The Perturbed Ion Equation of Motion Using Vectors

The dynamics of wave phenomena in plasma physics are significantly influenced by two primary factors: ion inertia, which fosters wave growth, and pressure gradients, which induce decay. The pressure gradient contributes to the diffusion of density fluctuations, leading to the gradual smoothing of these variations over time. If left unchecked, this process results in the decay of density structures within the plasma.

In contrast, ion inertia plays a critical role in the accumulation of positive ions at the trailing edge of a density blob, where positive charge density is already heightened. This accumulation increases the amplitude of the structures as the inertia amplifies the electric field perturbation, denoted as $delta E$. The underlying mechanism involves the interaction between polarized structures and the ions in their vicinity. As these structures move through a region, ions attempt to catch up with electrons to neutralize the electric field. However, the acceleration of ions is often insufficient to keep pace, resulting in their accumulation in areas of increased positive charge density. This feedback loop can become pronounced when the structures are fast-moving or sufficiently narrow, causing the background ions to amplify the electric field rather than diminish it. This positive feedback is central to the instability mechanism; however, it must occur rapidly enough to counteract diffusion, which otherwise leads to wave decay.

To mathematically describe these processes, we must consider second-order effects from both ion inertia and pressure gradient forces. For a wave propagating in a plane perpendicular to the magnetic field vector $mathbf{B}$, diffusion acts in both the x and y directions. Thus, it is beneficial to express the perturbed ion equation of motion in vector form:

begin{equation}
frac{partial mathbf{delta V_i}}{partial t} - frac{e delta mathbf{E}}{m_i} = -nu_i delta mathbf{V_i} - frac{nabla p_i}{n_0 m_i}
label{ion_cont_1}
end{equation}

Recognizing that $Omega_i = frac{eB}{m_i}$ and $p = n k T_i$, we can reformulate the equation as:

begin{equation}
frac{partial delta mathbf{V_i}}{partial t} - frac{delta mathbf{E}}{B} Omega_i = nu_i delta mathbf{V_i} - C_i^2 nabla frac{delta n}{n_0}
label{ion_cont_2}
end{equation}

Here, $C_i = sqrt{frac{k T_i}{m_i}}$ represents the thermal speed of the ions. By taking the divergence of equation ref{ion_cont_2}, we obtain:

begin{equation}
nabla cdot Bigg[ frac{partial delta mathbf{V_i}}{partial t} + nu_i delta mathbf{V_i} Bigg] = Omega_i nabla cdot frac{delta mathbf{E}}{B} - C_i^2 nabla^2 frac{delta n}{n_0}
label{div_ion_cont_1}
end{equation}

Utilizing the perturbed ion continuity equation, $frac{partial delta n}{partial t} = -n_0 nabla cdot delta mathbf{V_i}$, we derive:

begin{equation}
- frac{partial^2}{partial t^2} frac{delta n}{n_0} - nu_i frac{partial}{partial t} frac{delta n}{n_0} = Omega_i nabla cdot frac{delta mathbf{E}}{B} - C_i^2 nabla^2 frac{delta n}{n_0}
label{div_ion_cont_3}
end{equation}

To find an expression for the electric field perturbation $delta mathbf{E}$, we can reference the electron continuity equation:

begin{equation}
frac{partial}{partial t} delta n = -mathbf{V_e} cdot nabla delta n - n_0 nabla cdot delta mathbf{V_e}
label{partial_conti}
end{equation}

Incorporating diffusion and weak electron Petersen currents, we adjust the relevant terms and arrive at:

begin{equation}
delta mathbf{V_e} = -frac{nu_e}{Omega_e} frac{delta mathbf{E}}{B} - frac{nu_e}{Omega_e^2} frac{nabla p_e}{n_0 m_e}
label{delta_ve_cont}
end{equation}

Taking the divergence of equation ref{delta_ve_cont} and applying equation ref{partial_conti}, we find:

begin{equation}
nabla cdot frac{delta mathbf{E}}{B} = frac{Omega_e}{nu_e} Big[ frac{partial}{partial t} + mathbf{V_e} cdot nabla Big] frac{delta n}{n_0} - frac{1}{Omega_e} C_e^2 nabla^2 frac{delta n}{n_0}
label{nabla_E_B}
end{equation}

where $C_e^2 = frac{k T_e}{m_e}$. Combining this with equation ref{div_ion_cont_3} leads us to:

begin{equation}
- frac{psi}{nu_i} frac{partial^2}{partial t^2} frac{delta n}{n_0} = (1 + psi) frac{partial}{partial t} frac{delta n}{n_0} + mathbf{V_e} cdot nabla frac{delta n}{n_0} - frac{psi}{nu_i} C_s^2 nabla^2 frac{delta n}{n_0}
end{equation}

In this equation, $C_s^2 = left( frac{Omega_i}{Omega_e} right) C_e^2 + C_i^2 = frac{k(T_e + T_i)}{m_i}$ represents the square of the ion-acoustic speed of the plasma.

Table 1: Summary of Key Variables

In conclusion, the perturbed ion equation of motion encapsulates the dynamics of wave propagation in plasma, capturing the intricate balance between growth mechanisms driven by ion inertia and decay processes resulting from pressure gradients. Understanding this balance is crucial for advancing our knowledge in plasma physics and its applications.