\section{Introduction}

The recent increased interest in the analysis of hydrodynamic disc
flows is motivated, on one hand, by the study of turbulent
processes, and, on the other,  by the investigation of regular
structure formation in protoplanetary discs. Indeed, many
astrophysical discs are thought to be neutral or having ionization
rates too low to effectively couple with magnetic field. Among these
are cool and dense areas of protoplanetary discs, discs around young
stars, X-ray transient and dwarf nova systems in quiescence (see
e.g. Gammie and Menou 1998, Sano et al. 2000, Fromang, Terquem and
Balbus 2002). Observational data shows that astrophysical discs
often exhibit radial gradients of thermodynamic variables (see e.g.
Sandin et al. 2008, Issela et al. 2007). To what extent these
inhomogeneities affect the processes occurring in the disc is still
subject open to investigations. It has been found that strong local
entropy gradients in the radial direction may drive the Rossby wave
instability (Lovelace et al. 1999, Li et al. 2000) that transfers
thermal to kinetic energy and leads to vortex formation. However, in
astrophysical discs, radial stratification is more likely weak. In
this case, the radial entropy (temperature) variation on the global
scale leads to the existence of baroclinic perturbations over the
barotropic equilibrium state. This more appropriate situation has
recently become a subject of extensive study.

Klahr and Bodenheimer (2003) pointed out that the radial
stratification in the disc can lead to the global baroclinic
instability. Numerical results show that the resulting state is
highly chaotic and transports angular momentum outwards. Later Klahr
(2004) performed a local 2D linear stability analysis of a radially
stratified flow with constant surface density and showed that
baroclinic perturbations can grow transiently during a limited time
interval. Johnson and Gammie (2005) derived analytic solutions for
3D linear perturbations in a radially stratified discs in the
Boussinesq approximation. They find that leading and trailing waves
are characterized by  positive and negative angular momentum flux,
respectively. Later Johnson and Gammie (2006) performed numerical
simulations, in the local shearing sheet model, to test the radial
convective stability and the effects of baroclinic perturbations.
They found no substantial instability due to the radial
stratification. This result reveals a controversy over the issue of
baroclinic instability. Presently,  it seems that nonlinear
baroclinic instability is an unlikely development in the local
dynamics of sub-Keplerian discs with weak radial stratification.

Potential vorticity production, and the formation and development of
vortices in radially stratified discs have been studied by Petersen
et al. (2007a,b) by using pseudospectral simulations in the
anelastic approximation.  They show that the existence of thermal
perturbations in the radially stratified disc flows leads to the
formation of vortices. Moreover, stronger vortices appear in discs
with higher temperature perturbations or in simulations with higher
Reynolds numbers, and the transport of angular momentum may be both
outward and inward.

Keplerian differential rotation in the disc is characterized by a
strong velocity shear in the radial direction. It is known that
shear flows are non-normal and exhibit a number of transient
phenomena due to the non-orthogonal nature of the operators (see
e.g. Trefethen et al.  1993). In fact, the studies described above
did not take into account the possibility of mode coupling and
energy transfer between different modes due to the shear flow
induced mode conversion.  Mode coupling is inherent to shear flows
(cf.  Chagelishvili et al.  1995) and often, in many respects,
defines the role of perturbation modes in the system dynamics and
the further development of nonlinear processes. Thus, a correct
understanding of the energy exchange channels between different
modes in the linear regime is vital for a correct understanding of
the nonlinear phenomena.

Indications of the shear induced mode conversion can be found in a
number of previous studies. Barranco and Marcus 2005 report that
vortices are able to excite inertial gravity waves during 3D
spectral simulations. Brandenburg and Dintrans (2006) have studied
the linear dynamics of perturbation SFH to analyze nonaxisymetric
stability in the shearing sheet approximation. Temporal evolution of
the perturbation gain factors reveal a wave nature after the radial
wavenumber changes sign. Compressible waves are present, along with
vortical perturbations, in the simulation by Johnson \& Gammie
(2005b) but their origin is not particularly discussed.

In parallel, there are a number of papers that focus on the
investigation of the shear induced mode coupling phenomena. The
study of the linear coupling of modes in Keplerian flows has been
conducted in the local shearing sheet approximation (Tevzadze et al.
2003,2008) as well as in 2D global numerical simulations (Bodo et
al. 2005, hereafter B05). Tevzadze et al. (2003) studied the linear
dynamics of three-dimensional small scale perturbations (with
characteristic scales much less then the disc thickness) in
vertically (stably) stratified Keplerian discs. They show, that
vortex and internal gravity wave modes are coupled efficiently. B05
performed global numerical simulations of the linear dynamics of
initially imposed two-dimensional pure vortex mode perturbations in
compressible Keplerian discs with constant background pressure and
density. The two modes possible in this system are effectively
coupled: vortex mode perturbations are able to generate
density-spiral waves. The coupling is, however, strongly asymmetric:
the coupling is effective for wave generation by vortices, but not
viceversa. The resulting dynamical picture points out the importance
of  mode coupling and the necessity of considering compressibility
effects for processes with characteristic scales of the order or
larger than the disc thickness. Bodo et al. (2007) extended this
work to nonlinear amplitudes and found that mode coupling is an
efficient channel for energy exchange and is not an artifact of the
linear analysis. B05 is particularly relevant to the present study,
since it studies the dynamics of mode coupling in 2D unstratified
flows and is a good starting point for a further extension to
radially stratified flows. Later, Heinemann \& Papaloizou (2009a)
derived WKBJ solutions of the generated waves and performed
numerical simulations of the wave excitation by turbulent
fluctuations (Heinemann \& Papaloizou 2009b).

In the present paper we study the linear dynamics of perturbations
and analyze shear flow induced mode coupling in the local shearing
sheet approximation. We investigate the properties of mode coupling
using qualitative analysis within the three-mode approximation.
Within this approximation we tentatively distinguish vorticity,
entropy and pressure modes. Quantitative results on mode conversion
are derived numerically. It seems that a weak radial
stratifications, while being a weak factor for the disc stability,
still provides an additional degree of freedom (an active entropy
mode), opening new options for velocity shear induced mode
conversion, that may be important for the system behavior. One of
the direct result of mode conversion is the possibility of linear
generation of the vortex mode (i.e., potential vorticity) by
compressible perturbations. We want to stress the possibility of the
coupling between  high and low frequency perturbations, considering
that high frequency oscillations have been often neglected during
previous investigations in particular for protoplanetary discs.

Conventionally there are two distinct viewpoints commonly employed
during the investigation of hydrodynamic astrophysical discs. In one
case (self gravitating galactic discs) the emphasis is placed on the
investigation of the dynamics of spiral-density waves and vortices,
although normally present in numerical simulations, are thought to
play a minor role in the overall dynamics. In the other case
(non-self-gravitating hydrodynamic discs) the focus is on the
potential vorticity perturbations and density-spiral waves are often
thought to play a minor role. Here, discussing the possible (multi)
mode couplings, we want to draw attention to the possible flaws of
these simplified views (see e.g. Mamatsashvili \& Chagelishvili
2007). In many cases,  mode coupling makes different perturbation to
equally participate in the dynamical processes despite of a
significant difference in their temporal scales.

In the next section we present mathematical formalism of our study.
We describe three mode formalism and give schematic picture of the
linear mode coupling in the radially sheared and stratified flow.
Numerical analysis of the mode coupling is presented in Sec. 3. We
evaluate mode coupling efficiencies at different radial
stratification scales of the equilibrium pressure and entropy. The
paper is summarized in Sec. 4.

\section{Basic equations}

The governing ideal hydrodynamic equations of a  two-dimensional,
compressible disc flows in polar coordinates are:
\begin{equation}
{\partial \Sigma \over \partial t}  + {1 \over r} {\partial \left( r
\Sigma V_r \right) \over
\partial r}  + {1 \over r} {\partial \left( \Sigma V_\phi \right)\over
\partial \phi}  = 0~,~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
\begin{equation}
{\partial  V_r \over \partial t}  +  V_r{\partial  V_r \over
\partial r}+ {V_\phi \over r}{\partial V_r \over
\partial \phi}  - {V_\phi^2 \over r} = -{1 \over \Sigma}
{\partial P \over \partial r}  - {\partial \psi_g \over \partial r}
~,~~~~~~
\end{equation}
\begin{equation}
{\partial  V_\phi \over \partial t}  +  V_r{\partial  V_\phi \over
\partial r}+ {V_\phi \over r}{\partial V_\phi \over
\partial \phi} +
{V_r V_\phi \over r} = -{1 \over \Sigma r}{\partial P\over
\partial \phi}~,~~~~~~~~~
\end{equation}
\begin{equation}
{\partial  P \over \partial t}  +  V_r{\partial  P \over
\partial r}+ {V_\phi \over r}{\partial P \over
\partial \phi}
= - {\gamma P} \left( {1 \over r} {\partial (r V_r) \over
\partial r}  + {1 \over r} {\partial V_\phi \over \partial
\phi}  \right)~,
\end{equation}
where  $V_r$ and $V_\phi$ are the flow radial and azimuthal
velocities respectively. $P(r,\phi)$, $\Sigma(r,\phi)$ and $\gamma~$
are respectively the pressure, the surface density and the adiabatic
index. $\psi_g$ is the gravitational potential of the central mass,
in the absence of self-gravitation $~(\psi_g \sim -{1 / r})$. This
potential  determines the Keplerian angular velocity:
\begin{equation}
{\partial \psi_g \over \partial r} = \Omega_{Kep}^2 r ~,~~~~
\Omega_{Kep} \sim r^{-3/2};
\end{equation}

