Physics Letters A 374 (2010) 4872–4875
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Physics Letters A
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Geodesic acoustic modes and zonal flows in toroidally rotating tokamak plasmas
V.P. Lakhin
a
, V.I. Ilgisonis
a
, A.I. Smolyakov
a
,
b
,∗
a
RRC “Kurchatov Institute”, 123182, Moscow, Russian Federation
b
University of Saskatchewan, 116 Science Place, Saskatoon, S7N 5E2, Canada
article info abstract
Article history:
Received 25 July 2010
Accepted 6 October 2010
Available online 12 October 2010
Communicated by F. Porcelli
The effect of equilibrium toroidal rotation on the rotational eigen-modes in large aspect ratio tokamak
is studied. The case of equilibrium with uniform plasma density on magnetic surfaces is considered.
It is shown that the toroidal rotation results in a frequency up-shift of ordinary Geodesic Acoustic
Modes. A new unstable low frequency branch of the continuum modes is found. This mode appears
as a consequence of the non-uniform plasma pressure created by the centrifugal force on the magnetic
surfaces. This mode represents a linear eigen-mode counterpart of Zonal Flow modes. It is shown that
the growth rate of such a mode increases with the increase of the angular velocity of toroidal rotation.
©
2010 Elsevier B.V. All rights reserved.
Fluctuations of m
=
n
=
0 component of the electrostatic poten-
tial in a tokamak lead to the perturbations of poloidal and toroidal
plasma rotation. Such rotational perturbations spontaneously oc-
cur in the form of zonal flows (ZFs) and geodesic acoustic modes
(GAMs) and have been observed experimentally in many tokamaks
[1–3]. Such modes are considered to play an important role in reg-
ulating the anomalous, turbulent transport in modern tokamaks
(see [4] and references therein) either due to energy sink from
turbulence to rotational modes and/or due to the shearing effect of
the rotation on the turbulent transport. In a static toroidal plasmas,
geodesic acoustic modes represent the linear eigen-modes with the
frequency of the order of
ω
∼
c
s
/
R
0
, where c
s
is the sound velocity
and R
0
is the major radius of the magnetic axis. These modes are
linearly stable in the framework of ideal MHD. On the contrary,
the low frequency zonal flows have no linear counterpart and are
driven non-linearly due to energy transfer from turbulence.
Plasmas in present day tokamak experiments often have finite
rotation velocity. In experiments with an unbalanced tangential
neutral beam injection the equilibrium toroidal velocities may ap-
proach the sound velocities. Therefore, it is interesting and impor-
tant to study the influence of toroidal plasma rotation on ZFs and
GAMs. It is shown in this Letter that toroidal rotation, in addition
to the modification of the standard GAM modes existing in the
static case of non-rotating plasma, leads to the appearance of a
new low frequency mode which can be classified as a linear eigen-
mode of the zonal flow.
In the previous studies the effect of toroidal rotation on ZFs
and GAMs has been investigated within an MHD description of
*
Corresponding author at: University of Saskatchewan, 116 Science Place, Saska-
toon, S7N 5E2, Canada.
E-mail address: andrei.smolyakov@usask.ca (A.I. Smolyakov).
plasma [5,6]. The new GAM branch induced by toroidal rotation
has been found for a particular equilibrium in which the plasma
temperature is assumed constant on the magnetic flux surfaces.
The frequency of the new GAM is always lower than the frequency
of ordinary GAM [7] and tends to zero when the angular velocity
of toroidal rotation tends to zero. The effect of toroidal rotation on
the Alfvén continua, which are the GAM modification to the case
of finite poloidal and toroidal wave numbers, has been also studied
in [8,9] in the frame of MHD approach. Different plasma equilibria
have been considered. In Ref. [8] the equilibrium with isothermal
magnetic surfaces has been taken, and in Ref. [9] – the equilibrium
with uniform plasma density on the magnetic surfaces. In the lat-
ter Letter the existence of unstable Alfvén continua has been noted.
Also, the effect of equilibrium mass flow parallel to the magnetic
field on the Alfvén continua has been studied in Ref. [10] by the
kinetic approach.
Approximation of the constant temperature on the magnetic
flux surface is based on the assumption of high thermal conduc-
tivity along magnetic field lines. However, the parallel heat con-
ductivity can be not too effective in low aspect ratio tokamaks
with large fraction of trapped particles or in tokamaks with inten-
sive auxiliary heating, where the power absorption profiles can be
rather complicated (see, e.g., [11,12]). As a result, the other types
of equilibria of flowing plasmas with temperature, which is not
constant along the magnetic flux surfaces, are also possible. In par-
ticular, the equilibria, in which either the plasma mass density,
ρ
0
,
or the plasma entropy, p
0
/
ρ
Γ
0
(p
0
is the equilibrium pressure,
Γ
is the adiabatic index), are constant on the magnetic flux surface,
are also known (see, e.g., [13] and references therein). In this Let-
ter we will consider the properties of GAMs and ZFs in the case of
equilibrium, in which the mass density is uniform on the flux sur-
faces and will show that they are qualitatively different from the
case of equilibrium with isothermal flux surfaces. Another equilib-
0375-9601/$ – see front matter
©
2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2010.10.012