3 Ejectional flow

This case corresponds to $\sigma=A_1=A_2=0$. We can introduce a vector ${\vec C}\equiv[C_x \equiv C_1,C_y \equiv C_2,C_z=0]$, which is proportional to the position vector ${\vec r}_0= (x_0, y_0, 0)$ connecting the point A(x₀,y₀,z₀) (the centre of our local frame) with the Z-axis in a z=z₀ plane. Its absolute value is equal to ${\rm d}U_z/{\rm d}r$. The wave number dynamics is given by simple linear solutions: k_i(t)=k_i(0)-C_ik_zt, with i=x, y, while the vertical component k_z stays constant. The symmetry of the flow implies that the vector product of ${\vec C}$ and ${\vec k}$ has a constant vertical component ${\Delta}\equiv- ({\vec C}{\times}{\vec k})_z=C_yk_x(t)-C_xk_y(t)={\rm const}(t)$, which, in its turn, is a linear combination of transverse components of the wave number vector. This fact suggests the decomposition $k_{\parallel}(t)\equiv({\vec C} \cdot {\vec k})/\vert{\vec C}\vert$, $k_{\perp}\equiv({\vec C} \times {\vec k})_z/\vert{\vec C}\vert$, i.e.,

$${\vec k}(t)=(k_{\parallel}(t),~k_{\perp},~k_z),$$ (20)

with the entire time dependence of ${\vec k}(t)$ contained in

$$k_{\parallel}(t)=k_{\parallel}(0)-\vert{\vec C}\vert k_zt.$$ (21)

The system admits a surprisingly efficient and complete analytic description if we introduce, first, vectors of hydrodynamic ( ${\vec{\Omega}_h}\equiv{\vec k}{\times}{\vec v}$) and magnetic ( ${\vec{\Omega}_m}\equiv{\vec k}{\times}{\vec b}$) vorticity and consider the following scalar products: $V\equiv({\vec C}{\cdot}{\vec v})$, ${\cal V}\equiv({\vec C}{\cdot}{\vec{\Omega}}_h)$, $B\equiv({\vec C} {\cdot}{\vec b})$, and ${\cal B}\equiv({\vec C}{\cdot}{\vec{\Omega}}_m)$. The usefulness of these variables becomes apparent when we check that they obey the following set of first-order equations:

$$V^{(1)}=-\left[ln\left(\vert{\vec k}(t)\vert^2\right)\right]^{(1)}V+C_{\rm A}^2k_zB,$$ (22a)

$${\cal V}^{(1)}={\Delta}V+C_{\rm A}^2k_z{\cal B},$$ (22b)

B⁽¹⁾=-k_zV, (23a)

$${\cal B}^{(1)}=-{\Delta}B-k_z{\cal V},$$ (23b)

which is remarkably simple with only one time-dependent coefficient (time dependence exclusively due to shear) in Eq. (22a).

These variables give complete description of the system, because "physical" variables ${\vec v}$ and ${\vec b}$ are readily expressed by means of V, $\cal V$, B, and $\cal B$ via the following vector identities:

$${\vec v}={1 \over{\vert{\vec C}{\times}{\vec k}\vert^2}}{\biggl[{... ...vec C}{\times} {\vec k})+V({\vec k}{\times}({\vec C}{\times}{\vec k}))\biggr]},$$ (24a)

$${\vec b}={1 \over{\vert{\vec C}{\times}{\vec k}\vert^2}}{\biggl[{... ...vec C}{\times} {\vec k})+B({\vec k}{\times}({\vec C}{\times}{\vec k}))\biggr]}.$$ (24b)

While for the kinetic and magnetic energies of the system we also have rather transparent expressions:

$$E_{\rm k}={1 \over 2}\vert{\vec v}\vert^2={1 \over{2\vert{\vec C}{\times}{\vec k}\vert^2}} {\biggl[{\cal V}^2+V^2\vert{\vec k}\vert^2\biggr]},$$ (25a)

$$E_{\rm m}={C_{\rm A}^2 \over 2}\vert{\vec b}\vert^2={C_{\rm A}^2 ... ...C}{\times} {\vec k}\vert^2}}{\biggl[{\cal B}^2+B^2\vert{\vec k}\vert^2\biggr]},$$ (25b)

$$E_{\rm tot}\equiv E_{\rm k} + E_{\rm m}.$$ (25c)

From (22) and (23) we can further notice that the Alfvén waves, sustained by this system are "splitted" by the presence of the velocity shear.

