Hermite Closures for Numerical Simulations of Kinetic
Turbulence in Strongly Magnetized Plasmas
Maria Luı́sa da Silva Vilelas
Thesis to obtain the Master of Science Degree in
Engineering Physics
Examination Committee
Chairperson:
Supervisor:
Member of the Committee:
Dr. Horácio João Matos Fernandes
Dr. Nuno Filipe Gomes Loureiro
Dr. Vasco António Dinis Leitão Guerra
November 2013
Acknowledgements
To Dr. Nuno Loureiro and to my mother.
iii
Resumo
Quando se está na presença de um plasma pouco colisional, não se pode descrever o plasma como um fluı́do
condutor (teoria MHD) e outra descrição é necessária: a descrição cinética.
Em certos casos, é possı́vel reduzir o tamanho e complexidade do formalismo cinético e encontrar modelos
cinéticos reduzidos. Nesta tese de mestrado usa-se um modelo cinético reduzido que é válido quando se está
na presença de plasmas fortemente magnetizados e com um β pequeno, onde β é o rácio das pressões do
plasma e magnética e é da ordem do rácio das massas do electrão e ião, me /mi . A utilização de β pequeno
e da forte magnetização do plasma como parâmetros de expansão assimptótica permite a simplificação da
descrição cinética geral (6D) para um modelo reduzido com apenas quatro dimensões.
A equação cinética pode ser representada através de polinómios de Hermite, transformando-a num sistema
acoplado de equações do tipo fluı́do, sendo mais simples de resolver numericamente. Contudo, a formulação
de Hermite introduz uma dificuldade: a equação para o momento de ordem M depende do momento de
ordem M + 1, originando um problema de truncatura das equações.
Nesta tese de mestrado testam-se diferentes tipos de truncaturas possı́veis (polinomiais, linear e não
linear). Comparam-se também dois operadores de dissipação, o operador colisional de Lénard-Bernstein e o
operador de hiper-colisões, de onde se conclui que com o segundo existe uma convergência mais acentuada,
fazendo com que a dinâmica do sistema se capture com um reduzido número de momentos.
Palavras-chave: Plasmas fortemente magnetizados, plasmas com beta pequeno, turbulência, métodos
numéricos e analı́ticos, truncaturas de Hermite, Kinetic Reduced Electron Heating Model.
v
Abstract
When we are in the presence of a weakly collisional plasma, the plasma cannot be described as a conducting
fluid (MHD theory) and another theory is needed: the kinetic theory.
In certain cases, it is possible to reduce the size and the complexity of the kinetic formalism in order to
find reduced kinetic models. In this master thesis, a reduced kinetic model which is valid when we are in the
presence of strongly magnetized and low-β plasmas is used, where β is the ratio of the plasma pressure to the
magnetic pressure and is ordered similiar to the electron-ion mass ratio, me /mi . The use of the low-β and
the strong magnetization as asymptotic expansion parameters allows the simplification of the six-dimensional
kinetic theory to a four-dimensional reduced model.
The kinetic equation can be presented using Hermite polynomials, transforming it in a set of coupled fluidlike equations, much easier to solve numerically. However, the Hermite formulation introduces a difficulty:
the equation for the M th Hermite moment depends on the (M + 1)st moment, introducing a problem which
concerns the closure of the equations.
In this master thesis different possible closures (polynomial, linear and nonlinear) are tested. Two dissipation operators are also compared, the Lénard-Bernstein collision operator and the hyper-collisions operator,
from which it is possible to conclude that the latter converges in a more accentuated way, allowing the capture
of the system dynamics using a reduced number of moments.
Keywords: Strongly magnetized plasmas, low-beta plasmas, turbulence, numerical and analytical
methods, Hermite closures, Kinetic Reduced Electron Heating Model.
vii
Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Contents
ix
List of Tables
xi
List of Figures
xiii
1 Introduction
1
1.1
Fluid Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Hermite Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Theoretical Formulation
9
2.1
Gyrokinetic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Low-beta ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.1
Spatial scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.2
Time scales and perturbation amplitudes . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.3
Alfvénic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.4
Resistivity and collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
The Kinetic Reduced Electron Heating Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.5
Hermite Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3 Closures
25
3.1
Polynomial closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.2
Linear closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3
Nonlinear closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
ix
4 Linear Tests
29
4.1
The isothermal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.2
Closure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.2.1
The Lénard-Bernstein Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.2.2
The Hyper-Collisions Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.2.2.1
Closure gM +1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2.2.2
Closure gM +1 = gM −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.2.2.3
Closure gM +1 = gM + gM −1 − gM −2 . . . . . . . . . . . . . . . . . . . . . . .
39
4.2.2.4
Closure gM +1 = gM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2.2.5
Closure gM +1 = 2gM − gM −1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.2.2.6
Closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 . . . . . . . . . . . . . . . .
p
Closure gM +1 = − M/ (M + 1)gM −1 . . . . . . . . . . . . . . . . . . . . . .
√
Closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM . . . . . . . . . . . . . .
43
4.2.2.7
4.2.2.8
44
45
5 Conclusions
51
Bibliography
53
x
List of Tables
4.1
Normalized values for the plasma parameters for the Lénard-Bernstein collision operator. . . . .
33
4.2
Normalized values used for the plasma parameters for the hyper-collisions operator. . . . . . . .
37
xi
List of Figures
1.1
Energy cascade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Spectral representation of the energy cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Infinite set of masses coupled together by springs . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1
Illustration of the guiding center coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Graphical representation of the dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3
Graphical representation of the damping as function of the frequency for equation (2.59). . . . .
18
2.4
Graphical representation of the Hermite spectrum Em as function of m. . . . . . . . . . . . . . .
22
2.5
Graphical representation of the Hermite spectrum Em as function of m for different values of
νei τA and for the Lénard-Bernstein collision operator. . . . . . . . . . . . . . . . . . . . . . . . .
23
4.1
Graphical representation of the isothermal closure. . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.2
Error as function of M for the closure gM +1 = 0 with the Lénard-Bernstein collision operator. . .
33
4.3
Error as function of M for the closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 with the Lénard34
4.4
Bernstein collision operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
Error as function of M for the closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM with the
Lénard-Bernstein collision operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.5
4.6
Graphical representation of the damping as function of the frequency for gM +1 = 0 and gM +1 =
√
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the Lénard-Bernestein collision operator. . 35
√
Graphical representation of the damping as function of the frequency for gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
at k⊥ = 2/de for the Lénard-Bernestein collision operator using M = 9 and M = 19. . . . . . . .
35
4.7
Graphical representation of the dispersion relation with the closure gM +1 = 0 and M = 15. . . .
38
4.8
Error as function of M for the closure gM +1 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.9
Graphical representation of the dispersion relation with the closure gM +1 = gM −1 and M = 15. .
39
4.10 Error as function of M for the closure gM +1 = gM −1 . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.11 Graphical representation of the dispersion relation for the closure gM +1 = gM + gM −1 − gM −2 and
M = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.12 Error as function of M for the closure gM +1 = gM + gM −1 − gM −2 . . . . . . . . . . . . . . . . . .
41
xiii
4.13 Graphical representation of the dispersion relation for the closure gM +1 = gM and M = 15. . . .
42
4.14 Error as function of M for the closure gM +1 = gM . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.15 Graphical representation of the dispersion relation for the closure gM +1 = 2gM − gM −1 and M = 15. 43
4.16 Error as function of M for the closure gM +1 = 2gM − gM −1 . . . . . . . . . . . . . . . . . . . . . .
44
4.17 Graphical representation of the dispersion relation for the closure gM +1 = 4gM −6gM −1 +4gM −2 −
gM −3 and M = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.18 Error as function of M for the closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 . . . . . . . . . . .
p
4.19 Graphical representation of the dispersion relation for the closure gM +1 = − M/ (M + 1)gM −1
45
and M = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
p
4.20 Error as function of M for the closure gM +1 = − M/ (M + 1)gM −1 . . . . . . . . . . . . . . . . 46
√
4.21 Graphical representation of the dispersion relation for the closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
and M = 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
4.22 Error as function of M for the closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM . . . . . . . . .
47
47
4.23 Graphical representation of the damping as function of the frequency for gM +1 = 0 and gM +1 =
√
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the hyper-collisions operator. . . . . . . . . 48
√
4.24 Graphical representation of the damping as function of the frequency for gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
at k⊥ = 2/de for the hyper-collisions collision operator using M = 9 and M = 19. . . . . . . . . .
xiv
49
Chapter 1
Introduction
Plasma is a ubiquitous form of matter in the Universe. It is nearly always found to be magnetized and
turbulent. Very interesting examples include turbulence in the interstellar medium [1], which is stirred by
violent events like supernovae explosions; turbulence in accretion flows around black holes [1]; turbulence
in the solar wind streaming outward from the Sun [1]; and turbulence in nuclear fusion experiments in
laboratories [2]. Put simply, we can say that we are dealing with a turbulent system if chaotic fluctuations of
some field(s) over a broad range of scales are detected. In a plasma, these fluctuating fields are the electric
and magnetic fields and the distribution functions of the particles. So, turbulence is multiscale disorder [3].
The comprehension of the dynamics of turbulent systems remains one of the greatest problems in theoretical
physics. Indeed, Nobel Laureate Richard Feynman described turbulence as the most important unsolved
problem of classical physics[4].
At macroscopic scales, magnetized plasma turbulence is well described by the magnetohydrodynamics
(MHD) theory, where the plasma is described as an electrically conducting fluid. Within the context of
MHD, there has been significant progress in understanding how eddies (an eddy is the swirling of a fluid) at
large scales break up into smaller and smaller ones, but important questions remain regarding the nature of
turbulent fluctuations at scales smaller than a collisional mean free path. At these scales, fluid theories such
as MHD cannot be used to describe the dynamics of the systems and a alternative approach must be used:
the kinetic theory.
Kinetic theory is used when the plasma is weakly collisional, that is, the collision frequency between the
particles of the plasma is much smaller than the characteristic frequencies of the turbulence and when the
collisional mean free path is much larger than the characteristic spatial scales. Kinetic theory takes into
account the motions of all of the particles. The particles are thought as point particles, each with a given
charge and mass where each one of them has an orbit Xi (t) in the three-dimensional configuration space
x. Likewise, each particle has an orbit Vi (t) in the three-dimensional velocity space v. Combining the
three-dimensional configuration space x and the three-dimensional velocity space v, it is possible to get a
6N -dimensional phase space (x, v) where all the possible trajectories of the N particles in the system are
1
represented.
We could follow all of the particle orbits but it is far too detailed for any practical purpose [5]. So,
instead, we follow the species distribution function fs (x, v, t), which can be thought of as the ensemble
averaged number of point particles per unit six-dimensional phase space. The evolution of the distribution
function fs (x, v, t) is described by the Vlasov equation [5].
The kinetic representation of a plasma is naturally much more complex that the fluid representation.
This means that, whilst at macroscopic scales the understanding of turbulence remains elusive, in weakly
collisional plasmas, it is still very incipient. Given the analytical difficulties of the mathematical treatment
of turbulence, one tool that is extremely useful to clarify the dynamics of turbulent plasmas is the direct
numerical simulation.
Recall kinetic theory. For each species, we would have to solve a six-dimensional system of equations
coupled with the Maxwell’s equations, which involves a wide range of temporal and spatial scales. In spite
of the several recent advances in the available computational power, the direct numerical simulation of this
problem remains a computational challenge. However, in certain cases, it is possible to reduce the size
and complexity of the problem, which leads to reduced kinetic models. One example is the gyrokinetic
representation of strongly magnetized plasmas.
Gyrokinetics is a rigorous low-frequency anisotropic limit of kinetic theory [6, 7, 8, 9], which systematically averages out the cyclotron motion of particles about the magnetic field, resulting in a five-dimensional
equation describing the drift motion of charged rings.
The direct approach of solving the gyrokinetic equation remains computationally challenging as well.
