Efecto del cobre y el fenantreno en la generación de turbidez y la alimentación 51

A plasma wave is an interconnected set of particles and fields that propagate in a periodically repeating fashion. The presence of charge particles with different masses and energies, together with the presence of external magnetic fields in plasmas, creates a

complex system that can support a variety of wave modes that do not exist in free space. Waves in plasmas can be classified as electromagnetic or electrostatic according to whether there exists an oscillating magnetic field. Plasma waves can further be classified based on the oscillating species (i.e. electrons or ions) and based on whether they propagate in an unmagnetized plasma or based on their propagation direction in a magnetized plasma. Plasma wave modes are often uniquely identified by their dispersion relation, which is the relation between the frequency (𝜔) and the wave number (𝑘) of the wave mode. In what follows, we provide a background on the standard procedure that is often followed in order to derive the dispersion relations for various wave modes and introduce the two modes of plasma that are relevant to the research presented in this thesis. This section is not meant to be inclusive. Interested readers are referred to standard plasma physics text books (e.g. Chen [1984]) for detailed discussions.

Interactions of charged particles and fields, which plasma waves are a specific form of, are governed by the Boltzmann equation and the Maxwell’s equations. The Boltzmann equation (equation 2.1) determines how the distribution of particles in physical space and velocity space evolves over time in the presence of a force acting on the particles.

𝜕𝑓(𝑟,v,𝑡) 𝜕𝑡 + v ∙ 𝜕𝑓(𝑟,v,𝑡) 𝜕𝑟 + 𝐹 𝑚∙ 𝜕𝑓(𝑟,v,𝑡) 𝜕v = ( 𝜕𝑓 𝜕𝑡)_𝐶 (2.1)

In this equation 𝑓(𝑟, v, 𝑡) represents the distribution of a population of particles (i.e. electrons or ions) as a function of the three physical (spatial) components 𝑟 = (𝑥, 𝑦, 𝑧), the three velocity components v = (v_𝑥, v_𝑦, v_𝑧), and time t. In this equation F is the force

acting on the particles, m is the mass of the particles, and (𝜕𝑓_𝜕𝑡)

𝐶 is the time rate of change of f due to collisions. Also symbols _𝜕𝑟𝜕 and _𝜕v𝜕 stand for gradients in spatial (physical) space and velocity space, respectively, and are given by:

𝜕 𝜕𝑟≡ 𝜕 𝜕𝑥𝑥̂ + 𝜕 𝜕𝑦𝑦̂ + 𝜕 𝜕𝑧𝑧̂ ≡ ∇ (2.2) 𝜕 𝜕v≡ 𝜕 𝜕v𝑥𝑥̂ + 𝜕 𝜕v𝑦𝑦̂ + 𝜕 𝜕v𝑧𝑧̂ ≡ ∇v (2.3)

𝑓(𝑟, v, 𝑡) means that the number of particle per 𝑚3 _{at position r and time t with velocity} components between v𝑥 and v𝑥+ 𝑑v𝑥, v𝑦 and v𝑦+ 𝑑v𝑦, and v𝑧 and v𝑧+ 𝑑v𝑧 is 𝑓(𝑟, v, 𝑡) dv_x 𝑑v_y dv_𝑧. If we ignore the term due to collisions and further assume that the force F is purely electromagnetic, equation 2.1 takes the form:

𝜕𝑓(𝑟,v,𝑡) 𝜕𝑡 + v ∙ 𝜕𝑓(𝑟,v,𝑡) 𝜕𝑟 + 𝑞 𝑚(E + v × B) ∙ 𝜕𝑓(𝑟,v,𝑡) 𝜕v = 0 (2.4)

This is called the Vlasov equation. The derivation and the meaning of the Vlasov equation can be found in plasma physics text books (e.g. Chen [1984], Chapter 7) in sections that discuss the kinetic theory of plasma.

Motion of charged particles, in turn, may lead to generation of electric current J and accumulation of electric charge 𝜌. These are given by:

𝜌 = ∑ 𝑞𝑗 _𝑗𝑛_𝑗 = ∑ 𝑞𝑗 _𝑗∭ 𝑓𝑗(𝑟, v, 𝑡)dvx 𝑑vy dv𝑧 (2.5)

Where subscript j represents different populations or species of charged particles (i.e. electrons and ions) and 𝑛 and q are the number density and electric charge of each species. According to the Maxwell’s equations (equation 2.7–2.10) accumulation of electric charge and current generate electric and magnetic fields which, in turn, modifies the electromagnetic force that acts on the particles in the Vlasov equation.

∇ × E =−𝜕B_𝜕𝑡 (2.7) ∇ × B = 𝜇₀J +_𝑐1₂𝜕E_𝜕𝑡 (2.8) ∇ ∙ E =_𝜖𝜌

0 (2.9)

∇ ∙ B = 0 (2.10)

The Vlasov equation and the Maxwell’s equations are, therefore, self-consistent equations that describe evolution of particles and fields where particles produce fields as they move along their trajectories and the fields cause the particles to move in those exact trajectories. These equations can be used to identify various wave modes of plasma and derive their dispersion relations.

This approach that takes into account the distributions of electrons and ions in velocity space is called the kinetic theory. Kinetic theory is very successful in describing a variety of plasma processes; however, the level of complexity in this approach limits the extent of analytical solutions that one can derive. Moreover, this approach is often computationally highly expensive. In many cases, the Vlasov equation can be replaced by its moments, the continuity equation and the equation of motion for particles. This approach provides simplifications that enable one to treat the equations analytically and

derive the dispersion relations. Reducing the Vlasov equation to its moments is based on the fluid theory of plasma physics and fluid mechanics in which the identity of the individual particles is neglected as electron and ion populations move as a whole in a form of fluids. In the fluid theory, assuming a constant Maxwellian distribution in velocity, the distribution is uniquely described by a single number, temperature, through averaging. This approach is sufficiently accurate to describe the majority of plasma physics processes.

The lowest moment of the Vlasov equation is obtained by integrating equation 2.4 with respect to velocity, i.e. ∭(𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2.4)𝑑vx𝑑vy𝑑vz. It can be shown that the first moment yields

𝜕𝑛(𝑟,𝑡)

𝜕𝑡 + ∇ ∙ (𝑛𝑢) = 0 (2.11)

Where 𝑛(𝑟, 𝑡) = ∭ 𝑓(𝑟, v, 𝑡)𝑑v𝑥 𝑑v𝑦 𝑑v𝑧 is the number density for the corresponding fluid and 𝑢(𝑟, 𝑡) is the averaged fluid velocity by definition. This equation means that the total number of particles in a volume can change only if there is a net flux of particles across the surface that bounds the volume.

The next moment of the Vlasov equation is obtained by multiplying equation 2.4 by mv and integrating with respect to velocity. This yields the fluid equation of motion that describes the flow of momentum.

In the above equation ∇P is the Stress Tensor that takes into account the thermal motions of the particles relative to the fluid velocity. This term represents the pressure gradient force in thermodynamics. For the wave modes whose existence depends on the presence of this term, an additional equation (equation of state in thermodynamics) is needed that relates P to n and, therefore, a closed system of equations with equal number of equations and unknowns is derived (see Chapter 3 of Chen [1984]).