The effects that cause wave growth and decay are ionic inertia and a pressure gradient, respectively. The pressure gradient leads to diffusion, i.e. it works to smooth out density fluctuations. If left to itself it therefore creates a decay in the density of the structures over time. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get Original EssayIn contrast, ionic inertia causes slow positive ions to accumulate at the trailing edge in a mass where the positive charge density has already increased, so that inertia acts to make $delta E $ larger, which increases the width of the structures. The basis of this phenomenon is the fact that when a polarized structure passes over a particular region, the ions in that region try to reach the electrons to reduce the electric field. However, the acceleration of the ions is such that they take too long to reach the electrons. Instead, they end up accumulating in a region with an already increased positive charge density. This effect will gain importance if the structures are fast and/or sufficiently 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 diffusion, which causes the wave to decay. For a mathematical description of the processes described above we must consider the second order effects due to ionic inertia and the pressure gradient force. For a wave propagating in the plane perpendicular to $mathbf B$, diffusion will still act in the x and y directions. Therefore it is convenient to write the equation of motion of the 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}Remembering 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 thermal velocity of the ions. Taking the divergence of eqn. ref{ion_cont_2}, we getstart{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 ionic continuity, $partialdelta n/partial t=-n_0nablacdotdeltamathbf V_i$, we 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 perturbation of the electric field $deltamathbf E$ by observing the electronic continuity equation,begin{equation}frac{partial}{partial t}delta n=-mathbf V_ecdotnabladelta n-n_0nablacdotdeltamathbf V_elabel{partial_conti}end{equation } Since we now allow for diffusion and weak Petersen currents of electrons, we must 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 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 witheqn.spaceref{div_ion_cont_3} to get 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}dove $C_s^ 2=(Omega_i/Omega_e)C_e^2+C_i^2=[k(T_e+T_i)]/m_i$ is the square of the plasma ion-acoustic velocity . Therefore, we havebegin{equation}Big[frac{partial}{partial t}+frac{mathbf V_e}{1+psi}cdotnablaBig]frac{delta n}{n_0}=-frac{psi}{nu_i(1+psi ) }Large[frac{partial^2}{{partial t}^2}frac{delta n}{n_0}-C_s^2nabla^2frac{delta n}{n_0}Big]label{diffusion_eqn}fine{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 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 $psiomega$), $A$ constitutes the right-hand side of the equations (ref{convect_derivative} ) small. To solve this it is therefore possible to use a multiscale expansion in time. This is equivalent to assuming a solution with two independent time scales, one ($tau=t$) for the fast oscillations of the wave and one ($tau_g=epsilon t$) for the growth of the wave. The small constant $epsilon$ means that $tau$ must be long enough to make $tau_g$ feel. We can then split the equation (ref{convect_derivative}) into two parts, one for each timescale, recovering in the process the zeroth-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{tau_gpartial}{partial t}=frac{partial}{partial t}+epsilonfrac{partial}{partial tau_g}end {equation }The 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 the growth or decay, which means that the right-hand side must be zero for $ t to infty$ , or $f_1$ would grow indefinitely, so we should 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 it describes a simple spread, but in the x direction there is a big difference. In the case of structures that propagate only in the x-direction, the diffusion coefficient is $D=A(C_s^2-V^2)$, and can be positive or negative depending on $V$. If $V C_s$ the diffusion constant is negative, then we have anti-diffusion (ionic inertia increases the amplitude of the structure, essentially acting as a diffusion in reverse). From this expression we can only see that if a structure does notis sufficiently elongated, diffusion in the y direction becomes important. Therefore, the fastest growing structures are those for which the y derivatives in the density perturbations are initially much smaller than the x derivatives in the density perturbations. Once antidiffusion is strong enough, however, a stretched structure will continue to stretch. This is illustrated in Fig. ref{tilt_tower}where smooth diffusion stretches the structure in the y direction, but in the x direction, antidiffusion acts inward 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 "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 that propagates according to an angle $alpha$ with respect to the drift of $mathbf Etimes mathbf B$, but always in the xy plane. This angle is called the "flow angle". If we use a wave decomposition the wave vector is given bybegin{equation}k_x=k cos alpha hspace{1cm} k_y=k sinalphaend{equation}Now we can write eqn. ref{time_scale_1} in the form $frac{partial ln f_0}{partial tau_g}=gamma_{FB}$with growth rate $gamma_{FB}$beingbegin{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 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 linear 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 electric field component of the wave along the magnetic field. However, as the amplitude grows and becomes large, it is necessary to include nonlinear corrections which, in one way or another, should limit the amplitude. Furthermore, as will be shown in the next chapter, the nonlocal effects are such that the parallel electric field of the waves will grow monotonically over time, meaning that at some point the amplitude of the wave will have to decay. Note that the parallel electric field has not been included in this 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 might grow very fast to a small amplitude or very slowly to a large amplitude. This allowed researchers to think that a wave could change its frequency or phase velocity as the amplitude grows. The presence of nonlinear and nonlocal effects means that the growth rate is positive only for a limited period of time, until nonlinear effects come into play or until nonlocal effects take over. In either case, the waves will reach a peak amplitude and stop growing, or they will simply decay after reaching maximum amplitude. Observations indicate that the largest amplitudes are typically found at the acoustic velocity of the ions, i.e., according to the (ref{growth}) equation, 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 rate of a structure as it grows, until it can no longer grow, at which point it may or may not undergo decay. The fact that the spectra at the threshold velocity are not really narrow means that the structures actually have a finite lifeafter reaching a maximum amplitude. They must therefore undergo decay after reaching a maximum amplitude.section{Gradient Drift Growth Mechanism}If a wave propagates through a background plasma that 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 perturbative 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 higher density region at the bottom of the increment to a relatively lower density region at the top, thus increasing the relative density perturbation $delta n/n_0$. On the contrary, in a region of reduced density, the $delta mathbf E$ field would be directed in the opposite way, with a $deltamathbf Etimes mathbf B$ drift of the electrons in the $-y$ direction, thus bringing the lower density plasma into an already impoverished region, which again increases the value of $delta n/n_0$. newlineFor simplicity, here we focus on the effects of the environmental density gradient, keeping in mind that ultimately GD and FB can act together. We take into account the additional term of the perturbed electron velocity, i.e. $delta V_{e,y}=frac{delta E}{B}$. The continuity equation for the perturbed density from Eqn. ref{partial_ne} can therefore 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 still be valid and we find 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}Neglecting again the y derivatives in the perturbed quantities and assuming that the unperturbed $V_{e, 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}Let's rewrite Eqn. 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}e, substituting this result back into Eqn. ref{second_order}, we getstart{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 takes into account the fact that we have considered a gradient in the -y direction 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 antiparallel. Finally, we arrive abegin{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 slow time scales related to wave growth , with a growth rate given bybegin{equation}gamma_{GD}=frac{nu_i}{Omega_i}frac{V_d}{(1+psi)}frac{1}{L} label{final_growth}end{equation}La More general derivation shows that this growth rate competes with diffusion-associated decay for smaller-scale structures and with chemical recombination for larger-scale structures.})