M3DC1 

Research  Curriculum Vitae  Publications  M3DC1  
Reference Documents 
IntroductionM3DC1 is a code that solves the extendedmagnetohydrodynamic (MHD) equations, which is a model that describes plasma as electrically conducting fluids of ions and electrons. This code is primarily used for calculating the equilibrium, stability, and dynamics of fusion plasmas, but has also been used for a variety of other applications, including astrophysical applications. In particular, M3DC1 is designed to address some of the most critical challenges confronting tokamak plasmas: largescale instabilities, which significantly degrade thermal confinement; and disruptions, which cause complete loss of energy confinement and which may cause damage to reactorscale tokamaks such as ITER. M3DC1 builds upon some of the design principles pioneered by M3D, but the two codes are developed independently and do not share source code. The "C1" in M3DC1 refers to the $C^1$ property of its finite elements, which ensures that both the value and the derivatives of fields are continuous across mesh element boundaries. Advanced numerical methods are employed in M3DC1 to permit the efficient solution of its numerical model over a broad range of temporal and spatial scales. These methods include the use of highorder finite elements on an unstructured mesh; fully implicit and semiimplicit time integration options; physicsbased preconditioning; and parallelization via domain decomposition and the use of scalable parallel sparse linear algebra solvers. Documents describing how to use and run the code, and publications describing the numerical methods and physics results in more detail, can be found in the reference documents and the bibliography below. ApplicationsVertical Displacement Events (VDEs)
A description of 2D VDE calculations with M3DC1 is given in NM Ferraro, et al. Phys. Plasmas 23:056114 (2016). Sawteeth and Helical Core Equilibria
In most tokamak discharges, the plasma current slowly peaks until it becomes unstable to internal kink modes. This socalled "sawtooth" instability flattens the central current; at which time the process repeats. While these oscillations are generally benign, they are likely best avoided in burning plasmas: highenergy fusion products ($\alpha$particles) can increase the oscillation amplitude, which may excite neoclassical tearing modes and/or edge localized modes, potentially causing major disruptions. Using the M3DC1 code another mechanism for preventing the current from peaking that involves a selforganized, dynamo action was found, which avoids these problems. Numerical simulations and analytic theory have determined that, for certain, global parameters, the plasma will selforganize with the central safetyfactor, $q$, slightly above unity and constant in a central volume (as shown in above figure, click to enlarge), where $q$ is defined as how many times a magnetic field line travels around the torus the long way compared to the short way. Such a large, "shear"free region is unstable to pressuredriven "interchange" modes. The linearlyunstable, interchange eigenfunction extends throughout the shearfree region and drives a strong, stationary, $(1,1)$helical flow that then drives a stationary $(1,1)$component of the electrostatic potential, $\phi$, (shown) and of the magnetic field (shown). These combine to create a $(0,0)$"dynamo" voltage that prevents the current density from peaking in the center, hence maintaining the shearfree region with $q$ slightly above unity in the central region, and with a broad current profile. This mechanism could explain the physics behind the nonsawtoothing "hybrid" discharges observed in DIIID and in many other tokamaks. For more details, see SC Jardin, N Ferraro, I Krebs. Phys. Rev. Lett. 115:215001 (2015). EdgeLocalized Modes (ELMs)The highconfinement mode (Hmode) of tokamak plasma operation is characterized by steep thermal gradients and large current densities at the edge of the plasma. While Hmode is advantageous from a confinement standpoint, the large pressure gradients and current densities lead to periodic instabilities known as Edge Localized Modes (ELMs) that intermittently deliver significant heat loads to plasmafacing components of the confinement vessel. In reactorscale devices like ITER, these heat loads may be large enough to rapidly erode these components, and therefore must be mitigated. M3DC1 has been used to calculate how various nonideal effects, including resistivity, rotation, and twofluid terms change ELM stability. For example, see NM Ferraro (2010). M3DC1 has also been used to simulate the nonlinear evolution of ELMs, in order to better understand the thermal transport that these modes cause. Linear Response to Applied 3D Fields
