A Heuristic Exploration of Zonal Flow-like Structures in the Presence of Toroidal Rotation in a Non-Inertial Frame

Abstract

The mechanisms by which rotation influences zonal flows (ZFs) in plasma are incompletely understood, presenting a significant challenge in the study of plasma dynamics. This research addresses this gap by investigating the role of non-inertial effects—specifically centrifugal and Coriolis forces—on Geodesic Acoustic Modes (GAMs) and ZFs in rotating tokamak plasmas. While previous studies have linked centrifugal convection to plasma toroidal rotation, they often overlook the Coriolis effects or inconsistently incorporate non-inertial terms into magneto-hydrodynamic (MHD) equations. In this work, we derive self-consistent drift-ordered two-fluid equations from the collisional Vlasov equation in a non-inertial frame, and we modify the Hermes cold ion code to simulate the impact of rotation on GAMs and ZFs. Our simulations reveal that toroidal rotation enhances ZF amplitude and GAM frequency, with Coriolis convection playing a critical role in GAM propagation and the global structure of ZFs. Analysis of simulation outcomes indicates that centrifugal drift drives parallel velocity growth, while Coriolis drift facilitates radial propagation of GAMs. This work may provide valuable insights into momentum transport and flow shear dynamics in tokamaks, with implications for turbulence suppression and confinement optimization.

1. Introduction

The study of non-inertial effects in rotating plasmas has gained significant attention in fusion research, particularly concerning flow-shear generation and the behavior of Geodesic Acoustic Modes (GAMs) and Zonal Flows (ZFs) [1,2]. Although centrifugal convection has been identified as crucial in toroidally rotating plasmas [2], current approaches often lack self-consistency by simply adding non-inertial terms to magnetohydrodynamic (MHD) equations [3]. This limitation becomes particularly evident when examining Coriolis force effects, which are theorized to cause parallel transport asymmetry, but remain understudied [4].

Recent advances in plasma rotation research demonstrate that toroidal rotation significantly modifies equilibrium profiles and plasma behavior through centrifugal and Coriolis forces [3], while geodesic acoustic modes (GAMs) and zonal flows exhibit notable sensitivity to rotation profiles under electron cyclotron resonance heating (ECRH) conditions [5]. Despite these findings, theoretical models remain incomplete: some oversimplify rotation effects as basic convection flows [2], while others focus exclusively on centrifugal corrections [4]. Three critical limitations hinder progress: the absence of self-consistent incorporation of all non-inertial terms in drift-ordered fluid models, insufficient validation against experimental results from ASDEX-U [6] and HL-2A [5], and unresolved controversies regarding frame transformation versus effective force treatments [3]. Plasma rotation and flow shear prove crucial for suppressing edge-localized modes (ELMs) in tokamaks [7], although external momentum sources such as neutral beam injection (NBI) [8] face implementation challenges in reactor-scale devices such as ITER [9]. The observation of spontaneous rotation without external torque [6,10] has renewed the focus on intrinsic mechanisms, particularly the enigmatic origins of poloidal rotation [11].

Theoretical frameworks propose multiple rotation mechanisms: $\mathbf{E}\times \mathbf{B}$ flow and diamagnetic effects dominate in boundary regions, neoclassical effects involving finite-orbit physics and residual stress contribute significantly, while Hassam’s Stringer spin-up mechanism [12] demonstrates how heating-induced poloidal density asymmetries drive parallel flows through effective gravity coupling. Liu’s extension [1] reveals that radiofrequency wave heating enhances this asymmetry through resonance localization, suggesting that electron cyclotron waves could actively control poloidal rotation, a promising ELM mitigation strategy for reactors. This spin-up process shares mathematical similarities with Hasegawa–Wakatani dynamics [13], with contemporary interpretations linking it to centrifugal modifications of GAMs and zonal flows [14], creating theoretical synergies that underscore the need to investigate toroidal rotation’s non-inertial effects on GAM/ZF dynamics in rotating frames. This connection motivates our focus on toroidal rotation’s impact on GAM/ZF dynamics, particularly through non-inertial effects in rotating frames.

This paper is organized as follows: in Section 2, two-fluid equations ordered by self-consistent drift from the collisional Vlasov equation in a non-inertial frame are derived. These are then implemented in the Hermes code [15] to simulate rotation effects (Section 3). Analytical study of centrifugal/Coriolis effects on GAM/ZF is carried out in Section 4. Finally, the summary, discussion and limitations are addressed in Section 5.

2. Drift-Ordered Fluid Equations in a Non-Inertial Frame

The time evolution of the plasma distribution function is well described by the collisional Vlasov equation [16]:

$${\displaystyle \frac{\partial f_a}{\partial t}}+\mathbf{v}·\nabla f_a+{\displaystyle \frac{e_a}{m_a}}(\mathbf{E}+\mathbf{v}\times \mathbf{B})·{\displaystyle \frac{\partial f_a}{\partial \mathbf{v}}}=C_a\left(f_a\right).$$

(1)

where $f_a(x,v,t)$ represents the distribution function, a is the species of particles, and the right term of the equality is the collision operator. With integrals and moments operating on this equation, some standard descriptions of the plasma such as the drift kinetic equation and fluid equations can be obtained. Nevertheless, since this Vlasov equation is formulated in an inertial frame, modifications to the equation are required for rotating systems. Thyagaraja proved that the Vlasov equation has the same homogeneous expression in rotating systems [3]. However, tokamak profiles are measured in laboratory coordinates. Therefore, a distribution equation in a non-inertial frame with regard to the laboratory’s electrostatic potentials is needed.

For a moving frame, there is a coordinate transformation from the laboratory frame to the moving frame $\mathbf{q}(\mathbf{r},\mathbf{v},t)\to \mathbf{Q}(\mathbf{r},{\mathbf{v}}_{*},t)$, where ${\mathbf{v}}_{*}=\mathbf{v}-{\mathbf{V}}_{*}$,${\mathbf{V}}_{*}$ is the velocity of the moving frame. Then, the derivative transform is written as follows:

$$\begin{array}{c}\hfill \mathbf{q}=S_t\mathbf{Q}\end{array}$$

(2)

$S_t$ refers to the Lorentz transformation matrix from $\mathbf{q}$ to $\mathbf{Q}$. $\mathbf{q}$ is the position vector expressed in the laboratory (inertial) frame. It is the “absolute” position that an external observer can measure directly. $\mathbf{Q}$ denotes the position vector expressed in the body-attached or moving frame. The equality $S_t^T=S_t^{-1}$ is then obtained, assuming that the transformations between $\mathbf{q}$ and $\mathbf{Q}$ are reversible. The derivatives of both sides are related by the following expression:

$$\begin{array}{c}\hfill \dot{\mathbf{q}}=\dot{S_t}\mathbf{Q}+S_t\dot{\mathbf{Q}}\end{array}$$

(3)

