Navigation Princeton Plasma Physics Laboratory THEORY DEPARTMENT   Theory seminars, 2004-2005   Standard Location: Theory Conference Room Standard Time: Thursday Refreshments are at 10:30am Seminar is at 10:45am   Please contact mailto:Gwangson Choeif you would like to present a seminar, suggest a speaker or would like to be notified of seminars by email. Please note: All visitor arrangements, including Site Access Notification , are the responsibility of the PPPL host. Future seminars are subject to changes due to speakers’ availability. Local, flexible speakers maybe asked to reschedule their seminars to give opportunity for guests to deliver talks.   Page last updated March 28, 2005.

Date

Speaker

# Title

September 23

September 30

October 7

October 14

October

21

Roscoe White

PPPL

Zonal Flow Dynamics and anomalous Transport

October

28

Ben Chandran

University of Iowa

Thermal conduction in turbulent magnetized plasmas, and its effects in clusters of galaxies.

November 4

November 11

November 18

no speaker

APS Week

November23

Bedros Afeyan

Polymath Research, Inc.

KEEN Waves: Long Lived Non-stationary Nonlinear Coherent Structures in

the Spectral Gap of the Vlasov-Poisson System

Note-Tuesday

December 3

Roberto Torasso

NYU

"Stability of ballooning modes in the Hall-MHD Model"

Note-friday

December 9

T.S. Hahm

PPPL

Review of IAEA  Theory papers

December 16

Theory meeting

December 28

No speaker

Lab holiday

December 30

no speaker

Lab Holiday

January 4

Prof. Baofeng Feng

University of Texas - Pan American

Stable solitary waves in two-dimensional stabilized

Kuramoto-Sivashinsky systems

Note tuesday

January

13

Hyeon Park

PPPL

Study of unsolved issues of m=1 oscillation (ìSawtoothî) via 2-D ECE Imaging System on TEXTOR*

January 20

January 27

Theory Meeting

February 3

February 10

David J Strozzi

MIT

Electron Trapping in Raman Scattering from Inhomogenous Plasmas

February 17

Theory meeting Guoyong Fu and Nikolai Gorelenkov

PPPL

Theory/energetic particle SFG micro-seminar

February 24

Mikhail Sitnov

Structure and dynamics of thin non-Harris current

sheets"

February 28

Greg Hammett

PPPL

Calculation of Particle Noise-induced Diffusion and Its Effect on ETG Simulations

Special Monday Theory seminar located in the Display Wall Room at 1:30

March 3

Jay Johnson

PPPL

The magnetospheric response to the solar wind

March 10

Improvement of confinement in tokamaks by weakening of

poloidal magnetic field near boundary.

March 17

Theory meeting

March 24

March 31

John Krommes

PPPL

The Fluctuation--Dissipation Theorem (and beyond)

April 7

Igor Kaganovich

PPPL

Dynamics of Ion Beam Interaction with Background Plasma

April 15

D. VanEster

Laboratory for Plasma Physics, Association “EURATOM – Belgian State”, ERM/KMS, Trilateral Euregio Cluster, Brussels, Belgium

A simple method to account for drift orbit effects when modeling radio frequency heating in tokamaks

Please note, this is a special Friday Seminar, to be held in the Theory Seminar room at 11:00 am.

April 18

Michael Hesse

Goddard Space Flight Center/NASA

Mechanisms of electron demagnetization in collisionless magnetic reconnection

Please note, this is a special Monday Seminar, to be held in the Display Wall room at 10:45 am.

April 21

Theory meeting

April 28

Harry Mynick

PPPL

Tutorial on Stellarator Transport I

May 5

Masaaki Yamada and Russell Kulsrud

PPPL

Study of Two-Fluid MHD Physics of Magnetic Reconnection in Laboratory and Space Plasmas

May 10

Scott Parker

University of Colorado, Boulder

Gyrokinetic simulation of the collisionless and semi-collisional tearing mode instability

This is a special Tuesday seminar, to be held at 2:00 pm in the Theory Seminar Room.