• The first branch comprises just a simple, constant amplitude and constant frequency Alfvén mode, which is governed by the second order equation for $\cal B$
  

  $${\cal B}^{(2)}+{\omega}_{\rm A}^2{\cal B}=0,$$ (26)

  
  
  with ${\omega}_{\rm A}\equiv C_{\rm A}k_z$ and with obvious solution ${\cal B}(t)={\cal C}\sin({\omega}_{\rm A}t+{\phi}_0)$.
• The second branch is heavily modified by the presence of the velocity shear. Its temporal evolution is described by the second order equation for B:
  

  $$B^{(2)}+\left[\ln\left(\vert{\vec k}(t)\vert^2\right)\right]^{(1)}B^{(1)}+{\omega}_{\rm A}^2B=0.$$ (27)

  
  
  We can further simplify this equation in terms of the variable ${\Psi}\equiv\vert{\vec k}\vert B$ which obeys
  

  $${\Psi}^{(2)}+{\biggl[{\omega}_{\rm A}^2-{{k_z^2\vert{\vec C}{\times}{\vec k}\vert^2} \over{\vert{\vec k}\vert^4}}\biggr]}{\Psi}=0.$$ (28)

  
  
  Introducing the dimensionless parameter
  

  $$a\equiv{{C_{\rm A}}\over{\vert{\vec C}\vert}}\left[k_{\perp}^2+ k_z^2\right] ^{1/2},$$ (29)

  
  
  and the dimensionless "time" variable:
  

  $$T\equiv-(C_{\rm A}/\vert{\vec C}\vert)k_{\vert\vert}(t)=C_{\rm A}[k_zt-k_{\parallel}(0)/\vert{\vec C}\vert],$$ (30)

  
  
  we may transform (28) into the remarkably simple differential equation:
  

  $${{{\rm d}^2{\Psi}}\over{{\rm d}T^2}}+{\biggl[1-{{a^2}\over{{\left(a^2+T^2 \right)}^2}} \biggr]}{\Psi}=0.$$ (31)

  
  
  Solutions of this differential equation ${\Psi }_a(T)$ give the complete solution of the system, because all physical variables are readily expressed through this function, its first derivative and the trivial solution for the first, non-modified branch given by ${\cal B}(t)$. It means that the full solution of the initial value problem is expressible by certain combinations of these two Alfvén modes: one that does not "feel" the velocity shear and the another one that is affected by it.

Note that the symmetry of (31) implies that its solutions must be invariant with respect to the inversion operation. Namely, if a pair of initial values ${\Psi}_a(-T_0)$ and ${\Psi}^{(1)}_a(-T_0)$gives a certain solution then ${\Psi}_a(T_0)$ and $-{\Psi}^{(1)}_a(T_0)$ gives inversion-symmetric solution.

Figure 1: The plots of the function ${\Omega }^2_{\rm eff}(T)$ for different values of the a parameter. Solid line corresponds to the case a=0.9, dashed line to a=0.7, dotted one to a=0.5, and dashed-dotted to a=0.3.

Analysis of (31) begins by studying the temporal behavior of the "effective frequency"

$${\Omega}_{\rm eff}(T)\equiv{\sqrt { 1-{{a^2}\over{{\left(a^2+T^2 \right)}^2}}}}\cdot$$ (32)

For sufficiently large absolute values of T, ${\Omega}_{\rm eff}(T)\simeq 1$, implying that at the corresponding periods of time, the system supports just the usual constant frequency and constant amplitude Alfvén waves. However, when |a|<1 there are two moments of time $T_{\pm}\equiv\pm {\sqrt {a(1-a)}}$ when the "effective frequency of shear-modified Alfvén waves" becomes zero and in the interval T_-<T< T₊it stays imaginary. Therefore, within this time interval (with the width determined by $\Delta T\equiv 2T_{+}$), swift and sudden changes in the temporal evolution of Alfvén waves may be expected.

Quantitative picture of this behavior is illustrated by Fig. 1 where we plotted the ${\Omega }^2_{\rm eff}(T)$ for different (but all |a|<1) values of a. One sees that with the decrease of a the minimum of ${\Omega }^2_{\rm eff}(T)$ becomes sharper: the depth of the minimum in Fig. 1 (which accounts for maximum increments of the transient instability) steadily increases with the decreasing a. The width of the time interval, in which this function stays negative (and, correspondingly, ${\Omega}_{\rm eff}(T)$ is imaginary) is maximum for a=1/2 and tends to zero when $a\simeq 1$ and $a \ll 1$.

At a first glance, from (31), it seems that decreasing a we would have the continuous increase of the transient amplification rate. However decreasing a we also make smaller the scaling factor between the physical time t and the variable T, which appears in (31). Besides, the small values of a imply smaller values of the Alfvén speed and the vertical component of the wavenumber vector k_z. Therefore, we can conclude that transient amplification factor for lower frequency Alfvén waves is higher, but for the amplification to occur the system needs a longer time interval. This means, in turn, that outflows with shorter/longer lifetime values are expected to amplify higher/lower frequency Alfvén waves.

Figure 2: The numerical solution of Eq. (31) featuring the function ${\Psi }_a(T)$ for a=0.1 case. The initial values are: ${\Psi }_a(-100)=0.01$ and ${\Psi }_a^{(1)}(-100)=0$.

Figure 3: The numerical solution of Eq. (31), which is inversion-symmetric to the one given in Fig. 2. Initial values here are: ${\hat{\Psi}}_a(-100)=-0.1354$ and ${\hat{\Psi}}_a^{(1)}(-100)=-0.2186$.

