Download lecture 3 `Rapidly rotating convection, dynamos, and scaling laws`

3. Rapidly rotating convection,
dynamos, and scaling laws
Chris Jones
Department of Applied Mathematics
University of Leeds, UK
Nordita Winter School January 18th Stockholm
1
Lecture 3 Outline
1. Linear theory of rotating convection: numerics
Onset of convection in rapidly rotating spherical geometry
2. Linear theory of rotating convection: asymptotics
Asymptotic theory for convection in a rotating sphere in the E → 0 limit
3. Linear theory of rotating convection: effect of magnetic fields
4. Waves in the core
Fast and slow waves in the core, and the Braginsky-Meytlis theory
5. JB Taylor’s constraint
Torsional oscillations, zonal flows and thermal winds
6. Scaling laws
Estimating the typical velocities and magnetic fields in terms of the heat
flux using core dynamics
Nordita Winter School January 18th Stockholm
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Linear theory of onset of convection
We start by finding the critical Rayleigh at which convection first sets in. Since
velocities initially are small, there will be no dynamo and so no magnetic field.
Rotation has a strong influence on the onset of convection. The key parameter
is the Ekman number, Ω/νd2, d being the gap-width as usual.
The convection pattern at onset may be similar to that at higher Ra, so the
dynamo properties of the flow are of interest.
The effect of a magnetic field on the onset of convection is also important, as
this gives a guide to the effects of Lorentz force.
Magnetic field itself can also affect the dynamo properties of the flow, discussed
in next lecture.
1. Linear theory of rotating convection: numerics
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Spherical geometry
Ω
g
Flux
z
s
Flux
Rotating about z-axis. Gravity radially
inward, g = g0r.
Centrifugal
acceleration small. Length scale d
is gap-width from inner to outer
boundary. Convection usually onsets
outside tangent cylinder.
tangent
cylinder
Radius ratio inner/outer core for Earth is 0.35. Codes formulated in spherical
polars r, θ, φ, but cylindrical coordinates s, φ, z also useful.
Inner core boundary at r = 7/13 = 0.538, CMB at r = 20/13 = 1.538.
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Spectral formulation
There are two methods for the onset of rapidly rotating convection.
(i) Use spectral methods to investigate numerically small but finite E. Its
possible to get down to E ∼ 10−7 doing this.
(ii) Develop an asymptotic theory of convection valid in the E → 0 limit.
For the spectral approach, we set
u = ∇ × (T r) + ∇ × ∇ × (Pr)
and insert this into the curl and double curl of the momentum equation and
the temperature equation. Scale time on the viscous time-scale d2/ν.
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Spectral formulation
∂
∂T
2
2
E
−∇ L T −2
+ 2CP = 0
∂t
∂φ
∂
∂ 2
2
2 2
E
− ∇ L ∇ P − 2 ∇ P − 2CT + ERa L2T 0 = 0,
∂t
∂φ
T 0 being the temperature perturbation,
1 ∂
∂
2
L =−
sin θ
sin θ ∂θ
∂θ
1 ∂2
− 2
sin θ ∂φ2
1 2
2
C = 1z · ∇ − L 1z · ∇ + 1z · ∇L .
2
The temperature equation is
∂T 0
= QL2P + ∇2T 0,
Pr
∂t
where Q = 1 for internal heating and Q = 1/r3 for differential heating.
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ν
gαd4 dT0
ν
E=
,
Ra = −
,
Pr = .
2
Ωd
κν dr
κ
Here dT0/dr is the temperature gradient at the CMB.
Lowest critical Rayleigh number solutions at small E have ur , uφ and ωz
symmetric about the equator, and uθ and uz antisymmetric.
We can expand
P=
L
X
m
Pl(r)P2l+m
(cos θ) exp i(mφ − ωt),
l=0
and similarly for T and T 0.
Expand Pl(r) as a sum of Chebyshev polynomials, and formulate the equations
by choosing collocation points and boundary conditions.
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Insert expansions into the equations to get matrix eigenvalue equation
Ax = iωBx
with eigenvalues Ra and real frequency ω. Minimise critical Rayleigh number
over m.
Key point:
P2` only couples to P2`+1 and P2`−1
because of form of operator C from the Coriolis terms
Gives Banded matrices.
Allows ∼ 50 sph. harmonics and ∼ 50 radial functions, not demanding
computationally.
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Rapidly rotating convection at onset: varying E
Contours of axial vorticity.
Sequence with E = 3×10−5,
10−5,E = 3 × 10−6, 10−6.
