Stochastic Forcing Research Articles

Distributions and extreme value analysis of critical response rate and split-time metric in nonlinear oscillators with stochastic excitation

A single-degree-of-freedom oscillator with a nonlinear restoring force and stochastic external forcing is studied with the goal of understanding the distributions of certain response rates. Motivated by applications to ship motions, the response rates concern the minimum response rate leading to capsizing (the critical response rate) and a measurement comparing an observed response rate to the critical response rate (the split-time metric), whose distributions are investigated both analytically and numerically. Three nonlinear restoring forces are considered: piecewise linear (experiencing linearly softening stiffness above a “knuckle” point), doubly piecewise linear (experiencing piecewise linearly softening stiffness above a “knuckle” point), and the cubic softening restoring force of the Duffing oscillator. In the first two cases, an efficient numerical simulation of the critical response rate and split-time metric is proposed from a derived distribution; in the latter case, the density of the critical response rate is approximated assuming white noise excitation. A key interest is in the nature of the right tail of the split-time metric, specifically as it relates to its extrapolation through extreme value analysis in estimating the probability of capsizing. The distribution is found to have a “light” tail, which motivates the use of exponential rather than the generalized Pareto distribution for exceedances above threshold in extreme value analysis. Finally, threshold selection through a prediction error criterion for the exponential distribution is examined, and the Weibull distribution tail is suggested as a useful means for a more refined examination of the distribution tail.

Improving the Forecasting of El Niño Amplitude Based on an Ensemble Forecast Strategy for Westerly Wind Bursts

Abstract The forecasting of El Niño amplitude remains uncertain, with false alarms or underestimations often occurring. It has been suggested that westerly wind bursts (WWBs) are crucial atmospheric stochastic forcing that affects the development, amplitude, and predictability of El Niño. Effectively capturing the influence of WWBs in El Niño forecasting systems to mitigate El Niño forecast uncertainties continues to be a widely discussed topic. In this work, two El Niño ensemble forecast scenarios incorporating WWBs were devised utilizing an intermediate coupled model capable of simulating the rational features of El Niño–Southern Oscillation and a refined WWB parameterization scheme. To start with, both the seasonal variations and the dependence on the oceanic background state were considered to improve the parameterization of WWBs. Furthermore, considering the short intrinsic predictability limit of WWBs with respect to the interannual scale of El Niño evolution, a WWB ensemble forecast strategy was designed to obtain their occurrence probability and the statistical features of their magnitude and central location. With this in mind, an ensemble forecast of El Niño events occurring during 1979–2021 based on a WWB ensemble forecast was established (termed WWB-based). For comparison, a conventional El Niño ensemble forecast based on initial condition (IC) perturbations was also conducted (termed IC-based). Results indicated that the WWB-based approach shows better performance in forecasting the El Niño amplitude than the IC-based one. The present approach suggests that an ensemble forecast with proper consideration of the scale interaction between the fast WWBs and interannual variations is more physically consistent. Significance Statement Westerly wind bursts (WWBs) are believed to influence the growth and amplitude of El Niño events. Unfortunately, the short intrinsic predictability limit of WWBs brings considerable uncertainty to the forecasting of El Niño amplitude. In this study, a refined parameterization scheme for WWBs, along with an ensemble forecast strategy designed to obtain the occurrence probability of WWBs and the statistical features of their magnitude and central location, was employed in an intermediate coupled model to produce an ensemble forecast for the El Niño events occurring during 1979–2021. The results showed that an ensemble forecast of El Niño based on WWB ensemble forecast can fully consider WWBs’ potential impact on the El Niño system and improve El Niño forecasts.

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin’s method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

Two-stage stochastic programming model of US Army aviation allocation of utility helicopters to task forces

US Army aviation units often organize into task forces to meet mission requirements. The manner in which they allocate assets affects their long-term capabilities to provide aviation support. We propose a model to allocate utility helicopters across geographically separated task forces to minimize the total time of flight and unsupported air movement air mission requests (AMRs) by priority level. We model the allocation problem with a two-stage stochastic program, with the first-stage problem allocating a fleet’s helicopter teams to task forces. The stochastic demand for each task force is then revealed. The second-stage US Army aviation air movement operations planning problem is modeled as a stochastic mixed integer linear program (MILP). A practical application uses the air movement operations planning heuristic to solve the second-stage problem at scale and generate an optimal stochastic solution task force allocation. This paper provides evidence for the practical use of the proposed two-stage stochastic programming model for US Army aviation asset allocation by military decision-makers. Furthermore, this research provides a novel first formulation of a stochastic programming dial-a-ride problem with multinode refuel and a sound framework for military aviation asset allocation decision-making.

