Linear n=1 TAE structure and stability
Questions can be sent to N.N. Gorelenkov: email@example.com.
We fix simple case with analytical plasma profiles
n=1, R0=3m (major
radius of the last closed surface), a=1m,
circular, zero beta tokamak
where psi is a
poloidal flux varying from 0 at axis to 1
at the edge.
Plasma density profile is constant. Equilibrium profiles are given in this file. In
this file all the variable notations are standard for Grad-Shafranov
equation. F is normalized
in such way that B=1 at the geometrical center so that F=B*R/R0. Fp is
its prime with the
regard of the poloidal flux, which is also presented.
Energetic ion parameters:
EP beta profile ~ exp(-psi/0.37)
Distribution function is taken either Maxwellian or slowing down. In
the latter case we have
f=1/(v**3+v_crit**3) * exp(-<psi>/0.37)
where velocities are normalized to the injection velocity v_h and
Note that <Psi> in this
expression is considered averaged over the particle guiding center
orbit. In this case it is a function of particle constants of motion as
it should be. One can show that for alpha particles the distribution
function is a function of <Psi>. We used <Psi>
computing the critical velocity and the scattering frequency. One
simple and relatively accurate workaround of how to go from <Psi>
to particle integrals of motion is following.
If we define P_phi = e Psi
/mc - v|| R, we find that for trapped particle with the accuracy up to
ratio of the orbit width to the minor radius times epsilon^2 we have
P_phi = e <Psi> /mc. For passing particles with the same accuracy
we find P_phi = e*<Psi> /mc - sigma*v*sqrt( 1 -mu*B0/E )R0, where
sigma is the sign of v||, B0 is the magnetic field at the center of the
magnetic surface, and R0 is the major radius of the surface, mu is the
These parameters can correspond to deuterium plasma and EP mass and
m_i = m_f = 2
z_i = z_f = 1
EEP0 = 173keV
B0 = 1T
ne = 4.142 1013 cm-3
Te = 3.14 keV (used for v_crit value)
NOVA results for the TAE frequency are omega = 0.654 vA/
q1R0 where q1=q(a)=2.1.
The structure of this n=1 TAE is shown as radial dependence of r
times normal component of
the plasma displacement poloidal harmonics (below left) and the same
function in R,Z plane (below right).
Dominant harmonics are saved for comparisons in this
file for the case with the equal arc poloidal
angle choice and in this
file for case with Boozer coordinates.
Stability properties are computed first with the Maxwellian
distribution function of EPs.
Results are summarized in the following figure, where "effective"
particles charge is changed
during the scan. Growth rates are normalized to fast ion beta in the
center (beta=8pi pressure/B0^2,
where B0 is the magnetic field taken in the geometrical center of the
plasma) is plotted against zf-1.
The data for the following graph (Maxwellian distribution) are in
this file. The data structure used in
this file is presented via 5 columns as follows:
For slowind down similar scan produces following dependencies:
In the latter case theory curve was obtained for the model distribution
function with some v_crit term. Thus
there is some disagreement. It is much better agree with theory for n=2
case and slowing down distribution.
Theoretical expressions are derived for zero orbit EPs. The expressions
are documented in this FILE.