The creation of magnetic structures from
velocity shear
(Constraints for the solar dynamo)
Nic Brummell
Applied Mathematics,
University of California Santa Cruz
Geoff Vasil, Kelly Cline
JILA/APS, University of Colorado
Lara Silvers, Mike Proctor
DAMTP,
University of Cambridge, UK
UCSD Feb 2009
The Sun
Major puzzle: solar magnetic activity cycle
Solar magnetic activity is VARIABLE
but remarkably ORDERED.
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How does it work? A DYNAMO!
But how does the dynamo work?
Theories driven by observations
Things we know surprisingly well -- solar interior structure (thanks to helioseismology)
Solid body rotation
in the core
Interface layer – the TACHOCLINE
Differential rotation in
the convection zone
Large-scale dynamo: Theory
Existing poloidal field is
stretched by latitudinal
(and/or radial) differential
rotation into the toroidal
direction – an -like
mechanism
Strong toroidal field rises due to
magnetic buoyancy and, under
the influence of global rotation,
twists into the poloidal direction
– an a-like mechanism
Strong toroidal field rises as structures?
MAGNETIC BUOYANCY: the standard explanation
• Magnetic field exerts a magnetic pressure (Lorenz force JxB can be split into pressure
and tension
(  B)  B ~ (B 2 /2)    BB
)
• Concentrated B contributes to the total pressure

• Isothermal
pressure balance implies density lower in tube
Pg = density*temperature
Outside:
Inside:
Pt = Pg1
Pg2 = Pt-Pm
Pm ~ B2 => Pg2 < Pg1
densityinside < densityoutside
Magnetic buoyancy instabilities 101
"PARCEL ARGUMENT"
Tayler (1973; Moffat 1978; Acheson 1979)
Assume adiabatic (confined dynamics) and
no diffusive effects
 B,   Area 
A  A B  B   
B 




(*)
A
B

B

p
p

 const 

(*)


p

B  B,  

B  dB,  d

B2
2 0
Assume slow rise  pressure equilibrium maintained
Inside parcel :
Total pressure pT  pgas  pmagnetic  pgas 
(B  B) 2
B 2  2BB
pT  p  p 
 p  p 
(linearising)
2 0
2 0
Outside parcel:

(B  dB) 2
B 2  2BdB
pT  p  dp 
 p  dp 
(linearising)
2 0
2 0
BB
BdB
Equilibrium  p 
 dp 
(*)
0
For instability, need   d
0
B,
B,


Magnetic buoyancy instabilities 101
Combining the (*) relations, after some messy algebra
can rearrange
into the convenient form :
a 2 d  B 
2
g 2
ln   N
c dz   
where
B2
a2 
(Alfven speed)
 0
p
c2 
(sound speed)



g d

ln p
(Brunt - Vaisala frequency)
 dz

Alternatively :
B 2 1 d 1 dB 


 (stuff )
 0  dz B dz 
N2 
B

decreasing sufficiently quickly with height
 UNSTABLE (necessary condition)
(i.e. destabilises a convectively stable atmosphere
where N 2 > 0)
It's as if there is the magnetic field is supporting
the gas in a " top heavy" situation  unstable.
Magnetic buoyancy instabilities 101
More details:
Including diffusion?
Newcomb (1961) :
d
g
 2
(interchange modes)
dz
a  c2
d
g
 2
(3D undulating modes; k x  0; easier!)
dz
c
 small  good, maintains B gradients
 large  good, erodes stabilising thermal gradients
Acheson 1979 : k x  0,  0
a2 d

g 2
ln B  N 2

c dz

Thomas and Nye (1975) : equivalently
Stars : laminar values
2
a d
ln B  N 2

2
c dz
 sufficiently fast decrease in B only
is good enough now.
g

 1, turbulent values ~ O(1)?


