Acoustic waves traveling in both directions around the torus. Sheared flow changes the population density of waves in one direction relative to the other, breaking the symmetry and creating an imbalance between the two counterpropagating wave populations.
Shift of the intensity fluctuation profile due to shear.
Shifts in the Landau resonance points that are in the same direction resulting in a shift in the center of the fluctuation spectrum, which results in a net imbalance in wave momentum deposition.
Profiles of density, pressure, parallel flow, and the radial electric field shear for the case. The directions of the evolution of the profiles in time are denoted by the arrows. First few time steps are plotted using dashed lines and the final steady state with a thick solid line. We have plotted intermediate time steps via thin solid lines as long as they can be distinguished from the steady state. Here, the usual normalization is used such that is in units of , is in units of , and is in units of . Here, the left-hand side is an open boundary and does not correspond to the actual origin. Therefore, even though a macroscopic scaling (e.g., ) is in fact used, the domain should actually be taken as a finite annulus in the outer core region. Almost all the figures (except some cases in parameter scans) correspond to ITG-like parameters (i.e., , , etc.).
Profiles of density, pressure, parallel flow, and the radial electric field shear for the case, where the effect of toroidal flow is also included in the radial force balance (i.e., ) resulting in an outward flow proportional to . See Fig. 4 for a key.
Profiles of density, pressure, parallel flow, and the radial electric field shear for the case with an edge momentum source similar to particle source ( case). See Fig. 4 for a key.
Scaling of toroidal flow velocity with parameters (a) , the value of at the right hand boundary (i.e., boundary condition at ); (b) , the external momentum input; (c) ; and (d) . In all the figures, the dashed lines correspond to the case , whereas the solid lines are . Here is the parallel flow velocity normalized to at the left boundary, which is open. Note that this point does not always correspond to the maximum of the velocity profile especially when the radial momentum flow terms are turned on (i.e., ).
The case with and . See Fig. 4 for a key.
Profile of parallel flow for the , case for Eq. (42). See Fig. 4 for a key.
Direction of various terms in Reynolds stress for different parameter regimes. Here, those cases marked with an are the cases where the velocity is negative. Thus, for these cases, “outward,” for instance, means an “outward flux of negative momentum” (or an inward flux of positive momentum). The convention is such that the diffusive term is always “outward.” The last column denotes the effect of the term in the radial force balance. Here, “oppose” means that term reduces the magnitude of the shear and hence opposes the tendencies given in the previous two columns, while “promote” means the flow term enhances those tendencies. Note that here “R” means ; “Co” and “Ctr” mean the momentum drive is parallel and antiparallel to the direction of rotation, respectively, and the cases where the direction of the drive is not specified has .
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