1. 1
Self-organisation in magnetically confined plasmas- from noise to
large-scale sheared flows– Final Report
1110801
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
(Dated: March 11, 2015)
A reduced model for Zonal Flow (ZF) generation in magnetically confined fusion plasmas
has been theorised and verified. Small-scale instabilities in the fusion plasma have been
modelled using white noise and the coupling of the ZFs with a geodesic acoustic mode will
be modelled with a sinusoidal driving. The model is verified for stationary 1D solutions of
the Cahn-Hilliard equation and the phenomenon of stochastic resonance is explored before
a full temporal solution in 1D is obtained.
1. Introduction
With an ever increasing population and
substantial economic development, there is a
growing energy demand that must be met.
Allied with the increasing concern over
global warming it is more important than
ever to find a sustainable and powerful
energy source that meets the energy budget
and drastically reduces the carbon emissions
of the world’s energy production.
Nuclear fusion is one such energy source
that could provide an environmentally
friendly method of producing energy with a
virtually inexhaustible fuel supply.
Magnetically confined fusion plasmas are
very complex systems with many modes of
interaction between the particles and the
fields. Turbulent transports driven by
micro-instabilities make it extremely
difficult to confine the plasma magnetically
in a Tokamak for instance. The turbulence
causes the plasma to leave the confinement
area making it harder to reach and maintain
the Lawson criteria which govern plasma
ignition and sustenance [1].
It has however been noticed that upon
reaching a certain amplitude these micro-
instabilities can self-organise into stable
zonal flows (ZFs) which act as a sink to the
available energy for turbulence thus helping
to improve plasma confinement. During this
work the origin of these large scale shear
flows will be understood through a 1D
stationary model before moving onto a 1D
temporal model.
The models will contain stochastic drive
which will represent the micro-instabilities
in the fusion plasma. Numerical recipes will
be applied to find solutions of the model for
different stochastic forcing e.g. Gaussian
white noise and coloured noise. The

2. 2
phenomena of stochastic resonance will also
be explored and interpreted in relation to the
underlying physics.
Despite the presence of large-scale sheared
flows throughout nature, their structure and
evolution is not yet fully understood. A. P.
Newton, E. Kim and H. Liu attempted to
elucidate the complex linkage between
large-scale sheared flows and micro-
instabilities [2]. B. McNamara and K.
Wiesenfield began to investigate the
phenomenon of stochastic resonance in bi-
stable systems [3].
1.1 Nuclear Fusion
Nuclear fusion power has the potential to
solve a large amount of the world’s energy
production problems. Fusion in comparison
to other energy sources has a lot of benefits.
There is a large amount of fuel available and
at a cheap price, the process is CO2 neutral
and yields a relatively small amount of high
level radioactive waste. There is no risk of
uncontrolled energy release and a small
threat to non-proliferation of weapons
material [4].
Fusion refers to the controlled process by
which two light atoms are fused together
generating a heavier atom with the aim of
releasing energy. Nuclear fusion occurs
when two nuclei approach close enough for
the nuclear strong force to pull them
together into a larger nucleus. The binding
energy is the energy released when a nucleus
is created from protons and neutrons. The
higher the binding energy per nucleon the
more stable the atom is [4].
An electrostatic force opposes the
decreasing separation between the two
nuclei and must be overcome to achieve
fusion. The energy to overcome this force is
called the Coulomb barrier. The most
promising nuclei to give achievable fusion
are Deuterium and Tritium fusing into
Helium. They give the largest ratio of
binding energy to Coulomb barrier. The
fusion reaction is:
D + T → He2
4
1
3
1
2
+ n0
1
+ 17.6MeV. (1)
Nuclear fusion is an incredibly powerful
energy source releasing around a million
times more energy per kilo of fuel than fossil
fuels. As it has already been stated turbulent
transport in the plasma is one of the major
limiting factors of magnetically confined
fusions success.
1.2 Turbulence
Turbulence is a ubiquitous phenomenon in
fluids. Turbulence has a long history of
study on Earth in the hydrodynamic sense
due to its presence in the atmosphere, rivers
and oceans. Turbulence is often associated
with fluid dynamics and forms part of the
Navier-Stokes equation which is as follows:

3. 3
𝜌 (
𝜕𝒖
𝜕𝑡
+ (𝒖 ⋅ 𝛁)𝒖) = −𝛁𝑝 + 𝜈𝛁2
𝒖. (2)
∇𝒖 = 0. (3)
Where ρ is the density, u is the velocity, p is
the pressure and ν is the dynamic viscosity.
It is convenient to introduce a control
parameter known as the Reynolds number:
𝑅 =
𝐿𝑈
𝜈
, (4)
where R is the Reynolds number, L and U
are the characteristic length and velocity
scales of the system and ν is the kinematic
viscosity. In general high Reynolds numbers
indicate a turbulent system and low
Reynolds numbers give a laminar system.
This is because the Reynolds number is a
measure of the strength of the non-linear
term (𝒖 ⋅ 𝛻)𝒖 compared to the dissipative
term 𝜈𝛻2
𝒖 i.e. at low Reynolds number the
Navier-Stokes equation is effectively linear
whereas at high Reynolds number the non-
linear term is important in the evolution of
the system [5].
Turbulence is a seemingly chaotic state of
motion that consists of eddies at all scales.
Larger scales are determined by the driving
of the system whereas the small scales gain
their energy from the larger scales. Energy
cascades from large scales to small scales as
the larger eddies break down into smaller
ones which in turn break down further. This
continues until the eddies are small enough
that viscous effects dominate and their
energy is dissipated by viscous diffusion [6].
The inertial range is the range of scales
between the energy injection and the
dissipation scales. In this range the energy is
transferred from smaller wavenumbers to
larger wavenumbers without dissipation.
Kolmogorovs 1941 theory showed that for
3D turbulence the power spectrum of the
system goes as [7]:
𝐸(𝑘)~𝑘−
5
3 . (5)
Many systems can be described with 2D
turbulence rather than 3D turbulence. In 2D
turbulence not only is the energy conserved
but enstrophy (vorticity) is also conserved.
The origin of this phenomenon is the
vanishing in 2D of the vortex stretching term
that appears as a forcing term in 3D where it
is responsible for the unbounded growth of
enstrophy in the limit of R →∞. As a result
of this secondary conservation law another
energy cascade is introduced known as the
inverse energy cascade. The inverse energy
cascade delivers energy to scales with small
wavenumbers allowing large scale structures
to grow such as the ZFs in fusion plasmas.
The direct enstrophy cascade still transfers
energy to the higher wavenumbers as the
energy cascade did before. Figure 1 shows
the Power Spectral Density for two and three
dimensional turbulences. From figure 1 it
can be seen that in 2D turbulent systems
large scale structures can develop whereas in
the 3D case there is no cascade

