2. INSTABILITY ANALYSIS

Consider plasma with equilibrium electron and ion densities being given as n _eo and n _io, respectively, immersed in a static magnetic field B_s in the z-direction. The charge, mass, and temperature of the electrons and ions are denoted by (-e, m _e, T _e) and (e, m _i, T _i), respectively. We consider a low frequency electrostatic wave, such as a lower hybrid mode, propagating nearly perpendicular to the external magnetic field with propagation vector k lying in the x-z plane. An ion beam is considered propagating along x-axis perpendicular to the magnetic field with density n _bo and equilibrium beam velocity v _bo$\hat {\rm x}$. The quasi-neutrality condition at equilibrium is given by en _io + en _bo = en _eo. The equilibrium is perturbed by an electrostatic perturbation with potential

(1)$${\rm {\rm \phi} } = {\rm {\rm \phi} }_0 {\rm e}^{{\rm - i\lpar {\rm \omega} t - k}_{\rm x} {\rm x - k}_{\rm z} {\rm z\rpar }}.$$

The response of plasma electrons to the perturbation is governed by the equations of motion and continuity, which on linearization yield velocity and density perturbations

(2)$${\rm v}_{{\rm x1}}=- \displaystyle{{{\rm ek}_{\rm x} {\rm {\rm \omega} {\rm \phi} }} \over {{\rm m}_{\rm e} \left({{\rm {\rm \omega} }^2 - {\rm {\rm \omega} }_{{\rm ce}}^2 } \right)}}\comma \;$$

(3)$${\rm v}_{{\rm y1}} = - \displaystyle{{{\rm ek}_{\rm x} {\rm {\rm \phi} {\rm \omega} }_{{\rm ce}}^{} } \over {{\rm im}_{\rm e} \left({{\rm {\rm \omega} }^2 - {\rm {\rm \omega} }_{{\rm ce}}^2 } \right)}}\comma \;$$

(4)$${\rm v}_{{\rm z1}}=- \displaystyle{{{\rm ek}_{\rm z} {\rm {\rm \phi} }} \over {{\rm m}_{\rm e} {\rm {\rm \omega} }}}\comma \;$$

and

(5)$${\rm n}_{{\rm e1}}=- \displaystyle{{{\rm n}_{{\rm eo}} {\rm e{\rm \phi} }} \over {{\rm m}_{\rm e} }}\left[{\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm {\rm \omega} }^2 - {\rm {\rm \omega} }_{{\rm ce}}^2 }}+\displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm {\rm \omega} }^2 }}} \right]\comma \;$$

where ${\rm {\rm \omega} }_{\rm ce} \left(= \displaystyle{{\rm eB}_{\rm s} } \over {{\rm m}_{\rm e} {\rm c}} \right)$ is the electron cyclotron frequency and subscript 1 refers to perturbed quantities.

The response of the plasma ions can be obtained from Eq. (5) by replacing –e, m _e, ω_ce by e, m _i, ω_ci, respectively

(6)$${\rm n}_{{\rm i1}}=\displaystyle{{{\rm n}_{{\rm io}} {\rm e{\rm \phi} }} \over {{\rm m}_{\rm i} }}\left[{\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm {\rm \omega} }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 }}+\displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm {\rm \omega} }^2 }}} \right].$$

The response of beam ions to the perturbation is governed by the equation of motion

(7)$$\displaystyle{{\partial {\bf v}} \over {\partial {\rm t}}} + \lpar {\bf v} \cdot {\bf \nabla }\rpar {\bf v}=\displaystyle{{\rm e} \over {{\rm m}_{\rm i} }}\lpar {\bf E+}\displaystyle{1 \over {\rm c}}{\bf v} \times {\bf B}_{\rm s} \rpar.$$

On linearization, Eq. (7) yields the perturbed beam velocities

(8)$${\rm v}_{{\rm bx1}} = \displaystyle{{{\rm ek}_{\rm x} {\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar {\rm \phi}+v}_{{\rm b0}} {\rm {\rm \omega} }_{{\rm ci}}^2 } \over {{\rm m}_{\rm i} {\rm \lsqb \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 \rsqb }}\comma \;$$

