1. Introduction
The request for the existence of a linear unstable mode spectrum in a two-dimensional axially symmetric configuration clearly emerges from the theory of accretion on a compact astrophysical object. In fact, the Shakura idea (in Shakura Reference Shakura1973; Shakura & Sunyaev Reference Shakura and Sunyaev1973) that angular momentum transport across a stellar accretion disk is realized via an effective viscosity, naturally leads to search for an instability mechanism, able to trigger turbulence in the plasma profile. In the case of a differentially rotating disk, embedded in the central object gravitational field only, the axial symmetry of the configuration prevents any linear unstable mode arising (see Bisnovatyi-Kogan & Lovelace Reference Bisnovatyi-Kogan and Lovelace2001).
However, as a weak magnetic field is involved in the background disk morphology, the Velikhov–Chandrasekhar instability (Velikhov Reference Velikhov1959; Chandrasekhar Reference Chandrasekhar1960), also known as the magneto-rotational instability (MRI), is triggered. The basic mechanism underlying the emergence of such an instability is the direct coupling between the differential rotation of the disk and the magnetic field tension. In other words, the plasma disk inhomogeneity transforms the Alfvenic waves into growing modes.
The detailed nature of the MRI and its role in triggering turbulence across an accretion disk has been extensively discussed in Balbus & Hawley (Reference Balbus and Hawley1998). The condition for obtaining the growing modes is that the Alfven frequency (the wavenumber amplitude times the Alfven speed) is small compared to a frequency-like term containing the radial gradient of the disk angular velocity.
In Balbus (Reference Balbus1995), the study of MRI is extended to a stratified disk (characterized by an inhomogeneous thick profile), outlining the role that the vertical derivative of the angular velocity plays in the morphology of the unstable modes. The study suggests that the driving force of the instability is to be identified with the spatial gradient of the angular velocity, which is different to the non-magnetized case where the dominant contribution comes from the specific angular momentum gradients. We stress how the analysis of the stratified disk, performed in Balbus (Reference Balbus1995), relies on a vector formulation of the magneto-hydrodynamics (MHD) ideal equations, i.e., no use is made of the magnetic flux function as a dynamical variable. Furthermore, this study does not rely on the validity of the co-rotation theorem, which is disregarded in the construction of the dispersion relation.
In the present paper, we perform a study of the stratified disk, similar to that discussed in Balbus (Reference Balbus1995), but based on the magnetic flux function dynamics and directly accounting for the co-rotation theorem (the background disk angular frequency must depend on the flux function (Ferraro Reference Ferraro1937)) holding for the background magnetic configuration. Without a significant loss of generality, we simplify the analysis of the dispersion relation for the linear mode spectrum by considering a vanishing azimuthal component of the background magnetic field (as typically true for the magnetic field of a compact astrophysical object) and we also consider perturbations which propagate along the background magnetic field only (this feature is intrinsic for Alfvenic disturbances of the background, as MRI results to be). More specifically, we deal with a local perturbation scheme based on the construction of linear dynamics near to fiducial background coordinates, as allowed by the assumption that the perturbations have a sufficiently short wavelength to explore a limited portion of the steady profile. Here, we consider an incompressible plasma at any order of approximation coherently with the so-called Boussinesq approximation (Balbus & Hawley Reference Balbus and Hawley1998). The motivation for such an approximation must be recognized in the search for an Alfvenic instability, which is simplified by removing the magneto-acoustic waves. Indeed, we are assuming that the sound speed is much larger than the Alfven velocity and the
${\it\beta}$
-parameter of the plasma is large. It is also worth noting that in the local approach addressed below for the perturbations, the mass conservation equation states that the plasma is incompressible once we require that the relative mass density fluctuations are small in comparison with the ratio between the spatial plasma shift and the perturbation wavelength (a rather natural assumption for a linear regime in the limit of small-scale perturbations).
