- Difference in machine descriptions for for EC launcher 3 and 4.
- Force the toroidal mode number in Cyrano
- Cyrano use only the power from the first antenna - to be fixed!!
- Fokker-Plank / wave consistency is approximately imposed via iterations in Cyrano.
Alexei ask how the fast ion pressure was included in equilibrium calculations. The correct way to do it, from gyro-kinetics, is (p_par + p_perp)/2.
Action: Implement p=(p_par + p_perp)/2 in eqinput.
Reference from Alexei:
L. E. Zakharov and V. D. Shafranov, Equilibrium of current carrying plasmas in toroidal configurations, in Reviews of Plasma Physics volume 11, p. 153., edited by M. A. Leontovich, Consultants Bureau, New York (1986).
Details from Bruce:
... you have
p = (p_par + 2 p_perp)/3 delta p = p_par - p_perp
The term in the force balance is not this, but
(1/eB) [(curl b) z^2 + (b cross grad B) w] dot grad f
where z and w are the velocity space coordinates. Upon velocity space
integration, multiply by charge and sum over species, this yields
(curvature operator on) (p_par + p_perp)/2
under conventional tokamak ordering where you can combine the curvature
and grad-B drifts, as most codes do it.
If anisotropy is present, then you have in the curvature term
(p_par + p_perp)/2 = p + (delta p)/6
and in the parallel force balances you have just grad_par p_par, or
p_par = p + (2/3) (delta p)
If you set (delta p)/6 == G then you get 4G in the other equation
(see below for references).
In gyrokinetic theory, you take the gyrokinetic Poisson equation (without
FLR)
div (rho_M/B^2) grad_perp phi = n_e e - n_i e
where n_e and n_i are the velocity space integrals of f for each species
-- these are the gyrocenter densities. Take the time derivative, plug in
for df/dt with the force terms, and balance the parallel current with the
drift terms. You have
P_parallel = \intW z^2 f = p_par + nm u_par^2
p_perp = \intW wB f
Ordering u_par << c_s this forms (p_par + p_perp)/2 summed over species.
Then the MHD equilibrium sets flows and polarisation small, so that
B grad_par (J_par/B) + div (1/B^2) b cross grad (p_par + p_perp)/2 = 0
is your equilibrium condition. This is analogous to Strauss's RMHD
equation with just p_total under the grad in the second term. In a
tokamak, you can iterate on this equation to get equilibria (the
purpose of Strauss 1977 finite beta RMHD was stated to find equilibria),
which is what I do in GKMHD.
For the gyrokinetic/MHD correspondence, see
Miyato et al (J Phys Soc Japan 2009) Section 5.
In Eq 108 I call these "magnetic drifts" but the above is what goes in
there.
For the term involving G from parallel viscosity see Rogers and Drake (PRL
1997 229) and for how that comes from pressure anisotropy in a gyrofluid
model see B Scott Phys Plasmas (2007) 102318, which is the paper about
correspondence to reduced Bragisnkii equations in their limit.
The Strauss finite beta paper is Phys Fluids (1977) 1354.