\subsection{Equilibrium state}

We consider an axisymmetric $(\partial / \partial \phi \equiv 0),~$
azimuthal $(\bar {V}_{r} = 0)~$ and differentially rotating  basic
flow: $\bar {V}_{\phi}= \Omega(r)r$. In the 2D radially stratified
equilibrium (see Klahr 2004), all variables are assumed to follow a
simple power law behavior:
\begin{equation}
\bar {\Sigma}(r) = \Sigma_0 \left( {r \over r_0}
\right)^{-\beta_\Sigma},~~~~\bar {P}(r) = P_0\left( {r \over r_0}
\right)^{-\beta_P} ~,
\end{equation}
where overbars denote equilibrium and $\Sigma_0$ and $P_0$ are the
values of the equilibrium surface density and pressure at some
fiducial radius $r = r_0$. The entropy can be calculated as:
\begin{equation}
\bar S  =  \bar P \bar \Sigma^{-\gamma} = - \left(r \over r_0
\right)^{-\beta_S} ~,
\end{equation}
where
\begin{equation}
\beta_S \equiv \beta_P - \gamma \beta_\Sigma ~.
\end{equation}
$S$ is sometimes called potential temperature, while the physical
entropy can be derived as $\bar S = C_V \log S + \bar S_0$.

This equilibrium shows a deviation from the Keplerian profile due to
the radial stratification:
$$
\Delta \Omega^2(r) = \Omega^2(r) - \Omega^2_{Kep}= {1 \over r {\bar
{\Sigma}(r)}} {\partial {\bar{P}(r)}\over \partial r} =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$$
\begin{equation}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = - {P_0
\over \Sigma_0} {\beta_P \over r_0^2}\left( {r \over r_0}
\right)^{\beta_\Sigma-\beta_P-2} ~.
\end{equation}
Hence, the described state is sub-Keplerian or super-Keplerian when
the radial gradient of pressure is negative ($\beta_P>0$) or
positive ($\beta_P<0$), respectively. Although these discs are
non-Keplerian, they are still rotationally supported, since the
deviation from the Keplerian profile is small: $~\Delta
\Omega^2(r)\ll \Omega^2_{Kep}$.

\subsection{Linear perturbations}

We split the physical variables into mean and perturbed parts:
\begin{equation}
\Sigma(r,\phi) = {\bar {\Sigma}(r)} + {{\Sigma}^\prime(r,\phi)} ~,
\end{equation}
\begin{equation}
P(r,\phi) = {\bar{P}(r)} + P^\prime(r,\phi) ~,
\end{equation}
\begin{equation}
V_r(r,\phi) =  V_r^\prime(r,\phi) ~,
\end{equation}
\begin{equation}
V_\phi(r,\phi) = \Omega(r) r + V_\phi^\prime(r,\phi) ~.
\end{equation}
In order to remove background trends from the perturbations we
employ the global radial power law scaling for perturbed quantities:
\begin{equation}
\hat \Sigma(r)  \equiv \left({r \over r_0}\right)^{-\delta_\Sigma}
\Sigma^\prime(r) ~,
\end{equation}
\begin{equation}
\hat P(r) \equiv \left({r \over r_0}\right)^{-\delta_P} P^\prime(r)
~,
\end{equation}
\begin{equation}
\hat {\bf V}(r)  \equiv \left({r \over r_0}\right)^{-\delta_V} {\bf
V}^\prime(r) ~.
\end{equation}

After the definitions one can get the following dynamical equations
for the scaled perturbed variables:
\begin{equation}
\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} {\hat \Sigma \over \Sigma_0} +
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
$$
\left( {r \over r_0} \right)^{-\beta_\Sigma-\delta_\Sigma+\delta_V}
\left[ {\partial \hat V_r \over \partial r} + {1 \over r} {\partial
\hat V_\phi \over
\partial \phi} + {1+\delta_V-\beta_\Sigma \over r} \hat V_r \right]
= 0 ~,
$$
\begin{equation}
\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} \hat V_r - 2\Omega(r) \hat V_\phi +
\end{equation}
$$
{c_s^2 \over \gamma} \left({r \over r_0}
\right)^{\beta_\Sigma+\delta_P-\delta_V} {\partial \over \partial r}
{\hat P \over P_0} + c_s^2 {\delta_P \over \gamma r_0} \left( {r
\over r_0} \right)^{\beta_\Sigma+\delta_P-\delta_V-1} {\hat P \over
P_0} +
$$
$$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ c_s^2 {\beta_P \over \gamma r_0}
\left( {r \over r_0}
\right)^{2\beta_\Sigma+\delta_\Sigma-\beta_P-\delta_V-1} {\hat
\Sigma \over \Sigma_0} = 0 ~,
$$
\begin{equation}
\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} \hat V_\phi + \left( 2 \Omega(r) +
r {\partial \Omega(r) \over \partial r} \right) \hat V_r +
~~~~~~~~~~~~~~~
\end{equation}
$$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {c_s^2 \over \gamma r_0}
\left( {r \over r_0} \right)^{\beta_\Sigma+\delta_P-\delta_V-1}
{\partial \over \partial \phi} {\hat P \over P_0} = 0 ~,
$$
\begin{equation}
\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} {\hat P \over P_0} +
\end{equation}
$$
\gamma \left( {r \over r_0} \right)^{-\beta_P+\delta_V-\delta_P}
\left[ {\partial \hat V_r \over
\partial r} + {1 \over r} {\partial \hat V_\phi \over \partial
\phi} + {1+\delta_V-\beta_P/\gamma \over r} \hat V_r \right] = 0 ~,
$$
where $c_s^2 = \gamma P_0/\Sigma_0$ is the squared sound speed at
$r=r_0$.

\subsection{Local approximation}

The linear dynamics of perturbations in differentially rotating
flows can be effectively analyzed in the local co-rotating shearing
sheet frame (e. g., Goldreich \& Lynden-Bell 1965; Goldreich \&
Tremaine 1978). This approximation simplifies the mathematical
description of flows with inhomogeneous velocity. In the radially
stratified flows the spatial inhomogeneity of the governing
equations comes not only from the equilibrium velocity, but from the
pressure, density and entropy profiles as well. In this case we
first re-scale perturbations in global frame in order to remove
background trends from linear perturbations, rather then use
complete form of perturbations to the equilibrium (see Eqs. 14-16).
Hence, using the re-scaled linear perturbation ($\hat P$, $\hat
\Sigma$, $\hat {\rm \bf V}$) we may simplify local shearing sheet
description as follows. Introduction of a local Cartesian
co-ordinate system:
\begin{equation}
x \equiv r - r_0~,~~~~ y \equiv r_0 (\phi - \Omega_0 t)~,~~~~{x
\over r_0} ,~ {y \over r_0} \ll 1~,
\end{equation}
\begin{equation}
{\partial \over \partial x} = {\partial \over \partial r}~,~~~
{\partial \over \partial y} = {1 \over r_0}{\partial \over
\partial \phi}~,~~~ {\partial \over \partial t} =
{\partial \over \partial t} -  r_0 \Omega_0 {\partial \over
\partial y},
\end{equation}
where $\Omega_0$ is the local rotation angular velocity at $r=r_0$,
transforms global differential rotation into a local radial shear
flow and the two Oort constants define the local shear rate:
\begin{equation}
A \equiv {1 \over 2} r_0 \left[ {\partial \Omega(r) \over \partial
r}\right]_{r=r_0}~,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
\begin{equation}
B \equiv - {1 \over 2} \left[ r{\partial \Omega(r)\over \partial r}
+ 2\Omega(r) \right]_{r=r_0}= -A - \Omega_0~.
\end{equation}
Hence, the equations describing the linear dynamics of perturbations
in local approximation read as follows:
\begin{equation}
\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat P \over \gamma P_0} +
\end{equation}
$$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left[ {\partial \hat V_x \over
\partial x} + {\partial \hat V_y \over \partial y} +
{1+\delta_V-\beta_P/\gamma \over r_0} \hat V_x \right] = 0 ~,
$$
\begin{equation}
\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat V_x} - 2\Omega_0 \hat V_y +
\end{equation}
$$
~~~~~~~~~~~~~~~~~~~~~ c_s^2 \left[ {\partial \over \partial x} {\hat
P \over \gamma P_0} + {\delta_P + \beta_P/\gamma \over r_0} {\hat P
\over \gamma P_0} - {\beta_P \over \gamma r_0} {\hat S \over \gamma
P_0}\right] = 0 ~,
$$
\begin{equation}
\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat V_y} - 2B \hat V_y + c_s^2
{\partial \over \partial y} {\hat P \over \gamma P_0} =0 ~,
\end{equation}
\begin{equation}
\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat S \over \gamma P_0} - {\beta_S
\over \gamma r_0} \hat V_x = 0 ~,
\end{equation}
where $\hat S $ is the entropy perturbation:
\begin{equation}
\hat S \equiv \hat P - c_s^2 \hat \Sigma ~.
\end{equation}
Now we may adjust the global scaling law of perturbations in order
to simplify the local shearing sheet description (see Eqs. 25,26):
\begin{equation}
1 + \delta_V - \beta_P/\gamma = 0 ~,
\end{equation}
\begin{equation}
\delta_P + \beta_P/\gamma = 0 ~.
\end{equation}