4. 4
PSD
Wavenumber (k)
Energy Injection
in the small wavenumber direction and
hence no large scale structures can grow and
turbulence is purely an energy sink. It is
useful to consider the Navier-Stokes
equation in Fourier space instead of real
space. The linear terms are straight forward
to Fourier transform whereas the non-linear
term requires more care but yields important
information about the non-linear
interactions. In real space the non-linear
term is a product operation so in Fourier
space it is a convolution. Using Einstein
notation it can be shown:
ℱ((𝒖 ⋅ 𝛁)𝒖) = ℱ (𝒖𝑗
𝜕𝒖 𝑖
𝜕𝑥 𝑗
) = ℱ (
𝜕𝒖 𝑖 𝒖 𝑗
𝜕𝑥 𝑗
) =
𝑖𝑘𝑗 ∫ 𝑢𝑖̃𝒑+𝒒=𝒌
(𝒑)𝑢𝑗̃(𝒒)𝑑𝒑 (6)
where q=k-p. From this transform it can be
seen that interactions occur between triads of
wave vectors. Non-linear interactions can be
split into two distinct classes; 1) local
interactions where the three wave vectors are
of similar magnitude and 2) nonlocal
interactions where one wave vector is
largely different from the other two [8].
Turbulence in fusion plasmas is one such
system that can be described with 2D
turbulence rather than 3D. This is due to
rapid particle conduction along the confining
magnetic field. This suppresses any parallel
gradients meaning that the turbulence in a
tokomak is essentially 2D and contains the
dual energy cascade which is vital for ZFs
generation [9]. In tokomaks, turbulence is
driven by radial gradients in the plasma
density and temperature. The turbulence is
treated with a drift instability formulation.
The instabilities are formed on very small
scales but can couple to form turbulent
flows. The instabilities can be considered as
a fluctuation in density causing a small scale
charge build up. An E x B drift is
Figure.1. A comparison of the power spectral density for 2 and 3D turbulences, showing the key differences
and the respective powers of the different parts of the spectrum. The 2D turbulence is shown in black and the
3D turbulence is shown in red.

5. 5
established from the background magnetic
field and the electric field associated with
the increase in potential. The E x B drift will
act to remove the instabilities however as it
does it increases the momentum of the
plasma and causes it to overshoot the
unperturbed state in a periodic motion. This
on its own will not lead to a loss of
confinement however if the charge and
density perturbations are out of phase this
sets up an unstable system and the
perturbation will grow and can lead to a loss
of confinement [10]. The presence of the
dual energy cascade in the plasma produces
some important effects.
The energy transferred to the high
wavenumbers via the enstrophy cascade is
dissipated out into thermal processes via
particle collisions and viscosity whereas the
inverse energy cascade leads to the
development and growth of ZFs. Turbulent
eddies merge forming larger structures such
as ZFs until the structures are comparable to
the plasma dimensions. ZFs in magnetically
confined plasmas are defined as radially
localised (𝑘 𝑟 ≠ 0), axisymmetric and
azimuthally symmetric (m=n=0, where m
and n are the poloidal and toroidal mode
numbers respectively) electrostatic potential
modes with frequency 𝛺 𝑍𝐹 = 0 [11]. ZFs are
of interest as they can help to confine the
plasma. The total energy in the ZFs and the
turbulence is conserved so as the ZFs grow
the energy available for turbulence
decreases. ZFs can also regulate turbulence
via vortex shearing e.g. stretching, twisting
and breaking larger eddies into smaller ones
where they are dissipated more easily. In
toroidal geometry ZFs can couple to other
low frequency modes such as the geodesic
acoustic mode (GAM). This results from the
compressibility of the ZFs in toroidal
geometry, coupling to an acoustic mode with
m=±1, through the geodesic curvature of the
confining magnetic field [11]. The GAM has
a finite frequency described by
𝜔 𝐺𝐴𝑀 =
𝑐 𝑠
2
𝑎
. (7)
Where 𝑐 𝑠 is the sound speed in the plasma
and a is the minor radius of the tokomak.
The GAM also regulates turbulence and can
dissipate energy out of the system through
collisional and non-collisional landau
damping.
The plasma throughout this work will be
treated as a fluid. When dealing with fluids a
key tool is the Navier-Stokes equation (2).
Initially however it is sufficient to consider a
1D Navier-Stokes equation subject to a
forcing and under the constraint that it is
incompressible. The equation reads:
𝜕𝒖
𝜕𝑡
+ (𝒖 ⋅
𝝏
𝝏𝒛
) 𝒖 = 𝜈 (
𝜕
𝜕𝑧
)
2
𝒖 + 𝐹. (8)
Where F is an arbitrary forcing and z is the
one dimensional direction of interest.

