Planetary interiors:
Magnetic fields, Convection and Dynamo Theory
4. Core dynamics: Rotation and Magnetic fields
Chris Jones, Department of Applied Mathematics
University of Leeds UK

FDEPS Lecture 4, Kyoto, 30th November 2017

Section 4.
Core dynamics: Rotation and Magnetic fields
4.1 Rotating magnetoconvection(4.1) Rotating magnetoconvection

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Lorentz force
We have seen how magnetic fields can be created by dynamo
action, but how do they affect the fluid flow?
A region of fluid in which there is a current density j flowing in amagnetic field B experiences a body force j × B. This force iscalled the Lorentz force. It arises because electrons and ionsmoving in a magnetic field experience a force.
The Navier-Stokes equation in a rotating frame then becomes∂u
1
1+ 2Ωẑ × u = − ∇p + g αT 0 ẑ + j × B + ν∇2 u.
∂tρρ

(4.1.1)

Note that since µj = ∇ × B,
1
1
j×B=(∇ × B) × B.
ρµρ(4.1) Rotating magnetoconvection(4.1.2)

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Effects of Lorentz force
The Lorentz force can impede convection when it is strong. This is
why sunspots are darker than the solar photosphere. Strong
magnetic fields in the sunspot block convective heat transport
there making the sunspot relatively dark.
The Lorentz force can act as a restoring force, giving rise to waves,called Alfvén waves.
When combined with the Coriolis force, the Lorentz force gives riseto new waves, MAC waves (MAC: Magnetic, Archimedean andCoriolis forces).
Kinematic dynamos grow exponentially: the Lorentz force stops
them growing by changing the flow, causing the dynamo tosaturate at a finite field strength.

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Rotating magnetconvection: plane layer
The rotating plane layer with a vertical magnetic field

Rotation and magnetic field parallel to gravity, centrifugal
acceleration negligible.
A Rayleigh-Bénard convection cell with a liquid metal (Gallium or
liquid sodium) is put on a rotating turntable. Strong verticalmagnetic field imposed externally.

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Plane layer model
Chandrasekhar, 1961. The equation of motion is∂u11+ 2Ωẑ × u = − ∇p + g αT 0 ẑ + j × B + ν∇2 u.
∂tρρ

(4.1.3)

The temperature equation is∂T+ u · ∇T = κ∇2 T ,∂t(4.1.4)and the induction equation is∂B= ∇ × (u × B) + η∇2 B,∂t(4.1.5)with∇ · u = 0,∇·B=0(4.1.6)

since the fluid is incompressible.
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Linearisation

Since the imposed field is uniform, B = B0 ẑ, which has no current.
Then j × B = j × B0 ẑ, where j is the current induced byconvection, and we assume the magnetic field generated byconvection is b, which is small compared to B0 ẑ.
The induction equation is then∂(B0 + b)= ∇ × (u × (B0 + b)) + η∇2 (B0 + b),∂t(4.1.7)

which linearises to∂u∂b= ∇ × (u × B0 ẑ) + η∇2 b = B0
+ η∇2 b.
∂t∂z(4.1) Rotating magnetoconvection(4.1.8)

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Non-rotating case 1.
With Ω = 0 the linearised equations are then1∂u1
= − ∇p + g αT 0 ẑ + j × B0 ẑ + ν∇2 u,
∂t
ρ
ρ
∂T 0
= βuz + κ∇2 T 0 ,∂t
The z component of the induction equation is
∂bz
∂uz
= B0
+ η∇2 bz .
∂t∂z

(4.1.9)(4.1.10)(4.1.11)The z-component of the double curl of the momentum equation isthen∂ 2
B0 ∂ 2∇ uz = g α∇2H T 0 +∇ bz + ν∇4 uz .
∂t
µρ ∂z

(4.1.12)The simplest case is to look for steady neutral solutions with notime-dependence, so the time-derivative terms are zero.
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Non-rotating case 2.
We can then eliminate ∇2 bz from (4.1.12) using (4.1.11), to get
0 = g α∇2H T 0 −B02 ∂ 2 uz
+ ν∇4 uzµρη ∂z 2(4.1.13)and we eliminate T 0 using the temperature equation (4.1.10). We
non-dimensionalise as usual, and two dimensionless parameters
emerge.
B 2d 2g α∆Td 3,
Q= 0 ,(4.1.14)R=κνµρνη
are the usual Rayleigh number and Q is called the Chandrasekhar
number, which measures the field strength. With stress-free,constant temperature boundaries, uz = sin πz exp i(kx x + ky y ) is asolution, and if a2 = kx2 + ky2 , we get
Rsteady =π 2 + a2 2[(π + a2 )2 + π 2 Q],
a2(4.1.15)as the dispersion relation.
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Non-rotating magnetoconvection results
This compares with the rotating case result, (2.5.16), with Q inplace of E −2 . A similar analysis shows that as Q → ∞, minimum
Rsteady occurs when

a∼

π4
2

1/6

Q 1/6 ,

Rasteady ∼ π 2 Q.

