Search NASA⌕ Search

SEARCH · Search NASA

Results for “Nonlinear approximation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 721 records · Page 40
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

HYDRODYNAMIC ANALYSIS OF MACROALGAE LOCAL MODEL USING COMPUTATIONAL FLUID DYNAMICS

In this article, a local scale, fully nonlinear coupled fluid-structural interaction (FSI) sugar kelp model has been developed using a computational fluid dynamics (CFD) method. In this model, to be consistent with available experimental data, the sugar kelp is approximated as elongated rectangles with smoothed isosceles triangles at the ends and a single kelp model with one end fixed in a channel with constant current model is developed. Several different current speeds are simulated, and the resulting drag forces and calculated drag coefficients are validated by comparison with experimental data from the literature. In a previous study, a global scale model was developed using a computational structural dynamics (CSD) method to simulate macroalgae farming system and guide the system configuration design. In the global scale model, the hydrodynamic forces are calculated using Morison’s equation and the kinematics and dynamics of the sugar kelp are simplified and the group of kelps attached to the long line is modeled as a slender structure with the same length and an effective diameter such that the volumes are consistent with the real physical system. This simplified model matches the weight and buoyancy but adjusting the hydrodynamic properties when the general hydrodynamic coefficients are employed. Therefore, optimal hydrodynamic coefficients used in global scale model were determined to obtain the hydrodynamic force more accurately. The validated local scale model is then be applied to determine the hydrodynamic coefficients of the simplified sugar kelp model for global dynamic analysis.

hydrodynamic modeling, macroalgae, CFD model↗
DOE OSTI2016

Examination of the temperature dependent electronic behavior of GeTe for switching applications

The DC and RF electronic behaviors of GeTe-based phase change material switches as a function of temperature, from 25 K to 375 K, have been examined. In its polycrystalline (ON) state, GeTe behaved as a degenerate p-type semiconductor, exhibiting metal-like temperature dependence in the DC regime. This was consistent with the polycrystalline (ON) state RF performance of the switch, which exhibited low resistance S-parameter characteristics. In its amorphous (OFF) state, the GeTe presented significantly greater DC resistance that varied considerably with bias and temperature. At low biases (<1 V) and temperatures (<200 K), the amorphous GeTe low-field resistance dramatically increased, resulting in exceptionally high amorphous-polycrystalline (OFF-ON) resistance ratios, exceeding 10{sup 9} at cryogenic temperatures. At higher biases and temperatures, the amorphous GeTe exhibited nonlinear current-voltage characteristics that were best fit by a space-charge limited conduction model that incorporates the effect of a defect band. The observed conduction behavior suggests the presence of two regions of localized traps within the bandgap of the amorphous GeTe, located at approximately 0.26–0.27 eV and 0.56–0.57 eV from the valence band. Unlike the polycrystalline state, the high resistance DC behavior of amorphous GeTe does not translate to the RF switch performance; instead, a parasitic capacitance associated with the RF switch geometry dominates OFF state RF transmission.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. In this work, we defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.

42 ENGINEERING↗
DOE OSTI2018

Investigation of rotating mode behavior in BWR out-of-phase limit cycle oscillations – Part 1: Reduced order model

