tokens: is small compared with the speed of the flow, and the inertial term in Equation ( 2 ) can be ignored. This eliminates the variable, v θ from the induction equation, so it can be rewritten as: Therefore, expansion of the plasma () corresponds to attenuation while compression () corresponds to amplification of b θ . From the preceding discussion it follows that the latter situation is more likely to occur in downdrafts. Equation ( 4 ) looks remarkably similar to the vorticity equation for an incompressible fluid:where ω denotes vorticity in the z direction. The right-hand side of Equation ( 5 ) represents the well-known vortex stretching effect: the angular velocity of a vortex tube increases when it is stretched and decreases when it is compressed (Spurk Aksel 2008 , p. 100). In contrast, Equation ( 4 ) shows that the θ component of the magnetic field increases when the flow decelerates and decreases when it accelerates. The fact that the twist amplification stems from the induction equation emphasizes the magnetic nature of the process. Twist amplification may be thought of as a magnetic analogue of vortex stretching. In a more general situation, when the magnetic field is not weak, the twist evolution requires a more detailed treatment. In the following sections we study the twist amplification analytically as an eigenvalue problem and numerically, and analytically. We also discuss the energy source, and the physics of the amplification process. 3. UNSTABLE TWISTS IN AN EXPONENTIALLY DECAYING DOWNDRAFT The set of governing Equations ( 1 ) and ( 2 ) can be reduced to a single second order PDE for the magnetic field perturbation, b θ (Taroyan 2008 ):where , and . In deriving the above equation, we have used the equation of mass conservation. In a static medium with , the well-known wave equation can be derived from Equations ( 1 ) and ( 2 ): Equation ( 7 ) was first analyzed by Ferraro ( 1954 ) in the context of Alfvén wave propagation in the solar atmosphere. Solutions in terms of the Bessel functions were constructed for an isothermal atmosphere with an exponential density profile:where z 0 is the scale height. The corresponding Alfvén speed is given by:Note that in our notations z denotes depth. Equation ( 7 ) with the profile ( 8 ) was subsequently analyzed by different authors. They found standing wave solutions in terms of the Bessel functions (Ferraro 1954 ; An et al. 1989 ) or propagating wave solutions in terms of the Hankel functions (Hollweg 1978 ; Cally 2012 ) depending on the imposed boundary conditions. An advantage of the smooth exponential profile is the absence of any model dependent artificial reflections that may arise due to discontinuities in the Alfvén speed or its derivatives (Cally 2012 ). One of the drawbacks is the finite Alfvén travel time to (An et al. 1989 ). In studies of the convection zone, it is common to adopt a polytrope with a linear temperature. Here, we adopt the exponential profiles ( 8 ) and ( 9 ) with
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