2. Evidence of enhanced plasma rotation in JET DTE1 experiments with improved plasma performance

We selected the same JET discharges of the first deuterium–tritium-experiment campaign (DTE1) which were analysed in Kolesnichenko et al. (Reference Kolesnichenko, Lutsenko, Tyshchenko, Weisen and Yakovenko2018) (see also Thomas et al. Reference Thomas, Andrew, Balet, Bartlett, Bull, de Esch, Gibson, Gowers, Guo and Huysmans1998; Thomas Reference Thomas, Giroud, Lomas, Stubberfield, Rimini, Testa and Zastrow2001; Weisen et al. Reference Weisen, Sips, Challis, Eriksson, Sharapov, Batistoni, Horton and Zastrow2014). They have two important features. First, the overall confinement time was slightly higher at the largest fusion power in D–T experiments. Second, the central ion temperature in D–T discharges was higher than that in deuterium discharges where the ICRH was applied with the heating power approximately that of alpha particles (fusion products) in DT discharges. These facts indicated the presence of some anomalous heating mechanism because mainly electrons are heated during slowing down of 3.5 MeV alpha particles by Coulomb collisions.

It was revealed in Kolesnichenko et al. (Reference Kolesnichenko, Lutsenko, Tyshchenko, Weisen and Yakovenko2018) that the cyclotron resonance in the tokamak magnetic field can provide simultaneous interaction of high-frequency FMMs (having frequencies above the ion gyrofrequency) with MeV alpha particles at the plasma periphery and near-axis thermal ions. Due to this, inward SC of alpha particle energy can occur. The analysis carried out in Kolesnichenko et al. (Reference Kolesnichenko, Lutsenko, Tyshchenko, Weisen and Yakovenko2018) has shown that reasonable mode amplitudes are sufficient for efficient energy transfer during SC.

On the other hand, estimates show that the momentum SC can lead to a significant toroidal torque, its value weakly depending on the mode frequency (Kolesnichenko et al. Reference Kolesnichenko, Tykhyy and White2020). Therefore, one could expect a considerable influence of the momentum SC on the plasma rotation in the described experiments.

In order to see whether this was the case. in the experiment we analysed the relevant JET database. Two D–T discharges (#42847 and #42856) and three deuterium discharges (#41067, #41068 and #41069) were considered. All these discharges were heated by NBI with the power $\mathcal {P}_{{\rm nbi}}=10$ MW. In addition, the ICRH power was $\mathcal {P}_{{\rm icrh}}=0.9$ MW and $\mathcal {P}_{{\rm icrh}}=2$ MW in discharges #41067 and #41068, respectively; the alpha power was ${\mathcal {P}_{\alpha }=1.4}$ MW and $\mathcal {P}_{\alpha }=1.34$ MW in discharges #42847 and #42856, respectively. The rotation frequencies are shown in figure 1. We observe that the rotation is largest in the D–T discharge with highest performance (#42847). The exception is deuterium discharge #41067 where rotation is very large in the near-axis region. Plasma parameters in this discharge and in discharge #41068 were very similar, although ICRH power in discharge #41067 was small, $P^{{\rm icrh}}_{\#41067}/P^{{\rm icrh}}_{\#41068} \approx 1 / 2$. These facts demonstrate enhanced efficiency of plasma heating when the rotation is strong.

Figure 1. Radial dependence of rotation frequency in the JET DTE1 deuterium discharges (#41067, #41068, #41069) and D–T discharges (#42847, #42856). Time evolution of these discharges for 52–55 s was considered (see also Kolesnichenko et al. Reference Kolesnichenko, Lutsenko, Tyshchenko, Weisen and Yakovenko2018), it clearly shows that the improved performance (i.e. high Ti thereby high fusion performance) is associated with the high rotation frequency.

3. Change of momentum of fast ion population by destabilized modes

3.1. Qualitative consideration of the momentum transfer from energetic ions to a destabilized mode

We begin with a qualitative analysis of the wave–particle momentum exchange and concomitant transport processes. We will follow the approach of Kolesnichenko (Reference Kolesnichenko1980) and Kolesnichenko et al. (Reference Kolesnichenko, Yakovenko, Lutsenko, Weller and White2010b).

Using an analogy with quantum mechanics, we introduce plasmons (other names are quasiparticles and quantums) with the density $n_k$ and momentum $\boldsymbol {\mathcal {M}}=\boldsymbol {k}n_k$, $\boldsymbol {k}$ is the wavevector. The plasmon density is defined by $n_k=W_k/\omega$, with $W_k$ the wave energy density ($n_k$ has the dimension $h/V$, where $h$ is the Planck constant and $V$ is the volume). When the waves are destabilized by fast ions with the growth rate $\gamma _\alpha$, $n_k$ decreases (plasmons are emitted by fast ions) with the rate $\dot {n}_k=-2\gamma _\alpha n_k$, and the momentum evolves correspondingly, $\dot {\boldsymbol {\mathcal {M}}}_\alpha = \boldsymbol {k}\dot {n}_k$, where dot over $n_k$ denotes time derivative. This implies that the emission of plasmons generates the force acting on fast ions,

(3.1)\begin{equation} \boldsymbol{f}_\alpha= \dot{\boldsymbol{\mathcal{M}}}_\alpha ={-}2\gamma_\alpha\boldsymbol{k}n_k={-}\boldsymbol{k}\frac{2\gamma_\alpha}{\omega} W_k, \end{equation}

where subscript $\alpha$ labels fast ions.

Note that quantum mechanics analogy is known in plasma physics, especially in nonlinear plasma theory (see, e.g. overviews Sagdeev & Galeev (Reference Sagdeev and Galeev1969), Fukai & Harris (Reference Fukai and Harris1971), Tsytovich (Reference Tsytovich1977) and Kadomtsev (Reference Kadomtsev1982)). In particular, the density of plasmons, $n_k$, represents the basic quantity in the theory of weak turbulence, the kinetic equation for waves is an equation for $n_k$, the conservation of the plasmon energy and momentum leads to conditions of the decay instability of the waves, etc.

Equation (3.1) agrees with the known relation between the energy and momentum of a travelling wave:

(3.2)\begin{equation} \mathcal{M}= \frac{\text{wave energy}}{\text{phase velocity}}. \end{equation}

A question, however, arises whether (3.1) obtained by means of a simple qualitative consideration, by using quantum mechanics terminology, is applicable to realistic toroidal plasmas. The answer is given in subsequent sections, where it is shown that (3.1) correctly reflects the main features of the momentum transfer from resonant particles to the mode. Being simple, (3.1) enables one to make important conclusions immediately. Therefore, below we continue our qualitative analysis, applying (3.1) to tokamaks.

It follows from (3.1) that $\boldsymbol {f}_\alpha$ is directed along the wavevector but in the opposite direction. The wavevector is defined by ${\rm i}\boldsymbol {k} \tilde {X} =\boldsymbol {\nabla } \tilde {X}$, where $\tilde {X}$ is a wave perturbation which we take in the form $\tilde {X} =\hat {X}(r) \exp (-{\rm i}\omega t +{\rm i}m\vartheta - {\rm i}n\varphi )$, with $r$ the flux radial coordinate, $\vartheta$ and $\varphi$ poloidal and toroidal angles, respectively. Then $k_\vartheta =m/r$, $k_\varphi =-n/R$, and $k_\| =(m\iota -n)/R=k_\varphi +\iota \epsilon k_\vartheta$, where $k_\|$ is the wavenumber along the magnetic field, $m$ and $n$ are the poloidal and toroidal mode numbers, $\iota =q^{-1}$, $\iota$ is the rotational transform, $q$ is the tokamak safety factor, $\epsilon =r/R$, $R$ is the major radius of the torus. The longitudinal wavenumber in many cases is small. For instance, this is the case for high frequency FMM responsible for suprathermal ICE. This is true also for low frequency modes, in particular, for Alfvén gap modes, such as TAE modes. To see it, let us take into account that the frequency of TAEs can be approximated as $\omega =|k_{\|*}|v_{A*}$, where the star subscript means that magnitudes are taken at $r_*$ defined by $nq_* =m \pm 1/2$. Then

(3.3a,b)\begin{equation} \left |\frac{k_\|}{k_\varphi}\right | = \frac{1}{2|n|q}< 1, \quad \left |\frac{k_\|}{k_\vartheta}\right | = \frac{\epsilon}{2|m|q}< 1, \end{equation}

and $k_\vartheta$ dominates.

