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Subsections

4 Helical flow

For more complicated MHD flows, the time evolution of the wavenumber vector  ${\vec k}(t)$, governed by Eqs. (14), becomes nonlinear and it makes the temporal behavior of linear perturbations much more complex. A relatively simple, 2-D hydrodynamic system of such complexity was recently investigated (Mahajan & Rogava 1999). For the MHD helical flow, specified by the shear matrix (11) and considered in this section, we may deduce from (15) and (13c) that the perturbations may grow either exponentially (when differential rotation rate n>1), or vary in time in a periodic way (when n<1). Below we develop the mathematical formalism for the study of perturbation evolution for these two classes of solutions.

4.1 General formalism

The starting set of equations is again (16)-(19). Taking time derivative of (16) we find out that

$\displaystyle ({\vec k}{\cdot}{\vec
v})^{(1)}=-({\cal S}^T{\cdot}{\vec k}){\cdo...
...c k}{\cdot}
({\cal S}{\cdot}{\vec v})-\vert{\vec k}\vert^2(P+C_{\rm A}^2b_z)=0,$      

which after noticing that $({\cal S}^T{\cdot}{\vec k}){\cdot}{\vec v}= {\vec
k}{\cdot}({\cal S}{\cdot}{\vec v})$ implies that the following algebraic relation holds:
$\displaystyle P+C_{\rm A}^2b_z=-2{\vec k}{\cdot}({\cal
S}{\cdot}{\vec V})/\vert{\vec k}\vert^2=2{\vec k}^{(1)} {\cdot}{\vec v}/\vert{\vec
k}\vert^2.$     (33)

Note that this relation follows from (16), which, in turn, is direct consequence of the "incompressibility'' condition. The relation (33), applied to (17), helps to reduce the initial system to the following pair of the first-order ODE's:
                            $\displaystyle {\vec v}^{(1)}+{\cal S}{\cdot}{\vec v}=C_{\rm A}^2k_z{\vec b}-2\left({\vec
k}^{(1)}{\cdot}{\vec v}\right) {\vec k}/\vert{\vec k}\vert^2,$ (34)
    $\displaystyle {\vec b}^{(1)}={\cal S}{\cdot}{\vec b}-k_z{\vec v}.$ (35)

It is worthwhile also to note that $ {\vec k}({\vec
k}^{(1)}{\cdot}{\vec v})={\vec v}{\times}({\vec k}{\times}{\vec
k}^{(1)})$, which allows to rewrite (34) in the following form:
$\displaystyle {\vec v}^{(1)}+{\cal S}{\cdot}{\vec v}=C_{\rm A}^2k_z{\vec b} +{\vec
R}{\times}{\vec v},$     (36)

with ${\vec R}$ defined as:
$\displaystyle {\vec R}(t)\equiv\left({\vec k}{\times}{\vec k}^{(1)}\right)/\vert{\vec k}\vert^2.$     (37)

The interesting property of this form of the Eq. (34) is that the only time-dependent coefficient is just represented by one vector R(t)!

We can reduce (34) and (35) to the following explicit second-order equation for the magnetic field:

$\displaystyle {\vec b}^{(2)}+{\left[{\omega}_{\rm A}^2-{\cal S}^2 \right]}{\vec...
...t[{\vec k}^{(1)}{\cdot}{\vec
b}^{(1)}+{\vec k}^{(2)}{\cdot}{\vec b} \right]}=0.$     (38)

This equation is quite informative, because it shows explicitly the effects of the helical flow imposed upon the evolution of perturbations. Note that the term containing ${\vec k}^{(1)}{\cdot}{\vec b}^{(1)}$is also present in the pure outflow case (see for comparison Eq. (27)). While terms containing ${\cal S}^2{\vec b}$ and ${\vec k}^{(2)} \cdot {\vec b}$ are characteristic to the case of helical flow and they reflect different aspects of shear-induced processes in helical flows.