However, with the constant improvements in computational power and algorithms, there has been an increasing amount of work employing this approach, leading to a better understanding of turbulence in strongly
magnetized plasmas [10, 11].
In this thesis, a reduced kinetic model (Kinetic Reduced Electron Heating Model [12]) will be used, which
is valid for strongly magnetized plasmas with a low-β, where β is the ratio of the plasma pressure to the
magnetic pressure, β = 8πP/B 2 . For example, the solar corona is a plasma with these characteristics [13].
Using the low-β and the strong magnetization of the plasma as asymptotic expansion parameters allows
the simplification of the six-dimensional kinetic theory to a four-dimensional reduced model (the threedimensional configuration space plus the component parallel to the magnetic field of the velocity vector).
This reduced kinetic model is naturally much simpler than the six-dimensional kinetic description, allowing
computational studies that in the general case are prohibitive.
1.1
Fluid Turbulence
When the plasma is described as a fluid, a simple extension of Kolmogorov’s 1941 theory (K41) [7, 14, 15] is
impossible due to the fact that two of the key assumptions of this theory (isotropy and locality of interactions)
are incorrect for the MHD turbulence, being this one described by the Goldreich-Sridhar (GS) theory [1, 16].
2
When there is an injection of energy into the system, that energy is dissipated into heat. Since the energy
injection happens at larger scales and the energy dissipation happens at much smaller ones, there will be
a transference of energy from larger to smaller scales which occurs in a range known as the inertial range,
which is believed to be present for all turbulent systems. To better understand it, consider the motion of an
incompressible fluid, which is described by the Navier-Stokes equation:
∂u
+ u · ∇u = −∇p + ν∇2 u + f ,
∂t
(1.1)
where p is the pressure determined from the incompressibility, ν is the viscosity due to material properties
of the fluid (assumed constant for simplicity), f is an external forcing and u is the characteristic velocity.
Here, the external forcing is a stand in for whatever makes the fluid move, i.e., energy injection. In
physical systems, turbulence is stirred up by some source of energy, which can be more complicated than a
simple body force: energy can come from background gradients, shear, etc (i.e., it is converted from some
external field, such as, for example, gravity), mechanical movement (e.g., spoon in a tea cup) or boundary
conditions (e.g., airplane wing). Now we can ask, what do all these energy injection mechanisms have in
common?
Consider the parameters of the system: the characteristic velocity u, the characteristic scale L and the
viscosity ν. From these parameters, it is possible to find a dimensionless number,
Re =
uL
,
ν
(1.2)
which is the Reynolds number. Note that
Re =
uL
u/L
fluid motion rate
=
=
ν
ν/L2
viscous dissipation rate
|u · ∇u|
nonlinear term
∼
=
,
2
|ν∇ u|
viscous term
which is how the Reynolds number appears. When Re is small, the flow is linear. However, when we have
Re 1, the nonlinear term is important and we are in the regime of fully developed turbulence. In this
regime, u is very irregular in space and time-fluctuating at each point around its mean value U and varying
rapidly in space. So, we have
u = U + δu,
(1.3)
where U is the mean value of u and δu is its fluctuating part.
The energy goes into fluctuations at the system scale L, also known as the outer scale or energy-containing
scale. At this scale, fluctuations are the same order as the change in the mean flow (δuL ∼ δu) and we can
redefine:
Re =
δuL L
.
ν
3
(1.4)
What happens to the energy E =
1
2
R
d3 r|u|2 ? From the Navier-Stokes equation, equation (1.1), and
knowing that div u = 0 for an incompressible fluid, we arrive at
Z
Z
dE
= − ν d3 r|∇u|2 + d3 r u · f ,
dt
|
{z
} | {z }
1
(1.5)
2
where 1 is the dissipation term and 2 is the energy injection term. If we consider now the stationary state,
both dissipation and energy injection terms on the right-hand side of equation (1.5) must balance. The term
represented by 2 is finite and the viscosity is small, that is, the viscous term in equation (1.1) is neglegible at
the outer scale. The balance is accomplished by transferring kinetic energy to small scales, when the velocity
gradients must be large so they can compensate for the viscosity’s smallness.
Dimensionally, the viscous (inner) scale to which the energy has to travel in order to be dissipated is
3 1/4
ν
∼ LRe−3/4 ,
(1.6)
lν ∼
ε
where ε = δu3L /L is the injected power per volume and Re is given by equation (1.2). As was said before,
the system becomes turbulent when Re 1, i.e., lν L, which means that fluctuations arise over a broad
band of scales. This can be described with figure 1.1.
Scale
L
l
lν
Energy injected
Energy transported
Energy dissipated
Figure 1.1: Energy cascade.
For scales such that L l lν , we have the inertial range. The ability to make further progress lies on
the assumption that the physics of the inertial range is universal for all systems.
In 1922, Richardson [17] conjectured that the energy transfer is local in the scale space, that is, it occurs
via a cascade:
L → L/2 → L/4 → etc.
to the viscous scale. It is possible to build a spectral representation of this idea, which is described by figure
1.2.
Like in the fluids, the transference of energy to smaller scales is also present in the kinetic description
of turbulence. This transference of energy, which is described as the energy cascade, can be translated as
4
log (E (k))
k −5/3
energy
injected
energy cascaded
energy
dissipated
log (k)
1/L
inertial range
1/lν
Figure 1.2: Spectral representation of the energy cascade.
a kinetic cascade when we are speaking of kinetic description of turbulence. The kinetic cascade occurs in
phase space, reaching towards small scales both in physical space and velocity space. So, the inertial range
is also present in the velocity space because the emergence of small scales in velocity space is intertwined
with a cascade to small scales in physical space. Indeed, while things become more complicated than for the
Navier-Stokes equation (e.g., multiple fields and species), the basic principle remains the same: small-scale
structure is generated so that the energy injected at the outer scale can be transferred to the smaller scales
and converted into heat, thus achieving a statistical steady-state.
In most astrophysical contexts, the turbulence is driven in the plasma at scales much larger than the
Larmor radii, giving rise to a turbulent cascade through an inertial range in which the injected energy is
cascaded by nonlinear interactions to the dissipative scales at which the energy is converted into particle
thermal energy.
1.2
Hermite Formulation
An easy and intuitive mathematical tool to study kinetic turbulence is given by the Hermite formulation.
In the same way that the Fourier representation allows the spectral decomposition of kinetic turbulence in
the configuration space, the Hermite representation will do the same but in the velocity space. So, this
formulation is a very powerful tool in the comprehension of kinetic turbulence. Furthermore, it has the
advantage of transforming one kinetic equation into a set of coupled fluid-like equations, much easier to solve
numerically.
However, as we will see, the Hermite formulation introduces a difficulty: the equation for the mth Hermite
moment depends on the (m+1)st moment, introducing a problem which concerns the closure of the equations.
To better understand this difficulty, consider the Vlasov equation with only the convective term (neglect the
5
terms with the electromagnetic fields):
∂f
∂f
+v
= 0.
∂t
∂x
The exact solution for this equation subject to the initial condition f = fM (v) exp (ikx) is [2]
f = fM (v) exp [ik (x − vt)] ,
(1.7)
(1.8)
where fM is a Maxwellian. Note that as t → ∞, the solution gets progressively more oscillatory eikvt ;
this is the commonly known process of phase-mixing. Equation (1.7) can be arranged by using Hermite
polynomials and Fourier transforming in x [2]:
1
1
∂ 2 α`
+ ` (` − 1) α`−2 + ` +
α` + α`+2 = 0,
∂t2
2
4
(1.9)
which can take the form
∂ 2 x`
= k` (x`+1 − x` ) − k` (x` − x`−1 ) .
(1.10)
∂t2
This equation can be thought of a mechanical analog of equation (1.9) without the Fourier transform in x in
M`
terms of an infinite set of masses M` at positions x` (` ∈ N), each coupled to its nearest neighbors M`−1 and
M`+1 by springs with spring constant k` and k`+1 [2], which can be seen in figure 1.3.
k`−1 M`−1
k`
M`
k`+1 M`+1 k`+2
Figure 1.3: Infinite set of masses coupled together by springs, the mechanical analog of the Hermite formulation.
Now imagine an initial perturbation in the position x0 of the first mass in the mechanical system (this
is the same as an initial perturbation in the lowest moment α0 ). This perturbation produces a wave in the
coupled springs that propagates at high `. Taking Hermite moments of the exact solution, equation (1.8), it
is possible to get [2]
"
2
(kvt t)
α` (t) ∝ t` exp −
2
#
.
The peak of the wave reaches the `th moment, or the `/2 mass, at t =
(1.11)
√
`/(kvt ), where after that time
most of the wave’s energy will have propagated to higher `’s. Truncating the Hermite expansion at ` = L
corresponds to replacing the L/2 mass by something (e.g., if we set αL = 0, we are replacing the L/2 mass
by a wall, i.e., x`/2 = 0) that will be different from the initial system. So, one can ask what is the best way
to truncate the system, keeping its characteristics unchanged.
1.3
State of the art
Numerical simulations of plasmas allows us to study nonlinear effects in plasma physics. One way to do so
is to look for solutions of the Vlasov equation:
∂fs
∂fs
e
v × B ∂fs
+v·
+
E(x, t) +
·
= 0,
∂t
∂x
m
c
∂v
6
(1.12)
where fs , the distribution function of species S, lives in the phase space (x, v). Numerical simulations of fs
are not a trivial matter [18]. A commonly used numerical technique to perform a transformation of variables
is a Fourier transformation for the configuration space and a Hermite transformation for the velocity space
[18].
Grant and Feix (1967) [18] introduced the Hermite polynomials to solve the one-dimensional Vlasov
equation. Since the Hermite formulation transforms the initial equation into a set of coupled fluid-like
equations, to solve the closure condition problem, they introduced the closure gM +1 = 0, where gm is the
Hermite polynomial and (M + 1) is its highest moment. They found that this closure condition assumed that
the distribution function fs (x, v, t) was well behaved, which was not the fact. So, there was a time limit for
the validity of the use of this closure approximation. However, the time limit increased with M , meaning that
they could expect to follow the system accurately through sufficiently long times to exhibit all interesting
phenomena. They also discovered that introducing a small amount of collisions would damp the effect of
the neglected term and prevent the truncation errors from feeding back, which could eventually disrupt the
calculation, and still being able to capture the Landau damping. They also applied the Fourier-Hermite
representation for the case of the nonlinear cold plasma and the results indicated that this representation
could be used to study nonlinear plasmas.
In 1970, Joyce et al [19] reviewed two methods of integrating the nonlinear Vlasov equation, being one
of them the Hermite expansion. Using the gM +1 = 0 closure resulted in numerical instabilities and to solve
this problem, they were forced to solve a large number of Hermite polynomials. Due to computer limitations,
another method of truncating the system of equations that would require a smaller number of Hermite
polynomials was necessary and they believed that approximating the (M + 1)st coefficient by a polynomial
extrapolation was a good option. This approach worked very well for the linear Vlasov equation, meaning
that the difficulty of the closure condition could be avoid, contrarily to what Grant and Feix [18] believed.
Concerning the nonlinear Vlasov equation, they found out that it depended on the case treated but the
truncation would always introduce instabilities. They also shown that the numerical instabilities were based
on approximating a continuous eigenvalue spectrum by a discrete spectrum.
With the development of gyrofluid theory, Hammett et al (1993) [2] reintroduced the Hermite formulation
and the closure approximations problem. Gyrofluid equations attempt to extend the range of validity of fluid
equations to a more collisionless regime of tokamaks by developing fluid moment models of important kinetic
effects such as Landau damping. In 1990, Hammet and Perkins [20] introduced a linear closure approximation
to close fluid moment equations that successfully modeled kinetic resonances, such as Landau damping, in
one dimension and in slab geometry.