6. 6
Unfortunately the non-linearity of the
Navier-Stokes equation makes it very
difficult or impossible to solve analytically.
There are however techniques that can be
employed to simplify the problem at hand.
1.4 Eddy Viscosity
The concept of eddy viscosity arises when a
perturbation of a given size affects the
distribution of momentum and vorticity
within the flow. The coupling of momentum
and vorticity perturbations means the
Navier-Stokes equation never reduces to an
advection-diffusion equation even in 2D
where vorticity is a scalar. This can lead to a
large-scale momentum gradient leading to a
momentum flux in the same direction as the
gradient. These flows show large scale
instability. Flows such as these are referred
to as having a negative eddy viscosity.
Kraichnan (1976) derived an expression for
eddy viscosity within a closure framework
and obtained negative values in 2D. He then
interpreted this as the reverse flow of energy
in the inverse cascade of 2D turbulence [12].
This concept of eddy viscosity has been
useful in the history of turbulence study in
parameterising turbulent transport processes
and the passage of turbulent energy through
different scales.
When studying turbulence in fluids it is
common to ignore small scale vortices and
to calculate a large scale motion with an
eddy viscosity that characterises the
transport and dissipation of energy in the
smaller scale flow. Smagorinsky proposed a
useful formula for the eddy viscosity in
numerical models based on the local
derivatives of the velocity field and the local
grid size, the formula he proposed is as
follows
𝜈𝑆𝑀 = ∆𝑥∆𝑦√(
𝜕𝑢
𝜕𝑥
)
2
+ (
𝜕𝑣
𝜕𝑦
)
2
+
1
2
(
𝜕𝑢
𝜕𝑦
+
𝜕𝑣
𝜕𝑥
)
2
.(9)
Where ∆𝑥 and ∆𝑦 represent the local grid size.
This is the simplest and most commonly used
eddy viscosity model. It is widely used because
of its manageability, computational stability and
the simplicity of its formulation [13]. For many
1D treatments of turbulence and in the 1D work
that follows the eddy viscosity will be
represented as follows:
𝜈 𝐸𝐷 = 𝜆 (
𝜕𝒖
𝜕𝑧
)
2
, (10)
where λ is a constant. The use of eddy
viscosity is paramount to simplifying the
non-linear term in the 1D Navier-Stokes
equation. Equation (8) can now be rewritten
with the eddy viscosity as
𝜕𝒖
𝜕𝑡
=
𝜕
𝜕𝑧
{[𝜈 + 𝜆 (
𝜕𝒖
𝜕𝑧
)
2
]
𝜕𝒖
𝜕𝑧
} + 𝐹. (11)
Another technique that can be used to
simplify the situation at hand is rewriting
equation (8) in terms of a potential velocity
field and utilising a negative eddy viscosity.

7. 7
1.5 Cahn-Hilliard equation
A stream function ψ can be defined such
that:
𝒖 = (−
𝜕𝜓
𝜕𝑥
,
𝜕𝜓
𝜕𝑦
). (12)
This allows the vorticity to be expressed as:
ω= 𝜵 x 𝒖 = ∇2
𝜓 𝒆 𝒛̂ . (13)
Where ω is the vorticity. Equation (8) can be
expressed in terms of the stream function ψ
yielding:
𝜕
𝜕𝑡
∇2
𝜓 + 𝐽(∇2
𝜓, 𝜓) = ν∇4
𝜓 + 𝐹 (14)
where J is the Jacobian. In this work
attempting to elucidate the connection
between small-scale instabilities and large-
scale shear flows it is useful to consider slow
and fast time scales differently
corresponding to large and small-scale
perturbations respectively. Let the stream
function become ψ+Ψ where Ψ represents
the large scale perturbation. Multi-scale
analysis will be undertaken assuming the
slow and fast variable are independent of
each other. The space-time variables of the
basic flow (fast variables) are denoted x and
t and the slow variables are denoted X=εx
and T=𝜀2
𝑡 where ε is small. Due to the
assumption of independence the derivatives
can be expressed as:
∇→ ∇ + 𝜀𝜕𝑖 ;
𝜕
𝜕𝑡
→
𝜕
𝜕𝑡
+ 𝜀2 𝜕
𝜕𝑇
(15)
where derivatives with the fast spaced
variables are denoted by ∇ and slow space
variables 𝜕𝑖. The solution is then sought as a
series in ε:
𝜓 = 𝜓(0)
+ 𝜀𝜓(1)
+ 𝜀2
𝜓(2)
+. .. (16)
The full derivation of the multi-scale
analysis can be found in [14]. It is sufficient
in this case to consider a negative eddy
viscosity instability which is amplifying the
large-scale flow up to a point where the non-
linearities are relevant and linearization is
not permitted [14]. For a negative eddy
viscosity the solution is O(𝜀2
) and the large
scale dynamics are governed by the 1D
Cahn-Hilliard equation:
𝜕𝜓
𝜕𝑇
= −𝜕𝑖{[1 − (𝜕𝑖 𝜓)2]𝜕𝑖 𝜓} − 𝜕𝑖
4
𝜓, (17)
This can then be written in terms of the
velocity rather the stream function:
𝜕𝑣
𝜕𝑇
= −𝜕𝑖
2
{[1 − (𝑣)2]𝑣} − 𝜕𝑖
3
𝑣. (18)
In the work that follows a generic form of
the Cahn-Hilliard equation in 1D and subject
to forcing will be used, it is as follows:
𝑑𝑣
𝑑𝑡
= −𝐷
𝑑2
𝑑𝑧2
(−𝑎𝑣 + 𝑏𝑣3
− 𝛾∇2
𝑣 + 𝐹)(19)
where D, a, b and 𝛾 are constants.
Linearized stability analysis is always the
first step in understanding the stability of a
non-linear equation such as the Cahn-
Hilliard equation. Linearizing the Cahn-

8. 8
Hilliard equation and substituting a trial
solution of the form:
𝑣 = 𝑣̅ 𝑒 𝑖𝑘𝑧−𝜔𝑡
(20)
into it, a dispersion relation can be found.
The dispersion relation is as follows:
𝜔 = 𝐷(𝛾𝑘4
− 𝑎𝑘2). (21)
The dispersion relation solutions indicate
when the solutions of the linearized equation
are oscillatory, exponentially growing or
exponentially decaying. The differing types
of solution arise from whether 𝜔 is real or
imaginary and on the signs of the constants a
and 𝛾. The most stable modes can be found
by differentiating the dispersion equation
and equating to zero as is standard. This
obtains two (𝜔, 𝑘) pairs as follows:
𝜔 = −
𝐷𝑎2
4𝛾
, 𝑘 = ±√
𝑎
2𝛾
. (22)
The dispersion relation clearly follows a
double-well potential form as shown in
figure 2.
Figure.2. The dispersion relation for the linearised
Cahn-Hilliard equation. The minima are at𝜔 =
−
𝐷𝑎2
4𝛾
, 𝑘 = ±√
𝑎
2𝛾
.
The stationary solutions of the Cahn-Hilliard
equation are vital as they give information
on how the full temporal solution should
behave after a given transient time period.
Setting
𝑑𝑣
𝑑𝑡
= 0 equation (19) can be written
as:
𝑑2 𝑣
𝑑𝑧2
= 𝑎𝑣 − 𝑏𝑣3
+ 𝐹 (23)
.Here F is still a forcing term. It is known
that the non-linear term should suppress the
forcing term after the transient time period
so solutions of (23) with F=0 are important.
By inspection it can be seen they are of
arctangent nature.
Figure.3. The plot of z=arctan(x).
In the work that follows this forcing term
will be associated with the drift instabilities
present in a fusion plasma and will be
modelled by noise.
1.6 Noise
Drift instabilities in the plasma are
responsible for driving turbulent flows
which in turn can lead to the generation of
ZFs. This forcing will be modelled as noise
𝜔
𝑘