(9)$$\eqalign{{\rm v}_{{\rm by1}}&=\displaystyle{{{\rm v}_{{\rm b0}} {\rm {\rm \omega} }_{{\rm ci}}^{} } \over {{\rm i\lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }}}+\displaystyle{{{\rm ek}_{\rm x} {\rm {\rm \phi} {\rm \omega} }_{{\rm ci}}^{} } \over {{\rm im}_{\rm i} {\rm \lsqb \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 \rsqb }} \cr & \quad +\displaystyle{{{\rm v}_{{\rm b0}} {\rm {\rm \omega} }_{{\rm ci}}^3 } \over {{\rm i\lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar \lsqb \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 \rsqb }}\comma \; }$$

and

(10)$${\rm v}_{{\rm bz1}}=\displaystyle{{{\rm ek}_{\rm z} {\rm {\rm \phi} }} \over {{\rm m}_{\rm i} {\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar }}.$$

Substituting the perturbed velocities given by Eqs. (8), (9), and (10) in the equation of continuity

(11)$$\displaystyle{{\partial {\rm n}} \over {\partial {\rm t}}}+{\bf \nabla }{\rm \cdot }\lpar {\rm n}{\bf v}{\rm \rpar } = 0\comma \;$$

we obtain the perturbed beam density as

(12)$$\eqalign{{\rm n}_{{\rm b1}}&=\displaystyle{{{\rm n}_{{\rm b0}} {\rm ek}_{\rm x}^2 {\rm {\rm \phi} }} \over {{\rm m}_{\rm i} {\rm \lsqb \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 \rsqb }}+\displaystyle{{{\rm n}_{{\rm b0}} {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm {\rm \omega} }_{{\rm ci}}^2 } \over {{\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar \lsqb \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} {\rm \rpar }^2 - {\rm {\rm \omega} }_{{\rm ci}}^2 \rsqb }} \cr & \quad +\displaystyle{{{\rm n}_{{\rm b0}} {\rm ek}_{\rm z}^2 {\rm {\rm \phi} }} \over {{\rm m}_{\rm i} {\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2 }}.}$$

Using Eqs. (5), (6), and (12) in the Poisson's equation

(13)$$\nabla ^2 {\rm {\rm \phi} }=4{\rm {\rm \pi} e}{\rm n}_{{\rm e1}} - 4{\rm {\rm \pi} e}{\rm n}_{{\rm i1}} - 4{\rm {\rm \pi} e}{\rm n}_{{\rm b1}}\comma \;$$

we obtain

(14)$$1{\bf } + \left({\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm {\rm \omega} }_{{\rm pe}}^2 } \over {{\rm {\rm \omega} }_{{\rm ce}}^2 }} - \displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm {\rm \omega} }_{{\rm pe}}^2 } \over {{\rm {\rm \omega} }^2 }}} \right)- \displaystyle{{{\rm {\rm \omega} }_{{\rm pi}}^2 } \over {{\rm {\rm \omega} }^2 }}=\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 } \over {{\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2 }}\comma \;$$

where ${\rm {\rm \omega} }_{{\rm pe}} = \left({\displaystyle{{4{\rm {\rm \pi} n}_{{\rm eo }} {\rm e}^2 } \over {{\rm m}_{\rm e} }}} \right)^{{1 / 2}} $, ${\rm {\rm \omega} }_{{\rm pi}} =\left({\displaystyle{{4{\rm {\rm \pi} n}_{{\rm io }} {\rm e}^2 } \over {{\rm m}_{\rm i} }}} \right)^{{1 / 2}}$ are the electron and ion plasma frequencies, respectively, and we have taken ω_ci≪ω≪ω_ce and k_z≪k for beam ions.

Eq. (14) can be rewritten as

(15)$${\rm {\rm \omega} }^{\rm 2} - \displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm k}^2 }}\displaystyle{{\rm 1} \over {\rm K}}{\rm {\rm \omega} }_{{\rm pe}}^2 - \displaystyle{{\rm 1} \over {\rm K}}{\rm {\rm \omega} }_{{\rm pi}}^2 = \displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{1 \over {\rm K}}\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 {\rm {\rm \omega} }^2 } \over {{\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2 }}\comma \;$$

where

$${\rm K} = 1{\bf } + \displaystyle{{{\rm {\rm \omega} }_{{\rm pe}}^2 } \over {{\rm {\rm \omega} }_{{\rm ce}}^2 }}\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}.$$