Let us introduce spatial Fourier harmonics (SFHs) of perturbations
with time dependent phases:
\begin{equation}
\left( \begin{array}{c} {\hat V}_x({\bf r},t) \\ {\hat V}_y({\bf r},t) \\
{{\hat P}({\bf r},t) / \gamma P_0} \\ {{\hat S}({\bf r},t) / \gamma
P_0} \end{array} \right) =
\left( \begin{array}{r} u_x({\bf k}(t),t) \\ u_y({\bf k}(t),t) \\
-{\rm i} p({\bf k}(t),t) \\ s({\bf k}(t),t)
\end{array} \right) \times
\end{equation}
$$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\exp \left( {\rm i} k_x(t) x + {\rm i} k_y y \right) ~,
$$
with
\begin{equation}
k_x(t) = k_x(0) - 2Ak_y t~.
\end{equation}

Using the above expansion and Eqs. (27-30), we obtain a compact ODE
system that governs the local dynamics of SFHs of perturbations:
\begin{equation}
{{\rm d} \over {\rm d} t} p - k_x(t) u_x - k_y u_y = 0 ~,
\end{equation}
\begin{equation}
{{\rm d} \over {\rm d} t} u_x - 2 \Omega_0 u_y + c_s^2 k_x(t) p -
c_s^2 k_P s = 0 ~,
\end{equation}
\begin{equation}
{{\rm d} \over {\rm d} t} u_y - 2 B u_x + c_s^2 k_y p = 0 ~,
\end{equation}
\begin{equation}
{{\rm d} \over {\rm d} t} s - k_S u_x = 0 ~.
\end{equation}
where
\begin{equation}
k_P = {\beta_P \over \gamma r_0} ~~~ k_S = {\beta_S \over \gamma
r_0} ~.
\end{equation}
The potential vorticity:
\begin{equation}
W \equiv k_x(t)u_y - k_y u_x - 2B p ~,
\end{equation}
is a conserved quantity in barotropic flows: $W = {\rm const.}$ when
$k_P=0$.

\subsection{Perturbations at rigid rotation}

The dispersion equation of our system can be obtained in the
shearless limit ($A=0$, $B=-\Omega$). Hence, using Fourier expansion
of perturbations in time $\propto \exp({\rm i} \omega t)$, in the
shearless limit, we obtain:
\begin{equation}
\omega^4 - \left( c_s^2 k^2 + 4 \Omega_0^2 - c_s^2 \eta \right)
\omega^2 - c_s^4 \eta k_y^2 = 0 ~,
\end{equation}
where
\begin{equation}
\eta \equiv k_P k_S = {\beta_P \beta_S \over \gamma^2 r_0^2} ~.
\end{equation}
Solutions of the Eq. (40) describe a compressible density-spiral
mode and a convective mode that involves perturbations of entropy
and potential vorticity. For weakly stratified discs $(\eta \ll
k^2)$, we find the frequencies are:
\begin{equation}
\bar \omega_{p}^2 = c_s^2 k^2 + 4 \Omega_0^2 ~,
\end{equation}
\begin{equation}
\bar \omega_{c}^2 = - {c_s^4 \eta k_y^2 \over c_s^2 k^2 + 4
\Omega_0^2} ~.
\end{equation}
High frequency solutions ($\bar \omega_{p}^2$) describe the
density-spiral waves and will be referred later as the P-modes. Low
frequency solutions ($\bar \omega_{c}^2$), instead, describe radial
buoyancy mode due to the stratification. In barotropic flows
($\eta=0$) this mode is degenerated into stationary zero frequency
vortical solution. Therefore, we may refer to it as a baroclinic
mode. The mode describes instability when $\eta>0$. In this case the
equilibrium pressure and entropy gradients point in the same
direction. Klahr (2004) has anticipated such result, although worked
in the constant surface density limit ($\beta_\Sigma=0$). The same
behavior has been obtained for axisymmetric perturbations in Johnson
and Gammie (2005). For comparison, in our model baroclinic
perturbations are intrinsically non-axisymmetric. Hence, our result
obtained in the rigidly rotating limit shows that the local
exponential stability of the radial baroclinic mode is defined by
the Schwarzschild-Ledoux criterion:
\begin{equation}
{{\rm d} \bar P \over {\rm d} r} {{\rm d} \bar S \over {\rm d} r}
> 0 ~.
\end{equation}

The dynamics of linear modes can be described using the modal
equations for the eigenfunctions:
\begin{equation}
\left\{ {{\rm d}^2 \over {\rm d} t^2} + \bar \omega_{p,c}^2 \right\}
\Phi_{p,c}(t) = 0 ~,
\end{equation}
where $\Phi_p(t)$ and $\Phi_c(t)$ are the eigenfunctions of the
pressure and convective (baroclinic) modes, respectively. The form
of these functions can be derived from Eqs. (34-41) in the shearless
limit:
\begin{equation}
\Phi_{p,c}(t) = (\bar \omega_{p,c}^2+c_s^2 \eta) p(t) - 2 \Omega_0
W(t) - c_s^2 k_P k_x s(t) ~.
\end{equation}
All physical variables in our system ($p$, $u_x$, $u_y$, $s$) can be
expressed by the two modal eigenfunctions and their first time
derivatives ($\Phi_{p,c}$, $\Phi_{p,c}^\prime$). Hence, we can fully
derive the perturbation field of a specific mode individually by
setting the eigenfunction of the other mode equal to zero.

As we will see later, the Keplerian shear leads to the degeneracy of
the convective buoyancy mode. In this case only the shear modified
density-spiral wave mode eigenfunction can be employed in the
analysis.

\subsection{Perturbations in shear flow: mode coupling}

It is well known that velocity shear introduces non-normality into
the governing equations that significantly affects the dynamics of
different perturbations. In this case we benefit from the shearing
sheet transformation and seek the solutions in the form of the
so-called Kelvin modes. These originate from the vortical solutions
derived in seminal paper by Kelvin (1887). In fact, as it was argued
lately (see e.g., Volponi and Yoshida 2002), the shearing sheet
transformation leads to some sort of generalized modal approach.
Shear modes arising in such description differ from linear modes
with exponential time dependence in many respects. Primarily, phases
of these continuous spectrum shear modes vary in time through
shearing wavenumber; their amplitudes can be time dependent; and
most importantly, they can couple in limited time intervals. On the
other hand, shear modes can be well separated asymptotically, where
analytic WKBJ solution for the each mode can be increasingly
accurate. In the following, we will simply refer to these shearing
sheet solution as ``modes''.

The character of shear flow effects significantly depend on the
value of velocity shear parameter. To estimate the time-scales of
the processes we compare the characteristic frequencies of the
linear modes $|\bar \omega_p|$, $|\bar \omega_c|$ and the velocity
shear $|A|$. In order to speak about the modification of the linear
mode by the velocity shear, the basic frequency of the mode should
be higher than the one set by shear itself: $\omega^2 > A^2$.
Otherwise the modal solution can not be used to calculate
perturbation dynamics, since perturbations will obey the shear
induced variations at shorter timescales.

In quasi-Keplerian differentially rotating discs with weak radial
stratification:
\begin{equation}
\bar \omega_p^2 \gg A^2 ~~~~ {\rm and}  ~~~ \bar \omega_{c}^2 \ll
A^2 ~, {~~~ \rm when ~~~} {\beta_P \beta_S \over \gamma^2} \ll 1 ~.
\end{equation}
In this case the convective mode diverges from its modal behavior
and is strongly affected by the velocity shear: the thermal and
kinematic parts obey shear driven dynamics individually. Therefore,
we tentatively distinguish shear driven vorticity (W) and entropy
(S) modes. On the contrary, the high frequency pressure mode is only
modified by the action of the background shear. Hence, we assume the
above described three mode (S, W and P) formalism as the framework
for our further study.

For the description of the P mode in differential rotation we define
the function:
\begin{equation}
\Psi_p(t) = \omega_p^2(t) p(t) - 2\Omega_0 W(t) - c_s^2 k_P k_x(t)
s(t) ~,
\end{equation}
where
\begin{equation}
\omega_p^2(t) = c_s^2 k^2(t) - 4B \Omega_0 ~.
\end{equation}
This can be considered as the generalization of the $\Phi_P(t)$
eigenfunction for the case of the shear flow, by accounting for the
temporal variation of the radial wavenumber.

