2 Main consideration

Our aim is to develop a theory of collective phenomena in helical flows of magnetized plasmas. We adopt the standard MHD model and consider incompressible perturbations in an axisymmetric, cylindrical, steady flow of a plasma with uniform density ( ${\rho}_0={\rm const.}$), embedded in a vertical, homogeneous ${\vec B}_0=(0,~0,~B_0)$ magnetic field. With $\rho = {\rho}_0+ \varrho$, P=P₀+p, ${\vec V}={\vec U}+{\bf u}$, ${\vec B}={\vec B}_0+ {\vec B}^{'}$, the basic set of MHD equations for linearized perturbations is:

$$\nabla {\vec u}=0,$$ (1)

$$D_t {\vec u}+({\vec u} \cdot {\nabla}){\vec U}=-{1 \over {\rho}_0... ...a}p-{{{\vec B}_0}\over {4{\pi}{\rho}_0}}{\times}({\nabla}{\times}{\vec B}^{'}),$$ (2)

$$D_t{\vec B}^{'}=({\vec B}^{'} \cdot {\nabla}){\vec U} + ({\vec B}_0 \cdot {\nabla}){\vec u},$$ (3)

$$\nabla {\vec B}^{'}=0,$$ (4)

where $D_t\equiv{\partial}_t+({\vec U} \cdot {\nabla})$ is the convective derivative operator and ${\vec U}(r)$is the vector field of the steady state flow. Unfortunately the actual kinematic portrait of astrophysical helical flows is largely unknown. There is no credible data for recently discovered solar macrospicules with helical plasma motion (solar tornados), for accretion columns related to X-ray pulsars and cataclysmic variables, for accretion-ejection flows and galactic and extragalactic jets. Therefore, we need to adopt a phenomenological model for ${\vec U}$, that has to be general enough to encompass different possible sorts of real helical flows. Preferably the model must be three-dimensional and it should include a possibility of both outflowing/inflowing and rotational modes of motion. The model we choose is (Rogava & Poedts 1999):

$${\vec U}(r)\equiv[0,~r{\Omega}(r),~U(r)],$$ (5)

where r=(x²+y²)^1/2 is a distance from the rotation axis. For the angular velocity we take:

$${\Omega}(r)={\cal A}/r^n,$$ (6)

with ${\cal A}$ and n as constants. This model implies the following two limiting cases:

• Rigid rotation, with n=0 and ${\cal A}={\Omega}_0$.
• Keplerian rotation, with n=3/2 and ${\cal A}=(GM)^{1/2}$.

The corresponding Cartesian components of the linear azimuthal velocity $U_{\phi}=r{\Omega}(r)$ are:

$$U_x(x,y)=-{\cal A}y/r^n,$$ (7a)

$$U_y(x,y)={\cal A}x/r^n.$$ (7b)

The basic role in our analysis is played by the shear matrix ${\cal S}_{ik} \equiv{\partial}U_i/{\partial}x_k$. Evidently rotational part of the flow velocity generates the following four nonzero components of the shear matrix:

$${\sigma}\equiv{\cal S}_{xx}=2{\cal A}nx_0y_0/r_0^{n+2},$$ (8a)

$${\cal S}_{yy}=-{\sigma},$$ (8b)

$$A_1\equiv{\cal S}_{xy}=-{\cal A}\left[x_0^2+(1-n)y_0^2\right]/r_0^{n+2},$$ (8c)

$$A_2\equiv{\cal S}_{yx}={\cal A}\left[(1-n)x_0^2+y_0^2\right]/r_0^{n+2}.$$ (8d)

As regards the outflow component of the velocity we can use the model (Rogava et al. 2000):

$$U_z(x,y)=U_m\left[1-(r/R)^2\right],$$ (9a)

comprising a parabolically sheared "outflow" along the Z-axis, being similar to the well-known Hagen-Poiseuille flow. The outflow model may be modeled also in a number of other ways. In astrophysical jet literature, for instance, the model (Ferrari 1998):

$$U_z=U_m/{\rm cosh}[r^m],$$ (9b)

is often used. One can easily see that the outflow component, given by (9), generates the following two components of the shear matrix:

$$C_1\equiv{\cal S}_{zx}=-2U_mx_0/R^2,$$ (10a)

$$C_2\equiv{\cal S}_{zy}=-2U_my_0/R^2.$$ (10b)

