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start [2014/04/17 11:21] – [Documentation] simonw7start [2022/07/21 06:59] (current) – external edit 127.0.0.1
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-====== CORSICA ======+====== WSI Wiki ======
  
 +Here you'll find all of the codes and tutorials for our regular [[http://woodruffscientific.com/shortcourse|Scientific Computing Bootcamp]].  Please register for the training [[http://www.woodruffscientitic.com/apply|here]], or take a look at some of our past modeling/simulation work [[http://www.woodruffscientific.com/physics|here]].
  
 +[[https://woodruffscientific.us10.list-manage.com/subscribe?u=f4b64a3a93dfc6088f685111d&id=92569d866b | Sign up to our newsletter]] to keep up to speed!
  
-Welcome to the CORSICA wiki. +Since 2005 we have designed, built and tested [[http://www.woodruffscientific.com/magnets|custom magnetic coils]], [[http://www.woodruffscientific.com/pulsedpower |pulsed power systems]] and [[http://www.woodruffscientific.com/physics|simulated plasmas]] for a variety of applications. 
-This wiki will be populated with information about the CORSICA code suite. The purpose is to bring together CORSICA resources that cannot currently be found in a central location. +
- +
----- +
- +
-==== What CORSICA is ==== +
- +
-The LLNL CORSICA code provides a comprehensive predictive capability for axisymmetric toroidal plasmas. It has been applied successfully to many tokamaks, to the SSPX spheromak, and to the reversed-field pinches MST and RFX. At the heart of CORSICA is a 1.5-D, time-dependent plasma simulation code which solves the Grad-Hogan problem: self-consistent evolution of free-boundary plasma equilibria and internal profiles, including external conductors and magnetic diffusion, with a variety of available transport models. +
- +
----- +
- +
-==== Components of the code ==== +
-==1-1/2 D Core Transport Equations== +
- +
-In this section we derive the transport equations solved by CORSICA. +
-We start with the low-frequency limit of Maxwell's equations: +
- +
-\begin{equation}\label{FaradaysLaw} +
-  \Grad\times\Evec  =  - \ddt{\Bvec}  +
-\end{equation} +
-\begin{equation}\label{AmperesLaw} +
-  \Grad\times\Bvec  =  \mu_0\,\Jvec    +
-\end{equation} +
-\begin{equation}\label{QuasiNeutral} +
-  \sum_j q_j n_j    =  0 \qquad (q_j = Z_j e)  +
-\end{equation} +
-\begin{equation}\label{DivV} +
-  \Grad\cdot\Bvec   =  0 +
-\end{equation} +
-and the zeroth order moments of the Boltzman equation, including +
-fluctuations: +
-\begin{equation}\label{Continuity} +
-  \ddt{n_j} + \Grad\cdot(n_j \uvec_j + \Gamvec_{Aj}) = S_{n,j} +
-\end{equation} +
-\begin{equation}\label{FullMomentum} +
-  \ddt{}(m_j n_j \uvec_j) +  +
-    \Grad\cdot(m_j n_j \uvec_j \uvec_j + p\; \mathrm{I} + \Tens{\Pi}) +
-     n_j q_j (\Evec + \uvec_j\times\Bvec) + \Fvec_j + \Vec{S}_{\text{momentum},j} +
-\end{equation} +
-  +
-\begin{equation}\label{Energy} +
-  \frac32 \ddt{p_j} + \Grad\cdot(q_j + q_{A,j} + \frac52 p_j \uvec_j ) +
-    = Q_j + \uvec_j \cdot (\Fvec_j + q_j n_j \Evec) + S_{E,j} + S_{EA,j} +
-\end{equation} +
- +
-  +
-where the index $j$ refers to all ion species plus electrons, and the +
-quantities with subscript $A$ are anomalous transport terms resulting +
-from turbulent fluctuations.  In addition, note that the viscous terms +
-have been lumped into the source term. +
- +
-==Quasi-equilibrium== +
- +
-We are interested in modeling core transport in toroidal axisymmetric +
-plasmas.  By `core' we mean the part of the plasma characterized by +
-closed magnetic surfaces.  Given this situation it is convenient to +
-define a coordinate system consisting of $\psi$, a magnetic surface +
