arXiv subjects

Ö. D. Gürcan

Publications and source records attributed to Ö. D. Gürcan.

12 recordsLinked to original sources

arXiv2026

Solutions of the Navier-Stokes Equation Through Affine Transformations: The Triad Triplet

Considering triads in helical decomposition of three dimensional Navier-Stokes turbulence, three consecutive triads of the same shape and class, where the reference wave-number appears as the smallest, the middle and the largest wave-numbers respectively, constitutes an interesting object called the triad triplet that allows tracing the local transfer in wave-number space due to a given shape and class of triads. It is shown that, evolution equations for the triad triplet can be obtained from the equations of the three legs of a single triad that are put together via affine transformations involving scaling, rotation and reflection. Known power law solutions corresponding to constant flux of energy and helicity appear as exact solutions of a chain of triad triplets, with particular implications for the triadic phases. More interestingly, an exact, time dependent solution is available on an isolated triad triplet, that can be expressed using what appears to be straightforward generalization of Jacobi elliptic functions. These solutions, constitute novel exact nonlinear solutions of the inviscid limit of the Navier-Stokes equations.

physics.flu-dyn↗

arXiv2026

On Phase Transition of ITG Turbulence in the Dimits shift

The transition between turbulent and zonal flow dominated states is investigated by varying the ion temperature gradient in nonlinear gyrokinetic simulations. Independent gradient scans reveal three distinct regimes: a turbulent regime at high gradients, a zonal flow dominated regime with strongly reduced heat transport at low gradients, and an intermediate regime characterized by intermittent switching between these two states. These different regimes can be classified using an order parameter, defined as the fraction of the zonal to total free energy in the system. To assess the memory effects, the temperature gradient is first lowered gradually from high values that result in fully developed turbulence, to lower values in the Dimits shift region that form strong zonal flows, and then is slowly increased back. Once the zonal flows form, and efficiently suppress turbulence, they can persist at higher gradients, leading to an asymmetric response implying a hysteresis loop. It is observed that, in the zonal flow dominated state, the free energy is mostly condensated in the largest radial scale, with a steep slope of the $k_x$ spectrum, while in the turbulent state, it exhibits a wider spectrum with two distinct slopes.

physics.plasm-ph↗

arXiv2025

Self-organisation through layering of $β$-plane like turbulence in plasmas and geophysical fluids

Staircase formation and layering is studied in simplified, potential vorticity conserving models of plasmas and geophysical fluids, by investigating turbulent self-organisation and nonlinear saturation with different mechanisms of free energy production -- forcing or linear instability -- and with standard or modified zonal flow responses. To this end, staircase formation in both the standard and modified Charney-Hasegawa-Mima equations with stochastic forcing, along with two different simple instability driven models -- one from a plasma and from a geophysical context -- are studied and compared. In these studies, it is observed that $β$-plane turbulence that does not distinguish between zonal and non-zonal perturbations (i.e., standard zonal response) gradually forms large-scale, elliptic zonal structures that merge progressively, regardless of whether it is driven by forcing (though it should be slow enough to allow wave couplings) or by the baroclinic instability, using for example a two-layer model. Conversely, the plasma system, with its modified zonal response, can rapidly form straight, stationary jets of well-defined size, again regardless of the way it is driven: by stochastic forcing or by the dissipative drift instability. Furthermore, the instability-driven plasma system exhibits a phase transition between a zonal flow dominated state and an eddy dominated state. In both states, saturation is possible without large-scale friction.