(4.1.16)

So since a becomes large as magnetic field increases we get tallthin columns.
Since the critical Rayleigh number increases as Q → ∞, magnetic
field suppresses convection. This impeding effect of the magnetic
field is why sunspots are dark, the heat transport being reduced.
If κ < η, normally the case in planetary cores, steady modes are
the first to onset, but if κ > η oscillatory modes are possible.
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Alfvén waves
In the limit of no diffusion and uniform temperature T 0 = 0,
equations (4.1.11) and (4.1.12) are
∂uz
∂bz
= B0
,
(4.1.17)∂t∂z
and the z-component of the double curl of the momentum
equation is then∂ 2
B0 ∂ 2∇ uz = +∇ bz .
∂tµρ ∂z(4.1.18)

Look for wave like solution ∼ exp i(kx x + ky y + kz z − ωt), and wefind wave solutions with
B2
(4.1.19)
ω 2 = 0 kz2µρ
which are Alfvén waves travelling with the Alfvén speedc=(4.1) Rotating magnetoconvection

ω
B0
=√ .
kz
µρ

(4.1.20)
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Magnetic field lines as stretched strings

Since the waves travel along the field lines, it is helpful to think ofmagnetic field lines like stretched strings.
If you give the field an initial perturbation, it responds by sendind
waves along the field lines, just as a stretched string would.
The Lorentz force can be written
1(∇ × B) × B = (B · ∇)B/µ − ∇B2 /2µ.
µ(4.1.21)

The first term is called the curvature force. It is active when fieldlines are curved.
The second term is called magnetic pressure, and it adds to thegas pressure.

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Convection with rotation and magnetic field 1.
We go back to equations (4.1.3) to (4.1.6) and linearise1∂u1+ 2Ωẑ × u = − ∇p + g αT 0 ẑ + j × B0 ẑ + ν∇2 u, (4.1.22
∂t
ρ
ρ
∂T 0
= βuz + κ∇2 T 0 ,
∂t
∂b
∂u
= B0
+ η∇2 b.
∂t∂z

(4.1.23)(4.1.24)

Vorticity equation is
∂ζ
∂u
B0 ∂j
− 2Ω
= g α∇ × T 0 ẑ ++ ν∇2 ζ,
∂t
∂z
ρ ∂z

(4.1.25)

∂j
∂ζ
= B0
+ η∇2 j.
(4.1.26)∂t∂z
We might expect rotation and magnetic together to be even morestabilising, but this is not true.
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Convection with rotation and magnetic field 2.
We look for steady solutions as before, with the usual horizontal
dependence ∼ exp i(kx x + ky y ) with kx2 + ky2 = a2 ,and non-dimensionalise using d, d 2 /η as the unit of length andtime.
The dimensionless parameters appearing are now
Λ=

B02
,
µρΩη

E=

ν
,
Ωd 2

R∗ =

g α∆Td.
κΩ(4.1.27)

Λ is called the Elsasser number, and Λ ∼ 1 in planetary cores. E is
the Ekman number we had before (small in planets).
The modified Rayleigh number R ∗ = ERa. This is still large inplanets, but not as large as Ra.

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Convection with rotation and magnetic field 3.

If time-dependence is retained, two other Prandtl numbers appear,
Pr =

ν
,
κ

Pm =

ν
.
η

(4.1.28)

The hydrodynamic Prandtl number and magnetic Prandtl number
respectively.
The magnetic Prandtl number is Pm ∼ 10−5 in planetary cores,
whereas Pr ≈ 0.1.
If the buoyancy is due to light material rather than temperature,
the effective Prandtl number (the Schmidt number ν/κdiff ) islarge.

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Ordinary differential equations in z
Our equations now become
E (D 2 − a2 )ζ + 2Duz + ΛDjz = 0,

(4.1.29)E (D 2 − a2 )2 uz − 2Dζ + Λ(D 2 − a2 )Dbz − R ∗ a2 T 0 = 0, (4.1.30)(D 2 − a2 )bz + Duz = 0,

(4.1.31)(D 2 − a2 )jz + Dζ = 0,

(4.1.32)(D 2 − a2 )T 0 + uz = 0,

(4.1.33)

where ζ = ζ · ẑ, D = d/dz and a is the horizontal wavenumber.
The case with stress-free, electrically insulating and constanttemperature boundaries,
D 2 uz = uz = Dζ = jz = T 0 = 0 on z = 0, 1,(4.1.34)

gives uz = sin πz solutions.
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Rotating, magnetic dispersion relation
The expression for the modified Rayleigh number is now
π 2 + a2 [E (π 2 + a2 )2 + Λπ 2 ]2 + 4π 2 (π 2 + a2 )∗R =. (4.1.35)
a2
E (π 2 + a2 )2 + Λπ 2
If Λ is very small, this reduces to the purely rotating case, for whichat small E we have a ∼ E −1/3 and R ∗ ∼ E −1/3 , which is large.
But if Λ is O(1), then the E terms are negligible, and then a andR ∗ are also O(1), which is smaller than O(E −1/3 ). So the effect ofthe magnetic field is to reduce the critical Rayleigh number, i.e. it
helps convection.
The magnetic field is breaking the Taylor-Proudman constraint,
and so allowing convection to occur on larger horizontal
length-scales. This could be important in planetary dynamos.