9. 9
due to the random nature of both and the
very fast timescales involved.
1.6.1 White Noise
White noise is a random signal with constant
power spectral density across all
frequencies. It is statistically uncorrelated,
that is, two values of the noise at any pair of
times are identically distributed and
statistically independent
〈𝑓(0)𝑓(𝑡)〉 = 2𝐷𝛿(0) (24)
where D is the noise strength and δ is the
delta function [15]. It is common to use
Gaussian white noise where the Gaussian
nature refers to the probability of the signal
falling in a range of amplitudes. It should be
noted that whilst in general these
correlations relate to time during the work in
1D on the stationary solutions of the Cahn-
Hilliard equation there is no time domain as
such and so these correlations are in the
spatial domain.
In addition to driving with just white noise a
forcing term comprised of white noise and a
sinusoidal term ~ sin(𝛺𝑡) will be used. The
sinusoidal term is physically motivated by
the coupling of the ZFs with the GAM.
1.6.1 Coloured Noise
It is expected that replacing the forcing
combination of white noise and the
sinusoidal term with coloured noise will lead
to the same behaviour if the correlation
times of both are the same. Physically
coloured noise represents the finite
correlation between the instabilities due to
their physical size. Coloured noise is
generally understood as a characteristic of its
power spectrum. Coloured noise is
correlated and the correlation is assumed to
be:
𝑃𝑆𝐷(𝜈)~𝑃0 𝜈 𝛼
(25)
It is known that forcing the stationary
solutions of the Cahn-Hilliard equation with
a white noise forcing will cause random
switching of the velocity between positive
and negative values. This is subject to the
constraint that the amplitude of the noise has
to be large enough to cause this. With the
addition forcing from the sinusoidal term a
phenomenon called stochastic resonance can
be introduced.
1.7 Stochastic Resonance
The idea of stochastic resonance was first
coined by Benzi et al when modelling the
switching of the Earth’s climate from
periods of relative warmth to ice ages. The
period is around 100 000 years which is the
same period as the eccentricity of the Earth’s
orbit however current theories suggest this is
not enough to cause these transitions in
climate. Benzi et al introduced a bi-stable
“climatic potential” and a co-operative
phenomenon was suggested between the

10. 10
weak periodic forcing (eccentricity) and
random fluctuations (noise) in the system
that might account for the periodicity
observed [16].
Despite this being called a ‘resonance’ it is
not strictly a resonance in the sense of a
certain response increasing when the
frequency is tuned to a natural frequency but
rather the signal to noise ratio (SNR) is
maximised when the input noise is tuned to a
certain value.
Stochastic resonance manifests itself in non-
linear systems where a weak input
(sinusoidal forcing) can be amplified and
optimised with assistance from noise. For a
system that is linear the SNR output must
equal the SNR at input therefore an increase
in noise amplitude will result in an SNR
decrease at the output. However in
stochastic resonance an increase in the noise
amplitude can result in a SNR increase at the
output hence the non-linear nature is
important.
The mechanism can be considered as a
particle moving in a symmetrical double
well potential. Subject to no forcing a
damped particle will come to rest at one of
the two minima of the potential (±c). With a
small amount of random forcing the particle
will occasionally transition between wells.
As the variance of the noise (D) is increased,
the rate of transitions W increases. W grows
rapidly to begin with but once the barrier
becomes very easy to surmount, W grows
more slowly [17]. If position z(t) is the
output of the system and focussing on the
power spectrum S(Ω). Define S such that:
𝑆(𝛺) = |Z(Ω)|2
. (26)
Where Z(Ω) is the Fourier transform of z(t).
S(Ω) has a Lorentzian shape given by:
𝑆(𝛺)~
2𝛼
(𝛼2+𝛺2)
. (27)
Where 𝛼 is a shortest time scale or fastest
rate imposed upon the motion z. The weak
periodic forcing effectively tilts the potential
well asymmetrically up and down,
periodically raising and lowering the
potential barrier. It is assumed that the
periodic forcing is too weak to cause
transitions alone. The periodic forcing
allows the introduction of 𝑊±(𝑡) which is
the rate of transitions out of the positive and
negative wells respectively. The
characteristic rate imposed on the motion 𝛼
can be shown to be:
𝛼 = 𝑊+ + 𝑊− (28)
As the noise strength D is increased both
𝑊+(𝑡) and 𝑊−(𝑡) increase and thus 𝛼 also
increases. As expected from the Lorentzian
shape the output noise power increases
until 𝛼 = 𝜔𝑠, where 𝜔𝑠 is the periodic
driving frequency. What is somewhat
surprising is that the signal output also