May 12

Harry Mynick

PPPL

Tutorial on Stellarator Transport III

May 19

Theory meeting

May 26

Hong Qin

PPPL

A footnote on the adiabatic invariants

June 2

June 9

Shuanghui Hu

UC Irvine

Discrete Alfven Eigenmodes Excited by Energetic Particles in High-Beta Tokamaks

June 16

June 23

June 30

Bruce Scott

IPP

Theory and Computation in Full-F Gyrokinetics

July 28

Matthew Hole

Australian National University

Stepped Pressure Profile Equilibria in Cylindrical Plasmas via Partial Taylor Relaxation

August 10

Xianzhu Tang

Los Alamos National Lab

Magnetic Relaxation in Laboratory and Astrophysical Plasmas

August 18

Hiroshi Naitou

Yamaguchi University

Gyro-Reduced MHD Simulation of Kinetic Internal Kink Modes

Special Friday Seminar.

Thursday, October 21 10:45 am

Roscoe White

PPPL

#### Zonal Flow Dynamisc and Anomalous Transport

Nonlinear equations for the slow space-time evolution of the

(DW-ITG) envelope and zonal flow (ZF) amplitude have

been derived  within a coherent 4-wave drift wave-zonal flow model.

In the local limit this model demonstrates

spontaneous generation of zonal flow and nonlinear drift

wave-zonal flow dynamics in toroidal plasmas.  The model

allows slow temporal and spatial

variations of the DW-ITG radial envelope, incorporating the

effects of equilibrium variations, $ie$ turbulence spreading and

size-dependence of the saturated wave intensities and transport

coefficients.

The competition between linear drive/damping and

drift wave spreading due to linear and nonlinear group velocity

and nonlinear energy transfer between DW and ZF

determines the saturation levels of the fluctuating fields. The

turbulence intensity level exhibits a transition from Bohm

scaling at small system size ($L/\rho$) to gyro Bohm for large

system size.

This system exhibits chaotic behavior and intermittency, depending on

system size and proximity to marginal stability.  The model is

explored using Symbolic Dynamics, which is shown to provide a

useful means of analyzing

turbulence levels using a single time sequence data stream.

 Thursday, October 28 10:45 am   Ben Chandran   Department of Physics & Astronomy Universtiy of Iowa   Thermal conduction in turbulent magnetized plasmas, and its effects in clusters of galaxies.   Clusters of galaxies are the largest gravitationally bound objects in the universe. They host a wealth of interesting phenomena, from star formation on a massive scale to powerful extragalactic jets powered by supermassive black holes.  In this talk I will describe an outstanding theoretical puzzle in the study of galaxy clusters, namely the need to explain the observationally inferred heating of galaxy-cluster plasmas. One of the most important heating mechanisms is thermal conduction, which is modified by turbulent intracluster magnetic fields.  I will describe how the Rechester-Rosenbluth theory of heat conduction in stochastic fields can be applied to this case, in which the mean field is negligible. I will also present recent analytic and numerical results on electron diffusion and field-line trajectories in strong magnetohydrodynamic turbulence, as well as results on two other important heating mechanisms: turbulent intracluster motions and active galactic nuclei at the centers of clusters.

 Tuesday,  November 28 10:45 am       KEEN Waves: Long Lived Non-stationary Nonlinear Coherent Structures in the Spectral Gap of the Vlasov-Poisson System   Bedros Afeyan, Polymath Research Inc., Pleasanton, CA     We will discuss theoretical, computational and experimental results concerning ponderomotively driven (and released) Kinetic Electrostatic Electron Nonlinear (KEEN) waves. Direct high resolution Vlasov-Poisson simulations, nonlinear coupled mode theory in phase space, and optical mixing experiments on the Trident laser facility at LANL will be described whereby KEEN waves' existence was discovered, further explained and experimentally verified, respectively. Implications to laser-plasma interaction physics and the large set of unresolved anomalies in SRS spectra, for instance, will be touched upon. Mutual interaction of KEEN waves as well as their interactions with EPWs will also be described which open up new vistas of plasma physics in the spectral gap that was thought to exist in plasma physics based on linear theory, quasilinear reasoning and small amplitude nonlinear theories. KEEN waves exist deep in the nonlinear regime with remarkable stability properties. We will show links to Vlasov-Maxwell simulations and general nonlinear paradigms of instability saturation which have to be revisited in light of the existence of KEEN waves.

