Role of Analytic Theory in the US Magnetic Fusion Program

P.W. Terrya, Peter Cattob, Nikolai Gorelenkovc, Jim Myrad, Dmitri Ryutove, Phil Snyderf, and F. Waelbroeckg

aUniversity of Wisconsin-Madison bMassachusetts Institute of Technology cPrinceton Plasma Physics Laboratory dLodestar eLawrence Livermore National Laboratory fGeneral Atomics gUniversity of Texas at Austin Abstract: The essential role of analytic theory in fusion research is described. Key attributes show that analytic theory is critical to understanding experiment and complementary to computational approaches. It continues to develop and evolve, and will be crucial even as numerical techniques increase in sophistication. Five critical capabilities of analytic theory are reviewed to demonstrate its essential role in discovery, its complementarity with computation, and its importance in attacking gaps in knowledge. Because future capacity to engage in analytic theory is threatened by demographic trends, the need to develop strategies to maintain a healthy analytic theory capability is called out, and some initial ideas for addressing the problem are given.

I. Introduction

Analytic theory has played a decisive and foundational role in the development of magnetic confinement plasma physics, as evident in numerous contributions that have become standard material in textbooks. Examples include plasma fluid theory like magnetohydrodynamics and Braginskii two-fluid theory; plasma kinetic theory in its numerous manifestations, from Fokker-Plank to gyrokinetics; neoclassical transport theory; rf wave propagation theory; and many others. These models and theories are not restricted to a phenomenological arena, although they have revealing and well-established connections to phenomenology, rather they are derived hierarchically from exact descriptions using ordered expansions to delineate conditions under which they are valid.

Indeed, a hallmark of fusion plasma theory is its unusual rigor, rooted in the widespread use of asymptotic analysis and ordering techniques. This rigor is seen, for example, in the contrast with analytic theory in astrophysics, which tends to be much more heuristic and phenomenological. This rigor gives analytic theory in fusion plasma physics a number of powerful attributes: •In many instances, the task of model qualification, an important element of verification and validation, already has a formal expression in the derivation of the model. •The ordering that establishes the validity of a model has generally been carried through a hierarchy of models from complex to reduced, allowing a basic understanding of elemental workings to be mapped onto the complexities of a comprehensive model. •Solutions of rigorous models can be assumed to be realizable in plasmas, even if observations in no current experiment agree with the solution. This creates a palette of physically possible, verifiable behaviors, which can be drawn upon to understand new observations, be they from experiment or numerical simulations. •Analytic theory, under appropriate circumstances, is predictive.

It is important to recognize that analytic theory in fusion plasma physics is not like a dead language. It continues to evolve and develop in potent ways as a response to needs that arise in

research. This includes the response to new observations from experiment and simulation that are not readily accommodated within existing frameworks and understanding. It includes the need to develop new models for describing physics that is not included in existing models or for modeling observations in regimes beyond the validity of existing models. Consequently, as numerical approaches increase in reach and sophistication, tackling ever more comprehensive issues, the role of analytics will also be enhanced by its ability to formulate new procedures to solve problems thought to be unsolvable just a few years ago.

Analytic theory has capabilities that are essentially orthogonal to those of numerical simulation, but in a highly complementary fashion. These include the way analytic theory deals with the fundamental behaviors and properties of systems, often expressed in scalings and symmetries, the capacity to conceptualize the workings of complex systems in ways that differentiate component processes, and the ready access through conceptual underpinnings to basic understanding. The complementary strengths of analytic theory and simulation will be illustrated in more detail in the examples that follow. Because of this complementarity it is crucial that the capability to pursue analytic theory in the US fusion program remain viable, healthy, and vigorous in the future. In particular the drive toward predictive capability, with its prima facie base in comprehensive numerical algorithms, is largely viewed as an exclusively numerical enterprise. However the complementary essential roles of analytic theory and numerical simulation in fact increase the necessity for analytical theory as numerical efforts grow in sophistication. This holds true for virtually all gaps of the Greenwald Report that are addressable by theory. II. Critical Capabilities of Analytic Theory