Because $|k_\||<|k_\varphi |$, it follows from the relation $k_\| =k_\varphi + \epsilon \iota k_\vartheta$ that ${\rm sgn}\,k_\vartheta =-{\rm sgn}\,k_\varphi$ (${\rm sgn}\,m ={\rm sgn}\,n$) and, hence, ${\rm sgn}\,f_\vartheta =-{\rm sgn}\,f_\varphi$. Moreover, when $|m|q\gg 1$ and/or $|n|q\gg 1$, $k_\varphi \approx -\epsilon \iota k_\vartheta$, and

(3.4a,b)\begin{equation} f_\varphi \approx{-}\epsilon \iota f_\vartheta, \quad |\,f_\||\ll |\,f_\varphi|. \end{equation}

Correspondingly, the toroidal torque, $\mathcal {T}_\varphi =Rf_\varphi$, and poloidal torque, $\mathcal {T}_\vartheta =rf_\vartheta$, are connected in this case by relation

(3.5)\begin{equation} \mathcal{T}_\varphi \approx{-}\iota \mathcal{T}_\vartheta. \end{equation}

Due to relation $\epsilon \iota = B_\vartheta / B_\varphi$ ($B_\vartheta$ and $B_\varphi$ are components of the equilibrium magnetic field, $\boldsymbol {B}$), (3.4a,b) implies that the product $\boldsymbol {B}\boldsymbol {\cdot } \boldsymbol {f}_\alpha$ is relatively small. Nevertheless, it plays an important role because the power density lost by fast ions ($P_\alpha$) is determined by $f_\|$:

(3.6)\begin{equation} P_\alpha ={-} f_\| v_{{\rm res}}, \end{equation}

where $v_{{\rm res}}=\omega /k_\|$ is the longitudinal velocity of resonant particles. This equation together with longitudinal component of (3.1) yields the expected result, $P_\alpha =2\gamma _\alpha W_k$.

The binormal force leads to the flux of fast ions, $\varGamma _\alpha$, across the magnetic field,

(3.7)\begin{equation} \varGamma_\alpha = \frac{c}{e_\alpha B^2} (\boldsymbol{f} \times \boldsymbol{B})_r =\frac{c}{e_\alpha B}(f_\vartheta b_\varphi - f_\varphi b_\vartheta )=\frac{c}{e_\alpha B}\,f_b, \end{equation}

where $\boldsymbol {b} =\boldsymbol {B}/B$, the subscript ‘b’ labels the binormal component of a vector. This flux generates the radial electric field and the concomitant plasma rotation, both toroidal and poloidal. In addition, the binormal force directly affects the toroidal plasma rotation. Its toroidal component approximately equals to $f_{\varphi }$ [$\boldsymbol {k}_b =(k_\varphi -k_\|b_\varphi )\boldsymbol {e}_\varphi +(k_\vartheta -k_\|b_\vartheta )\boldsymbol {e}_\vartheta$]. Therefore, it well exceeds toroidal component of the longitudinal force when $k_\|$ is small.

A more rigorous analysis below, while confirming the conclusions drawn, shows that the obtained relations are not exact. The reason is that the force $\boldsymbol {f}$ is applied to resonant particles, whereas (3.1) does not take into account specific resonances in toroidal plasmas. It will be shown that the number $m$ should be corrected by the resonance numbers (in stellarators both $m$ and $n$ should be corrected for resonances associated with the lack of axial symmetry of the magnetic configurations, see e.g. Kolesnichenko et al. (Reference Kolesnichenko, Könies, Lutsenko and Yakovenko2011)). This is not the only reason why a more rigorous analysis should be done: it is not clear what is the phase velocity in (3.1) when the mode consists of several Fourier harmonics. In addition, (3.1) cannot describe effects of finite mode width.

3.2. Basic equations

Let us proceed to a more rigorous description of the influence of destabilized modes on a fast ion population.

We employ a quasilinear equation for the distribution function of these ions ($F$) in a tokamak magnetic field,

(3.8)\begin{equation} \frac{\partial F}{\partial t} =\mathcal{Q}(F), \end{equation}

where $\mathcal {Q} (F)$ is a transit-time-averaged quasilinear (QL) operator determined by (39), (43), (55) and (56) of Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1982).

We restrict our analysis to shear Alfvén waves with $\omega \ll \omega _B$ ($\omega _B$ the ion gyrofrequency) and passing particles with standard orbits, $\Delta r \ll r$ ($\Delta r$ is the orbit width). The variables the particle energy ($\mathcal {E} =M_\alpha v^2/2$), the magnetic moment ($\mu =\mathcal {E}_\perp /B$) and the coordinate of the particle guiding centre ($r$) will be used. The perturbed quantities will be labelled with a tilde and taken in the form $\tilde {X}=\mbox {Re}\sum _m X_m (r) \exp ({\rm i}\psi )$, where $\psi = -{\rm i}\omega t + {\rm i}m\vartheta -{\rm i}n\varphi$. Using the ideal magnetohydrodynamics (MHD) approximation, we assume the longitudinal component of the perturbed electric field to vanish, $\tilde {E}_\| =0$. In addition, as we are interested in Alfvén waves, we take $\tilde {B}_\| =0$. This enables us to take vanishing transverse vector potential of the electromagnetic field, $\tilde {\boldsymbol {A}}_{\perp } =0$, and to write the following relations:

(3.9a,b)\begin{gather} \tilde{\boldsymbol{E}}={-}\boldsymbol{\nabla}_\perp\tilde{\varPhi},\quad \tilde{\boldsymbol{B}}=\boldsymbol{\nabla}_\perp{\times}\tilde{A}_\parallel\boldsymbol{b}, \end{gather}

(3.10)\begin{gather}\tilde{A}_\parallel{=}\frac{c}{{\rm i}\omega}\boldsymbol{\nabla}_\parallel \tilde{\varPhi}, \end{gather}

where $A_\parallel$ is the longitudinal component of the vector potential, and $\tilde {\varPhi }$ is scalar potential. Thus, the electromagnetic field is expressed through potential $\tilde {\varPhi }$.

Due to these assumptions the QL operator of Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1982) reduces to

(3.11)\begin{equation} \mathcal{Q}(F) =\frac{1}{\tau_b}\sum_{m,s}\hat{\varPi} \tau_b D_{m,s}\hat{\varPi} F. \end{equation}

Here

(3.12)\begin{gather} D_{m,s}=\frac{{\rm \pi} e_\alpha^2}{2} |\mathcal{J}_{m,s}|^2\delta (\varOmega_s), \end{gather}

(3.13)\begin{gather}\varOmega_s=\omega -k_{\|s}v_\|, \end{gather}

(3.14)\begin{gather}|\mathcal{J}_{m,s}|^2= \frac{v_D^2}{4} \left |s\varPhi^\prime_m -\frac{m}{r}\varPhi_m\right |^2, \end{gather}

where $v_D=(v^2+v_\|^2)/(2\omega _{B\alpha }R)$ is the particle drift velocity, $\tau _b=2{\rm \pi} qR/|v_\||$ is the particle transit time, $m\Delta r /r \lesssim 1$, $k_{\|s} =(m_s-nq)/(qR)$, $m_s =m+s$, $s=\pm 1$, $k_\| (m,n)=(m-nq)/(qR)$, $\varPhi _m$ is a component of the scalar potential of the electromagnetic field, $\varPhi ^\prime ={\rm d}\varPhi /{\rm d}r$,

(3.15)\begin{gather} \hat{\varPi}= \frac{\partial}{\partial \mathcal{E}} + \mathcal{L} \frac{1}{r}\frac{\partial}{\partial r}, \end{gather}

(3.16)\begin{gather}\mathcal{L} =\left (\frac{\omega qR}{v_\|} +nq\right )\frac{1}{M_\alpha\omega\omega_{B\alpha}}, \end{gather}

$\mathcal {L} (v_{{\rm res}}) \equiv \mathcal {L}_s={m_s/(M_\alpha \omega \omega _{B\alpha })}$, $v_{{\rm res}}$ is the resonance longitudinal velocity determined by equation $\varOmega _s =0$. Note that the operator $\hat {\varPi }$ given by (3.15), (3.16) differs from that in Belikov & Kolesnichenko (Reference Belikov and Kolesnichenko1982) but it reduces to the latter at the resonance $\varOmega _s=0$.

