Download Dynamo Theory

Astrophysical Dynamo Theory
David Hughes
.
Department of Applied Mathematics, University of Leeds
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Astrophysical Dynamos
• What are they?
• Why do we need them?
• Why are they difficult to study?
• What are the outstanding problems?
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The Geomagnetic Field
The Earth has had a magnetic field for at least 3 billion years.
The field retains one polarity
for long periods of time, interspersed with fairly rapid reversals.
The temporal distribution of
the reversals appears to be random.
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Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
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Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
At the most basic level, there are two ways of explaining the continued existence of a
magnetic field in an astrophysical body.
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Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
At the most basic level, there are two ways of explaining the continued existence of a
magnetic field in an astrophysical body.
1. The field has been there since the body was formed.
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Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
At the most basic level, there are two ways of explaining the continued existence of a
magnetic field in an astrophysical body.
1. The field has been there since the body was formed.
or .......
2. It hasn’t!
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Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
At the most basic level, there are two ways of explaining the continued existence of a
magnetic field in an astrophysical body.
1. The field has been there since the body was formed.
or .......
2. It hasn’t!
In the absence of motion, magnetic field satisfies the diffusion equation
∂B
= η∇2 B,
∂t
with Ohmic timescale of decay T = L2 /η (here L is characteristic length, η is
magnetic diffusivity).
4/47
Why do we need dynamo theory?
Consideration of the Earth’s magnetic field provides one of the clearest answers to this
question.
At the most basic level, there are two ways of explaining the continued existence of a
magnetic field in an astrophysical body.
1. The field has been there since the body was formed.
or .......
2. It hasn’t!
In the absence of motion, magnetic field satisfies the diffusion equation
∂B
= η∇2 B,
∂t
with Ohmic timescale of decay T = L2 /η (here L is characteristic length, η is
magnetic diffusivity).
For the Earth, T ∼ 104 years, whereas the field has existed for 3 × 109 years.
Thus the field cannot be a fossil field. Clearly, inductive motions are crucial for the
maintenance of the field.
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The Solar Magnetic Field
1919: Joseph Larmor: ‘Brief
Communication’ to the British
Association for the Advancement of Science, How Could
a Rotating Body Such as the
Sun Become a Magnet?
This might be thought of as the starting point for astrophysical dynamo theory.
The Sun’s magnetic field can be observed over a wide range of scales and is
responsible for most solar dynamic phenomena.
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Large-Scale Solar Field
Magnetogram of the entire Sun. Black and white regions denote regions of strong
positive and negative polarity — in visible light these regions are sunspots.
Suggestive of a strong toroidal (east-west) field bursting through the solar surface.
Note that the field is antisymmetric about the equator.
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Solar Cycle
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Solar Cycle
Temporal variation of sunspot number since 1600.
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Solar Cycle
The field reverses sign approximately every 11 years.
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Large-Scale Solar Field: Summary of Key Observations
• The field is predominantly dipolar.
• It has a strong toroidal (azimuthal) component, antisymmetric about the equator.
• The azimuthal field is confined to lower latitudes and propagates towards the
equator, with an approximate period of 11 years.
• The field reverses every 11 years, so a full magnetic cycle is 22 years.
• The amplitude of the field is modulated in time in a complicated manner (e.g. the
Maunder minimum).
A full theory of the solar magnetic field must explain these observations.
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Large-Scale Solar Field: Summary of Key Observations
• The field is predominantly dipolar.
• It has a strong toroidal (azimuthal) component, antisymmetric about the equator.
• The azimuthal field is confined to lower latitudes and propagates towards the
equator, with an approximate period of 11 years.
• The field reverses every 11 years, so a full magnetic cycle is 22 years.
• The amplitude of the field is modulated in time in a complicated manner (e.g. the
Maunder minimum).
A full theory of the solar magnetic field must explain these observations.
It is of interest to note that the Ohmic decay time for the Sun is long, unlike that of
the Earth. Indeed L2 /η ∼ 109 years, comparable to the lifetime of the Sun. Thus we
cannot rule out a fossil field explanation for the Sun simply on these grounds.
