Computational Modeling and Software Development


CQL3D Code

Description | Purpose | Algorithms | Results | Publications | Documentation

Short Description

The relativistic Collisional/QuasiLinear 3D code CQL3D solves a Bounce-Averaged Fokker-Planck equation to obtain the 3 1/2-D distributions of electrons and multispecies ions, resulting from the balance between collisions, RF/neutral beam sources and applied toroidal electric field, in toroidal geometry. (3 1/2-D refers to 2 velocity, 1 radius, and implicit treatment of the poloidal variation through the bounce-averaging.) Steady-state and time-dependent solutions are supported.

Dates/Active Use

1985 to present


R.W. Harvey and M.G. McCoy, Yu. Petrov



Purpose/Function/Special Features

The overall aim for CQL3D is to create a general facility for the accurate calculation of heating and current drive in tokamaks. To the extent possible, all the physics effects which are known to be important in such calculations should be included. CQL3D is a multi-species, 2D-in-momentum-space, 1D in noncircular plasma radial cooordinate, fully relativistic, bounce-averaged, collisional/quasilinear Fokker-Planck equation solver. It is run in combination with LH, FW, EC and EBW ray tracing or full-wave (AORSA and TORIC-LH) rf data, the FREYA neutral beam deposition package, and a given toroidal electric field, thereby providing a general model for the distortion of the electron and ion distribution functions resulting from auxiliary heating and current drive injected from the plasma periphery. The distributions are taken to be toroidally symmetric and independent of azimuthal angle about the ambient magnetic field. Radial drifts are neglected. With the bounce-average, account is taken of variations as a function of (non-circular) radial coordinate, poloidal angle, and two momentum-space directions. A kinetic bootstrap current calculation is included. The code may be run with separate 2D momentum space solves on each flux surface, on in 3D mode including radial transport according to prescribed diffusion and pinch terms.

Although the focus of the code has been on electrons, it is a multispecies code, i.e., it can treat electron and multiple ion distributions simultaneously. The NFREYA neutral beam deposition module is coupled in for modeling of neutral beam current drive.

Further description of the code is available in Ref. [1], essentially The CQL3D Manual, below. It is founded on the 2D-in-velocity bounce-averaged Fokker-Planck CQL code[2], with RF/ray tracing methods first appearing in [3] . An early application of the code, indicating its versatility, was synergy studies between fast wave, lower hybrid, and electron cyclotron waves [4], and below.

Basic Algorithms

The steady state distributions and the radial rf absorption profile are obtained by iteration between (1) the Guassian elimination solution of the Fokker-Planck equation for the steady state on each flux surface, and (2) the rf energy transport equation integrated along a ray.

The computational scheme for time advancement consists of an alternating direction implicit scheme, between the two momentum-space variables and the radial variable. The momentum-space equation is solved in a fully implicit, direct gaussian elimination scheme previously developed for the 2D in momentum-space CQL code. Similarly, the radial variable is advanced fully implicitly, presently by a splitting algorithm alternating with the velocity-space advancement. More recently, sparse-matrix iterative techniques have been used for a fully-implicit solution of the 3D (2V,1R) equations.

Coupled Diagnostics

The distribution functions from CQL3D are coupled to calculations of:
  • X-ray Bremsstrahlung energy spectra along specified sight-lines.
  • Electron cyclotron [and EBW] microwave emission, solving the plasma wave energy transport equation along rays terminating at a detector [with GENRAY]. Emission and absorption is based upon general 2D velocity distributions.
  • Fusion reaction rates for standard interactions, including DD and DT neutron rates
  • NPA charge exchange spectra [with A. Bader, MIT].

Key Results

CQL3D results have strongly influenced the course of several important rf current drive experiments and proposals, specifically (1) experiments at GA and collaboration with Kurchatov Institute, Moscow[4], (2) implementation and interpretation of the lower hybrid current profile control experiments on the ASDEX experiment[5] and the associated planning for the PBX lower hybrid current profile control- MHD second stability region-experiment[6], (3) interpretation of fast wave current drive (FWCD) and ECCD experiments at GA[7,8], and (4) consideration of FWCD synergy with electron cyclotron resonance heating (ECRH)[9], and electron cycltron CD [10] in tokamak reactor environments.

An ECCD result first obtained unambiguously by CQL3D([11]), is that current drive by outside (i.e., outboard side) launch of the rf power is strongly preferred to inside launch, based on current drive efficiency. For inside launch, ECCD efficiency at low power is about one-half of that for the outside launch; as rf power increased, the inside launch efficiency can decrease and even pass through zero. In addition, outside launch efficiency doubles at high rf power. As a result of CQL3D, a new outside launch configuration was implemented in DIII-D.