Applying $\mathbf{Q}=S_t^{-1}\mathbf{q}$, Equation (3) is rewritten as the following, since $S_t{S}_t^{-1}$ is an antisymmetric matrix:

$$\begin{array}{c}\hfill \dot{\mathbf{q}}=\dot{S_t}{S}_t^{-1}\mathbf{q}+S_t\dot{\mathbf{Q}}\\ \hfill =\mathsf{\omega }\times \mathbf{q}+S_t\dot{\mathbf{Q}}\\ \hfill =S_t(\mathsf{\omega }\times \mathbf{Q}+\dot{\mathbf{Q}})\end{array}$$

(4)

where $\mathsf{\omega }$ is the eigenmatrix of $\dot{S_t}{S}_t^{-1}$. After differentiating in time, we obtain the following from Equation (4):

$$\ddot{\mathbf{q}}=S_t(\ddot{\mathbf{Q}}+2\mathsf{\omega }\times \dot{\mathbf{Q}}+\mathsf{\omega }\times (\mathsf{\omega }\times \mathbf{Q})+\dot{\mathsf{\omega }}\times \mathbf{Q})$$

(5)

Combined with the Newton–Lorentz law, Equation (4) can be further rewritten as follows:

$$\ddot{\mathbf{Q}}=-2\mathsf{\omega }\times \ddot{\mathbf{Q}}-\mathsf{\omega }\times (\mathsf{\omega }\times \mathbf{Q})-\dot{\mathsf{\omega }}\times \mathbf{Q}+{\displaystyle \frac{e_a}{m_a}}{S}_t^{-1}(\mathbf{E}+\mathbf{v}\times \mathbf{B})$$

(6)

Supposing a uniformly rotating frame, where the rotation frequency is ${\Omega }_0$, the Lorentz transformation has the following form from $\mathbf{q}(ct,R,\Phi ,Z)$ to $\mathbf{Q}(c{t}_{*},R_{*},{\Phi }_{*},Z_{*})$ [17]:

$$S_t=\left[\begin{array}{cccc}\gamma & 0& -\gamma \beta & 0\\ 0& 1& 0& 0\\ -\gamma \beta & 0& \gamma & 0\\ 0& 0& 0& 1\end{array}\right]$$

where $\gamma ={(1-{(r{\Omega }_0/c)}^2)}^{-1/2}$ is the Lorentz factor, $\beta =r{\Omega }_0/c$. Both $\mathbf{q}$ and $\mathbf{Q}$ are defined in the major radius coordinate system $(R,\Phi ,Z)$, where R denotes the major radius (the distance from the center of the torus), $\Phi$ is the toroidal angle (the angle around the torus), and Z represents the vertical coordinate (height above or below the midplane of the torus), as shown in Figure 1.

In the non-relativistic limit, the electric and magnetic fields in the laboratory and rotating frames are related by the following expressions [18]:

$$\begin{array}{c}\hfill {\mathbf{B}}_{*}=\mathbf{B}\\ \hfill {\mathbf{E}}_{*}=\mathbf{E}+({\mathsf{\Omega }}_{\mathbf{0}}\times \mathbf{R})\times \mathbf{B}\end{array}$$

(7)

${\mathsf{\Omega }}_{\mathbf{0}}$ is the rotation frequency, defined as ${\mathsf{\Omega }}_0{\mathbf{e}}_{\mathbf{z}}$ in the tokamak $(R,\varphi ,Z)$ coordinate [19]. From the definition of the transformation matrix $S_t$, it can be seen that $\mathsf{\omega }={\Omega }_{\mathbf{0}}$, the toroidal rotation rate. Therefore, the collisional Vlasov equation in uniformly rotating frames is obtained by combining Equations (1), (6), and (7).

$${\displaystyle \frac{\partial f_a}{\partial t}}+{\mathbf{v}}_{*}·\nabla f_a+{\displaystyle \frac{e_a}{m_a}}\left({\mathbf{E}}_{*}+{\mathbf{v}}_{*}\times \mathbf{B}-2{\displaystyle \frac{m_a}{e_a}}{\Omega }_{\mathbf{0}}\times {\mathbf{v}}_{*}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{m_a}{e_a}}\nabla {\Omega }_0^2{R}^2\right)·{\displaystyle \frac{\partial f_a}{\partial {\mathbf{v}}_{*}}}=C_a\left({f_a}_{*}\right)$$

(8)

Evidently, the metric tensor should be changed in a different coordinate system. In order to make it easier in the following integration and simulation, the metric tensor in this uniformly rotating system should be analyzed. Referring to [20], the rotating frame transformation matrix can be expressed as follows (assuming time in the two coordinates coincides at 0).

$$(d{R}_{*},R_{*}d{\Phi }_{*},d{Z}_{*},icd{t}_{*})=\left[\begin{array}{cccc}{\displaystyle \frac{1}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}& 0& {\displaystyle \frac{-i\frac{R{\Omega }_0}{c}}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}\\ 0& 0& 1& 0\\ 0& 0& 0& {\displaystyle \frac{1}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}\end{array}\right]\left(\begin{array}{c}dR\\ Rd\Phi \\ dZ\\ icdt\end{array}\right)$$

Multiplying the transformation matrix with its transpose, we obtain the co-variant tensor.

$$g_{ij}=\left[\begin{array}{cccc}{\displaystyle \frac{1}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}& 0& {\displaystyle \frac{\frac{R{\Omega }_0}{c}}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}\\ 0& 0& 1& 0\\ 0& {\displaystyle \frac{\frac{R{\Omega }_0}{c}}{\sqrt{1-\frac{R^2{\Omega }_0^2}{c^2}}}}& 0& -1\end{array}\right]$$

Obviously, the contra-variant tensor is equal to the covariant one in the non-relativistic limit $g_{ij}=g^{ij}$. We can say that the metric tensors remain the same in the assumed uniformly rotating system. The right side of Equation (8) is the Fokker–Planck collision operator, and we assume that its form remains the same in the current frame. Following the process in reference [21], we can obtain fluid equations by integrating zero, first, and second (scalar) moments over Equation (8) for plasma species j, namely the continuity equation,

$${\displaystyle \frac{\partial n_j}{\partial t}}+\nabla ·n_j{\mathbf{V}}_{\mathbf{j}}=S_j^n$$

(9)

the momentum conservation equation,

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial m_j{n}_j{\mathbf{V}}_{\mathbf{j}}}{\partial t}}+\nabla p_j+\nabla ·\left(\overleftrightarrow{\pi }+m_j{n}_j{\mathbf{V}}_{\mathbf{j}}{\mathbf{V}}_{\mathbf{j}}\right)=Z_{j}e{n}_j(\mathbf{E}+{\mathbf{V}}_{\mathbf{j}}\times \mathbf{B})\\ \hfill +2m_j{n}_j{\mathbf{V}}_{\mathbf{j}}\times {\mathsf{\Omega }}_{\mathbf{0}}+m_j{n}_{j}R{\Omega }_0^2\nabla R+{\mathbf{R}}_{\mathbf{j}}+{\mathbf{S}}_{\mathbf{j}}^{\mathbf{m}}\end{array}$$

(10)

and the energy conservation equation,

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}({\displaystyle \frac{3}{2}}{p}_j+{\displaystyle \frac{1}{2}}{m}_j{n}_j{V}_j^2)+\nabla ·{\mathbf{Q}}_{\mathbf{j}}=W_j\\ \hfill +{\mathbf{V}}_{\mathbf{j}}·({\mathbf{R}}_{\mathbf{j}}+Z_{j}e{n}_j\mathbf{E}+m_j{n}_{j}R{\Omega }_0^2\nabla R)+S_j^E\end{array}$$

(11)

where $S_j^n,{\mathbf{S}}_{\mathbf{j}}^{\mathbf{M}},S_j^e$ are density, momentum, and energy sources for species j, respectively.