The gyrofluid equations provide the dynamics of a few moments (typically 4 to 6 moments) of the gyrokinetic equation. Nevertheless, a finite set of fluid moment equations represents an approximation that
breaks down in some regimes, and must be handled with care. As pointed out by Hammett et al in 1992
[21], some authors believed that fluid equations were inherently unable to model Landau damping or other
7
resonance effects because of the closure problem and should be abandoned in favor of a fully kinetic treatment
if those effects were important in the problem that was being studied. On the other hand, some other authors
suggested ways to model Landau damping by adding dissipative terms to the fluid equations. In the end,
Hammett et al [21] were able to show that the gyrofluid equations agreed very well with the kinetic results.
More recently, Ng et al (1999) [22] used the Hermite formulation to study the Landau damping in weakly
collisional plasmas. Concerning the closure problem, they matched backward and forward recurrence relations
for every eigenvalue, in order to avoid numerical instabilities during the iteration.
In 2013, Bratanov et al [23] examined in different scenarios the linearized problem of electrostatic plasma
oscillations. For the collisional system, they compared between different discretization schemes and realized
that the Hermite representation is much superior than a finite differences scheme using a equidistant velocity
grid, although the latter is quite common in most numerical simulations. With the Hermite formulation,
they produced more matching of Landau modes than the finite differences scheme, meaning that for the
latter, a large number of nonphysical modes appeared, which in extreme cases, were less damped than the
least damped Landau solutions, difficulting the analysis if the latter were unknown; for the Hermite case,
all nonphysical modes presented were more strongly damped than the physical Landau solutions. They also
discovered that increasing the collisional frequency led to a bigger improvement in the case of the Hermite
formulation, which was not the case for the finite differences scheme on an equidistant velocity grid. Bratanov
et al [23] also studied the role of hyperdiffusivity terms. These are used for purely numerical reasons but they
have shown that those terms reproduce correctly important physical affects, such as the Landau damping.
The main topic of this thesis is the analysis of different closing schemes for the Hermite hierarchy of
equations. Firstly, it is necessary to check if the chosen closure does not affect the linear physics of these
equations, namely, if it captures the Landau damping in a correct way. An ideal closure should be able to
capture the Landau damping rates with as few Hermite moments as possible. Finding a satisfying closure
is a fundamental step in the applications of this physical model in the study of the turbulence in strongly
magnetized plasmas. Different closure conditions are presented: polynomial, linear and nonlinear. With
the nonlinear closure, convergence is guaranteed if M → ∞. Besides that, two collision operators are also
compared: the Lénard-Bernstein collision operator and the hyper-collisions operator.
1.4
Outline
This thesis is organized as follows. In chapter 2 the gyrokinetic equation is introduced, as well as the set of
ordering assumptions that encode the physical limit in which the used model holds rigorously. Our kinetic
reduced model is introduced and its linearized form too. In the end of the chapter there is a section about
the Hermite formulation. In chapter 3, the studied closures are presented. We study three types of closures:
polynomial, linear and nonlinear. In chapter 4, the dynamics of the system using the different closures and
different dissipation operators (Lénard-Bernstein and hyper-collisions) is studied. It is introduced how the
linearization of the set of equations is done as well. Finally, in chapter 5 some conclusions are presented.
8
Chapter 2
Theoretical Formulation
In this chapter, we will present the Kinetic Reduced Electron Heating Model (KREHM) [12]. The starting
point is the gyrokinetic description of magnetized plasmas. After introducing the set of ordering assumptions
that contain the physical limit in which the model holds rigorously, the reduced model will be presented. Its
linear theory is presented as well. After that, we introduce a spectral representation of the electron kinetics in
terms of Hermite polynomials. This provides both a very simple computational approach and an intuitively
appealing physical interpretation of velocity-space dynamics as a cascade in Hermite space.
2.1
Gyrokinetic Formulation
Gyrokinetics is a method for treating low-frequency fluctuations in magnetized plasmas. We begin with the
simplest case, which is all we will require here. Consider a plasma in a straight, uniform mean magnetic field,
B0 = B0 ẑ, with a spatially uniform equilibrium distribution function, ∇F0 = 0, i.e., the slab limit.
The mean gradients in turbulent astrophysical plasmas are generally dynamically unimportant on length
scales comparable to the ion gyroradius which makes the slab limit of direct astrophysical relevance.
In order for the gyrokinetic equations to be applicable, some basic assumptions must be satisfied:
1. Weak coupling, which is the standard assumption of plasma physics
n0e λ3De 1,
(2.1)
where n0e is the mean electron number density and λDe is the electron Debye length. With this
approximation, it is possible to describe the kinetic evolution of all plasma species using the FokkerPlanck equation.
2. Strong magnetization means that the ion Larmor radius ρi must be much smaller than the macroscopic
length scale L of the equilibrium plasma:
ρi =
9
vthi
L,
Ωi
(2.2)
where vthi = (2T0i /mi )1/2 is the ion thermal speed and Ωi = qB0 /(ms c) is the ion cyclotron frequency.
3. Low frequencies refers to the fact that the characteristic frequency of fluctuations ω must be small
compared to the ion cyclotron frequency Ωi :
ω Ωi .
(2.3)
This assumption leads to one of the key simplifications allowed by the gyrokinetic theory: to average
all quantities over the Larmor orbits of particles.
4. Small fluctuations means that the fluctuating quantities are of order ε in the gyrokinetic expansion.
To derive the gyrokinetic equation is necessary to order the time and length scales in the problem in order
to separate the equilibrium and fluctuating quantities. Here, we will summarize the resulting equations; a
detailed derivation can be found in [6].
In addition to the previous assumptions, we will consider that the typical fluctuations are highly anisotropic
as well:
kk
∼ O(ε),
k⊥
(2.4)
where ε is the fundamental expansion parameter:
ε=
ρi
1,
l0
(2.5)
where ρi is the ion Larmor radius and l0 is the largest length scale (it is the typical parallel wavelength of
the fluctuations).
To derive the gyrokinetic equations, begin with the Fokker-Planck equation and Maxwell’s equations and
systematically expand the first one under the gyrokinetic ordering. The minus first, zeroth and first orders
are solved to determine both the form of the equilibrium distribution and the evolution equation for the
perturbed distribution function — the gyrokinetic equation.
Let us summarize the resulting equations. The plasma distribution function up to the first order in ε is
fs (r, v, t) = F0s −
qs ϕ(r, t)
F0s + hs (Rs , v⊥ , vk , t),
T0s
(2.6)
where the equilibrium (zeroth-order) distribution function is a Maxwellian with uniform density n0 and
temperature T0s :
2
vk2 + v⊥
n0s
F0s (v) =
exp
−
2
2 )3/2
vth
(πvth
s
s
!
.
(2.7)
vths = (2T0s /ms )1/2 is the thermal speed, qs ϕ/T0s = O(ε) is the Boltzmann response containing the electrostatic potential ϕ and hs , also O(ε), is a function of the guiding-center position Rs :
Rs = r +
10
v⊥ × ẑ
.
Ωs
(2.8)
y
v⊥ ×ẑ
Ωs
Gyrocenter
Particle
v⊥
Rs
r
x
Figure 2.1: Illustration of the guiding center coordinates, where the magnetic field is in the z direction.
It satisfies the gyrokinetic equation:
∂hs
∂hs
c + vk ẑ ·
hχiRs , hs −
+
∂t
∂Rs
B0
∂hs
∂t
= qs
coll
∂ hχiRs F0s
,
∂t T0s
(2.9)
where the electromagnetic field enters via the ring average of the gyrokinetic potential χ = φ − v · A/c,
∂ hχiRs
∂hs
hχiRs , hs = ẑ ·
×
,
(2.10)
∂Rs
∂Rs
is the Poisson brackets, (∂hs /∂t)coll is the gyroaveraged collision operator and
Z 2π
1
v⊥ × ẑ
dθχ(Rs −
hχ(r, v, t)iRs =
, v, t)
2π 0
Ωs
(2.11)
is the average of χ at fixed guiding-center Rs over the gyroangle θ.
2.2
Low-beta ordering
We would like to have a minimal model that describes a real physical situation, e.g., turbulence in the
presence of a mean magnetic field. For such model, we cannot just write the already known fluid equations.
Instead, we can take the gyrokinetic equations under an appropriate physically motivated ordering of all
spatial and time scales and of the perturbation amplitudes and we can perform an asymptotic expansion,
making sure that it retains all the physical effects that are essential in the understanding of turbulence in
weakly collisional, strongly magnetized plasmas: ion finite Larmor radius (FLR), electron inertia, electron
Landau damping and collisions.
2.2.1
Spatial scales
In the first place, in collisionless or weakly collisional plasma, a key mechanism for the flux unfreezing is the
electron inertia. Thus, we can order
k⊥ de ∼ 1,
11
(2.12)
√
where de = c/ωpe = ρe / βe is the electron inertial scale, ωpe = (4πn0e e2 /me )1/2 is the electron plasma
frequency, e = |qe | is the elementary charge, ρe = vthe /Ωe is the electron Larmor radius and βe = 8πn0e T0e /B02
is the electron beta.
In second place, we can decouple the electrons from the ions at the ion sound scale by ordering
k⊥ ρs ∼ 1,
where ρs = ρi
p
(2.13)
Z/2τ is the ion sound radius, ρi = vthi /Ωi is the ion Larmor radius, Z = qi /e and τ = T0i /T0e .
Considering the temperature ratio to be order unity, equation (2.13) immediately implies that
τ 1/2
τ ∼ 1,
k ⊥ ρi ∼
∼ 1.
Z
(2.14)
As consequence of this, we retain the ion Finite Larmor Radius along with the ion sound scale.
In order for equations (2.12) and (2.13) to be consistent, we must have within our ordering
d e ∼ ρs ,
(2.15)
which means that within this model, there is not a disparity between them. This implies
s
de
2Z
me
=
∼ 1,
ρs
mi
βe
(2.16)
and, in order to achieve this, we order
βe ∼ Z
me
1.
mi
(2.17)
βe , which is the ratio of the plasma to the magnetic pressure, β = 8πP/B 2 , is ordered similar to the
electron-ion mass ratio, me /mi . We are restricting our consideration to low-β plasmas and allowing both
species to have finite temperature.
Now, using equation (2.12), we conclude that
k⊥ ρe ∼
p
βe 1,
(2.18)
i.e., our model will not include electron FLR effects.
2.2.2
Time scales and perturbation amplitudes
The plasma mass flow can be ordered with the E×B velocity drift. Therefore, we order the fundamental time
scale on which we allow our fields to vary in such a way that the characteristic frequency is that associated
with the E × B velocity u⊥ :
2
ω ∼ k⊥ u⊥ ∼ k⊥
cϕ
.
B0
(2.19)
Note that ω Ωi,e is a requirement of the gyrokinetic approximation.
Being the electron kinetics an essential feature of our model, the parallel streaming frequency of the
electrons must be the same order as the rate at which our fields vary:
ω ∼ kk vthe .
12
(2.20)
Taking into account equations (2.19) and (2.20) it is possible to see that they should be consistent with
each other, imposing an ordering on the size of the scalar potential:
kk 1
eϕ
ε
∼
∼√ ,
T0e
k⊥ k⊥ ρe
βe
(2.21)
where equation (2.18) was used. Notice that the appearance of the gyrokinetic expansion parameter ε in the
ordering of the perturbation amplitudes confirms that the use of the gyrokinetic approximation is appropriate
in the considered physical circumstances.
We also require that the density perturbations are of the same order as the electrostatic perturbations
which are given by equation (2.21):
δne
Z eϕ
Z ε
∼
∼ √ .
n0e
τ T0e
τ βe
(2.22)
Physically this follows from the requirement that the physics associated with the ion sound scale is retained.
2.2.3
Alfvénic perturbations
Now we will order the magnetic perturbations. Recall the ordering given by equation (2.17). It implies that
the electron thermal speed is comparable to the Alfvén speed ,
vA = √
B0
.
4πmi n0i
(2.23)
Since n0i = n0e /Z (quasineutrality), we have
vthe
=
vA
r
mi
me
r
βe
∼ 1.
