The Perturbed Ion Equation of Motion Using Vectors

The effects which cause growth and decay of the wave are ion inertia and a pressure gradient, respectively. The pressure gradient leads to diffusion, namely, is working to smooth out the density fluctuations. If left to its own, it therefore creates a decay of the density structures over time.

By contrast, ion inertia makes the slow positive ions pile up at the trailing edge a blob where the positive charge density is already enhanced, so that inertia acts to make $delta E$ larger, which increases the amplitude of the structures. At the root of this phenomenon is the fact that when a polarized structure is passing over a particular region, the ions in that region are trying to catch up with the electrons in order to reduce the electric field. However, the ion acceleration is such that they take too long too catch up with the electrons. They end up piling instead in a region with an already increased positive charge density. This effect will gain in importance if the structures are fast and/or narrow enough in which case the background ions will increase the electric field instead of decreasing it. This positive feedback mechanism is at the heart of the instability mechanism. However, this process has to be fast enough to overcome diffusion, which makes the wave decay.

For a mathematical description of the above processes we have to consider second order effects due to ion inertia and the pressure gradient force. For a wave propagating in the plane perpendicular to $mathbf B$, diffusion will nevertheless act in the x and the y directions. Therefore it is convenient to write the perturbed ion equation of motion using vectors:

begin{equation}

frac{partialmathbf{ delta V_i}}{partial t}-frac{edelta mathbf E}{m_i}=-nu_ideltamathbf V_i-frac{nabla p_i}{n_0 m_i}

label{ion_cont_1}

end{equation}

Recalling that $Omega_i=eB/m_i$ and $p=nKT_i$, we get

begin{equation}

frac{partial delta mathbf V_i}{partial t}-frac{delta mathbf E}{B}Omega_i=nu_ideltamathbf V_i-C_i^2nablafrac{delta n}{n_0}

label{ion_cont_2}

end{equation}

where $C_i=sqrt{kT_i/m_i}$ is the thermal speed of the ions. Taking the divergence of eqn. ref{ion_cont_2}, we get

begin{equation}

nablacdotBigg[frac{partialdeltamathbf V_i}{partial t}+nu_ideltamathbf V_iBigg]=Omega_inablacdotfrac{deltamathbf E}{B}-C_i^2nabla^2frac{delta n}{n_0}

label{div_ion_cont_1}

end{equation}

and, with the perturbed ion continuity, $partialdelta n/partial t=-n_0nablacdotdeltamathbf V_i$, we have

begin{equation}

-frac{partial^2}{{partial t}^2}frac{delta n}{n_0}-nu_ifrac{partial}{partial t}frac{delta n}{n_0}=Omega_inablacdotfrac{deltamathbf E}{B}-C_i^2nabla^2frac{delta n}{n_0}

label{div_ion_cont_3}

end{equation}

We can find an expression for the electric field perturbation $deltamathbf E$ by looking at the electron continuity equation,

begin{equation}

frac{partial}{partial t}delta n=-mathbf V_ecdotnabladelta n-n_0nablacdotdeltamathbf V_e

label{partial_conti}

end{equation}

Because we now allow for diffusion and weak electron Petersen currents, we have to add the appropriate terms to eqn.spaceref{pert_ve_approx} and get

begin{equation}

deltamathbf V_e=-frac{nu_e}{Omega_e}frac{deltamathbf 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 eqn.spaceref{delta_ve_cont} and using eqn.spaceref{partial_conti}, we find that begin{equation}

nablacdotfrac{delta mathbf E}{B}=frac{Omega_e}{nu_e}Big[frac{partial}{partial t}+mathbf V_ecdotnablaBig]frac{delta n}{n_0}-frac{1}{Omega_e}C_e^2nabla^2frac{delta n} {n_0}

label{nabla_E_B}

end{equation}

where $C_e^2=k T_e/m_e$. This can be combined with eqn.spaceref{div_ion_cont_3} to give us