 Friday Dec 3,  10:45 am       "Stability of ballooning modes in the Hall-MHD Model"   Roberto Torasso    NYU   The equations of the ballooning modes are derived within the Hall~magnetohydrodynamics (HMHD) model and given a standard  Hamiltonian form. The Hamiltonian structure of the equations is used to derive sufficient conditions for stability. In most cases, ideal magnetohydrodynamics (MHD) stability of ballooning modes implies HMHD stability, as is the case for tokamak configurations as well as plasmas with constant entropy or incompressible plasmas. However,  in the case of closed-line systems such as the field-reversed configuration (FRC), or in a typical magnetospheric magnetic field,   MHD ballooning stability does not automatically guarantee HMHD  stability. For the explicitly solvable configuration of the Z-pinch it is shown that the sufficient condition derived here is also necessary for stability. Wednesday Jan 5  10:45 am   Title: Stable solitary waves in two-dimensional stabilized Kuramoto-Sivashinsky systems   Prof. Baofeng Feng University of Texas - Pan American   By linearly coupling generalized two-dimensional Benney equations to an extra linear dissipative equation, two-dimensional (2D) extensions of a stabilized Kuramoto - Sivashinsky system are developed. The models apply to the description of surface waves on 2D liquid layers in various physical settings. A perturbation theory is developed by treating dissipation and gain in the models as small perturbations. Stable solitary wave solutions are predicted and numerically confirmed. Thursday  Feb 3 10:45 am     Study of unsolved issues of m=1 oscillation (ìSawtoothî) via 2-D ECE Imaging System on TEXTOR*    Hyeon Park, PPPL,   A novel 2-D Electron Cyclotron Emission Imaging (ECEI) system for measuring electron temperature fluctuations applied to study sawtooth crash physics on TEXTOR. A 128-channel prototype imaging system, covering 8 cm (radial) by 16 cm (vertical), with high spatial (1 cm x 1 cm) and temporal (up to ~5 msec) resolution employs large aperture optics to form a spatially resolved image of several cyclotron layers simultaneously. The ECEI system, which includes a 16-channel vertical array of antennas and wide-band transmission line, has provided behaviours of the electron temperature fluctuations similar to the ìmagnetic reconnectionî process during crash time of m=1 (sawtooth) oscillations, revealing details not accessible through conventional methods (1-D ECE and/or tomography). Long history of theoretical and experimental study of m=1 oscillation left remnants of mysteries and unresolved issues of physics such as current sheet and reconnection time scale. Details of poloidal and toroidal asymmetries of the measured electron fluctuation by 2-D ECEI may provide a clue of these issues.

