Stochastic resonance of rotating particles in turbulence

Domain: physics
Published: 2025-11-24
Editorial outcome: major_revision

Abstract
The chaotic dynamics of small-scale vorticity plays a key role in understanding and controlling turbulence, with direct implications for energy transfer, mixing, and coherent structure evolution. However, measuring or controlling its dynamics remains a major conceptual and experimental challenge due to its transient and chaotic nature. Here we use a combination of experiments, theory, and simulations to show that small magnetic particles of different densities, exploring flow regions of distinct vorticity statistics, can act as effective probes for measuring and forcing turbulence at its smallest scale. The interplay between the magnetic torque, from an externally controllable magnetic field, and hydrodynamic stresses, from small-scale turbulent vorticity, uncovers an extremely rich phenomenology. Notably, we present the first experimental and numerical observation of stochastic resonance for particles in turbulence, where turbulent fluctuations, remarkably acting as an effective noise, enhance the particle rotational response to external forcing. We identify a pronounced resonant peak in the particle rotational phase lag when the applied rotating magnetic field matches the characteristic intensity of small-scale turbulent vortices. Furthermore, we reveal a novel symmetry-breaking mechanism: an oscillating magnetic field with zero mean angular velocity can counterintuitively induce net particle rotation in zero-mean vorticity turbulence. Our findings pave the way to developing flexible techniques for manipulating particle dynamics in complex flows. The discovered mechanism enables a novel magnetic resonance-based approach to be developed for measuring turbulent vorticity. In this approach, particles act as probes, emitting a detectable magnetic field that can characterize turbulence even under conditions that are optically inaccessible.

Introduction
Turbulent flows, characterized by chaotic multi-scale fluctuations, are central to a wide range of natural and industrial processes1,2. A key challenge in turbulence research is the direct measurement and control of the vorticity, which is primarily concentrated in small-scale vortex filaments3,4. Due to their transient nature and rapid chaotic evolution, capturing the rotational dynamics of these filaments remains a major conceptual and experimental challenge. Several methods have been developed to infer small-scale turbulence characteristics, including particle tracking methods (e.g., tracer particles, deformable particles, and pattern-coated particles), as well as thermal and optical anemometry5–7. However, these approaches often suffer from inherent limitations such as insufficient spatial and temporal resolution, signal occlusion in optically dense environments, and difficulties in directly resolving vorticity at the smallest turbulence scales. As a result, achieving real-time, high-fidelity measurements of rotational motion in turbulent flows remains an ongoing challenge, particularly in complex and optically inaccessible flows.

A promising alternative is the use of magnetic particles, which respond to both stochastic vorticity fluctuations of turbulence (hydrodynamic torques8) and deterministic torques imposed by external magnetic fields. The dynamics of buoyant magnetic particles, in particular, are closely linked to the local vorticity. Particles with different densities exhibit distinct rotational behaviors as they explore different regions of the flow, leading to preferential concentration: light (heavy) particles tend to accumulate in high- (low-) vorticity regions7,9,10. These particles offer a unique dual role: they can serve as candidates for probing rotational fluctuations, while also enabling active flow manipulation through external forcing. This ability to modulate particle rotation and measure its response in real time provides a new pathway for investigating turbulence at small scales and suggests an elegant strategy for measuring turbulent vorticity. However, in order to unlock these novel directions, an accurate understanding of the complex interplay between hydrodynamic and magnetic torques on particle dynamics is required.

Previous studies have extensively explored magnetic particles in quiescent fluids under rotating magnetic fields, revealing rich phenomena such as self-assembly into chain-like structures11,12, synchronization-selected structures13, and more complex arrays14, enhanced mixing11,12,15, turbulence design16–18 and phase locking, where particles follow the rotating magnetic field with a constant phase lag with respect to the magnetic field when the rotating magnetic angular velocity ωH is below a threshold ωcr19. Investigations have also extended to self-propelling active magnetic particles (e.g., magnetotactic bacteria)20,21, magnetic Janus particles13,22, particles with different shapes23,24, and particles in complex flows25–27. However, the rotational dynamics of magnetic particles in fully developed turbulence remain poorly understood, presenting a fundamental challenge in predicting and controlling particle behavior in turbulent environments.

Here, we explore this question by investigating the rotational dynamics of magnetic particles in turbulence under a rotating magnetic field. While deterministic synchronized and oscillation (back-and-forth) rotational regimes have been observed in zero-noise systems (e.g., compasses or rolling particles in quiescent fluids28–30), our work extends this understanding into the complex environment of fully developed turbulence. We systematically explore the complex interplay between deterministic external magnetic forcing and turbulent hydrodynamic torques, revealing a significantly richer phase space of particle rotational dynamics. We show that the effect of turbulence (vorticity) acts as an effective noise (Supplementary Information) and reveal the first signature of the emergence of stochastic resonance (SR)—a counterintuitive phenomenon where internal fluctuations non-linearly cooperate with external forcing to amplify the system response31–39—of particles in turbulence, which can be used as the mechanism of magnetic particles acting as effective vorticity probes for a novel type of turbulence “microscope”. While the concept of SR, initially observed in bistable systems subjected to noise and weak periodic signals, has since been extended to a wide range of physical and biological systems, as well as signal processing and computer science40–45, its role in turbulence-driven rotational dynamics remains mostly unexplored.

To identify potential SR signatures, we set out to leverage turbulent vorticity fluctuations as a source of noise and examine the response of the magnetic particle dynamics to the rotating magnetic field, by means of experiments, simulations, and theoretical modeling. Our experimental platform (Fig. 1a and e) involves a dilute collection of approximately spherical light magnetic particles (produced by Styrofoam core coated with magnetic paint, Fig. 1b, with a mean density of 20% with respect to water and with a mean diameter of 0.7 mm, see “Methods” for details) which are immersed in a Von Kármán-type turbulent flow (producing homogeneous and isotropic turbulence (HIT) in the center region of the cell46–49) in an octagonal water-filled chamber. Magnetic particles are designed to be light and of a size comparable to the Kolmogorov scale, allowing them to preferentially enter and interact with vortex filaments. The flow is generated by two counter-rotating bladed disks and the chamber is placed in a system of Helmholtz coils, which is programmed to generate a uniform magnetic field in a restricted measurement volume, H = Hh, rotating in the xoy-plane with angular velocity vector ωH = ωHz^ where z^ is the unit vector along the z-axis (“Methods” and Supplementary Information). As the induced magnetic dipole, m, of a particle tends to align with the magnetic field direction, h, the particle experiences a magnetic torque balanced by the rotational drag and spins around an axis in the particle body frame parallel to z^ along its trajectory (as shown by typical stroboscopic time-lapse trajectories in Fig. 1c, d). The rotational motion is measured by tracking particle surface patterns (“Methods”).