In order to analyze the mode coupling in the considered limit, we
rewrite Eqs. (34-39) as follows:
\begin{equation}
\left\{ {{\rm d}^2 \over {\rm d} t^2} + f_p {{\rm d} \over {\rm d}
t} + \omega_p^2 - \Delta \omega_p^2 \right\} \Psi_p = \chi_{pw} W +
\chi_{ps} s ~,
\end{equation}
\begin{equation}
\left\{ {{\rm d} \over {\rm d} t} + f_s \right\} s = \chi_{s p 1}
{{\rm d}  \Psi_p \over {\rm d} t} + \chi_{s p 2} \Psi_p + \chi_{s w}
W ~,
\end{equation}
\begin{equation}
{{\rm d} W \over {\rm d} t}  = \chi_{ws} s ~,
\end{equation}
where $f_p$ and $\Delta \omega_p^2$ describe the shear flow induced
modification to the P-mode
\begin{equation}
f_p =  4 A { k_x k_y \over k^2}  - 2 {(\omega_p^2)^\prime \over
\omega_p^2 } ~,
\end{equation}
\begin{equation}
\Delta \omega_p^2 =  {(\omega_p^2)^{\prime \prime} \over \omega_p^2}
+ f_p {(\omega_p^2)^\prime \over \omega_p^2 } + 8AB{k_y^2 \over k^2}
~,
\end{equation}
parameter $f_s$ describes the modification to the entropy mode
\begin{equation}
f_s = c_s^2 \eta {k_x^2 (\omega_p^2)^\prime \over k^2 \omega_p^4} ~,
\end{equation}
and $\chi$ parameters describe the coupling between the different
modes:
\begin{equation}
\chi_{pw} =  2 \Omega_0 \Delta \omega_p^2(t) + 4A {k_y^2 \over k^2}
\omega_p^2 ~,
\end{equation}
\begin{equation}
\chi_{ps} = c_s^2 k_P k_x \left( \Delta \omega_p^2 + 4B {k_y \over
k_x} {(\omega_p^2)^\prime \over \omega_p^2} - 8AB {k_y^2 \over k^2}
\right) ~,
\end{equation}
\begin{equation}
\chi_{s p 1} = {k_S k_x \over k^2 \omega_p^2 }~,
\end{equation}
\begin{equation}
\chi_{s p 2} = -{k_S k_x \over k^2 \omega_p^2 } \left(
{(\omega_p^2)^\prime \over \omega_p^2} + 2B {k_y \over k_x} \right)
~,
\end{equation}
\begin{equation}
\chi_{s w} = -{2 \Omega k_S k_x \over k^2 \omega_p^2} \left(
{(\omega_p^2)^\prime \over \omega_p^2} + 2B {k_y \over k_x} + {k_y
\omega_p^2 \over 2 \Omega k_x } \right) ~,
\end{equation}
\begin{equation}
\chi_{w s} = -c_s^2 k_P k_y ~.
\end{equation}
Here prime denotes temporal derivative.

Equations (50-52) describe the linear dynamics of modes and their
coupling in the considered three mode model. In this limit, our
interpretation is that the homogeneous parts of the equations
describe the individual dynamics of modes, while the right hand side
terms act as a source terms and describe the mode coupling. This
tentative separation is already fruitful in a qualitative
description of mode coupling.

Dynamics of the density-spiral wave mode in the differential
rotation can be described by the homogeneous part of the Eq. (50).
Homogeneous part of Eq. (51) describes the modifications to the
entropy dynamics. Inhomogeneous parts of the Eqs. (50-52) reveal
coupling terms between the three linear modes that originate due to
the background velocity shear and radial stratification. We analyze
the mode coupling dynamics numerically, but use the coupling $\chi$
coefficients for qualitative description.

The sketch illustration of the mode coupling in the above described
three-mode approximation can be seen in Fig. \ref{coupling}. The
figure reveals a complex picture of the three mode coupling that
originates by the combined action of velocity shear and radial
stratification.

The temporal variation of the coupling coefficients during the swing
of the perturbation SFHs from leading to trailing phases is shown in
Fig. \ref{chi}.  The relative amplitudes of the $\chi_{pw}$ and
$\chi_{ps}$ parameters reveal that potential vorticity is a somewhat
more effective source of P mode perturbations when compared to the
entropy mode. On the other hand, it seems that S mode excitation
sources due to potential vorticity ($\chi_{sw}$) can be stronger
when compared with the P-mode sources ($\chi_{sp1}$, $\chi_{sp2}$).

The effect of the stratification parameters on the mode coupling is
somewhat more apparent. First, we may conclude that the excitation
of the entropy mode, which depends on the parapeters $\chi_{sp1}$,
$\chi_{sp2}$ and $\chi_{sw}$ is generally a stronger process for
higher entropy stratification scales $k_S$ (see Eqs. 58-60). Second,
we see that the generation of the potential vorticity depending on
the $\chi_{ws}$ parameter proceeds more effectively at hight
pressure stratification scale $k_P$. And third, we see profound
asymmetry in the three-mode coupling: P-mode is not coupled with the
W-mode {\it directly}.

A quantitative estimate of the mode excitation parameters can be
done using a numerical analysis. In this case, the amplitudes of the
generated W and S modes can be estimated through the values of
potential vorticity or entropy outside the coupling area. In order
to quantify the second order P mode dynamics we define its modal
energy as follows:
\begin{equation}
E_P(t) \equiv |\Psi_p(t)^\prime|^2 + \omega_p(t)^2 |\Psi_P(t)|^2 ~.
\end{equation}
This quadratic form is a good approximation to the P mode energy in
the areas where it obeys adiabatic dynamics: $k_x(t)/k_y \gg 1$.

The presented qualitative analysis suggests that perturbations of the
density-spiral waves can generate entropy perturbations not only due
to the flow viscosity (not included in our formalism), but also
kinematically, due to the velocity shear induced mode coupling. The
generated entropy perturbations should further excite potential
vorticity through baroclinic coupling. Hence, it seems that in
baroclinic flows, contrary to the barotropic case, P-mode
perturbations are able to generate potential vorticity through a
three-mode coupling mechanism: P $\to$ S $\to$ W. We believe that
traces of the described mode coupling can be also seen in Klahr
2004, where the process has not been fully resolved due to the
numerical filters used to remove higher frequency oscillations.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{Coupling.eps}
\end{center}
\caption{ Mode coupling scheme. In the zero shear limit two second
order modes P-mode and buoyancy mode with eigenfunctions $\Phi_p$
and $\Phi_c$ are uncoupled. In the shear flow, when the
characteristic time of shearing is shorter then the buoyancy mode
temporal variation scale ($A^2 > \bar \omega_c^2$), we use three
mode formalism. In this limit we consider the coupling of the P, W,
and S modes. $\chi$ parameters describe the strength of the coupling
channel. Asymmetry of the mode coupling is revealed in the fact that
compressible oscillations of the pressure mode are not able to
directly generate potential vorticity, but still do so via
interaction with S-mode and farther baroclinic ties with W-mode.}
\label{coupling}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{chi.eps}
\end{center}
\caption{The coupling $\chi$ parameters vs. the ratio of radial to
azimuthal wavenumbers $k_x(t)/k_y$ when latter passes through zero
value during the time interval $\Delta t = 4 \Omega_0^{-1}$. Here
$k_y = H^{-1}$, $k_P = k_S = 0.5 H^{-1}$.  } \label{chi}
\end{figure}

\section{Numerical Results}

In order to study the mode coupling dynamics in more detail we
employ numerical solutions of  Eqs. (34-37). We impose initial
conditions that correspond to the one of the three modes and use a
standard Runge-Kutta scheme for numerical integration (MATLAB ode34
RK implementation). Perturbations corresponding to the individual
modes at the initial point in time are derived in the Appendix A.

\subsection{W-mode: direct coupling with S and P-modes}

In this subsection we consider the dynamics of SFH when only the
perturbations of potential vorticity are imposed initially. As it is
known from previous studies (see Chagelishvili et al. 1997, Bodo et
al. 2005) vorticity perturbations are able to excite acoustic modes
nonadiabatically in the vicinity of the area where $k_x(t)=0$. Here
we observe a similar, but more complex, behavior  of mode coupling.
The W-mode is able to generate P and S-modes simultaneously. Fig.
\ref{SFH_w1} shows the evolution of the W-mode perturbations in a
flow with growing baroclinic perturbations ($\eta>0$). The results
show the excitation of both S and P-mode perturbations due to mode
coupling that occurs in a short period of time in the vicinity of
$t=10$. The following growth of the negative potential vorticity is
due to the baroclinic coupling of entropy and potential vorticity
perturbations.

Fig. \ref{SFH_w2} shows the evolution of potential vorticity SFH in
flows with negative $\eta$. After the mode coupling and generation
of P and S-modes, we observe a decrease of potential vorticity. This
represents the well known fact that stable stratification (positive
Richardson number) can play a role of ``baroclinic viscosity'' on
the vorticity perturbations.

Numerical calculations show that the efficiency of the mode coupling
generally decreases as we increase  the azimuthal wavenumber $k_y$
corresponding to an increase of the density-spiral wave frequency:
lower frequency waves couple more efficiently.