 Thursday Feb   10 10:45 am   David strozzi     Electron Trapping in Raman Scattering from Inhomogenous Plasmas Thursday Feb   24 10:45 am     Title: Structure and dynamics of thin non-Harris current sheets     Mikhail I. Sitnov   Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742       Recent multi-probe observations of the current sheets in the tail of Earth's magnetosphere and laboratory experiments, such as MRX, revealed that the sheet properties become quite unusual when its thickness decreases to a few thermal ion gyroradii. Such a thin current sheet may be embedded into a thicker plasma sheet or split into two sheets, in contrast to the classical equilibrium theory and in the absence of any conventional reconnection signatures. Thin current sheets in the tail are often very dynamic, flapping in the north-south direction. However, their flapping waves are also unusual, as they propagate too slowly or even in the direction opposite to the main current flow, at variance with presently dominating theories and simulation results. To address these issues, we present a steady-state Vlasov theory of thin current sheets, which generalizes the well-known isotropic Harris [1962] equilibrium (a Cartesian geometry analog of the Bennett pinch) by assuming anisotropic and non-gyrotropic plasmas and takes into account an additional invariant of particle motion, an analog of the magnetic moment, applicable for regions with strong gradients. We consider the dynamics of this new class of non-Harris equilibria using a full-particle code. The new equilibrium theory explains the effects of the current sheet embedding and bifurcation, while simulations confirm the structural stability of these new equilibria. The theory reveals an important role of non-gyrotropic effects arising from the figure-of-eight ion orbits in thin sheets. We also discuss the distinctive features of very thin sheets, with the thickness less than the thermal ion gyroradius, which is of relevance to the MRX experiment. Thursday March 3  10:45 am     Title: The magnetospheric response to the solar wind   Jay Johnson, PPPL   Abstract. Understanding the dynamical evolution of the Earth’s magnetosphere is of practical interest because the magnetosphere occasionally evolves into a disturbed state that can affect the quality of life through large scale damage to power grids, loss of communications, and disruption of satellite-based defense strategy.  The magnetospheric dynamics are ultimately driven by the solar wind while various dissipative processes cause the magnetosphere to evolve toward a quiescent state in the absence of strong driving.  The magnetospheric dynamics are commonly characterized with various information-dynamical measures to understand dimensionality as well as the most important dependencies among observed plasma and electromagnetic field variables in the coupled solar wind/magnetosphere system.  We identify nonlinear dependencies using mutual information and cumulant-based cost as discriminating statistics and discuss implications for modeling the magnetosphere and predicting its evolution.  Application of the techniques to understand the dynamics of solar flares and neurosystems is also discussed. Thursday  March 10  10:45 am     Title: Improvement of confinement in tokamaks by weakening of poloidal magnetic field near boundary.   V. Yankov.   Abstract. Theory of turbulent equipartition and experiment both indicate that density, pressure, and temperature profiles follow to poloidal magnetic field profile. An example is TFTR current ramp-down experiments. Therefore it is suggested to change magnetic geometry between core and boundary by toroidal conductors and/or plasma current. As a result density and temperature gradients will become steeper, and stored energy will be higher with low boundary plasma parameters. Suggested new mode of confinement may essentially simplify achieving of ignition. Stellarator applications will be discussed. Thursday, March 31, 10:45 am   John A. Krommes, PPPL, Princeton University The Fluctuation--Dissipation Theorem (and beyond) A pedagogical introduction to the classical Fluctuation--Dissipation Theorem (FDT) is given.  No new results are presented.  Rather, the following topics of current relevance are reviewed in an elementary fashion: intuition behind the FDT; derivation for an unmagnetized plasma; the role of normal modes; extension to gyrokinetics; application to the $\delta f$ simulation algorithm; relationship to steady-state turbulence. Additional topics to be discussed if time permits include the Entropy Paradox and the possible use (and abuse) of thermostats in stabilizing the $\delta f$ sampling noise. Thursday, April 7, 10:45am Speaker: Igor Kaganovich, PPPL Title: Dynamics of Ion Beam Interaction with Background Plasma   Abstract: The present concept for heavy ion fusion is based on the compression of intense ion beam pulses by means of ballistic focusing. To overcome the ion beam space-charge force, the ion beams are transported and focused in a background plasma. This should enable more than 10,000 times compression of ion beam pulse (100 times transversely, and 100 longitudinally.) Currently, 2,000 times compression has been already achieved (100 times transversely, and 20 longitudinally.) Theory and simulations of the plasma response to the propagation of an intense ion beam pulse will be reviewed. Visualization of the electron dynamics reveals the complex nature of the physical processes. Particular attention will be paid to an analysis of common misconceptions and difficulties encountered in studies of collective phenomena in ion beam-plasma interactions. A simple method to account for drift orbit effects when modeling radio frequency heating in tokamaks   D. Van Eester   Laboratory for Plasma Physics, Association “EURATOM – Belgian