Results
Theoretical representation
The particle rotation is characterized by a preferred magnetization direction, n (Fig. 2a), whose trajectory can be tracked in turbulence under the influence of a rotating magnetic field. By balancing the magnetic torque and drag torque induced by turbulence19,50 (“Methods” and Supplementary Information), the equations governing the particle rotational motion can be obtained, in the overdamped limit, as

dn/dt = ωa(n×h)(h×(n×h)n) + ωf × n, (1a)

ξr dωp,k/dt = ξr(ωf·n)n + ωf × n, (1b)

where ωa = μVpH2(χ∥ − χ⊥)/ξr is the magnetic field strength normalized by the rotational drag coefficient ξr, with μ, Vp, and χ∥,⊥ the particle permeability, volume, and anisotropic magnetic susceptibilities, respectively19,50,51 (“Methods”). The term ωp,∥ represents the particle spinning rate. The perpendicular rotational component, ωp,⊥, can be obtained by ωp,⊥ = n × n_ which represent the particle tumbling rate. The turbulent vorticity experienced by the particle is ωf, which is defined as ωf = 1/2(∇×u)O with the subscript O implying evaluation at the center of mass of the particle. The total particle angular velocity is thus ωp = ωp,∥ + ωp,⊥.

In a noiseless system, i.e., ωf = 0, the critical frequency is ωcr = ωa/2 (details in Supplementary Information). For ωH ≤ ωcr, the particle is synchronized with the rotating magnetic field (“phase-locked”), i.e., ωp,z = ωH. Otherwise, when ωH > ωcr, the particle undergoes back-and-forth motion (“back-and-forth”) when the maximum magnetic torque cannot balance the response drag torque: the phase lag angle β between n (the projection of n on xoy-plane) and h changes periodically over time20. These two regimes can be observed experimentally (Fig. 1f–i). In the “phase-locked” regime (Supplementary Movie 1), the particle rotates synchronously with the magnetic field, as evidenced by the normalized angular velocity, ωp,z/ωH ≈ 1, in Fig. 1g. In the “back-and-forth” regime (Supplementary Movie 2), the particle oscillates periodically (Fig. 1i).

As illustrated in the schematic diagram of Fig. 2a, the dynamics of the particle preferred magnetization direction, n, is fundamentally characterized by the phase lag angle, β, with respect to h on the xoy-plane. The governing equations, Eqs. (1), can be simplified and explicitly written as (derivations in “Methods”)

β_ = ωH − ωa/2 sin 2β − ωf,z. (2)

Geometrically, we have β = ωHt − ψ. The phase lag angle β represents the particle phase lag with respect to the magnetic field, while ψ is the angle of the particle rotation in the laboratory frame with respect to the x-axis, which describes the particle rotational displacement in the laboratory frame. The turbulent vorticity signal, ωf, represents the vorticity experienced by the particle center along its trajectory. This simplified representation, Eq. (2), separates different contributions and clearly shows that the rotational motion of the particle results from the interplay between the magnetic field rotational drag, the magnetic torque, and the turbulence drag torque (Fig. 2a).

Switching on turbulence
Now we wonder how this picture will change when turbulence is taken into consideration. The turbulence induces fluctuations on the small particles of a typical magnitude ωη = 1/τη, the Kolmogorov characteristic frequency. Experimentally, we turn on the turbulence and vary the intensity of the magnetic field. By statistically analyzing the probability density function (PDF) of the particle angular velocity ωp, a dominant peak is observed in the z-direction at approximately ωp,z ≈ ωH (Fig. 2b–d, purple circles), while the angular velocities in the x and y directions are similar to each other but significantly different from that in the z-direction (Fig. 2b–d, yellow and green circles).

To understand the influence of turbulence on the particle dynamics further, we carry out a systematic numerical investigation. To model the motion of light particles in HIT, we first perform direct numerical simulations using a pseudo-spectral method10,52,53 to obtain ωf (see “Methods” for details). The resulting vorticity data is then incorporated into the theoretical model, Eqs. (1), which is integrated using a third-order Runge–Kutta method. The results (Fig. 2b–d and Supplementary Movies 3–5) validate the theoretical model against the experimental results. Notably, the theoretical model (squares) accurately captures the essential statistical physics (peak positions) of the particle stochastic rotational dynamics in the experiments (circles). The slight discrepancies in the PDF shapes arise from polydispersity in particle shape and magnetic properties in experiments which influences how individual particles respond to an external magnetic field, whereas the simulations assume idealized spherical particles with the same magnetic characteristics for each particle (“Methods” and Supplementary Information). Further comparisons between simulation and experimental results all demonstrate good qualitative agreement (Supplementary Information, Supplementary Movies 9 and 10).

Additionally, to gain deeper insight into the stochastic nature of particle rotation in turbulence, we analyze the waiting time, τ—the duration between successive transitions of β between its metastable states—to characterize the nature of turbulence-induced noise in the absence of external forcing (i.e., ωH = 0). The Poisson-like distribution of τ confirm that, within the investigated parameter space, stochastic transitions induced by turbulent (vorticity) fluctuations can be effectively modeled using white noise with an equivalent intensity (Supplementary Information).

Three distinct regimes of particle dynamics
With the theoretical model capturing the core dynamics of this complex system, and with simulations allowing exploration of a broader parameter space and finer details of particle rotation, we systematically investigate how turbulence-induced vorticity fluctuations and the deterministic magnetic forcing shape particle rotational dynamics.