The viscous stress tensor $\overleftrightarrow{\pi }$ is given in Ref. [22], and the friction force between species ${\mathbf{R}}_{\mathbf{j}}$ has the same expression as in Ref. [4]:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathbf{e}}=-{\mathbf{R}}_{\mathbf{i}}=e{n}_e\left({\displaystyle \frac{{\mathbf{J}}_{\left|\right|}}{{\sigma }_{\left|\right|}}}+{\displaystyle \frac{{\mathbf{J}}_{\perp }}{{\sigma }_{\perp }}}\right)-0.71n_e{\partial }_{\left|\right|}{T}_e+{\displaystyle \frac{3\nu }{2{\Omega }_e}}{n}_e\mathbf{b}\times \nabla T_e\end{array}$$

(12)

with ${\sigma }_{\perp }=0.51{\sigma }_{\left|\right|}=e^2{n}_e/m_e{\nu }_e$. The divergence of energy flux is written as the following expression:

$$\begin{array}{c}\hfill \nabla ·{\mathbf{Q}}_{\mathbf{j}}=\nabla ·[({\displaystyle \frac{5}{2}}{n}_j{T}_j+{\displaystyle \frac{1}{2}}{m}_j{n}_j{V}_j^2){\mathbf{V}}_{\mathbf{j}}+\overleftrightarrow{\pi }·{\mathbf{V}}_{\mathbf{j}}+{\mathbf{q}}_{\mathbf{j}}]\\ \hfill =\nabla ·\int \int \int {\displaystyle \frac{1}{2}}{m}_j{v}_{*}^2{\mathbf{v}}_{*}{f}_a{d}^3{v}_{*}+\int \int \int {\displaystyle \frac{1}{2}}{m}_j{v}_{*}^2{C}_a\left(f_a\right)d^3{v}_{*}\end{array}$$

(13)

where ${\nu }_e$ is the electron classical collision frequency, and ${\Omega }_e$ is the gyro-frequency of the electron. $T_j$ and ${\mathbf{q}}_{\mathbf{j}}$ represent species’ thermal temperature and heat flux, respectively [4].

$$\begin{array}{c}\hfill {\mathbf{q}}_{\mathbf{e}}=-{\kappa }_{\left|\right|e}{\partial }_{\left|\right|}{T}_e+{\displaystyle \frac{5p_e}{2m_e{\Omega }_e}}\mathbf{b}\times \nabla T_e-{\kappa }_{\perp e}{\nabla }_{\perp }{T}_e\\ \hfill -0.71{\displaystyle \frac{T_e{\mathbf{J}}_{\left|\right|}}{e}}+{\displaystyle \frac{3{\nu }_e}{2{\Omega }_e}}{\displaystyle \frac{T_e\mathbf{b}\times \mathbf{J}}{e}}\end{array}$$

(14)

Here, ${\kappa }_{\left|\right|e}=3.16{\displaystyle \frac{p_e}{m_e{\nu }_e}},{\kappa }_{\perp e}=4.66{\displaystyle \frac{p_e{\nu }_e}{m_e{\Omega }_e^2}}$. Since the derivative remains unchanged in a uniform rotation frame, the viscosity terms are assumed to be the same as that in the inertial frame [22], assuming the equilibrium distribution remains approximately Maxwellian. From Equation (11), one can find that the Coriolis force has no contribution to the fluid energy.

Taking the same ordering and steps as Simakov’s work [4], we obtain the continuity equations under the cold-ion plasma assumption:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}+\nabla ·\left[n({\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}+{\mathbf{V}}_{\mathbf{mag}}+{\mathbf{V}}_{\left|\right|\mathbf{e}}+{\mathbf{V}}_{\mathbf{pe}}+{\mathbf{V}}_{\mathbf{Co}-\mathbf{e}}+{\mathbf{V}}_{\mathbf{Cf}-\mathbf{e}})\right]=S^n\end{array}$$

(15)

where ${\mathbf{V}}_{\left|\right|\mathbf{e}}$ is the parallel velocity of electrons, the velocities of electron $\mathbf{E}\times \mathbf{B}$, and diamagnetic and polarization drift are defined by crossing $\mathbf{B}$ in Equation (10).

$$\begin{array}{c}\hfill {\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}={\displaystyle \frac{\mathbf{b}\times \nabla \Phi }{B}},{\mathbf{V}}_{\mathbf{mag}}=-{\displaystyle \frac{\mathbf{b}\times \nabla p_e}{enB}},{\mathbf{V}}_{\mathbf{pe}}={\displaystyle \frac{\mathbf{b}\times {\mathbf{S}}_{\mathbf{e}}^{\mathbf{M}}}{enB}}.\end{array}$$

(16)

The velocities due to Coriolis and centrifugal forces are given as follows:

$$\begin{array}{c}\hfill {\mathbf{V}}_{\mathbf{co}-\mathbf{e}}=-{\displaystyle \frac{2m_e{V}_{\left|\right|e}}{eB}}{\mathsf{\Omega }}_{\mathbf{0}\perp },{\mathbf{V}}_{\mathbf{cf}-\mathbf{e}}={\displaystyle \frac{m_e{\Omega }_0^{2}R}{eB}}\mathit{b}\times \nabla R\end{array}$$

(17)

Dotting Equation (10) with $\mathbf{b}$ and dividing by $m_{e}n$, we can use the continuity Equation (15) to rewrite the electron momentum conservation equation:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial V_{\left|\right|e}}{\partial t}}+{\mathbf{V}}_{\mathbf{e}}·\nabla V_{\left|\right|e}-({\displaystyle \frac{\partial \mathbf{b}}{\partial t}}+{\mathbf{V}}_{\mathbf{e}}·\nabla \mathbf{b})·{\mathbf{V}}_{\mathbf{e}}=\\ \hfill -{\displaystyle \frac{e}{m_e}}{E}_{\left|\right|}+{\displaystyle \frac{R_{\left|\right|e}-{\partial }_{\left|\right|}{p}_e}{m_{e}n}}-{\displaystyle \frac{\mathbf{b}·(\nabla ·\overleftrightarrow{\pi })}{m_{e}n}}+2\mathbf{b}·{\mathbf{V}}_{\mathbf{e}}\times {\mathsf{\Omega }}_{\mathbf{0}}+R{\Omega }_0^2\mathbf{b}\nabla R\end{array}$$

(18)