To test the effect of background stratification parameters on the
mode coupling, we calculate the amplitude of the entropy and the
energy of the P-mode perturbations generated in  flows with
different pressure and entropy stratification scales. The amplitudes
are calculated after a $10 \Omega_0^{-1}$ time interval from the
change in sign of the radial wave-number.  In this case, modes are
well isolated and the energy of the P mode can be well defined.

Fig. \ref{surf_w} shows the results of these calculations. It seems
that the mode coupling efficiency is higher with stronger radial
gradients. In particular, numerical results generally verify our
qualitative results that the S-mode generation predominantly depends
on the entropy stratification scale $k_S$. Therefore, P-mode
excitation is stronger for higher values of $\eta$.

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{SFH_w1.eps}
\end{center}
\caption{Evolution of the W-mode SFH in the flow with
$k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing
baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Mode coupling occurs
in the vicinity of $t=10 \Omega_0^{-1}$, where $k_x(t)=0$.
Excitation of the P and S-modes are clearly seen in the panels for
pressure ($P$) and entropy ($S$) perturbations. Perturbations of the
potential vorticity start to grow due to the baroclinic coupling
with entropy perturbations. } \label{SFH_w1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{SFH_w2.eps}
\end{center}
\caption{Evolution of the W-mode SFH in the flow with
$k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with positive
$\eta$: $k_P=-0.2H^{-1}$, $k_S=0.2H^{-1}$. Interestingly, SFH
dynamics shows the decay of potential vorticity after the mode
coupling and excitation of S- and W-modes at $t=10 \Omega_0^{-1}$.
The latter fact is normally anticipated process in the flows that
are baroclinically stable. } \label{SFH_w2}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{surf_w.eps}
\end{center}
\caption{Surface graph of the generated S and P-mode amplitudes at
$ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and
$k_S$. Initial perturbations are normalized to set E(0)=1.
Excitation amplitudes of the entropy perturbations show stronger
dependence of the $k_S$ (left panel), while both entropy and
pressure scales are important (approximately $k_S k_P$ dependence)
for the generation of P-modes (right panel). See electronic edition
of the journal for color images.} \label{surf_w}
\end{figure}

\subsection{S-mode: direct coupling with W and P-modes}

Fig. \ref{SFH_s1} shows the evolution of the S-mode SFH in a flow
with growing baroclinic perturbations.  Here we observe two shear
flow phenomena: mode coupling and transient amplification. Entering
the nonadiabatic area (around $t = 10$) the entropy SFH is able to
generate the P-mode, while undergoing transient amplification
itself. The transient growth of entropy is unsubstantial and the
growth rate decreases with the growth of $k_y$. The W-mode is
instead constantly coupled to entropy perturbations through
baroclinic forces, although higher entropy perturbations at later
times give an  higher rate of growth of potential vorticity. The
total energy of perturbations is however dominated, at the end, by
the P mode.

Fig. \ref{surf_s} shows the dependence of the W and P-mode
generation on the pressure and entropy stratification scales. As
expected from qualitative estimates, P-mode excitation depends
almost solely on the pressure stratification scale $k_P$, while the
generation of potential vorticity generally grows with $\eta$.

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{SFH_s1.eps}
\end{center}
\caption{Evolution of the S-mode SFH in the flow with
$k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing
baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Perturbations of the
potential vorticity are coupled grow from the beginning due to the
baroclinic coupling with entropy perturbations. Excitation of the
P-mode is clearly seen in the panel for pressure ($P$), while the
panel for entropy perturbations ($S$) shows swing amplification in
the nonadiabatic area around $k_x(t)=0$. Change of the amplitude of
the entropy SFH affects the growth factor of potential vorticity
SFH.} \label{SFH_s1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{surf_s.eps}
\end{center}
\caption{Surface graph of the generated W and P-mode amplitudes at
$ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and
$k_S$. Initial perturbations are normalized to set E(0)=1.
Excitation amplitudes of the entropy perturbations show predominant
dependence on the $k_S$ (left panel), while only pressure
stratification scale $k_P$ is important for the generation of
P-modes (right panel). See electronic edition of the journal for
color images.} \label{surf_s}
\end{figure}

\subsection{P-mode: direct coupling with S-mode and indirect
coupling with W-mode}

Fig. \ref{SFH_p1} shows the evolution of an initially imposed P-mode
SFH in a flow with growing baroclinic perturbations.  The {\it
oscillating} behavior of the entropy perturbation for $t < 10$ is
given by the P-mode. This oscillating component has a zero mean
value when averaged over time-scales longer than the wave period.
The existence of the {\it aperiodic} S-mode is instead characterized
by a nonzero mean value.  When the azimuthal wavenumber $k_y(t)$
changes sign at $t = 10$, we can observe the appearance of a nonzero
mean value (marked on the plot by the horizontal dashed line),
indicating that the high frequency oscillations of the P-mode are
able to generate the aperiodic perturbations of the S-mode.  The
aperiodic part of the entropy perturbation is than able to generate
potential vorticity perturbations. However, as we see from Eq. (54)
and Fig. \ref{coupling}, there is no direct coupling between P and
W-modes. Therefore, the P-mode generates the S-mode by shear flow
induced mode conversion, while the W-mode is further generated
because of its baroclinic ties with the entropy SFH. We describe
this situation as the three-mode coupling or in other words,
indirect coupling of the P to the W-mode. Note, that although the S
and W-mode generation is apparent from the dynamics of entropy and
potential vorticity SFH, energetically it plays a minor role as
compared to the compressible energy carried by the P-mode.

Fig \ref{SFH_p2}. shows that P-mode generates potential vorticity
with a positive sign. However, the sign of the generated potential
vorticity depends on the initial phase of the P-mode. Hence, our
numerical results show generation of the W-mode with both positive
and negative signs.

It is interesting also to look at the P-mode dynamics in flows
stable to baroclinic perturbations (see Fig. \ref{SFH_p2}). The
initially imposed P-mode is able to generate the S-mode and
consequently the W-mode, that gives a growth of the potential
vorticity with time. Apart from the intrinsic limitations (the
dependence of the sign of the generated potential vorticity on the
initial phase of the P-mode and the low efficiency of the W-mode
generation), this process demonstrates the fact that potential
vorticity can be actually generated in flows with positive radial
buoyancy ($\eta<0$) and Richardson number.

Fig. \ref{surf_p} shows the dependence of the S and W-mode
generation on the pressure and entropy stratification scales. In
good agreement with qualitative estimates, the S-mode excitation
depends strongly on the entropy stratification scale $k_S$, while
the generation of the potential vorticity generally grows with
$\eta$.

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{SFH_p1.eps}
\end{center}
\caption{Evolution of the P-mode SFH in the flow with
$k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing
baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Mode coupling occurs
in the vicinity of $t=10\Omega_0^{-1}$, where W and S-modes are
excited. The amplitude of the generated aperiodic contribution to
the entropy perturbation is marked by the red doted line. Farther,
this component leads to the baroclinic production of potential
vorticity with negative sign.} \label{SFH_p1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{SFH_p2.eps}
\end{center}
\caption{Same as in previous figure but for $k_P=-0.2H^{-1}$ and $
k_S=0.2H^{-1}$. Perturbations are stable to baroclinic forces.
However, production of the potential vorticity with positive sign is
still observed.} \label{SFH_p2}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=84mm]{surf_p.eps}
\end{center}
\caption{Surface graph of the generated S and W-mode amplitudes at
$ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and
$k_S$. Initial perturbations are normalized to set E(0)=1.
Excitation amplitudes of the entropy perturbations mainly depend on
the $k_S$ (left panel), while both pressure and entropy
stratification scales are important for the generation of W-mode
perturbations (right panel). See electronic edition of the journal
for color images.} \label{surf_p}
\end{figure}

\section{Conclusion and Discussion}

We have studied the dynamics of linear perturbations in a 2D,
radially stratified, compressible, differentially rotating flow with
different radial density, pressure and entropy gradients. We
employed global radial scaling of linear perturbations and removed
the algebraic modulation due to the background stratification. We
derived a local dispersion equation for nonaxisymmetric
perturbations and the corresponding eigenfunctions in the zero shear
limit. We show that the local  stability of baroclinic perturbations
in the barotropic equilibrium state is defined by the
Schwarzschild-Ledoux criterion.

We study the shear flow induced linear coupling and the related
possibility of the energy transfer between the different modes of
perturbations using qualitative and a more detailed numerical
analysis. We employ a three-mode formalism and describe the behavior
of S W and P-modes under the action of the baroclinic and velocity
shear forces in local approximation.

We find that the system exhibits an asymmetric coupling pattern with
five energy exchange channels between three different modes. The
W-mode is coupled to S and P-modes: perturbations of the potential
vorticity are able to excite entropy and compressible modes. The
amplitude of the generated S-mode grows with the increase of entropy
stratification scale of the background ($k_S$) while the amplitude
of the generated P-mode perturbations grows with the increase of
background baroclinic index ($\eta$). The S-mode is coupled to the W
and P-modes: the amplitude of the generated P-mode perturbations
grows with increase of the background pressure stratification scale
($k_P$), while the amplitude of the W-mode grows with the increase
of baroclinic index. The P-mode is coupled to the S-mode: the
amplitude of generated entropy perturbations grows with the increase
of the background entropy stratification scale. On the other hand,
there is no direct energy exchange channel from P to W mode and,
therefore, no direct conversion is possible. Our results, however,
show that the P-mode is still able to generate W-mode through
indirect three mode P-S-W coupling scheme. This linear inviscid
mechanism indicates that compressible perturbations are able to
generate potential vorticity via aperiodic entropy perturbations.