Beyond the previously identified “phase-locked” and “back-and-forth” regimes, we uncover a third regime: the turbulence-dominated regime, observed at small ωa w.r.t. ωη, when the effect from the magnetic field is negligible. Here, the particle rotation is primarily driven by turbulence, and it exhibits a zero-mean normalized angular velocity (Fig. 3b), as evidenced by the random orientation distribution of the particle orientation, n (Fig. 3d and Supplementary Movie 6). Consequently, the phase lag β and its derivative β_ are randomized (Fig. 3c). In the “phase-locked” regime (ωa ≫ ωη, ωH < ωcr), the particle is synchronized with the magnetic field (Fig. 3e). In this regime, the particle angular velocity along the magnetic field rotation axis (z-axis) is locked to ωH (Fig. 3f, green line). The particle orientation is strongly confined, exhibiting 2D behavior (Fig. 3h and Supplementary Movie 7). In the “back-and-forth” regime (ωa ≫ ωη, ωH > ωcr), the particle undergoes periodic acceleration and deceleration (Fig. 3j, k, inset), forming distinct looping patterns in its orientation trajectory (Fig. 3l and Supplementary Movie 8). These regimes are summarized in a phase diagram, characterized by the normalized variance of the particle angular velocity in the z-direction, ⟨ω2p,z⟩ − ⟨ωp,z⟩2/ω2η, where ⟨ ⋅ ⟩ represents a time average. The noiseless critical frequency ωcr = ωa/2 is marked by a black dot-dashed line. Unlike in a quiescent fluid, the phase-locked regime (Fig. 3m, purple region) is bounded by the blue line, predicted by the scaling argument based on the simplified governing equation of Eq. (2): ωa = 2ωH + ωη (detailed reasoning can be found in “Methods”). This shift in the subcritical boundary, relative to the noiseless case (ωa = 2ωH), accounts for the influence of turbulence fluctuations, which have a standard deviation on the order of ωη.

Stochastic resonance
To determine whether SR occurs, we focus on the β and its behavior across different parameter settings. We summarize the time-averaged value of ⟨β_⟩ = ωη in one phase diagram (Fig. 4a), with ωη/ωa (vertical axis) quantifying the relative turbulent fluctuation intensity (noise) to the applied magnetic field strength, and ωH/ωa (horizontal axis) distinguishing noiseless subcritical, critical (i.e., ωH/ωa = 1/2, black dot-dashed line), and supercritical states.

For a fixed ωH/ωa in the subcritical (supercritical) range, ⟨β_⟩ = ωη exhibits a non-monotonic (monotonic) dependence on noise intensity (Fig. 4b), with a pronounced resonant peak at ωη/ωa ≈ 1, indicating that resonance arises at the transition from the “phase-locked” regime to the “turbulent-dynamics” regime. This is a clear suggestion that the observed resonance behavior results from the cooperation between turbulent fluctuations and magnetic forcing. SR is also clearly observed experimentally, as shown in Fig. 4c: diamonds denote the experimental results at ωH/ωa = 0.12 ± 0.01 (corresponding to the points marked in the phase diagram in Fig. 4a), while circles represent the simulation results at a comparable value of ωH/ωa for direct comparison. The experimental and simulation results agree reasonably well; at the resonance peak, the experimental curve appears flatter which can be attributed to effects of polydispersity of the particles used in experiments. The resonance mechanism can be interpreted via an effective potential well framework (Fig. 4d), where the magnetic field induces a cosine-type potential (i.e., −ωa cos 2β) modulated by a linear term (i.e., −ωH β), while turbulence acts as a stochastic noise.

For ωH/ωa < 1/2 (Fig. 4e), the phase lag angle is initially trapped in the potential minimum at weak noise levels (small ωη/ωa), exhibiting only mild fluctuations locally. As the turbulence-induced fluctuations become comparable to the magnetic field strength (ωη/ωa ≈ 1), preferential unidirectional escape occurs. This clearly identifies the observed resonant behavior as an example of SR: a cooperative interplay between stochastic forcing and the external deterministic forcing. This phenomenon can be thought of as the particle “surfing” the rotating magnetic field, aided by the turbulent vorticity. At even higher noise levels, randomization overrides the cooperative effect, causing bidirectional escape and a decline in ⟨β_⟩.

At the critical threshold ωH/ωa = 1/2, the potential well vanishes, rendering β critically sensitive to any perturbations (Fig. 4g). In the supercritical range (ωH/ωa ≫ 1/2), the potential well inherently favors persistent phase slipping, regardless of noise intensity (Fig. 4h), such that the SR mechanism vanishes in this regime.

In strongly subcritical regime (ωH/ωa ≪ 1/2), the resonance peak is purely set by the turbulent noise, making it sensitive to the preferential sampling effect7,9,10 (see Supplementary Information for neutral and heavy particles results). Here we introduce the relaxation time τ* = 1/√(ωa(ωa/4 − ωH)). Physically, τ* represents the characteristic time it takes for a perturbed particle rotational motion to return to its phase-locked state, i.e., β_ = 0, after a small disturbance (detailed derivations are in Supplementary Information). Close to the critical ratio ωH/ωa = 1/2, the significantly increasing particle relaxation time τ* shifts the resonance position, making it the dominant factor. Therefore, a longer τ* allows particles to escape the potential well even at lower noise levels.

Symmetry breaking mechanism
A striking consequence of SR is the symmetry-breaking effect, manifesting as a “zero plus zero is greater than zero” phenomenon. This leverages the fact that in the weak noise regime (ωη < ωa), the response of the system to the noise is non-linear (Fig. 4b, purple-shaded part).