Performing the same operations on Equation (10) for the ion species and neglecting $m_{i}n{\mathbf{V}}_{\mathbf{i}}·{\displaystyle \frac{\partial \mathbf{b}}{\partial t}}$ and $m_{i}n{\mathbf{V}}_{\mathbf{i}}·\nabla \mathbf{b}·{\mathbf{V}}_{\mathbf{i}}$ since they are comparatively small, we obtain the equation of parallel momentum of ions.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}{m}_{i}n{V}_{\left|\right|i}+\nabla ·\left[m_{i}n{V}_{\left|\right|i}({\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}+{\mathbf{V}}_{\left|\right|\mathbf{i}}+{\mathbf{V}}_{\mathbf{co}-\mathbf{i}}+{\mathbf{V}}_{\mathbf{cf}-\mathbf{i}})\right]=-{\partial }_{\left|\right|}{p}_e\\ \hfill -{\displaystyle \frac{3}{2}}{B}^{3/2}\partial ({\displaystyle \frac{{\pi }_{ci}}{B^{3/2}}})+2m_i{n}_i\mathbf{b}·{\mathbf{V}}_{\mathbf{i}}\times {\mathsf{\Omega }}_{\mathbf{0}}+m_i{n}_{i}R{\Omega }_0^2\mathbf{b}\nabla R+S_{\left|\right|i}^M+S_{\left|\right|e}^M\end{array}$$

(19)

where the Coriolis and centrifugal drift velocities for ions are given as follows:

$$\begin{array}{c}\hfill {\mathbf{V}}_{\mathbf{co}-\mathbf{i}}={\displaystyle \frac{2m_i{V}_{\left|\right|i}}{eB}}{\mathsf{\Omega }}_{\mathbf{0}\perp },{\mathbf{V}}_{\mathbf{cf}-\mathbf{i}}=-{\displaystyle \frac{m_i{\Omega }_0^{2}R}{eB}}\mathbf{b}\times \nabla R\end{array}$$

(20)

The parallel viscosity ${\displaystyle \frac{3}{2}}{B}^{3/2}\partial ({\displaystyle \frac{{\pi }_{ci}}{B^{3/2}}})\approx -1.28\sqrt{B}{\partial }_{\left|\right|}[{\displaystyle \frac{p_i}{{\nu }_{i}B}}{\partial }_{\left|\right|}(\sqrt{B}{V}_{\left|\right|i})]$ [23] goes to zero in the cold ion plasma assumption. At the same time, one can write the electron and ion fluid velocities in terms of the center-of-mass velocity [24]:

$$\begin{array}{c}\hfill {\mathbf{V}}_{\mathbf{i}}\approx \mathbf{V}+O(m_e/m_i)\\ \hfill {\mathbf{V}}_{\mathbf{e}}\approx \mathbf{V}-{\displaystyle \frac{\mathbf{J}}{n_{e}e}}+O(m_e/m_i)\end{array}$$

(21)

Then, Equations (18) and (21) can be combined to give a modified Ohm’s law:

$$\begin{array}{c}\hfill {\mathbf{E}}_{\left|\right|}\approx {\displaystyle \frac{R_{\left|\right|e}-{\partial }_{\left|\right|}{p}_e}{e{n}_e}}+{\displaystyle \frac{m_e}{e^2{n}_e}}{\displaystyle \frac{\partial J_{\left|\right|}}{\partial t}}-{\displaystyle \frac{m_e}{n_e^2{e}^2}}({\mathbf{V}}_{\mathbf{i}}·\nabla )J_{\left|\right|}\\ \hfill -2{\displaystyle \frac{m_e}{e}}\mathbf{b}·{\mathbf{V}}_{\mathbf{e}}\times {\mathsf{\Omega }}_{\mathbf{0}}-{\displaystyle \frac{m_e}{e}}R{\Omega }_0^2\mathbf{b}\nabla R\end{array}$$

(22)

The last two terms are relatively small compared to the other terms ($m_e/m_i$) and will be neglected in the following simulation. Further, following from the plasma momentum conservation equation (Equation (10)), we find

$$\begin{array}{c}\hfill \mathbf{J}={\mathbf{J}}_{\left|\right|}+{\displaystyle \frac{1}{B}}\mathbf{b}\times \nabla p_e+en({\mathbf{V}}_{\mathbf{pi}}-{\mathbf{V}}_{\mathbf{pe}})\\ \hfill +en({\mathbf{V}}_{\mathbf{co}-\mathbf{i}}+{\mathbf{V}}_{\mathbf{cf}-\mathbf{i}})-en({\mathbf{V}}_{\mathbf{co}-\mathbf{e}}+{\mathbf{V}}_{\mathbf{co}-\mathbf{e}})\end{array}$$

(23)

Applying this equation into the ambipolarity condition $\nabla ·\mathbf{J}=0$, the vorticity equation can be obtained using the Boussinesq approximation [25]:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \omega }{\partial t}}=\nabla ·\left(\omega {\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}\right)+{\nabla }_{\left|\right|}{J}_{\left|\right|}\\ \hfill -e\nabla ·(n{\mathbf{V}}_{\mathbf{mag}}-n{V}_{co-i}-n{\mathbf{V}}_{\mathbf{cf}-\mathbf{i}})-e\nabla ·(n{\mathbf{V}}_{\mathbf{co}-\mathbf{e}}+n{\mathbf{V}}_{\mathbf{cf}-\mathbf{e}})\end{array}$$

(24)

Using the expressions in Equations (11), (12), and (14), we notice that the conservative form of the electron energy equation can be written as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{3}{2}}{p}_e\right)+\nabla ·[{\displaystyle \frac{5}{2}}{p}_e({\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}+{\mathbf{V}}_{\left|\right|\mathbf{e}}+{\mathbf{V}}_{\mathbf{co}-\mathbf{e}}+{\mathbf{V}}_{\mathbf{cf}-\mathbf{e}})\\ \hfill -\nabla ·\left({\kappa }_{\left|\right|e}{\partial }_{\left|\right|}{T}_e\right)-{\displaystyle \frac{0.71T_e}{e}}{\mathbf{J}}_{\left|\right|}]=-W_i+{\displaystyle \frac{J_{\left|\right|}^2}{{\sigma }_{\left|\right|}}}-{\displaystyle \frac{0.71{\partial }_{\left|\right|}{T}_e}{e}}{J}_{\left|\right|}\\ \hfill +({\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}+{\mathbf{V}}_{\left|\right|\mathbf{e}}+{\mathbf{V}}_{\mathbf{co}-\mathbf{e}}+{\mathbf{V}}_{\mathbf{cf}-\mathbf{e}})·(\nabla p_e-{\mathbf{S}}_e^M+m_{e}nR{\Omega }_0^2\nabla R)+S_e^E\end{array}$$

(25)

In short, 2-fluid 5-field equations are obtained in a non-inertial frame. Then, it is possible to simulate transport physics nonlinearly.