Previous neutronic/thermal-hydraulic (TH) coupled numerical simulations using full-core TRACE/PARCS and SIMULATE-3K BWR models have shown evidence of a specific “rotating mode” behavior (steady rotation of the symmetry line, i.e. constant phase shift of approximately 90° between the first two azimuthal modes) in out-of-phase limit cycle oscillations, regardless of initial conditions and even if the first two azimuthal modes have different natural frequencies. This suggests a nonlinear coupling between these modes; otherwise, the phase shift between these modes would change at a constant rate during the limit cycle. The goal of the present work is to gain further insights on the rotating mode behavior using a simplified mathematical model which contains all of the important physics for this application while providing sufficient flexibility and simplicity to allow for in-depth understanding of the underlying phenomena. This was accomplished using a multi-channel, multi-modal reduced-order model, using a modification of the fixed pressure drop boundary condition to simulate channel coupling via the inlet and outlet plena, in order to destabilize the out-of-phase mode over the in-phase mode. Examination of the time-dependent solution of the nonlinear system showed a clear preference for rotating mode behavior in the four-channel model under stand-alone TH conditions and for conditions with weak neutronic feedback. Furthermore, when neutronic feedback was strengthened (i.e., larger reactivity feedback coefficients), the side-to-side mode (stationary symmetry line) was favored instead. Additional analyses using higher-fidelity numerical modeling, as well as a physical explanation for the rotating behavior seen in both sets of analyses, will be provided in a companion paper (“Part 2”).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Learning high-dimensional parametric maps via reduced basis adaptive residual networks

We propose a scalable framework for the learning of high-dimensional parametric maps via adaptively constructed residual network (ResNet) maps between reduced bases of the inputs and outputs. When just few training data are available, it is beneficial to have a compact parametrization in order to ameliorate the ill-posedness of the neural network training problem. By linearly restricting high-dimensional maps to informed reduced bases of the inputs, one can compress high-dimensional maps in a constructive way that can be used to detect appropriate basis ranks, equipped with rigorous error estimates. A scalable neural network learning framework is thus to learn the nonlinear compressed reduced basis mapping. Unlike the reduced basis construction, however, neural network constructions are not guaranteed to reduce errors by adding representation power, making it difficult to achieve good practical performance. Inspired by recent approximation theory that connects ResNets to sequential minimizing flows, we present an adaptive ResNet construction algorithm. This algorithm allows for depth-wise enrichment of the neural network approximation, in a manner that can achieve good practical performance by first training a shallow network and then adapting. We prove universal approximation of the associated neural network class for $L^2_v$ functions on compact sets. Our overall framework allows for constructive means to detect appropriate breadth and depth, and related compact parametrizations of neural networks, significantly reducing the need for architectural hyperparameter tuning. Numerical experiments for parametric PDE problems and a 3D CFD wing design optimization parametric map demonstrate that the proposed methodology can achieve remarkably high accuracy for limited training data, and outperformed other neural network strategies we compared against.

42 ENGINEERING↗
DOE OSTI2025

Structured light approaches in laser-based plasma diagnostics

There is a growing demand for plasma diagnostics suitable for industrial plasma reactors employed in semiconductor nanofabrication, especially relevant to microelectronics and quantum information systems. Such reactors typically have limited optical access and pose considerable diagnostic challenges, including intense background emission, significant thermal loads, and contamination of optical viewports. In this study, we outline research into structured light techniques (laser beams with tailored spatial, temporal, or phase characteristics) that effectively overcome these issues using laser-induced fluorescence (LIF) as an example. The focus of presented diagnostics is on ion kinetics analysis within an industrial plasma source, although this approach is broadly applicable to other plasma systems and diagnostic contexts. We present a confocal LIF implementation using an axicon-generated Bessel annular beam, achieving spatial resolutions of approximately 5 mm at a focal distance of 300 mm, with potential improvements to about 1 mm. This approach matches conventional orthogonal LIF performance but requires only one optical port. Wavelength-modulation LIF employs nonlinear laser wavelength tuning to measure spectral line derivatives, suppressing background emission and enhancing details of spectral line shape. Additionally, we present new results on applying vortex beams (laser beams carrying orbital angular momentum, OAM) for LIF measurements in an industrial plasma device. These measurements enable simultaneous axial and tangential velocity determination using a single laser beam and have been tested with xenon ion transition. Initial quantification of results was performed. Together, these structured-light approaches provide robust, background-resilient, multi-dimensional diagnostics for complex plasma environments.