To reveal this effect, instead of applying a constant-frequency rotating magnetic field (ωH), we now impose a time-dependent field rotation with a zero-average, ωH(t) (Fig. 5a). Each oscillation cycle consists of a forward and a reverse rotation phase, characterized by angular velocities ω+H and ω−H with corresponding durations T+H and T−H. These parameters are scaled as ω+H = ω−H = T−H = T+H = γ, ensuring that the net angular velocity over a full cycle remains zero. Two distinct response regimes of the oscillation in frames of fixed ωη/ωa are observed for varying γ (Fig. 5b). In the nonlinear regime (low noise), ⟨β_⟩ = ωη decreases monotonically, while in the linear regime (strong noise, where system responses collapse, see Fig. 4b, inset), it remains nearly unchanged and close to zero. This behavior aligns well with a superposition model, which predicts the response as ⟨β_⟩ = ⟨β_(ω+H)⟩T+H − ⟨β_(ω−H)⟩T−H / TH, (3) with TH = T+H + T−H, where ⟨β_(ω+H)⟩ and ⟨β_(ω−H)⟩ are interpolated from the numerical results in Fig. 4b. The predicted trends (Fig. 5b, dashed lines) show excellent agreement with the frequency-modulated field results (Fig. 5b, symbols).

We emphasize that when combining turbulence and the frequency-modulated magnetic field in this way, this leads to particle motion directly in the lab frame. To evidence this, we consider the lab-frame angular particle motion ψ(t) (Fig. 5c). With turbulence alone, the particle exhibits no net angular movement (green dotted line). Under a purely frequency-oscillatory magnetic field, the particle undergoes periodic motion with zero net angular displacement (blue dashed line). However, when both effects act concurrently, a net, non-zero mean angular drift remarkably emerges (black solid line), as the turbulent vorticity aids the particle in “surfing” the magnetic field, which reveals an unexpected symmetry-breaking mechanism in turbulent flows. This stands in contrast to the case of a constant ωH (Fig. 5c, inset), where a purely magnetic field-driven particle, instead of showing zero net angular movement as in the oscillating ωH scheme, follows ψ = ωHt (black solid line).

Methods
Experimental setup
The experimental setup comprises three fundamental components: a turbulence generator, a magnetic field generator, and magnetic particles. Detailed specifications of the experimental apparatus, along with a comprehensive description of the particle rotation tracking methodology, are not relevant to the core physics of current work, and hence they are provided in our dedicated instrumentation and methodology paper61. Here, we present only the essential information pertinent to the current study.

Turbulence generator: The experiments are conducted in a confined Von Kármán-type turbulent flow, which serves as a model system for studying HIT, particularly in the context of Eulerian and Lagrangian statistics48,49,62–64.

The working fluid is water, contained within an octagonal-shaped vessel with an internal diameter of 2R = 150 mm and a height of 220 mm. This specific geometry is chosen to enhance optical accessibility for visualization. The flow is driven by two counter-rotating disks, each with a diameter of 150 mm, separated by a distance of 150 mm, and rotating at the same angular velocity magnitude in opposite directions. Each disk is equipped with six straight blades, each 10 mm in height, designed to enhance inertial stirring. The disks are powered by two calibrated DC motors operating at a constant voltage, ensuring that the angular velocity Ω remains stable over time with a precision of approximately 1%. The motor rotation frequency can be adjusted within the range of 0.25–8 Hz. Under these conditions, a fully developed turbulent flow with up to Reλ = 447 can be generated in the central region of the container with small-scale statistics close to local isotropy46–49,62–64. This region, measuring approximately 20 × 20 × 20 mm, serves as the primary domain for measurements. To enable particle rotational motion tracking, a Photron NOVA S6 high-speed camera, operating at 3000 frames per second with a resolution of 1024 × 1024 pixels, is employed to record particle motion within the flow field.

Magnetic field generator: A uniform rotating planar magnetic field within a restricted measurement volume, H = Hh, with an angular frequency ωH, is generated using a system of two pairs of mutually perpendicular Helmholtz coils. The vertical pair (marked in red in Fig. 1a) has an interior diameter of 300 mm, while the horizontal pair (marked in blue in Fig. 1a) has an interior diameter of 245 mm. For each pair, the separation between the two coils is equal to the corresponding coil diameter.

Each Helmholtz coil pair generates a magnetic field along a single axis. By driving the two perpendicular coil pairs with AC currents that have a phase difference of π/2, a resultant magnetic field is produced that rotates at a constant angular velocity ωH. This system is capable of generating rotating magnetic fields with frequencies of up to 50 Hz.

Magnetic particles: The magnetic particles are produced by coating spherical Styrofoam cores with a layer of magnetic paint, which is paramagnetic. The Styrofoam cores are initially arranged in a fixed, evenly spaced configuration on a substrate. The magnetic paint is then applied from the top of the substrate to the Styrofoam spheres through a spraying process, ensuring that all particles are coated in a consistent manner; however, at the single-particle level the coating is not perfectly homogeneous, and small variations in thickness or distribution lead to slight differences in magnetic properties between particles. Finally, the coated particles are detached from the substrate. The particle volume fraction used in the experiments is 0.17%, ensuring that the system remains in the dilute regime. This low concentration minimizes inter-particle interactions, allowing us to focus on the dynamics of particle response to the complex interplay between turbulent fluctuations and the external magnetic field, without significant effects from collective dynamics.

These particles exhibit anisotropy in three key aspects: (1) Shape: The particles are approximately spherical, with a mean diameter of 0.762 ± 0.066 mm. (2) Density: The magnetic particles have a mean density of ρparticle = 208 ± 14 kg/m3. (3) Magnetic Properties: Due to the non-uniform distribution of the magnetic paint on the particle surface, the magnetic properties vary between particles. Specifically, the magnetic moment magnitude ∣m∣, the orientation of the magnetic moment, and the spatial distribution of magnetization across the surface differ from particle to particle. This variability influences how individual particles respond to an external magnetic field. All these three aspects of particle polydispersity can introduce minor differences between the simulation results and the experimental observations. However, these differences remain small, and overall, the simulations show good agreement with the experiments. More importantly, our theoretical modeling and the simulations successfully capture key physical features observed in the experiments, such as the location of peak responses.

