The effects that cause the wave to grow and decay are ion inertia and a pressure gradient, respectively. The pressure gradient leads to diffusion, that is, it works to smooth out density fluctuations. If left to its own devices, this therefore creates a degradation of density structures over time. Say no to plagiarism. Get a tailor-made essay on “Why Violent Video Games Should Not Be Banned”? Get the original essay. In contrast, ion inertia causes slow positive ions to accumulate at the trailing edge, a drop where the positive charge density is already enhanced, so inertia acts to enlarge the $delta E$, this which increases the amplitude of the structures. At the origin of this phenomenon is the fact that when a polarized structure passes over a particular region, the ions in that region try to catch up with the electrons in order to reduce the electric field. However, the acceleration of the ions is such that they take too long to catch up with the electrons. They eventually accumulate in a region with an already increased positive charge density. This effect will become more significant if the structures are sufficiently fast and/or narrow, 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 must be fast enough to overcome scattering, which causes waves to decay. For a mathematical description of the above processes, we must consider second-order effects due to ion inertia and pressure gradient strength. For a wave propagating in the plane perpendicular to $mathbf B$, diffusion will nevertheless act in the x and y directions. Therefore, it is convenient to write the equation of motion of perturbed ions using the 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 obtainbegin{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 speed thermal ions. By taking the divergence of Eq. ref{ion_cont_2}, we obtainbegin{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 disrupted ionic continuity, $partialdelta n/partial t=-n_0nablacdotdeltamathbf V_i$, we havebegin{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 disturbance $deltamathbf E$ by looking at the electronic continuity equation, begin{equation}frac{partial}{partial t}delta n=-mathbf V_ecdotnabladelta n-n_0nablacdotdeltamathbf V_elabel{partial_conti}end {equation }Because we now allow for diffusion and weak electronic Petersen currents, we need to add the appropriate terms to eqn.spaceref{pert_ve_approx} and getbegin{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 start{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 usbegin{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{deltan}{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 the ionic-acoustic plasma velocity. Then we started{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 formbegin{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}flabel{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 tracking the wave with speed $V=(E/B)/(1+psi)$. The constant A is given by begin{equation}A=frac{psi}{nu_i(1+psi)}end{equation}Since $psiomega$), $A$ makes the right side of the equations (ref{convect_derivative} ) small. A multi-scale temporal expansion can therefore be used to solve it. This is equivalent to assuming a solution with two independent time scales, one ($tau=t$) for the rapid oscillations of the waves and one ($tau_g=epsilon t$) for the growth of the wave. The small constant $epsilon$ means that $tau$ must be long enough for $tau_g$ to be felt. Therefore, we can split the equation (ref{convect_derivative}) into two parts, one for each time scale, thus recovering the zero-order description discussed in the previous section. That is, writebegin{equation}f=f_0(tau,tau_g)+epsilon f_1(tau,tau_g)end{equation}and the time derivative becomesbegin{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}Fast time scale is just the wave equation without Growthbegin{equation}Big[frac{partial}{partial tau}+Vfrac{partial}{partial x}Big]f_0=0end {equation}But for $f_1$ we also have growth, and we arrive at 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_0end{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_0end{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 the right side must be zero for $t to reach infty$. , or $f_1$ would increase indefinitely, so we need to requirebegin{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_0label{time_scale_1}end{equation}This is the equation for the growth of the structure. In the y direction this describes simple diffusion, but in the x direction there is an important difference. In the case of a 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 we have anti-diffusion (ionic inertia increasesthe amplitude of the structure, functioning essentially like reverse diffusion). We only see from this expression that if a structure is not elongated enough, diffusion in the y direction gains importance. Thus, the fastest growing structures are those for which the y derivatives in the density disturbances are initially much smaller than the x derivatives in the density disturbances. However, once the anti-diffusion is strong enough, 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 the anti-diffusion acts inwards and compresses it.begin{figure}[H]includegraphics[scale=1.0]{ pictures/tilt_tower.pdf}centeringcaption { {Evolution of a structure in the xy plane under the influence of diffusion in the y direction and the “anti-diffusion” of the Farley-Buneman instability in the x direction , illustrated at three different times.