Romadanov, Ivan [Princeton Plasma Physics Laborato↗
DOE OSTI2016

WE-AB-207B-11: Optimizing Tumor Control Probability in Radiation Therapy Treatment - Application to HDR Cervical Cancer

Purpose: The ultimate goal of radiotherapy treatment planning is to find a treatment that will yield a high tumor-control-probability(TCP) with an acceptable normal-tissue-complication probability(NTCP). Yet most treatment planning today is not based upon optimization of TCPs and NTCPs, but rather upon meeting physical dose and volume constraints defined by the planner. We design treatment plans that optimize TCP directly and contrast them with the clinical dose-based plans. PET image is incorporated to evaluate gain in TCP for dose escalation. Methods: We build a nonlinear mixed integer programming optimization model that maximizes TCP directly while satisfying the dose requirements on the targeted organ and healthy tissues. The solution strategy first fits the TCP function with a piecewise-linear approximation, then solves the problem that maximizes the piecewise linear approximation of TCP, and finally performs a local neighborhood search to improve the TCP value. To gauge the feasibility, characteristics, and potential benefit of PET-image guided dose escalation, initial validation consists of fifteen cervical cancer HDR patient cases. These patients have all received prior 45Gy of external radiation dose. For both escalated strategies, we consider 35Gy PTV-dose, and two variations (37Gy-boost to BTV vs 40Gy-boost) to PET-image-pockets. Results: TCP for standard clinical plans range from 59.4% - 63.6%. TCP for dose-based PET-guided escalated-dose-plan ranges from 63.8%–98.6% for all patients; whereas TCP-optimized plans achieves over 91% for all patients. There is marginal difference in TCP among those with 37Gy-boosted vs 40Gy-boosted. There is no increase in rectum and bladder dose among all plans. Conclusion: Optimizing TCP directly results in highly conformed treatment plans. The TCP-optimized plan is individualized based on the biological PET-image of the patients. The TCP-optimization framework is generalizable and has been applied successfully to other external-beam delivery modalities. A clinical trial is on-going to gauge the clinical significance. Partially supported by the National Science Foundation.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

EFT of large scale structures in redshift space [On the EFT of large scale structures in redshift space]

Here, we further develop the description of redshift-space distortions within the effective field theory of large scale structures. First, we generalize the counterterms to include the effect of baryonic physics and primordial non-Gaussianity. Second, we evaluate the IR resummation of the dark matter power spectrum in redshift space. This requires us to identify a controlled approximation that makes the numerical evaluation straightforward and efficient. Third, we compare the predictions of the theory at one loop with the power spectrum from numerical simulations up to ℓ = 6. We find that the IR resummation allows us to correctly reproduce the baryon acoustic oscillation peak. The k reach—or, equivalently, the precision for a given k—depends on additional counterterms that need to be matched to simulations. Since the nonlinear scale for the velocity is expected to be longer than the one for the overdensity, we consider a minimal and a nonminimal set of counterterms. The quality of our numerical data makes it hard to firmly establish the performance of the theory at high wave numbers. Within this limitation, we find that the theory at redshift z = 0.56 and up to ℓ = 2 matches the data at the percent level approximately up to k~0.13 hMpc –1 or k~0.18 hMpc –1 , depending on the number of counterterms used, with a potentially large improvement over former analytical techniques.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Stability of exact solutions of the nonlinear Schrödinger equation in an external potential having supersymmetry and parity-time symmetry

We discuss the stability properties of the solutions of the general nonlinear Schrödinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (${ \mathcal P }{ \mathcal T }$) symmetric superpotential $W(x)$ that we considered earlier, Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation $\{{\rm{i}}\,{\partial }_{t}+{\partial }_{x}^{2}-{V}^{-}(x)+| \psi (x,t){| }^{2\kappa }\}\,\psi (x,t)=0,$ for arbitrary nonlinearity parameter κ. We study the bound state solutions when ${V}^{-}(x)\,=(1/4-{b}^{2}){\text{sech}}^{2}(x)$, which can be derived from two different superpotentials $W(x)$, one of which is complex and ${ \mathcal P }{ \mathcal T }$ symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth b 2 of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov–Kolokolov (V–K) stability criterion. The numerical results of applying the V–K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for $\kappa \gt 2$ a new regime of stability for the exact solutions appears as long as $b\gt {b}_{{\rm{crit}}}$, where ${b}_{{\rm{crit}}}$ is a function of the nonlinearity parameter κ. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for $\kappa \gt 2$.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Code Demonstration: Approximation of nearly-periodic symplectic maps via structure-preserving neural networks