11. 11
S(Ω=𝜔𝑠) also grows and even faster than the
noise output power. This is evidence of a
cooperative phenomenon where incoherent
noise power can feed into a coherent output
signal. At different extremes in the cycle it is
more likely for a particle to be in the + or –
well due to the asymmetrical tilting of the
wells. The large excursions of the particles
position show up as an output with strong
periodic components and also an enhanced
spike at the frequency of the periodic driver
[3]. A full analytic derivation of the output
power spectrum 𝑆(𝛺) can be found in [3]
but will not be presented here. The final
result is:
𝑆(𝛺) = [1 −
𝛼1
2 𝜂0
2
2(𝛼0
2+𝜔 𝑠
2)
] [
4𝑐2 𝛼0
𝛼0
2+𝛺2
] +
𝜋𝑐2 𝛼1
2 𝜂0
2
𝛼0
2+𝜔 𝑠
2 𝛿(𝛺 − 𝜔𝑠). (29)
Where the rate is assumed to be of the form:
𝑊±(𝑡) = 𝑓(𝜇 ± 𝜂0 cos 𝜔𝑠 𝑡) (30)
where 𝜇 is the dimensionless measure of the
ratio of the potential barrier to noise
amplitude and 𝜂0 is the dimensionless
measure of the strength of modulation of 𝜇
by the signal. For a simple quartic potential
subject to a periodic driver such as:
𝑈(𝑧, 𝑡) = −
𝑎
2
𝑧2
+
𝑏
4
𝑧4
− 𝜀𝑧 cos 𝜔𝑠 𝑡. (31)
Where 𝜀 and 𝜔𝑠 are the amplitude and
frequency of the signal. The minima at ±c
are at ±√
𝑎
𝑏
when ε=0 and the height of the
potential barrier also at ε=0, is 𝑈0 =
𝑎2
4𝑏
. In
the absence of modulation (ε=0) and subject
to random forcing (in the case Gaussian
white noise) the mean first passage time is
given by the Kramer time:
𝑟𝑘 =
√2𝜋
𝑎
𝑒
2𝑈0
𝐷 (32)
The Kramers time formula is derived under
the assumptions that the probability density
of a particles position within a well is
Gaussian distributed centred about the
minima ±c, thus in order to use a Kramer
time approximation the driving oscillatory
frequency must be slower than the
characteristic rate for probability to
equilibrate within a well. This rate is given
by the curvature of the well at the minimum
i.e. 𝑈′′
(±𝑐). Thus the adiabatic
approximation is only valid under the
condition:
𝜔𝑠 ≪ 𝑈′′(±𝑐) = 2𝑎. (33)
A synchronisation between the noise
amplitude and the frequency of the
oscillatory driving can be achieved. . This
synchronisation occurs when the average
time between transitions 𝑇(𝜂) =
1
𝑟 𝑘
is
comparable to half the period 𝑇𝛺 of the
periodic forcing:
2𝑇(𝜂) = 𝑇𝛺. (34)

12. 12
Stochastic resonance like behaviour has
been observed in fusion plasmas when
subjected to external forcing. Figure 4 shows
power spectra data for different magnitudes
of magnetic perturbations driving large scale
instabilities in the fusion plasma.
Figure.4. The power spectra for different magnetic
pertubations in a fusion plasma. The stochastic
resonance spike is clearly visible as is the general
Lorentzian shape.
1.8 Numerical Methods
The problem at hand is a stochastic
differential equation (SDE). A typical SDE
is expressed as
𝑑𝑧 𝑡
𝑑𝑡
= 𝑓(𝑧𝑡, 𝑡)𝑑𝑡 + 𝑔(𝑧𝑡, 𝑡)𝑑𝑤𝑡 (35)
where 𝑧𝑡 is the realisation of a stochastic
process, 𝑓(𝑧𝑡, 𝑡) is the deterministic part of
the SDE characterising the local trend,
𝑔(𝑧𝑡, 𝑡) denotes the stochastic part which
influences the average size of fluctuations of
𝑧𝑡. The fluctuations originate from a
stochastic process 𝑑𝑤𝑡 which is called the
Wiener process. It is very hard or sometimes
impossible to deal with SDEs analytically
due to the non-differentiability of the
realisation of the Wiener process [18].
The stationary state Cahn-Hilliard equation
is a stochastic ordinary differential equation.
Numerically these are typically done with a
first order Euler type algorithm such as:
𝑦 𝑛+1 = 𝑦𝑛 + ℎ𝑓(𝑥 𝑛, 𝑦𝑛). (36)
Which advances the solution from 𝑥 𝑛 to
𝑥 𝑛+1 where 𝑥 𝑛+1=𝑥 𝑛 + ℎ. The formula
advances the solution through an interval h
but only uses derivative information at the
beginning of the interval. This means that
the step error is only one power of h smaller
than the corrections i.e O(h2
). The Euler
method is not recommended for practical use
due to this large step error when compared
to other techniques with the same step size
and also it is not very stable. Consider
instead a method that uses a step like above
to take a ‘trial’ step to the midpoint of the
interval, then use the value of both x and y at
the midpoint to compute ‘real’ step across
the whole interval.
By far the most commonly used method of
this type is the 4th
order Runga-Kutta
method. In equation form it is as follows:
𝑘1 = ℎ 𝑓(𝑥 𝑛, 𝑦𝑛) (37)
𝑘2 = ℎ𝑓(𝑥 𝑛 +
ℎ
2
, 𝑦𝑛 +
𝑘1
2
) (38)
𝑘3 = ℎ𝑓(𝑥 𝑛 +
ℎ
2
, 𝑦𝑛 +
𝑘2
2
) (39)

13. 13
𝑘4 = ℎ𝑓(𝑥 𝑛 + ℎ, 𝑦𝑛 + 𝑘3) (40)
𝑦 𝑛+1 = 𝑦𝑛 +
𝑘1
6
+
𝑘2
3
+
𝑘3
3
+
𝑘4
6
+ O(ℎ3
) (41)
The 4th
order Runge-Kutta method requires
four evaluations of the right hand side per
step h. This is superior to the midpoint
method if at least twice as large a step is
possible for the same accuracy. The 4th
order
method is an order of h more accurate than
the standard second order midpoint method.
Figure 5 shows a schematic representation of
the 4th
order method.
Figure.5. Schematic of the 4th
order Runga-Kutta
method. The derivative is evaluated four times for
each step at positions 1,2,3 and 4. The dotted line is
calculated from these derivatives.
The 1D temporal solutions are partial
differential equations with stochastic forcing
so a Runga-Kutta type algorithm will not
work. Instead a finite difference method is
required. The fundamental principle
involved in the finite difference method is to
replace the partial derivatives by
approximations obtained by the Taylor
expansions near the points of interest [19].
The 1D temporal solutions has two ∇2
terms
and a partial derivative with respect to time.
The finite difference method will be used to
deal with these terms.
Consider the equation:
𝜕𝑇
𝜕𝑡
= 𝛼
𝜕2 𝑇
𝜕𝑥2
(42)
The solution needs to be evaluated at every
grid point in the simulation for every time
step:
𝜕𝑇
𝜕𝑡
|𝑖 =
𝜕2 𝑇
𝜕𝑥2
|𝑖 (43)
The second derivative can be evaluated
using the standard finite difference
expression for second derivatives:
𝛼
𝜕2 𝑇
𝜕𝑥2
=
𝛼
(∆𝑥)2
(𝑇𝑖+1 − 2𝑇𝑖 + 𝑇𝑖−1). (44)
Where ∆𝑥 is the step size in the x direction.
The left hand side term is really:
𝜕𝑇
𝜕𝑡
|𝑖 =
𝜕𝑇(𝑥 𝑖,𝑡)
𝜕𝑡
=
𝜕𝑇 𝑖
𝜕𝑡
(45)
thus combining the two terms we obtain the
finite difference equation that will be used
when evaluating the 1D temporal solutions:
𝜕𝑇 𝑖
𝜕𝑡
=
𝛼
(∆𝑥)2
(𝑇𝑖+1 − 2𝑇𝑖 + 𝑇𝑖−1) (46)
The work undertaken will utilise these
numerical methods and assumptions made to
attempt to construct a simplified model of
zonal flow generation using a combination
of intuitive ideas of how to model the
underlying physical processes occurring and
some ideas from first principles.