Obviously (38) is a general equation and we can base our analysis on it. However a different approach may also be useful. Let us introduce "projection variables":

                      $\displaystyle U\equiv{\cal
S}_{ij}k_iv_j=-{\vec k}^{(1)}{\cdot}{\vec v},$ (39a)
    $\displaystyle H\equiv{\cal S}_{ij}k_ib_j=-{\vec k}^{(1)}{\cdot}{\vec b},$ (39b)
    $\displaystyle V\equiv{\cal S}^2_{ij}k_iv_j={\vec
k}^{(2)}{\cdot}{\vec v},$ (39c)
    $\displaystyle B\equiv{\cal
S}^2_{ij}k_ib_j={\vec k}^{(2)}{\cdot}{\vec b}.$ (39d)

Using these definitions and our basic set of equations we find:
                               $\displaystyle U^{(1)}=-2V+C_{\rm A}^2k_zH-{{2\left({\vec k}{\cdot}{\vec
k}^{(1)}\right)}\over{\vert{\vec k}\vert^2}}U,$ (40)
    H(1)=-kzU, (41)
    $\displaystyle V^{(1)}=-2{\Gamma}^2U+C_{\rm A}^2k_zB+{{2\left({\vec
k}{\cdot}{\vec k}^{(2)}\right)}\over{\vert{\vec k}\vert^2}}U,$ (42)
    B(1)=-kzV. (43)

From this set we can derive the following two second-order equations for H and B:
                                      $\displaystyle H^{(2)}+{{2\left({\vec k}{\cdot}{\vec
k}^{(1)}\right)}\over{\vert{\vec k}\vert^2}}H^{(1)}+{\omega}_{\rm A}^2H= -2B^{(1)},$ (44)
    $\displaystyle B^{(2)}+{\omega}_{\rm A}^2B=f(t)H^{(1)},$ (45)

where
$\displaystyle f(t)\equiv{{2\left({\vec k}{\cdot}{\vec k}^{(2)}\right)}\over{\vert{\vec
k}\vert^2}}-2{\Gamma}^2.$     (46)

Moreover, if we define
$\displaystyle {\Psi}\equiv\vert{\vec k}\vert H,$     (47)

we can rewrite (44) also as:
$\displaystyle {\Psi}^{(2)}+{\left[{\omega}_{\rm A}^2-\vert\vec R\vert^2-{{\left...
...ight)}\over {\vert{\vec k}\vert^2}}\right]}{\Psi}=-2\vert{\vec k}\vert B^{(1)}.$     (48)

Note the appearance of the vector ${\vec R}$ in this equation. Overall, (45) and (48) give rather transparent analytic presentation, which can be used for the analysis of interesting classes of solutions. Note also that in the absence of rotation f(t)=0 these equations become decoupled and they reduce to the "pure outflow" case. In that case, we had a splitting of the mode into the usual Alfvén and shear-modified ("hybrid" as we can also call it) Alfvén mode. Below we shall see that rotation not only gives birth to new instabilities but it couples these two branches of the Alfvén mode!

4.2 Exponential k(t) case

Depending on the actual type of the motion in the plane normal to Z there will be different regimes of the time variation of ${\vec k}(t)$ (see for details Mahajan & Rogava 1999). Instabilities are present both when the wavenumber varies exponentially and periodically.

In the former case, i.e., when the absolute value of ${\vec k}(t)$increases exponentially, the dissipation effects will eventually start to be important and try to damp the mode (Mahajan & Rogava 1999) despite the presence of the shear instability. The same argument is valid for those cases, too, when the temporal growth of the wavenumber is linear. Therefore, considering shear-induced nonmodal effects in inviscid fluids in cases when $\vert{\vec k}\vert$ is monotonously increasing, we should realize that the description is physically meaningful only for initial (finite) times.


  \begin{figure}
\par\includegraphics[angle=90,width=8.1cm,clip]{plot6.ps}\end{figure} Figure 6: The numerical solution for the case when temporal evolution of ${\vec k}$'s is exponential. The values of parameters are: kx=5, ky=10, kz=1, $\sigma = 10^{-2}$, A1=0.3, A2=10-2, C1=0.8, C2=0.9, $C_{\rm A}=0.5$.

It is worthwhile to make a look at some of these solutions. In Fig. 6 we present the results of numerical calculations, taken for the case, when kx=5, ky=10, kz=1, $\sigma = 10^{-2}$, A1=0.3, A2=10-2, C1=0.8, C2=0.9, $C_{\rm A}=0.5$. The value of ${\Gamma}^2=3.1{\times}10^{-3}$ is positive and the value of $\vert{\vec k}(t)\vert$ is exponentially increasing (see Fig. 6a). The characteristic amplification time scale ${\cal
T}\simeq{\Omega}^{-1}$ is of the order of the period of rotation. The resulting wave evolves accordingly, which is clearly visible from the figures. It is noteworthy to see that, analogous to the pure outflow case, shear flow energy is most efficiently absorbed by the longitudinal components of the velocity and the magnetic field perturbations. Still, now, the energy growth, plotted again for the $E_{\rm tot}(t)/E_{\rm tot}(0)$, is not algebraic but exponential and is entirely due to the exponential evolution of the perturbation wave number vector.