Particle rotation tracking: The particle rotational motion is tracked using an efficient particle rotation tracking algorithm. The algorithm initiates by pre-processing the experimental images through the application of Gaussian filtering to mitigate noise, contrast enhancement to ensure particle magnetization quality. Subsequently, a discrete candidate rotation angles is defined, and corresponding theoretical rotated images are generated utilizing 3D rotation matrices. For each temporal frame, the algorithm computes the cross-correlation coefficient between the actual experimental image and each theoretical rotated image to determine the optimal matching rotation angle. By tracking these optimal angles across the time series, the rotational motion trajectory of the particle is reconstructed and visualized. More details of the tracking methodology are reported in our dedicated instrumentation and methodology paper61.

Theoretical model
A paramagnetic particle moves in a homogeneous and isotropic turbulent flow while subjected to an external magnetic field H. The magnetic field rotates in the xoy-plane (laboratory frame) with an angular velocity ωH and intensity H, expressed as H = Hh = H(cos(ωHt), sin(ωHt), 0)T. The particle is modeled as a sphere with intrinsic magnetic anisotropy (same as that of a prolate spheroid of aspect ratio 1.2), characterized by a preferred magnetization direction n, which dynamically evolves over time. This instantaneous unit vector is given by n = (sin α cos ψ, sin α sin ψ, cos α)T, where α and ψ (defined in Fig. 2a) describe the orientation of n.

When the particle is placed in an external magnetic field, H, its induced magnetic moment takes the form19 m = VpH[χ⊥h + (χ∥ − χ⊥)(n·h)n], (4) where Vp = 4πr3/3 is the particle volume, with r denoting its radius. The anisotropic magnetic susceptibilities χ∥,⊥, arising due to the anisotropy of the particle shape (e.g., spheroidal) and the demagnetizing field factors N∥,⊥, and χ∥,⊥ = χ0/(1 + χ0N∥,⊥)51, causes the particle to respond differently to an external magnetic field depending on the direction. The subscripts parallel (∥) and perpendicular (⊥) refer to directions relative to the preferred magnetization direction n. Due to the tendency of the magnetic moment m to align with the magnetic field direction h, the rotating magnetic field induces the particle rotational motion. The magnetic torque acting on the particle is given by TM = μm × H, where μ is the magnetic permeability. The magnetic torque tries to align the particle’s preferred magnetic axis (n) with the magnetic field direction (h) and is balanced by the rotational drag torque (approximated in the Stokes regime) TD = −ξr(ωp − ωf)65, where ξr = 8πμr3 is the rotational drag coefficient and μ is the dynamic viscosity of water.

The particle rotational velocity ωp naturally decomposes into two components: a perpendicular component ωp,⊥ (tumbling) and a parallel component ωp,∥ (spinning) along n. The particle rotational motion is governed by the balance of torques, and we consider the overdamped condition50 TM + TD = 0, (5) with TM the magnetic torque and TD the rotational drag torque. By projecting this balance along and perpendicular to n, we obtain the equations governing tumbling and spinning. The tumbling dynamics, n_, obey dn/dt = ωa(n×h)(h×(n×h)n) + ωf × n, (6) where ωa = μVpH2(χ∥ − χ⊥)/ξr is the normalized magnetic field intensity, which is also called anisotropic frequency19,50, with ωa > 0 (ωa < 0) for a prolate (oblate) particle.

The particle spinning rate ωp,∥ satisfies ξr dωp,k/dt = ξr(ωf·n)n + ωf × n, (7) where for the overdamped particle, we have d(ξr ωp,k)/dt = 0. ωf is half of the vorticity of the background flow field, which is obtained through performing Direct Numerical Simulations (DNS) using a pseudo-spectral method. The validity of the overdamped model is further discussed in the Supplementary Information.

The total particle angular velocity is thus ωp = ωp,k + ωp,⊥ = (n·ωf)n + n × n_, with n_ = dn/dt (see Eq. (6)). Equations of motion (6) and (7) are numerically integrated using a third-order Runge–Kutta method.

The governing Eq. (5) can be reformulated in terms of the phase lag angle β and the orientation angle α, where α denotes the angle between the unit vector n and the z-axis, z^. To derive the governing equation for β, we project Eq. (5) onto z^-direction, yielding TM·z^ + TD·z^ = 0: (8) Here, the magnetic torque component along z^ is given by TM·z^ = ωa ξr sin 2α sin β cos β, while the drag torque contribution is expressed as TD·z^ = −ξr(ωp,z − ωf,z). Rewriting Eq. (8) explicitly in terms of β and α, we obtain β_ = ωH − ωa/2 κ sin 2β − ωf,z, (9) with the coefficient κ = sin 2α.

To establish the evolution equation for α, we introduce a unit vector N = (−sin ψ, cos ψ, 0)T, which denotes the direction perpendicular to nxoy, the projection of n onto the xoy-plane (see Fig. 2a for the definition). Projecting Eq. (5) along N leads to TM·N + TD·N = 0: (10) Here, the magnetic torque component along N is given by TM·N = 1/2 ωa ξr cos 2β sin 2α, while the drag torque contribution is TD·N = −ξr(ωp·N − ωf·N). Noting that ωp·N = α_, Eq. (10) can be rewritten explicitly as α_ = ωf·N + ωa/2 sin 2α cos 2β: (11) To capture the essential physics of the system, we assume κ ≈ 1, which is reasonable because κ follows a probability density function that is sharply peaked at 1 and decays rapidly to zero for values smaller than unity within the investigated parameter space (further details are provided in the Supplementary Information). The equation for β can be reduced to β_ = ωH − ωa/2 sin 2β − ωf,z, (12) which is controlled closely by the magnetic field. Equation (12) is used to predict the phase transition between the phase-locked regime and the turbulent-dynamics and back-and-forth regimes (blue solid line in Fig. 3m), and is ultimately describing the dynamics that gives rise to the mechanism of SR (Fig. 4c–g). Similar dynamical equations as Eq. (12) have been widely applied in multiple physical systems, including condensed matter physics, chemical and biological systems, optics and laser physics, electrical engineering, and financial econometrics66,67, with a range of related physical problems, such as Josephson junctions68–72, vortex diffusion and flux-line dynamics in superconductors68,73,74, reaction-rate characteristics relevant to chemistry, engineering, and biology75, Brownian motors76, and decision-making processes under uncertainty which resemble financial market fluctuations67,72. The universality of this equation suggests that our findings have broad implications beyond the specific context of magnetic particles in turbulence, offering deeper theoretical insights into nonlinear dynamical systems and facilitating the development of new applications in diverse scientific and technological domains.