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Field and rotation axes at different angles

The classic problem has the rotation vector and the magnetic fieldaligned with gravity. There have been studies where the rotation
vector and the magnetic field are no longer aligned.
In the case where they are mutually perpendicular, appropriate
near the equator of a planet, rolls can line up either with therotation, the field or neither, depending on the parameters.
These problems are complicated because of the many parameters
and angles involved, but they all reduce to systems of ODEs, which
are straightforward to solve numerically.

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Spherical geometry

The problem has also been considered in spherical geometry. Thisis more challenging numerically, but the overall picture is similar.
It is normally the case that there is a critical Elsasser number Λc .
For Λ < Λc rotation is dominant, and we get tall thin columns anda high critical Rayleigh number.
For Λ > Λc magnetic field breaks the Taylor-Proudman constraint,
and wider columns occur, with a lower critical Rayleigh number.
For more complicated magnetic fields, magnetically driven
instabilities can occur.

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4.2 Waves in rotating MHD(4.2) Waves in rotating MHD

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Geomagnetic field and Secular Variation
Geomagnetic field is described in terms of the Gauss coefficients ofthe spherical harmonic expansion outside the core B = −∇Ψ.
Ψ = rs∞ m=n
X
X  rs n+1
n=1 m=0r

Pnm (cos θ)(gnm cos mφ + hnm sin mφ).
Magnetic field Br measured at thesurface, but mantle insulating, so
can extrapolate field Br to
Core-Mantle boundary (CMB).
Radial geomagnetic field at CMB in
1980 and 2000. Wave-like behaviour
seen.
Small changes occur in 20 years.
Secular variation is firsttime-derivative of geomagnetic field.
Some Westward drift is visible.

(4.2) Waves in rotating MHD

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Types of waves in the coreP-waves
Inertial waves
Torsional waves
Magnetic Rossby waves

S-wavesInternal gravity waves
MAC waves
Dynamo waves

So many different waves! How do we make sense of them?
S-waves and P-waves are seismic waves, excited by earthquakes
and detected by seismometers. Travel very fast, 3-8 km/sec.
All except S-waves occur in a fluid. P-waves depend on
compressibility. All the other fluid waves occur in incompressible
fluid, ∇ · u = 0.
Inertial waves occur in rotating fluids, internal gravity waves indensity stratified fluids.
MAC waves, Torsional waves and magnetic Rossby waves are all
rotating MHD waves. Dynamo waves are the kinematic waves indynamo theory.
(4.2) Waves in rotating MHD

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Cartesian box model for core waves

We consider a small piece of core, rotating with angular velocityΩẑ, with uniform magnetic field B0 (any direction).
We then give the fluid a small velocity perturbation of the formu = uo exp i(k · x − ωt) wavevector k, frequency ω.
Wave travels in k-direction. ∇ · u = 0 so u0 · k = 0, transverse
waves.
We insert the wave form into the equation of motion and theinduction equation, then work out the dispersion relation between
ω and k.

(4.2) Waves in rotating MHD

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Equation of motion
Linearised equation of motion for adiabatic inviscid fluid∂u∂t+ 2Ω × u =Inertia

Coriolis accn−∇p 0 /ρ1
+ j × B0ρ

Pressure(4.2.1)

Lorentz force

If fluid is stably stratified (not adiabatic) there is an extra
buoyancy term.
Magnetic field equation (induction equation)∂b= (B0 · ∇)u + η∇2 b,µj = ∇ × b,
(4.2.2)∂tIgnore magnetic diffusion in simple theory (magnetic Reynolds
number 1000).

(4.2) Waves in rotating MHD

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Dispersion relation for rotating MHD waves 1.
Eliminate pressure by taking curl of equation of motion, to get thevorticity ζ equation.
1∂ζ− 2(Ω · ∇)u = (B0 · ∇)j,∂tρ

ζ = ∇ × u.

(4.2.3)

Take time-derivative and eliminate j using the curl of the induction
equation∂2ζ1∂u=(B0 · ∇)2 ζ,(4.2.4)− 2(Ω · ∇)
2
∂t
∂t
µρ
Now insert u = uo exp i(k · x − ωt) to get(B0 · k)2∂u2
− ω ζ = 2(Ω · ∇)(4.2.5)µρ∂t
Now take the curl of this, using ∇ × ζ = −∇2 u, to get(B0 · k)2∂ζ2−− ω ∇2 u = 2(Ω · ∇)µρ
∂t(4.2.6)and now use (4.2.5) to get(4.2) Waves in rotating MHD

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Dispersion relation for rotating MHD waves 2.
the dispersion relation
2(B0 · k)2
− ω 2 k2 ,µρp(Ω · k) ± (Ω · k)2 + k2 (B0 · k)2 /µρ
with solution ω =. (4.2.7)|k|4ω 2 (Ω · k)2 =

There are a lot of waves in this formula!
Start by setting B0 = 0. Thenω=2(Ω · k)= ωc .
|k|(4.2.8)

These are Poincaré waves or inertial waves. Period generally a bit
longer than 2π/2Ω = 12 hours.
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MC waves
Now restore B0 . Plus sign gives fast magneto-inertial waves.
These waves travel faster than the inertial waves, because themagnetic field adds to the restoring force.
Minus sign is more interesting, defineωM =(B0 · k).
(µρ)1/2(4.2.9)2π/ωm ∼ 6 years in the core. This is the Alfvén time, the time ittakes for an Alfvén to go back and forth in the core.
Generally in planets ωC >> ωM .
We can now writeω=(Ω · k) ±p(Ω · k)2 + k2 (B0 · k)2 /µρ
.
|k|

as
ωc
ωc
ω=
±
2
2
(4.2) Waves in rotating MHD

s
1+

2
4ωm
.
ωc2

(4.2.10)27/69

Slow MC waves

Then binomial expansion givesω = ωMC =

2
ωM
+ ...
ωC

(4.2.11)which in general corresponds to a period of thousands of years.
These are the slow MC waves. If the buoyancy term is included
they become MAC waves (M magnetic, A Archimedean, C
Coriolis). Too slow to see directly in the SV signal, but may be
important on dynamo time-scales.