However, it is very difficult to explain the short term variations in terms of a fossil field.
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Small-Scale Solar Magnetic Fields
Solar granulation. Bright spots are the sites of intense magnetic field.
Small-scale field has no preferred orientation and is not correlated with the solar cycle.
Suggests that this field is self-maintained, rather than being just a by-product of the
large-scale dynamo.
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Stellar Magnetic Fields
Magnetic field activity detected on
other solar-like stars via Ca II emission (e.g. Baliunas et al 1995).
Cyclic activity found on fairly slow
rotators (such as the Sun). More
vigorous, but less egular behavour
found on rapid rotators.
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Dynamo Experiments
Figure: Karlsruhe experiment side view
Figure: Karlsruhe top view
Figure: VKS experiment
Figure: Madison plasma experiment
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The Governing Equations
The dynamics of the magnetic field in stellar interiors is well described by the
equations of single fluid magnetohydrodynamics (MHD):
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The Governing Equations
The dynamics of the magnetic field in stellar interiors is well described by the
equations of single fluid magnetohydrodynamics (MHD):
Induction equation:
∂B
= ∇ × (u × B) + η∇2 B.
∂t
Momentum equation:
∂u
+ u · ∇u = −∇p + j × B + ρg + F other + F viscous .
ρ
∂t
Mass conservation:
∂ρ
+ ∇ · (ρu) = 0.
∂t
Energy equation:
D
pρ−γ = loss terms.
Dt
Equation of state:
p = RρT .
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Dynamo Terminology
Theoreticians find it helpful to classify dynamos into various categories — although
Nature has no need to obey this classification.
For example, dynamos can be classified as:
(i) Kinematic or dynamic
(ii) Slow or fast,
(iii) Small scale or large scale.
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Kinematic Dynamos
Kinematic dynamo theory addresses the following (simply stated) question:
Can we find a velocity field u(x, t) such that the magnetic field — governed solely by
the induction equation — grows?
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Kinematic Dynamos
Kinematic dynamo theory addresses the following (simply stated) question:
Can we find a velocity field u(x, t) such that the magnetic field — governed solely by
the induction equation — grows?
So here the problem is reduced to just one equation, linear in the magnetic field B:
∂B
= ∇ × (u × B) + η∇2 B.
∂t
R
Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic)
dynamo if M(t) 9 0 as t → ∞.
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Kinematic Dynamos
Kinematic dynamo theory addresses the following (simply stated) question:
Can we find a velocity field u(x, t) such that the magnetic field — governed solely by
the induction equation — grows?
So here the problem is reduced to just one equation, linear in the magnetic field B:
∂B
= ∇ × (u × B) + η∇2 B.
∂t
R
Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic)
dynamo if M(t) 9 0 as t → ∞.
Induction – leads to growth of energy through extension of field lines.
Dissipation – leads to decay of energy into heat through Ohmic loss.
Hydrodynamic dynamo works if induction by the fluid motions overpowers the
dissipative losses.
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Kinematic Dynamos
Kinematic dynamo theory addresses the following (simply stated) question:
Can we find a velocity field u(x, t) such that the magnetic field — governed solely by
the induction equation — grows?
So here the problem is reduced to just one equation, linear in the magnetic field B:
∂B
= ∇ × (u × B) + η∇2 B.
∂t
R
Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic)
dynamo if M(t) 9 0 as t → ∞.
Induction – leads to growth of energy through extension of field lines.
Dissipation – leads to decay of energy into heat through Ohmic loss.
Hydrodynamic dynamo works if induction by the fluid motions overpowers the
dissipative losses.
It is not straightforward to find rigorous examples of flows that act as dynamos.
Most of the early work demonstrated flows and fields that could not act as dynamos.
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Anti-Dynamo Results, or Necessary Conditions for Dynamo Action
These can be divided into three categories:
(i) Constraints on the symmetries of magnetic fields that cannot be generated by
dynamo action.
(ii) Constraints on the symmetries of velocity fields that cannot act as dynamos.
