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% pp_opava07
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% On the Papaloizou-Pringle instability in accretion tori
% Opava -- RAGtime no. 9. -- 09/2007
% JH
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% Mathematics:
\newcommand{\ii}{\mathrm{i}}                                   % imaginary 'i'
\newcommand{\dd}{\mathrm{d}}                                   % differentials
\newcommand{\ratio}[2]{{\textstyle \frac{\,#1}{\,#2}}}         % ratio of numbers
\newcommand{\vc}[1]{\mbox{\protect\boldmath$#1$}}              % four-vector
\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}         % partial derivative
\newcommand{\DD}[2]{\frac{\mathrm{D}#1}{\mathrm{D}#2}}         % Lagrangian derivative
\newcommand{\DDD}[2]{\frac{\mathrm{D}^2#1}{\mathrm{D}#2^2}}    % Second Lagrangian derivative
\newcommand{\der}[2]{\frac{\,\dd #1}{\,\dd #2}}                % total derivative
\newcommand{\cs}[1]{c_{\mathrm{s}#1}}                          % sound speed
\newcommand{\OmK}{\Omega_\mathrm{K}}
\newcommand{\ellK}{\ell_\mathrm{K}}
\newcommand{\ms}{\mathrm{ms}}

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\title{Kluzniak-Kita expansion at the inner edge}
\author{{Ji{\v r}{\'\i} Hor{\'a}k}}
\date{May 2009}