Now we show the details of predicting the boundary (Fig. 3m, blue solid line) that separates the phase-locked regime (Fig. 3m, purple region) from the back-and-forth and turbulent-dynamics regimes. The phase-locked regime exists if the equation β_ = 0 admits fixed points. Based on the simplified governing equation, Eq. (2) or Eq. (12), this condition translates to ensuring that the following equation has real solutions for any imposed ωH: ωa/2 sin 2β = ωH − ωf,z: (13) The left-hand side of Eq. (13) is bounded within [−ωa/2, ωa/2], while the right-hand side varies typically over [ωH − σ(ωf,z), ωH + σ(ωf,z)], where σ(ωf,z) is the variance of ωf,z. For Eq. (13) to have real solutions for any imposed ωH, the following condition must hold: ωH + σ(ωf,z) ≤ ωa/2, (14) which gives ωa ≥ 2ωH + 2σ(ωf,z): (15) Notice that the standard deviation of ωf,z is approximated to be of the order of the Kolmogorov frequency scale, i.e., σ(ωf,z) ≈ ωη, where ωη ∼ 1/τη. In our simulations, we observe that 2σ(ωf,z) ≈ ωη, which justifies this approximation. Based on this, we obtain the final criterion for the phase-locked regime from Eq. (15) as ωa ≥ 2ωH + ωη: (16) Direct numerical simulation of the Navier-Stokes equation
To accurately integrate the governing Eqs. (6) and (7), it is essential to correctly incorporate the Lagrangian vorticity signal of the particles ωf. We begin by performing DNS of statistically stationary homogeneous isotropic turbulence (HIT) to obtain the Lagrangian vorticity dynamics (ωf) of small, passively advected particles. The computational domain consists of a cubic box of size L = 2π with periodic boundary conditions. The simulations employ a pseudo-spectral method52 for spatial discretization and an Adams-Bashforth scheme for temporal integration, with the standard 2/3 dealiasing rule applied to ensure numerical accuracy53. The numerical framework has been validated in a large number of studies using similar schemes, interpolation methods77. The suspended particles are modeled as point-like, dilute tracers that interact with the turbulent flow through hydrodynamic forces alone. Their dynamics are characterized by the Stokes number (St) and the density contrast (βp), which describe their response time relative to the fluid and their mass relative to the surrounding medium, respectively. To capture the essential physics while simplifying the problem, we adopt a one-way coupled model, which has been extensively validated in both experimental and numerical studies9,10,78–82. The governing equation for particle motion is x€p = βp D[uf]O/Dt + 1/St (x_p − [uf]O), (17) where [uf]O denotes the fluid velocity with the subscript O indicating evaluation at the center of the particle, xp represents the particle position, and vp = x_p is the particle velocity. The Lagrangian vorticity is computed as ωf = 1/2(∇×u)O. The density contrast is defined as βp = 3ρf/(ρp + 2ρf), where ρp/ρf is the particle-to-fluid density ratio. The Stokes number is given by St = dp2/(12βp ντη), where dp is the particle diameter and ν is the fluid kinematic viscosity. In this study, we perform three-dimensional DNS at a resolution of N3 = 1283 with a Taylor-scale Reynolds number Reλ ≈ 6010,52,53,77 and collect 512 independent samples for three particle types: light (βp → 3.0, representing bubbles where ρp → 0), neutral (βp = 1.0), and heavy (βp = 0.01) particles, all with a Stokes number of St = 1.0. This specific Stokes number is chosen because light particles in HIT tend to accumulate preferentially in vortex filaments, leading to strong dynamic localization7,9,10,83. Extending our results to higher Reynolds number turbulence is the effect of intermittency, particularly the heavy-tailed distribution of vorticity fluctuations in fully developed turbulence. In principle, for very large Reynolds numbers, the intermittency of the small-scale vorticity field should be accounted for, as extreme vorticity events become more pronounced. However, previous studies have shown that while intermittency affects the fine-scale structure of turbulence, its influence on the rotational dynamics of small particles (compared to the Kolmogorov scales) is relatively weak, because the intense vorticity events, although frequent, are spatially localized and short-lived; thus, their cumulative effect on particle rotation is averaged out81,84,85. Our results and the underlying mechanism are in principle applicable in turbulent flows. Future research could explore the effects of finite-sized particles and beyond the dilute regime.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The article presents a compelling exploration of the rotational dynamics of light magnetic particles in turbulent flows under the influence of a rotating magnetic field. By combining experimental studies in a von Kármán turbulent flow, theoretical modeling, and direct numerical simulations (DNS) of homogeneous isotropic turbulence using a point-particle approximation, the authors deliver novel insights into the interplay between turbulence and magnetic forcing.\nThree main contributions, which I find particularly novel and impactful, stand out:\n- The first experimental observation of stochastic resonance (SR) in turbulent vorticity, offering a new mechanism for enhancing signal detection in noisy environments.\n- The identification of a symmetry-breaking phenomenon in the particles’ rotational response under zero-mean magnetic forcing, revealing a rich nonlinear dynamical behavior.\n- The proposal of a metrological technique for probing turbulent vorticity, using magnetic resonance of embedded particles, potentially valuable in microfluidic and rheological applications.\nThese findings are, in my opinion, not only important for the fundamental understanding of turbulence but also open new avenues in active microrheology, chiral fluid design, and metrological techniques. The demonstrated “surfing effect” highlights the nonlinear coupling between deterministic and stochastic forces, and the introduced phase-transition framework could inspire broader applications beyond turbulence research.\nThat said, several technical clarifications and improvements in presentation would strengthen the manuscript and ensure reproducibility. I believe the article has strong potential for publication in Nature Communications, and I will be happy to give a final recommendation after the authors address the points detailed below.\n1. Turbulence Generator Details\nThe description of the experimental facility is lacking. The authors should provide key flow characteristics, including the Reynolds number, energy injection/dissipation scales, and relevant integral/Kolmogorov timescales.\n2. Particle Characterization\nAdditional detail on the particles is necessary. Specifically:\n• What is their size relative to the turbulence length scales?\n• Can they be reasonably treated as point particles, or must finite-size effects be considered?