From Eq. (15), we obtain the dispersion relation as

(16)$$\left({{\rm {\rm \omega} }^{\rm 2} - {\rm {\rm \alpha} }^{\rm 2} } \right)= \displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{1 \over {\rm K}}\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 {\rm {\rm \omega} }^2 } \over {{\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2 }}.$$

In the limit of vanishing beam density Eq. (16) yields

(17)$${\rm {\rm \omega} }^{\rm 2} = {\rm {\rm \alpha} }^{\rm 2}={\rm {\rm \omega} }_{{\rm lh}}^2 \left({{\rm 1+}\displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm m}_{\rm i} } \over {{\rm m}_{\rm e} }}\displaystyle{{{\rm n}_{{\rm eo}} } \over {{\rm n}_{{\rm io}} }}} \right)\comma \;$$

where

(18)$${\rm {\rm \omega} }_{{\rm lh}}^2={{{\rm {\rm \omega} }_{{\rm pi}}^2 } {\left({1{\rm+}\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm {\rm \omega} }_{{\rm pe}}^2 } \over {{\rm {\rm \omega} }_{{\rm ce}}^2 }}} \right)}}.$$

For a plasma with n _eo = n _io, Eq. (17) yields

$${\rm {\rm \alpha} }^{\rm 2} = {\rm {\rm \omega} }_{{\rm lh}}^2 \left({1{\rm+}\displaystyle{{{\rm k}_{\rm z}^2 } \over {{\rm k}^2 }}\displaystyle{{{\rm m}_{\rm i} } \over {{\rm m}_{\rm e} }}} \right).$$

This result is the same as given by Papadopoulous and Palmadesso (Reference Papadopoulos and Palmadesso1976).

In Cerenkov interaction, (ω – k_x v_bo) ≈ 0 and under the resonance condition ω ≈ k_x v_bo. Rewriting Eq. (16), we obtain

$$\left({{\rm {\rm \omega} }^{\rm 2} - {\rm {\rm \alpha} }^{\rm 2} } \right){\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2=\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{1 \over {\rm K}}{\rm {\rm \omega} }_{{\rm pb}}^2 {\rm {\rm \omega} }^2$$

or

$${\rm \lpar {\rm \omega} } - {\rm k}_{\rm x} {\rm v}_{{\rm b0}} \rpar ^2= - \displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}\displaystyle{1 \over {\rm K}}\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 } \over {\left({{{{\rm {\rm \alpha} }^{\rm 2} } / {{\rm {\rm \omega} }^{\rm 2} }} - {\rm 1}} \right)}}.$$

Assuming perturbed quantity ω = k_x v_b0 + δ, where δ = δ_r + iγ we obtain the following expression for the instability growth rate γ (which is the imaginary part of the unstable frequency)

(19)$${\rm {\rm \gamma}} = {\rm {\rm \omega} }_{\rm pb} \displaystyle{{\rm k}_{\rm x} \over {\rm k}}\left[{\rm K}\left(\displaystyle{{\rm {\rm \alpha} }^{\rm 2} \over {{\rm k}_{\rm x}^2 {\rm v}_{\rm bo}^2 }} - 1 \right)\right]^{- 1/ 2}.$$

From Eq. (19), we find that the growth rate is maximum when ω = k_x v_bo, i.e., when the beam mode intersects with the lower hybrid mode. To obtain the maximum growth rate, we assume the perturbed quantities in Eq. (16)

$${\rm {\rm \omega} } = {\rm {\rm \alpha}+{\rm \delta} } \, \hbox{and} \, {\rm {\rm \omega} } = {\rm k}_{\rm x} {\rm v}_{\rm b0} + {\rm {\rm \delta} }\comma \;$$

where δ is the small frequency mismatch.

The maximum growth rate of the unstable mode is obtained as

(20)$${\rm {\rm \gamma} }_{\max } = {\rm Im} \ {\rm \delta} = \displaystyle{{\sqrt 3 } \over 2}\left[{\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 } \over {\rm K}}\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}{\rm {\rm \alpha} }} \right]^{1/3}.$$

The real part of the frequency of unstable wave scales as the square root of the beam energy and in terms of beam voltage V _b is given by

(21)$${\rm {\rm \omega} }_{\rm r} = {\rm k}_{\rm x} \left({\displaystyle{{2{\rm eV}_{\rm b} } \over {{\rm m}_{\rm i} }}} \right)^{1/2} - \displaystyle{1 \over 2}\left[{\displaystyle{{{\rm {\rm \omega} }_{{\rm pb}}^2 } \over {\rm K}}\displaystyle{{{\rm k}_{\rm x}^2 } \over {{\rm k}^2 }}{\rm {\rm \alpha} }} \right]^{1/3}.$$

3. RESULTS AND DISCUSSIONS

We have used the experimental parameters of lower hybrid beam plasma instability for our numerical calculations which are as follows: ion plasma density n _io = 10⁹cm⁻³, guide magnetic field B_S = 320 G, mass of ion m _i = 39 × 1836m _e (Potassium-plasma), temperature of electron T _e = 3 eV and temperature of ion T _i = 0.2 eV.