4.3 Periodic k(t) case

When values of ${\Gamma}^2$ are negative, the temporal evolution of ${\vec k}(t)$'s is periodic, which means that the absolute value of the wavenumber vector stays bounded within certain limits. It means that in this case shear-induced effects have potentially much wider durability and may potentially lead to strongly perceptible effects. The limits of this paper does not allow us to present full analysis of all possible classes of solutions. The most interesting feature of this case is that Alfvén waves become parametrically unstable in a helical flow. Figures 7-9 demonstrate this noteworthy fact. All figures are drawn for the case when A1=-A2=0.3, and ${\sigma }=0.01$, so that $W\equiv(-{\Gamma}^2)^{1/2}\simeq 0.2998$. Figure 7 shows the case when Alfvén frequency is ${\omega }_{\rm A}=0.29$. The system is still out of the parametric resonance and the system, therefore, exhibits just Alfvén waves, modulated by the periodic shear. The picture is drastically different when ${\omega }_{\rm A}=0.3$- Fig. 8 shows that there appears a robust instability of parametric (resonant) nature that leads to the strong exponential increase of the system energy in the same span of time as in Fig. 7. The resonance is quite sharp, because as soon as Alfvén frequency slightly exceeds the value of W the system becomes stable again, as it is visible from the Fig. 9, which is drawn for the case when ${\omega }_{\rm A}=0.31$.


  \begin{figure}
\par\includegraphics[angle=90,width=8.3cm,clip]{plot7.ps}\end{figure} Figure 7: The numerical solution for the case when A1=-A2=0.3, ${\sigma }=0.01$, C1=0.8, C2=0.9, $W\simeq 0.2998$ and ${\omega }_{\rm A}=0.29$.


  \begin{figure}
\par\includegraphics[angle=90,width=8.2cm,clip]{plot8.ps} \end{figure} Figure 8: The numerical solution for the case when $W\simeq 0.2998$ (the same set of parameters as in Fig. 7) but with ${\omega }_{\rm A}=0.3$.


  \begin{figure}
\par\includegraphics[angle=90,width=8.3cm,clip]{plot9.ps} \end{figure} Figure 9: The numerical solution for the case when $W\simeq 0.2998$ (the same set of parameters as in Fig. 7) but with ${\omega }_{\rm A}=0.31$.


  \begin{figure}
\par\includegraphics[angle=90,width=8cm,clip]{plot10.ps}\end{figure} Figure 10: The numerical solution for total energy of perturbations normalized on their initial values. The values of parameters are: kx=20, ky=5, kz=1, $\sigma = 0.6$, A1=0.5, A2=-0.9, Cx=0.01, Cy=0.05. In this case W=0.3, while ${\omega }_{\rm A}=0.3, 0.5, 0.6,$ and 0.8 on the plots labeled (a), (b), (c), and (d), respectively.

Further study shows that actually there are several isolated regions of instability, which happen to be centered around the values ${\omega}_{\rm A}=nW, n=1,2,...$ and usually first resonance $W={\omega}_{\rm A}$ is the most efficient one. This is a strong indication in favor of the resonant and parametric nature of this instability. Figures 10 and 11 illustrate this interesting property of the system. The plots are drawn for the total energy of a perturbation, normalized on its initial value: $E_{\rm tot}(t)/E_{\rm tot}(0)$. The simulation is made for the case when kx=20, ky=5, kz=1, $\sigma = 0.6$, A1=0.5, A2=-0.9, Cx=0.01, Cy=0.05. Note that in this case W=0.3. Making simulations for different values of the Alfvén velocity we see that perturbations are strongly unstable when ${\omega}_{\rm A}\simeq nW$. Namely the instability is present in Figs. 10a, c and Figs. 11a, c, where the values of ${\omega}_{\rm A}$ are 0.3, 0.6, 0.9, and 1.2, respectively. At the same time we see that for intermediate values of ${\omega}_{\rm A}$, when $nW<{\omega}_{\rm A}<(n+1)W$, the Alfvén waves stay stable. Unfortunately the complexity of the system does not allow us to perform strict analytic analysis and locate the actual width and size of instability regions. However, our results do show that there are several such regions and presumably the efficiency of the instability decreases with the increase of n.


  \begin{figure}
\par\includegraphics[angle=90,width=8cm,clip]{plot11.ps}\end{figure} Figure 11: The numerical solution for the total energy of perturbations. The values of parameters are the same as in Fig. 10. while ${\omega }_{\rm A}=0.9, 1.1, 1.2,$ and 1.4 on the plots labeled (a), (b), (c), and (d), respectively.


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