Toroidal Momentum Pinch

Investigations into the toroidal momentum pinch effect can be made by considering the quasi-linear fluxes. We look for an inward toroidal momentum flux as a result

of the Coriolis force (which appears in a co-moving frame)[96,91].

The toroidal momentum flux (Γζ) is approximated here with the parallel

momentum flux (Γk) and is normalized by the heat flux (Q). This approximation

is valid because we are performing investigations of a large aspect ratio device, for which the parallel direction is approximately the same as the toroidal direction. A different diagnostic routine may be required for investigations of the toroidal momentum flux in a spherical tokamak. Observations of the momentum flux are taken as a time average across the last half of the simulation and then averaged across a radial width of 0.2a, centred aroundρ= 0.5.

k θρi 0.1 0.2 0.3 0.4 0.5 0.6 Γ|| /Q (c s -1) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 v_tor/c_s = 0.00 v_tor/c_s = 0.26 v_tor/c_s = -0.26 v_tor/c_s = -0.52 v_tor/c_s = 0.52 v_tor/c_s = -0.78

(a) Γζ/Q as a function of kθρi for various toroidal velocities. v_tor/c_s -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Γ|| /Q (c s -1) -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 kθρi = 0.32

(b) Ratio of parallel momentum flux to heat flux as a function of toroidal velocity for mode withkθρi= 0.32.

Figure 5.11: Toroidal momentum flux is approximated as parallel momentum flux.

Figure 5.11a shows that as kθρi increases, the momentum transport tends

seen that a toroidal rotation has an effect upon the size and direction of momentum

transport. Momentum transport has a peak value inkθρi which moves to higherkθ

as toroidal velocity increase. Simulations withvtor/csi = 0.78 were also performed

but have not been plotted here due to their negative toroidal momentum flux with

a very large magnitude (approximately−2).

By again selecting the fastest growing mode (in a non-rotating case)kθρi =

0.32, we can develop a clearer image of the effects of a toroidal rotation upon momen-

tum transport. Figure5.11billustrates that a toroidal rotation (in either direction)

first decreases the momentum transport to zero and then leads to an inwards pinch effect on the toroidal momentum. It appears from these results that the gyro-fluid model predictions made by Peeters, which suggest that adiabatic electrons will re-

sult in no pinch, may not be completely accurate [93,94]. Although the pinch does

seem small, observations do show an increase in inwards pinch with higher rotation rates.

In a non-linear simulation this may result in an increase in the magnitude of

vtor towards the core region, and may introduce an E×B shear that will in turn

act to reduce heat transport in the system.

The reason for the asymmetry of the pinch effect is not immediately ap- parent, and the large inward momentum pinch for high positive toroidal velocity is an unexpected result which suggests some other unknown effect may be causing changes to the toroidal momentum transport. This asymmetry has not been ob- served in local gyrokinetic codes, which suggests that it is caused by the additional flows that arise from the profile gradients. This seems a reasonable conclusion when also considering the similar asymmetry observed for the frequency and growth rate of modes found previously.

5.8

Summary

In this section, linear investigations of strongly rotating plasmas have been per- formed in the global limit. To this end, the expected behaviour of rotational terms were first outlined by considering a co-moving frame. Predictions have previously

been made in a series of papers by Peeters [91,34] for a co-moving frame in the local

limit, which show that the rotational effects can largely be described by a centrifugal and Coriolis force acting on the plasma.

Linear simulations have been performed with parameters matching the CY- CLONE base case and measurements of the frequency and growth rate have been taken for individual toroidal modes. Reasonably good agreement was found for

changes of mode frequency between global simulations performed in ORB5 and pre- dictions made in the local limit, however, flows which arise from the profile gradients were found to cause substantial deviations of the growth rates from predictions. An already present stabilising effect was found as a result of the diamagnetic flows in global simulations which counteracted stabilising effects of the inertial terms.

Due to the existence of these profile gradient flows, investigations have con- cluded that there may be substantial differences between local and global codes in regards to the growth of linear modes for a strongly rotating plasma.

Comparisons between strong and weak-flow forms of ORB5 reveal a substan- tial difference between the scaling of mode frequency and growth rate with changes of toroidal velocity. However, it is important to note that a significant amount of this change may come from the introduction of a density correction term along with the strong-flow modifications.

Furthermore, an inwards toroidal momentum pinch effect was observed de- spite predictions that there should not be one for simulations with adiabatic elec- trons. However, it is not clear whether this pinch is entirely caused by the Coriolis force.