We have plotted the dispersion curves of lower hybrid waves in Figure 1, where the normalized frequency ω/ω_pi taken from Eq. (17) is plotted as a function of normalized wave number k_x/k. We have also plotted in Figure 1, the beam mode for beam velocity = 2.7 × 10⁶ cm/s. The velocity of the beam mode is chosen in such a way so that it intersects with the lower hybrid mode where k_x ≈ k or k_z≪k, a condition necessary for the existence of lower hybrid waves. It is observed that the wave frequency decreases with an increase in the value of k_x/k. The unstable wave number decreases and the unstable frequency increases with an increase in beam velocity, similar to the results of Idehara and Tomita (Reference Idehara and Tomita1986).

Fig. 1. (Color online) Dispersion curve of lower hybrid wave over a magnetized plasma and beam mode.

We have also plotted the dispersion curves of lower hybrid waves for different plasma parameters in Figure 2. It can be seen from the figure that the frequency decreases with the normalized wave number and therefore the beam velocity required for the excitation of lower hybrid wave decreases with an increase in the electron plasma density. However, a decrease in the electron plasma density from quasi-neutral condition does not alter the frequency too much. It is also found that an increase in the magnetic field increases the frequency and hence the beam velocity required for excitation increases with an increase in magnetic fields.

Fig. 2. (Color online) Dispersion curves of lower hybrid wave for (1) n_eo = 0.001 × n_io, (2) n_eo = 0.1 × n_io, (3) n_eo = n_io, (4) n_eo = 10 × n_io and (5) n_eo = 1000 × n_io.

In Figure 3, we have plotted the normalized frequency ω/ω_pi taken from Eq. (17) as a function of $\left( \left({\rm k}_{\rm z} / {\rm k} \right)\sqrt {{\rm m}_{\rm i} / {\rm m}_{\rm e}} \right)$ for the same parameters used for plotting Figure 1, so that we can compare our theoretical results quantitatively with the experimental observations of Chang (Reference Chang1975) on lower-hybrid instability.

Fig. 3. (Color online) Dispersion curve of lower hybrid wave over magnetized plasma: ω/ω_pi vs. $\displaystyle{{\rm k}_{\rm z} \over {\rm k}} \left({\displaystyle{{{\rm m}_{\rm i} } \over {{\rm m}_{\rm e} }}} \right)^{1/2}.$

Figure 4 shows the plots of growth rate γ versus $\left(\left({\rm k}_{\rm z} / {\rm k} \right) \sqrt {{\rm m}_{\rm i} / {\rm m}_{\rm e}} \right)$ for different values of beam velocities using Eq. (19). From Eq. (19), we find that the maximum growth rate occurs when the phase velocity ω/k_x is comparable to the ion beam velocity, as is observed in Figure 4. This is in good agreement with Figures 1 and 3, where the beam mode intersects with the lower hybrid wave mode to make it unstable. The point of intersection between the beam mode and the lower hybrid mode corresponds to ω/ω_pi = 0.573 and k_x/k = 0.999978, which gives the value of ω/k_z= 2.9 × 10⁸ cm/s. This value of ω/k_z is comparable to the electron thermal velocity v_te = (2T_e/m_e)^1/2 ≈ 1.03 × 10⁸ cm/s. We find that the maximum growth rate occurs when ω/k_z ≈ v_te in agreement with previous investigations (Chang, Reference Chang1975).

Fig. 4. (Color online) Growth rate γ of the unstable mode as a function of $\displaystyle{{{\rm k}_{\rm z} } \over {\rm k}}\left({\displaystyle{{{\rm m}_{\rm i} } \over {{\rm m}_{\rm e} }}} \right)^{1/2} $ for different ion beam velocities.