End result: Magnetic GRADIENTS that are important!
(gradients in the direction of gravity)
Standard concept of the Omega effect
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Large-scale dynamo: intuitive picture
1.
Generation/shredding of magnetic field
2.
Transport of magnetic field from CZ into tachocline
3.
 effect: Conversion of poloidal field to toroidal field -- DIFFERENTIAL ROTATION -- ORIGIN ?
4.
Formation of structures and magnetic buoyant rise
-- MAGNETIC BUOYANCY / SHEAR ?
5.
a mechanism: Regeneration of poloidal field
-- ROTATION, TURBULENT u, b CORRLNS??
6.
Recycling/breakdown of field
7.
Emergence of structures
-- TURBULENT TRANSPORT OF B ?
STUDY ELEMENTS OF THE DYNAMO
Heavily dependent on the tachocline
Boundary layer between
 the convection zone (differentially rotating)
 the radiative interior (solid body rotation)
Layer (plus sub-layers?) of
 strong radial shear
 weaker latitudinal shear
Almost certainly threaded with magnetic field
 weak poloidal?
 strong toroidal?
We wish to examine, via fully nonlinear MHD simulations:
 the interaction of velocity shear and weak background (poloidal) magnetic fields.
 the production of strong (toroidal) magnetic fields by the shear
 magnetic structures produced
 any magnetic buoyancy instabilities of the magnetic configuration
 the nonlinear evolution of the system
So … how does it work?
Cons of
mass
Cons of
mom
t     ( u)  0
Mmm … nonlinearity!
pressure
1/ 2 rotation
t u    (uu  aBB)  pT  g  Ck Ta (  u)
buoyancy
advection tension
forcing
1
+ Ck ( u  (  u)) + F
3
diffusion
Mmm … nonlinearity!
Ck
 (z)
t T    (uT)  (  2)T  u =
(
T)  H  H
Compression
Advection

 (0)
Ohmic &
2
Cons of
energy
Mmm … nonlinearity!
Induction
diffusion
t B    (u  B)  Ck B
Induction
B0
pT  pg  pm  T 
a  Ck 2Q
2
viscous
heating
diffusion
a | B |2 magnetic
pressure
2
Compressible MHD equations
(Navier-Stokes + induction)
Local simulations of elements of the dynamo
e.g. Brummell, Clune & Toomre, 2002
z=0
z=1
Layer 1 : Unstable m = m1 (=1)
• Full compressible MHD
(poloidal/toroidal)
• DNS
• Cartesian
Layer 2 : Stable
m=m2 (>1.5)
Thermal diffusivity z)
( not
• Pseudospectral / finite-difference
Ck(layer1)/Ck(layer2)=(m2+1)/(m1+1)
• Semi-implicit
“Stiffness”, S = (m2-mad)/(mad-m1)
• HIGHER Rq, Re, Pe, LOWER Pr than
global sims (resolves from diffusive scale UP)
z=zmx
,T:x,y,z) ) :
So … how do we solve these?
Blue Gene/L
Fastest machine in the world!
~ 213,000 cpus
596 Tflops peak, 478 sustained
The mega effect
Interaction between shear (gradients) in the azimuthal velocity and the poloidal
magnetic field produces strong toroidal magnetic field. TWO POSSIBILITIES:
(a) Latitudinal shear: (the usual)
Bf
q latitude
=
+
Bf
f
longitude
Bq
dVf/dq
(b) Radial shear: (the not-so-usual)
Bf
+
Z depth
=
f
longitude
dVf/dr
Br
Good reasons for the not-so-usual …
But really interested in gradients for magnetic buoyancy:
t BT = BP• δΩ + … , δΩ = r*sin(ϴ) ∇Ω
Differentiate MDI observational results:
Therefore reasons:
 Radial shear is stronger
 Need radial gradients for magnetic
buoyancy
 Some radial component likely
 Horizontal scale selection - better
option?
radial
latitudinal
Case (a): Latitudinal shear
(a) Latitudinal shear:
Bf
q latitude
=
+
Bf
f
longitude
dVf/dq
Bq
Model: Localised latitudinal velocity shear
Mimic some properties of the
tachocline :
• Use a convectively stable layer
• Force* a velocity shear in both the vertical
(z) and one horizontal (y) direction.
e.g. U(y,z) = f(z) cos(2 p y/ym)
where f(z) is a polynomial function chosen to confine the
shear to a particular layer between zu and zl (and to be
sufficiently continuous)
• Shear flow is hydrodynamically stable
Then add an initial magnetic field:
B0 = (0, By , 0) with By = 1
* Add
term in the equations that induces desired flow in absence of magnetic effects
By +
Induction of strong toroidal field by shear
Increasing Rm: Magnetic buoyancy instability
A more interesting movie!
• Instability
• Cyclic activity
• Two out-of-phase sequences of identical
but oppositely-directed magnetic structures.
• Broken the y-reflectional symmetry
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Latitudinal shear: conclusions
First demonstration that buoyantly-rising strong (toroidal) magnetic structures can be
spontaneously generated by the action of shear on a background (poloidal)
magnetic field.
BUT …
Radial gradients stronger => stronger toroidal field?
Currently the structures have scale of the latitudinal shear = BIG (too big)
SO …
Try radial gradients?
So consider … Case (b): Radial shear
One might expect:
Uo(z)
BZ
+
Forced
Strong layer
of Bx created
Initial
BX
Final
Yes!
… but turns out that this is much harder to make work than you might
think!
Radial shear: Magnetic buoyancy
Magnetic buoyancy
instability: volume
rendering of abs(B)
768^3
200,000 cpu hours … yikes!
Measured Taylor microscale
Reynolds number ~ 30
Pm=0.625
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Radial shear: Looks good, but problems …
Bx
z
Time
Not much flux transport.
Inefficient
Radial shear: Problems
The forcing required
to make this work is
HUGE!
Target velocity of forcing is
about Mach 15! Yikes!
If you run the same case
without magnetic field,
keeping the same large forcing,
get some serious KelvinHelmholtz turbulence!
WHY SUCH A LARGE
FORCING?
RE-CAST THIS
QUESTION …
Is the tachocline this strongly maintained?
Is it maintained by external processes independent of tachocline dynamics? (USUAL POV)
Or must maintaining processes take into account internal dynamics?
(UNUSUAL POV)
Must the tachocline be STRONGLY MAINTAINED against the tachocline dynamics?
In other words: is the shear we oberve the target shear of a a particular forcing or a modified
shear returned at the end of a complex nonlinear process?
Two simple 1-D analytic models to try and answer this question:
1.
Completely unmaintained shear
2.
Weakly maintained shear
If these don’t work in some sense, then answer is shear is strongly-maintained.
We evaluate under what conditions a magnetic buoyancy instability can manifest
itself …
Stability condition
Condition for instability (Newcomb 1961):
d g
 2
dz c
B2
around a mechanical equiilibrium
d(P 
)
20
 g