The main issue for our analysis is showing how the vertical derivative of the disk angular velocity cancels out from the dispersion relation, as far as the features of the co-rotation theorem are retained in the perturbation scheme. Indeed, our dispersion relation implies the same morphology of the MRI, (i.e., the same condition on the background parameters in order to trigger the instability), exactly as in the thin disk scenario.
This fact has a relevant physical implication for the accretion mechanism onto a compact object since it states that only the radial differential rotation of the disk accounts for the instability property of the plasma and hence, only the radial steady disk profile need be considered when the transport processes are analysed.
The paper is organized as follows. In § 2 we provide the general ideal MHD scheme to describe the plasma disk evolution in the formalism of the magnetic flux functions. All the basic equations are provided and the main implications of their structure are discussed. In § 3, we briefly describe the background plasma configuration setting the basic force balance relations. In § 4, we develop the linear perturbation equations, as written in the Fourier (plane wave) representation. The dispersion relation is then properly derived and the implications of its morphology are then discussed. Finally, in § 5, we present brief concluding remarks.
2. Two-dimensional axisymmetric dynamics
We now provide the basic equations governing the two-dimensional axisymmetric dynamics of a plasma in the magneto-hydrodynamical representation. Having in mind the specific application of such a dynamical system to the morphology and stability problem of an accretion disk, we write the magnetic and velocity fields, making use of the magnetic flux function
${\it\psi}$
and the angular velocity
${\it\omega}$
, respectively:
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{B}=-\frac{1}{r}\partial _{z}{\it\psi}\hat{\boldsymbol{e}}_{r}+\frac{\bar{B}_{{\it\phi}}}{r}\hat{\boldsymbol{e}}_{{\it\phi}}+\frac{1}{r}\partial _{r}{\it\psi}\hat{\boldsymbol{e}}_{z} & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}=\boldsymbol{v}_{p}+{\it\omega}r\hat{\boldsymbol{e}}_{{\it\phi}}. & \displaystyle\end{eqnarray}$$
$\boldsymbol{v}_{p}=v_{r}\hat{\boldsymbol{e}}_{r}+v_{z}\hat{\boldsymbol{e}}_{z}$
is the poloidal component of the velocity field and all of the dynamical variables depend on
$t$
,
$r$
and
$z$
only (the
${\it\phi}$
dependence being suppressed because of the axial symmetry).
The dynamics of the two magnetic variables
${\it\psi}$
and
$\bar{B}_{{\it\phi}}$
can be easily fixed by the ideal MHD equations involving the magnetic structure of the plasma. In particular, the azimuthal component of the electron force balance yields
$$\begin{eqnarray}\partial _{t}{\it\psi}+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\psi}=0.\end{eqnarray}$$
It is worth noting that this equation is equivalent to the poloidal component of the induction equation in the MHD scheme. Analogously, the azimuthal component of the induction equation yields the dynamics of
$\bar{B}_{{\it\phi}}$
as
$$\begin{eqnarray}\partial _{t}\bar{B}_{{\it\phi}}+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\bar{B}_{{\it\phi}}+\bar{B}_{{\it\phi}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}_{p}-2\frac{\bar{B}_{{\it\phi}}v_{r}}{r}=r(\partial _{z}{\it\omega}\partial _{r}{\it\psi}-\partial _{r}{\it\omega}\partial _{z}{\it\psi}).\end{eqnarray}$$
The evolution of the angular velocity
${\it\omega}$
and of the poloidal component
$\boldsymbol{v}_{p}$
is described by the momentum conservation system, whose azimuthal component reads
$$\begin{eqnarray}{\it\rho}r\left(\partial _{t}{\it\omega}+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}\right)+2{\it\rho}v_{r}{\it\omega}=\frac{1}{4{\rm\pi}r^{2}}(\partial _{r}{\it\psi}\partial _{z}\bar{B}_{{\it\phi}}-\partial _{z}{\it\psi}\partial _{r}\bar{B}_{{\it\phi}}),\end{eqnarray}$$
while the poloidal ones provide
$$\begin{eqnarray}\displaystyle {\it\rho}\left(\partial _{t}\boldsymbol{v}_{p}+\boldsymbol{v}_{p}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}_{p}-{\it\omega}^{2}r\hat{\boldsymbol{e}}_{r}\right) & = & \displaystyle -\boldsymbol{{\rm\nabla}}p-\frac{1}{4{\rm\pi}r}\left[\partial _{r}\left(\frac{1}{r}\partial _{r}{\it\psi}\right)+\frac{1}{r}\partial _{z}^{2}{\it\psi}\right]\boldsymbol{{\rm\nabla}}{\it\psi}\nonumber\\ \displaystyle & & \displaystyle -~\frac{1}{8{\rm\pi}r^{2}}\boldsymbol{{\rm\nabla}}\bar{B}_{{\it\phi}}^{2}+\boldsymbol{F}_{p}^{e}.\end{eqnarray}$$
Above
$\boldsymbol{F}_{p}^{e}$
is the external poloidal force acting in the disk and in what follows, we will identify it with the gravity of the central astrophysical object.