Chaotic Type I migration in turbulent discs

ABSTRACT By performing global hydrodynamical simulations of accretion discs with driven turbulence models, we demonstrate that elevated levels of turbulence induce highly stochastic migration torques on low-mass companions embedded in these discs. This scenario applies to planets migrating within gravito-turbulent regions of protoplanetary discs as well as stars and black holes embedded in the outskirts of active galactic nucleus (AGN) accretion discs. When the turbulence level is low, linear Lindblad torque persists in the background of stochastic forces and its accumulative effect can still dominate over relatively long time-scales. However, in the presence of very stronger turbulence, classical flow patterns around the companion embedded in the disc are disrupted, leading to significant deviations from the expectations of classical Type I migration theory over arbitrarily long time-scales. Our findings suggest that the stochastic nature of turbulent migration can prevent low-mass companions from monotonically settling into universal migration traps within the traditional laminar disc framework, thus reducing the frequency of three-body interactions and hierarchical mergers compared to previously expected. We propose a scaling for the transition mass ratio from classical to chaotic migration q ∝ αR, where αR is the Reynolds viscosity stress parameter, which can be further tested and refined by conducting extensive simulations over the relevant parameter space.

Insights into ENSO Diversity from an Intermediate Coupled Model. Part II: Role of Nonlinear Dynamics and Stochastic Forcing

Abstract In this study, we investigate how a single leading linear El Niño–Southern Oscillation (ENSO) mode, as studied in Part I, leads to the irregular coexistence of central Pacific (CP) and eastern Pacific (EP) ENSO, a phenomenon known as ENSO spatiotemporal diversity. This diversity is fundamentally generated by deterministic nonlinear pathways to chaos via the period-doubling route and, more prevailingly, the subharmonic resonance route with the presence of a seasonally varying basic state. When residing in the weakly nonlinear regime, the coupled system sustains a weak periodic oscillation with a mixed CP/EP pattern as captured by the linear ENSO mode. With a stronger nonlinearity effect, the ENSO behavior experiences a period-doubling bifurcation. The single ENSO orbit splits into coexisting CP-like and EP-like ENSO orbits. A sequence of period-doubling bifurcation results in an aperiodic oscillation featuring irregular CP and EP ENSO occurrences. The overlapping of subharmonic resonances between ENSO and the seasonal cycle allows this ENSO irregularity and diversity to be more readily excited. In the strongly nonlinear regime, the coupled system is dominated by regular EP ENSO. The deterministic ENSO spatiotemporal diversity is thus confined to a relatively narrow range corresponding to a moderately unstable ENSO mode. Stochastic forcing broadens this range and allows ENSO diversity to occur when the ENSO mode is weakly subcritical. A close relationship among a weakened mean zonal temperature gradient, stronger ENSO activity, and more (fewer) occurrences of EP (CP) ENSO is noted, indicating that ENSO–mean state interaction may yield ENSO regime modulations on the multidecadal time scale.

Investigation of the dynamic response of a floating wind-aquaculture platform under the combined actions of wind, waves and current

Floating wind-aquaculture platforms are drawing increasing attention from the academic and engineering communities due to their potential to fully exploit and utilize marine space and its resources. However, these platforms integrate both the hydrodynamics and aerodynamics of floating wind turbines and aquaculture cages, making their mechanical properties more complex. This study aims to evaluate the effects of three different hydrodynamic and aerodynamic damping components on and the contribution of the stochastic environmental loads to the dynamic response of a floating wind-aquaculture platform. A coupled hydro-aero-servo method is established. Decay and forced oscillation tests of the platform in still water are firstly numerically performed, followed by simulation of the dynamic behavior under different combinations of environmental loads, including the fluctuating wind load of the blades, stochastic wave excitation forces on the floating body and viscous force of the aquaculture cage system. The aerodynamic damping of the wind turbine and the hydrodynamic damping of the floating body are dominant in low- and wave-frequency range, respectively. Regarding the environmental load components, the second-order wave force and the turbulent wind load are dominant in the surge direction in the low-frequency range. The dynamic response of the platform in the wave-frequency range is mainly induced by the first-order wave force. Fish net can suppress the low-frequency motion but has almost no influence on the wave-frequency motion.

Fractional SDEs with stochastic forcing: Existence, uniqueness, and approximation

In this article, we are interested in fractional stochastic differential equations (FSDEs) with stochastic forcing, i.e., to FSDE we add a stochastic forcing term. The conditions for the existence and uniqueness of solutions of such equations are obtained, and the convergence rate of the implicit Euler approximation scheme for them is established. Such types of equations can be applied to the consideration of FSDEs with a permeable wall.