3. Simulation Set-Up and Results

To investigate the global parallel momentum transportation under the non-inertial effects, we carry out asymmetric simulations using the Hermes cold ion code within the BOUT++ framework. This code has good performance on simulations of global flux transport and edge turbulence [15]. We used the same parameters as in Ref. [15] to normalize the continuity Equation (15), modified Ohm’s law (parallel) (Equation (22)), vorticity equation (Equation (24)), ion parallel momentum equation (Equation (19)), and electron energy conservative equation (Equation (25)). Including classical perpendicular diffusion, we obtain the modified 5-field reduced 2-fluid plasma equations.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}=-\nabla ·\left(n{\mathbf{V}}_{E\times B}+n{\mathbf{V}}_{mag}\right)-{\nabla }_{\left|\right|}\left(n{v}_{\left|\right|e}\right)+\nabla ·(D_n{\nabla }_{\perp }n)\\ \hfill {\displaystyle \frac{3}{2}}{\displaystyle \frac{\partial p_e}{\partial t}}=-\nabla ·({\displaystyle \frac{3}{2}}{p}_e{\mathbf{V}}_{E\times B}+p_e{\displaystyle \frac{5}{2}}{\mathbf{V}}_{mag})\\ \hfill -p_e\nabla ·{\mathbf{V}}_{E\times B}-{\displaystyle \frac{5}{2}}{\nabla }_{\left|\right|}\left(p_e{v}_{\left|\right|e}\right)+v_{\left|\right|e}{\partial }_{\left|\right|}{p}_e+{\nabla }_{\left|\right|}\left({\kappa }_{e\left|\right|}{\partial }_{\left|\right|}{T}_e\right)\end{array}$$

(26)

$$\begin{array}{c}\hfill +0.71{\nabla }_{\left|\right|}\left(T_e{j}_{\left|\right|}\right)-0.71j_{\left|\right|}{\partial }_{\left|\right|}{T}_e+{\displaystyle \frac{\nu }{n}}{j}_{\left|\right|}(j_{\left|\right|}-j_{\left|\right|0})+\nabla ·\left({\displaystyle \frac{3}{2}}{D}_n{T}_e{\nabla }_{\perp }n\right)+\nabla ·\left(\chi n{\nabla }_{\perp }{T}_e\right)\end{array}$$

(27)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \omega }{\partial t}}=-\nabla ·\left(\omega {\mathbf{V}}_{E\times B}\right)+{\nabla }_{\left|\right|}{j}_{\left|\right|}\\ \hfill -\nabla ·\left(n{\mathbf{V}}_{mag}-n{\mathbf{V}}_{co-i}-n{\mathbf{V}}_{cf-i}\right)+\nabla ·\left({\mu }_i{\nabla }_{\perp }\omega \right)\end{array}$$

(28)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(n{v}_{\left|\right|i}\right)=-\nabla ·\left[n{v}_{\left|\right|i}\left({\mathbf{V}}_{E\times B}+{\mathbf{V}}_{\mathbf{co}-\mathbf{i}}+{\mathbf{V}}_{\mathbf{cf}-\mathbf{i}}\right)\right]+\nabla ·\left(D_n{v}_{\left|\right|i}{\nabla }_{\perp }n\right)\\ \hfill -{\partial }_{\left|\right|}{p}_e+2n\mathbf{b}·[({\mathbf{V}}_{E\times B}+{\mathbf{V}}_{co-i}+{\mathbf{V}}_{cf-i})\times {\mathsf{\Omega }}_0]+nR{\Omega }_0^2\mathbf{b}·\nabla R\end{array}$$

(29)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left[{\displaystyle \frac{m_e}{m_i}}(v_{\left|\right|e}-v_{\left|\right|i})+{\displaystyle \frac{1}{2}}{\beta }_e\psi \right]=\nu j_{\left|\right|}/n_e+{\partial }_{\left|\right|}\varphi -{\displaystyle \frac{1}{n_e}}{\partial }_{\left|\right|}{p}_e-0.71{\partial }_{\left|\right|}{T}_e\\ \hfill +{\displaystyle \frac{m_e}{m_i}}({\mathbf{V}}_{E\times B}+\mathbf{b}{v}_{\left|\right|i}+{\mathbf{V}}_{co-i}+{\mathbf{V}}_{cf-i})·\nabla (v_{\left|\right|i}-v_{\left|\right|e})\end{array}$$

(30)

Here, ${\beta }_e$ represents the electron beta with the expression ${\beta }_e={\mu }_0{p}_e/B^2$, and electromagnetic potential $\psi$ evolves as part of the parallel Ohm’s law. A field-aligned coordinate system is used for asymmetric simulations [26]:

$$\begin{array}{c}\hfill x=\psi ,y=\theta ,z=\zeta -{\int }_{{\theta }_0}^{\theta }{\displaystyle \frac{B_{\psi }{h}_{\theta }}{B_{0}R}}d\theta \end{array}$$

(31)

A field-aligned coordinate system (shown in Figure 2) is a special way to describe positions in a tokamak such that one coordinate always follows the magnetic field lines. This system is designed to match the geometry of the plasma and the shape of the magnetic field, which is usually twisted around the torus.

In the simulations, the electron inertial effects $n{\mathbf{V}}_{co-e}+n{\mathbf{V}}_{cf-e}$, $p_e{\displaystyle \frac{5}{2}}{\mathbf{V}}_{co-e}+p_e{\displaystyle \frac{5}{2}}{\mathbf{V}}_{cf-e}$ and $-n({\mathbf{V}}_{\mathbf{co}-\mathbf{e}}+{\mathbf{V}}_{\mathbf{co}-\mathbf{e}})$ in Equation (26), Equation (27), and Equation (28), respectively, and the electron rotation potential energy ${\displaystyle \frac{m_e}{m_i}}nR\nabla R{\Omega }_0^2·{\mathbf{V}}_{\mathbf{E}\times \mathbf{B}}+{\displaystyle \frac{m_e}{m_i}}nR{\Omega }_0^2\nabla R·{\mathbf{v}}_{\left|\right|\mathbf{e}}$ in Equation (27), are removed. All the inner boundary conditions of the variables are Neumann, and the outer boundaries have Dirichlet boundary conditions. We applied a shifted-circle shape equilibrium to set up the simulation. The grid file used in this paper is named cbm18, which has emerged as a popular choice due to its JET-like shifted-circular geometry [27]. The grid is based on a shifted-circular geometry, representative of conditions in modern tokamak devices such as JET. The grid is discretized in a field-aligned manner, which aligns the numerical grid with the magnetic field lines and facilitates the study of directional plasma dynamics. Figure 3 shows the equilibrium of cbm18. The resolution of the grid is $n_x=260$, $n_y=128$, the magnetic field at axis $B_0=2$ T, the major radius is $R_0=3.4$ m, and the minor radius is $r_a=1.2$ m. In Figure 4, the initial electron density (a), temperature (b), pressure (c), and parallel current density (d) profiles are shown.

Plasma potential, vorticity, density, and ion parallel velocity are obtained from the simulation. Using a new axisymmetric field solver, the resulting evolution of the parallel velocity (located at normalized $\psi =0.6{\psi }_0$) is shown for two poloidal locations (top and bottom) on the same flux surface in Figure 5. Oscillations at the top and bottom of the plasma are out of phase; they dampen with time to a quasi-steady state around 3 to 4 ms, then they keep increasing slowly. The major reason for the increase in the electrostatic potential is that the electric field is related to rotation in the non-inertial system (Equation (7)). In the later physical analysis section, an interactive relationship between parallel velocity and electrostatic potential can be seen based on Equation (49). Because the toroidal rotation rate is a constant number, rotation plays a role as a kind of source, contributing to the growth of velocity and potential.