In this notebook, we use the symplectic gyroceptron architecture from [Duruisseaux et al., 2022] to learn a surrogate map for the nearly-periodic symplectic flow map associated to a nearly-periodic Hamiltonian system composed of two nonlinearly coupled oscillators, where one of them oscillates significantly faster than the other: $\Big\lbrace$ $ {\dot{q}_1 = p_1 \hspace{5mm} \dot{p}_1 = -q_1 - \epsilon \partial_{q1}U(q_1 , q_2 )} \atop {\dot{q}_2 = \epsilon p_2 \hspace{4mm} \dot{p}_2 = -\epsilon q_2 - \epsilon \partial_{q2}U(q_1 , q_2 )}$. These equations of motion are the Hamilton's equations associated to the Hamiltonian $H_{\epsilon}(q_1,q_2,p_1,p_2)=\frac{1}{2}(q^2_1+p^2_1)+\frac{1}{2}\epsilon(q^2_2+p^2_2)+\epsilon$$U(q_1,q_2)$.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Year End Progress Report on Rattlesnake Improvements

Rattlesnake is a MOOSE-based radiation transport application developed at INL to support modern multi-physics simulations. At the beginning of the last year, Rattlesnake was able to perform steady-state, transient and eigenvalue calculations for the multigroup radiation transport equations. Various discretization schemes, including continuous finite element method (FEM) with discrete ordinates method (SN) and spherical harmonics expansion method (PN) for the self-adjoint angular flux (SAAF) formulation, continuous FEM (CFEM) with SN for the least square (LS) formulation, diffusion approximation with CFEM and discontinuous FEM (DFEM), have been implemented. A separate toolkit, YAKXS, for multigroup cross section management was developed to support Rattlesnake calculations with feedback both from changes in the field variables, such as fuel temperature, coolant density, and etc., and in isotope inventory. The framework for doing nonlinear diffusion acceleration (NDA) within Rattlesnake has been set up, and both NDA calculations with SAAF-SN-CFEM scheme and Monte Carlo with OpenMC have been performed. It was also used for coupling BISON and RELAP-7 for the full-core multiphysics simulations. Within the last fiscal year, significant improvements have been made in Rattlesnake. Rattlesnake development was migrated into our internal GITLAB development environment at the end of year 2014. Since then total 369 merge requests has been accepted into Rattlesnake. It is noted that the MOOSE framework that Rattlesnake is based on is under continuous developments. Improvements made in MOOSE can improve the Rattlesnake. It is acknowledged that MOOSE developers spent efforts on patching Rattlesnake for the improvements made on the framework side. This report will not cover the code restructuring for better readability and modularity and documentation improvements, which we have spent tremendous effort on. It only details some of improvements in the following sections.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