State”, ERM/KMS, Trilateral Euregio Cluster, Brussels, Belgium Monday, April 18, 10:45 am, Display Wall Room   Speaker:  Michael Hesse, Goddard Space Flight Center/NASA Title:  Mechanisms of electron demagnetization in collisionless magnetic reconnection   Abstract: Magnetic reconnection relies on the violation of the frozen flux constraint in a localized region of space. In this region, fluid elements of all individual plasma species relinquish their ties to the magnetic field, and they exchange magnetic connections. This phenomenon occurs on characteristic scales that depend on the nature of each plasma species. In a collisionless system, these scales are determined by certain kinetic processes that permit the scattering of individual particles off magnetic flux tubes. Since electrons are the lightest of species in classical plasmas, we will focus in this presentation on the electron dynamics in the inner reconnection region. Specifically, we will investigate the mechanisms that foster electron demagnetization in anti-parallel and guide-field reconnection cases. For both cases, we will present results from 2.5D and 3D kinetic simulations, as well as from analytic theory that is applied to the determination of demagnetization scale sizes. Finally, we will discuss the implications the electron results have on the dynamics of heavier species. Thursday, April 28, 10:45 am, Theory Seminar Room   Speaker: Harry Mynick, PPPL Title: Tutorial on Stellerator Transport I.   Abstract: An introductory presentation on stellarator neoclassical transport will be given with a discussion on the various transport mechanisms, ambipolarity constraint, ion & electron roots, etc. The talk will be light on formalism, emphasizing the basic physics. Thursday, May 5, 10:45 am, Theory Seminar Room   Speakers: Masaaki Yamada and Russell Kulsrud, PPPL Title: Study of Two-Fluid MHD Physics of Magnetic Reconnection in Laboratory and Space Plasmas   Abstract: In the past few years, the MRX experiment has generated key data to understand the physics of collisionless reconnection. We will highlight the most recent findings of the MRX (Magnetic Reconnection Experiment) laboratory experiments which address the two-fluids MHD physics of magnetic reconnection [1] and the results are compared with the recent space observations [2]. With the recent upgrade of MRX, our experimental operation regime has moved from the collisional to the collisionfree regime, and two-fluid effects have become more evident. The recent development from the one-fluid MHD to the two-fluid MHD formulations is presented to illuminated the physics of the Hall MHD in a collisionfree reconnection layer. In particular, a clear experimental verification of an out-of-plane Hall quadrupole field has been made in a Harris-like neutral sheet, with the width comparable to the ion skin depth, during magnetic reconnection [3]. High frequency fluctuations observed in the reconnection layer [4] also exhibit two fluid effects demonstrating different kinematics for electrons and ions. The recently developed theory investigate the causal relationship between the observed fast reconnection rate, magnetic turbulence and the Hall quadrupole fields are discussed in this talk. In collaboration with H. Ji, S. Gerhardt, A. Kuritsyn, Y. Ren, Y. Wang. Tuesday, May 10, 2:00 pm, Theory Seminar Room   Speaker: Scott Parker, University of Colorado, Boulder Title: Gyrokinetic simulation of the collisionless and semi-collisional tearing mode instability   Abstract: The nonlinear evolution of the collisionless and semi-collisional tearing mode instability is studied using an electromagnetic gyrokinetic particle-in-cell simulation. Drift-kinetic electrons are used. Simulation results show excellent agreement with linear eigenmode analysis. Collisionless nonlinear saturation compares well with existing theory in terms of saturation level and electron bounce oscillations. Electron-ion collisions are included to study the semi-collisional regime. The algebraic growth stage is observed and compares favorably with theory. Nonlinear island saturation is found to depend on collisionality. Thursday, May 12, 10:45 am, Theory Seminar Room   Speaker: Harry Mynick, PPPL Title: Tutorial on Stellarator Transport III   Abstract: Stefan Gerhardt's talk last week on flows in stellaratorsintroduced some of the approaches to transport optimizationwhich are now being implemented in the present generation ofstellarator experiments in the US and abroad. This 3rd and final tutorial talk on stellarator transport will address in more depth the physics underlying these and other transport optimization approaches, discussing as part of this energetic particle confinement and mitigating turbulent transport in stellarators. Thursday, May 26, 10:45 am, Theory Seminar Room   Speaker: Hong Qin,PPPL Title: A footnote on the adiabatic invariants   Abstract: It turns out that the adiabatic invariant for the time-dependent oscillator equation is just an asymptotic approximation to an exact invariant, the Courant-Snyder invariant. A thorough study of the symmetry and invariance of the related dynamics reveals many deeper, interesting structures that have important implications. For example, we can show that the adiabatic invariant is actually a stronger invariant than that proved by Arnold (1978), and a more general one than that proved by Kulsrud (1957) and Kruskal (1961). Thursday, June 9, 10:45 am, Theory Seminar Room   Speaker: Shuanghui Hu, UC Irvine   Title: Discrete Alfven Eigenmodes Excited by Energetic Particles in High-Beta Tokamaks Abstract: A new type of high-n discrete Alfven eigenmodes (termed alpha-TAE) is found in the high-beta second ballooning-mode stable toroidal plasmas. Here, n is the toroidal wavenumber, alpha denotes the ballooning drive due to pressure gradient