Differences

This shows you the differences between two versions of the page.

--- corsica [2014/04/17 12:39] – created wsi
+++ corsica [2022/07/21 06:59] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== CORSICA ======
+Welcome to the CORSICA wiki.
+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
+:\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.html]]
+----
+==== Documentation ====
+ {{::main.pdf| CORSICA user manual}}
+ {{::tokamak_ds.pdf| Tokamak deadstart}}
+{{::sspx_guide.pdf| SSPX manual }}
+ {{::d3d_demo.pdf| DIII-D manual }}
+{{:basismanual.pdf| Basis manual }}
+{{:finalreport.pdf| CORSICA final report }}
+{{:report_on_creating_iter_fiducial_states_44ddcn_v1_0.pdf| Fiducial generation manual}}
+{{::managing_device_configurations_in_corsic_449t78_v1_0.pdf| Device configuration management}}
+{{:code_builds_june4.pdf| Code builds }}
+{{:dc.pdf| Data collection for Ufiles }}
+{{:tutorial.pdf| Tutorial }}
+{{:haney.pdf| Scott Haney's Variational Procedure for vertical stabiility in Tokamaks }}
+{{::limit-line_1_.pdf| Superconductor limit line}}
+----
+==== Bibliography ====