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- 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).
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