dz
can be recast as
2
B
d(
)
g
20
2

N
P  dz
where N is the Brunt-Vaisalla frequency
Thermodynamics: Adiabatic => N = N0, const. (Reasonable but we’ll come back to this)
Note that NONE of this is really valid in our case, of course! :

x No mechanical equilibrium : background state evolves!
x Shear flow on top
BUT look to see if this is even CLOSE to being satisfied as an indication of LIKELIHOOD
of magnetic buoyancy …
Model 1:
No maintenance of the shear (slow rundown)
 things happen fast so can ignore complex nonlinear back reactions on the shear
 maintainence of shear independent of tachocline dynamics
Examine time-dependent buildup of the toroidal field:
No forcing, linearisation …
P  P0  P
   0  
u  (u(z),0,0)
B  (Bx (z),0,B0 )
 t Bx  B0 z u

 0 t u  B0 z Bx
dP0
 0 g
dz
Induction eqn: stretching production of tor field by shear on pol field
 eqn: linear back reaction on the shear
Momentum
Easy: assume background density const (but don’t have to)
Initial conditions: Bx=0 and a given profile u=U0(z) => solutions
u  (U 0 (z  v A t)  U 0 (z  v A t)) /2
Bx  00 (U 0 (z  v A t)  U 0 (z  v A t)) /2
=> Alfven wave dynamics
v A  B0 / 00
Alfven wave dynamics
Analytical: constant wave
speed
U(t,z)
Initial condition:
U0 = tachocline-like jump,
Magnitude U0, width z
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u  (U 0 (z  v A t)  U 0 (z  v A t)) /2
Bx  00 (U 0 (z  v A t)  U 0 (z  v A t)) /2
Bx(t,z)
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are needed to see this picture.
Full 3D simulations: Failures :(
We have plenty of examples of the
types of dynamics we describe …
U
Forcing not large enough:
System heads to diffusively-balanced
state plus Alfven waves.
Bx
Amplitude of Bx
Bx grows but saturates.
Really is just superposition of two waves, and only “grows” when they are interfering.
Bx (z  0)  00U0 (v A t)
Bx grows whilst interference is in shear zone, then halts in the constant parts of U0
i.e. Bx grows until
vA tA  z /2
i.e. grows for Alfven crossing time tA across half layer.
Two phases of dynamics:
1.
“Growth” (actually interfering waves) for t < tA

Bx grows linearly like Bx ~ tB0zU0 and u ~ U0
2.
“Wave propagation” for t > tA with

Bx ~ 00 U /2
Linear growth up
to a saturation max
value
Can Bx get big enough for magnetic buoyancy instability?

