Abstract
In this last chapter, we will go back to the same laboratory plasmas discussed in Chap. 6 to look for our last example of complex behaviour in plasmas. In particular, we will discuss the dynamics of turbulent transport across sheared flows. Sheared (or shear) flow is a label used to refer to any kind of flow that varies in space. Naturally, small-scale sheared flows are always present in any turbulent medium. For instance, any turbulent vortex (or ”eddy”) is a region where the local flow (mostly directed around the center of the eddy) varies quickly as one moves away from the center. The action of turbulence also tends to generate local patterns of flow that vary quickly in space and time to facilitate the dissipation of energy at the (usually very small) viscous scales. These are not the kind of sheared flows that interest us, though. We will be concerned about large-scale sheared flows instead. That is, flows that maintain their coherence over scales much larger and longer than any local turbulent scales. Large-scale sheared flows are somewhat rare in nature because shear very often drives instabilities of the Kelvin-Helmholtz (KH) type[1]. It is only in special conditions that shear flows can be kept stable, at least for sufficiently long times as to have an strong impact in the system dynamics. Shear flows are stabilized, usually, by either differential rotation or magnetic fields. For that reason, large-scale sheared flows often appear in stars, planetary atmospheres and oceans or in magnetized fusion toroidal plasmas, to name just a few (see Fig. 9.1). In the context of magnetic confinement fusion (MCF), they are usually referred to as zonal flows
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In MCF, it is sometimes usual to distinguish between mean (over time) and fluctuating shear flows; in that case, the term “zonal flow” is used to denote the fluctuating part of the shear flow. In this chapter, however, we will not make that distinction.
Fig. 9.1 
Left: latitudinal shear flow patterns around the northern pole of Jupiter’s atmosphere. Right: isocontours of turbulent electrostatic potential from a tokamak Ion-Temperature-Gradient (ITG) turbulence simulation showing the presence of a strong, radially-shear poloidal flow. Credits: Jupiter image (Ⓒ ESA/NASA - Hubble)
- 2.
As it is often the case, in some systems this reduction might be bad news; but in the case of fusion toroidal plasmas, this reduction of radial transport becomes extremely handy!
- 3.
The vorticity of a flow is defined as the curl of the local velocity vector, w ≡∇×v. Thus, the vorticity is zero everywhere for a uniform flow. For a solid-rotation in two-dimensions, though, it can be shown that its vorticity vector w ∝ Ω, being Ω the angular velocity vector that satisfies v = r × Ω. In a more general two-dimensional flow, the relation between vorticity and angular velocity changes, but regions where the vorticity is large are still indicative of an intense local rotation. In particular, turbulent eddies can be seen as regions where vorticity accumulates, with the orientation of the local rotation (clockwise or counterclockwise) being consistent by the sign of the local vorticity.
- 4.
It should be remembered that the tokamak and stellarator magnetic topology can be approximated by a family of nested, toroidal magnetic surfaces to which the magnetic field is tangent. The radial direction is the one perpendicular to these magnetic surfaces. The two directions on the surface are the poloidal and toroidal ones. The magnetic field has, therefore, both a poloidal and toroidal component, being the latter much larger. The flow motion that constitutes a zonal flow in tokamaks appears to be of electrostatic origin, thus flowing in the direction of the E ×B drift, that is contained in the magnetic surface but perpendicular to the magnetic field.
- 5.
This type of argument, that heavily relies on a diffusive description of the transport process, has however been challenged recently, as we will discuss in Sect. 9.2.3.
- 6.
In these cases, the transport dynamics were heavily reminiscent of those of the running sandpile, the poster child of self-organized criticality (see Chap. 1).
- 7.
In the case of tokamaks, it is the radial direction; for a sandpile, it is down the slope.
- 8.
- 9.
In addition, access to the H mode also requires a divertor configuration, as shown in Fig. 9.3, that permits to insulate the plasma from the wall sufficiently well.
- 10.
Other MCF toroidal devices, such as stellarators, have also produced H-modes and ETBs [13].
- 11.
ETBs have also been externally induced in tokamak plasmas, mostly via plasma biasing [15]. In this technique, a potential difference is forced between the plasma and the edge, that results in a radial electric field and an associated E ×B flow shear that acts on edge turbulence, reducing its levels and associated transport, and causing a steepening of the plasma profiles.