(iii) Lower bounds on the magnetic Reynolds number (or similar quantities) that must
be exceeded for dynamo action.
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Anti-Dynamo Results, or Necessary Conditions for Dynamo Action
These can be divided into three categories:
(i) Constraints on the symmetries of magnetic fields that cannot be generated by
dynamo action.
(ii) Constraints on the symmetries of velocity fields that cannot act as dynamos.
(iii) Lower bounds on the magnetic Reynolds number (or similar quantities) that must
be exceeded for dynamo action.
The most famous anti-dynamo theorem (which is in category (i)) is Cowling’s (1934)
theorem:
An axisymmetric magnetic field cannot be maintained by dynamo action.
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Anti-Dynamo Theorems (ii)
1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional
planar motions. (Also true for motions on a spherical surface, but not for motions on
a cylindrical surface.)
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Anti-Dynamo Theorems (ii)
1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional
planar motions. (Also true for motions on a spherical surface, but not for motions on
a cylindrical surface.)
Proof
Consider flows u(x, t) with u · ẑ = 0; for simplicity suppose the flow is
incompressible. Write B = B · ẑ . Then
DB
= η∇2 B
Dt
and so B decays everywhere to zero (multiply by B and integrate over the domain).
So we need consider B only in the xy -plane. Write B = ∇ × Aẑ . Then the induction
equation becomes
DA
= η∇2 A.
Dt
Thus it follows also that A must decay. Thus no magnetic field can be maintained by
a purely two dimensional planar flow. Note that this result holds even if B is
z-dependent.
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Anti-Dynamo Theorems (ii)
1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional
planar motions. (Also true for motions on a spherical surface, but not for motions on
a cylindrical surface.)
Proof
Consider flows u(x, t) with u · ẑ = 0; for simplicity suppose the flow is
incompressible. Write B = B · ẑ . Then
DB
= η∇2 B
Dt
and so B decays everywhere to zero (multiply by B and integrate over the domain).
So we need consider B only in the xy -plane. Write B = ∇ × Aẑ . Then the induction
equation becomes
DA
= η∇2 A.
Dt
Thus it follows also that A must decay. Thus no magnetic field can be maintained by
a purely two dimensional planar flow. Note that this result holds even if B is
z-dependent.
Moral of the story: Too much symmetry is a bad thing.
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Anti-Dynamo Theorems (iii)
Backus (1958):
A necessary condition for dynamo action driven by a steady flow in a sphere of radius
R is that
em R 2 /η > π 2 ,
where em is the maximum rate of strain.
The left hand-side could be regarded as a magnetic Reynolds number (here defined
unconventionally using the rate of strain rather than the velocity).
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Anti-Dynamo Theorems (iii)
Backus (1958):
A necessary condition for dynamo action driven by a steady flow in a sphere of radius
R is that
em R 2 /η > π 2 ,
where em is the maximum rate of strain.
The left hand-side could be regarded as a magnetic Reynolds number (here defined
unconventionally using the rate of strain rather than the velocity).
Childress (1969) proved a further necessary condition , namely
Um R/η > π 2 ,
where here Um is the maximum velocity in the flow.
So using the conventional definition of magnetic Reynolds number, Rm = UL/η, we
could write this as
max(Rm) > π 2 .
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First Working Dynamo: Herzenberg (1958)
Spheres radius a, separation R,
rotation axes inclined at angle
φ.
Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially,
poloidal.
It is then wound up to give a toroidal field at (2).
Field diffuses to (1) — where it has a poloidal component since rotation axes not
parallel — and can then be wound up by rotation of (1) to give toroidal component.
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First Working Dynamo: Herzenberg (1958)
Spheres radius a, separation R,
rotation axes inclined at angle
φ.
Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially,
poloidal.
It is then wound up to give a toroidal field at (2).
Field diffuses to (1) — where it has a poloidal component since rotation axes not
parallel — and can then be wound up by rotation of (1) to give toroidal component.