Z
(2.24)
Consequently, it is possible to find
ω ∼ kk vthe ∼ kk vA ,
(2.25)
i.e., Alfvén waves can propagate along the guide field with the same characteristic frequency as the electrons
stream and plasma flows. Notice that in view of equations (2.13) and (2.14), this ordering holds both for the
magnetohydrodynamic Alfvén waves (ω = kk vA ) and for the kinetic Alfvén waves (ω ∼ kk vA k⊥ ρs ).
We can now deduce the ordering for the perpendicular magnetic field δB⊥ = −ẑ × ∇⊥ Ak , by stipulating
that Alfvénic perturbations are accomodated by our ordering:
δB⊥
u⊥
∼
∼ ε,
B0
vA
(2.26)
where equations (2.21) and (2.14) have been used.
Finally, and completing the ordering of the perturbation amplitudes, from the perpendicular component
of Ampère’s law [6],
p
δBk
eϕ
∼ βe
∼ ε βe ,
B0
T0e
(2.27)
where we used equation (2.21). This will cause the parallel perturbations of the magnetic field, which are
negligible in the low-β ordering, to fall out of the final set of equations.
13
2.2.4
Resistivity and collisions
A good model must have a smooth transition from collisionless to semicollisional regimes. In our model,
electron-ion collisions, Ohmic resistivity and electron heat conduction will be retained. Resistivity can be
retained by ordering the electron-ion and electron-electron collision frequencies as comparable to the characteristic frequency of all the other processes:
νei = Zνee ∼ ω.
(2.28)
Since the Ohmic magnetic diffusivity (also known as resistivity) is η ∼ νei d2e , the previous ordering means
that the diffusive effects are retained in our ordering:
2
2 2
ηk⊥
∼ νei k⊥
de ∼ νei ∼ ω,
(2.29)
where equations (2.12) and (2.28) have been used. The resistivity can be considered useful in a model in
which a smoth transition between the collisional and collisionless regimes is possible, although it can be
neglected as well, when in the presence of a subsidiary collisionless expansion.
Our ordering of the electron collision immediately implies an ordering of the ion collisions:
Z2
νii = 3/2
τ
me
mi
νie =
1/2
3/2
p
Z
νei ∼
βe ω,
τ
Zme
νei ∼ βe ω,
mi
(2.30)
(2.31)
where equations (2.18) and (2.28) have been used. Thus, we are effectively assuming ions to be collisionless,
i.e., no ion viscosity.
As was said before, our model provides an electron heating channel, working even with very weak collisions,
which means that the electron Landau damping is included in it.
Finally, we can order the particle mean free path. Using equations (2.20) and (2.28), we have
kk λmfpe =
2.3
2
kk vthe
Z
kk λmfpi =
∼ 1.
τ
νei
(2.32)
The Kinetic Reduced Electron Heating Model
We are now ready to introduce the reduced model, which is applicable to low-β plasmas.
In the first place, recall equation (2.11). This gyroaveraging operator can be written in the Fourier space
and takes a simple mathematical form in terms of Bessel functions J0 and J1 . So, the Fourier transform of
hχ(r, v, t)iRs with respect to Rs can be written as
2
vk Akk
T0s 2v⊥
J1 (as ) δBkk
hχiRs ,k = J0 (as ) ϕk −
+
,
2
c
qs vths as
B0
(2.33)
where as = k⊥ v⊥ /Ωs and ϕk , Akk and δBkk are Fourier transforms (with respect to r) of the scalar potential, parallel component of the vector potential and parallel component of the perturbed magnetic field,
14
respectively. These fields are determined via Maxwell’s equations, namely, the quasineutrality and Ampère’s
law, where particle densities and currents are calculated from the gyrocenter distribution hs .
Now recall the gyrokinetic equation (2.9) for electrons (s = e). Retaining only the lowest order in the
expansion with respect to βe means that the Bessel functions in the expression for hχiRe (equation (2.33))
√
can be expanded in small argument ae ∼ k⊥ ρe ∼ βe 1 :
J0 (ae ) '
2J1 (ae )
= 1 + O(a2e ).
ae
(2.34)
Still in equation (2.9), the ϕ and Ak terms are the same order while the δBk term is one order of βe smaller.
This allows us to write the gyrokinetic equation (2.9) up to corrections of order O (βe ),
v k Ak
dhe
eF0e ∂
∂he
+ vk b̂ · ∇he = −
ϕ−
+
,
dt
t0e ∂t
c
∂t coll
(2.35)
where
∂he
c
dhe
=
+
{ϕ, he }
dt
∂t
B0
(2.36)
is a “convective” time derivative incorporating the E × B motion and
1 ∂he
−
Ak , he
∂z
B0
b̂ · ∇he =
(2.37)
is the parallel spatial derivative along the perturbed field line.
We can now define the total perturbed electron distribution function to lowest order in
p
me /mi ∼
√
βe
and in the gyrokinetic expansion [6] as
vk uke
δne
+2 2
n0e
vthe
δfe = ge +
!
F0e ,
where F0e is the equilibrium Maxwellian (see equation (2.60)), vthe =
(2.38)
p
2T0e /me is the electron thermal
speed (with T0e as the mean electron temperature), vk is the parallel velocity coordinate,
Z
δne
1
=
d3 vδfe ,
n0e
n0e
(2.39)
by definition, is the electron density perturbation (the zeroth moment of δfe ) normalized to its background
value n0e , and
uke =
e 2 2
d ∇ Ak ,
cme e ⊥
(2.40)
or, by the definition,
uke =
1
n0e
Z
d3 vvk δfe ,
(2.41)
is the parallel electron flow (the first moment of δfe ; Ak is the parallel component of the vector potential and
both are related via the parallel component of Ampère’s law [6]:
ẑ · (∇⊥ × δB⊥ ) = −∇2⊥ Ak =
=
4π
jk
c
4πen0e
uki − uke
c
15
(2.42)
where uki = 0 because, under the adopted ordering, the parallel current is carried predominantly by the
electrons; and de = c/ωpe is the electron skin depth).
The definition of the density and parallel electron flow velocity are consistent provided we demand


Z
1
 ge = 0,
d3 v 
(2.43)
vk
i.e., all moments of δfe higher than δne and uke are contained in ge . For example, the parallel electron
temperature perturbation is
Z
vk2
δTke
1
(2.44)
=
d3 v2 2 ge .
T0e
n0e
vthe
Now, the reduced model will be introduced. The dynamics of the plasma is described by the evolution
equations for ϕ, Ak and ge , which are all functions of time and three spatial coordinates; ge is also a function
of vk and v⊥ , although the v⊥ dependence can be ignored (or integrated out) if such a dependence is not
introduced by the collision operator.
eϕ
d Z
e 2 2
1 − Γˆ0
= b̂ · ∇
d ∇ Ak ,
dt τ
T0e
cme e ⊥
eϕ
δTke
Z
∂ϕ cT0e
d
Ak − d2e ∇2⊥ Ak = η∇2⊥ Ak − c
−
b̂ · ∇
1 − Γˆ0
−
,
dt
∂z
e
τ
T0e
T0e
!
2vk2
δTke
dge
e 2 2
+ vk b̂ · ∇ ge −
F0e b̂ · ∇
F0e = C[ge ] + 1 − 2
d ∇ Ak ,
dt
T0e
vthe
cme e ⊥
(2.45)
(2.46)
(2.47)
where the following short-hand notation is used
Z
2vk2
δTke
1
d3 v 2 ge ,
=
T0e
n0e
vthe
∂
c
d
=
+
{ϕ, . . .} ,
dt
∂t B0
1 ∂
−
Ak , . . . ,
b̂ · ∇ =
∂z
B0
(2.48)
(2.49)
(2.50)
η = νei d2e is the Ohmic diffusivity, C[ge ] is the collision operator and the integral operator Γ̂0 expresses the
average of the electrostatic potential over rings of radius ρi . In Fourier space, this operator becomes
Γ0 (b) = exp(−b)I0 (b),
(2.51)
2 2
where b = k⊥
ρi /2 and I0 is the modified Bessel function of the zeroth order [26]. What this means is that
within the gyrokinetic context, as the particle orbits the magnetic guide field, it is allowed to experience
different values of the electrostatic potential. The purpose of the Γ0 operator is to average ϕ over such orbits
and over the Maxwellian velocity distribution.
Equations (2.45 - 2.47) constitute a minimal physically realizable paradigm for low-frequency nonlinear
plasma dynamics with a strong guide field, including all effects we expect to be important: ion sound physics,
ion Finite Larmor Radius, electron inertia, electron collisions, Ohmic resistivity, and electron temperature
perturbation determined by a kinetic equation. These equations are referred to as Kinetic Reduced Electron
Heating Model (KREHM).
16
2.4
Linear Theory
In this section the linearized form of equations (2.45 - 2.47) is presented, for more detail please read [12].
The linearization is done around an equilibrium which contains both the guide field B0 = B0 ẑ and some
in-plane field which is part of the small perturbation of the guide field. After some manipulations and Fourier
transforming with respect to y, z and time, the result is
eϕ
Z
e 2
1 − Γ̂0
=
de kk (x)∇2⊥ Ak − ky f 00 (x)Ak ,
(2.52)
τ
T0e
cme
Z
1
− ω Ak − d2e ∇2⊥ Ak = −iη∇2⊥ Ak + ky cd2e f 00 (x)ϕ − kk (x)c
1+
ϕ − δTke ,
(2.53)
τ
e
!
2vk2
δTke
e 2
F0e
F0e = −iC[ge ] + 1 − 2
de kk (x)∇2⊥ Ak − ky f 00 (x)Ak , (2.54)
− ωge + kk (x)vk ge −
T0e
vthe
cme
−ω
where the function f (x) contains all the information about the in-plane equilibrium, kk (x) = kz + f (x)ky
and ∇2⊥ = ∂x2 − ky2 .
The simplest approach to linear theory in a kinetic plasma is to consider the purelly collisionless case,
which allows us to make contact with the existing theories. The collisionless limit is also a useful route
to certain linear results, such as Landau damping, that do in fact depend on an infinitesimal amount of
velocity-space dissipation.
Now set η = 0 and C[ge ] = 0 in equations (2.53) and (2.54), respectively. It is possible to solve equation
(2.54) for ge explicitly and then, after some algebra and using equation (2.52), the following simplification
appears (please, read [12] for more detail):
1
δTke
e
Z(ζ) + ζZ 0 (ζ) Z = −2ζ
1
−
Γ̂
0
Z 0 (ζ)
τ
2
Z
=
1 − 2ζ 2 + 0
1 − Γ̂0
Z (ζ) τ
(2.55)
where ζ(x) = ω/|kk (x)|vthe , Z(ζ) is the plasma dispersion function and Z 0 (ζ) = −2 [1 + ζZ(ζ)]. This can be
substituted into equation (2.53), whereupon equations (2.52) and (2.53) form a close set.
Assume that
kz ky f (x),
(2.56)
kz ky d2e f 00 (x),
(2.57)
kz ∂ 2 /∂x2 ky f 00 (x).
(2.58)
We find ourselves in a homogeneous plasma, where all terms containing f (x) can be neglected and kk = kz .
This means that we can now also Fourier transform in x and from equations (2.52), (2.53) and (2.55) we get
the following dispersion relation:
ζ2 − τ
2 2
1 2 2
k⊥
de /2
2 ρ2 /2) [1 + ζZ(ζ)] = 2 k⊥ de ,
1 − Γ0 (k⊥
i
17
(2.59)
Figure 2.2: Graphical representation of the dispersion relation given by equation (2.59) for ρi = 1 and
de = 0.2. The frequency is given by the blue line whislt the damping is given by the red one. Dashed lines
identify the locations where k⊥ ρi = 1 and k⊥ de = 1.
2
where k⊥
= kx2 + ky2 . This dispersion relation agrees with the gyrokinetic dispersion relation at low-β derived
in [6], equation (D17). Its graphical representation can be seen in figure 2.2.