Since the growth rate is maximum when the parallel-wave phase velocity of the lower hybrid wave is approximately of the order of the electron thermal velocity, we expect efficient energy transport from the perpendicular ion beam to the bulk of the electrons via Landau damping, accelerating the electrons or heating them. The emission caused by this energetic electron component could explain the observed X-ray spectra (Bingham et al., Reference Bingham, Dawson and Shapiro2002). The spectrum of instability is quite broad, extending from the region $\left(\left({\rm k}_{\rm z} / {\rm k} \right)\sqrt {{\rm m}_{\rm i} / {\rm m}_{\rm e}} \right)\simeq0.3$ to 1.2, and damps out after this value. It can be seen from Figure 4 that the beam modes with velocities 2.65 × 10⁶ cm/s [cf. Fig. 4(1)] and 2.68 ×10⁶ cm/s [cf. Fig. 4(2)] do not show any growth rate. As the velocity increases from 2.69 × 10⁶ cm/s [cf. Fig. 4(3)] to 2.70 × 10⁶ cm/s [cf. Fig. 4(5)], the maximum value of the growth rate increases from 8.6 × 10⁷ rad/s to 13.7 × 10⁷ rad/s. The growth rate is maximum for $\left(\left({\rm k}_{\rm z} / {\rm k} \right)\sqrt {{\rm m}_{\rm i} / {\rm m}_{\rm e}} \right)\simeq0.84$, which gives k_x/k = 0.999974 and which corresponds to the point of intersection of beam mode and lower hybrid mode (cf. Fig. 1). For any further increase in the beam velocity, the beam does not interact with the plasma mode in the lower hybrid range. A similar explanation goes for beam velocities less than or equal to 2.68 × 10⁶ cm/s. These results are similar to the experimental results of Chang (Reference Chang1975), where the maximum growth rate has been observed for $\left(\left({\rm k}_{\rm z} / {\rm k} \right)\sqrt {{\rm m}_{\rm i} / {\rm m}_{\rm e}} \right)\simeq1.5$ and ω/k_z = 10⁸ cm/s.

It is found that the beam velocities required for the excitation of lower hybrid wave using an ion beam are comparatively 10 to 100 times lower than the beam velocities required for excitation using an electron beam. Allen et al. (Reference Allen, Owens, Seiler, Yamada, Ikezi and Porkolab1978) have biased the beam-ionizer plate at 12 V in their experiment on parametric excitation of electrostatic lower-hybrid and ion-cyclotron modes by modulated electron-beam, giving a beam velocity of 2.05 × 10⁸ cm/s. Papadopoulos and Palmadesso (Reference Papadopoulos and Palmadesso1976) have considered electron beam velocity of 3 × 10⁹ cm/s. Idehara and Tomita (Reference Idehara and Tomita1986) have considered beam velocities ranging from 2.4 × 10⁶ cm/s to 3.3 × 10⁶ cm/s in an ion beam-plasma system and a test wave was found to become unstable near the lower-hybrid frequency in this velocity range. The velocity range observed in this article for the growth of lower hybrid wave also lies in this range.

This lower hybrid instability may also be relevant to the enhanced backscatter from the Space Shuttle exhaust as given in Bernhardt et al. (Reference Bernhardt, Ganguli, Kelley and Swartz1995). Assuming an oxygen plasma with density of O⁺ ions in the plume of order of 2 × 10⁵cm⁻³ and magnetic field B_S = 0.35 G, we have ω_pi/ω_ci = 702 and with ω_pe/ω_ce = 4.1 (for n_eo = n_io), we have a lower hybrid frequency of about 3.5 × 10⁴ rad/s. The corresponding beam velocity for maximum growth rate is 3.4 × 10³ cm/s.

It is interesting to notice that the maximum growth rate increases with an increase in perpendicular ion beam velocity but it decreases as the parallel electron beam velocity increases (Papadopoulos & Palmadesso, Reference Papadopoulos and Palmadesso1976). It is because the maximum growth rate is directly proportional to the unstable wave frequency in both the cases and the unstable frequency decreases with an increase in parallel electron velocity but increases with an increase in perpendicular ion beam velocity.

We have plotted in Figure 5, the maximum growth rate of the lower hybrid wave instability as a function of beam density. The maximum growth rate of the unstable mode increases with the beam density and scales as the one-third power of the beam density.

Fig. 5. (Color online) Maximum growth rate γ_max of the unstable mode as a function of n_bo.