Nonlinear saturation and oscillations of collisionless zonal flows

Abstract In homogeneous drift-wave turbulence, zonal flows (ZFs) can be generated via a modulational instability (MI) that either saturates monotonically or leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is often attributed as the predator–prey oscillations induced by ZF collisional damping; however, similar dynamics is also observed in collisionless ZFs, in which case a different mechanism must be involved. Here, we propose a semi-analytic theory that explains the transition between the oscillations and saturation of collisionless ZFs within the quasilinear Hasegawa–Mima model. By analyzing phase-space trajectories of DW quanta (driftons) within the geometrical-optics (GO) approximation, we argue that the parameter that controls this transition is N ∼ γ MI / ω DW , where γ MI is the MI growth rate and ω DW is the linear DW frequency. We argue that at N ≪ 1, ZFs oscillate due to the presence of so-called passing drifton trajectories, and we derive an approximate formula for the ZF amplitude as a function of time in this regime. We also show that at N ≳ 1, the passing trajectories vanish and ZFs saturate monotonically, which can be attributed to phase mixing of higher-order sidebands. A modification of N that accounts for effects beyond the GO limit is also proposed. These analytic results are tested against both quasilinear and fully-nonlinear simulations. They also explain the earlier numerical results by Connaughton et al (2010 J. Fluid Mech. 654 207) and Gallagher et al (2012 Phys. Plasmas 19 122115) and offer a different perspective on what the control parameter actually is that determines the transition from the oscillations to saturation of collisionless ZFs.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Investigation of rotating mode behavior in BWR out-of-phase limit cycle oscillations – Part 2: TRACE/PARCS model and physical explanation

Previous neutronic/thermal-hydraulic (TH) coupled numerical simulations using full-core TRACE/PARCS and SIMULATE-3K boiling water reactor (BWR) models have shown evidence of a specific “rotating mode” behavior (steady rotation of the symmetry line, i.e. constant phase shift of approximately 90° between the first two azimuthal modes) in BWR out-of-phase limit cycle oscillations, regardless of initial conditions and even if the first two azimuthal modes have different natural frequencies. This suggests a nonlinear coupling between these modes; otherwise, the phase shift between these modes would change at a constant rate during the limit cycle. The previous paper (“Part 1”) presented a series of results to examine this rotating behavior with a reduced-order model. The goal of the present study is to provide additional analyses of the predicted rotating mode behavior using higher-fidelity numerical modeling, as well as a physical explanation for why this mode is favored over side-to-side or other oscillatory behaviors from a TH perspective. Results are presented using TRACE and TRACE/PARCS for a small number of parallel channels, which confirmed that the conclusions developed from the reduced-order model remain applicable when applying a full two-fluid, six-equation, finite-volume modeling approach. From these results, a physical explanation has been put forth to explain why the rotating symmetry line behavior is preferred from a TH standpoint, demonstrating that predominantly out-of-phase unstable systems are most unstable when the variation in the total inlet flow rate is minimized (which minimizes the effective single-phase to two-phase pressure drop ratio) and that the rotating mode is the most successful in minimizing this total flow rate variation as compared with the side-to-side case or any other oscillation pattern. The conclusion is that the rotating mode will be favored for any out-of-phase unstable system of parallel channels with no neutronic feedback or relatively weak neutronic feedback. Here, previous analyses have indicated that systems with sufficiently strong neutronic coupling may favor the side-to-side oscillation mode over the rotating mode; this topic is left as a subject of future investigation.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media

We develop and analyze a model for the interaction of a quasi-Newtonian free fluid with a poroelastic medium. The flow in the fluid region is described by the nonlinear Stokes equations and in the poroelastic medium by the nonlinear quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. We establish existence and uniqueness of a solution to the weak formulation and its semidiscrete continuous-in-time finite element approximation. We present error analysis, complemented by numerical experiments.

Mathematics↗
DOE OSTI2024

Nonadiabatic transitions during a passage near a critical point

The passage through a critical point of a many-body quantum system leads to abundant nonadiabatic excitations. Here, we explore a regime, in which the critical point is not crossed although the system is passing slowly very close to it. Here, we show that the leading exponent for the excitation probability can then be obtained by standard arguments of the Dykhne formula, but the exponential prefactor is no longer simple and behaves as a power law on the characteristic transition rate. We derive this prefactor for the nonlinear Landau–Zener model by adjusting Dykhne’s approach. Then, we introduce an exactly solvable model of the transition near a critical point in the Stark ladder. We derive the number of excitations for it without approximations and find qualitatively similar results for the excitation scaling.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Relativistic weak lensing from a fully non-linear cosmological density field

