David Schaffner

The End of the Turbulent Cascade
Exploring possible signatures of MHD turbulent dissipation
beyond spectra in a magnetically-dynamic laboratory plasma
David Schaffner
Bryn Mawr College
APS-DPP – November 16, 2015
Savannah, GA
Thank you to my collaborators
Michael Brown – Swarthmore College
Slava Lukin - NSF
Swarthmore Undergrads – Adrian Wan, Peter Weck,
Emily Hudson, Ariel Rock, and Holden Parks
Thank you as well to the organizing committee
Funding support for this work from DOE-OFES and NSF CMSO
This talk is dedicated to my grandfather,
Jerry Schaffner (1927-2015),
Who passed on his love of science
to his son and his grandson
(And hopefully his great-granddaughter)
Defining turbulence
The term “turbulence” has taken on a variety of
meanings and connotations in plasma physics:
“turbulent fluctuations” “turbulent transport” “turbulent system”
Edge/E-static Turbulence
Candy/Waltz (General Atomics)
 Interactions of
many modes
 Energy injected/
dissipated at
multiple scales
 Driven by gradients
MHD Turbulence
 Fluid turbulencelike
 Clear separation of
energy injection
and dissipation
scales
 Drive by large scale
Log (Kinetic Energy per scale)
Kolmogorov Picture of Fluid Turbulence
Flow/Kinetic
Energy Spectrum
“turbulent cascade”
Log (k)
Kolmogorov Picture of Fluid Turbulence
Log (Kinetic Energy per scale)
Kinetic energy Injection ONLY at Large Scales
Kinetic energy transferred
between scales inertially
Power law scaling ~5/3
“Injection Scale”
“Inertial Range”
Clear Scale
Separation!
“Dissipation Range”
Log (k)
Kinetic energy
converted into heat at
small scales
Log (Kinetic Energy per scale)
Kolmogorov Picture of Fluid Turbulence
Transition occurs due to increasing
effect of viscosity on flow eddies
“the end of the turbulent cascade”
“Injection Scale”
“Inertial Range”
Clear Scale
Separation!
“Dissipation Range”
Log (k)
This manifests as a
steepening of the
energy spectrum
reflecting loss of
kinetic energy to heat
Kolmogorov Picture extended to MHD Plasmas
Log(KE and ME per scale)
1) Energy includes BOTH Flow and
Magnetic
2) Thermal dissipation due to both
flow viscosity AND plasma resistivity
Collisional Physics
However, other plasma
physics can contribute to
dissipation including
Log (k)
magnetic reconnection,
wave damping,
or shocks
The Solar Wind as an MHD Turbulence System
Magnetic Energy Spectra
As expected, a
steepening of
the energy
spectrum
indicates
transition from
inertial to
dissipation
range
Cluster FGM and STAFF-SC Data:
Sahraoui, PRL 2009
HOWEVER
Solar wind is
not a collisional
plasma!
What other metrics can be used to observe
dissipation in an MHD turbulent plasma?
Given the complexity of plasma versus fluid dynamics, is
spectra enough to characterize dissipation physics?
Alternate metrics discussed during this talk:
1) PDF of increments and structure functions
2) Permutation Entropy and Statistical Complexity
MHD Turbulence in the laboratory
SSX
@
Swarthmore College
A plasma gun source
generates a magnetically
dynamic plasma in a
flux-conserving column
without a background
field
Brown and Schaffner PSST 2014
Brown and Schaffner JPP 2015
MHD Turbulence in the laboratory
Flow
Spectrum
SSX Spectra show power
law scaling, but is steeper
than Kolmogorov scaling
(~3.0 vs 1.66)
Magnetic
Spectrum
Schaffner ApJ 2014
MHD Turbulence in the laboratory
Flow
Spectrum
Steepening observed—is
this indication the onset of
dissipation?
 Need to look for other
signatures to confirm
Magnetic
Spectrum
Schaffner ApJ 2014
A potential dissipation mechanism in MHD
plasmas is magnetic reconnection
Magnetic Energy In
Thermal
Energy
Out
Current Sheet
Look for jumps in signal  Distribution of increments
Non-Gaussian PDF of increments observed
Schaffner PPCF 2014
Intermittency or “Fat Tails”
PDF
Likely indicates presence
of current sheets
BUT
for signature of dissipation,
need sign of heating
Fluctuation Level / Stan. Dev
Some indirect correlations of intermittency
and ion temperature are observed
Mag
Increment
Ion T
(eV)
Schaffner
PRL 2014
Time (us)
Ion heating observed with
magnetic jump, but only in
time, not space
Temperature intermittency
correlates with changing
magnetic intermittency
Different Approach—look at statistics of PDF of
increments using structure functions
(i.e. look for a distinction between scales in the nature of the fluctuations at those scales)
Take p-moments for
different time steps
Δτ
Schaffner and Brown ApJ 2015
Different Approach—look at statistics of PDF of
increments using structure functions
Next, compute slope in different regions of
structure function (before and after spectral
break), for each moment, p
Schaffner and Brown ApJ 2015
Different Approach—look at statistics of PDF of
increments using structure functions
“Dissipation range” exhibits
monofractal scaling
(linear with order)
“Inertial range” exhibits
multi-fractal scaling
(non-linear with order)
These results are consist with
inertial to dissipation range
transitions seen in fluid
turbulence and the solar wind
See e.g.
Frisch & Vergassola 1991
Kiyani 2009
Different Approach—look at statistics of PDF of
increments using structure functions
Initial Conclusion:
Structure function analysis shows that the
nature of fluctuations before and after the
spectral break exhibit clear differences—
this could be due to a dissipation
mechanism associated with reconnection
Difference in fractal scaling motivates use of
permutation entropy/statistical complexity analysis
Scan through data
set recording
ordinal patterns
at various time
separations, τ
Permutation entropy/complexity analysis is a patternrecognition technique
N=5 positions  n!=5!=120 possible patterns
Permutation entropy/complexity analysis is a patternrecognition technique
Permutation Entropy, S[P]
PE reflects how many of the possible N
patterns a time series exhibits
Statistical Complexity, C[P]
SC reflects how far from the uniform
distribution is the dataset’s distribution
For more details on this technique, see Poster by Ariel Rock, Tuesday Afternoon: JP12.00032
Regions of PE vs SC plot indicate relative complexity
Chaotic
Stochastic
Normalized Entropy/ Complexity
PE/SC analysis is most useful as a function of increment
Local Maximum of
Complexity at inertial/
dissipation scale
transition
Entropy
Complexity
Δτ (us)
This suggests some
type of transition
process is occurring
which may indicate
evolution from inertial
scale to dissipation
scale physics—
dissipation structures
chaotic?
Spectra can indicate dissipation, other metrics can
illuminate underlying physics of process
Increments implies
reconnection’s role in
dissipation
Structure functions and
PE/SC analysis can
identify a process as
chaotic or stochastic
We’re nowhere near the end of the cascade…
Next analysis steps:
1) Continue to find
new metrics
2) Compare and
contrast metrics
to illuminate
dissipative
processes
Next experimental steps:
Design and construction of a new MHD turbulence
laboratory at Bryn Mawr College
BM2X (Bryn Mawr MHD EXperiment)
-Focused on sustained MHD turbulence
-Improved scale separation
-More diagnostics access
~16in
~48in
Thank you for your attention