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\begin{document}

\frame{\titlepage}

%-------------------------------------------------------------------

% Assumptions

\frame
{
  \frametitle{Assumptions}
  \begin{itemize}
    \item Stationary and axisymmetric disk: $\partial_t = \partial_\phi \equiv 0$
    \item Equatorial plane symmetry (odd/even functions of $z$)
    \item Polytropic euqation of state: $P = K\rho^{1+1/n}$ with $n=3/2$
    \item Heigh-dependent $\alpha$-viscosity: 
    \begin{equation}
      \nu = \frac{\alpha}{\Omega_\mathrm{K}}\frac{c_\mathrm{s}^2}{(1+1/n)},
      \quad
      \eta = \nu\rho
      \nonumber
    \end{equation}
    \item Geometrically thin disk: $|z|\ll r$  
    \item Spherically symmetric gravitational potential $\Phi(\sqrt{r^2+z^2})$:
    \begin{equation}
      \Phi(r,z) = \Phi(r,0) + \frac{1}{2}\Omega_\mathrm{K}^2(r) z^2 + \mathcal{O}(z^2/r^2),
      \quad\quad
      |z|\ll r
      \nonumber
    \end{equation}
  \end{itemize}
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{Perturbative approach}
  Rescale all quantities by their typical values $Q\rightarrow\tilde{Q}\equiv Q/Q_\ast$:
  \begin{itemize}
    \item radius: $\tilde{r} = r/r_\ast$
    \item azimuthal velocities: $\tilde{\Omega} = \Omega/\Omega_\mathrm{K}(r_\ast)$
    \item poloidal velocity: $\tilde{c_\mathrm{s}} = c_\mathrm{s}/c_{\mathrm{s}\ast}$, $\tilde{v^r} = v^r/c_{\mathrm{s}\ast}$, $\tilde{v^z} = v^z/c_{\mathrm{s}\ast}$
    \item vertical coordinate: $\tilde{z} = z/H_\ast$
    \item density and pressure: $\tilde{\rho}=\rho/\rho_\ast$, $\tilde{p} = p/(c_\mathrm{s\ast}^2\rho_\ast)$
    \item viscosity: $\tilde{\nu}=\nu/(c_{\mathrm{s}\ast} H_\ast)$, $\tilde{\eta}=\eta/(c_{\mathrm{s}\ast} H_\ast\rho_\ast)$
  \end{itemize}
  \vspace{0.2cm}\hrule\vspace{0.2cm}
  \noindent
  Disk heigh: $H\sim c_\mathrm{s}/\Omega_\mathrm{K}$ $\rightarrow$ Thin disk: 
  \begin{equation}
    \textcolor{blue}{
    \boxed{\epsilon=\frac{H_\ast}{r_\ast}=\frac{c_{\mathrm{s}\ast}}{r_\ast\Omega_\ast}\ll 1}
    }
    \nonumber
  \end{equation}
  $\rightarrow$ a small parameter. We assume $\tilde{Q}=\mathcal{O}(1)$ as $\epsilon\rightarrow 0$.
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{Hydrodynamical equations}
  {\tiny
  radial:
  \begin{eqnarray}
    \epsilon^2 v^r \partial_r v^r + \epsilon v^z \partial_z v^r - r(\Omega^2 - \Omega_\mathrm{K}^2) + 
    \epsilon^2 n\partial_r \cs{}^2 + \frac{1}{2}\epsilon^2\OmK^\prime z^2 =  
    \nonumber\\
    = \epsilon^3\frac{1}{r\rho}\partial_r(2r\eta\partial_r v^r) - 2\epsilon^3 \frac{\eta v^r}{\rho r^2} + 
    \epsilon^2\frac{1}{\rho}\partial_z(\eta\partial_z v^r) + \epsilon^2\frac{1}{\rho}\partial_z(\eta\partial_r v^z) 
    \nonumber\\
    + \epsilon^3\frac{1}{\rho}\partial_r\left[\left(\xi-\frac{2}{3}\eta\right)\frac{\partial_r(r v^r)}{r}\right] +
    \epsilon^2\frac{1}{\rho}\partial_r\left[\left(\xi-\frac{2}{3}\eta\right)\partial_z v^z\right],
  \end{eqnarray}
  azimuthal:
  \begin{eqnarray}
    \epsilon\frac{v^r}{r^2}\partial_r(r^2\Omega) + v^z\partial_z\Omega = \epsilon^2\frac{1}{\rho}\partial_r(r^3\eta\partial_r\Omega)
    +\frac{1}{\rho}\partial_z(\eta\partial_z\Omega),
  \end{eqnarray}
  vertical:
  \begin{eqnarray}
    \epsilon v^r \partial_r v^z + v^z \partial_z v^z  + 
    n\partial_z \cs{}^2 + \OmK^2 z =  
    \nonumber\\
    = \epsilon^2\frac{1}{r\rho}\partial_r(r\eta\partial_r v^z) + \epsilon^2\frac{1}{r\rho}\partial_r(r\eta\partial_z v^r) +
    \frac{1}{\rho}\partial_z(2\eta\partial_z v^z) 
    \nonumber\\
    + \epsilon^3\frac{1}{\rho}\partial_z\left[\left(\xi-\frac{2}{3}\eta\right)\frac{\partial_r(r v^r)}{r}\right] +
    \epsilon^2\frac{1}{\rho}\partial_z\left[\left(\xi-\frac{2}{3}\eta\right)\partial_z v^z\right],
  \end{eqnarray}
  continuity equation:
  \begin{eqnarray}
    \epsilon\frac{1}{r}\partial_r(r\rho v^r) + \partial_z(\rho v^z) = 0
  \end{eqnarray}
  }
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{Succesive solution}
  All quantities are expanded into series: 
  \begin{equation}
    Q(r,z) = \sum_i \epsilon^i Q_i(r,z)  
    \nonumber
  \end{equation} 
  Equations are solved in each order of $\epsilon$....
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{Zeroth order results}
  \begin{itemize}
    \item Radial equation $\rightarrow$ angular velocity: $\Omega_0(r,z) = \OmK(r)$
    \item continuity equation $\rightarrow$ vertical velocity: $v^z_0(r,z) = 0$
    \item vertical equation $\rightarrow$ speed of sound: 
    \begin{equation}
      \cs{0}^2(r,z) = \frac{1}{2n}\OmK^2\left[H^2(r) - z^2\right]=\frac{1}{3}\OmK^2\left[H^2(r) - z^2\right]
      \nonumber
    \end{equation}    
  \end{itemize}
  \vspace{0.2cm}\hrule\vspace{0.2cm}
  Other quantities (using standard expressions): 
  \begin{itemize}
    \item density: 
    \begin{equation}  
      \rho_0 = \left(\frac{\cs{0}^2}{1+1/n}\right)^n = 
      \left[\frac{\OmK^2}{5}\left(H^2 - z^2\right)\right]^{3/2}
      \nonumber
    \end{equation}  
    \item viscosity:
    \begin{equation}
      \eta_0 = \nu_0\rho_0 = 
      \frac{2}{3}\alpha\OmK^4\left[\frac{1}{5}\left(H^2 - z^2\right)\right]^{5/2}
      \nonumber
    \end{equation}  
  \end{itemize}
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{First order results}
  \begin{itemize}
    \item Radial equation $\rightarrow$ angular velocity: $\Omega_1(r,z) = 0$
    \item continuity equation $\rightarrow$ vertical velocity: $v^z_1(r,z) = 0$
    \item azimuthal equation $\rightarrow$ radial velocity: $v^r_0(r,z) = 0$
    \item vertical equation $\rightarrow$ speed of sound: $\cs{0}^2 (r,z)= 0$
  \end{itemize}
}