The dynamics of radially stratified discs have been already studied
by both, linear shearing sheet formalism and direct numerical
simulations. However, previous studies focus on the baroclinic
stability and vortex production by entropy perturbations, neglecting
the coupling with higher frequency density waves.

The most vivid signature of  density wave excitation in radially
stratified disc flows can be seen in Klahr (2004). The numerical
results presented on the linear dynamics of perturbation SFH show
high frequency oscillations after the radial wavenumber changes
sign. However, focusing on the energy dynamics, the author filters
out high frequency oscillations from the analysis.

The purpose of numerical simulations by Johnson and Gammie (2006)
was the investigation of the velocity shear effects on the radial
convective stability and the possibility of the development of
baroclinic instability. Therefore, no significant amount of
compressible perturbations is present initially, and it is hard to
judge if high frequency oscillations appear later in simulations.
Petersen et al. (2007a), (2007b) employed the anelastic
approximation that does not resolve the coupling of potential
vorticity and entropy with density waves. Moreover, if produced,
high frequency density waves soon develop into  spiral shocks (see
e.g., Bodo et al 2007). The anelastic gas approximation does
intentionally neglect this complication and simplifies the
description down to low frequency dynamics.

Numerical simulations of hydrodynamic turbulence in unstratified
disc flows showed that the dominant part of  turbulent energy is
accumulated into the high frequency compressional waves (see, e.g.,
Shen et al. 2006). On the other hand, it is vortices that are
thought to play a key role in hydrodynamic turbulence in accretion
discs, as well as planet formation in protoplanetary disc dynamics.
Therefore, any link and possible energy exchange between high
frequency compressible oscillations and aperiodic vortices can be an
important factor in the above described astrophysical situations.

Based on the present findings we speculate that density waves can
participate in the process of the development of regular vortical
structures in discs with negative radial entropy gradients.
Numerical simulations have shown that thermal (entropy)
perturbations can generate vortices in baroclinic disc flows (see
e.g., Petersen et al. 2007a, 2007b). Hence, vortex development
through this mechanism depends on the existence of initial regular
entropy perturbations, i.e., thermal plumes, in differentially
rotating baroclinic disc flows.

It seems that compressional waves with linear amplitudes can heat
the flow through two different channels: viscous dissipation and
shear flow induced mode conversion. However, there is a strict
difference between the entropy production by the kinematic shear
mechanism and viscous dissipation. In the latter case, compressional
waves first need to be tightly stretched down to the dissipation
length-scales by the background differential rotation to be  subject
of viscous dumping. As a result, the entropy produced by viscous
dissipation of compressional waves takes a shape of narrow stretched
lines. This thermal perturbations can baroclinically produce
potential vorticity of similar configuration. However, this is
clearly not an optimal form of potential vorticity that can lead to
the development of the long-lived vortical structures. On the
contrary, entropy perturbations produced through the mode conversion
channel can have a form of a localized thermal plumes. These can be
very similar to those used in numerical simulations by Petersen et
al. (2007a), (2007b). In this case compressional waves can
eventually lead to the development of  persistent vortical
structures of different polarity. Hence, high frequency oscillations
of the P mode can participate in the generation of anticyclonic
vortices that further accelerate dust trapping and planetesimal
formation in protoplanetary discs with equilibrium entropy
decreasing radially outwards.

Using the local linear approximation we have shown the possibility
of the potential vorticity generation in flows with both, positive
and negative radial entropy gradients (Richardson numbers). In fact,
the standard alpha description of the accretion discs implies
\emph{positive} radial stratification of entropy and hence, weak
baroclinic decay of existing vortices. In this case there will be a
competition between the ``baroclinic viscosity'' and potential
vorticity generation due to  mode conversion. Hence, it is not
strictly overruled that a significant amount of compressional
perturbations can lead to the development of anticyclonic vortices
even in flows with positive entropy gradients. In this case, radial
stratification opens an additional degree of freedom for velocity
shear induced mode conversion to operate. Although, the viability of
this scenario needs further investigation.

This paper presents the results obtained within the linear shearing
sheet approximation. At nonlinear amplitudes, the P mode leads to
the development of shock waves. These shocks induce local heating in
the flow. Therefore, a realistic picture of entropy production and
vortex development in radially stratified discs with significant
amount of compressible perturbations needs to be analyzed by direct
numerical simulations.

\section*{Acknowledgments}

A.G.T. was supported by GNSF/PRES-07/153. A.G.T. would like to
acknowledge the hospitality of Osservatorio Astronomico di Torino.
This work is supported in part by ISTC grant G-1217.


\section{Introduction}

Analytic approximations and perturbation methods
are of common use in different branches of general relativity theory.
The most common methods are the post-Newtonian approximations,
able to deal with the gravitational field of any system
in the non-relativistic limit,
the post-Minkowskian approximations,
which are appropriate for relativistic systems
in the weak gravitational field regime,
and the perturbation formalisms, which expand
(at linear order in general)
around some exact solution of the Einstein field equations.
The B3 session {\em Analytic approximations,
perturbation methods, and their applications}
of GR18 conference was dominated by the issues
more or less directly related to the problem of detecting gravitational waves
by currently operating or planned to be built in the near future detectors.
One of the most important targets for gravitational-wave detectors
are coalescing binary systems made of compact objects of different kinds.
These sources produce ``chirps'' of gravitational radiation
whose amplitude and frequency are increasing in time.
For successful detection of the chirp signals
and extraction of their astrophysically important parameters
it is crucial to theoretically predict
gravitational waveforms with sufficient accuracy.

One of the important sources for the future space-based LISA detector
are extreme-mass-ratio binaries
consisting of a small compact body (of stellar mass)
orbiting around a supermassive black hole.
The need for accurate modelling of the orbital dynamics of such binaries
motivates some recent work on the problem of calculating
the gravitational self-force experienced by a point particle moving
in the background space-time of a more massive body.
Several talks in the session reported the progress
on different aspects of the gravitational self-force computations.
These were talks by {\em S~Detweiler}, {\em M~Favata},
{\em J~L~Friedman}, {\em W~Hikida}, {\em N~Sago}, and {\em B~F~Whiting}.

For comparable-mass binaries,
computations based on the post-Newtonian approximation of general relativity
are useful for constructing gravitational-wave templates
for data-analysis purposes.
The post-Newtonian approximation describes
with great accuracy the inspiral phase of these systems.
The analytic post-Newtonian results are currently matched
to recent numerical calculations of the merger and ring-down phases
of black hole binaries \cite{Buonanno&Cook&Pretorius2007,Boyle&al2007}.
In the contributed talks
(by {\em L~Blanchet}, {\em B~R~Iyer}, and {\em M~Vas\'uth})
recent results on incorporating the spin-orbit effects
(also within the accuracy beyond the leading-order effects)
as well as the generalization to eccentric orbits
(most of the explicit analytic results concern circular orbits)
were presented.

Recently a new approach to the perturbative solution
of the problem of motion and radiation in general relativity was developed.
This is the approach pioneered
by Goldberger and Rothstein \cite{Goldberger&Rothstein2006}
in which effective field theory methods
were used to describe the dynamics
of nonrelativistic extended objects coupled to gravity.
{\em B~Kol} discussed this approach
as well as some of its applications.
{\em A~Tartaglia} presented a new semi-analytic method
for computing the emission of gravitational waves
by a compact object moving
in the background of a more massive body.

Several other topics were tackled in the contributed talks.
{\em C~L\"ammerzahl} discussed the influence of the cosmic expansion
on the physics in gravitationally bound systems.
{\em D~Singh} presented an analytic perturbation approach
for classical spinning particle dynamics,
based on the Mathisson-Papapetrou-Dixon equations of motion.
{\em H~Sotani} studied gravitational radiation
from collapsing magnetized dust.
{\em A~S~Kubeka} remarked on the computation
of the Ricci tensor for non-stationary axisymmetric space-times.

In the rest of this article
all talks contributed to the B3 session are sketched in more detail
(in the order in which they were presented at the conference).

\section{Contributed talks}

\subsection{{\em Self-force analysis in extreme-mass-ratio inspiral}
by S~Detweiler and I~Vega (reported by S~Detweiler)}

The motion of a small object of mass $m$ orbiting a supermassive black
hole of mass $M$ deviates slightly from a geodesic and has an acceleration
that scales as the ratio $m/M$ of the masses. This acceleration includes
the dissipative effects of radiation reaction and is said to result from
the gravitational self-force acting on $m$ \cite{Poisson2004}. As an
alternative, the effects of the self-force may be described as geodesic
motion in an appropriately regularized metric of the perturbed spacetime
\cite{Detweiler2005}.
The LISA effort requires accurate gravitational wave templates for data analysis.
For extreme-mass-ratio inspirals the templates should include
both the dissipative and conservative effects of the self-force.