physics.plasm-ph↗

arXiv2025

Flux Jamming, Phase Transitions and Layering in Turbulent Magnetized Plasma

This paper discusses transport barrier formation and layering as consequences of jam formation. Extensive use is made of analogies with the theory of traffic flow in one dimension. The relation of flux jamming to motility induced phase separation (MIPS) is explained. Two routes to heat flux jamming are identified. The first is due to a rollover in the heat flux-pulse size relation, i.e. $dQ_T(δT)/dδT<0$, and is similar to the condition of flux-gradient bistability. The second occurs when the delay time between pulse and heat flux exceeds a critical value. This does not require bistability and tends to occur near marginality. This analysis yields an estimate of the answer to the eternal question of 'how near is "near"?'. Staircase development is shown to follow jamiton train formation. The relation of jamming of avalanches to phase transitions in drift wave-zonal flow turbulence is elucidated. The formation of outward propagating blob trains and inward propagating void trains is demonstrated. The important role of turbulence spreading is identified.

physics.plasm-ph↗

arXiv2022

Phase and amplitude evolution in the network of triadic interactions of the Hasegawa-Wakatani system

Hasegawa-Wakatani system, commonly used as a toy model of dissipative drift waves in fusion devices is revisited with considerations of phase and amplitude dynamics of its triadic interactions. It is observed that a single resonant triad can saturate via three way phase locking where the phase differences between dominant modes converge to constant values as individual phases increase in time. This allows the system to have approximately constant amplitude solutions. Non-resonant triads show similar behavior only when one of its legs is a zonal wave number. However when an additional triad, which is a reflection of the original one with respect to the $y$ axis is included, the behavior of the resulting triad pair is shown to be more complex. In particular, it is found that triads involving small radial wave numbers (large scale zonal flows) end up transferring their energy to the subdominant mode which keeps growing exponentially, while those involving larger radial wave numbers (small scale zonal flows) tend to find steady chaotic or limit cycle states (or decay to zero). In order to study the dynamics in a connected network of triads, a network formulation is considered including a pump mode, and a number of zonal and non-zonal subdominant modes as a dynamical system. It was observed that the zonal modes become clearly dominant only when a large number of triads are connected. When the zonal flow becomes dominant as a 'collective mean field', individual interactions between modes become less important, which is consistent with the inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation is discussed for the same parameters and various forms of the order parameter are computed. It is observed that nonlinear phase dynamics results in a flattening of the large scale phase velocity as a function of scale in direct numerical simulations.

nlin.CD↗

arXiv2019

Generalized curvature modified plasma dispersion functions and Dupree renormalization of toroidal ITG

A new generalization of curvature modified plasma dispersion functions is introduced in order to express Dupree renormalized dispersion relations used in quasi-linear theory. For instance the Dupree renormalized dispersion relation for gyrokinetic, toroidal ion temperature gradient driven (ITG) modes, where the Dupree's diffusion coefficient is assumed to be a low order polynomial of the velocity, can be written entirely using generalized curvature modified plasma dispersion functions: Knm's. Using those, Dupree's formulation of renormalized quasi-linear theory is revisited for the toroidal ITG mode. The Dupree diffusion coefficient has been obtained as a function of velocity using an iteration scheme, first by assuming that the diffusion coefficient is constant at each v (i.e. applicable for slow dependence), and then substituting the resulting v dependence in the form of complex polynomial coefficients into the Knm's for verification. The algorithm generally converges rapidly after only a few iterations. Since the quasi-linear calculation relies on an assumed form for the wave-number spectrum, especially around its peak, practical usefullness of the method is to be determined in actual applications.

physics.plasm-ph↗

arXiv2019

Spiral chains in wavenumber space of two dimensional turbulence

Abstract Self-similar, fractal nature of turbulence is discussed in the context of two dimensional turbulence, by considering the fractal structure of the wave-number domain using spirals. In loose analogy with phyllotaxis in plants, each step of the cascade can be represented by a rotation and a scaling of the interacting triad. Using a constant divergence angle and a constant scaling factor, one obtains a family of such fractals depending on the distance of interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the supergolden ratio or some small Pisot numbers. While spiral chains can represent mono-fractal models of self-similar cascade, which can span a very large range in wave-number domain with good angular coverage, it is also possible that spiral chains or chains of consecutive triads play an important role in the cascade. As numerical models, the spiral chain models based on decimated Fourier coefficients have problems such as the dual cascade being overwhelmed by statistical chain equipartition due to almost stochastic evolution of the complex phases. A generic spiral chain model based on evolution of energy is proposed, which is shown to recover the dual cascade behavior in two-dimensional turbulence.