Complicated analysis shows dynamo action possible if
Rm >
15(R/a)3
cos φ sin2 φ
if
R a.
19/47
First Working Dynamo: Herzenberg (1958)
Spheres radius a, separation R,
rotation axes inclined at angle
φ.
Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially,
poloidal.
It is then wound up to give a toroidal field at (2).
Field diffuses to (1) — where it has a poloidal component since rotation axes not
parallel — and can then be wound up by rotation of (1) to give toroidal component.
Complicated analysis shows dynamo action possible if
Rm >
15(R/a)3
cos φ sin2 φ
if
R a.
First conclusive demonstration of dynamo action — though not relevant to isolated
astrophysical bodies.
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Dynamo Classification
From an astrophysical point of view, we are often interested in the generation of
large-scale fields (such as that on the Sun).
Also, astrophysically the magnetic Reynolds number Rm is huge, so we are interested
in dynamo action at high Rm.
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Dynamo Classification
From an astrophysical point of view, we are often interested in the generation of
large-scale fields (such as that on the Sun).
Also, astrophysically the magnetic Reynolds number Rm is huge, so we are interested
in dynamo action at high Rm.
These two problems have essentially been studied separately.
(i) The large-scale dynamo problem has typically been studied using mean field
electrodynamics.
(ii) The high Rm limit has led to what is known as the fast dynamo problem.
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Mean Field Magnetohydrodynamics
The generation of large-scale fields is typically studied using Mean Field MHD.
This is, at heart, a kinematic (linear) theory, which is often then modified to include
nonlinear effects.
The linear theory is self-consistent (though is not without problems); the nonlinear
theory not necessarily so.
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Kinematic mean field electrodynamics
Magnetic field is governed by the induction equation
∂B
= ∇ × (u × B) + η∇2 B,
∂t
where B is magnetic field, u is fluid velocity, η is magnetic diffusivity.
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Kinematic mean field electrodynamics
Magnetic field is governed by the induction equation
∂B
= ∇ × (u × B) + η∇2 B,
∂t
where B is magnetic field, u is fluid velocity, η is magnetic diffusivity.
Standard formulation splits velocity and magnetic field into mean (large-scale) and
fluctuating (small-scale) parts:
U = U 0 + u,
B = B 0 + b.
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Kinematic mean field electrodynamics
Magnetic field is governed by the induction equation
∂B
= ∇ × (u × B) + η∇2 B,
∂t
where B is magnetic field, u is fluid velocity, η is magnetic diffusivity.
Standard formulation splits velocity and magnetic field into mean (large-scale) and
fluctuating (small-scale) parts:
U = U 0 + u,
B = B 0 + b.
Averaging the induction equation leads to the following equations for the mean and
fluctuating magnetic fields:
∂B 0
= ∇ × (U 0 × B 0 ) + ∇ × E + η∇2 B 0 ,
∂t
∂b
= ∇ × (U 0 × b) + ∇ × (u × B 0 ) + ∇ × G + η∇2 b,
∂t
where mean emf E = hu × bi and G = (u × b) − hu × bi.
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Relation between mean field and mean emf
Equation for the fluctuating field can be written as
L(b) = ∇ × (u × B 0 ) .
Linear relation between b and B 0 implies linear relation between E and B 0 .
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Relation between mean field and mean emf
Equation for the fluctuating field can be written as
L(b) = ∇ × (u × B 0 ) .
Linear relation between b and B 0 implies linear relation between E and B 0 .
Mean emf is then written as the expansion:
Ei = αij B0j + βijk
∂B0j
+ ··· ,
∂xk
where convergence is anticipated as a result of the large spatial scale of B 0 .
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Relation between mean field and mean emf
Equation for the fluctuating field can be written as
L(b) = ∇ × (u × B 0 ) .
Linear relation between b and B 0 implies linear relation between E and B 0 .
Mean emf is then written as the expansion:
Ei = αij B0j + βijk
∂B0j
+ ··· ,
∂xk
where convergence is anticipated as a result of the large spatial scale of B 0 .
