scholarly journals Statistical State Dynamics of Weak Jets in Barotropic Beta-Plane Turbulence

2019 ◽  
Vol 76 (3) ◽  
pp. 919-945 ◽  
Author(s):  
Nikolaos A. Bakas ◽  
Navid C. Constantinou ◽  
Petros J. Ioannou
Keyword(s):  
Second Order ◽  
Homogeneous Turbulence ◽  
Nonlinear Flow ◽  
Flow Forming ◽  
Ginzburg Landau ◽  
Linear Flow ◽  
Weakly Nonlinear ◽  
Zonal Jets ◽  
Statistical State ◽  
Wide Range

Abstract Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (zonostrophic instability), which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg–Landau (G–L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G–L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide range of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G–L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G–L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G–L dynamics that is able to capture the evolution of weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability.

Feedback control of weakly nonlinear Rayleigh–Bénard–Marangoni convection

2001 ◽  
Vol 440 ◽  
pp. 27-47 ◽  
Author(s):  
A. C. OR ◽  
R. E. KELLY
Keyword(s):  
Feedback Control ◽  
Nonlinear Flow ◽  
Control Action ◽  
Control Analysis ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Wide Range ◽  
Nonlinear Control Processes ◽  
Gas Layer ◽  
Rayleigh Benard

We study the effect of proportional feedback control on the onset and development of finite-wavelength Rayleigh–Bénard–Marangoni (RBM) convection using weakly nonlinear theory as applied to Nield's model, which includes both thermocapillarity and buoyancy but ignores deformation of the free surface. A two-layer model configuration is used, which has a purely conducting gas layer on top of the liquid. In the feedback control analysis, a control action in the form of temperature or heat flux is considered. Both measurement and control action are assumed to be continuous in space and time. Besides demonstrating that stabilization of the basic state can be achieved on a linear basis, the results also indicate that a wide range of weakly nonlinear flow properties can also be altered by the linear and nonlinear control processes used here. These include changing the nature of hexagonal convection and the amount of subcritical hysteresis associated with subcritical bifurcation.

scholarly journals Spectral Broadening in a Continuously Pumped Singly Resonant Second-Harmonic Cavity

2021 ◽  
Vol 11 (15) ◽  
pp. 7122
Author(s):  
Simona Mosca ◽  
Tobias Hansson ◽  
Maria Parisi
Keyword(s):  
Continuous Wave ◽  
Frequency Comb ◽  
Second Harmonic ◽  
Input Power ◽  
Second Order ◽  
Optical Frequency ◽  
Frequency Combs ◽  
Research Areas ◽  
Mixed Regime ◽  
Wide Range

Optical frequency comb synthesizers with a wide spectral range are an essential tool for many research areas such as spectroscopy, precision metrology, optical communication, and sensing. Recent studies have demonstrated the direct generation of frequency combs, via second-order processes, that are centered on two different spectral regions separated by an octave. Here, we present the capability of optical quadratic frequency combs for broad-bandwidth spectral emission in unexplored regimes. We consider comb formation under phase-matched conditions in a continuous-wave pumped singly resonant second-harmonic cavity, with large intracavity power and control of the detuning over several cavity line widths. The spectral analysis reveals quite distinctive sidebands that arise far away from the pump, singularly or in a mixed regime together with narrowband frequency combs. Notably, by increasing the input power, the optical frequency lines evolve into widely spaced frequency clusters, and at maximum power, they appear in a wavelength range spanning up to 100 nm. The obtained results demonstrate the power of second-order nonlinearities for direct comb production within a wide range of pump wavelengths.

