ESPERANDO POR QUE LA MALDITA PELEA EMPIEZE YA Diablos, Paul, ¿NO TIENES, MALDITA SEA, UNA CASA PROPIA?

When energetic particles are injected into a tokamak, their orbits are determined by the particle energy, pitch angle33_{, position, and the equilibrium magnetic field. These orbits are broadly classified}

as trapped or passing, and lost or confined [68]. Trapped particle orbits are restricted in either poloidal or toroidal angle, often on the outboard side of a torus, and do not encircle the magnetic axis. Passing particle orbits encircle the magnetic axis, without the poloidal / toroidal angle restrictions of the trapped orbit. Most trapped and passing orbits are confined orbits, that retain particles within the plasma. Lost orbits result in the ejection of the particle from the plasma [68]. The orbits of interest here are trapped orbits34_{. These orbits are referred to as banana orbits due to}

the shape of their guiding centre trajectories, as illustrated in Figure 4-3 [7]. Particles in trapped orbits experience a reflective force in their motion parallel to the magnetic field due to the magnetic mirror effect as they approach the inboard side near that top / bottom of the tokamak. The points of reflection are colloquially known as the ‘tips’ of the banana orbit.

Figure 4-3: An illustration of energetic particle orbit trajectories in tokamaks.

Here, a banana orbit on JET is demonstrated.

Original Source: http://www.efda.org/2013/09/banana-orbits/

Precessional-like fishbone modes35 _{are resonant with the toroidal precession frequency of the}

banana tips. This frequency is given by the resonance condition:

𝜔𝜔=𝑘𝑘 ∙ 𝜎𝜎= _𝑅𝑅𝑛𝑛𝜎𝜎𝜙𝜙+𝑚𝑚_𝑒𝑒𝜎𝜎𝜃𝜃 (4-13) where𝑅𝑅is the major radius, 𝑟𝑟 is the minor radius , 𝑘𝑘 is the wavenumber, 𝑛𝑛 is the toroidal mode number, 𝑚𝑚 is the poloidal mode number, 𝜔𝜔 is the mode frequency, and𝜎𝜎𝜙𝜙 and 𝜎𝜎𝜃𝜃 are the toroidal

33 _{The angle between the injected particle and the relevant orbit} 34 _{We are interested in orbits where particles have time to lose energy.} 35 _{Such as the infernal fishbones observed on MAST.}

59

and poloidal precession speeds of deeply trapped36 _{particle guiding centres for the fishbone}

perturbation.

In tokamaks, fast ions have a range of energies from their injection energy (the velocity described in Equation 4-8) down to zero energies. Within this range, there will be a range of energies resonant with the banana orbit tips.

Plasma in MAST rotates toroidally at velocities approaching ion sound speed near the core [69] causing experimentally observed fishbone frequencies, 𝜔𝜔𝑒𝑒𝑚𝑚𝑝𝑝𝑒𝑒𝑒𝑒𝑖𝑖𝑚𝑚𝑒𝑒𝑛𝑛𝑑𝑑, to be affected by Doppler shift. The frame frequency is the frequency of the fishbone perturbation harmonic after toroidal plasma rotation effects have been compensated for:

𝜔𝜔𝑒𝑒𝑚𝑚𝑝𝑝𝑒𝑒𝑒𝑒𝑖𝑖𝑚𝑚𝑒𝑒𝑛𝑛𝑑𝑑 =𝜔𝜔𝑓𝑓𝑒𝑒𝑎𝑎𝑚𝑚𝑒𝑒+𝑛𝑛𝜔𝜔𝑒𝑒𝑐𝑐𝑑𝑑𝑎𝑎𝑑𝑑𝑖𝑖𝑐𝑐𝑛𝑛 (4-14) where n is the toroidal mode number, and 𝜔𝜔_{𝑒𝑒𝑐𝑐𝑑𝑑𝑎𝑎𝑑𝑑𝑖𝑖𝑐𝑐𝑛𝑛} is the toroidal rotation frequency of the plasma. We have assumed that the precession frequency of the banana orbit tips are in resonance with fast ions at the frame frequency of the fishbone mode, 𝜔𝜔_{𝑓𝑓𝑒𝑒𝑎𝑎𝑚𝑚𝑒𝑒}.