Picture by Emmanuel Dormy.
Note convection onsets first
near tangent cylinder (where
field is strongest in the
dynamo).
Convection
columns get thinner at smaller
E.
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Rapidly rotating convection: equatorial sections
Contours of axial vorticity
at E = 10−7.
Top,
numerical
calculations.
Bottom asymptotic theory.
Left internal heating, right
differential heating.
For radius ratio 0.35, internal
heating sets in first in the
interior. Note good agreement
between asymptotics and
numerics.
Note also spiralling nature of
solution.
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Asymptotic theory of rapidly rotating convection
Plane layer theory and numerics indicate tall thin columns,
1∂
∂
∼
∼ O E −1/3 ,
s ∂φ ∂s
∂
= O(1) as E → 0
∂z
Local theory, disturbances ∼ exp [i(Ks + M φ − Ωt)] ,
2
M
2
∇2, ∇H
→ −K 2 − 2 = −A2
s
u = ∇ × Ψẑ + ∇ × ∇ × ξẑ = uH + W ẑ
ẑ · ∇× and ẑ · ∇ × ∇× momentum equation
dW
E(A − iΩ)(A Ψ) − 2
+ iM ER θ = 0
dz
dΨ
2
E(A − iΩ)W − 2
= zER θ
dz
(A2 − iP Ω)θ = iM Ψ + zW
2
2. Linear theory of rotating convection: asymptotics
2
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Scaled equations
We can now scale E out of the problem
Ra = E
− 34
R,
Ω=E
K=E
θ = θ,
− 13
Ψ=E
− 23
k,
− 31
ω,
M =E
A=E
ψ,
− 13
− 13
m,
a,
W =E
− 23
w.
We get the Roberts-Busse equation, a second
order ODE in z,
i
iRm
d2w h R(a2 − iω)(m2 + a2z 2)
2
2 2
+
−
−
(a
−
iω)
a w=0
2
2
2
dz
a − iP ω
a − iP ω
with boundary conditions
dw
im
+ a2(a2 − iω)wz = 0,
dz
2. Linear theory of rotating convection: asymptotics
on z = ±(r02 − s2)1/2.
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Dispersion relation
Solving this second order ODE numerically gives the dispersion relation for
complex ω
ω = ω(s, k, m, R).
The natural way to proceed is to enforce ω real (zero growth rate) and
minimise R over m, k and s.
We call this the local theory, but it only gives the correct result when
convection sets in at the tangent cylinder
If convection onsets internally, we need WKBJ solutions which decay to zero
in both directions away from the critical cylinder.
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Solving the Dispersion relation equations
To get such a solution, we need to find a point sc in the complex s-plane
(even though s is distance from the axis!) at which both
∂ω
∂ω
= 0 and
= 0,
∂k
∂s
i.e. at which the group velocity is zero and the phase-mixing is zero. This
provides the global solution, which gives the correct critical Rayleigh number.
To find sc we solve five nonlinear equations:
∂ωi/∂m = 0, ∂ωi/∂R = 0, ωi = 0
and the real and imaginary part of ∂ω/∂s = 0.
The solution gives the five unknowns mc, Rc, ωc, sc = sr + isi. The radial
wavenumber k = 0 at sc.
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The amplitude equation
We seek solutions of the form
E
2/3
W ∼ W(x)w(s, z) exp
i
E 1/3
Z
ωt
k(s)ds + mφ − 1/3
E
The amplitude equation is
2 2
2 1 ∂ ω d W 1 ∂ ω
2
−
+
x
W
2
2
2
2 ∂k c dx
2 ∂s c
∂ω
∂ω
+
R1 − ω1 −
m1 W = 0.
∂R c
∂m c
This equation has solutions which decay as x → ±∞, . with x = (s −
sc)/E 1/6. w(s, z) is the local solution of the ODE in the neighbourhood of
sc .
Because sc is the point where phase-mixing vanishes the amplitude equation
has no x term.
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Solving the amplitude equation (which is straightforward) gives the first order
corrections R1. There is a sequence of solutions corresponding to higher
eigenvalues spaced in the ratio 1,3,5,7 etc. These higher eigenmodes come in
at Rayleigh numbers just above the first mode of instability.
Here R = Rc + E 1/3R1, m = mc + E 1/3m1, and ω = ωc + E 1/3ω1,
where Rc, mc and ωc are the leading order critical values.
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Extending the solution to the Real Axis
Once the complex turning point sc has been found Rc, mc and ωc are now
fixed.