}}label{tilt_tower}end{figure }More generally, we can also consider a structure propagating at an angle $alpha$ with respect to the drift $mathbf Etimes mathbf B$, but always in the xy plane. This angle is called the "flow angle". If we use wave decomposition, the wave vector is given bybegin{equation}k_x=k cos alpha hspace{1cm} k_y=k sinalphaend{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 therefore with $omega=kVcosalpha$ we find the traditional growth rate of Farley-Buneman, start{equation}gamma_{FB}=frac{(omega^2-k^2C_s^2)psi}{(1+psi)nu_i}label{growth}end{equation}subsection{Nonlinear complications and nonlocal}In the linear In local theory, the growth rate expressed in the equation (ref{growth}) is independent of time. 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 increases and becomes large, we need to include non-linear corrections which in one way or another should limit the amplitude. Furthermore, as the next chapter shows, nonlocal effects are such that the parallel electric field of the waves will increase monotonically with time, meaning that at some point the amplitude of the waves will have to decrease. Note that the parallel electric field has not been included in the present derivation and 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 can grow very quickly to a small amplitude or very slowly to a large amplitude. This led researchers to believe that a wave can change frequency or phase speed as the amplitude increases. The presence of non-linear and non-local effects means that the growth rate is only positive for a limited time, until non-linear effects come into play or until non-local effects take over. . In all cases, the waves will either reach maximum amplitude and stop growing, or they will simply decrease after reaching maximum amplitude. Observations indicate that the largest amplitudes are generally found at the acoustic velocity of the ions, that is, according to the equation (ref{growth}), at the threshold velocity (zero growth rate condition), for a wave in the xy plane. This suggests that nonlinear and/or nonlocal effects must decrease the speed of a structure as it grows, until itcan no longer grow, in which case it may or may not undergo disintegration. The fact that the spectra at the threshold velocity are not truly 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{Drift-gradient growth mechanism}If a wave propagates through a background plasma which presents a gradient in the direction of the background electric field (-y on fig.spaceref{Field_direction} ), this gradient provides an additional destabilization mechanism. The reason is that the disturbance electric field $deltamathbf E$ creates a drift $deltamathbf Etimes mathbf B$ of the electrons in the $+y$ direction. As a result, electrons move from a region of higher density at the bottom of the enhancement to a region of relatively lower density at the top, thereby increasing the relative density disturbance $delta n/n_0$. Conversely, in a region of density depletion, the $delta mathbf E$ field would be directed in the opposite direction, with a $deltamathbf Etimes mathbf B$ drift of electrons in the $-y$ direction, bringing thus a lower density plasma in an already depleted region, which again improves the value of $delta n/n_0$. newlineFor simplicity, we focus here on ambient density gradient effects, keeping in mind that ultimately GD and FB can act together. We take into account the additional term of the perturbed electronic velocity, namely $delta V_{e,y}=frac{delta E}{B}$. The continuity equation for the perturbed density of Eqn. ref{partial_ne} can then be written asbegin{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 always be valid and we see thatbegin{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}By again neglecting the y derivatives in the perturbed quantities, and assuming that the $V_{ e unperturbed, y}$ is negligible, we find thatbegin{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 the equation. ref{delta_ne_ni} in the formbegin{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, replacing this result in Eqn. ref{second_order}, we obtainbegin{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 set 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 explains the fact that we considered a gradient in the -y direction and so $L$ is actually positive here. In general, $L$ is considered positive if the gradient is parallel to the direction of the background electric field, 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 side describes, as before, the fast time scales associated with wave propagation, while the right side is associated with the time scales slow waves linked to the growth of waves, with a given growth rate})