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}+ V_ecdotnablafrac{delta n}{n_0}-frac{psi}{nu_i}C_s^2nabla^2frac{delta n}{n_0}

end{equation}

where $C_s^2=(Omega_i/Omega_e)C_e^2+C_i^2=[k(T_e+T_i)]/m_i$ is the square of ion-acoustic speed of the plasma. Then, we have

begin{equation}

Big[frac{partial}{partial t}+frac{mathbf V_e}{1+psi}cdotnablaBig]frac{delta n}{n_0}=-frac{psi}{nu_i(1+psi)}Big[frac{partial^2}{{partial t}^2}frac{delta n}{n_0}-C_s^2nabla^2frac{delta n}{n_0}Big]

label{diffusion_eqn}

end{equation}

Because $mathbf V_e$ is in the x direction, we can write eqn. ref{diffusion_eqn} in the form

begin{equation}

frac{D}{Dt}f=-ABigg[frac{partial^2}{{partial t}^2}-C_s^2frac{partial^2}{{partial x}^2}Bigg]f+A C_s^2frac{partial^2}{{partial y}^2}f

label{convect_derivative}

end{equation}

where $frac{D}{Dt}=frac{partial}{partial t}+Vfrac{partial}{partial x}$ is the convective derivative, or the time derivative when following the wave with velocity $V=(E/B)/(1+psi)$. The constant A is given by

begin{equation}

A=frac{psi}{nu_i(1+psi)}

end{equation}

Since $psiomega$), $A$ makes the right-hand-side of equations (ref{convect_derivative}) small. A multi-scale expansion in time may therefore be used to solve it. This is equivalent to assuming a solution with two independent time scales, one ($tau=t$) for the fast wave oscillations and one ($tau_g=epsilon t$) for the growth of the wave. The small constant $epsilon$ means that $tau$ has to be long enough for $tau_g$ to be felt. Therefore we can split equation (ref{convect_derivative}) into two parts, one for each time scale, recovering in the process the zeroth order description discussed in the previous section. That is to say, write

begin{equation}

f=f_0(tau,tau_g)+epsilon f_1(tau,tau_g)

end{equation}

and the time derivative becomes

begin{equation}

frac{partial}{partial t}=frac{partial}{partial tau}frac{partial tau}{partial t}+frac{partial}{partial tau_g}frac{partial tau_g}{partial t}=frac{partial}{partial t}+epsilonfrac{partial}{partial tau_g}

end{equation}

The fast time scale is just the wave equation without growth

begin{equation}

Big[frac{partial}{partial tau}+Vfrac{partial}{partial x}Big]f_0=0

end{equation}

But for $f_1$, we also have growth, and we get to first order in $epsilon$

begin{equation}

Big[frac{partial}{partial tau}+Vfrac{partial}{partial x}Big]f_1+frac{partial}{partialtau_g}f_0=ABig[C_s^2frac{partial^2}{{partial x}^2}-frac{partial^2}{{partial tau}^2}Big]f_0+A C_s^2frac{partial^2}{{partial y}^2}f_0

end{equation}

begin{equation}

Big[frac{partial}{partial tau}+Vfrac{partial}{partial x}Big]f_1=-frac{partial}{partialtau_g}f_0+ABig[C_s^2frac{partial^2}{{partial x}^2}-frac{partial^2}{{partial tau}^2}Big]f_0+A C_s^2frac{partial^2}{{partial y}^2}f_0

end{equation}

However, $f_0$ is an eigenvalue of $f_1$ because $f_1$ also contains the wave motion itself in addition to growth or decay, which means that the right hand side has to be zero for $ t to infty$, or $f_1$ would grow indefinitely, so that we have to require

begin{equation}

frac{partial}{partial tau_g}f_0=A(C_s^2-V^2)frac{partial^2}{{partial x}^2}f_0+A C_s^2frac{partial^2}{{partial y}^2}f_0

label{time_scale_1}

end{equation}

This is the equation for the growth of the structure. In the y direction it describes a simple diffusion, but in the x direction there is one important difference. In the case of structure propagating only in the x direction, the diffusion coefficient is $D=A(C_s^2-V^2)$, and it can be either positive or negative depending on $V$. If $V C_s$ the diffusion constant is negative, so that we have anti-diffusion (ion-inertia increases the amplitude of the structure, essentially working like diffusion in reverse). We can only see from this expression that is a structure is not elongated enough, diffusion in the y direction gains in importance. Thus, the fastest growing structures are those for which the y derivatives in the density perturbations are much smaller initially than the x derivatives in the density perturbations. Once anti-diffusion is strong enough, however, an elongated structure will continue to be elongated. This is illustrated in Fig. ref{tilt_tower} where regular diffusion elongates the structure in the y direction, but in the x direction, anti-diffusion acts inward and compresses it.