(4.2) Waves in rotating MHD

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Slow magnetic Rossby waves
Recall that quasi-geostrophic motions with kz small can haveperiods of months rather than days, i.e. smaller ωC .
2 /ω is larger than usual, so periods
So for these waves ωMC = ωMCmay be hundreds of years or maybe even just decades.

These are the slow magnetic Rossby waves, a special QG class ofthe slow MC waves.
Hide showed that in sphericalgeometry these slow magnetic
Rossby waves have aquasi-geostrophic columnar
structure.
Recently shown (Hori et al. 2015,2017) that these occur in dynamo
simulations and may be seen in the
SV signal.
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Magnetic Rossby Waves

Magnetic Rossby waves in a sphere can be solved exactly for the
special Malkus field B0 = B0 s φ̂. Cylindrical coordinates (s, φ, z).
Not very Earth-like field, but gives idea of what they look like.
The solution is too complex to go through here: details in Malkus
1967, J Fluid Mech. 28, p793.

(4.2) Waves in rotating MHD

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Magnetic Rossby Waves

1
0.8
0.8
0.6
0.6
0.4

0.4

0.2

0.2

Meridional plane section of ur
with m = 8 (exp imφ). Malkus
field B0 = B0 s φ̂. Cylindricalcoordinates (s, φ, z).

0
0
−0.2

For each m there are modes with
Ns rolls in the s-direction and Nz
rolls in the z-direction. Magnetic
Rossby waves have Nz = 1.
Modes which have Ns large and m moderate have only a small
amount of vortex stretching, so have frequencies faster than the
ten thousand year period of ‘normal’ MC waves.
−0.2

−0.4

−0.6

−0.8

−1
−1

−0.4

−0.6

−0.8

−0.6

−0.4

−0.2

0

(4.2) Waves in rotating MHD

0.2

0.4

0.6

0.8

1

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4.3 Torsional waves in the core(4.3) Torsional waves in the core

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Rotating MHD waves: Torsional Oscillations
There is a special class of waves with Ω · k = 0. They are
geostrophic waves, with the Coriolis force balanced entirely bypressure gradient. These modes don’t stretch the vortex lines. The
restoring force is just Lorentz force, so these are torsional
oscillations (TOs). In the Earth’s core, they have a period of about
6 years.

Torsional waves transport angular momentum. This gets
transferred to the mantle, resulting in small but observable lengthof day changes.
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Torsional oscillation observations(a) Gillet et al 2010:(b) Holme & de Viron 2013(a) The axisymmetric part of uφ constructed from observations of
the geomagnetic field.
(b) Length-of-day variations. Since total angular momentum isconserved, torsional waves change the mantle rotation rate.
An approximately six year period is seen in both data sets. Waves
propagate out from the tangent cylinder, the cylinder surrounding
the Earth’s solid inner core.
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Dynamo model equations∂uPm+ (u · ∇)u = −[∇p + 2ẑ × u − (∇ × B) × B]∂t
E
Pm2 Ra+T r + Pm∇2 u(4.3.1)
Pr
∂T
Pm 2
∂B
+ (u · ∇)T =∇ T + ,− ∇ × (u × B) = ∇2 B∂t
Pr∂t
(4.3.2, 3)∇ · u = 0, ∇ · B = 0(4.3.4)g α||D 5
ν
ν
ν
,
Ra
=
, Pr = ,Pm = . (4.3.5)
2
ΩD
νκη
κ
η
Boundary conditions used: no-slip, fixed flux; insulating magnetic.
Zero compositional flux at CMB, uniform sink in interior.
Parameters used: E ∈ [10−6 , 10−4 ], Ra/Rac = 8.3, Pr = 1,
Pm ∈ [1, 5], radius ratio = 0.35 These give ‘Earth-like’ dynamo
models: basically dipolar but with an appropriate amount ofreversed flux patches.
E=(4.3) Torsional waves in the core

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Torsional Oscillations movie

Time-lapse movie of the axisymmetric part of uφ on a meridional
section. E = 10−5 .
The solid inner core of the Earth doesn’t take part in theoscillations in these simulations.
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Detecting Torsional Oscillations in the simulations
Alfvén wave equation in spherical geometry is u 
∂ 2  uφ 
1 ∂
φ
2 ∂
3
= 3
s hUA
∂t 2 ss h ∂s∂s s(4.3.6)

where UA 2 = hBs 2 /µρi, the bar denoting φ average and angle
brackets z-average. h(s) is height of cylinder.
We run the dynamo simulation, and when initial transients havegone, we run for a further time τ , and evaluate UA from Bs , thefield averaged over time τ .
We make plots in the t-s plane of the fluctuating part of uφ , i.e.
fφ where u
fφ is the time-average over the whole τ run.
uφ0 = uφ − u
We then look for features in uφ0 propagating with the Alfvén speed.
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Torsional Oscillations detected