The solution form is
∼ exp
i
E 1/3
Z
k(s)ds
and the dispersion relation is now
ω(s, k) = ωc
We can solve this to get k at any value of s.
For s real, k is in general complex and nonzero. At one point s = sM on the
real axis, Im(k) = 0; this is where the maximum amplitude occurs.
This is significantly different from the local theory prediction.
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Location of the onset of convection
The location of the onset of convection at Prandtl number 1 with uniform
internal heating. Local theory, shown here puts the columns at s = 0.5r0.
Global theory puts them at s = 0.6r0.
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The scale of variation in the s direction on the real axis is O(E 1/3), the same
as the azimuthal length scale.
The domain of validity of the solution is given by the Anti-Stokes lines, defined
by
Z
s
Im
k(s)ds
=0
sc
where sc is the double turning point.
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Effect of imposed magnetic field
Fields considered are azimuthal fields (Fearn, 1979, Jones et al. 2003) and
axial fields (Sakuraba, 2002). Also studies in annulus and plane geometry.
Magnetic fields can reduce the critical Ra by the Lorentz force counteracting
the Proudman-Taylor constraint. Indeed, strong enough magnetic fields can
destabilise even at zero Rayleigh number.
Magnetic fields reduce the critical value of m at onset. Important, as at very
low E columns would be extremely thin.
However, these results are for imposed fields: in a dynamo flux expulsion may
push the field into ropes, and outside the ropes could still get thin columns.
Magnetic fields can enhance dynamo action
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Magnetic fields reduce m
Λ = 0.01.
E = 10−6.
Pr = Pm = 1
Λ = 0.31.
E = 10−6.
Pr = Pm = 1
Onset of convection with magnetic field B = B0s φ̂.
Elsasser number Λ = B02/ρΩµη.
Note that even a modest azimuthal field greatly expands the columns. Axial
field behaves similarly, but requires stronger field.
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Magnetic fields reduce Racrit
Solid line and *,
P r = 1, P m = 0.5
Dashed lines, triangles and
squares, P r = 0.5, P m = 1.
R = RaE −4/3, E = 10−6
for points, E → 0 for curves
Onset of convection with magnetic field B = B0s φ̂.
Elsasser number Λ = λ/100 = B02/ρΩµη.
Strong fields reduce the critical Rayleigh number, so enhance convection.
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Waves in the core
With uniform magnetic field B0 and constant temperature gradient T00 we
look for local wave-like solutions
u = u0 exp i(k · x − ωt).
Linearised equations are
1
∂u
0
0
+ 2Ω × u = −∇p + gαT r̂ + j × B0 + ν∇2u
∂t
µρ
∂b
= (B0 · ∇)u + η∇2b
∂t
∂T0
= −ur T00 + κ∇2T0
∂t
Ignore buoyancy, magnetic field, viscosity, and take ζ = ∇ × u,
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we get
∂ζ
∂ 2
− 2(Ω · ∇)u = 0,
∇ u + 2(Ω · ∇)ζ = 0
∂t
∂t
which gives the dispersion relation for inertial waves,
2Ω · k
.
ω = ωC =
|k|
For general k these waves are fast, periods typically days. The waves in
rotating convection are essentially inertial waves, but they are slower, because
kz /|k| is only O(E 1/3).
These tall thin column modes are called Rossby waves. Slow motions with kz
small are called geostrophic, Coriolis and pressure forces being in balance.
Viscosity leads to slow decay of the waves.
Inertial waves have periods too short to be seen in magnetic data. It is likely
there are inertial waves in the core, but we don’t know.
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Alfven waves
Ignore buoyancy, rotation and diffusion, and we get Alfven waves,
∂ζ
1
= (B0 · ∇)j,
∂t
µρ
∂b
= (B0 · ∇)u
∂t
which gives the dispersion relation,
ω = ωM
B0 · k
=
.
1/2
(µρ)
These have a period of about 50 years.
Only waves with Ω·k = 0 can have periods as slow as this, these waves consist
of motion in the φ direction only, independent of z, torsional oscillations.
There is observational evidence for torsional oscillations in magnetic data, and
also they affect the length of the day.
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Slow magnetostrophic waves
In these waves, Lorentz force and Coriolis force are in exact balance, so inertia
is irrelevant.
1
2Ω × u = −∇p + j × B0,
µρ
0
∂b
= (B0 · ∇)u,
∂t
taking the curl and double curl of the equation of motion we obtain
2
(B0 · k)2|k| ωM
ω=
=
.