\n• Quantify diverse sources of polydispersity and provide an estimate of the magnetic susceptibility.\n• The roughness appears significant from images, has this been measured or characterized?\n3. Magnetic Field Details\nWhile magnetic field amplitudes are mentioned in figure captions, they should also be clearly specified in the Methods section, along with the typical frequency range used.\n4. Rotational Tracking Methodology\nPlease elaborate on the method used for tracking rotational motion:\n• What pattern or texture is exploited for angular tracking?\n• Is any post-processing or filtering applied?\n• What is the temporal resolution and signal-to-noise ratio on angular velocity. Is it sufficient to eventually hope extracting rotational accelerations for future studies?\n5. Rotational Dynamics Modeling\nThe authors rely on Stokes rotational drag. Given the aspect ratio ~1.2, have they estimated the role of inertial torques?\nThese can be non-negligible even for weakly anisotropic, sub-Kolmogorov particles.\n6. Validity of the Point Particle Approximation\nHow appropriate is the point-particle model used in simulations, considering the actual particle size in experiments?\n7. Neglecting Rotation-Translation Coupling\nThe assumption of decoupled translational and rotational dynamics may not hold for finite-size or anisotropic particles.\nCould the authors assess the possible impact of these couplings?\n8. Overdamped Approximation\nThe overdamped limit is central to the analysis. The authors should discuss its validity more thoroughly:\n• Under what conditions (e.g. magnetic field strength, frequency, particle size/moent inertia/magnetization) is the approximation justified?\n• Have they evaluated rotational acceleration statistics to confirm the absence of inertial effects beyond first moments?\n9. Effect of von Kármán Flow Properties\nWhile the von Kármán setup is a standard turbulence generator, it is not isotropic or homogeneous, particularly due to its central core with non-zero mean helicity. Could this large-scale anisotropy influence the observed symmetry breaking or other phenomena? A discussion on this point would improve the interpretation of results and their generalizability.", "reviewer_2": "The authors report on the rotational dynamics of a magnetic particle (subjected to a rotating magnetic field) in a turbulent von Kármán flow (generated by two counter-rotating impellers). They observe three particle-rotation regimes experimentally and numerically: A phase-locked regime synchronized with the field rotation, a back-and-forth particle motion, and a turbulence-dominated one. The main result of the manuscript concerns the numerical evidence of stochastic resonance of the particle angle occurring for intermediate stochastic noise of turbulence.\nTo do so, the authors built an elegant experimental system and developed a numerical method to probe the stochastic resonance. Although redundancies and back-and-forth are too frequent, the manuscript is rather well written, and the figures are presented very cleanly. However, I find that the experimental part is too reduced and concerns only the phase-locked regime and the back-and-forth regimes, which are not new in the literature, the latter being not sufficiently quoted. Moreover, I have several serious concerns about the claims made by the authors.\nMajor comments:\n1) The synchronized regime and the back-and-forth regime are known phenomena occurring without noise, as already observed experimentally for a compass (V. Croquette et al., J. Phys. Lett. 1981, A. Poyé et al., Phys. Scr. 2019) or a rolling particle (A. Kaiser et al., Sci. Adv. 2017) in an alternating magnetic field. On the other hand, the turbulent regime occurs when the effect of the magnetic field is negligible, which, in practice, corresponds to a nonmagnetic particle in a turbulent flow. Thus, only the results related to the intermediate noise level are new, including the stochastic resonance. It should be clearly stated in the manuscript, and these references should at least be discussed. The bibliography also fails to include key articles relevant to the topic of this manuscript (see below).\n2) The authors want to address an important question: Understanding the coupling between a particle’s rotation and the underlying vorticity field in a turbulent flow. The authors state that magnetic particles can control the vorticity of the flow. In turbulent flows, energy is generally injected at large scales (e.g., by an impeller) and cascades towards small scales. In this system studied by the authors, the energy contained at the particle size is therefore much smaller than the energy-containing range, and the particle rotation will surely influence only slightly the dynamics of the flow and its vorticity. The authors could comment on that point.\nSuch claims could have been supported by experimental measurements in the flow velocity field, but the manuscript appears to make limited use of the experimental facility. Most of the results are numerical simulations, in particular, the main result of stochastic resonance in turbulence, which I believe could have also been evidenced experimentally. Consequently, the significance of this work to the field is questionable, mainly due to the lack of experimental data to compare with the numerical resolution of a single particle equation. Reading the abstract, I would have expected more experimental results (only Figs. 1 and 2 are experimental), and the main result of the manuscript is numerical. Moreover, the authors use optical methods to track the particle and measure the flow vorticity. I believe measurements with PTV or PIV would also give very high-quality vorticity data and probably be less intrusive to the flow than using particles with a size above the Kolmogorov length scale.\n3) The main message of the manuscript is that vorticity fluctuation effects on a magnetic particle can be seen as “noise” without the need to solve complex hydrodynamics equations. However, I do not see any results showing the impact of the particle’s rotation on the flow via hydrodynamic interactions, which is stated in the introduction as a mechanism to control turbulence.\n4) There is a claim in the abstract, introduction, and conclusion that using magnetic particles to probe turbulent fluctuations can circumvent optical techniques. However, the experimental results in the manuscript solely rely on optical measurements.\nCan the author clarify what they have in mind?\n5) How do the authors measure the Kolmogorov timescale in their experimental setup, presented in Fig. 2? What is the experimental Stokes’ number?\n6) The title is misleading and should be explicit, something like “Stochastic resonance dynamics of rotational magnetic particles in turbulence”.