Left: Inside Tangent Cylinder North. Right: Inside Tangent
Cylinder South. huφ i0 , E = 10−4 , Pm = 5. White curves have
gradient UA . Similar pictures for E = 10−5 .
Plus points: torsional oscillations found in dynamo simulations.
Mostly (but not exclusively) travel outwards. When field is scaled
to observed Br at the CMB, travel time from tangent cylinder toequator is about 3 years, similar to Gillet et al. data.
Minus points: Origin inside the tangent cylinder rather than at theTC. Waves not as periodic as suggested by the Gillet et al. data.
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Magnetoconvection simulations
In geodynamo simulations the importance of the Reynolds stress isoverestimated: kinetic and magnetic energy is typically similar,
whereas magnetic energy is much larger than kinetic energy in thecore.
Also, the need to generate a magnetic field places severe
restrictions on the accessible parameter space, e.g. very expensive
to lower Pm.
So we used Magnetoconvection simulations. Same code, butchange magnetic boundary condition: impose a dipole magnetic
field at the core-mantle boundary and/or inner-core boundary.
Saves CPU time: don’t have to wait for field to build up. Allows
lower E , much lower Pm and can get into dominant magnetic
energy regime.
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Magnetoconvection results

E = 5 × 10−6 , Pm = 0.1, Ra = 5Rac , Λ ≈ 100.
Left: uφ0 contours. Right: FL0 contours. FR0 negligible.
In this strong field, low E case, TO’s are much more periodic, andoriginate from the tangent cylinder and propagate outwards. No
significant reflection.

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Summary on Torsional Oscillations

TOs observed in the core from secular variation and LOD signal.
Can be modelled using dynamo and magnetoconvection codes.
They originate from quite small scale (∼ 400 km) convection,
which has turnover time and period ∼ 6 years.
The convection is broad-band in azimuthal wavenumber and has
an m = 0 component.
The convection disturbs the TC shear layer, and this forces theTOs which propagate outwards.

(4.3) Torsional waves in the core41/69

4.4 Magnetic Rossby waves in the core?

(4.4) Magnetic Rossby waves in the core?

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Secular Variation: waves or flow?
We now consider the nonaxisymmetic modes and their connection
with the secular variation.
• Are these patches of magnetic field advectedby an approximately steady flow, or are theymagnetic Rossby waves?
• Most, but not all of the signal can be fittedwith a steady flow, the eccentric gyre, shownin equatorial plane• This flow is believed to be fairly
z-independent. Carries the equatorial flux
patches westward in the Atlantic hemisphere,
while flow is eastward under the Pacific.
Recent evidence of fast flow at high latitudes
under Siberia.
Atlantic hemisphere(4.4) Magnetic Rossby waves in the core?

• There are definite features which cannot be
explained by a steady flow.
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Magnetic Rossby waves in a thick shell

The axial z-component of the vorticity equation is
ρ

dζ
∂u 0
e
− 2ρΩ z = êz · ∇ × (j 0 × B)
dt∂z

(4.4.1)e is the mean field.
where ζ 0 = ∇ × u 0 is the vorticity and B
In the simulations the azimuthal wavelength is shorter than theradial or axial wavelengths, or the mean length scales. We
integrate over z, and use ζ ≈ −(1/s)(∂us0 /∂φ) to get
Z +H
Z +H f2 3 0
Bφ ∂ us
d d 1
1 ∂us02Ωs us01
dz − 2=dz
2dt dt 2H −H s ∂φro − s2ρµ0 H −H s 3 ∂φ3(4.4.2)(4.4) Magnetic Rossby waves in the core?

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Magnetic Rossby waves in a thick shell
Look for waves exp{i(mφ − ωt)}.
Magnetic Rossby waves have negligible inertial term, so the mostright-hand term can be neglected.
We need to remember that
Ûφ ∂
d
∂
=
+
dt
∂t
s ∂φ
so
2 2
2
2
Ûφ m (r0 − s )B̂φ
ω
=
−
.
m
s
2Ωµρs 4(4.4.3)Magnetic Rossby waves go westward relative to mean flow
advection speed Ûφ .
Advection speed similar to the wave speed in the Earth’s core.
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Magnetic Rossby wave frequencies
(a)

(b)

The left plot is the wave speed and the right figure is theazimuthal flow speed from a dynamo simulation with E = 10−5 ,Pm = 5, Ra = 8.32Rac . VMC is phase speed of magnetic Rossby
waves, ζ = Ûφ /s is advection angular velocity.
VMC =m2 (r02 − s 2 )Bφ22Ωµρs 4.