14. 14
2. Methodology
The numerical methods undertaken will use
MATLAB and c++.
2.1 Stationary solutions of the
Cahn-Hilliard equation
The stationary solutions of the Cahn-Hilliard
equation will be investigated first. This
approach uses the 4th
order Runga-Kutta
method as outlined in section 1.8. Equation
(23) will be inserted into equations (37)-(41)
describing the Runga-Kutta method. Initially
the forcing used will be purely stochastic
driving in the form of white noise. The
white noise is generated using a random
number generator that adds a random
number to the solution at every step in the
simulation. The random number generator
produces uniformly distributed numbers in
the range of 0 to 1. In the case of stochastic
driving it is important to remember that in
the case of Gaussian white noise the
variance scales as 𝜎2
~𝑡 therefore 𝜎~√ 𝑡 or
in this stationary 1D model, 𝜎~√ℎ, where h
is the step size in the simulation. The
stochastic forcing term must have a √ℎ𝐷
factor in front when changing the length
scales of the numerical simulation. With just
the addition of the white noise the model
will be run to check that it does indeed lead
to what can be interpreted as ZFs generation
or in the case of the Cahn-Hilliard equation a
particle traversing between two potential
wells. All initial conditions are set to 0 and it
is imperative that the value of b from
equation (23) is positive to stop the solution
from becoming unstable.
2.2 Kramers Rate
Through trial and error a noise variance
level that causes no transitions will be found.
A loop will then be set up that varies the
noise variance level from a level that causes
no transitions to a level that causes
transitions at every step in the simulation.
When the loop is run for every noise value
the amount of transitions between potential
wells will be counted. The number of
transitions will then be plotted against noise
amplitude to give a fit for the Kramers rate.
An ensemble average will be taken for 10
realisations of the loop to get an error
estimate. The errors will be treated with the
standard mean absolute error treatment.
2.3 Stochastic Resonance
The phenomenon of stochastic resonance
will be investigated using the data obtained
from section 2.2. A further sinusoidal
driving term will be added to the stochastic
drive. The ideal frequency that the
sinusoidal driving term should be is known
from equation (34). When the frequency and
noise variance are set accordingly from
equation (34) the system is ran as usual. This

15. 15
time however using the MATLAB function
‘pwelch’ a power spectrum of the position of
the particle will be obtained. The power
spectrum is expected to show the
characteristic peak of stochastic resonance in
the low frequency regime of the power
spectrum. The Fourier transform of the
sinusoidal driving term will be added to the
plot to indicate where on the spectrum the
peak should be theoretically.
As has been stated it is theorised that driving
with coloured noise can replace driving with
the combination of white noise and a
sinusoidal term as long as the correlation
times are equal. For the purposes of this
stationary investigation it is more proper to
refer to correlation lengths. The coloured
noise is generated using the following
algorithm. Firstly Fourier decompose a
white noise signal into its frequency
components. The power at a frequency ν by
𝑃0 𝜈 𝛼
and then inverse transforming this into
a new signal. The correlation length of the
coloured noise signal is obtained by looking
at the length in which the power spectrum
drops by a factor of 1000. The value of α
will be varied until the correlation length is
the same as the white noise and periodic
driver and the location of the resonant peak
will be obtained and compared to see what
extent the combination can be replaced with
coloured noise.
Parameter optimisation will be undertaken
during this work to optimise the appearance
of the peak in the power spectrum. The
spectrums can appear very noisy so the
MATLAB function ‘smooth’ will be used to
improve the appearance of the peak also. It
should be noted however that this function
will reduce the amplitude of the peak so
although it is easier to see the location it
may appear to have a smaller relative
amplitude in comparison to the background
signal than in reality.
2.4 1D temporal solution
The temporal solutions will utilise the finite
difference method as has been outlined
previously. The solutions will be driven with
a combination of white noise and sinusoidal
driving as before. The white noise is once
again generated using a random number
generator. Periodic boundary conditions will
be used and the initial conditions will be
sinusoidal with the addition of a noise like
term. A domain size of 40 will be used with
spatial steps of dz=0.1. It is important that
the spatial resolution is good enough to
resolve the expected feature of the solution.
It is equally important for the simulation to
contain enough time steps to fully evolve
and reveal information about the transient
behaviour as well as the final stationary
behaviour. 1x106
time steps will be used
and the time step will be 1×10-6
.