In its simplest, kinematic, formulation, the α coefficients can be calculated by
measuring E after imposing a uniform kinematic magnetic field on the small-scale flow.
The αij depend on the properties of u and on η and provide information on the
growth of large-scale magnetic fields.
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The Mean Field Induction Equation
Consider the simplest case of isotropic turbulence.
Then αij = αδij and βijk = βijk .
Substituting for the mean emf gives the mean induction equation
∂B 0
= ∇ × (U 0 × B 0 ) + ∇ × (αB 0 ) + (η + β)∇2 B.
∂t
So β (in this simplest case) can be regarded as a turbulent diffusivity.
The more interesting term is that involving α, which is of a completely different form
to any term in the unaveraged induction equation.
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How does the new term work?
Decompose the magnetic field into poloidal (B P ) and toroidal (B T ) components.
Toroidal field can be produced from poloidal field
by differential rotation
pulling out a poloidal field
(so-called Omega-effect
— the easy bit).
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How does the new term work?
Decompose the magnetic field into poloidal (B P ) and toroidal (B T ) components.
Toroidal field can be produced from poloidal field
by differential rotation
pulling out a poloidal field
(so-called Omega-effect
— the easy bit).
Mathematically the dynamo loop can be closed (B T → B P ) through the α term (the
‘α-effect’). Physically one may think of this picture:
A poloidal field is raised and
twisted, givng rise to a tooidal
current. Averaging over many
such events leads to a net J T
and hence a net B P (Parker’s
(1955) ‘cyclonic events’).
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Parker Dynamo Waves
Simplest possible dynamo model consists of a plane layer in Cartesian geometry, with
a velocity shear (to generate toroidal from poloidal field) and an α-effect (to close the
loop and generate poloidal from toroidal).
Suppose x points south, y points east and z vertically upwards.
Seek an ‘axisymmetric’ field
B = B(x, z, t)ŷ + ∇ × (A(x, z, t)ŷ ) .
Then the mean field induction equation becomes
∂A
= αB + η∇2 A,
∂t
∂B
∂A
= V0
+ η∇2 B,
∂t
∂x
where V is the mean azimuthal velocity (the differential rotation).
For simplicity, assume V 0 and α are constants. Then we can seek plane wave solutions
A = Â exp(pt + ikx),
B = B̂ exp(pt + ikx).
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Parker Dynamo Waves
Leads to the dispersion relation
√ p = −ηk 2 1 ± (1 + i) D ,
where the dynamo number D = αV 0 /2ηk 3 .
There are growing solutions (i.e. dynamo action) if |D| > 1.
Importantly, these modes propagate as dynamo waves.
Propagation is towards the equator if αV 0 < 0 and towards the poles if αV 0 > 0.
Such a dynamo is known as an αω- dynamo (driven by a combination of the α effect
and the differential rotation ω. It provides the simplest explanation of the solar
butterfly diagram.
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More Complicated Mean Field Modelling
Going beyond the simple Parker waves, by choosing suitable forms of α, β and the
differential rotation (possibly with ad hoc nonlinearities), it is possible to model
magnetic fields in a range of astrophysical bodies.
(Dikpati et al 2006)
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More Complicated Mean Field Modelling
Going beyond the simple Parker waves, by choosing suitable forms of α, β and the
differential rotation (possibly with ad hoc nonlinearities), it is possible to model
magnetic fields in a range of astrophysical bodies.
(Dikpati et al 2006)
But note that this is not the same as these results being rigorously derived, and is a
weakness of the mean field theory. One can obtain similar output with very different
models, and very different output with quite similar models.
Mean field models have been tuned to the last few solar cycles and predictions made
for the current solar cycle 24 — but these have proved to be very wide of the mark.
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Two other serious difficulties
1. Results obtained for small Rm or small S do not carry over to the true astrophysical
regime (Rm 1, S = O(1)).
(a) makes sense for short correlation times or lots of diffusion.
(b) (or something more complicated) holds otherwise.
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An example: mean emf from rotating convection
(Hughes & Cattaneo JFM, 2008).