Physical ModelThe extendedMHD model implemented in M3DC1 consists of a set of fluid equations which evolve the particle number density $n$, fluid velocity $\vec{u}$, total pressure $p$, and electron pressure $p_e$: \begin{eqnarray} \frac{\partial n}{\partial t} + \nabla \cdot (n \vec{u}) & = & 0 \\ n m_i \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} \right) & = & \vec{J} \times \vec{B}  \nabla p \color{blue}{ \mbox{}  \nabla \cdot \Pi + \vec{F}} \\ \frac{\partial p}{\partial t} + \vec{u} \cdot \nabla p + \Gamma p \nabla \cdot \vec{u} & = & \color{blue}{(\Gamma  1) \left[Q  \nabla \cdot \vec{q} + \eta J^2  \vec{u} \cdot \vec{F}  \Pi : \nabla u \right]} \\ & & \color{red}{\mbox{} + \frac{1}{n e} \vec{J} \cdot \left( \frac{\nabla n}{n} p_e  \nabla p_e \right) + (\Gamma1) \Pi_e : \nabla \left(\frac{1}{n e} \vec{J} \right)} \\ \color{red}{ \frac{\partial p_e}{\partial t} + \vec{u} \cdot \nabla p_e + \Gamma p_e \nabla \cdot \vec{u}} & \color{red}{=} & \color{red}{(\Gamma  1) \left[Q_e  \vec{q}_e + \eta J^2  \vec{u} \cdot \vec{F}_e  \Pi_e : \nabla u \right]} \\ & & \color{red}{ \mbox{} + \frac{1}{n e} \vec{J} \cdot \left( \frac{\nabla n}{n} p_e  \nabla p_e \right) + (\Gamma1) \left[ \Pi_e : \nabla \left(\frac{1}{n e} \vec{J} \right) + \frac{1}{n e} \vec{J} \cdot \vec{F}_e \right]} \end{eqnarray} a generalized Ohm's law, in which the electric field $\vec{E}$ is obtained from the equations describing conservation of electron momentum and charge: \begin{equation} \vec{E} =  \vec{u} \times \vec{B} \color{blue}{ \mbox{} +\eta \vec{J}} \color{red}{ \mbox{} + \frac{1}{n e} \left( \vec{J} \times \vec{B}  \nabla p_e  \nabla \cdot \Pi_e + \vec{F}_e \right)} \end{equation} and a reduced set of Maxwell's equations which relates the electrical current density $\vec{J}$ in the plasma to the magnetic field $\vec{B}$ and describes the evolution of the magnetic field: \begin{eqnarray} \vec{J} & = & \frac{1}{\mu_0} \nabla \times \vec{B} \\ \frac{\partial \vec{B}}{\partial t} & = & \nabla \times \vec{E} \end{eqnarray} The "IdealMHD" model is a subset of the one described above, which is obtained in the limit where external sources, dissipative terms (like viscosity and resistivity) and the ion Larmor radius are zero, leaving only the terms colored black. "SingleFluid MHD" is a larger subset, in which the ions and electrons are considered to have the same fluid velocity and temperature, but where dissipative terms and external sources are included, shown in blue. M3DC1 implements a "TwoFluid" model, which takes differences in the ion and electron fluid velocities into account, which introduces the terms colored red. M3DC1 may also be used for solving the singlefluid equations. The idealMHD equations are best solved using specialized codes such as DCON, IPEC, GATO, ELITE, VMEC, or SPEC, which are designed to take advantage of special properties of the idealMHD equations. $\vec{F} = \vec{F}_i + \vec{F}_e$, where $\vec{F}_i$ and $\vec{F}_e$ are external forces acting on the ions and electrons, respectively (e.g. gravity). Similarly, $Q = Q_i + Q_e$, where $Q_i$ and $Q_e$ are external sources of heat to the ions and electrons. In M3DC1, several processes are implemented that are represented by these sources, including neutral beam injection. The viscous stress tensor $\Pi = \Pi_i + \Pi_e$ is the sum of the ion and electron stress tensors. Several processes contributing to the ion stress tensor are implemented: \begin{equation} \Pi_i = \Pi_i^\perp + \Pi_i^\wedge + \Pi_i^\parallel \end{equation} where $\Pi_i^\perp$ is the perpendicular ion viscosity, which represents simple crossfield angular momentum diffusivity; $\Pi_i^\wedge$ is the ion gyroviscosity, which represents finite ion Larmor radius effects; and $\Pi_i^\parallel$ is the parallel ion viscosity, which represents pressure