and curvature, and beta is the ratio of plasma to magnetic pressures. Multiple branches of the alpha-TAEs are observed due to the existence of multiple alpha-induced potential wells and, correspondingly, the eigenmodes can either be marginally stable or experience small but finite Alfven-continuum damping. Due to their trapped features, the alpha-TAEs exist independently of the toroidal Alfven frequency gap, in contrast to the usual TAE (toroidicity-induced Alfven eigenmode). Both the quasi-marginally stable and the finitely damped alpha-TAEs can be readily destabilized by energetic particles via wave-particle resonances. For negative magnetic shear, the alpha-TAE can extend into the small-alpha regime and evolve into the low-beta TAE. A two-dimensional eigenmode analysis, employing WKB approximation in the radial direction, demonstrates that the global alpha-TAE can be formed around the radial location with the maximal alpha value. Thursday, June 30, 10:45 am, Theory Seminar Room   Speaker: Bruce Scott, IPP, Garching, Germany   Title: Theory and Computation in Full-F Gyrokinetics The theory behind the FEFI code is presented. The gyrokinetic model is reviewed and then the particular version used is motivated by correspondence and computational tractability. The field theory version of gyrokinetics is used to ensure the existence of conservation laws. Large amplitude dynamics can be treated if the wavelength is small, and vice versa, based upon the time scale ordering that the ExB vorticity must be small compared to any gyrofrequency. Energy transfer and numerical issues involved in collisions and Alfven dynamics are discussed. The shear Alfven damping result is shown. The FEFI code is expected to be finished within 2005, to be applied to problems such as self consistent pedestal physics. Thursday, July 28, 10:45 am, Theory Seminar Room   Speaker: Matthew Hole, Australian National University, Australia   Title: Stepped Pressure Profile Equilibria in Cylindrical Plasmas via Partial Taylor Relaxation Magnetohydrodynamic (MHD) equilibrium states in three-dimensions (3D) with smooth pressure profiles have long bedeviled containment theory. Magnetic islands are formed at rational surfaces, resulting in pressure flattening. In this work, a new model is presented: the stepped-pressure profile equilibrium. The system comprises multiple Taylor relaxed plasma regions, which are separated by ideal MHD barriers. Such a model is well posed mathematically, and follows rigorous existence proofs of MHD equilibria with stepped pressure in weakly toroidally asymmetric plasmas. In addition to a description of 3D equilibria, the model is also motivated by observations of internal transport barrier formation at irrational flux surfaces in magnetic confinement experiments. This formulation may provide a minimum energy explanation for the existence of ITB's. In cylindrical geometry, where analytic progress is possible, we have constructed equilibrium solutions, and analyzed stability by a variational formulation. We show the existence of tokamak-like equilibria, with increasing smooth safety factor and stepped-pressure profiles. Unlike reverse field pinch q profiles, only the plasma core necessarily has reverse magnetic shear. Recent observations of ITB formation at minima in the safety factor provide some preliminary guidance for these calculations. Wednesday, August 10, 10:45 am, Theory Seminar Room Speaker: Xianzhu Tang, Los Alamos National Laboratory Title: Magnetic Relaxation in Laboratory and Astrophysical Plasmas Magnetic relaxation is an extreme form of self-organization by which magnetic energy and helicity injected on small scale are transported to and accumulated on system scale magnetic fields by small scale plasma fluctuations. A large class of fusion concepts such as RFP, Spheromak, ST-PCC, and ST-CHI, relies on magnetic relaxation to achieve high efficiency in comparison with the usual Tokamak and stellarator devices. Naturally occurring plasma such as that in the giant radio lobes is also subject to magnetic relaxation. The physical implications of Taylor's helicity-conserving minimum energy state in laboratory and radio lobe plasmas are understood in terms of two classes of resonance phenomena, which are demonstrated using Chandrasekhar-Kendall force-free eigenmodes. The linear resonances are regularized by any of plasma inertia, finite pressure, and non-uniform normalized parallel current density. The emergence of bifurcated regularized solutions is shown, along with their physics interpretation and significance. Finally, the nonlinear dynamics that lead to magnetic relaxation is investigated by initial value 3D MHD simulations. The relaxation of the driven plasma is shown to follow a helical instability cascade. Friday, August 18, 10:45 am, Theory Seminar Room Speaker: Hiroshi Naitou, Yamaguchi University Title: Gyro-Reduced MHD Simulation of Kinetic Internal Kink Modes The simulation of kinetic internal kink modes using gyro-reduced-MHD equations will be presented. The basic equations are the moment equations obtained from the original gyrokinetic Vlasov-Poisson-Ampere system. It is found that a three-field model can accurately describe the linear and nonlinear evolution of these modes with the stabilizing electron diamagnetic effect. However, this stabilization is found to be incomplete and the residual unstable modes can generate vortices due to the Kelvin-Helmholtz-like secondary instablity. The strong coupling between the kinetic internal kink modes and the resulting vortices may explain the mysterious sawtooth related phenomena. The five-field model including the Landau closure is developed to estimate the effects of the ion-Landau-damping on these modes. The necessity of benchmarking between the present code and gyrokinetic particle codes will be discussed.