User:Schmit.paul/Math entries

Local quasineutrality condition: $$ \tilde{\mathbf{j}} = 0 $$

Cold plasma equations:

$$\nabla \cdot \left(n_\alpha \mathbf{u}_\alpha \right ) = 0 $$

$$\mathbf{u}_\alpha \cdot \nabla \mathbf{u}_\alpha = \left (q_\alpha / m_\alpha \right ) \left (\mathbf{E} + \mathbf{u}_\alpha \times \mathbf{B} \right ) $$

Axisymmetric B

$$ \mathbf{B} = -\dfrac{\hat{\theta} \times \tilde{\nabla}\Psi\left(r,z\right)}{r} $$

Meridional/azimuthal momentum equation

$$ \tilde{\mathbf{u}}_\alpha \cdot \tilde{\nabla} \tilde{\mathbf{u}}_\alpha = \dfrac{q_\alpha}{m_\alpha} \left (\mathbf{E} + \mathbf{u}_{\alpha\theta} \times \mathbf{B} \right ) + \dfrac{\mathbf{u}_\theta^2}{r}\hat{\mathbf{r}} $$

$$ \tilde{\mathbf{u}}_\alpha \cdot \tilde{\nabla} \left (mru_{\alpha\theta} + q\Psi \right ) = 0 $$

Characteristic parameterization

$$ \dfrac{dr}{dt} = u_r, \mbox{ }  \dfrac{dz}{dt} = u_z. $$

Conservation of azimuthal momentum:

$$ u_{\alpha\theta} = \dfrac{q_\alpha}{m_\alpha} \dfrac{P_{\alpha\theta} / q_\alpha - \Psi}{r} \equiv \dfrac{q_\alpha}{m_\alpha} \dfrac{\Psi_{\alpha 0}' - \Psi}{r} $$

Individual species meridional EoM:

$$ \tilde{\mathbf{u}}_\alpha \cdot \tilde{\nabla} \tilde{\mathbf{u}}_\alpha = \dfrac{q_\alpha}{m_\alpha}\mathbf{E} - \dfrac{q_\alpha^2}{m_\alpha^2} \tilde{\nabla}\left(\dfrac{\left(\Psi_{\alpha 0}' - \Psi \right)^2}{2r^2} \right) $$

Meriodional EoM:

$$ \tilde{\mathbf{u}} \cdot \tilde{\nabla} \tilde{\mathbf{u}} = d\tilde{\mathbf{u}} / dt = -\tilde{\nabla} \left ( \dfrac{e^2 \left ( m_e \left( \Psi_{i0}' - \Psi \right)^2 + m_i \left(\Psi_{e0}' - \Psi \right)^2 \right )}{2m_i m_e \left(m_i + m_e \right) r^2} \right ) $$

Effective Potential (form 1):

$$ U_{\mbox{eff}} = \dfrac{e^2 \left ( m_e \left( \Psi_{i0}' - \Psi \right)^2 + m_i \left(\Psi_{e0}' - \Psi \right)^2 \right )}{2m_h\left(m_i + m_e \right) r^2} $$

Hybrid mass:

$$ m_h \equiv \left(m_i m_e \right)^{1/2} $$

Ueff:

$$ U_{\mbox{eff}} $$

Ueff canonical form:

$$ U_{\mbox{eff}} = \dfrac{e^2}{2 m_h} \dfrac{1}{r^2} \left( \left(\Psi_m - \Psi \right)^2 + \xi \right) $$

$$ \Psi_m \equiv \Psi_0 + \dfrac{m_h^2 r_0^2}{e\left(m_i + m_e \right)} \left(\dot{\theta}_{i0} - \dot{\theta}_{e0} \right) $$

$$ \xi \equiv \dfrac{m_h^2 r_0^4}{e^2 \left(m_i + m_e \right)^2} \left(m_i \dot{\theta}_{i0} + m_e \dot{\theta}_{e0} \right)^2 $$

Ueff Hooper's result:

$$ U_{\mbox{eff}} = \dfrac{e^2 \left(\Psi_0 - \Psi \right)^2}{2m_h r^2} $$

xi=0 plume optimization criterion:

$$ \dfrac{\dot{\theta}_{i0}}{\dot{\theta}_{e0}} = -\dfrac{m_e}{m_i} $$

Current Ring model:

Normalized, dimensionless coords:

$$ \rho \equiv r/a $$

$$ \zeta \equiv z/a $$

Vector potential:

$$ \mathbf{A} = \dfrac{\mu_0 I}{4\pi} \dfrac{\rho}{\left(\left(\rho+1\right)^2+\zeta^2\right)^{3/2}}\hat{\theta} $$