Model 1: Instability?
Estimate gradients of Bx from this amplitude simply as (Bx(max)-0)/z/2
d Bx2
1
(
) ~ z 0 (zU 0 ) 2
dz 20
4
Differentiate
Plug into mag buoy instability condition (adiabatic version)
z
N 02

4 H (zU 02 )

or Richardson number of forced flow,
Small!!
Ri < z/H

Need highly shear unstable original flow for possibility of mag buoy instabilities!!
Rule of thumb: Ri < 1/4 => turbulent shear
Tachocline Ri ~ 1000?
Maybe NO maintenance is not enough! Try some weak maintenance? => Model 2
Model 2:
Weak maintenance of the shear
Examine time-independent state weakly forced state
z (
B0
00
Bx  z u  zU 0 )  0
z (B0 u  z Bx )  0
Momentum
Induction
Integrate, combine and figure out some bounds …
=> estimate of max amplitude of toroidal field
Bx2
 0 0

1
Pm (U) 2
4
Pm = magnetic Prandtl number
U 
 | U
z
0
| dz
Magnetic energy bounded by the kinetic energy of the the forcing
 (but with Pm factor)
Stability?

Ri/Pm < z/H or
EVEN WORSE!
Ri < Pm . z/H
Conclusions:
UNDER THE ASSUMPTIONS
-- WEAK OR NO MAINTENANCE OF SHEAR
=> SHEAR INDEPENDENT OF TACHOCLINE DYNAMICS
then to get magnetic buoyancy instabilities NEED
-- large forcing
(large = hydrodynamically unstable)
-- or high Pm
We have confirmed these simple ideas with massive
numerical simulations!
(buoyancy found at small Pm for large forcing, or at large
Pm for small forcing)
Ultimate conclusion:
The maintenance of the tachocline needs to know about
tachocline dynamics.
In other words, we have to do a more self-consistent
problem.
Pm=4000
U0=1.2
Radial shear: Thoughts
Action of radial shear on radial field -- conclusions:

For magnetic buoyancy to occur, must stretch the field fast enough that Alfven waves
cannot radiate the magnetic energy away.

Simple models, confrmed by numerical experiments, show that this requires a very
strong initial/target shear -- one that is probably hydrodynamically unstable (or a
high Pm which is not valid for the Sun).

The models assumed that the tachocline dynamics and the processes that maintained
the tachocline itself were independent. At this stage we concluded that this must not
be true and a more holistic, more complicated model of tachocline dynamics was
needed.

We had pondered one quirky caveat in this work however, and had only speculated on
it’s effect. That was the possibility of a DOUBLE-DIFFUSIVE INSTABILITY.
Double-diffusive instabilities 101
In the presence of TWO components that affect the density, can have an
instability even if the overall density stratification is convectively stable.
e.g. ocean water: warm salty water overlying cooler fresh water
warm
salty
cool
fresh
Overall density
stratification
stable
Thermal
stratification
stabilising
Salinity
stratification
destabilising
IF the diffusivity of the stabilising component is much LARGER than that of the
stabilising component, can have instability:
Parcel perturbed upwards should return downwards due to temperature deficit but
this diffuses away quickly, and so continues upwards due to relative freshness.
Double-diffusive magnetic buoyancy
In the earlier work , we assumed that the thermodynamics adjusted adiabatically.
If we assumed that the thermal diffusivity was sufficiently large, then we could
re-do the work assuming that the adjustment was isothermal.
Now we have to include the diffusivities via the ratio


(Note: not Pm!)