µρ(Ω · k)
ωC
These waves have periods of order 1000 years. These may be connected with
excursions and reversals of the magnetic field.
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Plate-like turbulence of Braginsky and Meytlis
Small-scale motions with Ω · k = 0 and B0 · k = 0 don’t excite inertial or
Alfven waves in the core and so are ’preferred’.
Since u · k = 0 these are motions in the Ω-B0 plane, the ’plate-like’ motions
of Braginsky and Meytlis.
They argue that turbulence in the core will primarily be driven by convective
instability, given that inertia plays a negligible roll on dynamo timescales.
The most unstable slow motions will be along these plates, and hence strongly
anisotropic. They therefore envisage turbulence in the core as being highly
anistropic, different from almost all other known turbulence.
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J B Taylor’s constraint
This concerns torsional oscillations. We take the φ-component of the full
equation of motion and integrate it over a geostrophic cylinder, that is a
cylinder of fixed radius s, i.e. centred on the polar axis.
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J B Taylor’s constraint
∂
∂t
Z
Z
ρuφ ds +
Z
2ρusΩ ds =
uφ(2E)1/2
(j × B)φ ds − 2πs
(1 − s2)1/4
The Coriolis term is zero, because no net flow across the cylinder (mass
conservation). Reynolds stress is ignored, because it is small in the core
(though not in many simulations).
The last term comes from the viscous friction at the boundaries, which is
produced by Ekman suction. It is small, because E is small.
On a long term average, the time-dependent term must be zero, so the Lorentz
force balances the Ekman suction. Since this is small, but B and j are not,
the magnetic field must be in a special configuration which makes the integral
almost zero.
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This constraint is called JB Taylor’s constraint. Low E simulations,
(particularly plane layer models which can reach lower E) do seem to satisfy
Taylor’s constraint.
The degree to which Taylor’s constraint is satisfied is monitored by the
’Taylorization parameter’
R
j × B · 1φ dS
C(s)
Tay = R
.
|j × B · 1φ| dS
C(s)
However, as it is an integral, quite complicated time-dependent fields can
satisfy the constraint, so it is not very restrictive.
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Zonal flows
In non-magnetic regions of fluid planets, Reynolds stresses can build up large
zonal flows (differential rotation), that is azimuthal (East-West) flows.
If spiralling convection patterns are avergaed in the φ direction, there is a net
acceleration produced by the average of the (u · ∇)u term.
Only the rather weak viscosity, enhanced by turbulence, can balance these
Reynolds stresses, which therefore lead to strong flows as seen on giant
planets.
Magnetic field can oppose the Reynolds stresses, which are anyway only weak
in slow moving cores.
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Thermal wind
A more important source of differential rotation in planetary cores is the
thermal wind equation, the φ component of the vorticity equation,
∂uφ gα ∂T 0
2Ω
=
.
∂z
r ∂θ
In the atmosphere, the strong pole equator temperature difference leads to
strong jetstreams.
In the core, the polar regions may be warmer, and compositionally lighter,
which could lead to an anticyclonic polar vortex just below the CMB.
Magnetoconvection plumes could also lead to anticyclonic vortices in the polar
regions.
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Scaling laws
Hope is that by decreasing E, P m we can identify what happens in the
asymptotic limit as E → 0 and P m → 0. Small length scales emerge in
these limits. How do they affect the behaviour?
The aim is to try and estimate typical velocities and magnetic field strengths in
the core, by seeing how they scale as we move out of the region of parameter
space that can be accessed by simulations.
Direct approach, Christensen and Aubert (2006), look at many dynamo model
runs to see if asymptotic laws are emerging.
Can also try to use physical arguments to see which terms are balancing in the
appropriate limit.
Inertia
Coriolis
Buoyancy
Lorentz
Viscous
We would like to know the typical velocity and magnetic field that results from
a given heat flux.
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Heat transport and typical velocity
In nonlinear theories, temperature fluctuation has to be estimated from heat
flux,
Z
Fconv =
ρcpUr T 0 dS/4πr2 ∼ ρcpU∗T∗
S
where U∗ and T∗ are root mean square velocity and temperature fluctuations,
and cp is the specific heat.
This assumes that there is a strong correlation between hot fluid and rising
fluid. In rotating convection this is not so clear!
In strongly nonlinear convection, balance is between inertia and buoyancy, the
mixing length theory, so
U∗2/d ∼ gαT∗ ∼ gαF/ρcpU∗
In compressible convection d is usually taken as the density scale height, in
Boussinseq convection as the distance between the boundaries.
