SCoPE

**Name of the code.
**

Abbreviation SCoPE comes from **S**elf- **Co**nsistent **P**lasma **E**volution.
Development of the theory and software for the code started in 1993 as a
joint project between Culham Laboratory , United Kingdom and Moscow
State University , Russian Federation .

**Main authors.**

The originators of the project were Dr. D.C. Robinson, Fellow of Royal Society and D.P. Kostomarov, Professor, correspondent member of Russian Academy of Science. Many first class specialists in plasma physics, mathematical methods and numerical analysis contributed to the code. Among them are Professor, DSC F.S. Zaitsev, Dr. A.G. Shishkin, Dr. M.R. O'Brien, Dr. R.J. Akers, Dr. M. Gryaznevich and many others.

The main reason for the project was that most previous models either gave too general a formulation, which was very difficult to implement, or concentrated on one or other particular aspect of the problem, with much less attention given to the others. Many existing codes gave poor numerical treatment of the problem. The main aim of the project was to formulate a general, but computationally tractable, self-consistent model of toroidal plasma equilibrium evolution, to develop an accurate numerical algorithm and perform computer implementation of the model and algorithm.

Joint efforts have resulted in prominent results [1-7]. In particular, SCoPE predicted numerically the existence of advanced regimes with non-monotonic safety factor profiles in START. Studies and discussion of this phenomena went for about two years. Now it is recognized that such regimes exist in spherical tokamaks. SCoPE made noticeable contribution to understanding of plasma behaviour in advanced regimes in START and MAST and optimization of Ohmic and NBI discharges. Recently extended SCoPE modelling allowed to draw the conclusion that plasma resistivity in spherical tokamaks is close to neo-classical.

**Brief details of the physical model and options.
**

A large number of processes with very different time scales develop in toroidal plasmas. Code SCoPE considers relatively slow processes of gradual transition from one quasi-equilibrium state to another, caused by self-consistent interactions between the plasma and the magnetic field. On the one hand, different processes in the plasma change the pressure and current and their profiles (i.e. how they are distributed across the plasma), which leads to a slow rearrangement of the magnetic field. On the other hand, these plasma processes are themselves influenced by the magnetic field and how it evolves.

It is very important to develop accurate models and codes to describe plasma evolution as these have many applications. These include predictions of how plasma properties change with time, studies of tokamak regimes with the potential for improved performance, and investigations of the influence of non-ohmic currents on the evolution of plasma equilibria. One should note that in many cases it is hard to predict the behaviour of plasma using simple models or physical intuition. In these cases numerical modelling plays a crucial role.

Details of the model are presented in [6]. Here we mention only its main features.

The model is based on the Grad-Shafranov equilibrium equation, parallel in respect to the magnetic field projection of the Ohm law, kinetic and transport equations. The main physical ideas for description of the magnetic field evolution are taken from [10]. This general approach is well known, however its implementations strongly vary.

The SCoPE model includes many advanced options for simulating plasma behaviour in real devices. These are:

- taking account of intrinsic pressure-gradient-driven currents (so-called bootstrap, Pfirsch-Schluter and diamagnetic currents), currents induced by varying magnetic field and additional non-inductive currents driven by radio-frequency electromagnetic waves or neutral beam injection (NBI);
- use of neo-classical resistivity;
- treatment of the effects of the vessel wall;
- specification of the time dependence of currents in the solenoid and poloidal field coils;
- direct coupling with kinetic and/or transport equations;
- solution of free or fixed plasma boundary problems;
- solution of control problems, for example, for maintaining a given plasma position, time variation of total plasma current, safety-factor profile, etc.
- solution of inverse problems for reconstruction of the equilibria using magnetic and optic diagnostics.

As an option the model can use data from experimental measurements and EFIT reconstructions.

Examples of the use of SCoPE can be found in Refs. [1-9]. Ref. [6] demonstrates one of the capabilities of the numerical algorithm and the SCoPE code to simulate plasma evolution in the presence of strongly localised currents driven by Lower Hybrid waves. In Refs. [1-4,8,9] it has been shown how optimised shear regimes can be achieved and sustained in spherical tokamaks START and MAST. The main results of calculations are in qualitative accord with experimental indications. Steady-state regimes in MAST with and without NBI are studied in Ref. [5]. Ref. [7] justifies that plasma conductivity in spherical tokamaks is close to neoclassical.

**Brief description of numerical techniques employed.**

Details of numerical techniques can be found in Ref. [6].

One of the new features of the mathematical model used in SCoPE is the formulation of the problem for magnetic field evolution in cylindrical (R,Z) co-ordinates. Here R is the major radius, and Z is the vertical co-ordinate, of a given spatial location. This is appropriate to an Euler description of the motion of the plasma column. In previous models Lagrangian or semi-Lagrangian approaches have been used. The numerical method also uses (R,Z) co-ordinates, which results, in particular, in a less complicated and more accurate algorithm for the solution of the free boundary problem than the use of the alternative inverse variables technique. (R,Z) co-ordinates allow a more precise description of plasma configurations with X-points, and easy treatment of physical effects related to the solenoid, control poloidal filed currents, currents induced in the wall of the vessel, etc. These co-ordinates also mean one can avoid the use of artificial viscosity in the numerical method and the necessity for special description of the plasma boundary motion.