- 12.
At least, not in the sense of cases such as the near-marginal transport discussed in Chap. 6, where transport was a consequence of the complex dynamics taking place in the system.
- 13.
Whether the origin of the fractal structure or the shear flow is due to complex dynamics or not, is another matter. In the case of the self-generated edge transport barrier that appears in tokamaks, it certainly is!
- 14.
To be more precise, UCAN2 is a so-called global δf code, that assumes closeness to an equilibrium distribution f 0 that contains the plasma equilibrium profiles (i.e., plasma density, flow and temperature), so that only the deviations from f 0 are followed in time. These type of setups are very useful to evolve turbulence over a large domain as long as the variations of the profiles are not too large. For that reason, they would not be good for turbulence simulations in near-marginal conditions, where intense profile evolution is expected at the fluctuation scales. The approach is however adequate for the kind of study carried out here.
- 15.
This means that electrons are considered to be extremely mobile, being able to move along the field lines to provide any force balance required. This approximation importantly simplifies the simulations, since the electron distribution function does not need to be computed [24].
- 16.
ITG instabilities are thought to be responsible for a large fraction of the radial ion heat transport in tokamaks [19].
- 17.
The usual phase space in which any kinetic equation is solved has six dimensions, three spatial ones corresponding to position and another three that correspond to the velocity vector. In gyro-kinetics, however, the averaging over the gyro-motion eliminates one velocity dimension [23]. The velocity part of phase space is thus reduced to two dimensions, one for v ⊥ and another for v ∥, respectively the perpendicular and parallel components of the velocity with respect to the local magnetic field. v ⊥ is no longer a vector since the gyro-phase, that gives its orientation, is gone after the averaging.
- 18.
We use the common practice of denoting v ⊥ and v ∥ as the components of the ion velocity perpendicular and parallel to the local magnetic field.
- 19.
When all other drifts are considered, in addition to parallel motion, the projection of the trajectory of any trapped particle on any toroidal cross section has a crescent moon or banana shape. For that reason, trapped ion orbits are usually known as banana orbits.
- 20.
The same goes for the numerical details of the simulation, such as the number of cells considered, the number of particles per cell included or the time resolution used.
References
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1961)
Townsend, A.A.: The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge (1980)
Terry, P.W.: Suppression of Turbulence and Transport by Shear Flows. Rev. Mod. Phys. 72, 109 (2000)
McIntyre, M.C., Palmer, T.N.: The ”surf zone” in the Stratosphere. J. Atmos. Terr. Phys. 46, 825 (1984)
Biglari, H., Diamond, P.H., Terry, P.W.: Influence of Sheared Poloidal Rotation on Edge Turbulence. Phys. Fluids B 2, 1 (1990)
Newman, D.E., Carreras, B.A., Diamond, P.H., Hahm, T.S.: The Dynamics of Marginality and Self-Organized Criticality as a Paradigm for Turbulent Transport. Phys. Plasmas 3, 1858 (1996)
Sanchez, R., Newman, D.E.: Self-Organized Criticality and the Dynamics of Near-Marginal Turbulent Transport in Magnetically Confined Fusion Plasmas. Plasma Phys. Controlled Fusion 57, 123002 (2015)
Ogata, D., Newman, D.E., Sanchez, R.: Investigation of the Interaction Between Competing Types of Nondiffusive Transport in Drift Wave Turbulence. Phys. Plasmas 24, 052307 (2017)