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Mixing length theory
Mixing length theory: U∗2/d ∼ gαT∗ ∼ gαF/ρcpU∗.
Gives the Deardorff velocity
U∗ ∼
gαF d
ρcp
1/3
which works well in laboratory experiments.
In the core, for 1TW of convective heat flux this gives U∗ about 10 times too
big. Suggests that rotation/magnetic field is slowing down the convection.
6. Scaling Laws
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Inertial theory of rotating convection
Vorticity equation
1
u · ∇ω − 2(Ω · ∇)u = ∇ × gαT r̂ + ∇ × (j × B),
ρ
0
vorticity eqn
U∗2
ΩU∗ gαT 0
Giving 2 ∼
∼
, ignoring Lorentz force.
L⊥
d
L⊥
Here U∗ is typical convective velocity. |ω| ∼ U∗/L⊥, d = rcmbx − ricb,
L⊥ is length scale perpendicular to z, the roll axis.
1/2
−4
6
1/2
U∗d
5 × 10 × 2 × 10
∼
∼ 4km
L⊥ ∼
−5
Ω
7 × 10
L⊥ is Rhines length, balance of inertia and Coriolis. On longer length scales,
inertia << Coriolis.
6. Scaling Laws
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Core flux heat estimate
Convective heat flux per square metre Fconv ∼ ρcpU∗T 0
Eliminate T 0 to get
2/5
gαFconv
U∗
2/5
= Ro ∼
=
(Ra
)
Q
Ωd
ρcpΩ3d2
For compositional convection, gαFconv replaced by buoyancy flux.
Fitting data from dynamo simulations, CA2006 obtained
Ro = 0.85Ra0.41
Q
very close to inertial scaling.
Taking typical core velocity as 15 km/year from the secular variation gives
Ro = U∗/Ωd = 2.9 × 10−6, giving
RaQ ∼ 2 × 10−14 → Fconv ∼ 3.9 × 10−3Wm−2 → Qconv ∼ 0.6TW
with usual estimates for cp etc.
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Reasonable, but ignores contribution from compositional convection. Mass
flux depends on the rate of growth of inner core, controlled by total heat flux
including conduction down the adiabat.
At ICB, usual estimates with a 1.5Gyr old inner core give mass flux of 31,000
Kg/sec, giving RaQ ∼ 8 × 10−13 at the ICB.
Typical velocity near the ICB then 60 km/year, rather large. Mass flux near
2/5
CMB much less, so if the formula Ro ∼ RaQ is interpreted locally it may
be OK.
Magnetoconvection theory suggests magnetic field increases roll size. Not
clearly seen in simulations, but may start to change results. Starchenko and
Jones (2002) suggested that L⊥ stops reducing with Ro at very low Ro.
Leads to (RaQ)1/2 law for U∗.
6. Scaling Laws
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Magnetic field strength
In the Earth’s core, magnetic energy is much greater than kinetic energy, so a
simple balance as used in astrophysics won’t work here.
We start with
Ohmic dissipation + Viscous dissipation = rate of working of buoyancy forces,
The rate of working work done by the buoyancy forces is
Z
ρgαT 0ur dv
which can be written in terms of the heat flux.
Ignore the viscous dissipation and we obtain
Z
Z
gαFconv
dv ∼ ηµj2 dv
cp
Now need to relate magnetic energy to magnetic dissipation.
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Christensen-Tilgner law
The dissipation time is the time taken for the magnetic energy to be dissipated
through ohmic loss,
Z
Z
τdiss
ηµj2 dv =
B2/2µ dv
Equivalently, the magnetic dissipation length
δB =
τdiss
η
1/2
.
Christensen and Tilgner (Nature, 2004) proposed that
δB ∼ dRm−1/2
mainly on the basis of simulations and laboratory experiments. It also has
some theoretical support, because at high Rm flux ropes of this thickness are
formed.
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Predicted field strength
We now have
2
2
(∇
×
B)
ηB
gαF
2
ηµj ∼ η
∼ 2 ∼
,
µ
µδB
cp
giving
1/2
gαFconv µd
B∗ ∼
U∗cp
With the inertial theory scaling for U∗ this gives
B∗ ∼ µ
1/2 2/5 1/5
d
ρ
Ω
1/10
.
gαF
cp
3/10
.
Remarkable feature is weak dependence of B on Ω. We are assuming though
that the planet is in the rapidly rotating low Ro regime.
6. Scaling Laws
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