The importance of analytic theory, its past contributions, the types of contributions that can

be anticipated in the future, and the complementary nature of analytic theory and numerical simulation, can be framed in terms of critical capabilities. The following critical capabilities will be discussed: 1) the development of hierarchies of models, from heuristic reduced models to comprehensive integrated models; 2) the discovery and elucidation of fundamental properties in plasma physics processes, including symmetries, scalings, and mechanisms, and the prediction of their role in magnetic confinement; 3) the conceptualization of the workings of systems and the integration of their component processes, both in developing new ideas and in understanding existing systems; 4) the verification and understanding of the results of numerical algorithms, both in the informal sense of understanding and confirming the validity of unexpected outcomes from simulation and the formal testing of numerical algorithms as a companion exercise to validation, and 5) the development of entirely new models, both for hierarchies, and for areas where existing models are missing important physics or capabilities.

Model Hierarchies - Analytic theory and reduced models contribute to a healthy balance in the fusion program. While large simulation codes are a practical necessity in fusion research today, it is absolutely essential that they be backed by a hierarchy of less complex approaches ranging from reduced numerical models, to purely analytic calculations, and finally even to cartoon models or back-of-the-envelope handwaving. Indeed, one can argue that a problem is not really solved until it has been understood at all these levels: 1. “first principles” numerical simulation 2. reduced numerical models 3. analytic models 4. heuristic, intuitive models

The hierarchical approach is necessary to establish the validity of intuitive concepts (up the hierarchy), identify important physics not in existing simulation models (up the hierarchy), and understand complex simulations (down the hierarchy), and experiments. Ultimately the fusion

program needs both quantitative answers and intuitive understanding. The emphasis today has shifted to the quantitative side, but clearly intuition is essential for the field to develop new concepts, be they to advance science or improve fusion component design, as discussed in more detail in subsequent sections.

There are many examples of where a hierarchy of models has served the program well. One particular example from the physics of blob filaments and scrape-off-layer (SOL) transport illustrates theoretical progress up the hierarchy. Blob filaments, or “blobs”. were first observed experimentally in the 1980’s [1, 2], but the significance of these turbulence structures was not fully appreciated until much later. In 2001, an intuitive, entirely analytic model describing the propagation of blobs in the SOL was devised [3]. This model provided a new paradigm for SOL transport, emphasizing convective motion of coherent turbulent structures, in contrast to turbulent diffusion. As recent reviews [4, 5] of blob physics attest, this simple picture has stimulated several hundred papers - experimental, theoretical, and numerical - which impact the important, ITER-relevant topic of the interaction of intermittent plasma events with plasma facing components. SOL transport can both broaden the heat flux channel impacting the divertor, and carry intermittent plasma structures to the radial walls of the device. Figure 1 illustrates a simple conceptual model of a blob circuit [4] and below it the results from a reduced-model numerical simulation of blobs in NSTX [6], which could be compared directly with a gas puff imaging diagnostic. Within the last year or so, the first gyro-kinetic simulations in realistic tokamak geometry [7] are providing simulations of the dynamics of these filamentary turbulent structures.

Fig. 1. Conceptual circuit model and model based simulation for blob dynamics in the SOL.

Fundamental Properties – Physics provides insight into the universe by focusing on the physical mechanisms that operate in nature. In the reductionist approach, physical objects, by the choice and design of the problem, have a simple connection to mechanisms. An example is a fundamental particle and its decay channels. In complex systems like plasmas, mechanisms remain essential to understanding, and generally relate to and reveal symmetries and scalings. The latter are among the most effective means of characterizing and representing complex behavior. Analytic theory is particularly well suited and adept at probing mechanisms, uncovering their fundamental properties, and deducing the way they impact systems.

In the identification of fundamental processes, analytic theory establishes direct ties to physics understanding. This understanding of mechanisms provides pathways for extending the concepts beyond and outside the original context, creating synergies that accelerate progress. Because the enabling physics can be made transparent in the analytic treatment, strategies for confirmation with experiment are naturally suggested. The capability of analytic theory to probe fundamental processes is strongly complementary with simulation. Fundamental processes and their direct effects may be masked by the complexity naturally present in solutions of comprehen-

sive models. On the other hand, simulation is essential to determine how the processes play out in the complex environments treated by comprehensive models.

