Sheared flow stabilisation

It is well known that sheared flows can suppress turbulence and stabilise the linear microinstabilities that drive it[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. The basic picture of stabilisation of linear instabilities by flow shear is as follows: the flow acts on opposite ends of the mode structure differentially. This tilts the mode, which reduces its radial wavenumber, which in turn reduces its growth rate. In ballooning theory, sheared flows stabilise small amplitude modes by advecting the mode structure in the ballooning angle, θ0[48], rotating from the outboard, “bad

samples the good curvature region, the growth rate of a bad curvature driven mode is necessarily smaller than one whoseθ0 stays fixed in the bad curvature region on the outboard side. Alternatively, this process can be viewed as coupling modes with

θ0 = 0 to those withθ0 =π. Ballooning theory predicts that the toroidal ITG mode

is stabilised when|θ0| ∼π/2[46].

The nonlinear picture is similar, with sheared flows tilting the eddies, which reduces their radial correlation lengths[47, 51]. The correlation length is roughly the distance between two turbulent eddies of comparable sizes[25]. A fluid parcel moved by one correlation length finds itself no longer in its original context - it is no longer the same eddy. Without flow, the time taken for this to happen is the turbulent correlation time,τcorr, or, equivalently, the eddy turnover time[34]. Flow with a shearing rate4greater thanτ_corr−1 stretch the turbulent eddies to the correlation length in less than one correlation time, essentially reducing their lifetimes. Shorter lifetimes are also accompanied by a reduction in turbulent intensity. Clearly, shorter lifetimes and reduced correlation lengths have the consequence that fluid parcels don’t move as far perpendicular to the flow direction during one correlation time, leading to a decrease in the transport of particles and heat. This can be seen simply by using a random-walk argument. A more complete review of flow shear stabilisation can be found in [34].

There are actually three other conditions required for flow shear to be able to suppress turbulence[34]. The sheared flow needs to be stable, the turbulence must stay within the flow for longer thanτcorrand the dynamics should be limited to two dimensions. These are all features typical of tokamak plasma turbulence, but not necessarily of other fluids. Shear flows can drive instabilities in non-ionised fluids, for example5.

Magnetic shear is also deeply intertwined with E×B shear[46, 49]. The existence of rational surfaces6 in tokamaks tends to localise turbulent fluctuations close to those surfaces, unless there is significant coupling between flux surfaces. Magnetic shear and flow shear then can have a synergistic effect on both linear microinstabilities and turbulent fluctuations. Magnetic shear can also change the stability of sheared flows by, for example, increasing the threshold for the Kelvin- Helmholtz instability, which is a flow shear-driven instability.

TheE×B drift has a central role in many theoretical frameworks of plasma

4

The shearing rate is the gradient of the flow velocity and has units of inverse time.

5

Gradients in the parallel velocity in tokamaks can also drive instabilities, but this effect is typically far outweighed by the suppression caused by perpendicular flow shear.

6

These are flux surfaces whereqis a rational number. That is, the number of complete toroidal

Turbulence

Gradients

Transport

Flow shear

ITG, TEM, etc.

D, χ, χ_φ

Γ, Q,

Γ_φ

Figure 3.3: Schematic of the interaction of turbulence, sheared flows and equilibrium quantities. The free energy in the density and temperature gradients drives linear microinstabilities which go on to nonlinearly interact and develop into turbulence. Turbulent fluctuations drive increased transport which affects the profile gradients. At the same time, Reynolds stresses from the fluctuating velocities can generate sheared flows (such as zonal flows) which can suppress turbulence and the microin- stabilities that drive it. These complex interactions form a synergistic self-regulation process whereby the plasma can self-organises into a more quiescent state.

physics as it affects all species equally - independent of mass or charge. Flow shear also has a kind of universality, reducing turbulent fluctuations and transport in almost all classes of magnetic confinement devices and plasma regimes, even though the driving microinstabilities change from case to case[49]. The combination of these facts has lead to shearedE×B flows being an area of intense study.

The reduction in transport due to flow shear suppression of turbulence can lead to steepening of gradients[53]. Often, this is confined to a small radial region of the tokamak, this region being referred to as a transport barrier. The first transport barriers were observed at the edge[27]. Later, experiments found the existence of internal transport barriers (ITBs). The transport in transport barriers can be reduced as far as neoclassical levels.

The H-mode (“high confinement” mode, in contrast to the “low confinement” L-mode) was discovered on ASDEX in 1982[54]. This is a regime with steep density gradients near the edge which spontaneously develops (given certain conditions). At the same time, density fluctuations die down and a radial electric field,Er, develops

at the edge. The spatial structure of theEr leads to strong shearedE×B flows[49].

The H-mode has since been found on every machine with external heating and a divertor[34]. H-mode plasmas are of great interest to the fusion programme, as they often have a doubling or more of the confinement time, as well as an increased

central density[35]. This combination leads to a greater fusion power in the core of the tokamak. However, the exact physical processes that underlie the L-H transition are still a matter of active research. There are a number of universal features of H-modes, with one of the central ones beingE×B flow shear.

There are a whole host of different mechanisms to generate sheared flows, for example by turbulent Reynolds stresses, momentum input from neutral beams, or symmetry breaking (so-called “intrinsic” rotation). MAST achieves high rotation rates from a combination of neutral beam injection (NBI) and its low moment of inertia.

3.6

Summary

In this chapter, we have discussed the origins of turbulence in tokamak plasmas and the main drives for the most common microinstabilities. The free energy in the equilibrium profile gradients can be tapped by various microinstabilities, which can nonlinearly interact, developing into chaotic turbulence. Strong turbulence in tokamaks is responsible for the anomalous transport, with levels well above that predicted by neoclassical theory. The turbulence may drive linearly stable zonal flows which act as a self-regulation mechanism, damping the turbulence through sheared flows and acting as a benign store of free energy.