16. 16
The system will be driven with wave vectors
of different magnitude to investigate
whether or not the system can evolve to a
stationary zonal flow like state regardless of
the driving wave vector amplitude and also
how quickly the solution becomes stationary
for different wave vectors.
3. Results
The model was firstly run with just the
stochastic driving. It was run with various
strengths of noise to investigate its
behaviour depending on the noise variance.
Figures 6-9 show the probability
distribution functions (PDFs) of position for
increasing the noise amplitude respectively.
A step size of h=0.0005 was used and
constants of a=2 and b=1 were used. The
noise variance values used were as follows:
D=0.001, D=0.5, D=1.4 and finally D=20
with respect to figures 6-9.
Figure.6. The probability distribution function of
position for a noise variance of D=0.001.
Figure.7. The probability distribution function of
position for a noise variance of D=0.5.
Figure.8. The probability distribution function of
position for a noise variance of D=1.4
Figure.9. The probability distribution function of
position for a noise variance of D=20
The position as a function of ‘time’ for
figure 8 is shown in figure 10. It should be
noted that this is not time in the true sense

17. 17
but rather as the system updated through
each step.
Figure.10. Position against time for the case where
the particle spends a random amount of time in each
well before switching with D=1.4.
The Kramer rate was determined by varying
the noise variance amplitude and counting
the number of transitions between wells. The
constants were given the values from before
and the noise was varied from D=0.03 to
D=4783. Figure 11 shows how the
transition frequency varies with the noise
amplitude. The non-linear fit gives a
Kramers rate of:
𝑟𝑘 = 𝑟0 + 𝐴𝑒 𝑅𝐷
. (47)
Figure.11. The transition frequency plotted against
noise amplitude when calculating the Kramer rate.
The fit is a non-linear curve fit.
Where 𝑟0=18±1, A=-16±1 and R=-
0.010±0.004 are constants.
The phenomena of stochastic resonance was
then investigated. A sinusoidal driving term
was added with a frequency 𝜔𝑠=6. The noise
variance amplitude selected was D=1.4.
Figure 12 and 13 show the power spectral
density of position for both the raw data and
smoothed data respectively.
Figure.12. The PSD for a white noise and sinusoidal
driving on a loglog scale.
Figure.13. The smoothed PSD for a white noise and
sinusoidal driving on a loglog scale.
The correlation time for the white noise and
sinusoidal driving was τc=7.861 and the
frequency of the resonant peak was
fr=0.0312.

18. 18
v(z)
Position, z
The procedure was then repeated for a
coloured noise driver. The value determining
the correlation was set to α=3.92 after some
trail and error. This gave a correlation time
for coloured noise of τc=7.862. Figure 14
shows the smoothed power spectrum density
for the coloured noise driving.
Figure.14. The smoothed PSD for a coloured noise
driving term.
The coloured noise driving gave a resonant
peak at fr=0.0319.
The 1D temporal solutions were then found.
The constants were prescribed the following
values: a=1, b=1, γ=0.5 and D=1. The
amplitude of the driving noise remains as it
was the stochastic resonance driving. The
system was first driven with a small wave
vector value of k=
2𝜋
25
. Figure 15 shows the
velocity as a function of position for this k
driving. With the same constants the system
was also driven with a wave vector of k=
2𝜋
4
.
Figure 16 shows velocity as a function of
position for this driving.
Figure.15. Velocity as a function of position for a
driving of k=
2𝜋
25
.
Figure.16. Velocity as a function of position for a
driving of k=
2𝜋
4
.
4. Discussion
After the construction of the model for the
1D stationary solution were complete it was
imperative to test whether it performed as
expected. Figures 6-9 show how the system
behaves when subject to just stochastic
forcing with different noise amplitudes.
Some trial and error was required to find
suitable amplitudes for demonstrating some
key features of the model. Figure 6 shows
the effect of driving with a D=0.001 noise
amplitude. It can be seen from the PDF that
Position, z
v(z)

19. 19
the particle undergoes no transitions
between wells and is fully confined. Another
feature is the well like structure at the top of
the PDF which is a nice representation of the
particle moving around the equilibrium
point, +c, due to the stochastic forcing. The
lack of transitions in this case can physically
be related to turbulent flows that do not
reach a large enough size to cause the
formation of ZFs. Figure 7 has a driving of
D=0.5 and shows a very small number of
transitions in the PDF. This can be attributed
to turbulent flows beginning to reach the
required scales that can cause large-scale
shear flow formation. The bi-modal PDF of
figure 8 is the key result in showing the
model works as expected. The bi-modal
PDF along with the evidence from figure 10
show the particle undergoes a transition
between the wells, spends a random amount
of time in that well and then transitions
back. This suggests the driving noise
amplitude of D=1.4 is ideal for the
parameters used when representing the
turbulent flows that are responsible for ZF
generation. The PDF in figure 9 shows that
when the noise amplitude is increased to a
large value, here D=20, the particle will
transition at virtually every time step rather
than spending a period in one well before
transitioning again. This is clearly not what
is required for this work so it is an important
fact to uncover that the driving noise does
have upper limits on what can be useful
physically. This work could be improved
further by more exploration into which
values are deemed critical i.e. at what value
the first transition will occur and which
value it can be assumed transitions occur at
every time step.
An expression for the Kramer rate was then
found by counting transitions when the noise
amplitude was raised through a series of
values. Figure 11 shows the data collected
and equation (47) the Kramer rate
determined. There are two successes to the
information achieved. The theory outlined in
section 1.7 suggest the rate of transitions
should grow rapidly with noise amplitude to
begin with before slowing down and then
remaining constant as the potential barrier
becomes easy to surmount i.e. transitions
occur at every time step. The data in figure
11 clearly agrees with this and also although
the expression for the Kramer rate obtained
is of a different functional form to that in
equation (47) they do give the same shape
when compared. The error estimates
obtained by the averaging procedure appear
to have been sufficient for capturing the
variability in running the programme as the
errors on values seem to be of realistic order
and coupled to the seemingly reliable nature
of data, it can be taken as an adequate
method.
The power spectrum shown in figures 12
and 13 for a white noise and sinusoidal