From the results of the simulation, we also find that a ZF and GAM-like structure appears in the presence of uniform toroidal rotation. As shown in Figure 6, at the quasi-steady state, the parallel velocity changes from a ‘frozen’ state to an alternative structure under the effects of non-inertial force and drift. The amplitude of the velocity also increases significantly compared to the non-rotating simulation. In the following figures, the velocity is normalized by ion acoustic velocity.

$$V_{cs}=\sqrt{{\displaystyle \frac{{\gamma }_B{k}_B(T_i+T_e)}{m_i}}}$$

(32)

Here, $k_B$ is the Boltzmann constant, and ${\gamma }_B={\displaystyle \frac{5}{3}}$ is the ratio of the specific heat capacities.

In addition, the results of simulations reveal the evolution of parallel flow with time. The spatial radial structure is obtained and the GAM-like structure propagates across flux surfaces, as shown in Figure 7. The harmonic oscillations grow from the core area of the equilibrium and propagate to the boundary regions.This kind of feature is quite similar to GAMs.

Comparing the results for different rotation rates, it is obtained that the toroidal rotation rate has an influence on the poloidal velocity structure, as shown in Figure 8. Combined with the velocity spatial spectrum in Figure 9, we can also conclude that a higher rotation rate decreases the number of parallel flow poloidal modes.

Additionally, the mean Mach numbers of parallel velocity grow continuously with increases in toroidal rotation (Figure 10).

However, it is still unknown whether this kind of structure is ZF or GAMs. Here, we assume these $m\approx 0$ and $m\approx 1$ modes (m denotes poloidal number) are obtained as zonal-flow and GAM. Since both of them depend on radial electrostatic fields, we then perform spatial Fourier transform on electrostatic fields and obtain the $m\approx 0$ and $m\approx 1$ components; the results are shown in Figure 11. From the figure, it can be concluded that the amplitude of the $m=0$ mode is excited, but its frequency is damped and is influenced by toroidal rotation. On the other hand, for the $m=1$ mode, the frequency of the electrostatic field increases with the influence of non-inertial effects. Therefore, this shows that ZF and GAM are triggered in this rotating frame. By performing a further Fourier transform in time on the above two components, we then further obtain the GAM frequency and ZF amplitude (Figure 12). From the figure, we can conclude that the velocity of zonal flow and GAM frequency increase with rotation rates, but GAM damps with time.

GAM formation is considered as the phase difference between density and electrostatic potential [14]. Some experimental methods such as ECRH and NBI are believed to have effects on ZF and GAMs, leading to asymmetry of the poloidal density [12]. In Appendix A, we develop a simple mode and verify that ECRH can trigger plasma density with in–out asymmetry poloidally. In the following simulation, we run the cases with the injection of an in–out asymmetric density source (cosine), shown in Figure 13. It is found that an asymmetric source can enhance the amplitude of oscillation but contribute to the mean of parallel flow only with the existence of toroidal rotation. As discussed in Section 4, the injections of NBI and ECRH can induce poloidal asymmetry. Here, we suggest that the influence of NBI and ECRH on ZF and GAM is caused not only by density asymmetry but also by the change in the toroidal velocity profile [6].

4. Physical Explanation

To understand the mechanism of simulation results further, we use simple MHD equations to study non-inertial effects analytically. From Equations (15), (19), and (24), we can obtain a simplified reduced MHD mode by eliminating electron momentum and diffusion terms:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}=-\nabla ·[n{\displaystyle \frac{\mathbf{b}\times \nabla \varphi }{B}}]\end{array}$$

(33)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \Omega }{\partial t}}=\nabla ·[p\nabla \times ({\displaystyle \frac{\mathbf{b}}{B}})]+{\nabla }_{\left|\right|}{j}_{\left|\right|}+\nabla ·(en{\mathbf{V}}_{co-i}+en{\mathbf{V}}_{cf-i})\\ \hfill {\displaystyle \frac{\partial n{v}_{\left|\right|}}{\partial t}}=-\nabla ·\left[n{v}_{\left|\right|}({\mathbf{V}}_{E\times B}+{\mathbf{V}}_{co-i}+{\mathbf{V}}_{cf-i})\right]+{\partial }_{\left|\right|}{p}_e\end{array}$$

(34)

$$\begin{array}{c}\hfill +2n\mathbf{b}·({\displaystyle \frac{\mathbf{b}\times \nabla \varphi }{B}}\times {\mathsf{\Omega }}_{\mathbf{0}})+nR{\Omega }_0^2\mathbf{b}·\nabla R\end{array}$$

(35)

A common approach to solving the continuity is to use the following [21]:

$$\nabla \times {\displaystyle \frac{\mathbf{b}}{B}}\approx {\displaystyle \frac{2}{B}}\mathbf{b}\times \mathsf{\kappa }$$

(36)

where $\mathsf{\kappa }$ is the curvature term, and to split the $\mathbf{E}\times \mathbf{B}$ advection term into a divergence-free advection term and a divergence term:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}=-{\displaystyle \frac{1}{B}}\mathbf{b}\times \nabla \varphi ·\nabla n-n\nabla ·({\displaystyle \frac{1}{B}}\mathbf{b}\times \nabla \varphi )\end{array}$$

(37)

then approximate

$$\begin{array}{c}\hfill \nabla ·({\displaystyle \frac{1}{B}}\mathbf{b}\times \nabla \varphi )\approx {\displaystyle \frac{2}{B}}\mathbf{b}\times \mathsf{\kappa }·\nabla \varphi \end{array}$$

(38)

In Clebsch coordinates, the $\mathbf{E}\times \mathbf{B}$ advection term appears as follows [21]:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{B}}\mathbf{b}\times \nabla \varphi ·\nabla n={\displaystyle \frac{\partial \varphi }{\partial x}}{\displaystyle \frac{\partial n}{\partial z}}-{\displaystyle \frac{\partial \varphi }{\partial z}}{\displaystyle \frac{\partial n}{\partial x}}\\ \hfill {\displaystyle \frac{B_{t}IR}{B}}(-{\displaystyle \frac{\partial \varphi }{\partial y}}{\displaystyle \frac{\partial n}{\partial z}}+{\displaystyle \frac{\partial \varphi }{\partial z}}{\displaystyle \frac{\partial n}{\partial y}})+{\displaystyle \frac{R{B}_p{B}_t}{B^2{h}_{\theta }}}({\displaystyle \frac{\partial \varphi }{\partial x}}{\displaystyle \frac{\partial n}{\partial y}}-{\displaystyle \frac{\partial \varphi }{\partial y}}{\displaystyle \frac{\partial n}{\partial x}})\end{array}$$

(39)

For axisymmetric flows, the z derivatives (toroidal angle) are zero, so this term vanishes, leaving only the compression term. This leaves

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}=-n{\displaystyle \frac{2}{B}}\mathbf{b}\times \kappa ·\nabla \varphi \end{array}$$

(40)