anisotropy and is responsible for ion sound wave damping and poloidal flow damping in the extendedMHD model. Presently, the $\Pi_e$ term is implemented as a hyperresistive term. The heat flux density $\vec{q} = \vec{q}_i + \vec{q}_e$ is the sum of the ion and electron heat flux densities, which themselves are represented as the sum of perpendicular and parallel parts: \begin{equation} \vec{q}_{(i,e)} = \kappa_{(i,e)}^\perp \nabla T_{(i,e)}  \kappa_{(i,e)}^\parallel \frac{\vec{B} \vec{B} \cdot \nabla T_{(i,e)}}{B^2} \end{equation} M3DC1's equations are formulated in $(R,\varphi,Z)$ coordinates, and do not make use of magnetic coordinates. Therefore there is no coordinate singularity at poloidal field nulls (such as the magnetic axis or xpoints of toroidal plasmas), and no need to impose boundary conditions there. The model implemented in M3DC1 contains far more options than can be reasonably described here. For a more detailed, complete, and uptodate description of the model implemented in M3DC1, please refer to the M3DC1 Model Reference. Reduced ModelsA useful feature of M3DC1 is that it represents the magnetic field and fluid velocity using stream functions and potentials, similar to a Helmholtz decomposition: \begin{eqnarray} \vec{B} & = & \nabla \psi \times \nabla \varphi + F \nabla \varphi \\ \vec{u} & = & R^2 \nabla U \times \nabla \varphi + R^2 \omega \nabla \varphi + R^{2} \nabla_\perp \chi \end{eqnarray} One advantage of this representation is that the magnetic field is manifestly divergencefree. Another advantage is that we obtain physically meaningful reduced models by restricting the set of scalar fields that are evolved. Specifically, we obtain the twofield reduced model by only considering the evolution of $\psi$ and $U$, and we obtain the fourfield reduced model by only considering the evolution of $\psi$, $U$, $F$, and $\omega$. These reduced models provide accurate solutions in certain physical limits, at a fraction of the computational cost of the full extendedMHD model. Resistive WallM3DC1 optionally includes a resistive wall model, which is treated as a spatially resolved region of arbitrary thickness. This capability enables the study of socalled "resistive wall modes" (RWMs) in tokamaks, which are external kink modes that would be stable in the presence of a perfectly conducting wall, but which are unstable when the wall has finite resistivity.
Numerical MethodsFinite ElementsIn its 2D and 2Dcomplex modes of operation, M3DC1 uses twodimensional, reducedquintic triangular finite elements on an unstructured mesh. These highorder, $C^1$ elements are described in detail in SC Jardin J. Comp. Phys. 200:133 (2004). In its 3D mode of operation, M3DC1 uses threedimensional triangularprism elements that are a tensor product of the triangular reducedquintic elements in the poloidal plane with cubic Hermite elements in the toroidal (or axial) direction. These threedimensional elements also have the $C^1$ property. Implicit and SemiImplicit Time AdvanceReference Documents
How to Obtain M3DC1To run or obtain the source code for M3DC1 please complete and submit the PPPL Theory Codes LicenseRelease form. Please note that this agreement requires that a designee of PPPL be allowed to review any output of M3DC1 prior to publication to help ensure that the code has been run correctly and that the results have been properly interpreted. AcknowledgmentsThe development of M3DC1 has been supported by the US Department of Energy through the Center for Extended MHD Modeling (CEMM) and Center for Tokamak Transient Simulations (CTTS) SciDAC activities. The numerical tools and methods relating to meshing and domain decomposition are developed through a collaboration with the RPI SCOREC group. PublicationsThe following is a list of publications that focus primarily on M3DC1 modeling or numerical methods. It does not include publications that feature M3DC1 calculations in support of understanding specific experiments. If you would like to cite M3DC1, please see the list of references by subject to find the best appropriate references.
References By SubjectThe following list includes some of the best references for the implementation of various models in M3DC1, as well as some references to the application of the code to physical phenomena.