The dynamics of the magnetized plasma, as described in the ideal MHD, is then closed by providing the mass conservation relation, i.e., the continuity equation for the mass density
${\it\rho}$
$$\begin{eqnarray}\partial _{t}{\it\rho}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{v}_{p})=0,\end{eqnarray}$$
and eventually assigning a suitable equation of state to express the pressure
$p$
, for example in the barotropic form
$p=p({\it\rho})$
. In the perturbation analysis below, we will not need to specify the plasma equation of state, simply requiring that it is incompressible.
We conclude by stressing how (2.4) and (2.5) correlate the two azimuthal components of the magnetic and velocity fields, respectively, making the gradient of one a source term for the generation of the other (the left-hand sides of these equations are linear homogeneous operators in the corresponding dynamical variable).
3. Background morphology
We consider the background plasma to be in a purely rotating steady configuration
$\boldsymbol{v}_{0}={\it\omega}_{0}r\hat{\boldsymbol{e}}_{{\it\phi}}$
(here the suffix
$0$
denotes background quantities). The rotation is clearly differential across the disk, i.e.,
${\it\omega}_{0}={\it\omega}_{0}(r,z)$
, but the validity of the co-rotation theorem (Ferraro Reference Ferraro1937) (holding for a stationary axisymmetric plasma) requires the condition
${\it\omega}_{0}={\it\omega}_{0}({\it\psi}_{0})$
,
${\it\psi}_{0}$
being the background magnetic flux function. Accounting for the validity of the co-rotation theorem is the basic feature of our study of two-dimensional MRI and constitutes the major difference compared to the analysis in Balbus (Reference Balbus1995).
When referred to the background equilibrium configuration, the dynamical system discussed in the previous section reduces to a force balance system. The gravity of the central body, around which the plasma disk is orbiting, is crucial in fixing the steady morphology, via the gravostatic equilibrium equation, involving also the background mass density
${\it\rho}_{0}$
and pressure
$p_{0}$
. Furthermore, since the plasma disk is embedded in the vacuum magnetic field of the central object, described via the function
${\it\psi}_{0}$
, the Lorentz force-free condition must hold. Thus, we respectively obtain the two equations
$$\begin{eqnarray}\displaystyle \boldsymbol{{\rm\nabla}}p_{0}={\it\rho}_{0}\left({\it\omega}_{0}^{2}({\it\psi}_{0})r\hat{\boldsymbol{e}}_{r}-{\it\omega}_{K}^{2}(r,z^{2})\boldsymbol{r}_{p}\right) & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{1}{4{\rm\pi}r}\left[\partial _{r}\left(\frac{1}{r}\partial _{r}{\it\psi}_{0}\right)+\frac{1}{r}\partial _{z}^{2}{\it\psi}_{0}\right]=0, & & \displaystyle\end{eqnarray}$$
${\it\omega}_{K}$
denotes the Keplerian frequency and
$\boldsymbol{r}_{p}$
is the vector radius in the meridian plane.