Top plot shows time sequence of already spatially averaged emf.
Bottom plot shows the cumulative average. Slow convergence to an extremely small
value of α. α here scales as urms /Rm, rather than with urms as one would expect
from a turbulent (fast) process.
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Small Scale Dynamo Growth
2.
L(b) = ∇ × (u × B 0 ) .
The standard formulation assumes there are no exponentially growing solutions to the
equation L(b) = 0, i.e. no small-scale dynamo action. However there may be — and
at high Rm there almost certainly will be.
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Small Scale Dynamo Growth
2.
L(b) = ∇ × (u × B 0 ) .
The standard formulation assumes there are no exponentially growing solutions to the
equation L(b) = 0, i.e. no small-scale dynamo action. However there may be — and
at high Rm there almost certainly will be.
This leads us on nicely to .....
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Fast Dynamo Theory
Astrophysically, the magnetic Reynolds number Rm is invariably extremely large.
Mathematically this has led to the idea of looking at kinematic dynamos in the limit
as Rm → ∞.
Suppose that a flow u(x, t) acts as a dynamo for some finite Rm. Then if dynamo
action persists as Rm → ∞, the dynamo is said to be fast. Otherwise it is said to be
slow.
All of the fast dynamos considered have been small-scale dynamos.
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Anti-dynamo theorem for fast dynamos
The main theorem, due to Vishik and Klapper & Young states that:
The dynamo growth rate is bounded above by the topological entropy of the flow.
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Anti-dynamo theorem for fast dynamos
The main theorem, due to Vishik and Klapper & Young states that:
The dynamo growth rate is bounded above by the topological entropy of the flow.
The topological entropy can here be identified as the exponential growth of material
lines — and importantly is zero for a non-chaotic flow. Thus, as a corollary, we have
the important anti-dynamo theorem:
Fast dynamo action cannot arise as a result of an integrable flow. i.e. a necessary
condition for fast dynamo action is that the flow be chaotic (i.e. have exponentially
separating trajectories).
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Chaotic Flows
Flows depending on just two variables, x and y say, are computationally helpful since
the z-dependence of the magnetic field is then just exp(ikz).
For a given k the problem involves only two, rather than three, spatial dimensions.
34/47
Chaotic Flows
Flows depending on just two variables, x and y say, are computationally helpful since
the z-dependence of the magnetic field is then just exp(ikz).
For a given k the problem involves only two, rather than three, spatial dimensions.
Steady 2-d flows are integrable and hence cannot be fast. But flows u(x, y , t) are
good candidates for study if they are chaotic.
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Chaotic Flows
Flows depending on just two variables, x and y say, are computationally helpful since
the z-dependence of the magnetic field is then just exp(ikz).
For a given k the problem involves only two, rather than three, spatial dimensions.
Steady 2-d flows are integrable and hence cannot be fast. But flows u(x, y , t) are
good candidates for study if they are chaotic.
The aim is to produce some kind of evidence for the growth rate as Rm → ∞.
Not easy numerically of course; 3d steady and unsteady flows are even more
demanding computationally.
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The Galloway Proctor flow
GP-flow (Nature, 1992):
u = ∇ × (ψ(x, y , t)ẑ ) + ψ(x, y , t)ẑ
ψ(x, y , t) = A (cos(x + cos t) + sin(y + sin t))
35/47
The Galloway Proctor flow
GP-flow (Nature, 1992):
u = ∇ × (ψ(x, y , t)ẑ ) + ψ(x, y , t)ẑ
ψ(x, y , t) = A (cos(x + cos t) + sin(y + sin t))
(GP flow movie)
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Chaos in the GP flow
Finite-time Lyapunov exponents
showing
the
stretching
of
an infinitesimal
element initially
located at the
marked location.
An infinitesimal element ξ satisfies
dξ
= ξ · u.
dt
The Lyapunov exponent is an average measure of the exponential stretching, along a
fluid trajectory.
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Growth rates
Growth rates versus k at fixed Rm.
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Growth rates
Growth rates versus k at fixed Rm.