In addition to (3.8), we will need a linear growth rate of instability. For Alfvénic perturbations we can write the following local growth rate (without damping mechanisms):

(3.17a,b)\begin{equation} \gamma_\alpha =\sum_{s} \gamma_s, \quad \gamma_s =\frac{{\rm \pi}^2}{2}\frac{e_\alpha^2v_A^2}{c^2}\int {\rm d}^3\,v v_D^2\delta (\varOmega_s)\hat{\varPi}F, \end{equation}

where $v_A$ is Afvén velocity. This relation directly follows from equation $\epsilon _{11} =c^2k_\|^2/\omega ^2$, where $\epsilon _{11}$ is a component of the dielectric tensor in a Maxwellian plasma with energetic ions, provided that $k_\vartheta v_D qR\ll v_\|$ and the resonance $\varOmega _s =0$ is responsible for the fast-ion interaction with Alfvén waves (Kolesnichenko Reference Kolesnichenko1980; Belikov, Kolesnichenko & Silivra Reference Belikov, Kolesnichenko and Silivra1992). It can also be obtained from an eigenmode equation when the mode is relatively narrow, see e.g. Kolesnichenko et al. (Reference Kolesnichenko, Lutsenko, Wobig and Yakovenko2002).

Note that normally, i.e. when the spatial inhomogeneity of energetic ions with density gradient ${\rm d}n_\alpha /{\rm d}r <0$ drives instability, $m_s <0$ and $n<0$.

3.3. ‘Quasilinear hydrodynamics’ of particles destabilizing the mode

Below we consider the influence of Alfvén modes on the fast-ion flux across the magnetic field, the rate of change of the fast-ion momentum and energy.

The flux surface averaged volume element in the velocity space in the tokamak magnetic field ($B=\bar {B}/h$, $h=1+\epsilon \cos \vartheta$) is ${\rm d}^3\bar {v} =\sum _{\sigma }\,{\rm d}\mathcal {E} \,{\rm d}\mu \tau _b \bar {B} / (M_\alpha ^2qR)$, where $\sigma ={\rm sign}\,v_\|$, and $\dot {\vartheta } =v_\| /(qR)$ was used. Taking this into account we calculate integrals $\int d^3 \bar {v}GQ$ for $G= 1$, $M_\alpha v_\|$, $\mathcal {E}$, and $\mathcal {E}_\|$. After integrating by parts the term with the energy derivative we obtain

(3.18)\begin{equation} \int {\rm d}^3\,\bar{v}GQ =\sum_{m,s}\int {\rm d}^3\,\bar{v} \left (-\frac{\partial G}{\partial \mathcal{E}}+ \frac{1}{r}\frac{\partial}{\partial r} G\mathcal{L}_s\right ) D_{m,s}\hat{\varPi}F. \end{equation}

In particular, for $G=1$ we obtain

(3.19)\begin{equation} \frac{\partial n_\alpha}{\partial t} ={-}\frac{1}{r} \frac{\partial }{\partial r}r\varGamma_\alpha, \end{equation}

where $\varGamma _\alpha =\sum _{s} \varGamma _{\alpha, s}$ with

(3.20)\begin{equation} \varGamma_{\alpha, s} ={-}\sum_m\frac{{\rm \pi} e_\alpha^2}{8r}\mathcal{L}_s \left |s\varPhi^\prime_m -\frac{m}{r}\varPhi_m \right |^2 \int {\rm d}^3\,v v_D^2\delta (\varOmega_s)\hat{\varPi}F \end{equation}

being the particle flux across the magnetic field, which is caused by the mode.

Due to (3.17a,b), (3.20) takes the form

(3.21)\begin{equation} \varGamma_\alpha ={-}\sum_{m,s}\frac{\gamma_{s}k_{\vartheta,s}}{M_\alpha\omega\omega_{B\alpha}}\frac{c^2}{4{\rm \pi} v_A^2} \left |s\varPhi^\prime_m -\frac{m}{r}\varPhi_m \right |^2, \end{equation}

where $k_{\vartheta, s }= m_s/r$. It follows from here that the flux of resonance ions destabilizing the modes ($\gamma _\alpha >0$) is directed outwards when the spatial gradient term dominates in (3.17a,b) and $\partial F/\partial r <0$ (in which case $k_{\vartheta,s} <0$). On the other hand, since the mode has finite width, $\varGamma _\alpha (r)$ is a non-monotonic function, having a maximum at a certain radius.

Using $\tilde {\boldsymbol {E}}=-\boldsymbol {\nabla }_\perp \tilde {\varPhi }$, in local approximation we obtain

(3.22)\begin{equation} \varGamma_\alpha ={-}\frac{2c}{e_\alpha B}\sum_{m,s}k_{\vartheta,s}\frac{\gamma_{s}}{\omega}W_m, \end{equation}

where

(3.23)\begin{equation} W_m =\frac{c^2}{8{\rm \pi} v_A^2}|\boldsymbol{E}_m|^2, \end{equation}

is the energy density of the $m$th harmonic of a mode (which includes both electric and magnetic perturbations, with $|\tilde {B}|^2 =(c^2/v_A^2)|\tilde {E}|^2$). Because $\varGamma _\alpha$ arises due to the $\boldsymbol {f}\times \boldsymbol {b}$ drift, as described by (3.7), (3.22) reads

(3.24)\begin{equation} f_b=f_\vartheta b_\varphi - f_\varphi b_\vartheta ={-}\sum_m k_{\vartheta,s}\frac{2\gamma_s}{\omega}W_m. \end{equation}

Now we proceed to calculations with $G=M_\alpha v_\|$ in (3.18). Using $\partial v_\| /\partial \mathcal {E} =1/(M_\alpha v_\|)$ and (3.8), we obtain

(3.25)\begin{equation} \frac{\partial}{\partial t}\left (M_\alpha n_\alpha \langle v_\| \rangle \right )=\sum_{m,s}\int {\rm d}^3\,\bar{v} \left (-\frac{1}{v_{{\rm res}}}+\frac{1}{r}\frac{\partial}{\partial r}r\frac{k_{\vartheta,s}v_{{\rm res}}} {\omega\omega_{B\alpha}}\right ) D_{m,s}\hat{\varPi}F, \end{equation}

where $v_{{\rm res}} =\omega /k_{\|s}$, $\langle v_\|\rangle =\int d^3 v v_\| F/(M_\alpha n_\alpha )$ the average velocity of fast ions. Taking into account (3.17a,b) we can write

(3.26)\begin{equation} \frac{\partial}{\partial t}\left (M_\alpha n_\alpha \langle v_\|\rangle\right)=f_\|^{(1)} +f_\|^{(2)}, \end{equation}

where

(3.27)\begin{gather} f_\|^{(1)}={-}\sum_{m,s} k_{\|s}\frac{2\gamma_s}{\omega}W_m, \end{gather}

(3.28)\begin{gather}f_\|^{(2)}=\sum_{m,s} \frac{{\rm \pi} e^2m_s}{8\omega_{B\alpha}\omega}\frac{1}{r}\frac{\partial}{\partial r}v_{{\rm res}} \left [\left |s\varPhi^\prime_m -\frac{m}{r}\varPhi_m \right |^2\int {\rm d}^3\,v v_D^2\delta (\varOmega_s)\hat{\varPi}F\right ]. \end{gather}