-label, $\theta$, a poloidal angle variable, and $\varphi$, a toroidal +
-angle variable.  Both $\theta$ and $\varphi$ vary between $0$ and +
-$2\pi$.  All physical quantities must be periodic in these angles to +
-ensure single-valuedness, and axisymmetry requires that all physical +
-scalars be independent of the $\varphi$.  The $(\psi, \theta, +
-\varphi)$ coordinates are described in more detail in +
-Sec.~\ref{ch:details}.\ref{sec:Coordinates}. +
- +
-The $\psi = \hbox{constant}$ surfaces are called magnetic flux +
-surfaces because the magnetic field lines lie in these surfaces. +
-As a result, we can write: +
-\begin{equation}\label{eq:BGradPsi} +
-   B^\psi = \Bvec\cdot\Grad\psi = 0 +
-\end{equation} +
- +
- +
-Given that the magnetic field lines lie in the flux surfaces, along +
-with the axisymmetry assumption, one can express the magnetic field in +
-the general form: +
-\begin{equation}\label{BRep} +
-    \Bvec = \Grad\varphi\times\Grad\psi + F\Grad\varphi +
-\end{equation} +
-where $F(\psi)$ is an arbitrary function (it is shown in +
-Sec.~\ref{ch:details}.\ref{app:GradShafranov} that axisymmetry implies +
-that $F$ is independent of $\theta$). It is easy to show that this +
-expression for $\Bvec$ guarantees that $\Bvec$ is divergence-free. +
- +
-We proceed by calculating the total momentum balance, summing +
-Eq.~\ref{Momentum} over all species. This gives: +
-\begin{equation} +
-  \Evec\sum_j q_j n_j + \Jvec\times\Bvec + \sum_j\Fvec_j = \Grad p +
-\end{equation} +
-Using quasi-neutrality, and the fact that net force due to interspecies +
-friction must be zero, we have +
-\begin{equation}\label{MHDEqForceBalance} +
-  \Jvec\times\Bvec = \Grad p +
-\end{equation} +
-This is the ideal \acro{MHD} equilibrium relation. Thus, as the plasma +
-evolves on the transport timescale it moves through a series of +
-quasi-static \acro{MHD} equilibrium. +
- +
-There are several implications of Eq.~\ref{MHDEqForceBalance}. First, +
-we see that the constant flux surfaces must also be constant pressure +
-surfaces since +
-\begin{equation} +
-  \Bvec\cdot\Grad p = 0  +
-\end{equation} +
-This implies that $p = p(\psi)$.  Next we see that the current must +
-also flow in these surfaces since +
-\begin{equation} +
-  \Jvec\cdot\Grad p = \Jvec\cdot\Grad\psi\: p'(\psi) = 0 +
-\end{equation} +
-which implies $J^\psi = 0$.  +
- +
-We can calculate an expression for the current by substituting our  +
-expression for the magnetic field, Eq.~\ref{BRep}, into Ampere's law. +
-The result is: +
-\begin{equation} +
-\label{eq:J} +
-\mu_0\, \Jvec  = \Grad\varphi\; \Delstarpsi -  +
-               \Grad\varphi\times\Grad\psi\; \ddpsi{F} +
-\end{equation} +
-where +
-\begin{equation} +
-  \Delstarpsi \equiv R^2 \Grad\cdot\frac1{R^2}\Grad\psi +
-\end{equation} +
- +
-Finally, substituting this result into Eq.~\ref{MHDEqForceBalance} +
-leads to the Grad-Shafranov equation for the magnetic flux function +
-$\psi(\Rvec)$: +
- +
-\begin{equation} +
-\label{eq:GradShafranov} +
-  \Delstarpsi = - \mu_0 R^2 \ddpsi{p} - F\ddpsi{F} +
-\end{equation} +
- +
-This is a quasilinear partial differential equation for $\psi$. Many +
-computer codes have been written to solve this problem, with a variety +
-of functional forms for $p(\psi)$ and $F(\psi)$, and with various +
-types of boundary conditions.  The twist here is that $p$ and $F$ +
-depend parametrically on time, being evolved consistent with the +
-flux-surface averaged transport equations that will be derived below. +
- +
-==Scalar Transport Equations== +
- +
-Next we derive a set of transport equations for the ion density, the +
-electron and ion energy, and the magnetic flux. +
- +
-Ion Density +
- +