CQL3D current drive results have since been shown to agree in detail with DIII-D experimental data near both the first and second cylcotron harmonics.[7,8].

The code has provided detailed estimates of EC current drive in ITER, contributing strongly to the advancment of this scheme for current profile control [10].

Application of the code has been made to several additional tokamak experiments: FT-1 (Joffe), Tore Supra, T-10, TdeV, Versator-II. The code has lead in devolpment of similar codes in Europe and Japan[12].

New applications are being made to the disruption/runaway electron problem[13].

Radial transport modeling proved to be a decisive factor in interpretation of the TCV ECCD experiment [14] and the MST reversed field pinch[15]. The CQL3D code incorporates a variant, CQLP, which solves the 3D Fokker-Planck collisional problem along magnetic field lines, including the parallel streaming term, one flux surface at a time[16, 17].

Further work is being carried out on the parallel-transport Fokker-Planck resulted in the FPET code (renamed STELLA, after further additions) which models parallel electron transportthe electron streaming term along field lines, and obtains the self-consistent parallel electric field under the condition of equal electron and ion fluxes to divertor plate[18]. Additional boundary conditions and RF quasilinear diffusion have been incorporated[23].

The fully nonlinear and fully relativistic collision operator option has been benchmarked[23]. Current work [19] includes addition of a finite-orbit-width (FOW) option in the code, based on theory developed by Kupfer[20]. Ref. [21] reports validation work with the NSTX spherical tokamak at Princeton Plasma Physics Laboratory. Report [24] provides a detailed description of the FOW enhancement of CQL3D, and is submitted for publication.

New work (2015) is aimed at a time-dependent continuum 2D-in-V, 2D-in-space Fokker-Planck code for axisymmetric mirror devices, coupling in neutral beam and RF heating, and a neutrals code. This work is extending Ref. [21] to multispecies, and is combined with the self-consistent parallel electric field and divertor boundary conditions as employed in Ref. [18].

