc in model
E, III
respectively.
Theradioδσ of the conducting region showsa quasiperiodic variation, corresponding the flow oscillations (fig. 2b). More notably,360

the evolution is almost perfectly anti-correlated with the change δσomtc of the14

outside MTC
0

at E = 1.5*10-5, Pm=3, Pr = 0.1, H=1.4& fixed entropy outer boundary

8.8

17.6

26.4

Dimensional time tJ [years]

35.2

44.1

Jovian LOD changes?
• The gas giant’s rotation rate
– System III (1965): 9h 55m 29.71s• relying on measurements of decametric radial emission fromthe magnetosphere (Burke & Franklin 1955)• the accuracy in O(10-2s) has been some debate– the true change (Higgins et al. 1996, 1997)– jovimagnetic
et al. 2001;
Ridley &PERIOD
Holme 2016)HIGGINS ET SV
AL.'(Russell
JUPITER'SROTATION

the same observatoryand the samefrequencyduring– what by
else??
o apparitions separated
about 24 years are shown
panels a and b. The crosscorrelation function, i.e.,e plot of crosscorrelationcoefficientsversusthe CMLft of the secondhistogramrelative to the first one, is

29.75

ownin panelc. The Cli/ILshiftfor maximumcorrela-

29.70

n, in degrees,is AA. The correctedrotation period,, is calculatedby meansof equation(1).

The averagingof these initially determined rotation
riod values was done in two stages. Values obtainedm different pairs of apparitionscan certainly be coneredto be statistically independent. We also consid(Higgins
et al. 1996)
d that values obtained during the same
apparition

2655

29.90

29.85

29.80

29.65
29.60
29.55

29.50
1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984

Mid-Date (year)

Summary
Axisymmetric, torsional Alfvén waves possibly excited in
Jupiter’s metallic H region• identified in Jovian dynamo simulations– implementing a smooth transition from the metallic to molecularregions, forming a magnetic TC• propagating in cylindrical radius with Alfvén speeds ~ Bs/r1/2– on timescales of O(100-1 yrs) for an equatorial field of 1-3 mT• Note: the dimensional values may vary

– reflections from MTC, also standing ‘oscillations’, may reveal the radius– angular momentum exchanges with the overlying molecular region,
fluctuating LOD– detectable in surface zonal flows beyond the metallic region

Thank you



@u
Pm
1
P m2 Ra dT+ u · ru =rp̂ + 2êz ⇥ u(r ⇥ B) ⇥ B
Sêr + P m F V (2)@t 2 Numerical
E simulations⇢
P r dr@B• dynamo(tabler1)⇥ (⌘r ⇥ B) , r · B = 0= rsimulations⇥ (u ⇥ B)(3)@t equations
–
governing@S • MHD dynamos
Pm 1Pr1 ⌘in rotatingbyanelastic
convection
2 spherical shells+ u · rS = ⇤ to modeldriven
r
·
⇢T
rS
+
H
+
(r
⇥
B)
+ hydrogen
QV(4)
the metallic hydrogen region and the overlying
molecular
layer
@t
P
r
E
⇢
⇢T
P
mRaT
– adopting the Lantz-Braginsky-Roberts formalism (Lantz & Fan 1999; Braginsky & Roberts

Anelastic spherical dynamo simulations

where⇤ anelastic approximation, compressible but omitting sound speeds
1995; also Jones+ 2011)⇤ convection is driven mostly?? by internal heating– dimensionless, governing equations about the reference state:
✓

◆

@uj
@
@ui
2 @ @uj
+
⇢
⇢
⇢ P@x
@xj
@xi 1
3⇢ @xi @xjP m2 Ra dT@umj+ u · ru =Sêr + P m F V✓ rp̂ + 2êz ⇥ u ⇢ (r ⇥ B) ⇥◆B
@t
E
P
r
dr
@uj
@ui @ui2
Q@B=+
ij r · uV=r⇥j (u ⇥
B)
r
⇥
(⌘r
⇥
B)
, r·B =0
@x
@x
@x
3
j
i
@t
r · ⇢ u = 01

FV,i =



2
@S
P
m
1
P
r
1⌘
J =
+ u ·QrS= ⌘(r ⇥ B)r · ⇢T rS + H +(r ⇥ B)2 + QV@t
P r ⇢TP mRaT E ⇢

(1)

(5)
(2)

(6)
(3)

(7)
(4)

Also, the dimensionlesswhere parameters are– with Ekman, kinetic/magnetic Prandtl,✓and Rayleigh◆ numbers with mid-depth values (Xm):
@uj
1 @
@u⌫i
2Tm
@ d2 @uS
j
⌫
⌫
F
=
⇢
+
⇢
(5)
V,i
E=
,
P
r
=
P
m
=
,
Ra
=
(8)
⇢
@x
@x
@x
3⇢
@x
@x
j
j
i
i
j
⌦d2

⌘m
✓
◆⌫
@uj
@ui @ui
2
-5
7)
Q
+
(6)
ij r · uV =(1.5-2.5)*10
0.1
3
O(10
@x
@x
@x
3
j
j
i
output parameters-7
O(10-18)
0.1-1
QJ = ⌘(r ⇥O(10
B)2 )
(7)
2
|B ⇤ |
d|u⇤ |
|u⇤ |
E
RmE
2
⇤=
=dimensionless|B | , Rmparameters== |u| , Ro =
= |u|
=
(9)
Also, theare⇢m µspherical⌘mbased on pseudo⌦dmPm• Leeds
dynamo code:
spectralP method
0 ⌘m ⌦⌫
E=,⌦d2
three models built by Jones (2014): table

⌫
Pr =


⌫
Pm =
,
⌘m

Tmid d2 SRa =⌫

(8)