Starting with the density equation, we look for solutions of the following form:

$$\begin{array}{c}\hfill n=n_0+n_1=n_0+n_{1,0}{e}^{i{k}_{\perp }x+i{k}_{\left|\right|2}y-i\omega t}\\ \hfill \varphi ={\varphi }_0+{\varphi }_1={\varphi }_0+{\varphi }_{1,0}{e}^{i{k}_{\perp }x+i{k}_{\left|\right|1}y-iwt}\\ \hfill v_{\left|\right|}=v_{\left|\right|0}+v_{\left|\right|1}=v_{\left|\right|0}+v_{\left|\right|1,0}{e}^{i{k}_{\perp }x+i{k}_{\left|\right|2}y-i\gamma t}\end{array}$$

(41)

In the equations, $k_{\Vert 1}$ and $k_{\Vert 2}$ denote the two different parallel (to the magnetic field) eigen wave numbers for electrostatic potential $\varphi$ and density n, separately. Linearizing Equation (40), assuming a simple circular cross-section, large aspect ratio, and keeping only the poloidal flow,

$$\begin{array}{c}\hfill \mathbf{b}\times \mathsf{\kappa }·\nabla \to {\displaystyle \frac{1}{R}}sin\theta {\displaystyle \frac{\partial }{\partial x}}\end{array}$$

(42)

we get

$$\begin{array}{c}\hfill n_{1,0}=n_0{\displaystyle \frac{2k_{\perp }}{BR\omega }}sin\theta {\varphi }_{1,0}\end{array}$$

(43)

Then, the linearized form of Equation (35) becomes as follows:

$$\begin{array}{c}\hfill -i\gamma v_{\left|\right|1}={\displaystyle \frac{m_i{R}^2{\Omega }_0^2{B}_p{B}_t}{e{B}^2{h}_{\theta }}}({\displaystyle \frac{B{h}_{\theta }}{B_p}}i{k}_{\perp }{v}_{\left|\right|1}sin\theta -{\displaystyle \frac{B}{R{B}_p^2}}i{k}_{\left|\right|2}{v}_{\left|\right|1}cos\theta )\\ \hfill +{\displaystyle \frac{2m_i{v}_{\left|\right|0}{k}_{\perp }R{\Omega }_0^2{B}_p{B}_t{\varphi }_1}{e\omega B^3{h}_{\theta }}}({\displaystyle \frac{B{h}_{\theta }}{B_p}}i{k}_{\perp }si{n}^2\theta -{\displaystyle \frac{B}{R{B}_p^2}}i{k}_{\left|\right|1}sin\theta cos\theta )\\ \hfill -{\displaystyle \frac{2i{m}_i{\Omega }_0{k}_{\perp }{v}_{\left|\right|0}}{eB}}{v}_{\left|\right|1}sin\theta -{\displaystyle \frac{v_{\left|\right|1}{m}_i{\Omega }_0^2}{e}}sin\theta cos\theta +e{T}_0{\displaystyle \frac{2i{k}_{\perp }{k}_{\left|\right|1}}{rBR\omega }}{\varphi }_{1}sin\theta \\ \hfill -{\displaystyle \frac{2i{n}_0{v}_{\left|\right|0}{k}_{\perp }}{BR\omega }}{\varphi }_{1}sin\theta +2i{\Omega }_0{\varphi }_1{k}_{\perp }sin\theta -{\displaystyle \frac{2{\varphi }_1{\Omega }_0^2{k}_{\perp }}{B_p\omega }}si{n}^2\theta \end{array}$$

(44)

The first two terms in Equation (44) apply Equation (39)’s rule.

$$n{\displaystyle \frac{m_{i}R{\Omega }_0^2}{eB}}\mathbf{b}\times \nabla R·\nabla V_{\left|\right|}=n{\displaystyle \frac{m_{i}R{\Omega }_0^2}{eB}}{\displaystyle \frac{R{B}_p{B}_t}{B{h}_{\theta }}}({\displaystyle \frac{\partial R}{\partial x}}{\displaystyle \frac{\partial V_{\left|\right|}}{\partial y}}-{\displaystyle \frac{\partial V_{\left|\right|}}{\partial x}}{\displaystyle \frac{\partial R}{\partial y}})$$

(45)

where

$${\displaystyle \frac{\partial R}{\partial x}}=-{\displaystyle \frac{BR}{R^2{B}_p^2}}{\displaystyle \frac{dZ}{ds}},{\displaystyle \frac{\partial R}{\partial y}}={\displaystyle \frac{B{h}_{\theta }}{B_p}}{\displaystyle \frac{dR}{ds}}$$

(46)

s is the distance along the field line. Since vorticity is approximately:

$$\begin{array}{c}\hfill \Omega ={\displaystyle \frac{m_i{n}_0}{B_0^2}}{\nabla }_{\perp }^2\varphi \end{array}$$

(47)

Then we can rewrite the vorticity equation into a linearized form:

$$\begin{array}{c}\hfill i\omega k_{\perp }^2{\displaystyle \frac{m_i}{B^2}}{\varphi }_1=4{\displaystyle \frac{i{k}_{\perp }^{2}e{T}_0}{\omega B^2{R}^2}}{\varphi }_{1}si{n}^2\theta +{\displaystyle \frac{2k_{\perp }{h}_{\theta }{m}_i{\Omega }_0^2}{\omega R{B}^2}}{\varphi }_{1}sin\theta cos\theta \\ \hfill +{\displaystyle \frac{2m_i{k}_{\perp }R{\Omega }_0^2{B}_t{\varphi }_1}{\omega B^2{h}_{\theta }}}(-i{h}_{\theta }{k}_{\perp }si{n}^2\theta +{\displaystyle \frac{1}{R{B}_p}}i{k}_{\left|\right|1}sin\theta cos\theta )\\ \hfill -2{\displaystyle \frac{i{\Omega }_0{m}_i{k}_{\perp }{h}_{\theta }}{B}}{v}_{\left|\right|1}(BR{k}_{\perp }sin\theta +k_{\left|\right|2}cos\theta )+{\nabla }_{\left|\right|}{j}_{\left|\right|}\end{array}$$

(48)

To remove the parallel current term from the vorticity equation, we average over a flux surface by defining $<·>=\oint ·d\theta$ for a large aspect ratio. When $k_1\approx 0,k_2\approx 0$, we obtain a simple equation operating based on Equation (44) and Equation (48):

$$i\gamma v_{\left|\right|1}={\displaystyle \frac{h_{\theta }{\Omega }_0^2{k}_{\perp }}{B_p\omega }}{\varphi }_1$$

(49)

Equation (49) gives the expression of the ZF’s ($k_1\approx 0,k_2\approx 0$) growth rate. When the perturbation frequencies dampen ($\omega$ = imaginary values), the parallel velocity can rise with the electrostatic potential. On the other hand, parallel velocity also contributes to the potential. That helps to explain why the potentials keep increasing at the end of the simulation instead of becoming horizontal in Figure 6. We can also obtain the frequency of ZFs from Equation (48):

$$\omega k_{\perp }^2<{\varphi }_1>{\displaystyle \frac{m_i}{B^2}}={\displaystyle \frac{2e{T}_0{k}_{\perp }^2}{\omega B^2{R}^2}}<{\varphi }_1>-{\displaystyle \frac{m_{i}R{\Omega }_0^2{B}_t{k}_{\perp }^2}{\omega B^2}}<{\varphi }_1>$$