On MAST, values of 𝜔𝜔𝑒𝑒𝑐𝑐𝑑𝑑𝑎𝑎𝑑𝑑𝑖𝑖𝑐𝑐𝑛𝑛 are inferred from spectroscopic measurements and calculated by TRANSP simulations, and have a strong dependence on flux surfaces. Our model used an existing profile for toroidal rotation [32], as TRANSP data was not available for all MAST discharges. These rotational frequencies fall off quickly as the last-closed flux surface is approached, demonstrated below for the normalised flux profile (𝜓𝜓_𝑁𝑁=_𝜕𝜕𝜕𝜕

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿) for #27920.

Figure 4-4: The rotation frequency profile for MAST #27920.

36 _{A deeply trapped orbit is one that has an inboard orbit path very close to the outboard orbit path, where the}

trajectory width shown as (3) in Figure 4-3 approaches zero. 60

The resonant-particle interactions for fishbones performed in this thesis utilises techniques similar to those used to analyse “bump-on-tail” behaviour in previous work by Breizman [70]. This approach assumes a combination of large aspect ratio approximation, a reduced dimensional framework, and isotropic thermal conditions. As outlined in White [68], this allows the toroidal precession frequency of a trapped orbit to be expressed in the simplified form:

𝜔𝜔𝑑𝑑≅2𝐸𝐸𝑞𝑞𝑓𝑓̂𝜅𝜅 2 𝑒𝑒 + 𝐸𝐸𝑞𝑞�1−𝜅𝜅2_� 𝑒𝑒 (4-15) where 𝜅𝜅= 𝜎𝜎∥𝑅𝑅

𝜎𝜎⊥√2𝑒𝑒 is the trapping/passing boundary value

37 _{[68], 𝑠𝑠̂(𝑟𝑟) =}𝑒𝑒

𝑞𝑞 𝑑𝑑𝑞𝑞

𝑑𝑑𝑒𝑒 is the shear, 𝐸𝐸 is the resonant energy of the particles in the trapped orbit, q is the safety factor, r is the distance from the magnetic axis, and R is the major radius at r (as shown in Figure 1.2 B)

Assuming deeply trapped orbits causes 𝜅𝜅 to approach zero38_{, the toroidal precession rate given in}

Equation (4-15) is further simplified to:

𝜔𝜔𝑑𝑑⇒ 𝜔𝜔𝑑𝑑0=_𝑚𝑚 𝐸𝐸𝑞𝑞

𝐷𝐷𝑅𝑅𝑒𝑒Ω𝑖𝑖𝑐𝑐 (4-16)

where Ω_{𝑖𝑖𝑐𝑐} is the ion cyclotron frequency39_.

With this expression for the toroidal precession rate of deeply trapped orbits, Equation (4-16) is rearranged to solve for the resonant energy of energetic particles in deeply trapped orbits.

𝐸𝐸=𝑚𝑚𝐷𝐷𝑅𝑅𝑚𝑚𝑒𝑒Ω𝑖𝑖𝑐𝑐𝜔𝜔𝑒𝑒0

𝑞𝑞 =

1

2𝑚𝑚𝐷𝐷𝜎𝜎𝑒𝑒2 (4-17) The fast ion velocity in resonance with the toroidal precession rate, 𝜎𝜎_𝑒𝑒, is obtained by converting Equation (4-17) to the kinetic energy equivalent40_{, and substitute the frequency at the start of the}

fishbone, 𝜔𝜔_{𝑓𝑓𝑒𝑒𝑎𝑎𝑚𝑚𝑒𝑒}(𝑟𝑟,𝑧𝑧,𝐶𝐶0), for the deeply trapped toroidal precession rate, 𝜔𝜔𝑑𝑑0: 𝜎𝜎𝑒𝑒0(𝑟𝑟,𝑧𝑧) =�2𝑅𝑅𝑒𝑒Ω𝑖𝑖𝑐𝑐𝜔𝜔𝑓𝑓𝑠𝑠𝑠𝑠𝑚𝑚𝑒𝑒(𝑒𝑒,𝑧𝑧,𝑑𝑑0)

𝑞𝑞(𝑒𝑒,𝑧𝑧) (4-18)

Equation (4-18) is the resonant velocity of the fishbone mode.