The presence of the imaginary effective frequency for a limited time interval means that Alfvén waves excited and maintained by the shear flow become subject to a certain, velocity shear induced instability, which is present only within the limited time interval. That is why we specify this phenomenon by the term "transient instability". Since the time interval is rather brief one can expect that the appearance of this instability will have a burst-like, explosive nature: initially at times T < T_- waves stay almost unaffected by the presence of the shear flow. However, as soon as the system will enter the transient instability interval T_-<T< T₊, the Alfvén waves undergo drastic and abrupt change in their amplitudes. Depending on the initial values, the mode of evolution will be a certain mixture of swiftly decaying and/or increasing modes within the $\Delta T$instability domain. But as soon as T > T₊ the waves become stable again and their amplitudes do not change anymore The resulting wave amplitude is enhanced/diminished in comparison with the initial amplitude depending whether transiently increasing/decreasing component was dominant for the initially excited wave.

Figure 4: The numerical solution of Eq. (28) presented for all components of ${\vec v}$ (solid lines on first three plots) and ${\vec b}$ (dashed lines on first three plots) vectors. Last plot shows time evolution of $E_{\rm tot}(t)/E_{\rm tot}(0)$. The values of parameters are: $C_{\rm A}=0.2$, kx=20, ky=10, kz=0.1, Cx=0.1, Cy=0.05, a=0.1789. Initial values are: ${\Psi }(0) =0.01$ and ${\Psi }^{(1)}(0)=0$.

Numerical results, represented by Figs. 2-5, fully confirm these qualitative expectations. Figures 2 and 3 illustrate the inversion symmetry of the functions  ${\Psi }_a(T)$. The first of these two plots shows the temporal evolution for a=0.1 and with initial conditions ${\Psi }_a(-100)=0.01$ and ${\Psi }_a^{(1)}(-100)=0$. The figure is plotted for the interval -100<T<100. Evidently the inversion symmetry implies that an another solution of this equation for the same value of a but with the initial values ${\hat{\Psi}}_a(-100)={\Psi}_a(100)$ and ${\hat{\Psi}}_a^{(1)}(-100)=-{\Psi}_a^{(1)}(100)$ will be exactly inverse-symmetric. In this particular example these initial values are ${\hat{\Psi}}_a(-100)=-0.1354$ and ${\hat{\Psi}}_a^{(1)}(-100)=-0.2186$. The inversion symmetry of these solutions is apparent. Physically this fact implies that the presence of the shear flow ensures burst-like and robust increase of amplitudes (energy) of some Alfvén waves, while there are always other waves which, on the contrary, sharply loose their energy under the influence of the shear flow.

Momentary appearance of the transient instability on the presented graphs is related to the narrowness of the transient instability interval, which is apparent from Fig. 1. Note that the plotting time interval in Fig. 1 is taken very narrow in order to give magnified portrait of the behavior of ${\Omega}_{\rm eff}$, while Figs. 2 and 3 are deliberately drawn for the much wider range in order to illustrate the behavior of waves on a large time span.

The family of solutions ${\Psi }_a(T)$ of (31) does not represent a physical variable of the problem, so in order to recover information about the temporal evolution of perturbations for physical variables it is more convenient to solve numerically (22-23) and to recover components of vectors ${\vec v}$ and ${\vec b}$ from (24). Besides, it is instructive to calculate the kinetic energy, the magnetic energy and the total energy of perturbations as given by (25). In order to track the temporal evolution of the shear-modified wave branch as such we set the amplitude of the unmodified component ${\cal B}(t)$ to be zero ${\cal C}=0$. Alternatively our numerical task was, first, to solve Eq. (28) in order to get functions ${\Psi}(t)$ and ${\Psi}^{(1)}(t)$. Second step was to calculate the physical variables by means of Eqs. (22)-(25).

This set of calculations was performed for different values of the system parameters and some representative examples are given in Figs. 4 and 5. Note that they are plotted as functions of the real time variable t and not the variable T used in (31). From these figures we readily see that when $\Delta = 0$( $k_{\perp}=0$), the shear flow efficiently "pumps" energy into the longitudinal components of the velocity and the magnetic field perturbations, while the transverse components stay basically unchanged aside from the transitory, "burst-like" increase of their amplitudes in the brief, transient amplification phase. This is another indication of the above-mentioned fact that the velocity shear primarily affects the incompressible limit of the fast magnetosonic waves. The asymptotic increase of the total energy, as shown in Fig. 4, being entirely due to the increase in v_z and b_z, is quite substantial (about two orders of magnitude for the given example).

Figure 5: The numerical solution of Eqs. (22)-(23) with the same parameters as in Fig. 4 except ky=10.1.

This is a typical mode of behavior for perturbations with $\Delta = 0$ and it is quite similar with the behavior of hydromagnetic waves in plane shear flows (Chagelishvili et al. 1993). By imparting a small but nonzero ${\Delta}$ (see Fig. 5), we can make all perturbation components grow. However, the overall increase of the total energy of perturbations in the latter case is somewhat smaller than in the former case.