The talk described a novel, efficient method
for simultaneously calculating both the gravitational self-force
as well as its effect on the gravitational waveform.
The Authors replaced the usual singular point source
with a distributed, abstract, analytically determined source.
The resulting perturbation in the field from this special distributed source
is guaranteed to be differentiable at the location of the particle
and to provide the appropriate self-force effect on the motion of the particle.
At the same time, the field from the distributed source
is identically equal to the actual perturbed field in the wave zone.
Thus, this abstract field simultaneously provides
both the self-force acting on a point source
and also the effect of the self-force on the waveform of the radiation.

\subsection{{\em The adiabatic approximation
and three-body effects in extreme-mass-ratio inspirals} by M~Favata}

Extreme-mass-ratio inspirals (EMRIs) are an important class of LISA sources
consisting of a compact object inspiralling into a supermassive black hole.
The detection of these sources and the precision measurement
of their parameters relies on the accurate modeling of their orbital dynamics.
A precise description of the binary's orbit requires
an evaluation of the compact object's self-force.
The adiabatic approximation (more appropriately
referred to as the {\em radiative approximation}
\cite{Pound&Poisson2007a,Pound&Poisson2007b}),
consists of computing the time-averaged rates of change
of the three conserved quantities of geodesic motion.
Its use greatly simplifies the computation of the orbital evolution.
However, the adiabatic approximation ignores corrections
to the conservative dynamics proportional
to the mass $m$ of the compact object.
These `post-adiabatic' corrections
will affect the binary's positional orbital elements,
for example, by causing $O(m)$ corrections to the pericenter precession rate.

Using a toy model
of an electric charge orbiting a central mass
and perturbed by the electromagnetic self-force,
Pound, Poisson, and Nickel \cite{Pound&Poisson&Nickel2005}
have called into question
the accuracy of the adiabatic approximation,
especially for eccentric orbits.
In order to estimate
the size of the post-adiabatic phase errors
in the gravitational case,
the Author presented an analytical computation,
accurate to second-post-Newtonian order,
of the small-eccentricity corrections
to the gravitational-wave phase.
These post-Newtonian, eccentricity corrections
to the phase can be significant
not only for EMRIs but for other binary sources as well.
Using this phase expansion it was found
that the post-adiabatic phase errors
are sufficiently small that waveforms
based on the adiabatic approximation
can be used for EMRI detection,
but not for precise parameter measurements.

The Author also discussed
the effect of a third mass orbiting the EMRI.
The analysis models the system as a hierarchical triple
using the equations of motion
of Blaes, Lee, and Socrates \cite{Blaes&Lee&Socrates2002}.
To have a measurable effect on the EMRI's waveform,
the distant mass must be sufficiently close to the compact object
that both the inner and outer binaries would be detected as EMRIs.
Such ``double-EMRI'' systems are rare.

\subsection{{\em Extreme-mass-ratio binary inspiral in a radiation gauge}
by~J~L~Friedman, T~S~Keidl, Dong-Hoon~Kim, E~Messaritaki and A~G~Wiseman
(reported by~J~L~Friedman)}

Gravitational waves from the inspiral of a stellar-size black hole of mass $m$
to a supermassive black hole of mass $M$ can be accurately approximated
by a point particle moving in a Kerr background.
To find the particle's orbit to first order in the mass ratio $m/M$,
one must compute the self-force. The computation requires a renormalization,
but the well-known MiSaTaQuWa prescription
\cite{Mino&Sasaki&Tanaka1997,Quinn&Wald1997} involves a harmonic gauge,
a gauge that is not well suited to perturbations of the Kerr geometry.
In a harmonic gauge, one solves ten coupled PDEs,
instead of the single decoupled Teukolsky equation
for the gauge invariant components ($\psi_0$ or $\psi_4$) of the perturbed Weyl tensor.
In the talk progress was reported
in finding the renormalized self-force from $\psi_0$ or $\psi_4$.
Following earlier work by Chrzanowski and by Cohen and Kegeles,
a radiation gauge was adopted
to reconstruct the perturbed metric from the perturbed Weyl tensor.
The Weyl tensor component is renormalized by subtracting a singular part
obtained using a recent Detweiler-Whiting version \cite{Detweiler&Whiting2003}
of the singular part of the perturbed metric
as a local solution to the perturbed Einstein equations.
The Authors' method relies on the fact that the corresponding renormalized $\psi_0$
is a {\em sourcefree} solution to the Teukolsky equation.
One can then reconstruct a nonsingular renormalized metric in a radiation gauge,
a gauge that exists only for vacuum perturbations.
More details can be found in Ref.\ \cite{Keidl&al2007}.

\subsection{{\em Adiabatic evolution of three `constants' of motion in Kerr spacetime}
by~W~Hikida, K~Ganz, H~Nakano, N~Sago and T~Tanaka (reported by~W~Hikida)}

General orbits of a particle of small mass around a Kerr black hole
are characterized by three parameters:
the energy, the angular momentum, and the Carter constant.
For energy and angular momentum, one can evaluate their change rates
from the fluxes of the energy and the angular momentum at infinity
and on the event horizon according to the balance argument.
On the other hand, for the Carter constant,
one can not use the balance argument
because the conserved current associated with it is not known.
Recently Mino proposed a new method
of evaluating the averaged change rate of the Carter constant
by using the radiative field.
The Authors developed a simplified scheme for practical evaluation
of the evolution of the Carter constant based on the Mino's proposal.
In the talk this scheme was described in some detail,
and derivation of explicit analytic formulae for the change rates
of the energy, the angular momentum, and the Carter constant was presented.
Also some numerical results for large eccentric orbits were shown.
For more details see Ref.\ \cite{Ganz&al2007}.

\subsection{{\em Gravitational self-force
on a particle orbiting a Schwarzschild black hole}
by~L~Barack and N~Sago (reported by~N~Sago)}

In the talk the calculation of the gravitational self-force
acting on a pointlike particle moving around a Schwarzschild black hole
was presented. The calculation was done in the Lorenz gauge.
First, the Lorenz-gauge metric perturbation equations
were solved directly using numerical evolution in the time domain.
Then the back-reaction force
from each of the multipole modes of the perturbation was computed.
Finally, the {\em mode sum} scheme was applied
to obtain the physical self-force.
The temporal component of the self-force
describes the rate of the loss of orbital energy.
As a check of their scheme,
the Authors compared their result for this component
with the total flux of gravitational-wave energy
radiated to infinity and through the event horizon.
The radial component of the self-force was also calculated.
From their result for the self-force,
the Authors computed the correction to the orbital frequency
due to the gravitational self-force,
taking into account of both the dissipative and the conservative effects.
More details can be found in Ref.\ \cite{Barack&Sago2007}.

\subsection{{\em Mobile quadrupole as a semianalytic method for gravitational-wave emission}
by~A~Tartaglia, M~F~De~Laurentis, A~Nagar and N~Radicella (reported by~A~Tartaglia)}

The quadrupole formula is the simplest approximation
for studying the gravitational-wave emission from a binary system.
The formula gives its best performance
for quasi-circular and quasi-stationary motion of the emitters.
Whenever the motion is far from the quasi-circular approximation,
other semi-analytic methods
or numerical calculations of growing complexity must be used.
In the talk a situation was studied
where the gravitational wave is emitted by a concentrated object
accelerating in the background field of a central mass.
Provided one knows the background metric,
the gravitational wave represents a first order perturbation on it,
so that the space-time trajectory of each object of the pair is almost geodesic.
Once the equations of a geodesic are written,
the motion can be thought of as an instantaneous rotation
around an (instantaneously at rest) curvature centre for the space trajectory.
In this condition the quadrupole formula is easily applicable
at each different place along the geodesic,
after calculating the curvature and the equivalent angular velocity
as the ratio between the three-speed and the curvature radius.
Everything is reduced to a problem of ordinary geometry.
The energy emission rate and the waveforms obtained by this way
must simply be converted from the local time
to the time of a far away inertial observer.
The approach was applied
to the capture of a mass by a Kerr black hole.
The method is much lighter
(from the computational point of view)
than numerical relativity, giving comparable results.
For the research results relevant to the presented approach see Refs.\
\cite{Dymnikova1977,Dymnikova&Popov1980,Schnittma&Bertschinger2004}.

\subsection{{\em The non-radiated multipoles in the perturbed Kerr spacetime}
by~L~R~Price and B~F~Whiting (reported by~B~F~Whiting)}

For the self-force problem in general relativity,
it has been shown that the perturbed metric
produced by a finite-mass test, point-particle,
has a singular part which exerts no influence on the particle,
while the self-force which the particle experiences
arises entirely due to a metric perturbation
which is smooth at the location of the particle \cite{Detweiler&Whiting2003}.
However, metric reconstruction from the perturbed Weyl tensor
is unable to yield perturbations
for the non-radiated multipoles in Petrov type II spacetimes,
such as that surrounding the Kerr black hole \cite{Whiting&Price2005}.
In the talk a new form of the perturbed Einstein equations,
developed by the Authors on the base of the Newman-Penrose formalism,
was presented. With its assistance,
progress towards filling the low multipole gap,
which will contribute to the calculation
of regularization parameters for the self-force problem, was discussed.