Growth rates versus Rm at fixed k.
(a) is the GP flow.
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Growth rates
Growth rates versus k at fixed Rm.
Growth rates versus Rm at fixed k.
(a) is the GP flow.
Good numerical evidence of self-similar behaviour with constant growth rate as
Rm → ∞.
But not a proof!
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Numerical Modelling
Most of what I have discussed so far either addresses the linear (kinematic) problem or
incorporates nonlinearities into the mean field coefficients.
The full dynamo problem of course requires a self-consistent solution to all the
governing equations, which requires a computational approach.
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Numerical Modelling
Most of what I have discussed so far either addresses the linear (kinematic) problem or
incorporates nonlinearities into the mean field coefficients.
The full dynamo problem of course requires a self-consistent solution to all the
governing equations, which requires a computational approach.
It should though be pointed out that astrophysical parameters are either very small or
very large, so it is impossible to perform totally realistic astrophysical simulations.
At the base of the solar convection zone, the various dimensionless parameters are:
Ra = g (∇ − ∇ad )L4 /(νκHp )
Re = UL/ν
Rm = UL/η
Pr = ν/κ
Pm = ν/η
β = pgas /pmag
M = U/cs
Ro = U/2ΩL
Rayleigh no.
Reynolds no.
magnetic Reynolds no.
Prandtl no.
magnetic Prandtl no.
plasma beta
Mach no.
Rossby no.
1020
1013
1010
10−7
10−3
105
10−4
10−1
(A similar problem arises in geodynamo modelling: the Earth is a rapid rotator, with
Ekman number E = ν/2ΩL2 = O(10−15 .)
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Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
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Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
For example, tracking sound waves numerically is very computationally expensive,
since the waves are fast and hence the timestep is small. If these are not dynamically
important for the physical process being considered then they can be removed, via an
asymptotic reduction of the equations.
The Boussinesq approximation neglects sound waves and treats the fluid as
incompressible.
The anelastic approximation, which is more general but also more complicated, also
neglects sound waves but retains the effects of density stratification.
Most numerical dynamo models solve the anelastic equation in a rotating spherical
shell.
39/47
Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
For example, tracking sound waves numerically is very computationally expensive,
since the waves are fast and hence the timestep is small. If these are not dynamically
important for the physical process being considered then they can be removed, via an
asymptotic reduction of the equations.
The Boussinesq approximation neglects sound waves and treats the fluid as
incompressible.
The anelastic approximation, which is more general but also more complicated, also
neglects sound waves but retains the effects of density stratification.
Most numerical dynamo models solve the anelastic equation in a rotating spherical
shell.
That said, dealing with the issues of high Re and Rm and low Pm remain in place. It
is worth noting that in the largest current simulations one can get to Re ∼ Rm ∼ 104
— still a long way short of the true values!
39/47
Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
40/47
Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
For example, tracking sound waves numerically is very computationally expensive,
since the waves are fast and hence the timestep is small. If these are not dynamically
important for the physical process being considered then they can be removed, via an
asymptotic reduction of the equations.
The Boussinesq approximation neglects sound waves and treats the fluid as
incompressible.
The anelastic approximation, which is more general but also more complicated, also
neglects sound waves but retains the effects of density stratification.
Most numerical dynamo models solve the anelastic equation in a rotating spherical
shell.
40/47
Numerical Modelling
Some of the difficulties of the extreme parameter values can be addressed through
modifications to the governing equations.
For example, tracking sound waves numerically is very computationally expensive,
since the waves are fast and hence the timestep is small. If these are not dynamically
important for the physical process being considered then they can be removed, via an
asymptotic reduction of the equations.
The Boussinesq approximation neglects sound waves and treats the fluid as
incompressible.
The anelastic approximation, which is more general but also more complicated, also
neglects sound waves but retains the effects of density stratification.
Most numerical dynamo models solve the anelastic equation in a rotating spherical
shell.
That said, dealing with the issues of high Re and Rm and low Pm remain in place. It
is worth noting that in the largest current simulations one can get to Re ∼ Rm ∼ 104
— still a long way short of the true values!