-The full evolution of the ion density is given by Eq.~\ref{Continuity}. +
- +
-Electron Density +
- +
-The electron density is found from quasi-neutrality; i.e. we can solve +
-Eq.~\ref{QuasiNeutral} for $n_e$ to give: +
-\begin{equation} +
-  n_e = \sum_k Z_k n_k +
-\end{equation} +
-Furthermore, since $J^\psi = 0$, the radial electron particle flux +
-must equal the weighted radial ion particle flux: +
-\begin{equation} +
-  \Gamma_e\cdot\Grad\psi = +
-      \sum_k Z_k (n_k \uvec_k + \Gamma_{\text{A}k})\cdot\Grad\psi +
-\end{equation} +
- +
-==Energy== +
- +
-The disparity between the electron and ion masses makes the +
-equilibration time between electrons and ions much longer than the +
-equilibration time amongst ion species: +
-\begin{equation} +
-  \tau_{ee}:\tau_{ii}:\tau_{ei} \sim  +
-    1:\frac1{Z^3}\sqrt{\frac{m_i}{m_e}}\left(\frac{T_i}{T_e}\right)^{3/2}:% +
-      \frac1{2Z}\frac{m_i}{m_e} +
-\end{equation} +
- +
-Given this disparity, we are justified in assuming that all ion species are characterized by the same temperature $T_i$ and the electrons have a different temperature $T_e$. +
- +
-The electron-ion equilibration time $\tau_{ei}$ is still short for most +
-fusion plasmas. However, preferential heating of one species can +
-result in a large difference between $T_e$ and $T_i$. +
- +
-We now write Eq.~\ref{Energy} for the electrons and ions, with the ion +
-equation obtained by summing the equations for the individual ion species: +
-\begin{equation} +
-  \frac32 \ddt{p_e} + \Grad\cdot(q_e + \frac52 p_e \uvec_e ) +
-    = Q_e + \uvec_e \cdot (\Fvec_e - e n_e \Evec) + S_{E,e} +
-\end{equation} +
-and +
-\begin{equation} +
-  \frac32 \ddt{p_i} + \Grad\cdot(q_i + \frac52 p_i \uvec_i ) +
-    = S_{E,i} + \sum_k Q_k + \uvec_k \cdot (\Fvec_k + q_k n_k \Evec)  +
-\end{equation} +
-where  +
- +
-\begin{equation} +
-  x_i = \sum_k x_k, \quad x = p, q, S_{E} \\ +
-  p_i \uvec_i = \sum_k p_k \uvec_k \label{eq:pui} +
-\end{equation} +
-and where the anomalous contributions to $q_i$ and $S_E$ have been +
-lumped in with the non-anomalous parts.  Note that Eq.~\ref{eq:pui} is +
-an implicit definition of $\uvec_i$.  This is consistent with +
-$\uvec_i$ being the total ion particle flux since all ion species +
-share the same temperature. +
- +
-Conservation of energy and momentum by collisional processes leads to +
-the following relationship between the collisional heating source and +
-the collisional friction term: +
-\begin{equation} +
-  \sum_j Q_j + \uvec_j \cdot \Fvec_j = 0 +
-\end{equation} +
- +
- +
----- +
- +
- +
-==== Online CORSICA Resources ==== +
- +
- +
-Currently, the CORSICA code is described online. [[https://fusion.gat.com/THEORY/caltrans/]] +
- +
-The Basis language can be found here: [[https://wci.llnl.gov/codes/basis/]] +
- +
----- +
- +
-==== Documentation ==== +
- +
-CORSICA user manual {{::main.pdf|}} +
- +
-Tokamak deadstart {{::tokamak_ds.pdf|}} +
- +
-SSPX manual {{::sspx_guide.pdf|}} +
- +
-DIII-D manual  {{::d3d_demo.pdf|}} +
- +
-Basis manual {{:basismanual.pdf|}} +
- +
-CORSICA final report {{:finalreport.pdf|}} +
- +
-Fiducial generation manual {{:report_on_creating_iter_fiducial_states_44ddcn_v1_0.pdf|}} +
- +
-OneTwo user manual {{:onetwo.pdf|}} +
- +
-Device configuration management {{::managing_device_configurations_in_corsic_449t78_v1_0.pdf|}} +
- +
-PF coil design tutorial {{::tutorial.pdf|}} +
- +
-Code builds {{:code_builds_june4.pdf|}} +
- +
-Final report v2 {{:finalreport2.pdf|}} +
- +
-Data collection for Ufiles {{:dc.pdf|}} +
----- +
- +
-==== Bibliography ==== +
- +
  
 +[[http://woodruffscientific.com/wiki/doku.php?id=Afterschool|Afterschool]]
start.1397755264.txt.gz · Last modified: 2022/07/21 06:59 (external edit)