Both similar, but flow dominates near equator, waves dominate at
higher latitudes. Higher latitudes where to seek waves? Large mbetter, but m > 13 can’t be detected.
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Rossby waves in simulations
(a)

(b)

E = 10−5 , Pm = 5, Ra = 8.32Rac
Technique same as for TOs: remove time average part and plot
the z-averaged part of us .
Left: φ − t plot of < us > at s = 0.5ro (latitude 60◦ ). Right:
same at s = 0.766ro .
White dashed: advection speed.
Left: Black solid is wave speed + advection speed for m = 5 andBlack dashed is m = 8. Right solid is m = 6.
At higher latitudes, waves + flow correlate well. At low latitudes,
hard to distinguish.
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Comparing with data

At 40◦ S, upper black line is drift
speed from observational model
gufm1, lower black line is drift
speed with core flow removed(Hulot et al. 2002). Difference ≈0.2◦ per year.
The coloured dashed lines are theexpected wave speeds from ourmodel as a function of thez-averaged toroidal field.
Indicates that the azimuthal field is a lot stronger than the poloidal
field of 3 mT if there is a wave component to the secular variation(4.4) Magnetic Rossby waves in the core?

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Conclusions on Magnetic Rossby waves• Magnetic Rossby waves can be seen in dynamo simulations• Waves are generally slow, so the wave drift will be dominated byadvection at lower latitudes.
• Best chance of separating the waves from the advection velocity
is at high latitudes, where the wave speed is significant comparedto the flow speed.
Difficulty is that waves are quite nonlinear.
• Need to analyse the data to see the wave speeds of the different
wavenumbers: dispersion relation depends on ratio ofwave/advection speed.

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4.5 Shallow water MHD model(4.5) Shallow water MHD model

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Stable layer in the Earth’s core?
Helffrich & Kaneshima (2010)
found a layer at the top of theliquid outer core where seismic
waves are slower than in rest ofouter core. Interpreted as astable layer few hundred km
thick.
Earth’s core with stable layer
Braginsky (1984, 1993) suggested light material released as thesolid inner core grows would accumulate under the core mantle
boundary: inverted ocean of the core. Even if the core iscompositionally mixed, there could be a thermally stably stratified
layer below the CMB.

Neutrallybuoyant interior

CMB

Inner
Core

Stable Stratified layer

MantleAxisymmetric core oscillations visible in both secular variation andlength of day observations. 6 year period due to torsional Alfvénoscillations in whole core, but Buffett (2015) suggested the 60 year
period could be MAC waves in the stably stratified layer.
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Waves in the tachocline?

McIntosh et al. 2017. Brightpoints in the corona show a large
scale slow wave-like motion, withvery slow wavespeed, 3 m/sec.
Bright points in the Solar Corona,viewed from N. Pole using Stereo

Slower than any waves in the corona. Suggestion is that there aremagnetic Rossby waves in the stably stratified tachocline.
These are coupled through the magnetic field to the bright points
in the corona.
These waves may be linked to dynamo action in the Sun.
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Shallow water MHD model

Perfectly conducting fluid, so the wavy boundary is a field line.
When applied to a stably stratified layer in the core, picture is
upside down, with wavy boundary being the interface between the
low density layer fluid and higher density outer core fluid. Rigid
boundary at the CMB is then above the wavy boundary.

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Magnetic field configuration

We concentrate on simplest problem, an azimuthal field symmetric
about the equator B = B0 sin θφ̂. Now have fairly complete
picture of this case.
Applications have radial fields as well, but these can transmit
energy in and out of the layer through Alfvén waves.
Also found significant differences if the azimuthal field has aninternal zero, e.g. B = B0 sin θ cos θφ̂.
Antisymmetric components of azimuthal field are expected in the
Earth and the Sun.

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Laplace Tidal theory with magnetic field

Spherical shell of radius R0 , coordinates (r , θ, φ), rotating withangular speed Ω0 , with gravity −gr̂ and azimuthal magnetic field
B = B0 (θ)φ̂.
The shell has a thin layer of electrically conducting fluid of depth
H, H  R0 .
We look at waves and instabilities in this system, using shallow
water MHD. Longuet-Higgins (1968) gave a full description of the
waves in the B0 = 0 case.
Mathematically equivalent to a thin stably stratified layer of fluid:
gH/R02 → N 2 , where N is the buoyancy frequency.

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MHD Shallow Water Model, Gilman(2000)∂B+ (u · ∇)B = (B · ∇)u∂t∂u
1+ (u · ∇)u + 2Ω × u =(B · ∇)B − g ∇H
∂t
µ0 ρ
∂H
+ ∇ · (Hu) = 0∂t∇ · (HB) = 0(4.5.1)(4.5.2)(4.5.3)(4.5.4)In these equations u and B represent the horizontal components ofthe velocity and magnetic field respectively. The operator ∇ is thehorizontal gradient, ρ is the density of the fluid, µ0 is thepermeability of free space and H is the thickness of the layer.
In (4), its horizontal divergence, so no contradiction with B being
divergence free in 3D.
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Linearised equations
Waves ∼ exp i(mφ − ωt). Scale frequency with 2Ω0 , so
λ = ω/2Ω0 .
H = H0 + h, h scaled as η = gh/4Ω2 R02 , change of variable:µ = cos θ. Differential operator D = − sin θ∂/∂θ = (1 − µ2 )∂/∂µ.
−λuθ + µuφ + Dη − mα2 bθ − 2α2 µbφ = 0
2(4.5.5)2

λuφ − µuθ − mη + mα bφ + 2α µbθ = 0(4.5.6)λ(1 − µ2 )η + Duθ − muφ = 0(4.5.7)λbθ + muθ = 0(4.5.8)λbφ + muφ = 0.