physics.flu-dyn↗

arXiv2017

A nested polyhedra model of isotropic MHD turbulence

A nested polyhedra model has been developed for magnetohydrodynamic (MHD) turbulence. Driving only the velocity field at large scales with random, divergence free forcing results in a clear, stationary $k^{-5/3}$ spectrum for both kinetic and magnetic energies. Since the model naturaly effaces disparate scale interactions, does not have a guide field and avoids injecting any sign of helicity by random forcing, the resulting three dimensional k-spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the time scale for the equipartition offset with those of the smallest scales shows a $k^{-1/2}$ scaling.

physics.flu-dyn↗

arXiv2017

Envelope modulations of kinetic plasma waves: A general framework

The general framework of envelope modulations for plasma turbulence in the presence of strong magnetic fields is discussed. It is shown that the problem can be formulated using a functional which is equivalent to the plasma dielectric function for monochromatic waves. Therefore expanding around its linear wave solutions, one can describe the weakly nonlinear evolution of waves. The resulting "quasi-linear" dielectric tensor, can describe the evolution of small scale fluctuations advected and modulated by background flows and corrugations, as well as the evolution of the large scale flows driven by small scales. As a concrete example of the framework, gyrokinetic evolution of zonal flows and drift waves is considered and the resulting nonlinear Schrödinger equation is derived.

physics.plasm-ph↗

arXiv2017

Linear gyrokinetic investigation of the geodesic acoustic modes in realistic tokamak configurations

Geodesic acoustic modes (GAMs) are studied by means of the gyrokinetic global particle-in-cell code ORB5. Linear electromagnetic simulations in the low electron beta limit have been performed, in order to separate acoustic and Alfvénic time scales and obtain more accurate measurements. The dependence of the frequency and damping rate on several parameters such as the safety factor, the GAM radial wavenumber and the plasma elongation is studied. All simulations have been performed with kinetic electrons with realistic electron/ion mass ratio. Interpolating formulae for the GAM frequency and damping rate, based on the results of the gyrokinetic simulations, have been derived. Using these expressions, the influence of the temperature gradient on the damping rate is also investigated. Finally, the results are applied to the study of a real discharge of the ASDEX Upgrade tokamak.

physics.plasm-ph↗

arXiv2017

A nested polyhedra model of turbulence

A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave-vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio, and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave-vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave-vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads aranged approximately in a spherical form. For each vertex (i.e. discretized wave-vector) in this space, there are either 9 or 15 pairs of vertices (i.e. wave-vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs. Transforming the equation in Fourier space, to a network of interacting nodes, that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. Such a model gives the usual Kolmogorov spectrum. Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range can be considered with a much lower number of degrees of freedom. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive three dimensional shell models.

physics.flu-dyn↗

arXiv2012

Characterization of predator-prey dynamics, using the evolution of free energy in plasma turbulence

A simple dynamical cascade model for the evolution of free energy is considered in the context of gyrokinetic formalism. It is noted that the dynamics of free energy, that characterize plasma micro-turbulence in magnetic fusion devices, exhibit a clear predator prey character. Various key features of predatory prey dynamics such as the time delay between turbulence and large scale flow structures, or the intermittency of the dynamics are identifed in the quasi-steady state phase of the nonlinear gyrokinetic simulations. A novel prediction on the ratio of turbulence amplitudes in different parts of the wave-number domain that follows from this simple predator prey model is compared to a set of nonlinear simulation results and is observed to hold quite well in a large range of physical parameters. Detailed validation of the predator prey hypothesis using nonlinear gyrokinetics provides a very important input for the effort to apprehend plasma microturbulence, since the predator prey idea can be used as a very effective intuitive tool for understanding.

physics.plasm-ph↗