In figure 2.3 we can see the damping as function of the frequency (there is an infinite number of solutions,
only a limited number of those are represented).
Figure 2.3: Graphical representation of the damping as function of the frequency for two values of k⊥ for
equation (2.59). In blue we have the damping as function of the frequency at k⊥ = 1/ρi , where ρi = 1, and
in orange we have k⊥ = 1/de , where de = 0.2. For k⊥ = 1/ρi and k⊥ = 1/de , the least damped solution is
(±1.53, −0.12) and (±4.73, −1.92), respectively.
18
2.5
Hermite Formulation
The velocity-space dynamics and the emergence of small-scale structure in vk are best understood in terms of
the expansion of the electron distribution function ge in Hermite polynomials [2, 18, 24, 25]. Recall equations
(2.45-2.47). Having these equations with ∂/∂z = 0 and η = 0 lead us to
1 dδne
1
e 2 2
=
Ak ,
de ∇⊥ Ak ,
n0 dt
Bz
cme
δTke
cT0e
d
δne
Ak − d2e ∇2⊥ Ak = −
Ak ,
+
,
dt
eBz
n0e
T0e
!
2vk2 F0e
vk
δTke
dge
e 2 2
−
Ak , ge −
= C[ge ] − 1 − 2
Ak ,
d ∇ Ak .
dt
Bz
T0e F0e
vthe Bz
cme e ⊥
(2.60)
(2.61)
(2.62)
Equation (2.62) does not contain an explicit dependence on the perpendicular velocity coordinate, v⊥ . If
we consider that v⊥ has been integrated out, we will have ge = ge (x, y, vk , t). Let
∞
X
Hm vk /vthe
√
ge (x, y, t, vk ) =
gm (x, y, t)F0e (vk ),
2m m!
m=2
(2.63)
where the Hermite polynomials are given by [26]
2
Hm vk /vthe = (−1)m exp vk /vthe
h
2 i
dm
m exp − vk /vthe
,
d vk /vthe
with gm the Hermite expansion coefficients (dimensionless), which can be calculated according to
Z +∞
Hm vk /vthe
1
gm =
dvk √
ge (vk ).
n0e −∞
2m m!
(2.64)
(2.65)
The g0 and g1 coefficients do not show up in the sum of equation (2.63) since g0 = g1 = 0 due to the
decomposition of δfe adopted before: δne and uke have been explicity separated from ge .
Recall our set of equations (2.60-2.62). Taking the Hermite transform of equation (2.62) and using the
recursive property of the Hermite polynomials,
1
vk
Hm vk /vthe = Hm+1 vk /vthe + mHm−1 vk /vthe ,
vthe
2
(2.66)
we arrive at
dgm
vthe
=
dt
Bz
r
m+1
Ak , gm+1 +
2
r
! √
m
2
e 2 2
Ak , gm−1
+
δm,2 Ak ,
d ∇ Ak − Dm gm ,
2
Bz
cme e ⊥
(2.67)
where δm,2 is a Kronecker delta and Dm is a dissipation operator.
We can now ask what form can this dissipation operator take. When we are in the presence of the
Lénard-Bernstein collision operator, we have
Dm = νei m,
(2.68)
where νei is the electron-ion collision frequency. However, the linear dependence on m of this operator forces
us to use a large number of Hermite polynomials in order to resolve the velocity-space cutoff as νei → 0, and
we would like to solve as few moments as possible. So, Dm will take another form instead.
19
A numerical tool that has been used for a long time in order to simulate high Reynolds number turbulence
is the hyperviscosity [27, 28], which has the purpose of introducing an artificial damping in the system. This
damping term is defined as:
γ = −νH
|k|
kc
α
,
(2.69)
where kc is the cuttoff wavenumber. Based on this, we can introduce the hyper-collisions operator as
Dm = νH
m α
M
,
(2.70)
where M is the highest moment. It introduces an artificial dissipation range in the system as well and at the
same time it does not introduce any non-physical effects. With it we will get a faster convergence, thus we
will solve less moments than using the Lénard-Bernstein collision operator.
We can now ask which values of the power α and the hyperviscosity νH can we choose. Usually the size of
the damping is set experimentally [27, 28], the only requirement is that the dissipation range appears within
the resolved modes. We have α 1 but it cannot be very large, however, the optimal choice depends on the
resolution of the simulation. Finally, we can ask about its upper limit. If it is too big, it would be the same
as having no damping for all the modes except for the ones near the value of M , which would be extremely
damped. Concerning the value of the hyperviscosity νH , it is chosen so the damping for the smallest modes
is smooth whilst near the value of M is extremely damped.
Now we summarize our system of partial differential equations in the two-dimensional position space:
1
e 2 2
1 dδne
=
Ak ,
de ∇⊥ Ak ,
(2.71)
n0 dt
Bz
cme
δTke
cT0e
d
δne
Ak − d2e ∇2⊥ Ak = −
Ak ,
+
,
(2.72)
dt
eBz
n0e
T0e
! √
r
r
vthe
dgm
m+1
m
2
e 2 2
=
Ak , gm+1 +
Ak , gm−1
+
δm,2 Ak ,
de ∇⊥ Ak − Dm gm . (2.73)
dt
Bz
2
2
Bz
cme
This is an attractive way of carrying out numerical simulations of collisionless (in fact, weakly collisional)
kinetic phenomena in strongly magnetized plasmas. This system is much simpler than the full kinetic or
gyrokinetic [29, 6] descriptions that have been used to study plasma turbulence [10, 36]. According to [2],
although Hermite polynomials provide a complete basis set, they converge if enough moments are kept, so,
one of the main questions is how many Hermite moments must be kept for any given collisionality. The first
numerical study using this system of equations can be seen in [30].
The Hermite formalism allows us to derive very concise derivations of important results [12], being one
of them the Hermite spectrum, which quantifies the velocity-space dissipation scale. Linearizing equation
(2.73) for a given ky and knowing that the Hermite spectrum is given by
Em =
we find that it evolves as
|gm |2
,
2
∂Em
By
∂ √
= −|ky | vthe
2mEm − 2Dm Em .
∂t
Bz
∂m
20
(2.74)
(2.75)
This partial differential equation can be solved and different solutions can be found [12, 30], it will depend
on the dissipation operator we are using.
In here, as an exercise, we will find a steady-state solution for equation (2.75) for the Lénard-Bernstein
α
collision operator, i.e., Dm = νei m [12] and for the hyper-collisions operator, Dm = νH (m/M ) [30]. For
the Lénard-Bernstein collision operator, equation (2.75) will then become
|ky |
By
∂ √
2mEm + 2νei mEm = 0.
vthe
Bz
∂m
(2.76)
After some straightforward algebra, we arrive at the solution of last equation:
" 3/2 #
C(ky )
m
Em = √
exp −
,
mc
m
(2.77)
where C(ky ) is some function of ky and can be determined by the dynamics of ϕ, Ak and δTe . mc stands for
the collisional cutoff and is given by
mc =
3 B v |k |
√ y the y
2 2 Bz νei
2/3
.
(2.78)
Concerning the hyper-collisions operator, equation (2.75) becomes
|ky |
By
∂ √
α
2mEm + 2νH (m/M ) Em = 0.
vth
Bz e ∂m
(2.79)
For any value of α, we arrive at the solution:
Em
" (2α+1)/2 #
C(ky )
m
= √
exp −
,
mc
m
(2.80)
where C(ky ) still is some function of ky and the collisional cutoff mc and is given by
mc =
(1/2 + α) By vthe |ky |M 4
√
Bz
νH
2
2/(2α+1)
.
(2.81)
The graphical representation of the Hermite spectrum in steady-state for both collision operators can be
seen in figure 2.4. The values of the collision frequencies are νei τA = 0.1 and νH τA = 50, these values will be
used later on chapter 4 and their motivation will be explained in that chapter.
As we can see, for the Lénard-Bernstein collision operator, that presents a linear dependence on m, the
spectrum evolves smoothly with m (blue line) whilst for the hyper-collisions operator, we can see clearly the
dissipation range (dashed orange line). We can also compute the values of mc , which is given by equations
(2.78) and (2.81) for the Lénard-Bernstein collision operator and hyper-collisions operator, respectively. For
the Lénard-Bernstein collision operator, with νei τA = 0.1, we have
mc = 17.8,
(2.82)
and for the hyper-collisions operator, with νH τA = 50, α = 4 and M = 25, we have
mc = 14.6.
21
(2.83)
Figure 2.4: Graphical representation of the Hermite spectrum Em as function of m. When m mc , the
energy spectrum scales as Em ∼ m−1/2 . In blue we have the Hermite spectrum for the Lénard-Bernstein
collision operator and in orange, for the hyper-collisions operator, with νei τA = 0.1 and νH τA = 50, α = 4
and M = 25.
From these values of mc and from the analysis of the behaviour of the dissipation range for both collision
operators, we can predict how many Hermite moments we need to solve. For the Lénard-Bernstein collision
operator, the spectrum has a smooth behaviour in the dissipation range, meaning that we would need M 18;
concerning the hyper-collisions operator, the spectrum shows a very abrupt dissipation range, meaning that
we would need M > 15.
In figure 2.5, the Hermite spectrum using the Lénard-Bernstein collision operator for three different values
of the collisional frequency is shown.
As an exercise, we also computed the values of mc for the three values of the collisional frequency. For
νei τA = 0.1 we have mc = 17.8, for νei τA = 0.01 we have mc = 82.5 and for νei τA = 0.001 we have mc = 383.1.
When the value of νei τA decreases, the value of mc increases, as expected.
22
Figure 2.5: Graphical representation of the Hermite spectrum Em as function of m for different values of νei τA
and for the Lénard-Bernstein collision operator. When m mc , the energy spectrum scales as Em ∼ m−1/2 .
23
Chapter 3
Closures
In this chapter different closure conditions will be derived. We will start with polynomial closures, i.e., that
are derived from the Taylor series and after that we will manipulate equation (2.73) in order to get other
closure conditions.
First recall equations (2.71-2.73) from chapter 2. Under proper normalization (for more detail see [30]),
these equations become
∂ne
+ {ϕ, ne } = Ak , ∇2⊥ Ak ,
∂t
n
o
√
∂
Ak − d2e ∇2⊥ Ak + ϕ, Ak − d2e ∇2⊥ Ak = η∇2⊥ Ak + ρ2s ne + 2g2 , Ak ,
∂t
√ ρs √ ∂g2
+ {ϕ, g2 } = 3
Ak , g3 + 2 Ak , ∇2⊥ Ak ,
∂t
de
√
√ ρs ∂gm
ρs + {ϕ, gm } = m + 1
Ak , gm+1 + m
Ak , gm−1 − Dm gm .
∂t
de
de
(3.1)
(3.2)
(3.3)
(3.4)
As it is possible to see, we are in the presence of a infinite set of fluid-like coupled equations which
means that at some finite number M the system must be truncated. Closures are necessary in the study
of physical systems with a large number of degrees of freedom and when it is only possible to compute a
small number of moments. An ideal closure should capture all the physics (e.g., Landau-damping) of the
problem with as few moments as possible. The equation for the mth Hermite moment is coupled to the
(m + 1)st Hermite moment and in order to close the system, the latter must be estimated. So, we need some
closure approximation (which introduces truncation errors). Analysing equation (3.4), we see that we have
two dependences on the (m + 1)st Hermite moment: we have the first term on the right-hand side of the
equation and we have the time derivative. Thus, it is obvious that a good estimative for the value of gm+1
would be an extrapolation from lower order moments [19, 24].
A closure condition that is usually adopted [2, 18, 24] does not depend on lower order moments:
gm+1 = 0.
