Diferencias entre machos y hembras en aprendizaje y memoria

We have so far explained two different possible descriptions of a plasma. When deal- ing with global quantities and equilibrium, looking at the plasma as one single fluid, thus using MHD, is appropriate. If one wants to study in detail what happens to single par- ticle orbits, one should consider a guiding centre drift description. There are, however, phenomena where a great number of particles are involved, but each particle is affected differently according to its position in the

(

x, v

)

phase space. For such phenomena, it is much more appropriate to use a statistical approach, such as kinetic theory. Here, it is necessary to know the distribution function in phase space, f

(

x, v, t

)

, which gives (depending on its normalisation) the total number of particles or unity when integrated

over the six-dimensional phase space. f must be a continuous function of its arguments,

positive and finite at any instant of time, and must tend to zero as the velocity becomes infinitely large. For instance, in a highly collisional plasma, the thermal equilibrium state takes the form of a Maxwellian distribution with density and temperature profilesn

(

s

)

and

T

(

s

)

of the form

fM

(

s, E

) =

n

(

s

)

m 2πT

(

s

)

e

−

E/T

(

s

)

_{. (2.49)}

Here, the normalisation is chosen such that the integral over velocity space yields the density. The evolution equation for the distribution function is the Boltzmann equation

d f dt

=

∂ f ∂t

+

v

· ∇

f

+

F m

·

∂ f ∂v

=

∂ f ∂t     c , (2.50)

where the LHS denotes the total time derivative and the RHS dissipation due to colli- sions. Again the force is the Lorentz force, F

=

Q

(

E

+

v

×

B

)

. If there are different species inside the plasma with different mass and charge, Eq. (2.50) has to be satisfied for each species separately. However, the electromagnetic fields in the Lorentz force are the same for all species, thus linking the state of the different species together into a set of equations. Often, the collisional term is neglected, and the resulting equation is the Vlasov equation d f dt

=

∂ f ∂t

+

v

· ∇

f

+

F m

·

∂ f ∂v

=

0. (2.51)

An important difference to MHD is the distinction between particles with different ve- locities. With the distribution function, the whole velocity range is covered, whereas the fluid description only takes into account an average velocity of a fluid element (consisting of many particles). The most striking difference is in the investigation of the stability of different excitations, where the kinetic theory can ”resolve” instabilities due to a difference in velocity of the particles within the plasma. The link between the two descriptions can be made through the global quantities like density, pressure and current. In the kinetic descriptions, these are defined as moments of the distribution function. Explicitly, we can

2.3. Kinetic description 25

write for every speciesi

ni

(

x, t

) =

Z fi

(

x, v, t

)

d3v, (2.52a) ji

(

x, t

) =

Qi Z v fi

(

x, v, t

)

d3v, (2.52b) pi

(

x, t

) =

mi 3 Z v2fi

(

x, v, t

)

d3v, (2.52c) p_⊥i

(

x, t

) =

mi 2 Z v2_⊥fi

(

x, v, t

)

d3v, (2.52d) p_ki

(

x, t

) =

m_i Z v2_kf_i

(

x, v, t

)

d3v. (2.52e)

The factors of one third and one half in the definitions of the thermal and perpendicu- lar pressures come from the fact that there are three indistinguishable directions in the isotropic, and two perpendicular and one parallel directions in the anisotropic case.

The evolution of the system due to heating effects can only be dealt with using kinetic theory, since the wave-particle interactions, and especially the so-called Landau damp- ing, depend on the relative velocities of the particles with respect to the wave propagation. As a consequence, the wave propagation and the time evolution of the system subject to ICRF wave injection is computed using distribution functions. However, solving the Boltz- mann equation (2.50) directly is very difficult, if not impossible, due to the large variety of spatial and temporal scales involved. Usually, the drift kinetic approach is applied, where the Boltzmann equation is averaged over the fast gyro motion. In velocity, space, a spherical coordinate system is then introduced with

(

v_⊥cos α, v_⊥sin α, v_k

)

, where α

is the gyroangle, over which the equation is averaged. Then, the distribution function is usually expanded, separating terms of different spatial scales, where mostly the gyrora- dius serves as small parameter compared to the orbit width.13,14 Especially for ICRF, the wave-particle interactions take place on a scale of the order of the gyroradius, and codes trying to solve the drift kinetic equation directly in terms of f therefore usually have to make approximations in longer spatial scales, as for example neglecting finite orbit width effects. However, these effects can be very important, and we therefore chose a different approach. If we want to retain full finite orbit width effects, we have to follow the parti- cles by integrating their equations of motion. However, the gyroradius is very small with respect to the orbit scale, and in a first approach we have to neglect gyro motion effects, and find equations of motion for the guiding centre. An unperturbed system will conserve energy, if no electromagnetic fields (other than the equilibrium magnetic field) or collisions are present. The wave kinetic equation for such systems reduces to15

v

· ∇

f

=

0, (2.53)

with the distribution function depending only on the constants of motion. The guiding cen- tre trajectories can be described in the Hamiltonian formulation. However, the equations of motion have to be in the correct form, such that the resulting drift velocity vsatisfies Liouville’s theorem. Thus, the evolution of the guiding centre drift orbits must be such

26 2. Theoretical background that16,17 v

=

vk B σB

+ ∇ × (

ρ_kσB

)

 1

+

ρ_kµ0K

·

B/B2 , (2.54)

where ρ_k

=

mv_k/QB is the parallel gyroradius and K and σ have been introduced in

Sec. 2.1.2. Most authors neglect the termρ_kµ0K

·

B/B2, departing from the strict Hamil-

tonian formulation. Additionally, the retention of pressure anisotropy by retainingσandK

is a complete novelty, and the isotropic relation can be found settingσ

=

1 andK

=

j.

The corresponding equations of motion implemented in VENUS are given in Chapter 6. Note that these equations of motion include terms related to electromagnetic perturba- tions happening in the same time and length scales as the guiding centre orbits. This represents a generalisation of Eq. (2.53), but those terms are not considered in this the- sis. In order to solve for the evolution of the distribution function, we have to add the per- turbations corresponding to the collisions and RF wave-particle interactions. This is done using a Monte Carlo approach, which will simulate the Coulomb collisions, i.e. the RHS of Eq. (2.50), and the action of the wave on the distribution, i.e. the termQ/mE

·

∂ f/∂v.

The latter operator will link the dynamics on the gyro time and spatial scale to the evolution on the orbit time scale, and will be derived in the frame of quasilinear theory in Chapter 6. Finally, our approach will be to represent the distribution function by a certain number of particles, then called markers, whereof each samples a part of the distribution function in phase space. Then, we can evolve the marker or particles’ guiding centre orbits together with the constraints coming from collisions with background plasma and wave-particle in- teractions, and simulate in this way the evolution of the distribution function by retaining full orbit width effects.

In document Colettis, Natalia Claudia (página 45-48)