40/47
Computational Dynamo Models
Simulations
of
anelastic convection, increasing
the rotation rate
(from Brown et
al 2008 Ap.J.).
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Computational Dynamo Models
Toroidal
and
radial magnetic
field for moderate rotation rate.
Note that the
field is essentially
small scale (from
Brun et al 2004
Ap.J.).
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Computational Dynamo Models
At very high
rotation rates the
field takes the
from of ‘magnetic
wreaths’
(from Brown et
al 2010 Ap.J.).
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Current Views on the Solar Dynamo
Internal solar rotation, deduced
from helioseismology. The thin
white region between the convective and radiative zones is the
tachocline, a region of strong radial shear.
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Current Views on the Solar Dynamo
Internal solar rotation, deduced
from helioseismology. The thin
white region between the convective and radiative zones is the
tachocline, a region of strong radial shear.
There are currently three models for the solar dynamo; all have problems.
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Current Views on the Solar Dynamo
Internal solar rotation, deduced
from helioseismology. The thin
white region between the convective and radiative zones is the
tachocline, a region of strong radial shear.
There are currently three models for the solar dynamo; all have problems.
(i) Dynamo action distributed throughout the convection zone.
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Current Views on the Solar Dynamo
Internal solar rotation, deduced
from helioseismology. The thin
white region between the convective and radiative zones is the
tachocline, a region of strong radial shear.
There are currently three models for the solar dynamo; all have problems.
(i) Dynamo action distributed throughout the convection zone.
(ii) An interface dynamo in which dynamo action takes place in, and just above, the
tachocline.
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Current Views on the Solar Dynamo
Internal solar rotation, deduced
from helioseismology. The thin
white region between the convective and radiative zones is the
tachocline, a region of strong radial shear.
There are currently three models for the solar dynamo; all have problems.
(i) Dynamo action distributed throughout the convection zone.
(ii) An interface dynamo in which dynamo action takes place in, and just above, the
tachocline.
(iii) Flux transport dynamo, in which dynamo action takes place in two widely
separated regions: the tachocline and the surface.
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Distributed Dynamo
• Here the poloidal field is generated throughout the convection zone by the action
of cyclonic turbulence.
• Toroidal field is generated by the latitudinal distribution of differential rotation.
• No role is envisaged for the tachocline
Pros
Scenario is theoretically possible wherever convection and rotation take place together.
Cons
Small-scale dynamo action may dominate. Indeed, computations at high Rm show
that it is hard to generate a large-scale field.
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Interface Dynamo
• The dynamo is thought to work at the interface of the convection zone and the
tachocline.
• The mean toroidal (sunspot field) is created by the radial differential rotation and
stored in the tachocline.
• The mean poloidal field (coronal field) is created by turbulence (some sort of
α-effect) in the lower reaches of the convection zone.
Pros
The radial shear provides a natural mechanism for generating a strong toroidal field.
The stable stratification enables the field to be stored and stretched to a large value.
As the mean magnetic field is stored away from the convection zone, the α-effect is
not suppressed by a strong field.
Cons
Relies on transport of flux (of just the right strength) to and from tachocline ? how is
this achieved?
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Flux Transport Dynamo
• Here the poloidal field is generated at the
surface of the Sun via the decay of active
regions with a systematic tilt
(Babcock-Leighton scenario) and
transported towards the poles by the
observed meridional flow.
• The flux is then transported by a conveyor
belt meridional flow to the tachocline
where it is sheared into the sunspot
toroidal field.
• No role whatsoever is envisaged for the turbulent convection in the bulk of the
convection zone.
Pros
Does not rely on turbulent α-effect.
Sunspot field is intimately linked to polar field immediately before.
Cons
Requires strong meridional flow at base of CZ of exactly the right form.
Ignores all poloidal flux returned to tachocline via the convection
Effect will probably be swamped by ‘α-effects’ closer to the tachocline.
Relies on existence of sunspots for dynamo to work (cf. Maunder Minimum).
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