(50)

Most terms cancel, leaving the following:

$${\omega }^2={\displaystyle \frac{e{T}_0}{m_i}}{\displaystyle \frac{2}{R^2}}-R{\Omega }_0^2{B}_t$$

(51)

Using the same method, we obtain Equations (44) and (48) for GAM ($k_1\approx 0,k_2\approx 1$):

$$\begin{array}{c}\hfill {\displaystyle \frac{m_i{R}^2{\Omega }_0^2{B}_p{B}_t}{e{B}^2{h}_{\theta }}}{\displaystyle \frac{B{h}_{\theta }}{B_p}}i{k}_{\perp }{v}_{\left|\right|1,0}exp(-i\gamma t)=-{\displaystyle \frac{2{\varphi }_{1,0}{\Omega }_0^2{k}_{\perp }}{B_p\omega }}exp(-i\omega t)\end{array}$$

(52)

$$\begin{array}{c}\hfill i\omega k_{\perp }^2{\displaystyle \frac{m_i}{B^2}}{\varphi }_{1,0}exp(-i\omega t)=2{\displaystyle \frac{i{k}_{\perp }^{2}e{T}_0}{\omega B^2{R}^2}}{\varphi }_{1,0}exp(-i\omega t)+m_{i}R{\Omega }_{0}B{k}_{\perp }^2{v}_{1,0}exp(-i\gamma t)\end{array}$$

(53)

Applying Equation (52) to Equation (53), we obtain the following:

$$\begin{array}{c}\hfill v_{1,0}exp(-i\gamma t)={\displaystyle \frac{2ieB}{m_i\omega B_p{B}_t{R}^2}}{\varphi }_{1,0}exp(-i\omega t)\end{array}$$

(54)

$$\begin{array}{c}\hfill {\omega }^2={\displaystyle \frac{e{T}_0}{m_i}}{\displaystyle \frac{2}{R^2}}+{\displaystyle \frac{2e{\Omega }_0{B}^4}{m_i{B}_p{B}_{t}R}}\end{array}$$

(55)

From the equations, we find that centrifugal force and drift contribute to the growth of parallel velocity and damping of the GAM; this result is qualitatively similar to Casson’s work [28]. On the other hand, the synergistic effect of centrifugal drift and Coriolis drift increases the frequency of GAM and makes this kind of $m=1$ structure propagate in the radial direction. This agrees qualitatively with the results seen in Figure 12.

Due to the gradient of the potential perturbation, the density perturbation is surrounded by an $\mathbf{E}\times \mathbf{B}$ drift vortex. Since the density is not homogeneous, the component of the corresponding particle flux parallel to the density gradient has divergence, leading to an increase of the initial perturbation, where ${\mathbf{V}}_{\mathbf{E}}$ is anti-parallel to $\nabla n$, and to a decrease where ${\mathbf{V}}_{\mathbf{E}}$ and $\nabla n$ are parallel. Consequently, the divergence of the $\mathbf{E}\times \mathbf{B}$ flow leads to a motion of the initial perturbation in the direction perpendicular to both the magnetic field ($\mathbf{B}$) and the density gradient ($\nabla n$). Similar to ITG turbulence, the evolution of the density perturbation $\tilde{n}$ is directly related to the compressibility of the $\mathbf{E}\times \mathbf{B}$ flow:

$${\displaystyle \frac{\partial \tilde{n}}{\partial t}}+\nabla ·\left(\tilde{n}{\mathbf{V}}_E\right)+\cdots =0$$

Here, the term $\nabla ·{\mathbf{V}}_E$ describes the radial propagation of perturbations. Here, centrifugal advections help to enhance the parallel perturbation. When the ion pressure gradient is negligible, the Coriolis drift acts analogously to the ion diamagnetic drift (shown in Figure 14).

$${\mathbf{V}}_{*\mathrm{i}}={\displaystyle \frac{T_i}{qB}}{\displaystyle \frac{\mathbf{B}\times \nabla n}{nB}}.$$

Both can drive a radial propagation of geodesic acoustic modes (GAMs), even when the pressure gradient term vanishes. Thus, in our model, the divergence of ${\mathbf{V}}_E$ (due to $E_r$ structure) combined with either the ion diamagnetic drift or Coriolis drift is crucial for the radial propagation of GAMs [29].

5. Conclusions and Discussion

In this study, drift-ordered two-fluid equations were derived from the collisional Vlasov equation in a non-inertial rotating frame under the cold ion approximation. These equations were implemented into the Hermes code, enabling simulations focused on axisymmetric ($n=0$) modes. The results reveal the emergence of zonal flows (ZFs) and geodesic acoustic modes (GAMs) influenced by centrifugal and Coriolis forces. Harmonic oscillations in momentum were observed to originate from the plasma core and propagate outward toward the edge, exhibiting a dynamic evolution of flow structures.

The simulations further demonstrate that both the amplitude of zonal flows and the frequency of GAMs increase with stronger toroidal rotation, consistent with previous theoretical and numerical findings [12]. To complement the simulation results, a simplified analytical model was developed to investigate the underlying physics of ZF and GAMs in a non-inertial frame. The analysis indicates that centrifugal and drift terms contribute to the growth of parallel velocity and the damping of GAMs, while the combined action of centrifugal and Coriolis drifts enhances GAM frequency and supports the radial propagation of the mode structure. Those simulation results can partly explain the flow induced by ECRH [5] on HL-2A.

While many previous studies have treated toroidal rotation primarily as a convection or centrifugal drift flow, the present work highlights the importance of including all relevant non-inertial terms—particularly the Coriolis effect—when modeling flow-driven phenomena. Notably, the phase difference between parallel momentum and electrostatic potential is identified as a possible driver of GAM activity. In the absence of ion magnetic drift (due to the cold ion assumption), Coriolis convection appears to play a central role in the radial transport and global structure of GAM and ZFs.

The non-inertial effects become significant in plasmas characterized by high rotation velocities, typically when the Mach number $M=V_{\varphi }/C_s$ approaches unity or higher, as well as in regions with pronounced radial gradients in flow velocity. Such conditions are often realized in the core or near the edge of tokamak plasmas during strong NBI or RF heating. The current model assumes fluid (non-kinetic) descriptions and neglects the influence of non-inertial terms on collisional dynamics and source/sink terms. Therefore, it is applicable for scenarios where kinetic effects, such as phase space trapping and Landau damping, are subdominant, and where equilibrium source and loss processes are not strongly modified by rotation-induced forces. Despite the insights gained, certain limitations of the cold ion model must be acknowledged. The absence of ion magnetic drift—an important contributor to GAM and ZF physics—as well as the exclusion of ion viscosity and heat flux effects, restricts the completeness of the model. Future work will focus on incorporating a hot ion model to more accurately capture the full range of rotation-driven phenomena, including the interplay between ion dynamics and electromagnetic effects.