(4.5.9)

Two key parameters=4Ω20 R02,
gH0(4.5) Shallow water MHD model

α2 =

vA2
B02
2
,
where
v
=
,
A
µ0 ρ
4Ω20 R02

λ=

ω
.
2Ω0
(4.5.10)57/69

Determining the Eigenvalues and Eigenvectors
The solutions for the dependent variables are expansions of
Associated Legendre Polynomials,

uθ =

∞
X

m
Am
n Pn (µ),

n=m

bφ =

bθ =

∞
X

Bnm Pnm (µ),

uφ =

n=m
∞
X

Dnm Pnm (µ),

n=m

∞
X

Cnm Pnm (µ),

n=m

η=

∞
X

Enm Pnm (µ).

(4.5.11)n=m

Insert truncated expansions into equations → matrix eigenvalue
problem for λ. Wave types found:
Magneto-Inertial Gravity waves :Magneto-Kelvin waves(MIG waves)
Fast magnetic Rossby waves:(4.5) Shallow water MHD model

Slow magnetic Rossby waves
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Second oder ODESThe equation for uθ can be written(1 − µ2 )∂ 2 uθ
2m2
∂uθ
+
µ
2
2
2
2
2
2
∂µ
(λ − α m )(1 − µ ) − m ∂µnm(λ + 2mα2 )(λ + 2mα2 )2 2+ (λ2 − α2 m2 ) −
− 2
µ
2
2
2
(λ − α m )(λ − α2 m2 )om22m(λ + 2mα2 )µ2−uθ = 0,(4.5.12)−
1 − µ2 (λ2 − α2 m2 )(1 − µ2 ) − m2
and the equation for η is2
(λ + 2mα2 )2 (1 − µ2 )∂η2 ∂ η
(1 − µ ) 2 + 2
−1 µ
2
2
2
2
2
2
2
∂µ
(λ − m α ) − (λ + 2mα ) µ∂µh−m(λ + 2mα2 )m2(λ + 2mα2 )2 µ2 i
22 2−
+  (λ − m α ) −+(λ2 − m2 α2 )1 − µ2λ2 − m 2 α 22m(λ + 2mα2 )(λ2 − m2 α2 )+ 2
η = 0.
(4.5.13)(λ − m2 α2 )2 − (λ + 2mα2 )2 µ2(4.5) Shallow water MHD model

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Asymptotic simplification
These equations look complicated, but for  and α small, there are
solutions with λ ∼ O(−1/2 ) which is large, then the equation for η2(λ + 2mα2 )2 (1 − µ2 )∂η2 ∂ η
(1 − µ ) 2 + 2
−1 µ
2
2
2
2
2
2
2
∂µ
(λ − m α ) − (λ + 2mα ) µ∂µh−m(λ + 2mα2 )m2(λ + 2mα2 )2 µ2 i
2
2 2
+
−
+

(λ
−
m
α
)
−
(λ2 − m2 α2 )1 − µ2λ2 − m 2 α 22m(λ + 2mα2 )(λ2 − m2 α2 )+ 2
η = 0.
(4.5.14)(λ − m2 α2 )2 − (λ + 2mα2 )2 µ2
simplifies down to(1 − µ2 )



∂2η
∂η
m2
2
+
λ
−
−
2µ
η = 0.
∂µ2∂µ
1 − µ2(4.5.15)

which is the standard associated Legendre equation withwell-known solutions.
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Magneto-Inertial Gravity waves
When , α are small, we have gravity waves, η ∼ Pnm (µ),
ppn(n + 1)gH0= ±N n(n + 1).
ω=±(4.5.16)
R0
If  (rotation) increases, waves slow down and become equatoriallytrapped. If α (magnetic field) increases, waves speed up, also
become equatorially trapped. Eastward and westward branches
found.
ηα = 0.001
=1

λ: 2.4316 α:0.001 and ε: 1

η
0.1

2

0.05

1

0

0

α=5
=1

λ: 6.5646 α:5 and ε: 1
0.1

2

0.05
1
0
0
−0.05

−0.05
−1

−1
−0.1
−0.1

−2

−2
−2

α = 0.001
 = 100

η

−1

0

1

2

−2
η

λ: 0.37963 α:0.001 and ε: 100
0.01

2

0.005
1
0

α=5
 = 100−10
1
2
λ: 5.3152 α:5 and ε: 100

2

0.02
0.01

1

0

0

0
−0.005

−0.01
−1

−1
−0.01
−2
−2

(4.5) Shallow water MHD model

−1

0

1

2

−0.015

−0.02
−2
−2

−1

0

1

2

−0.03

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Further asymptotic simplification

At large  with α ∼ O(1), there are solutions with λ ∼ O(−1/4 )which is small.
We now consider the magnitude of the terms in the uθ equation,and many terms are eliminated, leaving the Parabolic Cylinderfunction equation, which is a form of Hermite’s equation.