Many examples could be cited to illustrate these points. One is the suppression of turbulence by stable shear flows. This initially novel idea was first developed with purely analytical theory in an effort to understand the H mode [8] and its complex set of phenomena [9]. The use of asymptotic analysis in the theory allowed identification of a key property that could be tested experimentally, namely, that when the shearing time becomes smaller than the nominal nonlinear correlation time, the latter decreases to match the shearing time. This prediction of analytic boundary layer theory was observed a short time later in experiment, as shown for τsh and τc

lab in Fig. 2, confirming the original idea [10]. The successful initial comparison of theory and experi-

A)

B)

Fig. 2. A) Experimental observation of the balance of shearing time and correlation time in a shear layer. B) Shear suppression in a toroidal simulation.

ment stimulated many other experimental tests and confirmations [11]. The analytic understanding of the mechanism made it possible to couch the results in terms of fluid dynamics and show that they extended to non-ionized fluids, particularly in geophysical flows [12]. Because the initial theory used a simplified model that boiled down the physics to its essence, the fundamental mechanism had to be tested in more complete models. It was subsequently tested in toroidal models [13], and is now a standard phenomenon in magnetic fusion turbulence simulations.

A second example involves the area of energetic particle (EP) physics and the theory for nonlinear saturation of toroidal Alfvén eigenmodes (TAEs) [14]. Analytic theory has played a leading role in describing, understanding, and predicting a variety of phenomena involving energetic particles [15]. TAE stability is a central problem because of the deleterious effects on fusion plasma performance. TAE stability is hard to validate in present experiments because they have only specific EP species. Hence, analytic theory plays an important role. In Fig. 3 distilled complex behaviors from analytic theory were analyzed in order to identify the essentials behind the nonlinear TAE saturation observed in many experiments. A relatively simple formula was developed [14] to connect two limiting cases interpolated between near threshold conditions, i.e., (1) when the damping rate is less than but about equal to the growth rate 𝛾! ≤ 𝛾!; and (2) when the modes are strongly driven 𝛾! ≪ 𝛾!. In the first case the saturated amplitude expressed in terms of the particle trapping frequency gives 𝜔!! ≈ 𝜈!""! 𝛾! 𝛾! − 1, whereas in the second case it is 𝜔!! ≈ 𝜈!""! (!!

!!)! !. Numerically the TAE amplitude prediction was done using the

guiding center following code ORBIT [16] modified for the purpose of demonstrating the saturation of the Alfvénic modes in the presence of EPs. In general, simulation is important for confirming the hypotheses of the analytic theory, but by themselves, the simulation results are difficult to interpret without the guidance of the theory and its critical hypotheses. This illustrates a crucial issue that arises as advances in computational capability allow increasing complex simulations. While the ability to simulate such situations is desirable for realism, they can be as difficult to interpret as an experiment.Analytic theory, with its penchant for distillation down to essential mechanisms and hypotheses, provides essential inroads to interpretation.

Fig. 3. An analytic theory formula for the saturated amplitude of toroidal Alfvén eigenmodes compared with simulation results.

Conceptualization of Systems – Analytic theory enables broad conceptualizations of the workings of systems and integration of their component processes, both in developing new ideas and in understanding existing systems. This process can produce wholly new, unplanned for ideas. Given the inherent complexity of systems, analytic theory identifies key aspects of component parts and integrates them in a rudimentary conceptualization. Numerical approaches play a complementary role because they test the conceptualization with all of the details and pro- cesses intrinsic to the system. An example is the creation of the concept of the snowflake divertor [17]. This development in the problem of power handling in burning plasmas was first conceived as an analytic theory exercise in conceptualization. A near-second-order null of the poloidal field having a characteristic six-fold symmetry (Fig. 4A) is created by bringing two first-order nulls close to each other by manipulating currents in remote PF coils. Extremely strong flaring of the magnetic field near the second-order null leads to significant lowering of the heat flux to the divertor plates, combined with “activation” of additional strike points compared to the standard divertor (four instead of two). The theory integrated key elements of a new magnetic field configuration with a complex plasma response. As with prior examples, predictions were tested with detailed computation [18] and with experiment [19, 20]. Fig. 4B shows redistribution of the heat flux between strike points on the TCV device [21], Another important effect is a much easier access to detached operation observed on NSTX and DIII-D [20].