The first term in (3.26) corresponds to what is expected from analogy with quantum mechanics, as described by (3.1) with $m$ replaced by $m_s$. To see the nature of the second term let us combine (3.28) and (3.20). This yields

(3.29)\begin{equation} f_\|^{(2)}={-}\sum_{s} M_\alpha \frac{1}{r}\frac{\partial }{\partial r}rv_{{\rm res}}\varGamma_{\alpha, s}. \end{equation}

We observe that $f_\|^{(2)}$ is associated with the transverse particle flux generated due to the interaction of fast ions and the mode. It describes the radial redistribution of the momentum of resonant particles by the destabilized mode, not the momentum exchange between the mode and particles: $\int _0^a {\rm d}r\,r f_\|^{(2)} =0$. Due to (3.19), (3.29) can be written in the form

(3.30)\begin{equation} f_\|^{(2)}=\sum_s M_\alpha v_{{\rm res}}\frac{\partial n_{\alpha s} }{\partial t}= \sum_s M_\alpha v_{{\rm res}}\int {\rm d}^3\,v Q(F). \end{equation}

Because $\int {\rm d}^3\,x f_\|^{(2)} =0$, (3.30) implies that the redistribution of the momentum of resonance particles is just a consequence of quasilinear relaxation of distribution function. When the instability is driven by the spatial gradient of fast ions, the QL relaxation process decreases $n_\alpha$ at small radii and increases $n_\alpha$ at larger radii (for $n_\alpha ^\prime <0$). It is clear that the presence of sources and sinks of energetic ions is required to provide a steady state value of $\int {\rm d}^3\,v Q(F)$, in which case $\partial n_\alpha /\partial t =0$ but $\int {\rm d}^3\,v Q(F)\neq \partial n_\alpha /\partial t$ (the terms describing sources and sinks of fast ions should be added to (3.8) to make $\partial n_\alpha /\partial t =0$ in the steady state).

Thus, the modes can transport the local momentum of fast ions even in the absence of mismatch of the regions where damping and drive dominate. We refer to this effect as mode induced redistribution (MIR). The MIR is associated with $f_\parallel ^{(2)}$, whereas the SC is due to $f_b$ and $f_\parallel ^{(1)}$. Below we will see the role of poloidal and toroidal components of the forces.

Knowing $f_\|$ we can write

(3.31)\begin{equation} f_\vartheta b_\vartheta +f_\varphi b_\varphi ={-}\sum_{m,s} k_{\|s}\frac{2\gamma_s}{\omega}W_m +f_\|^{(2)}. \end{equation}

This equation should be combined with (3.24), which does not include effects of MIR and, therefore, contains components of $\boldsymbol {f}^{(1)}$. We find

(3.32a,b)\begin{gather} f_\vartheta =f_\vartheta^{(1)}+f^{(2)}_\|b_\vartheta,\quad f_\vartheta^{(1)}={-}\sum_{m,s} k_{\vartheta, s}\frac{2\gamma_s}{\omega}W_m, \end{gather}

(3.33a–c)\begin{gather}f_\varphi =f_\varphi^{(1)}+f^{(2)}_\|b_\varphi,\quad f_\varphi^{(1)}={-}\sum_{m,s} k_{\varphi}\frac{2\gamma_s}{\omega}W_m,\quad f_b^{(2)}=0, \end{gather}

where terms of the order $b_\vartheta ^2/b_\varphi ^2$ are neglected.

Equations (3.24), (3.26), (3.32a,b) and (3.33a–c) prove that relation (3.1) is in agreement with the QL theory (the toroidal components of (3.1) coincides with $f_\varphi ^{(1)}$, but $m$ should be replaced with $m_s$ in the binormal, poloidal and longitudinal components of (3.1)). On the other hand, the $\boldsymbol {f}^{(1)}$ coincides also with the force obtained within Hamiltonian approach, see Appendix A. It is clear, however, that the MIR force, $\boldsymbol {f}^{(2)}$, cannot be described by the single-particle approaches of § 3.1 and Appendix A.

In order to see the relative roles of the mode-momentum exchange and the momentum redistribution we have to estimate the ratio $f_\|^{(2)}/f_\|^{(1)}$. Assuming that the mode width is much less than characteristic lengths of inhomogeneity of the plasma and fast ions ($L$ and $L_\alpha$) we obtain

(3.34)\begin{equation} f_\|^{(2)}\sim \sum_{m,s} \frac{2\gamma_s k_{\vartheta,s} W_m}{\omega_{B\alpha}k_{\| s}\varDelta_w}, \end{equation}

where $\varDelta _w = ({\rm d}\ln W /{\rm d}r )^{-1}$ is a characteristic width of the mode. On the other hand, the drift term in $\hat {\varPi }F$, with $\hat {\varPi }$ given by (3.15), exceeds the term $\partial F/\partial \mathcal {E}$ with $F \propto \mathcal {E}^{-3/2}$ (which is stabilizing, $\partial F/\partial \mathcal {E} <0$) when

(3.35)\begin{equation} \zeta \equiv \frac{k_{\vartheta,s}v_{{\rm res}}}{3k_{\|s}\omega_{B\alpha}L_\alpha}>1, \end{equation}

where $(L_\alpha )^{-1} = {\rm d}\ln n_\alpha /{\rm d}r$. It follows from (3.34), (3.35) that the ratio $f_\|^{(2)}/f_\|^{(1)}$ well exceeds unity, being a product of $\zeta$ and a large multiplier,

(3.36)\begin{equation} \frac{f_\|^{(2)}}{f_\|^{(1)}}\sim \zeta \frac{3L_\alpha}{\varDelta_w} \gg 1. \end{equation}

Equations (3.8) and (3.18) with $G=\mathcal {E}$ reduce to

(3.37)\begin{equation} \frac{\partial}{\partial t}\left (M_\alpha n_\alpha \langle \mathcal{E}\rangle\right )=P^{(1)} +P^{(2)}, \end{equation}

where $P^{(1)}$ represents the fast-ion power sink because of the instability,

(3.38)\begin{equation} P^{(1)}={-}\sum_m 2\gamma_\alpha W_m, \end{equation}

and $P^{(2)}$ is the redistributed power

(3.39)\begin{equation} P^{(2)}=\frac{1}{r}\frac{\partial}{\partial r}\sum_{m,s} \int {\rm d}^3\,v\mathcal{E} D_{m,s}\hat{\varPi}F. \end{equation}

When $G=\mathcal {E}_\|$, we obtain

(3.40)\begin{equation} P_\|^{(1)}=P^{(1)} =\sum_sf_{\|,s}^{(1)}v_{{\rm res}}, \end{equation}

which is a consequence of the fact that the resonance mode–particle interaction does not affect particle transverse velocities distribution, and

(3.41)\begin{equation} P_\|^{(2)}=\sum_s \frac{M_\alpha v^2_{{\rm res}}}{2}\frac{\partial n_{\alpha s} }{\partial t}, \end{equation}

which is similar to (3.30).

3.4. Momentum transfer to gap modes

3.4.1. Simple gap mode

We restrict first our analysis to a gap mode consisting of two harmonics with the poloidal mode numbers $m$ and $m+\mu$ but with the same $n$, where $\mu =\pm 1$ for TAE modes, $\mu =\pm 2$ for EAE modes, etc.