Ueff for current ring:

$$ U_{\mbox{eff}} = U_0 \left(\dfrac{\Delta}{\rho} - \dfrac{\rho}{\left(\left(\rho+1\right)^2+\zeta^2\right)^{3/2}}\right)^2 $$

$$ \Delta \equiv \dfrac{4 \pi \Psi_m}{\mu_0 I a} $$

$$ U_0 \equiv \dfrac{\mu_0^2 I^2 e^2}{32 \pi^2 m_h} $$

$$ \xi = 0 $$

=Diagnostics Project=

$$ n_e = n_{\infty}\exp\left(\dfrac{e\Phi}{T_e}\right)$$

Boltzmann Distribution

$$\nabla\cdot\left(n_i\mathbf{v}\right)=0 $$

ion cont eq

$$\nabla\cdot\left(n_i m_i \mathbf{vv}\right) + \nabla n_i T_i + e n_i \nabla\Phi + \nabla\cdot\mathbf{\Pi} = 0$$ ion momentum eq

$$e\Phi = T_e\ln\left(n_i/n_{\infty}\right)$$

Boltzmann potential

$$\nabla_{\parallel}\left(n_i v_{\parallel}\right) = -\nabla_{\perp}\cdot\left(n_i \mathbf{v}_{\perp}\right) \equiv S $$

Parallel continuity

$$\nabla_{\parallel}\left(n_i m_i v_{\parallel} v_{\parallel}\right) + \left(T_i + T_e \right)\nabla_{\parallel}n_i = -\nabla_{\perp}\cdot\left(n_i m_i v_{\perp} v_{\parallel}\right) + \left(\nabla\cdot\mathbf{\Pi}\right)_{\parallel} \equiv S_m $$

parallel momentum

$$n_i\mathbf{v}_{\perp} = -D\nabla_{\perp}n $$

phenomenological perpendicular diffusion eq

$$\left(\nabla\cdot\mathbf{\Pi}\right)_{\parallel} = \nabla_{\perp}\cdot\left(\eta \nabla_{\perp} v_{\parallel}\right)$$

phenomenological shear viscous stress eq

$$\nabla_{\perp} n_i \sim \left(n_{\infty} - n_i\right)/a $$

$$\nabla_{\perp} v_{\parallel} \sim \left(v_{\infty\parallel} - v_{\parallel}\right)/a $$

$$\nabla_{\perp}^2 n_i \sim \left(n_{\infty} - n_i\right)/a^2 $$

$$\nabla_{\perp}^2 v_{\parallel} \sim \left(v_{\infty\parallel} - v_{\parallel}\right)/a^2 $$

Perpendicular gradient scale-length approximation

$$ \mathbf{r}_{\perp}/a \rightarrow \mathbf{r}_{\perp} $$

$$ \int\dfrac{D}{c_s a^2}dz \rightarrow z $$

dimensionless coordinates

$$n_i/n_{\infty} \rightarrow n $$

$$ v_{\parallel}/c_s \rightarrow M $$

Normalized field quantities

$$ c_s \equiv \sqrt{\left(T_e + T_i\right)/m_i} $$

sound speed and Mach number

$$ M\dfrac{dn}{dz} + n\dfrac{dM}{dz} = 1-n $$

$$ \dfrac{dn}{dz} + nM\dfrac{dM}{dz} = \left(M_{\infty} - M\right)\left(1-n+\alpha\right)$$

$$\alpha \equiv \dfrac{\eta}{m_i n_{\infty} D}$$

one dimensional system of equations for density and velocity fields

$$ \operatorname{det}\begin{vmatrix} M & n \\ 1 & nM \end{vmatrix} = n\left(M^2 - 1\right) $$

system determinant

$$ n = \dfrac{1}{1-M_{\infty}M+M^2}$$

solution if alpha=0

$$ \Gamma_i = \dfrac{n_{\infty}c_s}{2\mp M_{\infty}}$$

resulting ion flux at sheath edge

$$ \Gamma_i=f n_{\infty} c_s$$

$$ f\left(M_{\infty},\alpha\right) = \exp\left \{-1 - 1.1M_{\infty} + \left(1-\sqrt{\alpha}\right)\left(0.31 + 0.6M_{\infty}\right)\right\}$$

$$\approx \exp\left \{-1 - 1.1M_{\infty}\right\}$$

numerical solution for range of alphas

=AC-DC powerpoint=

$$ \displaystyle \left(\Delta v\right)^2 $$

$$\displaystyle \Delta J \sim \Delta v $$

$$\displaystyle v \Delta v $$

$$ \Delta v = \frac{eE}{m}\Delta t $$

$$ E_i = \frac{1}{2}mv^2 + \frac{1}{2}m\left(-v\right)^2 = 2\left(\frac{1}{2}mv^2\right) $$

$$ E_f = \frac{1}{2}m\left(v+\Delta v\right)^2 + \frac{1}{2}m\left(-v+\Delta v\right)^2 = 2\left(\frac{1}{2}mv^2\right) + m\Delta v^2$$

$$ \Delta E_{field} = -\left( E_f - E_i\right) = -m\Delta v^2 $$

$$ \Phi_P(x,t)= \frac{e^2}{4m\omega^2}\mathbf{E}^2 =\Phi_0\cos{[k\left(x-\Psi(t)\right)]} $$