In this paper we examine cosmological weak lensing on non-linear scales and show that there are Newtonian and relativistic contributions and that the latter can also be extracted from standard Newtonian simulations. We use the post-Friedmann formalism, a post-Newtonian type framework for cosmology, to derive the full weak-lensing deflection angle valid on non-linear scales for any metric theory of gravity. We show that the only contributing term that is quadratic in the first order deflection is the expected Born correction and lens-lens coupling term. We use this deflection angle to analyse the vector and tensor contributions to the E- and B- mode cosmic shear power spectra. In our approach, once the gravitational theory has been specified, the metric components are related to the matter content in a well-defined manner. Specifying General Relativity, we write down a complete set of equations for a GR+ΛCDM universe for computing all of the possible lensing terms from Newtonian N-body simulations. We illustrate this with the vector potential and show that, in a GR+ΛCDM universe, its contribution to the E-mode is negligible with respect to that of the conventional Newtonian scalar potential, even on non-linear scales. Thus, under the standard assumption that Newtonian N-body simulations give a good approximation of the matter dynamics, we show that the standard ray tracing approach gives a good description for a ΛCDM cosmology.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Learning Constrained Parametric Differentiable Predictive Control Policies With Guarantees

We present differentiable predictive control (DPC), a method for offline learning of constrained neural control policies for nonlinear dynamical systems with performance guarantees. We show that the sensitivities of the parametric optimal control problem can be used to obtain direct policy gradients. Specifically, we employ automatic differentiation (AD) to efficiently compute the sensitivities of the model predictive control (MPC) objective function and constraints penalties. To guarantee safety upon deployment, we derive probabilistic guarantees on closed-loop stability and constraint satisfaction based on indicator functions and Hoeffding’s inequality. We empirically demonstrate that the proposed method can learn neural control policies for various parametric optimal control tasks. In particular, we show that the proposed DPC method can stabilize systems with unstable dynamics, track time-varying references, and satisfy nonlinear state and input constraints. Our DPC method has practical time savings compared to alternative approaches for fast and memory-efficient controller design. Specifically, DPC does not depend on a supervisory controller as opposed to approximate MPC based on imitation learning. We demonstrate that, without losing performance, DPC is scalable with greatly reduced demands on memory and computation compared to implicit and explicit MPC while being more sample efficient than model-free reinforcement learning (RL) algorithms.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Nonlinear saturation and oscillations of collisionless zonal flows

In homogeneous drift-wave (DW) turbulence, zonal flows (ZFs) can be generated via a modulational instability (MI) that either saturates monotonically or leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is often attributed as the predator-prey oscillations induced by ZF collisional damping; however, similar dynamics is also observed in collisionless ZFs, in which case a different mechanism must be involved. Here, we propose a semi-analytic theory that explains the transition between the oscillations and saturation of collisionless ZFs within the quasilinear Hasegawa-Mima model. By analyzing phase-space trajectories of DW quanta (driftons) within the geometrical-optics (GO) approximation, we argue that the parameter that controls this transition is N ~ \gamma_MI/\omega_DW, where \gamma_MI is the MI growth rate and \omega_DW is the linear DW frequency. We argue that at N << 1, ZFs oscillate due to the presence of so-called passing drifton trajectories, and we derive an approximate formula for the ZF amplitude as a function of time in this regime. We also show that at N >~ 1, the passing trajectories vanish and ZFs saturate monotonically, which can be attributed to phase mixing of higher-order sidebands. A modification of N that accounts for effects beyond the GO limit is also proposed. These analytic results are tested against both quasilinear and fully-nonlinear simulations. They also explain the earlier numerical results by Connaughton et al. [J. Fluid Mech. 654, 207 (2010)] and Gallagher et al. [Phys. Plasmas 19, 122115 (2012)] and offer a revised perspective on what the control parameter is that determines the transition from the oscillations to saturation of collisionless ZFs.

collisionless zonal flows, modulational instabilit↗