Lp-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise F˙:∂tu=aijuxixj+biuxi+cu−b¯u1+β+ξu1+γF˙,(t,x)∈R+×Rd;u(0,⋅)=u0, where aij,bi,c,b¯ and ξ are C2 or L∞ bounded random coefficients. Here β>0 denotes the degree of strong dissipativity and γ>0 represents the degree of stochastic force. Under the reinforced Dalang's condition on F˙, we show the well-posedness of the SRDE provided γ<κ(β+1)d+2 where κ>0 is the constant related to F˙. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal Hölder regularity of the solution in time and space.

A dynamic capillarity equation with stochastic forcing on manifolds: A singular limit problem

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold ( M , g ) (M,g) : d ( u ε , δ − δ Δ u ε , δ ) + d i v f ε ( x , u ε , δ ) d t = ε Δ u ε , δ d t + Φ ( x , u ε , δ ) d W t , \begin{equation*} d \left (u_{\varepsilon ,\delta } -\delta \Delta u_{\varepsilon ,\delta }\right ) +div\mathfrak {f}_{\varepsilon }(\mathbf {x}, u_{\varepsilon ,\delta })\, dt =\varepsilon \Delta u_{\varepsilon ,\delta }\, dt + \Phi (\mathbf {x}, u_{\varepsilon ,\delta })\, dW_t, \end{equation*} where f ε \mathfrak {f}_{\varepsilon } is a sequence of smooth vector fields converging in L p ( M × R ) L^p(M\times \mathbb {R}) ( p &gt; 2 p&gt;2 ) as ε ↓ 0 \varepsilon \downarrow 0 towards a vector field f ∈ L p ( M ; C 1 ( R ) ) \mathfrak {f}\in L^p(M;C^1(\mathbb {R})) , and W t W_t is a Wiener process defined on a filtered probability space. First, for fixed values of ε \varepsilon and δ \delta , we establish the existence and uniqueness of weak solutions to the Cauchy problem for the above-stated equation. Assuming that f \mathfrak {f} is non-degenerate and that ε \varepsilon and δ \delta tend to zero with δ / ε 2 \delta /\varepsilon ^2 bounded, we show that there exists a subsequence of solutions that strongly converges in L ω , t , x 1 L^1_{\omega ,t,\mathbf {x}} to a martingale solution of the following stochastic conservation law with discontinuous flux: d u + d i v f ( x , u ) d t = Φ ( u ) d W t . \begin{equation*} d u +div\mathfrak {f}(\mathbf {x}, u)\,dt =\Phi (u)\, dW_t. \end{equation*} The proofs make use of Galerkin approximations, kinetic formulations as well as H H -measures and new velocity averaging results for stochastic continuity equations. The analysis relies on the use of a.s. representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.

General validity of the second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications

In this paper, we derive a generalized second fluctuation-dissipation theorem (FDT) for stochastic dynamical systems in the steady state and further show that if the system is highly degenerate, then the classical second FDT is valid even when the exact form of the steady state distribution is unknown. The established theory is built upon the Mori-type generalized Langevin equation for stochastic dynamical systems and hence generally applies to nonequilibrium systems driven by stochastic forces. These theoretical results enable us to construct a data-driven nanoscale fluctuating heat conduction model based on the second FDT. We numerically verify that our heat transfer model yields better predictions than the Green-Kubo formula for systems far from the equilibrium.

Generative Data‐Driven Approaches for Stochastic Subgrid Parameterizations in an Idealized Ocean Model

AbstractSubgrid parameterizations of mesoscale eddies continue to be in demand for climate simulations. These subgrid parameterizations can be powerfully designed using physics and/or data‐driven methods, with uncertainty quantification. For example, Guillaumin and Zanna (2021, https://doi.org/10.1029/2021ms002534) proposed a Machine Learning (ML) model that predicts subgrid forcing and its local uncertainty. The major assumption and potential drawback of this model is the statistical independence of stochastic residuals between grid points. Here, we aim to improve the simulation of stochastic forcing with generative models of ML, such as Generative adversarial network (GAN) and Variational autoencoder (VAE). Generative models learn the distribution of subgrid forcing conditioned on the resolved flow directly from data and they can produce new samples from this distribution. Generative models can potentially capture not only the spatial correlation but any statistically significant property of subgrid forcing. We test the proposed stochastic parameterizations offline and online in an idealized ocean model. We show that generative models are able to predict subgrid forcing and its uncertainty with spatially correlated stochastic forcing. Online simulations for a range of resolutions demonstrated that generative models are superior to the baseline ML model at the coarsest resolution.

