ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula
tokens: any fixed z , and , where . Note that f is zero at the boundaries of these intervals and similarly g is zero at r = 0 and r = R 0 , so the functions are continuous. We solved for V Z 2 in Equation ( 26 ), in order to study the nature of the vertical motion induced by the magnetic shear and potentially explore the relationship between this type of perturbation and the mass flux observed in the chromosphere in the form of jets. The solution for V Z 2 in the zero-beta case is simply proportional to the square of B θ 1 ,and zero otherwise—see Figures 2 and 3 . Hence V Z 2 will be of a similar form to B θ 1 , i.e., a pulse propagating in the z -direction. We suggest that a pulse in the form of this example could possibly influence plasma and contribute to the formation of a jet. Using our perturbation model we can approximate the vertical mass flux of the plasma within the locality of the pulse, φ z = ρ v z ≈ ε 2 ρ 0 V Z 2 . Figure 2. A 2.5D representation of the given example of a V Z 2 pulse from Section 3.5 , at particular times: t = 0.2 τ , t = 0.5 τ , and t = τ . The pulse appears as two peaks due to the axisymmetry around the z- axis of the tube, and has been rescaled so that the V Z 2 (vertical) axis is in units of , while the r- and z- axes are in units of R 0 and v A τ , respectively. An animated version of this figure, showing the evolution of the pulse from t = 0 to t = 2 τ , is available online. (An animation of this figure is available.) Figure 3. Rotated views of the 2.5D plot in Figure 2 at t = τ , showing the z dependence (left) and the r dependence (right). Interestingly, the nature of the perturbation also suggests that there could be plasma with lower velocity near the center of the flux tube, since we would have φ z = 0 at r = 0. Once the pulse is fully formed at t = τ , it will maintain its shape while it propagates in the positive z- direction. For t ≥ τ , φ z will be constant on concentric toroidal surfaces, which can be partially seen in Figure 4 . The toroids degenerate into a cylinder, bounded by the limits for V Z 2 stated in Equation ( 31 ), as the magnitude of φ z approaches zero; outside of this cylinder there is no perturbation. The maximum magnitude of φ z is reached where both f and g are maximized, on the circle with radius r = R 0 /(2 1/ λ ) at z = v A ( t − τ /2). Other forms of the B θ 1 pulse are of course possible as solutions to the first-order equations, but solutions that are suitable for the model are often similar to our example in form and behavior. Figure 4. Left: a 3D plot of the vertical mass flux φ z at time t = τ /2, for the example in Section 3.5 (with scaled Cartesian axes for simplicity). At this point the pulse is “half formed” and there is positive mass flux for 0 ≤ z ≤ v A τ /2 within the tube. The blue arrows at the lower boundary represent the driver for the torsional magnetic field perturbation. An animated version of this figure, showing the evolution of the pulse from t = 0 to t = τ /2,