We begin with a consideration of the forces arising due to resonance mode–particle interaction at the $r_*$ radius defined by $k_{\|1}(r_*) +k_{\|2}(r_*) =0$, where $nq_*=m+\mu /2$ and $k_{\|1}(r_*)q_*R =-k_{\|2}(r_*)q_*R =-\mu /2$. The subscripts $1$ and $2$ label magnitudes relevant to the $m$ and $m+\mu$ harmonics, respectively. The resonance numbers ($s=\pm 1$) should satisfy equation

(3.42)\begin{equation} s_1+s_2 =0, \end{equation}

in order to provide $k_{\|1,s_1}(r_*) +k_{\|2,s_2}(r_*) =0$. Then we obtain from (3.27) that $f_{\|*}^{(1)}=0$, provided that the difference between $W_{m_1}(r_*)$ and $W_{m_2}(r_*)$ is negligible. On the other hand, because $m+\mu /2 =nq_*$, the condition $k_{\|1}(r_*) +k_{\|2}(r_*) =0$ leads to $k_{\bar {m},_*}=0$, where $k_{\bar {m},_*}=(\bar {m}\iota _* -n)/R$ is the longitudinal wavenumber with $\bar {m} =0.5(m_1+m_2)= 0.5(m_{s_1}+m_{s_2})$ at $r=r_*$. Taking $k_\|$ with the poloidal mode number $\bar {m}$ we obtain from (3.1) the longitudinal force $f_{\|*}=0$. The poloidal component of (3.1) with $\bar {m}$ is

(3.43)\begin{equation} f_{\vartheta *}={-} \frac{\bar{m}}{r_*}\frac{2\gamma_\alpha}{\omega}W. \end{equation}

This corresponds to (3.32a,b) for $f^{(1)}_{\vartheta *}$. Thus, (3.1) with $k_\vartheta =\bar {m}/r$ correctly describes the rate of momentum exchange between the gap modes and fast ions.

The presence of the magnetic shear can strongly break the antisymmetry of the wavenumbers $k_{\|1,s_1}$ and $k_{\|2,s_2}$ at $r\neq r_*$, making $k_{\|s}$ of a gap mode not vanishing and considerable. To see this we approximate the mode frequency by $\omega =|k_{\|*}|v_{A *}$. Then the resonance condition ($\omega =k_{\|s} v_{{\rm res}}$) for these harmonics at the radius $r=r_* +\Delta r$, where $\iota =\iota _* +\Delta \iota$, can be written as follows:

(3.44)\begin{gather} \left [ \mbox{sgn}\,k_{\|1*} +\frac{2s_1}{\mu} +\frac{2(m+s_1)}{\mu}\frac{\Delta \iota }{\iota_*}\right ]v_{{\rm res}}^{(m)}=v_{A*}, \end{gather}

(3.45)\begin{gather}\left [ \mbox{sgn}\,k_{\|2*} +\frac{2s_2}{\mu} +\frac{2(m+\mu +s_2)}{\mu}\frac{\Delta \iota }{\iota_*}\right ]v_{{\rm res}}^{(m+\mu)}=v_{A*}, \end{gather}

where $\mbox {sgn}\,k_{\|1*}=- \mbox {sgn}\,k_{\|2*}=-1$, $s_1$ and $s_2$ satisfy (3.42). We observe that while $v_{{\rm res}}^{(m)}$ and $v_{{\rm res}}^{(m+\mu )}$ have different signs at $r_*$, the signs of the terms proportional to $\Delta \iota$ are the same for both harmonics. This implies that $|v_{{\rm res}}|$ of one of the harmonics grows, whereas $|v_{{\rm res}}|$ of another one decreases as $r$ moves away from $r_*$. For instance, taking $s_1=1$ we obtain for TAE modes,

(3.46)\begin{gather} v_{{\rm res}}^{(m)}=\frac{v_{A*}}{1+2(m+1)\Delta \iota /\iota_*}, \end{gather}

(3.47)\begin{gather}v_{{\rm res}}^{(m+1)}={-}\frac{v_{A*}}{1-2m\Delta \iota /\iota_*}. \end{gather}

In particular, at the radius $r_m$ where $q=m/n$ (3.46), (3.47) yield

(3.48)\begin{gather} v_{{\rm res}}^{(m)}=\frac{v_{A*}}{1+(m+1)/m}, \end{gather}

(3.49)\begin{gather}v_{{\rm res}}^{(m+1)}={-}\infty. \end{gather}

Here we took into account that $(\iota _m -\iota _*) /\iota _* =1/(2m)$, with $\iota _m=\iota (r_m)$.

Because of this, $|v_{{\rm res}}|$ can exceed the maximum velocity of the energetic ions, $v_\alpha$, at certain radii within the mode width, leading to $k_\|$ determined by another harmonic only, which decreases the growth rate but may increase of $f_\|$. In contrast, when $|v_{{\rm res} *}|< v_A$, the increase of the resonance velocity can destabilize the mode. Presumably, this was the case in the Large Helical Device stellarator where odd and even TAEs were observed (Kolesnichenko et al. Reference Kolesnichenko, Yamamoto, Yamazaki, Lutsenko, Nakajima, Narushima, Toi and Yakovenko2004).

Below we exclude from the consideration these particular cases to see effects of the shear only and employ (3.27) with two harmonics equally contributing to $\gamma _s$ and with $W_m=W_{m+\mu }$. Then the summation in (3.27) over $m$ reduces to calculation of

(3.50)\begin{equation} k_{\|s}^{\varSigma}\equiv \sum k_{\|s} =\omega \left (\frac{1}{v_{{\rm res}}^{(m)}} +\frac{1}{v_{{\rm res}}^{(m+\mu)}} \right ). \end{equation}

Combining (3.44) and (3.45) we obtain

(3.51)\begin{equation} \frac{k_{\|s}^{\varSigma}(\iota)}{|k_{\|1*}|} =\frac{4nq_*}{\mu}\frac{\Delta \iota}{\iota_*} \end{equation}

and

(3.52a,b)\begin{equation} k_{\|s}^{\varSigma} =\frac{2n}{R}\frac{\Delta \iota}{\iota_*},\quad \frac{k_{\|s}^{\varSigma}}{k_{\varphi}^{\varSigma}} ={-} \frac{\Delta \iota}{\iota_*}, \end{equation}

where $k_{\varphi }^{\varSigma }=2k_{\varphi 1}$, $|\Delta \iota |/\iota _*\lesssim 1/(2|m|)$. The force $f^{(1)}$ is maximum at the radius where the product $k_{\|s}^{\varSigma } \gamma _s W$ is largest.

Note that for narrow modes the ratio $\Delta \iota /\iota _*$ can be approximated by $\Delta \iota /\iota _* =-\hat {s} \Delta r /r_*$, where $\hat {s}$ is the magnetic shear.

3.4.2. Multiple-harmonic gap mode

Now we proceed to analysis for modes consisting of more than two harmonics, such as global TAE. Because the mode amplitudes of the pairs of coupled harmonics are localized around certain radii, each pair can be treated independently. In TAEs harmonics with $m$ and $m+1$ are coupled in the region between the $q_m=m/n$ and $q_{m+1}=(m+1 )/n$ rational flux surfaces; the width of this region is $\Delta q=1 /n$. Therefore, when the mode occupies a certain region $(\Delta q)_{mode}$, the number of coupled harmonics can be $n(\Delta q)_{mode} +1$. Using (3.52a,b), we can evaluate the longitudinal force $f^{(1)}$ produced by one pair of harmonics by (3.27) with $k_{\|s} \sim 1/(qR)$, the total force produced by the mode with $n\gg 1$ has $k_\|^{{\rm mode}}\sim n/(qR)$, with some average $q$. This overestimates $k_\|^{{\rm mode}}$ and, hence, $f^{(1)}_{{\rm mode}}$, when only a few harmonics have large local growth rate.

The wavenumber $k_{\|s}$ is not the only factor that determines $f^{(1)}$: it is of importance where maxima of $|\mathcal {J}_{m,s}|^2$ are located.

4. Plasma rotation and generation of the electric field

To study the rotation caused by destabilized eigenmodes we proceed from the equation obtained by summing equations of motion for the electrons, bulk plasma ions and energetic ions. As in the works of Kolesnichenko et al. (Reference Kolesnichenko, Yakovenko and Lutsenko2010a,Reference Kolesnichenko, Yakovenko, Lutsenko, Weller and Whiteb), we neglect several terms. First, the electron and fast-ion inertial terms, whose contribution is much less than that of the bulk plasma ions. Second, the pressure-gradient terms, which implies that we are considering effects superposed on the neoclassical transport. Third, the centrifugal term which is beyond applicability of our equations, being proportional to $\tilde {E}^4$.