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Asymptotic theory for MIG waves at large  or α
d 2 uθ
m(λ + 2mα2 )(λ + 2mα2 )2 2
2
2 2
+
(λ
−α
m
)−
u
−
µ ũθ = 0,θdµ2
(λ2 − α2 m2 )(λ2 − α2 m2 )(4.5.17)
Rescale
1 (λ2 − α2 m2 ) 1/4µ̂,(4.5.18)µ= √2 (λ + 2α2 m)2The solutions of this equation are the Parabolic Cylinder Functions.
Dispersion Relation 

1 (λ2 − α2 m2 ) 1/2m(λ + 2mα2 )122 2(λ − α m ) −=ν+ ,2 (λ + 2α2 m)2(λ2 − α2 m2 )2
(4.5.19)for ν = 0, 1, 2....
1/2
For large , with α ∼ O(1), λ ∼ ± (2ν+1)
,
1/4

1/3
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Fast Magnetic Rossby waves
At small  we have Rossby waves as well as MIG waves. These
have frequency O(Ω0 ), and spherical harmonic solutions. The term∂η/∂t drops out of the mass conservation equation.
They travel westwards, and at small α
ω=−

2Ω0 m
.
n(n + 1)

At large  they turn into equatorially trapped waves, described byparabolic cylinder functions.
As α increases, at α = 0.5, λ = −0.5m. For α > 0.5, they becomesuper-Alfvénic.
At large α they become polar trapped, described by Whittaker’s
equation, with associated Laguerre function solutions.
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Fast Magnetic Rossby wave eigenfunctionsη

α = 0.001
 = 0.01
m=1

λ −0.49988 α:0.001 and ε: 0.01

η
0.15

2

0.1
1

0.05
0

0

−0.05
−1

α = 0.1
 = 0.01
m=1

λ −0.49988 α:0.1 and ε: 0.01
0.1

2

0.05
1
0
0
−0.05
−1
−0.1

−0.1
−2

η

α = 0.001
 = 100
m=1

−2

−0.15
−2

−1

0

1

2

−2

λ −0.27096 α:0.001 and ε: 100

η
0.015

2

0.01
1

0.005
0

0

−1

−0.005−1

α = 0.1
 = 100
m=1

0

1

2

λ −0.28769 α:0.1 and ε: 100
0.015

2

0.01
1

0.005
0

0

−0.005
−1

−0.01
−2

η

α = 10
=1
m=2

−0.01
−2

−0.015−2−1
012
λ: −6.8105 α:10 and ε: 1

−0.015
−2

η

2

0.2

1

0.1

0

0
−0.1

−1

α = 10
=1
m=3

−1
0
1
2
λ: −19.795 α:10 and ε: 1

2

0.2

1

0.1

0

0
−0.1

−1

−0.2
−2

−0.2
−2

−2

(4.5) Shallow water MHD model

−1

0

1

2

−2

−1

0

1

2

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Slow Magnetic Rossby waves
These slow waves are in hydromagnetic equilibrium, so inertial
term in the equation of motion is negligible. Time-dependence
only in induction equation.
Frequency goes to zero as α goes to zero, so this branch doesn’t
exist in non-magnetic case.
When α and  are small, eigenfunction are spherical harmonics andω=

mva2(n(n + 1) − 2),2Ω0 R02so they travel eastwards. The n = 1 mode is anomalous, andtravels very slowly westwards.
Note that magnetic Rossby waves in a full sphere travel westwards.
At large α, slow magnetic Rossby waves become polar trapped,
and travel at nearly the same speed as the polar trapped fastmagnetic Rossby waves, going east rather than west.
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Magnetic instability when m = 1
4

0.4

3

0.2

1

n=1
m=1

0
−1

n=2
m=2

−2
−3
−4
0

ε=1
ε=10
ε=100
ε=10000λ=ω/2 Ω0

λ=ω/2 Ω0

2

α

−0.4
−0.6
−0.8

n=3
m=3
0.5

−0.2

1

1.5

−1
0

0.5

1

α

1.5

2

When m = 1, the fast and slow magnetic Rossby wave branches
coalesce and give rise to unstable waves. Similar behaviour in the
unstratified full sphere (Malkus 1967).
Instability requires α > 0.5. Right hand figure shows frequencies ofn = 2, m = 1 against α for varying .
The unstable waves can be found asymptotically at large α.
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Antisymmetric B = B0 sin θ cos θThis azimuthal basic state field is antisymmetric, so it is zero at
the equator
The MIG waves, Kelvin waves and fast magnetic Rossby waves
behave similarly to the symmetric case.
Surprisingly, slow magnetic Rossby waves are not found. Analytic
studies show a singularity near the equator for these modes.
Problem arises whenever the basic state field has a zero.
Need to restore magnetic diffusion to investigate the slow magnetic
Rossby wave case for this field.

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Summary on Shallow water MHD models
Combination of numerical and asymptotic methods result in a
fairly complete picture for the simple basic state field B = B0 sin θ.
Equatorial trapping and polar trapping can occur.
For strong magnetic fields, α > 0.5, m = 1 instability can occur,though field required is too large for planetary applications.
Slow magnetic Rossby waves are more sensitive to the exact field
configuration, and more work is needed here. Some features of thegeomagnetic field propagate eastward: could be connected towaves in a stably stratified layer?
Wave models in more realistic tachocline models could be
investigated, in the light of the slow waves detected by McIntosh
et al. 2017.

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