Dif-Pradalier, G., Hornung, G., Garbet, X., Ghendrih, Ph., Grandgirard, V., Latu, G., Sarazin, Y.: The E ×B Staircase of Magnetised Plasmas. Nucl. Fusion 57, 066026 (2017)
Wagner, F., Becker, G., Behringer, K., Campbell, D.: Regime of Improved Confinement and High-Beta in Neutral-Beam-Heated Divertor Discharges of the ASDEX Tokamak. Phys. Rev. Lett. 49, 1408 (1982)
Shimada, M., et al.: Progress in the ITER Physics Basis - Overview and Summary. Nucl. Fusion 47, S1 (2007)
Wagner, F.: A Quarter-Century of H-Mode Studies. Plasma Phys. Controlled Fusion 49, B1 (2007)
Hirsch, M., Amadeo, P., Anton, M., Baldzuhn, J., Brakel, R., Bleuel, D.L., Fiedler, S., Geist, T., Grigull, P., Hartfuss, H., Holzhauer, E., Jaenicke, R., Kick, M., Kisslinger, J., Koponen, J., Wagner, F., Weller, A., Wobig, H., Zoletnik, S.: Operational Range and Transport Barrier of the H-Mode in the Stellarator W7-AS. Plasma Phys. Controlled Fusion 40, 631 (1998)
Diamond, P.H., Itoh, S.I., Itoh, K., Hahm, T.S.: Zonal Flows in Plasmas - A Review. Plasma Phys. Controlled Fusion 47, R35 (2005)
Weynants, R.R., Van Oost, G.: Edge Biasing in Tokamaks. Plasma Phys. Controlled Fusion 35, B177 (1993)
Sanchez, R., Newman, D.E., Leboeuf, J.N., Decyk, V.K., Carreras, B.A.: Nature of Transport Across Sheared Zonal Flows in Electrostatic Ion-Temperature-Gradient Ggyrokinetic Plasma Turbulence. Phys. Rev. Lett. 101, 205002 (2008)
Sanchez, R., Newman, D.E., Leboeuf, J.N., Carreras, B.A., Decyk, V.K.: On the Nature of Radial Transport Across Sheared Zonal Flows in Electrostatic Ion-Temperature-Gradient Gyrokinetic Tokamak Plasma Turbulence. Phys. Plasmas 16, 055905 (2009)
Sanchez, R., Newman, D.E., Leboeuf, J.N., Decyk, V.K.: Nature of Transport Across Sheared Zonal Flows: Insights from Ggyrokinetic Simulations. Phys. Rev. Lett. 53, 074018 (2011)
Horton, W.: Drift waves and transport. Rev. Mod. Phys. 71, 735 (1999)
Newman, D.E., Terry, P.W., Diamond, P.H., Liang, Y.M.: The Dynamics of Spectral Transfer in a Model of Drift-Wave Turbulence with Two Nonlinearities. Phys. Fluids 5, 1140 (1993)
Sydora, R.D., Decyk, V.K., Dawson J.M.: Fluctuation-Induced Heat Transport Results from a Large, Global Toroidal Particle Simulation Model. Plasma Phys. Controlled Fusion 38, A281 (1996)
Leboeuf, J.N., Decyk, V.K., Newman, D.E., Sanchez, R.: Implementation of 2D Domain Decomposition in the UCAN Gyrokinetic PIC Code and Resulting Performance of UCAN2. Commun. Comput. Phys. 19, 205 (2016)
Brizard, A.J., Hahm, T.S.: Foundations of Nonlinear Gyrokinetic Theory. Rev. Mod. Phys. 79, 421 (2007)
Krall, N.A., Trivelpiece A.W.: Principles of Plasma Physics. McGraw Hill, New York (1973)
Balescu, R.: Transport Processes in Plasmas. North-Holland, Amsterdam (1988)
Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. Adam Hilger Publishers, Bristol (1988)
del-Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional Diffusion in Plasma Turbulence. Phys. Plasmas 11, 3854 (2004)
Mier, J.A., Sanchez, R., Newman, D.E., Garcia, L., Carreras, B.A.: Characterization of Nondiffusive Transport in Plasma Turbulence via a Novel Lagrangian Method. Phys. Rev. Lett. 101, 165001 (2008)
Metzler, R., Klafter, J.: The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach. Phys. Rep. 339, 1 (2000)
Zaslavsky, G.M.: Chaos, Fractional Kinetics, and Anomalous Transport. Phys. Rep. 371, 461 (2002)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Sánchez, R., Newman, D. (2018). Laboratory Plasmas: Dynamics of Transport Across Sheared Flows. In: A Primer on Complex Systems. Lecture Notes in Physics, vol 943. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1229-1_9
Download citation
DOI: https://doi.org/10.1007/978-94-024-1229-1_9
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-024-1227-7
Online ISBN: 978-94-024-1229-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)