We now stress how, consistent with the analysis developed in Balbus (Reference Balbus1995) and according to the original Shakura idea (Shakura Reference Shakura1973), we consider an unmagnetized background plasma disk, neglecting its backreaction to the vacuum external field of the central object. Different physical situations, in which a steady backreaction is considered (Coppi Reference Coppi2005) or a Grad–Shafranov-like equation for the background can be written (Ogilvie Reference Ogilvie1997), are clearly of interest in view of their stability properties. However, the present set-up for the background is the most natural one for a local model, in which the spatial scale of the perturbations is much smaller than the steady profile. In fact, the current density, due to the perturbations (associated to the second spatial derivatives of the perturbed magnetic flux function) are much greater than the background currents, and this hierarchy holds for the Lorentz force also. Thus, addressing the magnetization of the background disk would reliably correspond to a minor change in the following analysis with respect to the force-free condition above.
After solving the second of these equations to obtain the form of
${\it\psi}_{0}$
, for example in the form of a dipole contribution (typical in compact astrophysical objects), we can assign the function
${\it\omega}_{0}({\it\psi})$
and the equation of state for the background plasma to determine the profile of the mass density
${\it\rho}_{0}$
by the integration of the first. However, for the perturbation analysis here developed, such details are unessential (indeed we deal with a local approach) and we can directly proceed to the linear dynamics. Finally, we note that, while for a thin configuration the disk angular velocity is almost Keplerian (i.e.,
${\it\omega}_{0}\simeq {\it\omega}_{K}$
), the radial pressure gradient takes a relevant role for a thick disk profile (i.e.,
${\it\omega}_{0}\neq {\it\omega}_{K}$
), see Ogilvie (Reference Ogilvie1997).
4. Linear perturbation theory
We now address the linear perturbation approach, which is based on axisymmetric non-stationary corrections to the equilibrium, also assumed to be of a very small spatial scale (their wavevectors have large magnitude) with respect to the background quantities. In other words, we deal with a local perturbation approach in which the linear terms, denoted via the suffix
$1$
, are expanded in Fourier series, analysing the single monochromatic modes, i.e.,
$$\begin{eqnarray}(\cdots \,)_{1}(t,\boldsymbol{r}_{p})=\bar{(\cdots \,)}_{1}\exp \{\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}_{p}-{\it\Omega}t)\}\,\quad \,\bar{(\cdots \,)}_{1}=\text{const.}\end{eqnarray}$$
Treating the perturbations as local corrections (
$|\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}_{0}|\ll 1$
, being
$r_{0}$
the fiducial radius, around which the mode exists), does not mean that we deal with a homogeneous background. In fact, each background quantity is calculated at the fiducial coordinates
$\{r_{0},z_{0}\}$
and behave as a constant coefficient in the perturbation scheme. But this is true also for the spatial gradients and, in particular, for the derivatives of the background angular velocity (which are expected to trigger the MRI).
Furthermore, we consider an incompressible plasma, for which
$\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}_{p}=0$
and a vanishing background azimuthal magnetic field. Often (see Balbus & Hawley Reference Balbus and Hawley1998) such a requirement comes out as the so-called ‘Boussinesq approximation’ when a local approach is pursued, but here it must be regarded as a basic feature of the system, introduced to better select the Alfvenic signature of the MRI. This same point of view leads us to simplify our analysis, by eliminating the magnetic pressure with the requirement that the wavevector
$\boldsymbol{k}$
be parallel to the background magnetic field
$\boldsymbol{B}_{0}$
, i.e.,
$\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\psi}_{0}=0$
.
Since each monochromatic mode obeys the relations
$$\begin{eqnarray}\partial _{t}(\cdots \,)_{1}=-\text{i}{\it\Omega}(\cdots \,)_{1};\quad \boldsymbol{{\rm\nabla}}(\cdots \,)_{1}=\text{i}\boldsymbol{k}(\cdots \,)_{1},\end{eqnarray}$$
we can easily restate the dynamics of the perturbations (as deduced by the basic system of the ideal MHD evolution equations) in terms of a closed algebraic system, providing the dispersion relation for the mode spectrum.