The snowflake divertor is only one example of divertor innovation involving analytic theory. Other innovative concepts include a cusp/X-divertor [22], where the poloidal field is flared near the strike points, and a super-X divertor [23], where a strike point is moved to as large major radius as possible in order to increase an area of the wetted surface.

Fig. 4 A) An “ideal” snowflake configuration. B) Experimental demonstration of the plasma heat redistribution between several strike points (courtesy B. Labit, TCV). An overall experimental configuration is shown in the right inset. Understanding and Verifying Numerical Results – The complexity of numerical algorithms requires methods for testing the validity of their results. This applies in an informal sense, where it is necessary to understand and confirm the validity of unexpected outcomes from simulation. It also applies in a formal sense, where there are prescriptive procedures for ensuring that a numerical algorithm faithfully solves a mathematical model, and that the mathematical model is an accurate representation of nature. These procedures are referred to respectively as verification and validation (V&V). V&V is indispensible in magnetic fusion sciences for developing predictive capability via numerical models. The exercise of code-code comparisons and benchmarking has been an important first step in verification, but is insufficient, in and of itself. A successful outcome guarantees only that two codes give the same answer, not that the answer is a correct solution of the model they are solving. Consequently, comparisons with analytic theory are required for verification. Because it is not generally possible to solve model equations analytically for the complex cases for which numerical models are created, it is necessary to devise methods that test algorithms with analytic solutions that are different from the general problem. The method of manufactured solutions, for example, creates artificial problems with exact analytic solutions that are also solved by the numerical algorithm [24]. Because this type of exercise requires innovative analytic theory to devise the problem, find the solutions, and then carry out the tests, the capability to perform analytic theory is crucial. An example of verification using analytic theory is the residual flow calculation [25-26]. Residual flow is not an actual flow in a fusion plasma, but a flow response to an impulsive perturbation that instantaneously charges a rational surface. The response decays from the initial state with oscillations produced by geodesic acoustic modes to a residual, non zero level. The residual flow calculation was devised and carried out as a problem in analytic theory to test whether gyrokinetic and gyro Landau fluid codes correctly handled the guiding center drifts of trapped and passing particles that establish flows in tokamak plasmas. Because code developers thought that both types of codes properly treated the relevant physics, the demonstration that only the gyrokinetic codes reached the predicted residual flow under the appropriate impulsive perturbation was a dramatic test of code capability [27]. A modified flow-response calculation was recently developed for the situation in which an external stochastic magnetic field is applied once the residual flow level is reached after the impulsive perturbation [28]. The calculation was used to verify that the novel flow response of a gyrokinetic simulation in which the potential crosses zero was a correct solution of the equations [29]. The confirmation led to an understand-ing of the high β runaway [30], where transport rates increase to very high levels above a critical

β, as a bona fide physical process associated with the disabling of zonal flows by charge loss via streaming along stochastic fields. Figure 5 A) shows the flow response for the case with a stochastic magnetic field with two different magnitudes of A||. The time to zero crossing is dependent on A||. In Fig. 5 B), a comparison of the simulation with analytic theory predictions of the scaling of the time-to-zero-crossing provide verification of the simulation results. A)

B)

Fig. 5. A) Evolution of residual flow after application of an external stochastic magnetic field. B) Scaling of the time-to-zero-crossing with A||, for simulation (dots) and analytic theory (line).