%-------------------------------------------------------------------

\frame
{
  \frametitle{Disk thickness}
  Angular momentum conservation:
  \begin{equation}
    \dot{M}\left(\ell - \ell_{+}\right) = -2\pi r^3\int_{-\infty}^\infty \eta(\partial_r\Omega)\,\dd z,
    \nonumber    
  \end{equation}
  takes the form in the lowest order of approximation
  \begin{equation}
    \dot{M}_1\left[\ellK(r) - \ell_{+}\right] = -4\pi r^3\OmK^\prime(r)\int_0^H\eta_0\,\dd z
    \nonumber    
  \end{equation}
  \begin{equation}
    \dot{M}_1 = -4\pi r\int_0^H\rho_0 v^r_1\,\dd z.
    \nonumber    
  \end{equation}
  From that we can derive the disk thickness as
  \begin{equation}
    H(r) = \left[\frac{3\lambda}{4-\kappa}\frac{\ellK - \ell_{+}}{r^2\OmK^5}\right]^{1/6},
    \nonumber    
  \end{equation}
  \begin{equation}
    \textcolor{blue}{
    \kappa(r)\equiv 2\frac{\dd \ln \ellK}{\dd \ln r}},
    \quad
    \lambda\equiv 8\cdot 5^{3/2}\pi^{-2}\left(\frac{\dot{M}_1}{\alpha}\right)
    \nonumber    
  \end{equation}
  KK-solutions: $\kappa\equiv1$; $\kappa\rightarrow0$ as $r\rightarrow r_\mathrm{ms}$
}

\frame
{
  \frametitle{Second order results}
  \begin{itemize}
  \item Coupled radial and azimuthal equations $\rightarrow$ KK ansatz for $v^r_1$: 
  \begin{equation}
    v^r_1(r,z) = f_1(r)(H^2 - z^2) + f_2(r)
    \nonumber
  \end{equation}
  \item Solution for the radial velocity
  {\footnotesize
  \begin{equation}
    \boxed{
    v^r_1(r,z) = \alpha r\OmK\left(\frac{H}{r}\right)^2\left[
    \Lambda\left(1 - \frac{z^2}{H^2}\right) + \frac{32\alpha^2}{15}\frac{\Lambda}{\kappa} - 
    \frac{2(4-\kappa)}{3\kappa}\frac{\dd \ln H}{\dd \ln r}\right],}
    \nonumber
  \end{equation}}
  where
  \begin{equation}
    \Lambda(r)\equiv \frac{1}{15}\frac{(4-\kappa)(16-5\kappa)+2r\kappa^\prime}{\kappa + 64\alpha^2/25}
    \nonumber
  \end{equation}
  KK-solutions: $\Lambda(r)\equiv\Lambda_\mathrm{KK}$.
  \item accretion rate
  \begin{equation}
    \dot{M}_1 = \frac{\alpha\pi^2\OmK^4 H^6}{12\cdot 5^{3/2}\kappa}\left[
    r\kappa^\prime + (4-\kappa)\left(\frac{1}{2}(16-5\kappa)-6\frac{\dd \ln H}{\dd \ln r}\right)\right]
    \nonumber  
  \end{equation}
  \end{itemize}
}