We observe how the poloidal velocity
$\boldsymbol{v}_{p}$
is absent in the background equilibrium and it is therefore natural to express it by means of the poloidal shift vector
${\bf\xi}_{p}$
of the plasma elements, i.e., we have
$$\begin{eqnarray}\boldsymbol{v}_{p_{1}}=\partial _{t}{\bf\xi}_{p}=-\text{i}{\it\Omega}{\bf\xi}_{p}\Rightarrow \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\xi}_{p}=\text{i}\boldsymbol{k}\boldsymbol{\cdot }{\bf\xi}_{p}=0.\end{eqnarray}$$
First of all, we stress how the continuity equation (2.7) is reduced by the incompressibility constraint above to a vanishing of the perturbed mass density
${\it\rho}_{1}$
. Furthermore, (2.3) provides the perturbed magnetic flux function
${\it\psi}_{1}$
in the form
$$\begin{eqnarray}{\it\psi}_{1}=-{\bf\xi}_{p}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\psi}_{0}.\end{eqnarray}$$
Hence the poloidal component (2.6) of the momentum conservation system reads, in the linear approximation, as
$$\begin{eqnarray}{\it\Omega}^{2}{\bf\xi}_{p}+2r{\it\omega}_{0}\left(\dot{{\it\omega}}_{0}{\it\psi}_{1}+{\it\omega}_{1}^{+}\right)\hat{\boldsymbol{e}}_{r}=\text{i}\boldsymbol{k}\frac{p_{1}}{{\it\rho}_{0}}-\frac{k^{2}{\it\psi}_{1}}{4{\rm\pi}r^{2}{\it\rho}_{0}}\boldsymbol{{\rm\nabla}}{\it\psi}_{0}.\end{eqnarray}$$
Above, we have split the perturbed angular velocity
$$\begin{eqnarray}{\it\omega}_{1}=\dot{{\it\omega}}_{0}{\it\psi}_{1}+{\it\omega}_{1}^{+}\end{eqnarray}$$
in terms of its first-order co-rotation contribution
$\dot{{\it\omega}}_{0}{\it\psi}_{1}$
(where
$\dot{{\it\omega}}_{0}\equiv (\text{d}{\it\omega}/\text{d}{\it\psi})_{{\it\psi}={\it\psi}_{0}}$
) and a generic linear deformation
${\it\omega}_{1}^{+}$
.
Taking the scalar product of the system above with the wavevector
$\boldsymbol{k}$
, we easily obtain the perturbed pressure contribution as a consequence of preserving the incompressibility condition during the dynamics, i.e.,
$$\begin{eqnarray}k^{2}p_{1}=-2\text{i}{\it\rho}_{0}{\it\omega}_{0}k_{r}r(\dot{{\it\omega}}_{0}{\it\psi}_{1}+{\it\omega}_{1}^{+}).\end{eqnarray}$$
The scalar product of (4.5) with
$\boldsymbol{{\rm\nabla}}{\it\psi}_{0}$
, once (4.4) is used, yields the basic relation
$$\begin{eqnarray}\left({\it\Omega}^{2}-y_{r}-{\it\omega}_{A}^{2}\right){\it\psi}_{1}=2{\it\omega}_{0}r\partial _{r}{\it\psi}_{0}{\it\omega}_{1}^{+},\end{eqnarray}$$
where
${\it\omega}_{A}^{2}\equiv (\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0})^{2}/4{\rm\pi}{\it\rho}_{0}=k^{2}v_{A}^{2}$
(
$v_{A}$
being the Alfven velocity) and
$y_{r}\equiv 2{\it\omega}_{0}r\partial _{r}{\it\omega}_{0}$
.