Selected Publications

  1. R.W. Harvey and M.G. McCoy, The CQL3D Code, Proc. IAEA TCM on Advances in Sim. and Modeling of Thermonuclear Plasmas, pp. 489-526, Montreal, (1992), available through USDOC/NTIS No. DE93002962; see also, CQL3D Manual.
  2. G.D. Kerbel and M.G. McCoy, Phys. Fl. No. 28, p. 3629 (1985).
  3. F.X. Soeldner et al., [Combined paper with R.W. Harvey and M.G. McCoy] Review of Lower Hybrid Experiments on Asdex, in Plasma Physics and Controlled Thermonuclear Fusion Research (Proc. 13th Int. Conf. Washington, 1990) (IAEA, Vienna, 1991) Vol.I, p.613.
  4. R.W. Harvey, S.C. Chiu, M.G. McCoy, G.D. Kerbel, G.R. Smith and T.K. Mau, 3D Fokker-Planck Calculation of Combined Fast Wave/Lower Hybrid and Electron Cyclotron Current Drive in Tokamaks, Proc. of IAEA TCM on Fast Wave Current Drive in Reactor Scale Tokamaks, Ed. by D. Moreau, A. Becoulet, Y. Peysson, in Arles, France, September 23-25, 1991. (See below.)
  5. V.V. Alikaev et al, Electron Cyclotron Current Drive Experiment on T-10, Nucl. Fus. 32 p. 1811(1992), and 35 p.369 (1995).
  6. S. Bernabei, personal communication (1993).
  7. C.C.Petty, et al., Fast Wave and Electron Cyclotron Current Drive in the DIII-D Tokamak, Nucl. Fus., Vol. 35, p. 773 (1995).
  8. R.A.James, C.C.Petty, and R.W.Harvey, A Comparison of Fundamental and Second Harmonic Inside Launch ECCD in the Presence of a DC Electric Field, Proc. of EC-9 Workshop on Electron Cyclotron Emission and Electron Cyclotron Heating Conference, Borrego Springs, CA (1995).
  9. R.W. Harvey, et al., Calculation of Combined Fast Wave/Lower Hybrid and Electron Cyclotron Current Drive in Tokamaks, in Proceedings of IAEA TCM on Fast Wave Current Drive in Reactor Scale Tokamaks (Synergy and Complementarity with LHCD and ECRH), Arles, France, 1991 (IAEA, Vienna).
  10. Electron Cyclotron Heating and Current Drive in ITER, R.W.Harvey, W.M.Nevins, G.R.Smith, B.Lloyd, M.R.O'Brien, and C.D.Warrick, GA Report GA-A22141 (Submitted to Nucl. Fus., 1996)
  11. R.W. Harvey, M.G. McCoy, and G.D. Kerbel, Power Dependence of Electron-Cyclotron Current Drive for Low- and High-Field Absorption in Tokamaks, PRL,62 (1989) 426.
  12. E. Westerhof, Fokker-Planck Quasi-Linear Codes for the Study of Electron Cyclotron Resonance Heating and Electron Cyclotron Current Drive, in Proc. of 9th Joint Workshop on ECE and ECRH, Borrego Springs, 1995.
  13. R.W. Harvey, V.S. Chan, S.C. Chiu, T.E. Evans, M.N. Rosenbluth, and D.G. Whyte, Runaway Electron Production in the DIII-D Killer Pellet Experiments, Calculated with the CQL3D/KPRAD Model, Physics of Plasmas 7, 4590 (2000).
  14. R.W. Harvey, O. Sauter, R. Prater, and P. Nikkola, Radial transport and electron cyclotron current drive in the TCV and DIII-D tokamaks, Phys. Rev. Lett. 88, Article 205001 (May, 2002).
  15. R. O'Connell, D.J. Hartog, C.B. Forest, J.K. Anderson, S.C. Prager, J.S. Sarff, T.M. Biewer, S.D. Terry, R.W. Harvey, Transition from stochastic magnetic to electrostatic-like transport in the Reversed Field Pinch, Phys. Rev. Lett. 91, 045002 (2003).
  16. O. Sauter, R.W. Harvey, and F.L. Hinton, 3-D Fokker-Planck Code for Studying Parallel Transport in Tokamak Geometry with Arbitrary Collisionalities and Application to Neoclassical Resistivity, Contrib. Plasma Phys. 34, 169 (1994).
  17. O. Sauter, C. Angioni, and Y.R. Lin-Liu, Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime, Phys. of Plasmas 6, 2834 (1999).
  18. K. Kupfer, R.W. Harvey, O. Sauter, G. Staebler, and M.J. Schaffer, Kinetic Modeling of SOL Plasmas, Physics of Plasmas 3, 3644 (1996).
  19. Yu. V. Petrov, R. W. Harvey, “Finite Orbit Width Features in the CQL3D Code”, TH/P6-02, IAEA FEC, San Diego, CA, Oct. 8-12 (2012). [See below Petrov_Harvey_CQL3D-FOW_Theory_IAEA_FEC (2012)]
  20. K. Kupfer, Fokker-Planck Formulation for RF Current Drive, Including Wave Driven Radial Transport, IAEA TCM on FWCD in Reactor Scale Tokamaks, Arles, 1991. [Copy below, by permission of author.]
  21. R.W. Harvey, and Yu. V. Petrov, D. Liu, W.W. Heidbrink, G. Taylor, P.T. Bonoli, ``CQL3D-HYBRID-FOW Modeling of the Temporal Dynamics of NSTX NBI+HHFW Discharges'', Proc. of 20th Top. Conf. on Radiofrequency Power in Plasmas, Sorrento, Italy, AIP Conf. Collection 1580, AIP Publishing NY (2014). [See below Harvey_Petrov_Liu_CQL3D-FOW-Hybrid_NSTX_RF (2014].
  22. R.W. Harvey, Y.-R. Lin-Liu, O. Sauter, A.P. Smirnov, T. Luce, R. Prater, “Current Drive Due to Localized Electron Cyclotron Power Deposition in DIII-D”, Proc. of Radio Frequency Power in Plasmas, 13th Topical Conference, 12-14 April 1999, Annapolis, MD, USA, AIP Conference Proceedings 485, 253-6 (1999). [See below Harvey_etal_Localized_ECCD (2000)].
  23. Yu.V. Petrov, and R.W. Harvey, "Benchmarking the Fully Relativistic Collision Operator in CQL3D", CompX report CompX-2009-1 (2009). [See below Petrov Fully Relativistic (2009)].
  24. Yu.V. Petrov, and R.W. Harvey, "A Fully-Neoclassical Finite-Orbit-Width Version of the CQL3D Fokker-Planck code", CompX report CompX-2016-1 (2016). [See below Petrov_Harvey_CQL3D-FOW_CompX-2016-1 (2016)].


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