(3.5)
This closure approximation results in large oscillations in the distribution function [18, 24] due to the fact
that the gm+1 term of the time derivative of the gm is set to zero. To better understand this problem, we
25
can recall its mechanical analog, the infinite set os masses Ml at positions xl each coupled to its nearest
neighbors Ml−1 and Ml+1 by springs with spring constant kl and kl+1 . Setting (3.5) is the same as replacing
the L/2 mass with a fixed wall, which reflects the energy back to lower springs, i.e., to lower moments. In
this case there is no damping mechanism in this mechanical system and the wave energy produced by the
initial perturbation will be trapped and bounces around l = 0 and l = L/2 − 1. One way of solving this
problem is to include a dissipative term, the Dm dissipation operator, or to find a closure condition that is
a extrapolation from the lower order moments [24] in order to include some dissipation in the system.
In this chapter, we will analyse different closure conditions: polynomial, exact and nonlinear; all of them
of the form gM +1 = gM +1 (gM , gM −1 , . . . , gM −n ).
3.1
Polynomial closures
In this section, we take advantage of the Taylor series in order to build different closure approximations that
depend from lower order moments.
First consider the Taylor series:
gM +1 = gM
∂gm 1 ∂ 2 gm +
+
+ ···
∂m m=M 2 ∂m2 m=M
(3.6)
Imagine that we want the 0th order of the series, we will simply get
gM +1 = gM .
(3.7)
Now, if we expand it to the 1st order, and using the centered difference scheme to do it, we get another
closure condition:
gM +1 = 2gM − gM −1 .
(3.8)
We can arrive at another two closure conditions after some straightforward algebra using different difference
schemes:
gM +1 = gM −1 ,
(3.9)
gM +1 = gM + gM −1 − gM −2 .
(3.10)
To conclude this section, we introduce as well a closure condition based on the fourth order extrapolation
[19, 24]:
gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 .
3.2
(3.11)
Linear closure
We can now derive another closure condition by analysing the last equation of our system of equation,
equation (3.4) at m = M . Consider that M is in the m inertial range. Then we can think of a statistical
26
steady-state, ∂/∂t = 0, and we can neglect the collisional term, since it must be small in the inertial range,
leaving us with
√
√ ρs ρs ∂gM
+ {ϕ, gM } = M + 1
Ak , gM +1 + M
Ak , gM −1 ,
∂t
de
de
(3.12)
and then we can linearize it. For more details on the linearization, please see chapter 4 or [31].
√
√
ρs
ρs
− iωgM = i M + 1B0 ky gM +1 + i M B0 ky gM −1 .
de
de
From this equation, we find a new closure condition:
r
M
ρs B0 k y
gM +1 = −
gM −1 − ω √
gM .
M +1
de M + 1
(3.13)
(3.14)
For large M , the second term on the right hand side can be neglected when compared with the first one,
from which arises a new closure condition:
r
gM +1 = −
3.3
M
gM −1 .
M +1
(3.15)
Nonlinear closure
Recall equation (3.4). For any collision frequencies, there is allways a sufficiently large m such that
Dm
1
ω
(3.16)
is true since both Dm operators scale positively with m. Taking this into account, some terms of equation
(3.4) can be neglected, such as dgm /dt and Ak , gm+1 . The latter can be neglected due to the fact in the
dissipation range of the energy cascade, we have gm+1 /gm 1. By neglecting these two terms, we are left
with another two terms that depend on gm−1 and gm . Setting m = M + 1, we have a new closure condition,
√
ρs M + 1 gM +1 =
Ak , gM ,
(3.17)
de DM +1
which is a nonlinear closure and that is only valid in the limit Dm ω, where Dm = mνei if we are
α
working with Lénard-Bernstein collision operator and Dm = νH (m/M )
if we are considering a hyper-
collision operator. Notice that from the derivation of (3.17), we have the certainty that with this closure the
solution will converge if enough moments are kept.
27
Chapter 4
Linear Tests
In this chapter we test the linear physics of our set of equations with the closures from the previous chapter.
A good closure should be able to reproduce the result given by equation (2.59) (see figure 2.2) with as few
moments as possible. Recall our set of equations given by (3.1 - 3.4):
∂ne
+ {ϕ, ne } = Ak , ∇2⊥ Ak ,
∂t
n
o
√
∂
Ak − d2e ∇2⊥ Ak + ϕ, Ak − d2e ∇2⊥ Ak = η∇2⊥ Ak + ρ2s ne + 2g2 , Ak ,
∂t
√ ρs √ ∂g2
+ {ϕ, g2 } = 3
Ak , g3 + 2 Ak , ∇2⊥ Ak ,
∂t
de
√
√ ρs ∂gm
ρs + {ϕ, gm } = m + 1
Ak , gm+1 + m
Ak , gm−1 − Dm gm ,
∂t
de
de
(4.1)
(4.2)
(4.3)
(4.4)
where ne is the perturbed electron density, ϕ is the electrostatic potential, {P, Q} = ∂x P ∂y Q−∂y P ∂x Q is the
Poisson bracket and the in-plane magnetic field is given by B⊥ = −ẑ×∇⊥ Ak . We also have ∇⊥ = ∂x2 +∂y2 and
the dissipation is provided by the Dm operator, which can take the mνei form, where νei is the electron-ion
α
collision frequency, if we are in the presence of the Lénard-Bernstein collision operator or νH (m/M ) if we
are considering hyper-collisions. The perturbed electron density ne is given by
ne =
2 Γ̂
(b)
−
1
ϕ,
0
ρ2i
(4.5)
which is known as the gyrokinetic Poisson equation. ρi is the ion Larmor radius and the integral operator
Γ̂0 expresses the average of the electrostatic potential over rings of radius ρi .
We shall now procede with the linearization. We assume that all fields can be written as
χ = χeq + χ1 (x, y, t)
= χeq + χ1 (x) exp(iky y) exp(−iωt)
(4.6)
where χ1 represents small perturbations to the equilibrium. We consider an equilibrium described by B⊥,eq =
B0 ŷ, with B0 a constant and ne,eq = ϕeq = gm,eq = 0.
29
Considering what was said before, we find the following linearized equations:
2
−iωne = −iB0 k⊥
ky Ak ,
√
ρ2i
1
ρ2s
ρ2s
+
n
−
i
2
−iωAk = −iky B0
e
2
2 2 (Γ (b) − 1)
2 g2 ,
1 + d2e k⊥
1 + d2e k⊥
1 + d2e k⊥
0
√ ρs
√
2
−iωg2 = i 3 ky B0 g3 − i 2ky B0 k⊥
Ak ,
de
√
√ ρs
ρs
−iωgm = i m + 1 ky B0 gm+1 + i m ky B0 gm−1 − Dm gm .
de
de
(4.7)
(4.8)
(4.9)
(4.10)
In matrix form, this set of equations reads:

ne


 Ak


 g2
M

 g3

 ..
 .

gM







 = 0,






where M is a tridiagonal matrix given by the system of equations. Its entries are:
√ ρs
√
√ ρs
ρ2s
2
Mj,j+1 = iB0 k⊥ ky , i 2
2 , −i 3 d ky B0 , · · · , −i M d ky B0
1 + d2e k⊥
e
e
(4.11)
(4.12)
Mj,j = (−iω, −iω, −iω, −iω + D3 , · · · , −iω + DM )
(4.13)
√ ρs
√ ρs
√
ρ2i
1
ρ2s
2
2k
B
k
,
−i
3
M
,
i
Mj+1,j = iky B0
+
k
B
,
·
·
·
,
−i
k
B
y 0 ⊥
y 0
y 0 ,
2
2 2 (Γ (b) − 1)
1 + d2e k⊥
1 + d2e k⊥
de
de
0
(4.14)
where j = 1, · · · , M + 1. From now on, we work on the matrix form due to its simplicity. In order to find
the dispersion relation given for each closure, we need to compute Det M = 0 and solve it in order to find
ωN (k⊥ ), where the subscript N stands for N umerical. Notice that if we have a (M + 1) × (M + 1) matrix,
we will find (M + 1) solutions for the dispersion relation and we can ask which is the solution we are looking
for. As we were able to see in chapter 2, figure 2.3, there is an infinite number of solutions to the exact
linear dispersion relation but we are only interested in the least damped one. This is because it will be the
dominant mode as time goes to infinity.
Before we analyse the behaviour of the system in the presence of the different closures, we will consider
the isothermal closure.
All the computational results were obtained using the computational software program Mathematica
9.0.0.0.
4.1
The isothermal closure
The simplest closure possible is the isothermal closure, i.e.,
gm = 0, for m ≥ 2,
30
(4.15)
which is the same as saying that ge = 0 in our initial set of equations, (2.60-2.62). This is the popular
isothermal-electrons closure [33] and is not a solution of that equation unless Ak , ∇2⊥ Ak = 0. The use of
a fluid model (ge = 0) might be a useful simplification but it is not an interesting limit.
Our set of equations will be reduced to
2
−iωne = −iB0 k⊥
k y Ak ,
ρ2i
ρ2s
1
−iωAk = −iky B0
2 + 1 + d2 k 2 2 (Γ (b) − 1) ne ,
1 + d2e k⊥
0
e ⊥
or, in the matrix form,


2
iB0 k⊥
ky
−iω
iky B0
ρ2s
2
1+d2e k⊥
+
ρ2i
1
2 2(Γ (b)−1)
1+d2e k⊥
0
−iω


ne
Ak
(4.16)
(4.17)

 = 0.
(4.18)
Setting the determinant of the matrix to zero, we are able to find the solution for the dispersion relation:
1/2
B0 ky k⊥
ρ2i
2
ω = ±p
ρs −
.
2
2 (Γ0 (b) − 1)
1 + d2e k⊥
(4.19)
As we can see, for a 2 × 2 matrix, we have two solutions, whose real parts are symmetric and the imaginary
ones are equal to zero i.e., no damping. Its graphical representation can be seen in figure 4.1.
Figure 4.1: The isothermal closure. In blue we have the frequency and in orange we have the damping. The
full lines correspond to the analytical solution while the dashed ones correspond to the isothermal closure.
When k⊥ ρi 1, the shear Alfvén waves are recovered.
4.2
Closure Analysis
In this section we are going to analyse the different closures introduced in the previous chapter. Firstly
we will choose a closure from the previous chapter and see how the system behaves in the presence of the
31
Lénard-Bernstein collision operator, equation (2.68). After that, we switch to the hyper-collisions operator,
equation (2.70), and analyse how the system responds to each one of the closures.
The error introduced by each closure is quantified by
|ωA − ωN |
× 100,
ωA
γA − γN × 100,
εγ (%) = γA εω (%) =
(4.20)
(4.21)
and it will be always computed at k⊥ de = 2 where de is the smallest physically meaningful spatial scale. We
will declare a closure acceptable if, for m = M , it yields an error
εω,γ . 1%.
(4.22)
Finally, we note that we expect the error to decrease exponentially with M , as is characteristic of spectral
methods.
Now we are ready to proceed with the analysis of the dispersion relation for both collision operators.
4.2.1
The Lénard-Bernstein Collision Operator
In this section we will be using the Lénard-Bernstein collision operator, whose Hermite transform is Dm =
mνei . For this collision operator, we take three closures from the previous chapter:
gM +1 = 0,
(4.23)
gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 ,
√
gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM ,
(4.24)
(4.25)
and analyse what happens in the presence of this operator. gM +1 = 0 is chosen because it is the simplest
and usually adopted one; gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 is an example of a polynomial closure and
√
has been suggested to work well in a different context [24]; gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM is
chosen because is the only closure that is guaranteed to converge as M → ∞.
We know that for the Lénard-Bernstein collision operator we need a large M because of the linear
dependence on m, so, we begin to plot the error as function of M , for the three closures and see how it behaves.
However, due to computational limitations, we were only able to compute the error up to M = 25 for the three
√
closures: gM +1 = 0, gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 and gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
(Mathematica has trouble solving equation (4.11) for larger values of M ). The graphical representation of
the error as function of M can be seen in figures 4.2, 4.3 and 4.4, respectively.
