Energy density of the plasma

Now we want to examine how large the energy density associated with the plasma struc-tures presented in Sec. 2.1.4 is. This energy density will then be compared to the energy density of the corresponding homogeneous plasma. The following calculation is based on that of Ref. 12, but it takes into account the facts that the perturbed state has to be renormalized before it can be compared to the homogeneous state and that it has to be calculated in a new frame of reference. The result obtained in this section was first presented in Refs. 19 and 18.

When the energy density is calculated in the frame of reference in which the homoge-neous state has minimum energy, the Vlasov-Poisson system is said to possess potentially usable free energy if the perturbed state has an energy which is lower than that of the uniform state. The homogeneous state has minimum energy in the frame moving with the center of mass velocity vCM = _1+δ^δ vD. However, for the sake of simplicity, we will prefer the laboratory frame in which the ions are assumed to be at rest, thus ignoring effects of orderO(δ).

To find the energy densityw of a plasma wave in the laboratory frame we now assume a stationary wave with periodicity 2L and phase velocityv₀:

w = 1 andf_i(x⁰, u) in (2.58) when transformed to wave frame coincide with (2.7) renormalized by (2.56), (2.57). This means thatw is equal to the dimensional energy density, normalized bykT_en⁽ⁿ⁾_e0 , where n⁽ⁿ⁾_e0 is given by (2.57).

In the unperturbed homogeneous statew in (2.58) is found to be which follows simply from the Maxwellian distribution functions. There is no contribution from the electric field.

If a hole structure is present, the distribution functions in the laboratory framef_e(x⁰, v) and f_i(x⁰, u), which are related to the ones in the wave frame (2.7) by a Galilean trans-formation, have to be considered. The energy density can then be calculated as follows.

First, we calculate the quantity

w_e(Φ) = Z +∞

−∞

dv v²f_e(v) (2.60)

which is twice the kinetic energy density of the electrons in the laboratory frame (we dropped the x⁰-dependence). Next, w_e is written in terms of ˜v = v−v₀, the velocity in

The quantity wb_e introduced in Eq. (2.61) is again twice the kinetic energy density of the electrons, this time measured in the wave frame. To findw_ewe differentiate Eq. (2.61) by Φ, simplify and then integrate again. The differentiation yields

d can be obtained by integrating the stationary Vlasov equation in the wave frame over the velocity ˜v:

On the left hand side of Eq. (2.63) the differentiation∂_xis replaced by ^dΦ(x)_{dx dΦ}^d . Eq. (2.63) then reads the rhs of (2.62) vanishes.

Eq. (2.62) can therefore be simplified to the following expression:

d

dΦw_e(Φ) =n_e(Φ) +v²₀n⁰_e(Φ). (2.65)

ENERGY DEFICIT AND NEGATIVE ENERGY STRUCTURES 23 Integration of (2.65) over Φ yields w_e:

w_e(Φ) = Z Φ

0

n_e(Φ)dΦ +v²₀[n_e(Φ)−n_e(0)] +w_e(0) (2.66) The only quantity which remains to be determined is w_e(0). It is calculated using the definition (2.60), setting Φ = 0 in (5a) of Ref. 12. The result is

w_e(0) = (1 +K) 1 +v_D²

, (2.67)

where K = k²₀Ψ/2. The ionic term w_i can be calculated using similar arguments. In analogy to (2.65) one finds:

d

dΦw_i(Φ) =−n_i(Φ) + u²₀

θ n⁰_i(Φ), (2.68)

wherew_i refers to the second integral in (2.58), including the factor 1/θ(note thatw_i was defined without this factor in Ref. 12).

In this case, however, it is easier to calculate w_i(Ψ) than w_i(0). For this reason the limits of integration are chosen as follows:

w_i(Φ) = Z Ψ

Φ

n_i(Φ)dΦ + u²₀

θ [n_i(Φ)−n_i(Ψ)] +w_i(Ψ). (2.69) w_i(Ψ) is again found from the definition analogous to (2.60), setting Φ = Ψ in the distri-bution function (5b) of Ref. 12:

w_i(Ψ) = (1 +A)

θ , (2.70)

where Ais given by (21) in Ref. 12. Thus the contribution of the kinetic energy terms to the total energy density is known.

We now insert the contribution of the kinetic energy of the electrons, given by (2.66) and (2.67), as well as the contribution of the kinetic energy of the ions, given by (2.69) and (2.70) into the expression for the total energy density (2.58). The contribution of the field energy is rewritten by replacing Φ⁰(x)² by the classical potential V(Φ). Also, in the contribution of the ion kinetic energy, the integralRΨ

Φ is replaced byRΨ 0 −RΦ

0 . This allows us to use Poisson’s equationn_e−n_i= Φ⁰⁰(x), which in turn can be expressed by −V⁰(Φ).

After performing the (trivial) integration over Φ this contribution can be combined with the one from the field energy. Finally, the integration overxis performed for those terms not depending on space, and one gets an exact expression for the total energy density:

w= 1

Before the energy density (2.71) can be compared to the energy density of a homoge-nous plasma as given by (2.59), one has to make sure both expressions are given in the

same normalization. We remember that w is equal to the dimensional energy density, normalized bykT_en⁽ⁿ⁾_e0 , with n⁽ⁿ⁾_e0 given by (2.57). If the same standardized normalization kT_en_e0 is to be used instead,w has to be divided by M (2.55). It is found by inspection that M can be written as

M =: 1 + (2.72)

with of order O(Ψ). As we will consider only the small amplitude limit we make use of 1/(1 +) ≈ 1−. Thus the total energy density of a structured plasma in the same normalization as was used for the homogeneous plasma is given by

w_S ≈ 1−

Note that this renormalization was not taken into account in Ref. 12.

We now introduce the difference in the energy density, ∆ ˜w by

∆ ˜w :=w_S−w_H. (2.74)

We will show later that it is possible to find situations where ∆ ˜w is negative, which means that the perturbed state is then energetically lower than the unperturbed state. This is what we will call a negative energy state.

After insertion of (2.59) and (2.73) into the definition of ∆ ˜wthe small amplitude limit, Ψ 1, is taken, neglecting all contributions of order O(Ψ²) and higher. The third part of Eq. (2.73) can be transformed using

Z +L

By inspection it can be seen that the contribution of (2.75) to (2.73) is of orderO(Ψ²) and can be neglected. Note that this implies that the contribution of the electrostatic energy is of higher order than that of the nonlinear trapping effect. Thus particle trapping obviously is not a small correction, but the dominant effect even at infinitesimal wave amplitude!

Note that A, K and are of order O(Ψ). By inserting n_e(Φ) and n_i(Φ), given by (20a,b) in Ref. 12, into (2.73) we then get the final expression for the energy density difference:

ENERGY DEFICIT AND NEGATIVE ENERGY STRUCTURES 25 This expression was first presented in Ref. 19. It allows us to search for regions in the parameter space where ∆ ˜w is negative, i.e. where negative energy modes exist. This will be done in the following section.