Developing New Models – As mentioned in the first paragraph of the introduction, a major role of analytic theory has been development of models for describing plasma physics. While these models have been highly successful, none provide a complete description of plasma behavior, and none are exact. As experiments enter new regimes of operation, more advanced diagnostics reveal new details, and general understanding advances, it becomes necessary to develop new models or incorporate new capabilities into existing models. Such efforts may be to develop or complete a hierarchy of models, as described above, to extend models to new regimes, or to add new capabilities in terms of known limitations and missing physics. Given the sophistication of existing models and their basis in rigorous ordering techniques, extension of such models and the development of new models must employ analytic theory with an equal or greater degree of rigor and an understanding of all of the subtleties of previous work. This type of work thus requires significant analytic theory expertise. Virtually any scenario for the future development of fusion includes this type work. It is necessary for integrated simulation efforts and improvements to modules, and for predictive capability and the understanding that implies. A prominent example is the introduction of kinetic capabilities into fluid codes. This has been done to create reduced fluid models with key kinetic processes that run in a fraction of the time of kinetic models. It has also been done to introduce kinetic effects into models of global processes that are well treated by fluid codes. The improvements allow more accurate description of dissipative processes, pressure anisotropy, neoclassical effects, etc. Figure 6 illustrates progress in developing drift kinetic closures to extend the capabilities of workhorse MHD codes such as M3D-C1 and NIMROD. The first stage of this effort provides the evaluation of neoclas-sical transport coefficients in axisymmetric 3D geometry. The figure shows a comparison with an analytical model (red) of the neoclassical conductivity and bootstrap current coefficients associated with the pressure gradient, electron temperature gradient, and ion temperature gradient in the banana regime. Two equilibrium configurations are considered, one with large-aspect-ratio circular geometry (blue) and another with the geometrical parameters of NSTX (green) [31]. Gyrokinetic closures have also been developed [32] and are being implemented [33]. Important symmetries need to be understood to correctly perform momentum transport and profile evolution studies that properly retain intrinsic rotation [34].

Fig. 6. Comparison of analytic theory with a drift kinetic fluid closure for the neoclassical conductivity and bootstrap current coefficients for two toroidal configurations.

III. Addressing Gaps

The above examples demonstrate the essential and unique capabilities of analytic theory, and the complimentary role it plays with numerical simulation in discovery and understanding. Critically, the wide uses made of analytic theory in these examples provide ample demonstration of its crucial capabilities in bridging gaps between the known and the unknown. Because it is a fundamental approach in physics (since at least Newton), and not simply a technique, model, or algorithm, it can be applied to all gap areas in fusion that are amenable to theory, including those of the Greenwald Report. Having noted this breadth of applicability, we forego the exercise of listing Greenwald gaps. However, the detailed examples above are indicative of the kinds of approaches that can be applied to the Greenwald gaps. These types of gaps represent known shortcomings. It is also true that analytic theory is capable of making discoveries of things we don't even know about, and that are not therefore in the Greenwald Report.

IV. Looking to the Future

The above assertions are categorical but probably not controversial. There exists generally an

appreciation for analytic theory, and it is funded by DOE. However, despite this appreciation, there are forces that threaten to diminish and compromise future US capability for the analytic theory required to tackle critical gaps. If not addressed, these forces, which are largely demographic but could become structural, are poised to lead to a loss of capability in analytic theory. The pioneers who bequeathed to fusion plasma physics its remarkable analytic theory tradition, are mostly gone. The generation of scientists they trained to do analytic theory, and who account for the bulk of present efforts, are themselves nearing retirement. Younger generations of theorists entering the program have mostly been trained in numerical approaches. Students who are given a choice between a Ph.D. project in analytical theory and simulation, tend to choose the latter. This is partly because they perceive that numerical skills are more marketable, and partly because, for a generation that has grown up with computation, there is an allure for performing sophisticated simulations on ever more powerful computers. These kinds of incentives will only increase with the scale of projected simulation efforts and advances in computational science. It is essential that strategies be developed within the US program to counter these trends. Potential remedies should address at least three broad areas of concern. These are 1) providing

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FIG. 6. (Color online). A comparison between the output of the NIES code with the Sauter analytic fits in thezeroth-order collisionality limit. The Sauter model is plotted in red. Results from NIES for a large aspect ratio (LAR)equilibrium are in blue, while the results for an NSTX equilibrium are in green. In (a), the triangles correspond tothe calculated neoclassical conductivity ratio while the circles correspond to the calculated pressure gradient drivecoefficient. In (b), the triangles correspond to the calculated electron temperature gradient drive coefficient, whilethe circles correspond to the calculated ion temperature gradient drive coefficient.

frame of each species’ macroscopic flow, which sim-plifies the task of evaluating accurately the highergyrotropic moments needed for the fluid closure,namely the pressure anisotropy, the parallel heatflux, and the parallel collisional friction force. The2D NIES code provides a proof of principle that sucha formulation9,17 can be solved efficiently. We intendto continue this work in pursuit of an efficient neo-classical solver in three spatial dimensions. A first,near-term step would be to extend the code to finitebut still small collisionality, allowing the distributionfunction to vary poloidally. In the longer term, wewill work in nonaxisymmetric geometries, allowingus to solve for the bootstrap current around mag-netic islands. By coupling such a code with an ex-tended MHD solver (e.g., M3D-C1), we will be ableto study the hybrid evolution of various core plasmainstabilities, such as the neoclassical tearing modeor sawtooth instability.