begin{figure}[H]

includegraphics[scale=1.0]{pictures/tilt_tower.pdf}

centering

caption { {Evolution of a structure in the x-y plane under the influence of diffusion in the y direction and ``anti-diffusion" from the Farley-Buneman instability in the x-direction, shown at three different times.}}

label{tilt_tower}

end{figure}

More generally, we can also consider a structure propagating at an angle $alpha$ to the $mathbf Etimes mathbf B$ drift, but still in the x-y plane. This angle is called the ``flow angle". If we use a wave decomposition the wave vector is given by

begin{equation}

k_x=k cos alpha hspace{1cm} k_y=k sinalpha

end{equation}

We can now write eqn. ref{time_scale_1} in the form $frac{partial ln f_0}{partial tau_g}=gamma_{FB}$

with growth rate $gamma_{FB}$ being

begin{equation}

gamma_{FB}=-A(C_s^2-V^2)k_x^2-AC_s^2k_y^2=-Ak^2(C_s^2-V^2cos^2alpha)

end{equation}

and thus with $omega=kVcosalpha$ we find the traditional Farley-Buneman growth rate,

begin{equation}

gamma_{FB}=frac{(omega^2-k^2C_s^2)psi}{(1+psi)nu_i}

label{growth}

end{equation}

subsection{Nonlinear and nonlocal Complications}

In the linear local theory, the growth rate expressed in equation (ref{growth}) is time-independent. This means that a wave would grow indefinitely if $omega^2>k^2C_s^2$, at least in the absence of any wave electric field component along the magnetic field. However, as the amplitude grows and becomes large we need to include the non-linear corrections which one way or the other are expected to limit the amplitude. Furthermore, as is shown in the next chapter, nonlocal effects are such that the parallel electric field of the waves will grow monotonically with time, meaning that at some point the wave amplitude will have to decay. Note that the parallel electric field was not included in the present derivation and that it has an impact on $psi$. The inclusion of the parallel electric field in the derivation and a description of its impact will be presented in Chapter 4.

%A wave could grow very fast to a small amplitude or very slowly to a large amplitude. This let researchers think that a wave can change their frequency or phase velocity as the amplitude grew.

The presence of non-linear and non-local effects means that the growth rate is positive only for a limited amount of time, until either non linear effects come to play or until nonlocal effects take over. Either way, the waves will either reach a peak amplitude and stop growing, or they will simply decay after reaching a maximum amplitude. The observations indicate that the largest amplitudes are typically found at the ion acoustic speed, namely, according to equation (ref{growth}), at the threshold speed (zero growth rate condition), for a wave in the x-y plane. This suggests that the non-linear and/or nonlocal effects have to decrease the velocity of a structure as it grows, until it can no longer grow, at which point it may or may not undergo a decay. The fact that the spectra at the threshold speed are not really narrow means that the structures actually have a finite lifetime after reaching a maximum amplitude. They must therefore undergo decay after reaching a maximum amplitude.

section{Gradient-Drift Growth Mechanism}

If a wave is propagating through a background plasma which has a gradient in the direction of the background electric field (-y in fig.spaceref{Field_direction}), this gradient provides an additional destabilization mechanism. The reason is that the perturbation electric field $deltamathbf E$ creates a $deltamathbf Etimes mathbf B$ drift of the electrons in the $+y$ direction. As a result, electrons are moving from a higher density region at the bottom of the enhancement to a relatively lower density region at the top, thereby increasing the relative density perturbation $delta n/n_0$. Conversely, in a density depletion region, the $delta mathbf E$ field would be directed oppositely, with a $deltamathbf Etimes mathbf B$ drift of the electrons in the $-y$ direction, therefore bringing lower density plasma into an already depleted region, which again enhances the value of $delta n/n_0$. newline