20. 20
forcing display the key evidence for the
phenomenon of stochastic resonance
occurring. Although the data is figure 12 is
very noisy especially in the high frequency
regime this is expected due to the stochastic
driving, however the peak is still visibly
apparent. The peak appears much more
clearly in figure 13 after the smooth function
has been applied to the data. It can be seen
the peak has lost some amplitude as
expected but the SNR is greatly improved.
The agreement with the Fourier transform of
the sinusoidal term and the location of the
peak is excellent. This agrees with the theory
outlined in section 1.7 and the delta function
component of equation (29). A potentially
worrying feature of figures 12 and 13 is the
appearance of what could be taken for a
second peak. At this moment the cause of
this second peak is unknown although it has
been theorised that it could be a second
harmonic type phenomena occurring.
When driving with just coloured noise as is
the case in figure 14 it is clear to see that the
replacement of white noise with an
additional periodic forcing term with
coloured noise is a viable option provided
the correlation times are equal. The two τc
are not identically equal however at τc=7.861
and τc=7.862 it can be seen by comparing
figures 13 and 14 and the two values
obtained for the peak location, fr=0.0312 and
fr=0.0319, this small difference in τc is
relevant and it correct to say the stochastic
and periodic forcing can be replaced with
coloured noise. This is an important result as
it suggests that during the 1D temporal
solutions the behaviour obtained with the
stochastic and periodic driving will be
reproduced if just a coloured noised driving
was used.
The ability to obtain successful stochastic
resonance data is very important for the 1D
temporal solutions. The fact that given a
particular driving frequency and noise
amplitude leads to successful stochastic
resonance peaks is a mathematical statement
that given enough transient time the 1D
temporal solution will go to a stationary
state with a ZF like behaviour. The fact that
the resonance peak appear at low frequency
indicate that the switching in the 1D
temporal model will occur when it is driven
with low k wavenumbers.
The 1D temporal solutions were then found.
Firstly it was driven with a wave vector of
𝑘 =
2𝜋
25
as shown in figure 15. Comparing
this with the theoretical solution in figure…
there is good agreement of the functional
form. It can therefore be concluded that
when driving with this low k wave vector
after a certain transient period a stationary
system has been reached that has suppressed
the stochastic driving term and a reached a
smooth kink like solution that represents ZF

21. 21
generation. It should be remembered that the
stochastic and periodic forcing like that of
the initial conditions are added at every time
step. In light of this a remarkable feature can
be understood whereby the non-linearity of
the system is suppressing the stochastic
drive and establishing a stationary solution.
Driving with a wave vector of k=
2𝜋
4
gave
some very interesting dynamics as can be
seen in figure 16. As the time progresses the
solution decays away further and further.
This is clear indication that the driving has
to be in smaller wave vector regime if the
solution is to reach a satisfactory stationary
solution. This can be understood by
considering the amount of power inputted
from the forcing term. When it is driven by a
high wave vector regime there is not enough
power in the low wave vector scale to allow
the larger scales to grow. The diffusion term
in the right hand side of equation (19) is a
negative diffusivity term that should cause
things to coalesce into larger structures such
as ZFs. However when driving with large
wave vectors this flux cannot achieve large
enough amplitudes to create this negative
diffusion. That is to say the dissipation term
in the flux dominates the non-linear and
forcing terms causing the solutions to decay
away.
The solutions achieved with this 1D
temporal system have been successful in
replicating what is expected. It is known that
the driving has to be with a small wave
vector magnitude to establish stationary ZFs
and that driving with larger wave vectors
will lead to a decaying solution. Given more
time this work could be taken much further.
Time to optimise the parameters would be
ideal as although the figures give what is
expected they are not of the quality that
could be achieved through parameter
optimisation. A full investigation using a
wide range of k values would be able to
identify the cut off for which k vectors lead
to ZF generation and which lead to a
decaying solution as well as investigating
the time it takes for the stationary solution to
emerge for those k values that do lead to ZF
generation. If simulation time had not been
so limited a smaller spatial step would be
utilised to better resolve the features of the
graph.
5. Conclusions
The aim of this work was to produce a
simplistic model of ZF generation in fusion
plasmas. The suggested method involved
modelling the small scale micro-instabilities
that drive turbulent flows in plasma with
stochastic driving. This is suggested due to
the fast time scales involved in both
processes. It has been observed that ZFs can
couple was an acoustic GAM mode in the
plasmas. This coupling is equivalent to an
additional forcing on the plasma which is

22. 22
modelled with a sinusoidal driving term. The
model has been tested and proved to
successfully mimic ZF generation in the
simplified case that has been considered
both in a 1D stationary and 1D temporal
scenario. The model consistently matches
with theoretical predictions about the
underlying physics in the fusion plasmas
suggesting the simplifications have captured
the essence of the physics at hand. The work
could be furthered by first completing the
study in the 1D temporal case by extending
it to include coloured noise and investigate
the transient time dependence on the driving
wave vector before an extension into 2D is
performed. This will verify the model
performs as it should when another level of
complexity is introduced.
References
[1] K. Miyamoto, Plasma physics for nuclear
fusion, MIT press (1989).
[2] A.P.L. Newton, E. Kim, and H. Liu,
Phys.Plasmas 20, (2013).
[3] B. McNamara, K. Wiesenfeld, Physical
review A 39, 4854-4869 (1989).
[4] C. Braams, P. Scott, Nuclear Fusion:
Half a Century of Magnetic Confinement
Fusion Research, CRC press (2002).
[5] J.Mathieu, J. Scott, An Introduction to
Turbulent Flows, CUP (2000).
[6] H. Tennekes, J. Lumley, A First Course
in Turbulence, MIT press (1972).
[7] U. Frisch, P.L.Sulem, PhysFluids 8
(1984).
[8] C.R. Doering, J.D.Gibbon, Applied
Analysis of the Navier-Stokes equation,
CUP (1995).
[9] J. Kim, Turbulent Electron Thermal
Transport in Fusion Plasmas, Proquest
(2008).
[10] G.D.Conway, Plasma Phys. Control.
Fusion 50 (2008).
[11] J. Robinson et al., Plasma Phys.
Control. Fusion 54 (2012).
[12] R.Kraichan, Journal of the Atmospheric
Sciences 33 p.1521-1536 (1976).
[13] [7] M. Lesieur, Large Eddy Simulations
of Turbulence, CUP (2005).
[14] S.Gama et al., Fluid. Mech. 260 p.95-
126 (1994).
[15] W.C. Van Etten, Introduction to
Random Signals and Noise, J.Wiley and
Sons (2009).
[16] R. Benzi et al., Tellus 34, 10-16 (1982).
[17] L. Gammaitoni et al, Review of Modern
Physics 70 p.223-283 (1998).

23. 23
[18] W.H. Press, B.P. Flannery, Numerical
Recipes: The Art of Scientific Programming,
CUP (2007).
[19] R. J. LeVeque, Finite Difference
Methods for Ordinary and Partial
Differential Equations:- Steady State and
Time-Dependent Problems, SIAM (2006).