ACKNOWLEDGMENTS

This research was supported in part by an awardfrom the Department of Energy (DOE) Office of Sci-ence Graduate Fellowship Program (DOE SCGF).

The DOE SCGF Program was made possible in partby the American Recovery and Reinvestment Actof 2009. The DOE SCGF program is administeredby the Oak Ridge Institute for Science and Educa-tion for the DOE. ORISE is managed by Oak RidgeAssociated Universities (ORAU) under DOE con-tract number DE-AC05-06OR23100. All opinionsexpressed in this paper are the authors’ and do notnecessarily reflect the policies and views of DOE,ORAU, or ORISE.

This work was also sponsored in part by the U.S.Department of Energy under grant nos. DEFC02-08ER54969 and DEAC02-09CH11466 and the Sci-DAC Center for Extended MagnetohydrodynamicModeling (CEMM).

1P.H. Rebut, R.J. Bikerton, and B.E. Keen, Nucl. Fusion 25,1011 (1985).

2P.C. de Vries, M.F. Johnson, B. Alper, P. Buratti, T.C.Hender, H.R. Koslowski, V. Riccardo, and JET-EFDA Con-tributors, Nucl. Fusion 51, 053018 (2011).

3K. Ikeda, Nucl. Fusion, 47 (2007).4T.C. Hender, J.C Wesley, J. Bialek, A. Bondeson, A.H.Boozer, R.J. Buttery, A. Garofalo, T.P Goodman, R.S.Granetz, Y. Gribov, O. Gruber, M. Gryaznevich, G.Giruzzi, S. Gunter, N. Hayashi, P. Helander, C.C. Hegna,D.F. Howell, D.A. Humphreys, G.T.A. Huysmans, A.W.Hyatt, A. Isayama, S.C. Jardin, Y. Kawano, A. Kellman,

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funding incentives and appropriate environments for doing analytic theory, 2) safeguarding and enhancing the capacity to train analytical theorists so they have the kinds of skills described in this paper, and 3) finding ways to help students discover the excitement of doing analytic theory.

We cannot pretend to have all of the answers at this point, but provide some initial thoughts on the types of things that might be done in each of these areas.

Funding Incentives and appropriate environments • Create centers or institutional homes for analytic theory. One model might be the Institute for Fusion Studies under its original funding structure and mandate from 1980. • Stipulate that large-scale simulation projects carry a dedicated analytic theory component. • Establish postdoctoral fellowships for doing analytic theory. • Institute forums for reporting and discussing analytic theory work, either as special sessions within existing workshops, or in a new workshop.

Training • Given that universities are central in training graduate students, determine the future role of universities in DOE-funded fusion research in the context of the field’s transition to larger scale projects like ITER and the Fusion Nuclear Science Facility. To date, this topic has not been given adequate consideration. • Fund new faculty lines at universities in analytical theory for fusion sciences. One model for this idea is the recent NSF initiative “Faculty Development in the Space Sciences” that provides faculty salary and student support for five years for new hires in institutions that win an award. • Fund 5-6 graduate student positions at universities known to train strong analytic theorists to assure that at least one theory student is trained per year on average.

Attracting students to analytic theory • Designate an APS-DPP visiting lectureship slot for analytical theory with the idea of using these lectures to attract students to analytic-theory research. • Develop internet material, textbooks, etc., that portray the excitement and intellectual rewards of doing analytic theory.

In attempting to ensure that analytic-theory capabilities in the US fusion program remain healthy, some assessment must be made of the proper manpower balance between analytic theory and computation. A suitably defined metric for manpower balance, measuring a ratio of analytic theory manpower to that of numerical computation, would likely show an ongoing shift from near 100% in the early days of fusion to a smaller percentage in the present. It is also true that the ideal ratio, however that might be determined, is changing in time in response to the evolving landscape of computational and experimental needs and capabilities. Therefore, the appropriate balance will have to be determined by making a careful assessment of needs and tasks, an exercise that should be carried out jointly by DOE and the fusion community.

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