For simplicity, we focus on ambient density gradient effects here, keeping in mind that in the end GD and FB can act together. We take into account the additional term of the perturbed electron velocity, namely $delta V_{e,y}=frac{delta E}{B}$. The continuity equation for the perturbed density from Eqn. ref{partial_ne} can then be written as

begin{equation}

frac{partialdelta n}{partial t}=-frac{partial}{partial x}delta(n_0 V_{e,x})-frac{partial}{partial y}delta(n_0 V_{e,y})

label{conti_e_x_y_1}

end{equation}

However, in the x direction, Eqn. ref{delta_ne_ni} must still be valid and we find that

begin{equation}

frac{partialdelta n}{partial t}=-frac{partial}{partial x}delta(n_0 V_{i,x})-frac{partial}{partial y}(n_0 delta V_{e,y}-V_{e,y}delta n)

label{conti_e_x_y_2}

end{equation}

Neglecting the y derivatives in the perturbed quantities again, and assuming that the unperturbed $V_{e,y}$ is negligible, we find that

begin{equation}

frac{partialdelta n}{partial t}=-frac{partial}{partial x}delta(n_0 V_{i,x})-delta V_{e,y}frac{partial n_0}{partial y}

label{second_order}

end{equation}

We rewrite Eqn. ref{delta_ne_ni} in the form

begin{equation}

frac{delta E}{B}=frac{nu_i}{Omega_i}frac{V_d}{(1+psi)}frac{delta n}{n_0}=frac{nu_i}{Omega_i}Vfrac{delta n}{n_0}

label{growth_1}

end{equation}

and, substituting this result back into Eqn. ref{second_order}, we get

begin{equation}

frac{partialdelta n}{partial t}=-n_0Vfrac{partial}{partial x}frac{delta n}{n_0}-n_0frac{nu_i}{Omega_i}Vfrac{delta n}{n_0}frac{partial n_0/partial y}{n_0}

label{growth_sub}

end{equation}

Next we define the background density scale $L$ in the y direction via,

begin{equation}

L=-left(frac{1}{n_0}frac{partial n_0}{partial y}right)^{-1}

label{scale_length}

end{equation}

The negative sign accounts for the fact that we have considered a gradient in the -y direction and therefore $L$ is actually positive here. In general, $L$ is considered positive if the gradient is parallel to the background electric field direction, and negative if it is anti-parallel. Finally, we then arrive at

begin{equation}

left [frac{partial}{partial t}+Vfrac{partial}{partial x}right]frac{delta n}{n_0}=frac{nu_i}{Omega_i}Vfrac{delta n}{n_0}frac{1}{L}

label{scale_length_2}

end{equation}

The left-hand-side describes, as before, the fast time scales associated with wave propagation, while the right-hand-side is associated with the slow time scales related to the growth of the waves, with a growth rate given by

begin{equation}

gamma_{GD}=frac{nu_i}{Omega_i}frac{V_d}{(1+psi)}frac{1}{L}

label{final_growth}

end{equation}

The more general derivation shows that this growth rate competes with decay associated with diffusion for shorter scale structures and with chemical recombination for larger scale structures.citep{STM_2001}

section{Kinetic description of the linear theory}

A more general treatment of the plasma is based on kinetic theory, which deals with the velocity distribution of the particles. The fluid equations are based on velocity moments of the velocity distributions and are therefore less general, i.e., contain less information.

In kinetic theory, the velocity distribution of a particular species, $f(mathbf x,mathbf V,t)$, describes the probability of finding particles in a velocity interval ($mathbf V,mathbf V + delta mathbf V)$ and at a position interval ($mathbf x,mathbf x + delta mathbf x)$ at time $t$. The Boltzmann equation is used to describe $f$ and is written as

begin{equation}label{eq:boltzman}

frac{partial f}{partial t}+mathbf Vcdot mathbf nabla f+frac{q}{m}(mathbf E+mathbf Vtimes mathbf B)cdot nabla_V f=left . frac{partial f}{partial t}right )_c

end{equation}

where $ q $,$m$,$mathbf r$ and $mathbf V$ are the charge, mass, position and velocity of the particle, {bf B} is the magnetic field and $nabla_V f$ denotes the gradient of $f$ in velocity space. The term on the right-hand-side of equation space(ref{eq:boltzman})