We consider first the binormal component of this equation which contains the binormal force acting on fast ions due to their emission of the momentum: as shown in § 3, the binormal force leads to the transverse fast ions flux across, i.e. produces the radial electric current, $j_{\alpha r}$, which generates the radial electric field, $E_r$, and can affect plasma rotation. In the cylindrical approximation we can write

(4.1)\begin{equation} M_in_i\dot{u}_{ib} =\sum_{\sigma=e,i,\alpha}f_{\sigma b}^{(1)} + f_{{ib}}^{{\rm vis}}-\frac{1}{c} \overline{Bj_r}. \end{equation}

The following notations are used: ${\boldsymbol {u}}_i=\bar {\boldsymbol {V}}_i$ is the bulk plasma ion hydrodynamic velocity ($\boldsymbol {V}_i$) averaged over flux surface, a bar over letters denotes flux surface averaging (below can be omitted), $\boldsymbol {f}_\sigma ^{(1)}$ is the force acting on the $\sigma$ species due to emission/absorption of the momentum, $\boldsymbol {f}_i^{{\rm vis}}$ is associated with the ion viscosity or other mechanisms braking the rotation (discussed at the end of this section), a dot over letters denotes a time derivative, $\boldsymbol {j}$ is the overall current density ($\boldsymbol {j}=\sum _{\sigma =\alpha,e,i} \boldsymbol {j}_\sigma )$ induced by destabilized modes.

The radial current on the right-hand side of this equation can be expressed through a time derivative of the radial electric field, as follows from the curl-free Maxwell equation,

(4.2)\begin{equation} \dot{E}_r +4{\rm \pi} j_r=0. \end{equation}

On the other hand,

(4.3)\begin{equation} \dot{u}_{ib}={-}c\dot{E}_r/B \end{equation}

on the left-hand side, which is obtained from the radial component of the equation of motion for the ion component in the assumption of small $\dot {u}_{ir}$ and $\partial (\boldsymbol {\nabla }_r p_i)\partial t$. Due to these last two relations, (4.1) takes the form

(4.4)\begin{equation} \dot{E}_r ={-}\frac{B}{cM_in_i }\left (\sum_{\sigma=e,i,\alpha} f_{\sigma b}^{(1)} + f_{ib}^{{\rm vis}}\right )-\frac{v_A^2}{c^2} \dot{E}_r. \end{equation}

We observe that the last term on the right-hand side is much less than the left-hand side term. The reason is that the bulk plasma current arising in response to the fast ion current almost compensates the latter.

The physics of this phenomenon is the following: the resonant interaction of the mode and fast ions leads to the radial flux described by, for example, (3.7) which generates the radial electric field and concomitant radial current of the bulk plasma particles. We found that this current is almost equal to the fast-ion current. In other words, the overall current density, $j_r =\sum _{\sigma =e,i,\alpha }j_{\sigma r}$, is small, the term $Bj_r /c$ in (4.1) is less than the inertia term on the left-hand side of this equation by a factor of $v_A^2/c^2$.

Therefore, below we neglect the last term in (4.4).

Note that in the absence of SC, i.e. when the region driving the instability coincides with the damping region, $\sum _\sigma {f_{\sigma b}^{(1)} }=0$ (the diffusion of resonance particles is intrinsically ambipolar, see e.g. Kolesnichenko et al. (Reference Kolesnichenko, Yakovenko, Lutsenko, Weller and White2010b)) and (4.4) yields $E_r=0$.

The cylindrical approximation adopted in (4.1) underestimates the role of inertia. Coupling of the poloidal motion to the toroidal one in axisymmetric toroidal configurations enlarges the plasma inertia in the poloidal rotation by a factor $\mathcal {K} \approx 1+2q^2$, see Helander & Sigmar (Reference Helander and Sigmar2005) and Appendix B. Therefore, equations that determine poloidal and toroidal velocities are (the terms proportional to $j_r$ are neglected)

(4.5)\begin{gather} \mathcal{K}M_in_i\dot{u}_{i\vartheta} =\sum_{\sigma=e,i,\alpha}f_{\sigma \vartheta}^w + f_{i\vartheta}^{{\rm vis}}, \end{gather}

(4.6)\begin{gather}M_in_i\dot{u}_{i\varphi} =\sum_{\sigma=e,i,\alpha}f_{\sigma \varphi}^w + f_{i\varphi}^{{\rm vis}}, \end{gather}

where $\boldsymbol {f}^{w} =\boldsymbol {f}^{(1)} +\boldsymbol {f}^{(2)}$.

The electric field is determined by (4.2) which can be written as

(4.7)\begin{equation} c\dot{E}_r +\dot{u}_{i\vartheta}\bar{B}_\varphi -\dot{u}_{i\varphi}\bar{B}_\vartheta =0. \end{equation}

Assuming that $E_r=0$ before the instability (we consider effects superposed on the neoclassical magnitudes) we can remove dots over letters.

The $\boldsymbol {E}\times \boldsymbol {B}$ drift leads to velocities $u^E_\vartheta$ and $u^E_\varphi$ determined by

(4.8)\begin{gather} u^E_\vartheta ={-}c \frac{E_rB_\varphi}{B^2}, \end{gather}

(4.9)\begin{gather}u^E_\varphi =c \frac{E_rB_\vartheta}{B^2}, \end{gather}

so that the poloidal flow dominates. However, toroidicity and trapped particles decrease the poloidal velocity, whereas the toroidal force, $f_\varphi$, can increase the toroidal velocity. Below we consider this issue.

It follows from (4.7) that the radial electric field is associated with both poloidal rotation and toroidal rotation. The poloidal and toroidal motions are decoupled only in the limit cases $\mathcal {S} \gg 1$ and $\mathcal {S} \ll 1$, where

(4.10)\begin{equation} \mathcal{S} = \frac{ u_\varphi B_\vartheta}{ u_\vartheta B_\varphi}=\frac{\varOmega_\varphi}{q\varOmega_\vartheta}, \end{equation}

and $\varOmega _\vartheta =v_\vartheta /r$, $\varOmega _\varphi =v_\varphi /R$ the rotation frequencies. Note that non-averaged motion can be completely decoupled only in the toroidal direction, see Appendix B and Helander & Sigmar (Reference Helander and Sigmar2005). When $\mathcal {S} \ll 1$, (4.7) determines the poloidal flow which coincides with the $\boldsymbol {E}\times \boldsymbol {B}$ flow,

(4.11)\begin{equation} u_\vartheta ={-}c \frac{E_r}{B}\approx u_\vartheta^E. \end{equation}

However, $u_\varphi \gg u_\varphi ^E$ when $\mathcal {S}\gg 1$:

(4.12)\begin{equation} u_\varphi =c \frac{E_r}{B_\vartheta}=\frac{u^E_\varphi}{\varTheta^2}, \end{equation}

where $\varTheta =B_\vartheta /B_\varphi \ll 1$.

For $\mathcal {S} \sim 1$, although the poloidal and toroidal motion equally contribute to (4.7), toroidal velocity dominates, $u_\varphi /u_\vartheta \sim \varTheta ^{-1}\gg 1$, and the ratio of angular velocities is $\varOmega _\varphi /\varOmega _\vartheta \sim q$.

Note that in neoclassical theory $u_\vartheta ^{{\rm neo}} /v_{i,{\rm th}}\sim \rho _i /L$ ($v_{i,{\rm th}}$ the ion thermal velocity, $\rho _i$ the ion Larmor radius, $L$ a characteristic inhomogeneity length), therefore normally $\mathcal {S} \gg 1$ but

(4.13)\begin{equation} u_\varphi^{{\rm neo}} = \frac{c}{B_\vartheta} \left (E_r-\frac{1}{n_\sigma e_\sigma} \frac{{\rm d}p_\sigma}{{\rm d}r}\right ). \end{equation}

Thus, we have to evaluate $\mathcal {S}$, which requires estimates for $u_\varphi$ and $u_\vartheta$.