This is equivalent to looking for double-diffusive instabilities in this system.
Adiabatic


P  
  
P0  0 
P  
  
P0  0 

g
P
2
B
d(
2 0 )
 N 02
dz
Isothermal

g
P
2
B
d(
2 0 )
N 02

dz
( 1)  1
Clearly, if  is small, then this inequality is more likely satisfied.
Double-diffusive instability
Two simulations differing only in the thermal diffusivity  and therefore 
Fixed magnetic and viscous diffusivity (=> Prandtl number changes)
.,.5
Lower  is unstable!
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are needed to see this picture.
Double-diffusive instability
Fluctuations are induced in B and w
Double-diffusive instability
Added complexity:
Since this is not a bifurcation about a
static equilibrium but rather one about
a dynamically-evolving background
state, there is a condition on the growth
rate as well as just a “Rayleigh
number” criterion (Ratot~RaT-RaB)
Growth rate must be faster than the
evolution of the background for
instability to manifest.
Background evolves on the Alfven
timescale of the background field, so
now dynamics depends on background
field strength!
Instability occurs weaker and higher
up.
We are still working on the nonlinear
evolution of this instability …
Conclusions
Solar boundary layer -- the tachocline -- essential for solar dynamics.
Dynamics of the tachocline very mysterious.
In terms of generating buoyant magnetic structures:

TYPE 1: Latitudinal shear can produce buoyant structures but of the wrong scales.

TYPE 2: Radial shear struggles to produce buoyant structures at solar parameters
by regular magnetic buoyancy instabilities.

This is due to the fact that a system laced radially with magnetic field can radiate
away magnetic energy through Alfven waves. This will occur always be the case
because it is unlikely that you can have Type 1 alone (no poloidal field)

Differences between Type 1 and Type 2? Layers vs finite structures? etc

Is the tachocline shear strongly maintained against internal processes?

TYPE 3: Double-diffusive instead of regular magnetic buoyancy instabilities CAN
occur at solar parameter. It remains to be seen if this is a viable mechanism of
magnetic transport.

Are any of these mechanisms strong enough to transport magnetic field through a
convection zone?
The End
Latitudinal shear AND radial field
Very similar
dynamics except for
tilted geometry:
 Equilibria are
equilibria.
 Instabilities still
occur.
So why do layers
behave differently?
By=1, Bz=0 (no vertical field)
By=1, Bz=1 (vertical field)
Layer dynamics
Probably due to the fact that layers CANNOT have the following sort of
behaviour:
No horizontal pressure gradients: no poloidal flows driven in layers.
Wide but finite layers? Ends rise first, waves in middle?
Wide but finite layers
8x wider shear than previous latitudinal shear case
Shows some signs of what we expect:
• Ends rise first
• Middle spreads by waves
More work to come …
Another escape route?
A completely different type of instability:
DOUBLE-DIFFUSIVE INSTABILITIES
For the work we presented so far, arguments were made under an adiabatic parcel
argument in general.
Adiabatic => thermally-sealed.
Opposite extremely: isothermal => infinitely leaky (in heat)
High Pm, low forcing
Pm=4000
U0=1.2
Radial shear
Why was such a large forcing necessary?
Initial forcing of shear:
 dt uo ~ F
Initial induction of Bx:
dt Bx0 ~ Bz dz u0
Back-reaction on shear:
 dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 +  dzz u0 + F
magnetic
pressure
magnetic
tension
Question:
Does magnetic buoyancy act before magnetic tension destroys the shear?
Radial shear
Answer: Only if the forcing is LARGE
X
X
 dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 +  dzz u0 + F
Usually balance diffusion and forcing so that in the absence of magnetic
fields, a steady target velocity is produced.
 dtu1 ~ dx(p + a Bx0Bx0) + Bz dz Bx0 +  dzz u0 + F
Here, must also balance magnetic tension with a larger forcing otherwise
• tension acts back on the shear before the magnetic pressure becomes significant
• dynamics are dominated by Alfven waves
dt Bxn ~ Bz dz un
 dtun ~ Bz dz Bxn + F
Radial shear: Problems
Want to work out effectively whether it is possible to amplify the horizontal
field Bx fast enough to go buoyantly unstable before Alfven waves radiate
away the magnetic energy.
i.e. you have an Alfven crossing time of the layer to get Bx as big as possible!
Theory:
Unforced case (rundown problem from an initial velocity shear profile):
For instability, we need (ballpark based on Newcomb condition):
dPM/dz ~ P*N0(z)2/g
magnetic pressure, PM = Bx(z)2/2 ; N0(z) = buoyancy frequency of the initial background profile
The best that unforced shear can do after roughly one Alfven time is
Bx ~ Bz tA ∂zu
Afven time, tA ~ Δz/vA
Therefore, for instability, require
Δz (∂zu(z))2 ~ N0(z)2 HP
Δz/HP ~ Ri
i.e. for z small, Ri must be small => SHEAR FLOW IS (HYDRODYNAMICALLY) UNSTABLE!
Radial shear: Problems
Theory:
Forced case (forcing set up to balance dffusion in absence of B):
There are some steady solutions:
Bz∂zBx + μ∂zzu -μ∂zzu0
Bz∂zu + η∂zzBx
= 0
= 0
Combining these gives
Q Bx - ∂zzBx = (Bz/η) ∂zu0
Q = (Bz)2/ημ
In terms of the velocity jump ΔU = ∫ |∂zu0| dz and the magnetic Prandtl number σM = ν/η
there is a sharp bound:
sup(PM)≤ σM ρ0 ΔU2/8
Buoyancy needs:
Δz/HP ~ Ri/σM
Even worse for solar values: since σM is small, Ri must be even smaller.
Radial shear: Thoughts
Action of radial shear on radial field -- conclusions:

For magnetic buoyancy to occur, must stretch the field fast enough that Alfven waves
cannot radiate the magnetic energy away.

Simple models, confrmed by numerical experiments, show that this requires a very
strong initial/target shear -- one that is probably hydrodynamically unstable.

In the case where magnetic buoyancy is induced, simulations show that the presence
of magnetic field suppresses the hydrodynamic instability, so that the resultant MHD
velocity shear is reasonable. Adding the magnetic field back into the hydrodynamic
simulation seems to return the system to the same MHD state.

In the case where magnetic buoyancy is not induced, a diffusive balance results.

These results are true for forced AND slow rundown (very low diffusivity) cases.
Two big questions remain:

Even when buoyancy does occur, it is inefficient at transporting magnetic flux
vertically. Why?

The latitudinal shear problem worked just fine. Would adding vertical field to this
problem ruin things?
Radial shear: Problem avoidance
To avoid these theoretical restrictions, need:
1.
Large forcing
2.
Large magnetic Prandtl number
Both work:
1.
Seen already
2.
Large Pm
Pm = 4000
Latitudinal shear AND radial field
Very similar
dynamics except for
tilted geometry:
 Equilibria are
equilibria.
 Instabilities still
occur.
So why do layers
behave differently?
By=1, Bz=0 (no vertical field)
By=1, Bz=1 (vertical field)
Layer dynamics
Probably due to the fact that layers CANNOT have the following sort of
behaviour:
No horizontal pressure gradients: no poloidal flows driven in layers.
Wide but finite layers? Ends rise first, waves in middle?
Wide but finite layers
8x wider shear than previous latitudinal shear case
Shows some signs of what we expect:
• Ends rise first
• Middle spreads by waves
More work to come …
Conclusions
Solar boundary layer -- the tachocline -- essential for solar dynamics.
Dynamics of the tachocline very mysterious (Why is it so thin? Is it turbulent? -- Much
more to come from Steve Tobias later).
In terms of generating buoyant magnetic structures:

Radial shear acting on radial field is much less efficient than latitudinal shear acting
on latitudinal field

This is due to the fact that a system laced radially with magnetic field can radiate
away magnetic energy through Alfven waves.


Horizontal field => waves confined to structures => energy not radiated away?
Furthermore, the dynamics of layers are very different from the dynamics of tubes

No poloidal flows

Layers => “tethering” of structures by magnetic tension

What is the role of waves in the tachocline?

(Important for angular momentum transport too)
The End
Latitudinal shear with radial field
By=1, Bx=0 (old)
By=1, Bx=1 (new)
Very similar
dynamics except for
tilted geometry:
 Equilibria are
equilibria.
 Instabilities still
occur.
So why do layers
behave differently?
Radial shear: Magnetic buoyancy
High enough forcing: strong enough magnetic
layer created such that magnetic buoyancy can act.
However, things are much more
complicated and turbulent than
previous case …
Alfven wave dynamics
Initial condition: U0 = tachocline-like jump, jump U0, width z:
Bx(t,z)
U(t,z)
Analytical:
constant wave speed
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
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QuickTime™ and a
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QuickTime™ and a
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Toy Calculation:
with density stratification