As we can see from table 4.1, we chose νei τA = 0.1 for the electron-ion collision frequency. For comparison,
we note that in the solar wind νei τA = 4.2 × 10−6 , a much smaller value which, however, would require an
unfeasibly large number of Hermite moments in order to solve the velocity space dynamics.
As we can see from the three graphics, the damping converges exponentially, as expected, but the same
does not happen with the frequency, which presents a very oscillatory behaviour. We also see that, although
32
B0
ky
ρs
ρi
de
νei
1
1
1
1
0.2
0.1
Table 4.1: Normalized values used for the plasma parameters for the Lénard-Bernstein collision operator.
Figure 4.2: Error as function of M for the closure gM +1 = 0 with the Lénard-Bernstein collision operator,
Dm = mνei . The red line stands for the error of the frequency ω and the blue, dashed line corresponds to
the error of the damping γ. The error was computed at k⊥ de = 2. The convergence rate for the damping is
aγ = 0.0096.
the damping for the three closures converges at a similar rate, the nonlinear closure is the one that shows the
fastest convergence. Extrapolating, we would need at least M > 100 to get an error for the damping such
that εγ . 1%. Concerning the error for the frequency, it is hard to say how many M ’s are required to satisfy
the error condition due to its oscillatory behaviour. Recalling the Hermite spectrum in chapter 2, figure 2.4,
we saw that we need M 18. Since we are only solving the set of equations until M = 25, we can say that
the number of moments we are solving is not large enough to capture the convergence of the error of the
frequency, resulting in these large oscillations.
In figure 4.5 we can see the graphical representation of the damping as function of the frequency for
√
two closures, gM +1 = 0 and gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM , and for the analytical solution at
k⊥ = 2/de for M = 15. We see that for both closures, the least damped mode is too far away from the least
damped mode of the analytical solution. This agrees with the large errors found in the error graphics.
We can also compute the damping as function of the frequency for different M ’s and see how it behaves.
This result can be seen in figure 4.6 for the nonlinear closure with M = 9 and M = 19. As expected, the
33
Figure 4.3: Error as function of M for the closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 with the LénardBernstein collision operator, Dm = mνei . The red line stands for the error of the frequency ω and the
blue, dashed line corresponds to the error of the damping γ. The error was computed at k⊥ de = 2. The
convergence rate for the damping is aγ = 0.0097.
√
Figure 4.4: Error as function of M for the closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM with the
Lénard-Bernstein collision operator, Dm = mνei . The red line stands for the error of the frequency ω and
the blue, dashed line corresponds to the error of the damping γ. The error was computed at k⊥ de = 2. The
convergence rate for the damping is aγ = 0.0099.
approximated solution slowly converges for the analytical one, confirming that M 25 is required to capture
the system dynamics.
34
Figure 4.5:
of the damping as function of the frequency for gM +1 = 0 and gM +1 =
√ Graphical representation
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the Lénard-Bernstein collision operator and for M = 15.
In blue we have the analytical solution, which can also be seen in figure 2.2, in orange
we have thesolutions
√
for the closure gM +1 = 0 and in green we have the solutions for gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
at k⊥ = 2/de .
Figure √
4.6: Graphical representation
of the damping as function of the frequency for gM +1 =
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the Lénard-Bernestein collision operator using M = 9
and M = 19. In blue we have the analytical solution, which can also be seen in figure 2.2, in orange we have
the solutions for the M = 9 and in green we have the solutions for M = 19 at k⊥ = 2/de .
35
We can conclude that with the Lénard-Bernstein collision operator, a very large M is required to get
an acceptable value for the error for the damping γ, due to the linear dependence of this operator on m.
Concerning the frequency ω, due to the fact that we are computing a small number of moments, we do not
see any convergence, meaning that M 25 is required. One way to decrease the value of M is to increase
the value for the electron-ion collision frequency, νei . In the next section the hyper-collisions operator will
be introduced.
4.2.2
The Hyper-Collisions Operator
Now that we confirmed that with the Lénard-Bernstein collision operator we need a very large M to get a
solution that satisfies our error condition, we are going to use the hyper-collisions operator and see what
changes. But first, we need to search what values of α and νH should we choose.
We want some values of α and νH such that we have weak or no damping at low m, to garantee the
existence of an inertial range. This can be translated as
α
νH (m/M )
1,
ω
α
νH (m/M )
1,
ω
m M,
(4.26)
m ∼ M.
(4.27)
It is possible to see that if we say α has a constant value and M isn’t fixed, νH will vary such that those
conditions are satisfied. However, here, we will choose a value for M and use a suitable value for α and νH ,
all constant.
Concerning the value for α, making α 1 would progressively concentrate the damping in fewer and
fewer modes (at higher m) but it cannot be very large because of the possibility of bottleneck effects [34].
For that reason we choose
α = 4.
(4.28)
Relatively to the value of νH , we can make a small numerical exercise in order to find it. Doing the analysis of
the graphical representation of the dispersion relation (2.2), it is possible to see that we have ω M ax τA ∼ 6.5
and if we choose m = M/2 and m = M and replace it in equations (4.26) and (4.27), respectively, we get
4
νH (1/2)
1,
ω M ax
νH
1,
ω M ax
m = M/2,
(4.29)
m = M.
(4.30)
This means that we can choose any value for νH as long as it satisfies
ω M ax νH 2α ω M ax ,
(4.31)
where we have α = 4. Notice that α is the parameter that will change the range from which we can choose
the value for νH : this range increases with α.
36
If we choose νH τA = 50, the conditions given by equations (4.26) and (4.27) are satisfied. Concerning the
value for M , we want to capture the system dynamics with as few Hermite moments as possible, so let us
choose
M = 15,
(4.32)
which is the value found for the collisional cutoff for the Hermite spectrum in chapter 2. Now we can proceed
with the study of the behaviour of the dispersion relation with different closures.
B0
ky
ρs
ρi
de
νH
α
1
1
1
1
0.2
50
4
Table 4.2: Normalized values used for the plasma parameters for the hyper-collisions operator.
4.2.2.1
Closure gM +1 = 0
We will begin with the closure condition gM +1 = 0. Our system, when represented in matrix form, is given
by a tridiagonal matrix where the nonzero elements are given by:
√ ρs
√ ρs
√
ρ2s
2
Mj,j+1 = iB0 k⊥ ky , i 2
(4.33)
2 , −i 3 de ky B0 , · · · , −i 15 de ky B0
1 + d2e k⊥
!
4
4
14
3
, · · · , −iω + νH
, −iω + νH
(4.34)
Mj,j = −iω, −iω, −iω, −iω + νH
15
15
√ ρs
√
√ ρs
ρ2i
1
ρ2s
2
2k
B
k
,
−i
3
15
Mj+1,j = iky B0
+
,
i
k
B
,
·
·
·
,
−i
k
B
y 0 ⊥
y 0
y 0
2
2 2 (Γ (b) − 1)
1 + d2e k⊥
1 + d2e k⊥
de
de
0
(4.35)
where j = 1, · · · , 16.
Our matrix M is a 16 × 16 matrix, so we have 16 solutions and we can do as in chapter 2, that is, we can
represent graphically the damping as function of the frequency for a certain k⊥ . However, this will be done
in the end of the chapter, after analysing all the closures. We will take another closure and compare both
results for k⊥ = 2/de .
Now, we can solve the system for M = 15, the chosen M . The solution can be seen in figure 4.7.
For M = 15, the error at k⊥ de = 2 is
εω = 0.011% ,
εγ = 0.170%.
It is possible to see that for the chosen M , M = 15, the error is very inferior than 1%. This means that there
may be a M < 15 such that the error is approximately 1%. To justify this statement, we can plot the error
εω,γ as function of M . The graphical representation of the error is given by figure 4.8.
37
Figure 4.7: Graphical representation of the dispersion relation with the closure gM +1 = 0 and M = 15. The
hyper-collisions operator was used. The subscript A stands for Analytical Solution, which is represented in
figure 2.2, while the subscript N stands for Numerical Solution, i.e., the solution given by the system of
equations (4.7 - 4.10). ω is the frequency and γ is the damping, the errors for k⊥ de = 2 are εω = 0.011% and
εγ = 0.170%.
Figure 4.8: Error as function of M for the closure gM +1 = 0 at k⊥ de = 2. The red line stands for the error
of the frequency ω and the dashed blue line corresponds to the error of the damping γ. The condition given
by (4.22) is valid for M & 15.
We see an oscillatory behaviour. One justification for these oscillations is the fact that the system is
sensitive to even or odd values of M . We see as well that the error decreases exponentially (a = 0.31 for εω
and a = 0.19 for εγ ), as was said before this is one property of the spectral methods, and so this behaviour
38
is expected.
4.2.2.2
Closure gM +1 = gM −1
For this closure condition, our matrix M will present the 15th row as:
M15,. = 0, · · · , 0,
−i
√
15 +
√ 16 (ρs /de ) ky B0 , −iω + νH .
(4.36)
The graphical representation of the solution of this system for M = 15 is given by figure 4.9.
Figure 4.9: Graphical representation of the dispersion relation for the closure gM +1 = gM −1 and M = 15.
For k⊥ de = 2, the error for the frequency is εω = 0.53% and the error for the damping is εγ = 4.23%.
For this closure condition, the errors at k⊥ de = 2 are
εω = 0.53% ,
εγ = 4.23%.
As it is possible to see, M = 15 is not enough to get an error inferior than 1%, which means that a larger
M is necessary. To discover it, we analyse the behaviour of the error as function of M , which can be seen in
figure 4.10.
As for gM +1 = 0, with this closure the error also presents an oscillatory behaviour. Concerning its
convergence, it is exponential, as expected and equal to a = 0.16 for εω and a = 0.06 for εγ .
4.2.2.3
Closure gM +1 = gM + gM −1 − gM −2
With this closure, the 15th row of matrix M becomes
M15,. = 0, · · · , 0,
√
√
√ √
i 16 (ρs /de ) ky B0 , −i 15 + 16 (ρs /de ) ky B0 , −iω − i 16 (ρs /de ) ky B0 + νH .
(4.37)
For M = 15 and the hyper-collisions operator, we have the dispersion relation, which is represented in figure
4.11. The errors, at k⊥ de = 2, are
39
Figure 4.10: Error as function of M for the closure gM +1 = gM −1 at k⊥ de = 2. The red line stands for the
error of the frequency ω and the blue, dashed line corresponds to the error of the damping γ. An error that
satisfies condition (4.22) is given for M 25.
Figure 4.11: Graphical representation of the dispersion relation for the closure gM +1 = gM + gM −1 − gM −2
and M = 15. The error for the frequency is εω = 5.07% and the error for the damping is εγ = 5.86%, for
k⊥ de = 2.
εω = 5.07% ,
εγ = 5.86%,
which are distant from the required condition, equation (4.22). Plotting the error as function of M (see figure
4.12), we can search a new M such that εω,γ . 1% is satisfied.
Looking at the dispersion relation, we see a jump in the frequency around k⊥ de = 2. This happens
40
because, since we have 16 solutions (remember that we are working with M = 15), near that value of k⊥ de
the real part of the least damped mode, the frequency, is given by a solution that is not the same as for the
previous value of k⊥ de , performing what looks like a jump.
Figure 4.12: Error as function of M for the closure gM +1 = gM + gM −1 − gM −2 at k⊥ de = 2. The red line
stands for the error of the frequency ω and the blue, dashed line corresponds to the error of the damping γ.
An error that satisfies condition (4.22) is given for M 25.
Analysing the error as function of M , we see that we need M 25 to have an error inferior than 1%. We
also see that the error continues with the oscillatory behaviour and that it grows smaller as M grows bigger,
as expected. It also presents the exponential convergence, with a = 0.11 for both εω and εγ .
4.2.2.4
Closure gM +1 = gM
For this closure, the 15th row of matrix M becomes
√
√
M15,. = 0, · · · , 0, −i 15 (ρs /de ) ky B0 , −iω − i 16 (ρs /de ) ky B0 + νH .
