Poloidal tilting symmetry of high order tokamak flux surface shaping in gyrokinetics

Abstract

Abstract.A poloidal tilting symmetry of the local nonlinear δf gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal tilt of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by tilting the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using updown asymmetry.

Turbulent momentum transport due to the beating between different tokamak flux surface shaping effects

“…"Fast poloidal variation" on the other hand refers to variation in the flux surface shape that has a spatial scale much smaller than the minor radius. This distinction is important because tilting all shaping effects with large poloidal mode number also tilts any envelope, while tilting only the fast poloidal variation keeps any slowly-varying In order to make use of the tilting symmetry of [18] we must first introduce a fast poloidal coordinate z ≡ m c θ (7) and separate the two poloidal scales. Here m c is a characteristic mode number that indicates the boundary between fast and slow poloidal variation.…”

“…Building on this work, references [18] and [19] use gyrokinetics to argue that certain types of flux surface shapes may increase the intrinsic rotation driven by up-down asymmetry. To make the problem analytically tractable, both explore breaking updown symmetry using "fast" shaping effects, where "fast" refers to shaping with a small spatial scale (i.e.…”

“…In this work, we investigate a property of the momentum transport that went unexamined in the analysis of [18] and [19]: the impact of a slowly-varying envelope created by the beating of two or more fast flux surface shaping effects. This work will use much of the analysis from [18] and [19], but will explicitly study momentum transport generated by flux surfaces with slowly-varying envelopes. We will also provide an overview of the expected scaling of momentum flux with shaping mode number for all flux surface shapes.…”

“…In section 2.1 we detail the analytic models we use to both specify the magnetic equilibrium and calculate the turbulent momentum flux. Next, in section 2.2, we revisit the tilting symmetry of [18] in order to determine when flux surfaces with envelopes must have exponentially small momentum transport. Subsequently, in section 2.3 we clarify what we have learned by considering a few example flux surfaces.…”

Intrinsic rotation driven by turbulent acceleration

“…A comprehensive theory including all of these symmetry-breaking mechanisms is given in [16,17,18,15,19]. There have also been a number of studies dedicated to individual mechanisms, including the effect of diamagnetic flows [20,21,22,23,24], up-down asymmetry of flux surfaces [25,26,27,28,29,30,31,32,33,34], slow poloidal variation of fluctuations [35], and 'global' effects [36,37,38,39], which include radial profile variation mingled with the other effects mentioned. Here we consider the effect of turbulent particle acceleration along the mean magnetic field.…”