Sliding friction acoustics

Contact interactions generate acoustic energy in complicated ways, dictated by dozens of system parameters. The ability to accurately predict these stochastic friction forces and resultant acoustic energy from first principles is currently lacking. A summary of experimental tests with sliding and stick-slip of metal-on-metal contact are presented to provide insight into the influence of roughness, sliding velocity, and normal load on the generated acoustics. Normal load and normal displacement increase in dynamic amplitude under stick-slip conditions relative to smooth sliding. In the current configuration, stick-slip also produces an excess of up to 30 dB SPL, relative to smooth sliding. Relationships between normal load, surface roughness, tangential velocity, and acoustic output are explored, and parameters are tuned to fit these variables using an existing empirical relationship from the literature.

Stochastic Dynamical Modeling of Wind Farm Turbulence

Low-fidelity engineering wake models are often combined with linear superposition laws to predict wake velocities across wind farms under steady atmospheric conditions. While convenient for wind farm planning and long-term performance evaluation, such models are unable to capture the time-varying nature of the waked velocity field, as they are agnostic to the complex aerodynamic interactions among wind turbines and the effects of atmospheric boundary layer turbulence. To account for such effects while remaining amenable to conventional system-theoretic tools for flow estimation and control, we propose a new class of data-enhanced physics-based models for the dynamics of wind farm flow fluctuations. Our approach relies on the predictive capability of the stochastically forced linearized Navier–Stokes equations around static base flow profiles provided by conventional engineering wake models. We identify the stochastic forcing into the linearized dynamics via convex optimization to ensure statistical consistency with higher-fidelity models or experimental measurements while preserving model parsimony. We demonstrate the utility of our approach in completing the statistical signature of wake turbulence in accordance with large-eddy simulations of turbulent flow over a cascade of yawed wind turbines. Our numerical experiments provide insight into the significance of spatially distributed field measurements in recovering the statistical signature of wind farm turbulence and training stochastic linear models for short-term wind forecasting.

Multifractality and scale-free network topology in a noise-perturbed laminar jet

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in a globally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for peak multifractality produce a complex network whose node degree distribution obeys an inverse power-law scaling with an exponent of $2 &lt; \gamma &lt; 3$ , indicating scale-free topology and (iii) the Hurst exponent and the global clustering coefficient can serve as early warning indicators of global instability under specific operating and forcing conditions. By characterising the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics commonly observed in turbulent flows can also be observed in laminar flows under certain stochastic forcing conditions.

Stochastic fractional conservation laws

In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws with stochastic forcing. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.

The most predictable component of a linear stochastic model

It is known that special combinations of variables can have more predictability than any variable in the combination. While such combinations can be obtained numerically in specific cases, this leaves broader questions unanswered. For example, given the dynamics of a linear stochastic model, what is the maximum predictability? What is its structure? What stochastic forcing maximizes predictability? This paper answers these questions. Specifically, this paper derives an upper bound on the predictability of linear combinations of variables. This bound is achieved when stochastic forcing correlates perfectly between eigenmodes. In a certain limit, the structure of the most predictable combination can be derived analytically. This structure is called the Pascal Mode due to its relation to Pascal’s Triangle in a special case. These results provide a new perspective on observation-based climate predictability estimates. For instance, the most predictable component of monthly North Atlantic sea surface temperature is only modestly more predictable than the least-damped eigenmode. This aligns with the theoretical findings, as neither the stochastic forcing nor the dynamical eigenvalues are tailored to enhance predictability. The Pacific shows more predictability relative to the least-damped mode, but this increase is an order of magnitude smaller than the theoretical limit given the dynamics.

A numerically efficient output-only system-identification framework for stochastically forced self-sustained oscillators

Self-sustained oscillations are ubiquitous in nature and engineering. In this paper, we propose a novel output-only system-identification framework for identifying the system parameters of a self-sustained oscillator affected by Gaussian white noise. A Langevin model that characterizes the self-sustained oscillator is postulated, and the corresponding Fokker–Planck equation is derived from stochastic averaging. From the drift and diffusion terms of the Fokker–Planck equation, unknown parameters of the system are identified. We develop a numerically efficient algorithm for enhancing the accuracy of parameter identification. In particular, a modified Levenberg–Marquardt optimization algorithm tailored to output-only system identification is introduced. The proposed framework is demonstrated on both numerical and experimental oscillators with varying system parameters that develop into self-sustained oscillations. The results show that the computational cost required for performing the system identification is dramatically reduced by using the proposed framework. Also, system parameters that were difficult to be extracted with the existing method could be efficiently computed with the system identification method developed in this study. Pertaining to the robustness and computational efficiency of the presented framework, this study can contribute to an accurate and fast diagnosis of dynamical systems under stochastic forcing.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation.

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equationby means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equationfor the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equationby means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Continuous probabilistic solution to the transient self-oscillation under stochastic forcing: a PINN approach

Continuous probabilistic solution to the transient self-oscillation under stochastic forcing: a PINN approach