Therefore, the resulting helical flow is the superposition of a cylindrical outflow along the Z-axis with a differential motion in the transverse cross-section of the jet. The complete $3 \times 3$traceless shear matrix is:

$${\cal S}=\left(\begin{array}{lll} {\sigma}&A_1&0\cr A_2&-{\sigma}&0 \cr C_1&C_2&0\cr\end{array}\right).$$ (11)

This shear matrix has a number of interesting properties. Its square is equal to:

$$\vert\vert{\cal S}^2\vert\vert=\left(\begin{array}{lll} {\Gamma}^... ...-{\Gamma}^2 & 0 \cr {\varepsilon}_1& {\varepsilon}_2 &0 \cr \end{array}\right),$$ (12)

where we use the notation:

$${\varepsilon}_1\equiv A_2C_2+{\sigma}C_1,$$ (13a)

$${\varepsilon}_2\equiv A_1C_1-{\sigma}C_2;$$ (13b)

$${\Gamma}\equiv\left({\sigma}^2+A_1A_2\right)^{1/2}=\sqrt{n-1}~{\Omega}(r),$$ (13c)

and its cube is: $\vert\vert{\cal S}^3\vert\vert={\Gamma}^2\vert\vert{\cal S}\vert\vert$.

The nonmodal local method for studying the dynamics of linearized perturbations in kinematically complex flows (Lagnado et al. 1984; Craik & Criminale 1986; Mahajan & Rogava 1999) allows to reduce the initial set of partial differential equations for perturbation variables $F({\vec r},t)$, defined in the real physical space, to the initial value problem formulated for the perturbation variable amplitudes ${\hat F}({\vec k},t)$, defined in the space of wave numbers (k-space). The key element of this approach is the time variability of ${\vec k}$'s, imposed by the presence of the shear flow! This variability is governed by the following set of equations:

$${\vec k}^{(1)}+{\cal S}^T\cdot{\vec k}=0,$$ (14)

henceforth for an arbitrary function f we use the notation: ${\partial}_t^nf\equiv f^{(n)}$.

For the helical flow we find that ${\vec k}^{(3)}={\Gamma}^2{\vec k}^{(1)}$. The corresponding characteristic equation ${\rm det}[{\cal S}^T-{\lambda}{\times} {\cal I}]=0$ (with ${\cal I}$ being a unit matrix) yields the equation for the eigenvalues:

$${\lambda}({\lambda}-{\Gamma})({\lambda}+{\Gamma})=0,$$ (15a)

with the solutions:

$${\lambda}_1=0,~~~~{\lambda}_{2,3}={\pm}{\Gamma}.$$ (15b)

We see that the differential rotation parameter nplays a decisive role in determining the evolution scenario for the wave number vector ${\vec k}(t)$: when n<1 (including the rigid rotation case) $\Gamma$ is imaginary and the time evolution of ${\vec k}(t)$ is periodic, while when n>1 (including the Keplerian rotation regime), $\Gamma$ is real and makes the time behavior of ${\vec k}(t)$ exponential.

In the dimensionless notation $v_{x,y,z}\equiv{\hat u}_{x,y,z}$, ${\cal P}\equiv i{\hat p}/{\rho}_0$, $b_{x,y,z}\equiv i{\hat B}^{'}_{x,y,z}/B_0$, perturbation amplitudes evolve as:

$$({\vec k}{\cdot}{\vec v})=0,$$ (16)

$${\vec v}^{(1)}+{\cal S}{\cdot}{\vec v}=-{\vec k}{\cal P}-C_{\rm A}^2[{\vec k}b_z-k_z {\vec b}],$$ (17)

$${\vec b}^{(1)}={\cal S}{\cdot}{\vec b}-k_z{\vec v}, (i=x,y)$$ (18)

$$({\vec k}{\cdot}{\vec b})=0.$$ (19)

In the next two sections we separately consider the problem of a purely ejectional/injectional flow, and a flow with helical motion of plasma particles. We shall see that in both cases the Alfvén waves sustained by the flow are amplified. For the former case the amplification is transient (algebraic instability), while for the latter case the velocity shear makes Alfvén waves exponentially unstable.