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Acoustic Energy Balance During the Onset, Growth and Saturation of Thermoacoustic Instabilities

2020 ◽  
Author(s):  
R. Gaudron ◽  
D. Yang ◽  
A. S. Morgans
Keyword(s):  
Energy Balance ◽  
Network Models ◽  
Acoustic Energy ◽  
Dimensionless Numbers ◽  
Weakly Nonlinear ◽  
Acoustic Damping ◽  
Thermoacoustic Instabilities ◽  
Wide Range ◽  
Acoustic Absorption Coefficient ◽  
Flame Describing Function

Abstract Thermoacoustic instabilities can occur in a wide range of combustors and are prejudicial since they can lead to increased mechanical fatigue or even catastrophic failure. A well-established formalism to predict the onset, growth and saturation of such instabilities is based on acoustic network models. This approach has been successfully employed to predict the frequency and amplitude of limit cycle oscillations in a variety of combustors. However, it does not provide any physical insight in terms of the acoustic energy balance of the system. On the other hand, Rayleigh’s criterion may be used to quantify the losses, sources and transfers of acoustic energy within and at the boundaries of a combustor. However, this approach is cumbersome for most applications because it requires computing volume and surface integrals and averaging over an oscillation cycle. In this work, a new methodology for studying the acoustic energy balance of a combustor during the onset, growth and saturation of thermoacoustic instabilities is proposed. The two cornerstones of this new framework are the acoustic absorption coefficient Δ and the cycle-to-cycle acoustic energy ratio λ, both of which do not require computing integrals. Used along with a suitable acoustic network model, where the flame frequency response is described using the weakly nonlinear Flame Describing Function (FDF) formalism, these two dimensionless numbers are shown to characterize: 1) the variation of acoustic energy stored within the combustor between two consecutive cycles, 2) the acoustic energy transfers occurring at the combustor’s boundaries and 3) the sources and sinks of acoustic energy located within the combustor. The acoustic energy balance of the well-documented Palies burner is then analyzed during the onset, growth and saturation of thermoacoustic instabilities using this new methodology. It is demonstrated that this new approach allows a deeper understanding of the physical mechanisms at play. For instance, it is possible to determine when the flame acts as an acoustic energy source or sink, where acoustic damping is generated, and if acoustic energy is transmitted through the boundaries of the burner.

scholarly journals Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

2020 ◽  
Keyword(s):  
Magnetic Field ◽  
Angular Velocity ◽  
Landau Equation ◽  
Temperature Modulation ◽  
External Fields ◽  
Ginzburg Landau ◽  
Electrically Conductive ◽  
Weakly Nonlinear ◽  
Conductive Fluid ◽  
Ginzburg Landau Equations

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

scholarly journals Selection of Second Order Models’ Design Using D-, A-, E-, T- Optimality Criteria

2019 ◽  
pp. 1-15
Author(s):  
Wangui Patrick Mwangi ◽  
Ayubu Anapapa ◽  
Argwings Otieno
Keyword(s):  
Optimality Criteria ◽  
Second Order ◽  
Factorial Designs ◽  
Economical Design ◽  
Wide Range ◽  
Eigen Value ◽  
Average Variance ◽  
Rsm Technique ◽  
Central Composite ◽  
Selection Of

There are numerous designs for fitting second order models that can be used in conjunction with the response surface methodology (RSM) technique in optimization processes, be it in agriculture, industries and so on. Some of the designs include the equiradial, Notz, San Cristobal, Koshal, Hoke, Central Composite and Factorial designs. However, RSM can only be applied in conjunction with a single design at a time. This research aimed at choosing a design out of the most widely employed designs for fitting 2nd order models involving 3 factors for optimization of French beans in conjunction with the RSM technique. The most commonly used designs for second order models were first identified as Box-Behnken designs, Hoke D2 and Hoke D6 designs, 3k factorial designs, CCD face centred, CCD rotatable and CCD spherical. Design matrices for these 7 designs were formed and augmented with 5 centre points (chosen through lottery methods), and information and optimal design matrices were formed. Then, for each design, the analysis of D-, A- E-, T- optimality (D-Determinant, A-Average Variance, E-Eigen Value and T-Trace) was carried out according to Pukelsheim’s definitions. The results were ranked for each criterion and the ranks corresponding to each design were averaged. The design chosen was Hoke D2 with the least average- 1.75. The Hoke D2 was found to be optimal in minimizing the variance of prediction and the most economical design among the seven. The findings are in agreement with other researchers and scientist that a design may be optimal in one criterion but fails in another criterion. Further, Hoke designs are in the class of the economical designs. It is recommended that more optimality criteria be applied and a wide range of designs be involved to see whether the results would still agree with these findings.

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