(4.38)
For M = 15, its graphical representation can be seen in figure 4.13. We see that for k⊥ de = 2, the error for
the frequency does not satisfy the condition εω . 1%, although it is very close:
εω = 1.31% ,
εγ = 0.58%.
So, we can plot the error as function of M and look for the M that satisfies that condition. The graphical
representation of the error can be seen in figure 4.14 and it presents the same characteristics as the previous
error graphics. We have convergence values of a = 0.062 for εω and a = 0.067 for εγ .
41
Figure 4.13: Graphical representation of the dispersion relation for the closure gM +1 = gM and M = 15.
The error for the frequency is εω = 1.31% and the error for the damping is εγ = 0.58%, for k⊥ de = 2.
Figure 4.14: Error as function of M for the closure gM +1 = gM at k⊥ de = 2. The red line stands for the
error of the frequency ω and the blue, dashed line corresponds to the error of the damping γ. An error that
satisfies condition (4.22) is given for M ≥ 22.
4.2.2.5
Closure gM +1 = 2gM − gM −1
For this closure, the 15th row of M becomes
√
√ √
M15,. = 0, · · · , 0, −i 15 − 16 (ρs /de ) ky B0 , −iω − i2 16 (ρs /de ) ky B0 + νH .
42
(4.39)
The graphical representation of its solution can be seen in figure 4.15. For M = 15, the errors at k⊥ de = 2
are
εω = 3.40% ,
εγ = 8.06%,
which are much bigger than 1%. So, once again, we must find an M such that the error condition is satisfied
and for that we plot the error as function of M , which is represented in figure 4.16.
Figure 4.15: Graphical representation of the dispersion relation for the closure gM +1 = 2gM − gM −1 and
M = 15. The error for the frequency is εω = 3.40% and the error for the damping is εγ = 8.06%, for
k⊥ de = 2.
Once again the error decreases with M and presents an oscillatory behaviour. It presents the expected
exponential convergence, with a = 0.13 for εω and a = 0.05 for εγ .
4.2.2.6
Closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3
For this closure condition, the 15th row of matrix M is now given by:
M15,. = 0,
··· ,
0,
√
i 16 dρs ky B0 ,
e
√
−i4 16 dρs ky B0 ,
e
−i
√
√ 15 − 6 16
ρs
k B ,
de y 0
√
−iω − i4 16 dρs ky B0 + νH
e
.
(4.40)
We can plot its solution, which is represented in figure 4.17. For M = 15, we have the following errors at
k⊥ de = 2:
εω = 1.05% ,
εγ = 17.73%.
Both errors are superior than 1%, which means that we need a larger M to get an error inferior than 1%.
So, plotting the error as function of M (figure 4.18), we need M 25 to get an acceptable error. The error
presents the same characteristics as before, including the exponential convergence, with a = 0.08 for εω and
a = 0.13 for εγ .
43
Figure 4.16: Error as function of M for the closure gM +1 = 2gM − gM −1 at k⊥ de = 2. The red line stands
for the error of the frequency ω and the blue, dashed line corresponds to the error of the damping γ. An
error that satisfies condition (4.22) is given for M 25.
Figure 4.17: Graphical representation of the dispersion relation for the closure gM +1 = 4gM − 6gM −1 +
4gM −2 − gM −3 and M = 15. The error for the frequency is εω = 1.05% and the error for the damping is
εγ = 17.73%, for k⊥ de = 2.
4.2.2.7
Closure gM +1 = −
p
M/ (M + 1)gM −1
The 15th row of M, for this closure, becomes
M15,. = 0, · · · , 0, −iω + νH .
44
(4.41)
Figure 4.18: Error as function of M for the closure gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 at k⊥ de = 2.
The red line stands for the error of the frequency ω and the blue, dashed line corresponds to the error of the
damping γ. An error that satisfies condition (4.22) is given for M 25.
Its solution is represented in figure 4.19. The errors are
εω = 3.16% ,
εγ = 8.11%.
Once again, we need to search for a larger M such that we have an error inferior than 1%. For that, we plot
the error as function of M , as can be seen in figure 4.20. The error as function of M is similar to the previous
representations for the other closures, presenting the expected exponential convergence with a = 0.13 for εω
and a = 0.10 for εγ .
The error presents the same characteristics as the previous error graphics and it is possible to see that
we need M 25 to get an acceptable error.
4.2.2.8
√
Closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM
For this closure, the linearization needs to be redone, since we have two Poisson brackets, one inside another.
To remember how to linearize, please take another look at the first pages of this chapter. After some
straightforward algebra, we arrive at a new version of the 15th row of M:
2
√
ρ
M
+1
2
2
s
M15,. = 0, · · · , 0, −i 15 (ρs /de ) ky B0 , −iω + D
B0 ky + νH
de
M +1
(4.42)
where Dm is given by (2.70). We can now solve the system and its solution can be seen in figure 4.21.
The errors for M = 15 at k⊥ de = 2 are
εω = 0.26% ,
εγ = 0.84%,
45
p
Figure 4.19: Graphical representation of the dispersion relation for the closure gM +1 = − M/ (M + 1)gM −1
and M = 15. The error for the frequency is εω = 3.16% and the error for the damping is εγ = 8.11%, for
k⊥ de = 2.
p
Figure 4.20: Error as function of M for the closure gM +1 = − M/ (M + 1)gM −1 at k⊥ de = 2. The red line
stands for the error of the frequency ω and the blue, dashed line corresponds to the error of the damping γ.
An error that satisfies condition (4.22) is given for M 25.
which are inferior than 1%. This means that maybe there is M < 15 such that the error is closer to 1%. For
that, we plot the error as function of M , that can be seen in figure 4.22.
46
Figure √4.21:
Graphical
of the dispersion relation for the closure gM +1 =
representation
(ρs /de ) M + 1/ (DM +1 ) Ak , gM and M = 15. The error for the frequency is εω = 0.15% and the error for the damping is εγ = 0.45%, for k⊥ de = 2.
√
Figure 4.22: Error as function of M for the closure gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ de = 2.
The red line stands for the error of the frequency ω and the blue, dashed line corresponds to the error of the
damping γ. An error that satisfies condition (4.22) is given for M ≥ 14.
The error as function of M presents the same characteristics as the error for the previous closures.
Concerning the convergence, it continues to be exponential and has a = 0.30 for εω and a = 0.20 for εγ .
As was said in the section of the closure gM +1 = 0, we want to choose another closure and compare both
47
results, when representing graphically the damping as function of the frequency. This can be seen in figure
4.23. The chosen k⊥ was k⊥ = 2/de . We can see that the 16 solutions always come in pairs and that the
least damped mode for both closures and analytical solution coincide.
Figure 4.23:
of the damping as function of the frequency for gM +1 = 0 and gM +1 =
√ Graphical representation
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the hyper-collisions operator and M = 15. In blue
we have the analytical solution, which can also be seen in figure 2.2, in orange
for
√ we have the solutions
the closure gM +1 = 0 and in green we have the solutions for gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM at
k⊥ = 2/de .
As a final exercise, as in the previous subsection, we can plot the damping as function of the frequency
for a chosen closure and different M , to see how it evolves. Once again we chose the nonlinear closure,
√
gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM , with M = 9 and M = 19 and it can be seen in figure 4.24. The
least damped mode for both M ’s coincide with the least damped mode of the analytical solution. This did
not happened for the Lénard-Bernstein collision operator, which can be seen in figure 4.6. Comparing the
spectra for both collision operators, we see that they have a very differente behaviour.
To conclude this section, we can say that the hyper-collisions operator is by far the best operator, when
compared with the Lénard-Bernstein collision operator, to perform numerical simulations. With it we can
reproduce the analytical dispersion relation with a small number of moments and still being able to satisfy the
error condition: εω,γ . 1%. We were also able to observe the exponential convergence that is characteristic of
the spectral methods. With this operator, the number of moments depends on the closure, but the closures
√
that require the less number of moments are gM +1 = 0 and gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM .
Besides that, we can see as well that the convergence rates for both closures are very similar and between all
closures, these are the ones that present a faster convergence.
48
Figure √
4.24: Graphical representation
of the damping as function of the frequency for gM +1 =
(ρs /de ) M + 1/ (DM +1 ) Ak , gM at k⊥ = 2/de for the hyper-collisions operator using M = 9 and M = 19.
In blue we have the analytical solution, which can also be seen in figure 2.2, in orange we have the solutions
for the M = 9 and in green we have the solutions for M = 19 at k⊥ = 2/de .
49
Chapter 5
Conclusions
In this thesis we study different closure conditions in order to truncate the set of coupled fluid-like equations
originated by the expansion in Hermite polynomials of the electron distribution function ge of a reduced
kinetic model, the Kinetic Reduced Electron Heating Model [12]. This model constitutes a minimal physically
realizable paradigm for low-frequency nonlinear plasma dynamics with a strong guide field and includes
important effects, such as ion Finite Larmor Radius, electron inertia, electron Landau damping and collisions.
Three types of closure conditions were introduced: polynomial, linear and nonlinear. The closures were tested
using two dissipation operators: the Lénard-Bernstein collision operator and the hyper-collisions operator.
The latter is used for purely numerical reasons, it introduces an artificial dissipation range in the system,
leading us to a faster convergence.
We started with the Lénard-Bernstein collision operator, whose Hermite transform is Dm = νei m.
Due to its linear dependence on m, we knew that we needed to keep a large number of Hermite polynomials in order to resolve the velocity-space cutoff as νei → 0. Since we were limited up to M = 25,
we focused our attention on three closures: gM +1 = 0, gM +1 = 4gM − 6gM −1 + 4gM −2 − gM −3 and
√
gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM . The first one is very simple and is the usually adopted one, the
second one is an example of a polynomial closure and the third is guaranteed to converge as M → ∞. For
the three closures, we investigated the error for both damping and frequency as function of M . The damping
presented an exponential convergence, as expected, with a very slow convergence. Concerning the error for
the frequency, it presented a very oscillatory behaviour without any convergence. This was justified with the
fact that M = 25 was not enough for the frequency to show convergence. This fact was confirmed with the
collisional cutoff mc = 17.8 for the Hermite spectrum, meaning that M 18 was required.
α
Concerning the hyper-collisions operator, Dm = νH (m/M ) , it is used purely for numerical reasons but
it is able to reproduce important physical effects, such as the Landau damping. The error as function of M
for both the frequency and damping for all closures presented exponential convergence and there was a M
such that the error was inferior than 1%. The closures that presented the best results were gM +1 = 0 and the
√
nonlinear closure, gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM . We were able to reproduce the dispersion
51
relation (and so, the Landau damping) for M < 25. Indeed, we got an error εω,γ . 1% for M = 15, this
agreed with the collisional cutoff mc = 14.6 obtained for this operator.
The behaviour of the dissipation range in the Hermite spectrum allowed us to specify at least how many
M ’s were required. In the case of the hyper-collisions operator, we have gm+1 /gm 1 and so, M & mc was
required. For the Lénard-Bernstein collision operator, gm+1 /gm < 1, leaving us to M mc . Both these
statements agreed with the numerical simulations.
We also represented graphically the damping as function of the frequency for both dissipation operators
√
and the gM +1 = 0 and gM +1 = (ρs /de ) M + 1/ (DM +1 ) Ak , gM closures. The spectra for both operators
presented a very distinct point distribution. With the Lénard-Bernstein collision operator, as M increases,
it looked like the approximate modes converged to the analytical ones, i.e., not only the least damped mode
but also the other modes. Concerning the hyper-collisions operator, the only modes that coincided were the
least damped ones, the others did not looked like they were converging to the analytical ones. Thus, the
possibility of non-physical behaviour in nonlinear simulations (where modes other than the least damped
ones may be important) cannot be ruled out and warrants future investigation.
Future work on this topic includes the changing of the parameters of the hyper-collisions operator and
nonlinear tests for the different closures.
52
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