Modeling Kinematic
Dynamos in 2 and 3-D
Ryan Horton, Nicholas Featherstone, Mark Miesch
Motivation
• Convection on many
different scales
• Magnetic events on
many
different scales
• Long (~22 year) solar cycle
• How do all these different scales fit
together?
• How can we know how accurately our
models are working?
2
Granulation
The Dynamic Sun
Convection on Many Scales
Helioseismic Inference
Differential Rotation
SST
Meridional Circulation
Ω
NSSL
Tachocline
vθ
500 nHz
360 nHz
Howe et al. 2000;
Schou et al. 2002 Zhao et al. 2012
Supergranulation
SDO/AIA
The Magnetic Sun
Solar Cycle
(The Big Mystery)
Global-Scale
Magnetism
Small-Scale Magnetism
(Magnetic Carpet)
\
Lites et al (2008)
SDO/HMI
The Induction Equation
Dissipation
Generation
Always zero
Compression and Expansion
Stretching and Shear
Advection (Movement)
5
Magnetic Reynolds Number
Dissipation
Generation
•Ratio of generation to dissipation
•Essentially the reciprocal of the diffusivity
6
Methods
•Numerical solution to induction equation
•Periodic 3-D domain
Magneato
•Computational domain broken up into many
B2
small cubes
1
B
dA
1
dA 2
EMF 2
•Constrained transport (Evans & Hawley 1988)
•Preserves divergence free magnetic field by
EMF 1
integrating EMF around cell faces
7
Objectives
Assessing Dynamo Properties
of Numerical Diffusion
Examine the dynamo properties
of “solar-like” convective flows
There is always diffusion present
Some examples of flows in current solar models
Featherstone et al. 2009
Evans and Hawley 1988
8
Effects of Vortical Flows in 2-D
• Flow “winds up” the field
• Takes field in one direction and
generates field in another direction
• Eventually dissipation always wins as
fields in opposing directions come
together
Moffatt 1978
9
Code Validation
The Expulsion of Magnetic Fields by Eddies, Weiss 1966
10
Assessing Numerical Diffusion
Runs with Different Eta (128x128)
Average Field Strength
5e-7
5e-5
5e-6
5e-4
400
1000
200
100
40
11
Peak of Average Magnetic Energy
Magnetic Reynolds Number vs Peak of Magnetic Field for Different
Grid Spacings
128x128
~Rm^(1/3)
64x64
32x32
Magnetic Reynolds Number
“2.5-D” Flow
•Invariant in z-direction
•Vortices do not connect
Into Slide
Out of Slide
13
Peak Magnetic Energy
Different Grid Spacings with 2.5-D Flow
~Rm^(1/3)
256x256
~Rm^(1/3)
128x128
Magnetic Reynolds Number
14
Creating a Dynamo
• Impossible in 2-D (diffusion always wins)
• Field lines will wrap
• Opposing field comes together and
cancels
• Possible in 3-D
• “stretch, twist, fold”
15
Fully 3-D Flow
A-flow
B-flow
Total flow is a weighted sum of the two
16
128 Cubed 3-D Flow
Magnetic Energy
Field Lines
17
After Many Iterations...
Magnetic Energy
Field Lines
18
Magnetic Energy
Fully 3-D Dynamo
Iterations (*25)
19
Overview
• Characterized numerical diffusion in two
and three dimensional flows
• Showed that the numerical diffusion
depended on grid size
• Showed the numerical diffusion varied
with flow
• Creation of a 3-D dynamo motivated by
those flows present in solar models
20