For a monochromatic mode, the linearized (2.4) and (2.5) take respectively the form
$$\begin{eqnarray}\displaystyle r{\it\Omega}{\it\omega}_{1}^{+}=-2{\it\omega}_{0}{\it\Omega}{\it\xi}_{r}-\frac{\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}}{4{\rm\pi}{\it\rho}_{0}r}(\bar{B}_{{\it\phi}})_{1} & & \displaystyle\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle {\it\Omega}(\bar{B}_{{\it\phi}})_{1}=-r^{2}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}_{0}{\it\omega}_{1}^{+}, & & \displaystyle\end{eqnarray}$$
which combined together give the relation
$$\begin{eqnarray}r\left({\it\Omega}^{2}-\bar{{\it\omega}}_{A}^{2}\right){\it\omega}_{1}^{+}=-2{\it\omega}_{0}{\it\Omega}^{2}{\it\xi}_{r}.\end{eqnarray}$$
Finally, the radial component of (4.5), using (4.7), can be restated as
$$\begin{eqnarray}{\it\Omega}^{2}\partial _{r}{\it\psi}_{0}{\it\xi}_{r}=-({\it\alpha}y_{r}+v_{Az}^{2}k^{2}){\it\psi}_{1}-2{\it\alpha}{\it\omega}_{0}r\partial _{r}{\it\psi}_{0}{\it\omega}_{1}^{+},\end{eqnarray}$$
where
${\it\alpha}\equiv 1-k_{r}^{2}/k^{2}$
.
The system of (4.8), (4.11) and (4.12) is a closed algebraic set in the variables
${\it\psi}_{1}$
,
${\it\omega}_{1}^{+}$
and
${\it\xi}_{r}$
which can be easily combined to obtain the following dispersion relation
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\Omega}^{4}-b{\it\Omega}^{2}+c=0,\quad c\equiv {\it\omega}_{A}^{2}\left(y_{r}+{\it\omega}_{A}^{2}\right),b\equiv \left(K_{0}^{2}+2{\it\omega}_{A}^{2}\right)+4{\it\omega}_{0}^{2}({\it\alpha}-1). & \displaystyle\end{eqnarray}$$
Here
$$\begin{eqnarray}K_{0}^{2}\equiv \frac{1}{r^{3}}\partial _{r}\left(r^{4}{\it\omega}_{0}^{2}\right)=y_{r}+4{\it\omega}_{0}^{2}.\end{eqnarray}$$
It is immediate to check that the necessary condition to obtain the MRI is provided by the inequality
${\it\omega}_{A}^{2}<-y_{r}$
, which ensures
$c<0$
. Indeed, also the request
$b<0$
could provide unstable behaviour, however the expression of
$K_{0}^{2}$
implies that
$b=c/{\it\omega}_{A}^{2}+4{\it\omega}_{0}^{2}(k_{z}^{2}/k^{2})$
and hence
$b<0$
requires again
$c<0$
.
Thus we see how, when we take into account the validity of the co-rotation theorem, the condition for MRI is the same in the case of a stratified thick disk as in a thin disk configuration (Balbus & Hawley Reference Balbus and Hawley1991). Such an issue relies on the cancellation of the
$z$
-derivative of the angular velocity in the dispersion relation, and this fact is the main difference between the present analysis and that performed in Balbus (Reference Balbus1995), where a vector formulation is addressed instead using the magnetic flux function (disregarding the co-rotation theorem).
Let us now try to clarify the basic difference existing between the present analysis and that addressed in Balbus (Reference Balbus1995). In the following, we will maintain the notation used by the current authors and Balbus (Reference Balbus1995).
The key point is that in Balbus (Reference Balbus1995), the vertical derivative of the background angular velocity enters the dispersion relation (2.4), while we have no such a term in (4.13). In addition, we observe that, in the limit of a thin disk endowed with a vertical background magnetic field, while our dispersion relation reduces to the correct one (see formula (111) of Balbus & Hawley (Reference Balbus and Hawley1998)), the one in Balbus (Reference Balbus1995) does not.