\subsection{{\em Higher-order spin effects in the radiation field of compact binaries}
by~L~Blanchet, A~Buonanno and G~Faye (reported by~L~Blanchet)}

In the talk the investigation, motivated by the search for
gravitational waves emitted by binary black holes, of the gravitational
radiation field of compact objects with spins was discussed. The
approach is based on the multipolar post-Newtonian wave generation
formalism and on the formalism of point particles with spin
(Papapetrou-Dixon-Bailey-Israel). The Authors computed: (i) the
spin-orbit coupling effect in the binary's equations of motion one
post-Newtonian (PN) order beyond the dominant effect (confirming a
previous result by Tagoshi et al.\ \cite{Tagoshi&Ohashi&Owen2001}),
(ii) the spin-orbit coupling effects
in the binary's mass and current quadrupole moments at the same order,
(iii) the spin-orbit contributions in the gravitational-wave energy flux,
and (iv) the secular evolution of the binary's orbital phase up to 2.5PN order
(for maximally rotating black holes).
Previous results on the spin-orbit effect at the lowest order
were computed in Refs.\ \cite{Kidder&Will&Wiseman1993,Kidder1995}.
Crucial ingredients for obtaining the next-order 2.5PN contribution in the
orbital phase are the binary's energy and the spin precession equations.
These results provide more accurate gravitational-wave templates to be
used in the data analysis of rapidly rotating Kerr-type black-hole
binaries with the ground-based interfrometric detectors and the
space-based detector LISA.
Details of the presented results were published in Refs.\
\cite{Faye&Blanchet&Buonanno2006,Blanchet&Buonanno&Faye2006}.

\subsection{{\em The 3PN gravitational wave luminosity
from inspiralling compact binaries in~eccentric orbits}
by~K~G~Arun, L~Blanchet, B~R~Iyer and M~S~S~Qusailah (reported by~B~R~Iyer)}

Some details of the computation
of the complete gravitational-wave luminosity of inspiralling compact binaries
on quasi-elliptical orbits up to the third post-Newtonian (3PN) order
using multipolar post-Minkowskian formalism were presented.
There are two types of contributions
to the gravitational-wave luminosity at 3PN order:
the instantaneous type terms,
which depend on the dynamics of the binary only at the retarded instant,
and the hereditary terms,
which are sensitive to dynamics of the system in the entire past.
The new inputs for the calculation of the 3PN instantaneous terms
include the mass octupole and current quadrupole at 2PN for general orbits
and the 3PN accurate mass quadrupole.
Using the 3PN quasi-Keplerian representation of elliptical orbits
obtained recently \cite{Memmesheimer&Gopakumar&Schafer2004},
the flux is averaged over the binary's orbit.
The hereditary terms have a `tail', `tail of tail' and `tail-squared' contributions
which are computed using a semi-analytic procedure
extending the earlier work of Blanchet and Sch\"afer at 1.5PN
\cite{Blanchet&Schafer1993}.
This semi-analytic extension uses
the 1PN quasi-Keplerian parametrisation of the binary
and exploits the doubly periodic nature of the orbital motion.
The final 3PN accurate energy flux averaged over the binary's orbit was presented
in the modified harmonic (which contains no logarithmic terms) and the ADM coordinates.
Also a gauge invariant expression of the flux was provided
in terms of the orbital frequency and the periastron precession constant.
The results are consistent with those obtained by perturbation theory
in the test particle limit to order $e_t^2$
(where $e_t$ is the so-called time eccentricity)
and the 3PN circular orbit results.
These results form the starting input
for the construction of templates for inspiralling binaries in quasi-eccentric orbits,
an astrophysically possible class of sources
both for the ground-based and the space-based gravitational-wave interferometers.

\subsection{{\em On the influence of the cosmic expansion
on the physics in gravitationally bound systems}
by~C~L\"ammerzahl and H~Dittus (reported by~C~L\"ammerzahl)}

It is an old question whether the cosmological expansion influences
the dynamics of gravitationally bound systems
\cite{Lammerzahl&Preuss&Dittus2007}.
Though sometimes it has been claimed
that the expansion will tie apart gravitationally bound systems,
the majority of papers covering this issue derive no measurable influence.
In the talk some additional arguments for the latter were given.
It was shown that (i) the gravitational field created by an isolated body
will feel only a tiny influence,
(ii) the planetary orbits also are practically inert against the expansion,
and (iii) Doppler tracking of satellites in deep space
is also only marginally influenced by the cosmic expansion.

\subsection{{\em Spin evolution in binary systems}
by~M~Vas\'uth and J~Maj\'ar (reported~by~M~Vas\'uth)}

Gravitational waves emitted by compact binary systems are characterized
by different parameters of the binary. Among them the effects of rotation
of the orbiting bodies appear at 1.5 post-Newtonian (PN) order both in the
dynamical description and the wave generation problem. In the talk the
evolution of the individual spins of the bodies was discussed for compact
binaries in circular and general eccentric orbits.
For a 2PN description the spin precession equations
were analyzed up to 0.5PN order.
To the lowest order
the angles between the total angular momentum and spin vectors
are constant and spin-spin interaction
causes additional harmonic dependence.
The true anomaly parameterization
proved to be useful in the description of eccentric orbits.
In the circular and general cases
linear and harmonic dependence of the angles
describing the orientation of spins were found.

\subsection{{\em Matched asymptotic expansion
as a classical effective field theory} by~B~Kol}

In the talk it was explained
how the method of {\em classical effective field theory}
borrowed from {\em quantum field theory}
by Goldberger and Rothstein \cite{Goldberger&Rothstein2006}
in the context of the motion of a compact object
within a background whose typical length scale is much larger,
is equivalent to {\em matched asymptotic expansion},
and moreover it offers additional insight.
Feynman diagrams, divergences, (dimensional) regularization, counter-terms
and the Feynman gauge appeared.
The ideas were demonstrated by the case of caged black holes
(black holes within a compact dimension).
Within this method the source
is replaced by a "black box" effective action.
Another application of these ideas is to the inspiral problem of a binary system.
The Author presented a computation utilizing high energy physics methods
of the radiation reaction force for the case of a scalar field.

\subsection{{\em An analytic perturbation approach
for classical spinning particle dynamics} by~D~Singh}

The Author presented a perturbation method
to analytically describe the dynamics of a classical spinning particle,
based on the Mathisson-Papapetrou-Dixon equations of motion.
By a power series expansion with respect to the particle's spin magnitude,
it was demonstrated how to obtain an analytic representation
of the particle's kinematic and dynamical degrees of freedom
that is formally applicable to infinite order in the expansion parameter.
Within this formalism, it is possible to identify
a classical analogue of radiative corrections
to the particle's mass and spin due to spin-gravity interaction.
The robustness of this approach
was demonstrated by showing how to explicitly compute
the first-order momentum and spin tensor components
for arbitrary particle motion in a general space-time background.
Potentially interesting applications
based on this perturbation approach were also considered.
For more details see Ref.\ \cite{Singh2007}.

\subsection{{\em Gravitational radiation from collapsing magnetized dust}
by~H~Sotani, S~Yoshida and K~D~Kokkotas (reported by~H~Sotani)}

The Authors studied the influence of magnetic fields
on the axial gravitational waves emitted during
the collapse of a homogeneous dust sphere.
It was found that while the energy emitted
depends weakly on the initial matter perturbations
it has strong dependence on the strength and the distribution
of the magnetic field perturbations.
The gravitational wave output of such a collapse
can be up to an order of magnitude larger or smaller
calling for detailed numerical 3D studies of collapsing magnetized configurations.
More details are given in Ref.\ \cite{Sotani&al2007}.

\subsection{{\em Gravitational waveforms for compact binaries}
by~M~Vas\'uth and J~Maj\'ar (reported by~M~Vas\'uth)}

Among the promising sources of gravitational radiation
are binary systems of compact stars.
The detectable signal is characterized by different parameters of the system,
e.g., rotation of the bodies and the eccentricity of the orbit.
The Authors presented a method
to evaluate the gravitational wave polarization states
for inspiralling compact binaries
and considered eccentric orbits
and the spin-orbit contribution
in the case of two spinning objects up to 1.5 post-Newtonian order.
In the circular orbit limit the presented results
are in agreement with existing results.
For more details see Ref.\ \cite{Vasuth&Majar2007}.

\subsection{{\em On the Ricci tensor for non-stationary axisymmetric space-times}
by~A~S~Kubeka}

The results on Ricci tensor for non-stationary axisymmetric space-times
determined by Chandrasekhar \cite{Chandrasekhar1975}
have been found to be incorrect both in the linear and non-linear regimes.
However, the incorrectness of the Ricci tensor
does not affect the well-known results on linear perturbations of a Schwarzschild
black hole solution.

\section*{Acknowledgments}

MV contribution was supported by OTKA grant No.\ F049429.
The work of SD, LP, IV and BW was supported by NSF Grant No.\ PHY-0555484.

\section*{References}