Another new feature is the use of the current F (which gives the toroidal magnetic field B_tor=F/R), as one of the unknown functions instead of the toroidal flux. This avoids the necessity of calculating second order derivatives numerically and thus allows more accurate calculation of the right-hand-side of the Grad-Shafranov equation.

**Range of applicability, limitations.**

The code is applicable to description of evolutionary processes on time scale of the order of 1/10 -1/1000 of discharge duration.

**Software **

Graphical user-friendly interface SCoPEShell
has been developed for code SCoPE. The whole interface is written in
JAVA and thus is platform independent. The interface has been
successfully tested under MS Windows 98, ME, XP and X-Window Linux Red
Hat 7.0, 8.0, 9.0.

The interface allows one to

- set all input and output parameters of SCOPE;
- edit selected SCOPE source files;
- compile and run SCOPE;
- plot 2D and 3D SCOPE data;
- convert data to NetCDF format.

Up-to-date detailed description of the interface, which includes "Overview", "Tutorial", "SCoPeShell commands" and "SCoPeShell Windows" in HTML format is available through the button "Help" in SCoPeShell . This information can be viewed with any standard browser either within SCoPeShell or alone.

The code SCoPE was developed using elements of package technology and object-oriented approach. Some ideas of object-oriented techniques were used for programming of difference operators, sources of external current, creating spline approximation of grid functions, etc.

Package technology implemented in SCoPE includes a program generator, written in JAVA, which can construct a particular program from a cascading menu of options. The generator takes as input modules from the SCoPE program foundation, tunes them for the particular problem according to the user choice and builds ready-to-run programs. The latest version of SCoPE includes over 130 subroutines in Fortran-77. It is more than 40000 lines long and contains about 30 variants of physical models and several different numerical algorithms. These include

- Classical or neo-classical plasma conductivity; different radial transport models (either calculated as the solution of transport or kinetic equations, or input from experimental measurements of density and temperature);
- Various types and approximations for currents produced by external forces (for example, approximate formulae, numerically calculated or input from other codes);
- Fixed or free boundary problem;
- Options allowing different controls, such as of plasma position, total plasma current, safety factor profile, etc.
- Among the optional numerical algorithms are: modifications to method for time discretization with different approximations for derivatives; the coordinate descent method, gradient method and Tikhonov regularization for solution of control problems; various algorithms for data smoothing, etc.

SCoPE has an option to input data in the form of TRANSP-like u-file or use a data set internally in Fortran routines and functions. The following physical input data is typically used in SCoPE for calculations:

- Vessel geometry, i.e. position of walls.
- Current in the toroidal field coils (the rod current) as a function of time.
- Position and currents in the poloidal field coils (PFC). In the free boundary problem the input PFC currents are overwritten by the PFC currents calculated by SCoPE, which are adjusted to give the specified plasma position. PFC currents are also used to calculate the induced plasma current.
- Position of and current in the solenoid. The solenoid current can be either kept exactly as input or adjusted to give a prescribed total plasma current as a function of time. The solenoid current is also used for calculation of the induced plasma current.
- Position of the plasma boundary as function of time. In the free boundary problem SCoPE searches for the closest position to that prescribed. In the fixed (prescribed) boundary problem the input boundary limits the plasma directly.
- Total toroidal plasma current as function of time.
- Initial plasma toroidal current density.
- Mass, charge, density (as function of time and the half-width of the flux surface) and temperature for each particle species present in plasma. The total plasma pressure can be calculated either by the sum over all species or taken from the input file, where it is specified as function of time and radius of the flux surface. Alternatively SCoPE has an option for calculation of density and temperature using transport and kinetic equations.
- Z effective.
- NBI current drive and additional fast ion pressure arising from NBI.

SCoPE produces a large amount of important plasma and device characteristics, such as:

- Poloidal flux and the poloidal flux surface contours as functions of space and time.
- Plasma toroidal current density.
- Safety factor.
- R co-ordinate of the magnetic axis R_axis and toroidal magnetic field at R_axis as functions of time.
- Toroidal loop voltage.
- PFC, solenoid and wall currents as functions of time.

The software has been used successfully with various compilers and operating systems on SUN, IBM RISC and personal computers, indicating the reliability of the code. For a relatively simple calculation the typical time for the calculation of ~500 time steps on a grid 50x50 is less than 1 hour on a modern workstation. However, for the solution of control problems, the CPU time can substantially increase (up to several hours). The requirement for RAM is usually 30-60 MB which is relatively modest. The hard disk space required for the output files for one run is typically ~20 MB.