In the unstable region we can write

(4.14)\begin{gather} {u}_{i\vartheta}=\frac{1}{M_in_i\mathcal{K}}f_{\alpha \vartheta}^w(\Delta t)_\vartheta, \end{gather}

(4.15)\begin{gather}{u}_{i\varphi} = \frac{1}{M_in_i}f_{\alpha \varphi}^w (\Delta t)_\varphi, \end{gather}

where $(\Delta t)_{\varphi }\leqslant \tau ^{{\rm vis}}_{\varphi }$ and $(\Delta t)_{\vartheta }\leqslant \tau ^{{\rm vis}}_{\vartheta }$ are characteristic times. At the initial stage of instability, i.e. when the viscosity and other braking mechanisms are negligible, ${(\Delta t)_{\varphi }=(\Delta t)_{\vartheta }}$. Then for $f=f^{(1)}$ we obtain $\mathcal {S}\equiv \mathcal {S}_0$ and $(\varOmega _\varphi /\varOmega _\vartheta)_0$ with

(4.16a,b)\begin{equation} \mathcal{S}_0 = \mathcal{K}\varTheta\frac{f_{\alpha \varphi}^{(1)}}{f_{\alpha \vartheta}^{(1)}} = (1+2q^2)\varTheta^2 \frac{nq}{m},\quad \left (\frac{\varOmega_\varphi}{\varOmega_\vartheta}\right )_0= q \mathcal{S}_0 . \end{equation}

We conclude that $\mathcal {S}_0\sim \epsilon ^2$ and $(\varOmega _\varphi /\varOmega _\vartheta )_0 \sim q\epsilon ^2$ for $m\sim nq$. In this case the poloidal velocity represents the $\boldsymbol {E}\times \boldsymbol {B}$ drift and the poloidal frequency dominates when $q\epsilon ^2 \ll 1$. In the presence of the MIR force, such that $f_\varphi ^{(2)}\sim f_\vartheta ^{(1)}$, $\mathcal {S}_0 \sim \mathcal {K}\varTheta \sim 1$ for $q\sim 1.5$ and $\epsilon \sim 0.2$.

In the later stage of instability, rotation braking becomes important. The braking mechanisms have been extensively studied (see, e.g. the overview Ida & Rice (Reference Ida and Rice2014)). The poloidal rotation braking may be determined by neoclassical or anomalous processes. In both cases, the damping occurs via friction between trapped and passing ions. As a result, the poloidal velocity evolution after a change in the driving force is non-exponential because of the complicated rearrangement of the pitch-angle distribution (Morris, Haines & Hastie Reference Morris, Haines and Hastie1996). However, it seems to be reasonably approximated by an exponent with a characteristic time $\sim \tau _{ii}$ (Morris et al. Reference Morris, Haines and Hastie1996; Hinton & Rosenbluth Reference Hinton and Rosenbluth1999). For the toroidal rotation in axisymmetric configurations, the braking via friction between trapped and passing particles is impossible, whereas the neoclassical viscous braking is weak. Therefore, the toroidal braking results from turbulence and/or symmetry breaking of the magnetic field (ripple, resonant magnetic perturbations, etc.). The turbulent momentum transport is characterized by the Prandtl number (the ratio of ion thermal diffusivity to the perpendicular viscosity coefficient). In experiments, it was found to be less than unity in JET (Weisen et al. Reference Weisen, Camenen, Salmi, Versloot, de Vries, Maslov, Tala, Beurskens and Giroud2012) and larger than unity in TFTR (Scott et al. Reference Scott, Diamond, Fonck, Goldston, Howell, Jaehnig, Schilling, Synakowski, Zarnstorff and Bush1990) (experiments on other devices are mentioned in Ida & Rice (Reference Ida and Rice2014)), but it seems to be $\sim 1$. The toroidal flow damping due to field ripple was observed in experiments with artificially enhanced ripple (e.g. in JET de Vries et al. (Reference de Vries, Versloot, Salmi, Hua, Howell, Giroud, Parail, Saibene and Tala2010), see also another in Ida & Rice (Reference Ida and Rice2014)). The magnetic braking due to resonant magnetic perturbations may be strong but we will not consider this case here.

We conclude that in the steady state we can use (4.14) and (4.15) with $(\Delta t)_\vartheta \sim \tau _{ii}$ and $(\Delta t)_\varphi \sim \tau _E(\Delta r)^2/a^2$, where $\tau _E$ is the overall energy confinement time, and $\Delta r$ is a characteristic radial extent of the region where the momentum transport takes place. Then the toroidal velocity exceeds the poloidal one when

(4.17)\begin{equation} \frac{f_{\alpha \varphi}^w}{f_{\alpha\vartheta}^w}\frac{\tau_E}{\tau_{ii}}\left (\frac{\Delta r}{a}\right )^2> 1, \end{equation}

in which case the contribution of the $E\times B$ drift to toroidal velocity is negligible. It is clear that the estimate (4.17) is true when the unstable mode exists for $(\Delta t)_{{\rm mode}} >\max [(\Delta t)_\vartheta,(\Delta t)_\phi ]$.

5. Application to ITER

In order to see whether effects of the SC and MIR can be considerable, we have to make numerical estimates. We selected a global TAE in the ITER 15 MA baseline scenario, which is located in the region $0.6 \leqslant r/a\leqslant 0.8$ and has $n=20$ (Pinches et al. Reference Pinches, Chapman, Lauber, Oliver, Sharapov, Shinohara and Tani2015). This mode is destabilized by alpha particles and NBI deuterons with the local growth rate $\gamma _\alpha /\omega \sim 10^{-2}$. Relevant parameters are: $R=6.2$ m; $a=2$ m; $B=5.3$ T; plasma elongation $\kappa =1.7$; plasma volume $V_p=830\ {\rm m}^3$; NBI power $\mathcal {P}_{{\rm nbi}}=33$ MW; $v_{\alpha }=1.3\times 10^7\ {\rm m}\ {\rm s}^{-1}$; $v_{{\rm beam}}^D=10^7\ {\rm m}\ {\rm s}^{-1}$; $v_A=7\times 10^6\ {\rm m}\ {\rm s}^{-1}$; $q\sim 1.5$ at $r/a \sim 0.7$ (Pinches et al. Reference Pinches, Chapman, Lauber, Oliver, Sharapov, Shinohara and Tani2015). The mode amplitudes are not known. We take $\tilde {B}/B =10^{-4}$, which is realistic.

We infer from this the following. The poloidal mode numbers vary from $m\sim 20$ to $m\sim 30$. The toroidal force $f_{\alpha, \varphi }^{(1)}$ is

(5.1)\begin{equation} f_{\alpha, \varphi}^{(1)}= 2\frac{n}{R}\frac{\gamma_\alpha}{\omega}\left (\frac{\tilde{B}}{B}\right)^2 \frac{B^2}{8{\rm \pi}}=7.2\times 10^{{-}3}\frac{\mathrm{N}}{\mathrm{m}^3}. \end{equation}

Its ratio to the volume averaged NBI force, $\langle \,f_\varphi ^{{\rm NBI}}\rangle = \chi _{{\rm beam}}\mathcal {P}_{{\rm nbi}}/(V_pv_{{\rm beam}})$ with $\chi _{{\rm beam}} =v_\varphi /v_{{\rm beam}}$, is

(5.2)\begin{equation} \frac{f_{\alpha, \varphi}^{(1)}}{\langle \,f_\varphi^{{\rm NBI}}\rangle }= \frac{n}{4{\rm \pi}}\frac{\gamma_\alpha}{\omega} \frac{\tilde{B}^2V_pv_{{\rm beam}}}{\chi_{{\rm beam}}\mathcal{P}_{{\rm nbi}}R} =3.6. \end{equation}

It follows from here that the toroidal torque produced by SC can exceed that of NBI, especially at the plasma periphery, $r/a \sim 0.7$, where $f_\varphi ^{{\rm NBI}}<\langle \,f_\varphi ^{{\rm NBI}}\rangle$. Note that other mode-induced forces, except for $f_{\|}^{(1)}$, are even larger than $f_\varphi ^{{\rm NBI}}$:

(5.3a–c)\begin{equation} \frac{f_{\vartheta}^{(1)}}{f_{\varphi}^{(1)}}=\frac{m_s}{\epsilon n}\sim 5,\quad \frac{f_{\|}^{(1)}}{f_{\varphi}^{(1)}}=\frac{1}{2\epsilon nq}=\frac{1}{60}, \quad \frac{f_{\|}^{(2)}}{f_{\varphi}^{(1)}}\sim 4. \end{equation}

Knowing forces and assuming that $(\Delta t)_\varphi$ and $(\Delta t)_\vartheta$ are less than the mode duration, we can write the following estimates for the generated electric field and plasma rotation in the steady state due to SC:

(5.4a,b)\begin{equation} |E_r|\sim \frac{B}{cM_in_i\mathcal{K}}|\,f_\vartheta^{(1)}|(\Delta t)_\vartheta, \quad |u_\varphi|\sim \frac{1}{M_in_i}|\,f^{(1)}_\varphi |(\Delta t)_\varphi, \end{equation}

where $(\Delta t)_\vartheta =\tau _{ii}$ and $(\Delta t)_\varphi =\tau _E (\Delta r)^2 /a^2$ with $\Delta r$ a characteristic distance between the regions of drive and damping. We take $\tau _E =3$ s (Green et al. Reference Green2003), $\Delta r=0.5(\Delta r)_{{\rm mode}} =0.1a$. Then in the region of mode location $(\Delta t)_\vartheta \sim 0.01$ s and $(\Delta t)_\varphi \sim 0.03$ s, which leads to $E_r \sim 1 \ {\rm kV}\ {\rm m}^{-1}$ and $u_\varphi =500 {\rm m}\ {\rm s}^{-1}$.

For comparison, we calculate the toroidal velocity in JET. According to figure 1, ${\varOmega _\varphi =(1\unicode{x2013}2) \times 10^4\ {\rm rad}\ {\rm s}^{-1}}$ at $R\sim 3.4$ m. Then the flux surface averaged toroidal velocity can be evaluated as $u_\varphi =(3 \unicode{x2013} 6)\times 10^4\ {\rm m}\ {\rm s}^{-1}$. This well exceeds the calculated velocity in ITER. However, in reality the mode amplitude may exceed $\tilde {B}/B =10^{-4}$. For instance, for $\tilde {B}/B =10^{-3}$, we obtain that the rotation velocity in ITER approximately that in JET and $E_r \sim 100\ {\rm kV} {\rm m}^{-1}$.

We remind that the generated electric field and the rotation velocity as well have opposite directions in the driving region and the damping region during SC. This means that they produce sheared rotation within the mode width.

6. Summary

The results of this work can be summarized as follows.

The change of momentum of energetic ions because of MHD modes destabilized by these ions and the concomitant sheared plasma rotation in tokamaks are studied. Both a quasilinear theory and a Hamiltonian approach are used to consider the mode–particle momentum exchange. The plasma rotation is studied by employing a flux surface equation obtained by summing up plasma fluid equations of motion for different particle species, where the influence of toroidicity on the ion inertia term is taken into account.

In the framework of quasilinear theory, the analysis is carried out for Alfvénic modes destabilized due to spatial inhomogeneity of the energetic ions. The moments of a two-dimensional QL operator are calculated. This enabled us to formulate MHD-like equations for the particle density, momentum and power density, which contain the transverse flux of fast ions, the forces (in particular, the binormal force $f_b$ and the longitudinal forces $f_\|^{(1)}$, $f_\|^{(2)}$) acting on fast ions, and the power absorbed and radially redistributed due to the mode. As a result, two mechanisms of the influence of destabilized modes on the momentum of resonance particles are revealed. First, emission and absorption of momentum (resulting in the forces $f_b$, $f_\|^{(1)}$), which can lead to SC of the momentum of fast ions exciting the instability. Second, redistribution of the momentum of fast ions (resulting in the force $f_\|^{(2)}$), which is a consequence of finite mode width – a phenomenon called MIR.

The same forces (except for the MIR force) are obtained by applying a Hamiltonian approach. It is found that these forces persist even when the wave frequency is not small compared with the ion gyrofrequency.

It is concluded that the binormal force arising due to wave emission ($f_{b}$) leads to the transverse flux of fast ions and the concomitant radial electric current, $j_r^{(\alpha )}$, during SC, i.e. when the region driving the instability does not coincide with the damping region. The current $j_r^{(\alpha )}$ is compensated to a large extent by the arising plasma current. However, the resulting current is still sufficiently large to generate a considerable radial electric field according to equation $\dot {E}_r +4{\rm \pi} j_r =0$.

In contrast, the MIR force, $f_\|^{(2)}$, is not associated with the radial electric field; it can transport the local momentum of fast ions even in the absence of mismatch of the regions where damping and drive dominate.

Both $f_b$ and $f_\|$ affect toroidal rotation. However, when $k_\|/k\ll 1$ (typically, the case of TAEs), ${f}^{(1)}_\|$ is small. For this reason, the force $f_\varphi$ responsible for the toroidal rotation is mainly determined by the toroidal component of the binormal force $(\boldsymbol {f}_b)_\varphi$ and $f^{(2)}_\|$. It is shown that the binormal momentum is rapidly, on the time scale of approximately particle transit time, redistributed between fast ions, thermal ions and electrons, so that all the species are accelerated almost simultaneously. In contrast, the longitudinal momentum exchange between species takes place on much longer time scales.

Because $f^{(1)}_\vartheta \gg f^{(1)}_\varphi$, poloidal rotation dominates during the initial stage of instability being determined by the $\boldsymbol {E}\times \boldsymbol {B}$ drift. However, because of braking mechanisms (such as viscosity, magnetic ripple, collisions between passing and trapped ions), in the later stage it can be suppressed; then the toroidal velocity exceeds the poloidal one, and its magnitude well exceeds the $\boldsymbol {E}\times \boldsymbol {B}$ drift velocity.

Note that the mode amplitude is a free parameter in our theory. Due to this, our theory is applicable to experiments with any mode amplitudes, regardless of the mechanism which limits it. Various mechanisms are known that can determine mode amplitudes, see e.g. review papers (Gorelenkov, Pinches & Toi Reference Gorelenkov, Pinches and Toi2014; Chen & Zonca Reference Chen and Zonca2016; Todo Reference Todo2019).

Our estimates for ITER where a global TAE with $n=20$ and $m=20\unicode{x2013}30$ is predicted (Pinches et al. Reference Pinches, Chapman, Lauber, Oliver, Sharapov, Shinohara and Tani2015) show that the mode-induced forces can be significant, being even larger than the toroidal force caused by NBI. This result may produce an illusion that the expected influence of the mode-induced forces on plasma rotation can be unrealistically large, and a question then arises whether the obtained relations for these forces are correct. In connection with this, we remind that the forces arising due to emission of the momentum are evaluated in three independent ways: first, by using quantum mechanics analogy; second, by quasilinear theory; and third, by Hamiltonian formalism. All these three techniques lead to the same result: first, they give exactly the same relation for the toroidal force; second the presence of $m$ in the poloidal and other components of relation (3.1) instead of $m_s=m\pm 1$ predicted by (3.24), (3.26), (3.32a,b) and (A5), (A6) is not important for estimates because $m\gg 1$. The MIR force is found only in a quasilinear theory, but it has a clear physical meaning and thus it should be correct, too. In spite of large values of these forces, their overall effect can be moderate or even small. The matter is that the forces vary radially and can have opposite directions in the layers located very close to each other, in which case they tend to compensate each other because of viscosity. In seems, this is the situation in ITER with the MIR force produced by a multicomponent TAE. On the other hand, effects of the SC forces strongly depend on the radial location of driving and damping regions, they are weak when these regions considerably overlap. Therefore, although our estimates indicate strong forces produced by the destabilized TAE mode in ITER, a detailed information on the mode features (driving and damping mechanisms, the mode structure and amplitude, etc.) and plasma viscosity is required for a reliable prediction of sheared rotation caused by of SC and MIR in particular scenarios of ITER. This issue deserves further study.