$$ \dfrac{2\pi a}{kv_{rel}} \dfrac{1}{v_{rel}} \sim \dfrac{a}{k v^2_{rel}} \ll 1 $$

$$ \dot{x}_\pm(t) = \dot{\Psi}(t) \pm \left[\dfrac{2\Phi_0}{m}\left(\alpha - \cos{[k\left(x-\Psi(t)\right)]}\right)\right]^{1/2} $$

$$ \alpha \equiv \frac{1}{\Phi_0}\left[1/2m(\dot{x}-\dot{\Psi})^2 + \Phi_0 \cos[k(x-\Psi)] \right] $$

$$ \int p\, dq $$

$$ \mbox{J}_{sep}=\left[m\int^\lambda_0 \left(\dot{x}_{+}-\dot{x}_{-}\right)\, dx\right]_{\alpha = 1} = 16\dfrac{\sqrt{m\Phi_0}}{k} $$

$$ \mbox{J}_{sl}=\lambda m \dot{x} $$

$$ \mbox{J}_{sl}^f = \mbox{J}_{sl}^i + \mbox{J}_{sep} $$

$$ \left|\Delta \dot{x}\right| = \dfrac{8}{\pi}\sqrt{\dfrac{\Phi_0}{m}} $$

$$ \eta \equiv k\left(x-\Psi(t)\right) $$

$$ \ddot{\eta}=\tilde{\phi}\left[\sin{\eta}-\epsilon\right]=-\frac{df}{d\eta} $$

$$ \tilde{\phi} \equiv a_{crit}k $$

$$ \epsilon \equiv a/a_{crit} $$

$$ a_{crit} \equiv \frac{\Phi_0 k}{m} $$

$$ f(\eta) = \tilde{\phi}\left(\cos{\eta} + \epsilon\eta\right) $$

$$ \Psi\left(t\right) = \frac{1}{2}at^2 + v_0 t + x_0 $$

$$ s\equiv x-(1/2at^2+v_0t+x_0) $$

$$ \tau \equiv \int_{s_i}^{s_f}\dfrac{ds}{\dot{s}} + \dfrac{1}{a} \left(\dot{s}_f - \dot{s}_i\right) $$

$$ \Delta \dot{x} = a\tau $$

$$ \dot{s} = \left[\dfrac{2}{m}\left( E - \phi(s) - mas \right)\right]^{1/2} $$

$$ \Delta\dot{x}(\eta_t)\sim \frac{2\epsilon\tilde{\phi}^{1/2}}{k} \left \{\frac{\ln{\left[1+\Theta G + \left(\Theta^2 G^2 + 2\Theta G \right)^{1/2}\right]}}{\left(\cos{\eta_t}\right)^{1/2}} + \frac{1}{\gamma^{1/2}}\tan^{-1}{\left[\frac{\left(\gamma/2 \right)^{1/2}\Theta}{\left[M-\gamma/2\right]^{1/2}} \right]} - \frac{\left(2M\right)^{1/2}}{\epsilon} \right \} $$

$$ \Theta (\eta_t) = \frac{\eta_t - \eta_{min}}{2} $$

$$ \gamma = \left(1-\epsilon^2\right)^{1/2} $$

$$ G(\eta_t) = \frac{\cos{\eta_t}}{\epsilon-\sin{\eta_t}} $$

$$ M(\eta_t) = \cos{\eta_t}+\epsilon\eta_t + \gamma + \epsilon\left(\pi+\sin^{-1}{\epsilon} \right) $$

$$ \Delta\dot{x} \sim \frac{2\epsilon\tilde{\phi}^{1/2}}{k}\left[\ln{\left(\frac{\pi}{\epsilon-\eta_t}\right)}+\tan^{-1}{\frac{\pi}{2\sqrt{3}}}-\frac{2}{\epsilon}-\frac{\pi}{2}\right] $$

$$ \lim_{\epsilon\rightarrow 0} \Delta\dot{x} \sim -\frac{4\tilde{\phi}^{1/2}}{k} = -4\left(\frac{\Phi_0}{m}\right)^{1/2} $$

$$ \epsilon\rightarrow 0 $$

$$ \ddot{\Psi} \ll 1 $$

$$ \ddot{\Psi} > 0 $$

$$ \left |\ddot{\Psi}\right |\rightarrow 0 $$

$$ \displaystyle \epsilon \sim m $$

$$ \displaystyle v = \omega/k $$

$$ \displaystyle v $$

$$ \displaystyle f(v) $$

$$ \displaystyle f_0 $$

$$ f^\prime $$

$$\displaystyle \nu \sim 1/v^3 $$

$$ \Phi_P = \frac{e^2\left|E_0\right|^2}{4m\omega^2} $$

$$ \Phi_P = \sum_\nu \frac{e^2\left|E_\nu\right|^2}{4m\omega\left(\omega-\nu\Omega\right)} $$

$$ \displaystyle \Phi_P $$

$$ \displaystyle \omega_p \sim n^{1/2} \sim \epsilon^{-1/2} $$