**Integrated modelling.**

SCoPE code has links to codes FPP-3D (kinetics), TRANSP (transport and kinetics), EFIT (equilibrium reconstruction using experimental measurements), LOCUST (Monte-Carlo kinetics).

**Benchmarking and validation carried out. **

Theoretical results about the convergence of the presented method are not available. In this situation comparison with other models and codes becomes very important. The numerical algorithm and code SCoPE have been tested with a number of "physics" tests. One of the tests was to compare results from SCoPE and TOPEOL. The fixed time equilibrium code TOPEOL (which solves the Grad-Shafranov equation only, i.e. without the Ohm's law) was used to fit experimental data from the START spherical tokamak at different moments of time. Then the SCoPE code was used to evolve the plasma for the same time varying input parameters as in the experiment. The resulting sequence of equilibria was very close to that from TOPEOL. The difference in the poloidal flux, toroidal current density and safety factor was not greater than 15%. This test gives an example which validates the numerical method and SCoPE software. Plasma evolution with SCoPE was also close to the experimental observations.

Comparison with analytical results of neo-classical theory and other analytical results. Good agreement with code TRANSP. The code was successfully tested with different compilers in different operating systems and hardware platforms such as IBM RS, SUN, DEC, Intel and oth.

**Some references where the code's theory, methods and applications are
described.**

- F.S. Zaitsev. Mathematical modelling of toroidal plasma evolution. - Moscow: MAX Press Publishing Co., 2005, 524 p. (in Russian) http://www.zone-x.ru/showtov.asp?fnd=&cat_id=258662
- Kostomarov D.P., Zaitsev F.S., Shishkin A.G., O'Brien M.R., Gryaznevich M. Access to ``Advanced'' Regimes in Tight Aspect Ratio Plasmas. 25th European Conference on Controlled Fusion and Plasma Physics. Praha, 1998, P. 660. http://epsppd.epfl.ch
- Akers R.J., Mikkelsen D.R., Conway N.J., Counsell G.F., Nightingale M.P.S., Zaitsev F.S. Neutral Beam Ion Confinement and Current Drive in the Spherical Tokamak. 26th European Conference on Controlled Fusion and Plasma Physics. Maastricht, 1999, ECA v. 23J, p. 733-736. http://epsppd.epfl.ch
- Kostomarov D.P., Zaitsev F.S., Shishkin A.G., O'Brien M.R., Gryaznevich M., Akers R.J., Krastylev A.V. Access to Optimised Shear Equilibria in Spherical Tokamaks. 26th European Conference on Controlled Fusion and Plasma Physics. Maastricht, 1999, ECA v. 23J, p. 1745-1748. http://epsppd.epfl.ch
- Kostomarov D.P., Zaitsev F.S., Shishkin A.G., Robinson D.C., O'Brien M.R., Gryaznevich M. The Problem of Evolution of Toroidal Plasma Equilibriua. Comp. Phys. Comm. 2000, v. 126, p. 101-106.
- Akers R.J., Bond A., Buttery R.J., Carolan
P.G., Counsell G.F., Cunningham G., Fielding S.J., Gimblett

C.G., Gryaznevich M., Hastie R.J., Helander P., Hender T.C., Knight P.J., Lashmore-Davies C.N., Maddison G.P., Martin T.J., McClements K.G., Morris A.W., O'Brien M.R., Ribeiro C., Roach C.M., Robinson D.C., Sykes A., Voss G.M., Walsh. M.J., Wilson H.R., Zaitsev F.S. Steady State Operation of Spherical Tokamaks. Nucl. Fusion. 2000, v. 40, n. 6, p. 1223-1244. - Zaitsev F.S., Shishkin A.G., Kostomarov D.P., O'Brien M.R., Akers R.J., Gryaznevich M., Trefilov A.B., Yelchaninov A.S. The Numerical Solution of the Self-Consistent Evolution of Plasma Equilibria. Comp. Phys. Comm. 2004, v. 157/2, p. 107-120.
- Kostomarov D.P., Zaitsev F.S., Akers R.J.
and Shishkin A.G. Investigation of the Electrical Conductivity of

a Plasma in a Spherical Tokamak. Doklady Physics. 2004, v. 9, n. 6, p. 350-353. - .P. Kostomarov, F.S. Zaitsev, V.V. Nefedov, A.G. Shishkin, D.C. Robinson, M. Cox, M.R. O'Brien. Equilibrium Evolution of Tight Aspect Ratio Tokamak Plasma Including Non-Ochmic Currents. The 1996 Spherical Tokamak/Torus Workshop. UKAEA Culham Laboratory. Abingdon, December, 1996. Vol. I, p. 370-388.
- Zaitsev F.S., Trefilov A.B., Akers R.J. An Algorithm for Reconstruction of Plasma Parameters Using Indirect Measurements. 30th European Conference on Controlled Fusion and Plasma Physics. St. Petersburg, 2003, ECA v. 27A, P-2.70. http://epsppd.epfl.ch
- L.E. Zakharov, V.D. Shafranov, Reviews of Plasma Physics. 11(1987)118.