\frame
{
  \frametitle{Second order results}
  \begin{itemize}
    \item azimuthal velocity
    \begin{equation}
      \Omega_2(r,z) = \OmK\left(\frac{H}{r}\right)^2\left[
      \frac{2}{15}\alpha^2\Lambda\left(1-6\frac{z^2}{H^2}\right)-\frac{4-\kappa}{4}+
      \frac{1}{2}\frac{\dd \ln H}{\dd \ln r}\right].
      \nonumber
    \end{equation}
    \item vertical velocity
    \begin{eqnarray}
      v^z_2 &=& \frac{1}{6}\alpha z\OmK\left(\frac{H}{r}\right)^2\Big\{
      \left[2(4-\kappa)\Lambda - r\Lambda^\prime\right]\left(1 - \frac{z^2}{H^2}\right) +
      \nonumber\\ &\phantom{=}&
      \left(\frac{\lambda}{\OmK^4 H^6} - 5\Lambda\right)\frac{\dd \ln H}{\dd \ln r}\Big\},
      \nonumber
    \end{eqnarray}
  \end{itemize}
}

\frame
{
  \frametitle{Behavior of solutions close to the inner edge}
  Two singularities close to inner edge:
  \begin{itemize} 
    \item $\kappa(r)\rightarrow 0$ as $r\rightarrow r_\ms$
    \item $H(r)\rightarrow 0$ as $\ellK(r)\rightarrow\ell_{+}\approx\ell_\ms$ 
  \end{itemize}
   Behavior of the solutions:
  \begin{eqnarray}
    H(r)&\propto& (\ellK - \ell_{+})^{1/6}    \nonumber \\
    \rho(r) &\propto& (\ellK - \ell_{+})^{1/2}  \nonumber \\
    \cs{}^2(r) &\propto& (\ellK - \ell_{+})^{1/3}  \nonumber \\
    \eta(r) &\propto& (\ellK - \ell_{+})^{5/6}   \nonumber \\
    v^r(r) &\propto& \epsilon (\ellK - \ell_{+})^{-2/3} \nonumber \\
    v^z(r) &\propto& \frac{\epsilon^2 (r-r_\ms)}{(\ellK-\ell_{+})^{3/2}} \nonumber \\
    \Omega(r)-\OmK(r) &\propto& \frac{\epsilon^2 (r-r_\ms)}{(\ellK-\ell_{+})^{2/3}} \nonumber
  \end{eqnarray}  
}

\frame
{
  \frametitle{Solution (outline)}
  \begin{itemize}
  \item
  $\ell_{+}$ is an `eigenvalue' $\Rightarrow$ Assumption: $\ell_{+}\rightarrow\ell_\ms$ as $\epsilon\rightarrow 0$
  \begin{equation}
    \ell_{+}\equiv\ell_\ms + \delta_{+}^2(\epsilon)L_{+}, \quad L_{+} > 0
    \nonumber
  \end{equation}  
  \item disk solution breaks down when
  \begin{equation}
    r\rightarrow r_{+} = r_\ms + \delta_{+} X_{+}, \quad
    X_{+}^2 = 2 L_{+} (\dd^2\ellK/\dd r^2)_\ms
    \nonumber
  \end{equation}  
  \item
  Magnification: $r\equiv r_{+} + \delta_r(\epsilon) X$ $\Rightarrow$ $\partial_r = \delta_r^{-1}\partial_X$.
  \item
  Rescaling:
  \begin{equation}
    H = (\delta_r\delta_{+})\bar{H}, \quad
    \rho = (\delta_r\delta_{+})^{1/2} \bar{\rho}, \quad
    \cs{}^2  = (\delta_r\delta_{+})^{1/3} \bar{c}_\mathrm{s}^2, 
    \nonumber
  \end{equation}  
  \begin{equation}
    \eta = (\delta_r\delta_{+})^{5/6}\bar{\eta}, \quad
    v^r = \epsilon(\delta_r\delta_{+})^{-2/3}\bar{v}^r, \quad
    v^z = (\epsilon^2\delta_r^{-1/2}\delta_{+}^{3/2}) \bar{v}^z
    \nonumber
  \end{equation}  
  \item 
  Substitution into hdyn equations, balancing terms...
  \item
  \textcolor{blue}{Relation between $\epsilon$, $\delta_r$, $\delta_{+}$ ??}
  \end{itemize}  

}


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\end{document}