The reason for the discrepancy above is to be found in our implementation of the co-rotation theorem
${\it\omega}_{0}={\it\omega}_{0}({\it\psi}_{0})$
. To elucidate where such information enters the technical problem, we first observe that it is not in the form of the first-order azimuthal velocity. In Balbus (Reference Balbus1995), this quantity must read
$$\begin{eqnarray}{\it\delta}v_{{\it\phi}}=\partial _{t}{\it\xi}_{{\it\phi}}-r{\bf\xi}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}_{0},\end{eqnarray}$$
where
${\it\xi}_{{\it\phi}}$
is the plasma azimuthal shift and we consider that
$r{\it\omega}_{1}^{+}=\partial _{t}{\it\xi}_{{\it\phi}}$
. This expression can be easily mapped into (4.6), as soon as (4.4) and the co-rotation are taken into account.
The crucial point is that the co-rotation condition allows us to partially cancel out the term
$$\begin{eqnarray}-{\it\delta}\boldsymbol{B}\boldsymbol{\cdot }r\boldsymbol{{\rm\nabla}}{\it\omega}_{0}-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}{\it\delta}v_{{\it\phi}},\end{eqnarray}$$
which appears in (2.2g) of Balbus (Reference Balbus1995). In the local approximation, and expressing
${\it\delta}\boldsymbol{B}$
via
${\it\psi}_{1}$
, we rewrite the first term above as
$$\begin{eqnarray}-r{\it\delta}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\omega}_{0}=\text{i}{\it\psi}_{1}\left(\boldsymbol{k}\wedge \boldsymbol{{\rm\nabla}}{\it\omega}_{0}\right)\boldsymbol{\cdot }\boldsymbol{e}_{{\it\phi}}=\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{B}\dot{{\it\omega}}_{0}{\it\psi}_{1}.\end{eqnarray}$$
This term cancels in (4.16) the co-rotation part of
${\it\delta}v_{{\it\phi}}$
in the second term, reducing equation (2.2g) of Balbus (Reference Balbus1995) to our (4.4), as written in
$\boldsymbol{k}$
and
${\it\omega}$
.
It is just this cancellation in (2.2g) of Balbus (Reference Balbus1995) that removes the vertical derivative of
${\it\omega}_{0}$
from the final dispersion relation, the disappearance of such a derivative in (2.2d) of Balbus (Reference Balbus1995) is ensured by the above relation (4.15).
5. Concluding remarks
We have analysed the stability of a stratified and differentially rotating ideal plasma disk, as described in a two-dimensional axisymmetric scheme. After the set-up of the basic dynamical system, required to address the problem considered, we briefly characterize the morphology of the steady background configuration of the plasma disk. The steady configuration is assumed to be embedded in the gravitational and magnetic fields of the central object, according to the astrophysical implementation of our analysis in the behaviour and stability of stellar accretion structures. Then, we constructed the Fourier representation of the linear perturbation dynamics, allowed by the requirement of dealing with short wavelength deformation of the background, and hence by a local perturbation approach. The combination of such algebraic equations lead to the morphology of the dispersion relation, providing the structure of the spectral modes. As result of such a procedure, we arrive to study the profile of the MRI in the case of a stratified disk, evaluated with the help of two simplifying assumptions for selecting the Alfvenic nature of this instability: the plasma is required to be incompressible (so removing the acoustic modes) and the perturbations propagate along the background magnetic field (i.e., we deal with zero first-order magnetic pressure). As a basic issue of our analysis, we demonstrated how accounting for the co-rotation theorem on the background has significant implications for the linear dispersion relation. In particular, the co-rotation theorem accounts for the cancellation of the vertical derivative of the background disk angular velocity from the structure of the linear unstable modes. Such a feature does not emerge in the vector approach depicted in Balbus (Reference Balbus1995), where the co-rotation profile is not taken into account. Thus, the dispersion relation for a stratified thick accretion disk retains the same structure as in the thin disk approximation, i.e., the condition to trigger the MRI is associated to the same inequality. As a result, the stability of the disk, the corresponding turbulence and the associated angular momentum transport are essentially determined by the radial profile of the background configuration, which is fixed by the balance between the gravitational force, the pressure gradient and the centripetal force. The main difference in the MRI morphology between thin and thick accretion disks consists in the role that the radial pressure can take in the later case, while in the former, the disk angular frequency has essentially a Keplerian profile.