\n7) What is the Reynolds number? It should be clearly stated, as is standard in any turbulence study. Depending on its value, the rotational drag coefficient could be laminar (as used in the manuscript) or turbulent. Thus, it could change the conclusion of the manuscript. It should be relatively easy numerically to take this point into account.\n8) Why experimental PDFs are skewed in Figs. 2b-d, whereas the signals in Fig. 1g and Fig.1i look symmetric?\n9) It would have been interesting to include more experimental points in Fig. 3 (beyond the three already provided) to perform a reasonable comparison with simulations.\n10) The typical system size and magnetic field strength should appear in the main text (instead of the Methods Section). Generally speaking, the frequent back-and-forth between the main text, Methods section, and Supplemental Information disrupts the reading flow. Some redundancies also appear in these different parts.\n11) I believe there is a mistake in the magnetic particle density line 489. 0.2 kg/m^3 is a density 6 times lighter than air, please correct that. Why not choose neutrally buoyant particles instead of very light ones compared to water?\n12) The bibliography is extensive (73 references). However, several experimental studies, conducted in different research groups, have investigated 3D turbulence generated by magnetic particles (e.g., E. Falcon et al. PRL 2017, A. Cazaubiel et al. Phys. Rev. Fluids 2021, J.B. Gorce et al. Phys. Rev. Lett. 2024), or regimes reminiscent of 2D turbulence (e.g., N. Francois et al. Phys. Rev. X 2014, G. Kokot et al. PNAS 2017, M. Bourgoin et al. Phys. Rev. X 2020). Including some of these works in the bibliography would strengthen the contextual background.\n13) Is the magnetic coating uniform? There is an inconsistency on this point in the Methods section (lines 476 and lines 490-491)? What determines the direction of the magnetic moment of each particle?\n14) Is it in Nature Communications’ policies to publish the dedicated instrumentation and methodology in another journal, as indicated in lines 426-427?\nMinor comments\nIn line 489 and the Supplemental Information, the accuracy of the magnetic particle diameter is in micrometers. Is it significant?\nMovies 6, 7, and 8 are long (5 min each) and could be sped up.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to Reviewer 1\nWewouldliketothankthereviewerforthethoroughreadingofourmanuscript,forhis/hermanyinsightful\ncomments, and suggestions that allowed us to considerably improve our manuscript. We really appreciate\nthese efforts by the reviewer.\nSynopsis: The article presents a compelling exploration of the rotational dynamics of light magnetic\nparticles in turbulent flows under the influence of a rotating magnetic field. By combining experimental\nstudies in a Von Kármán turbulent flow, theoretical modeling, and direct numerical simulations (DNS) of\nhomogeneous isotropic turbulence using a point-particle approximation, the authors deliver novel insights\ninto the interplay between turbulence and magnetic forcing.\nThree main contributions, which I find particularly novel and impactful, stand out:\n- The first experimental observation of stochastic resonance (SR) in turbulent vorticity, offering a new\nmechanism for enhancing signal detection in noisy environments.\n- The identification of a symmetry-breaking phenomenon in the particles’ rotational response under zero-\nmean magnetic forcing, revealing a rich nonlinear dynamical behavior.\n- The proposal of a metrological technique for probing turbulent vorticity, using magnetic resonance of em-\nbedded particles, potentially valuable in microfluidic and rheological applications.\nThese findings are, in my opinion, not only important for the fundamental understanding of turbulence but\nalsoopennewavenuesinactivemicrorheology,chiralfluiddesign,andmetrologicaltechniques. Thedemon-\nstrated “surfing effect” highlights the nonlinear coupling between deterministic and stochastic forces, and the\nintroduced phase-transition framework could inspire broader applications beyond turbulence research.\nThatsaid,severaltechnicalclarificationsandimprovementsinpresentationwouldstrengthenthemanuscript\n1\nand ensure reproducibility. I believe the article has strong potential for publication in Nature Communi-\ncations, and I will be happy to give a final recommendation after the authors address the points detailed\nbelow.\nResponse: Weareverygratefultothereviewerforallthesuggestionsonhowtoimproveourmanuscript.\nIn line with the comments of the reviewer, we have made a sincere and dedicated effort for the revision\nof the manuscript. In addition, a point-by-point response to all comments/suggestions is provided below.\nPleasenotethatallmodificationsmadeintherevisedmanuscriptandSupplementaryMaterialbasedonthe\ncomments of the reviewer are highlighted in blue. For convenience, here we have also quoted the modified\ncontent in the revised main paper and the revised Supplementary Material and highlighted them in blue.\nIn the following, we answer/clarify all points raised by the reviewer.\nThe description of the experimental facility is lacking. The authors should provide key flow characteris-\ntics, including the Reynolds number, energy injection/dissipation scales, and relevant integral/Kolmogorov\ntimescales.\nResponse: In the following, we provide a description of the key flow characteristics relevant to our exper-\niments.\nFirst, we note that the detailed description of the experimental setup, –including the turbulence gen-\nerator (a modified French washing machine), the magnetic field generator (Helmholtz coils), the magnetic\nparticles, and the integrated design of the facility– has been thoroughly reported and discussed in a com-\npanion paper (Wu et al., 2025), focusing on the experimental design.\nFor clarity and completeness in the present work, we now include the essential flow characteristics,\nwhich are directly relevant to the phenomena investigated here.\nThe measured dissipation rate ε is determined from the driving torque on the two impellers as ε =\nTΩ/(ρV )(Mordant,1997;Labbéetal.,1996),whereT isthesumofthedrivingtorquesfromtheimpellers\nw\nat the top and bottom of the water tank and V = 2.8×10−3 m3 is the volume of the water tank. The\nw\ndriving torques are measured using two strain gauge torque meters mounted on the motors.\nIn our experiments, we investigated two turbulence intensities: (1) the weak-turbulence condition,\nfor which the results are shown in Fig. 2(b–d) of the main text; and (2) the strong-turbulence condition, for\nwhich the results are presented in Supplementary Fig. 1(b).\nFor the weak-turbulence case, the impellers rotate at 0.83 Hz. The corresponding mean energy dis-\nsipation rate is ε = 0.037±0.002 m2/s3, which yields a Taylor-scale Reynolds number of Re = 398, a\nλ\nKolmogorov length scale of η =0.072 mm, and a Kolmogorov time scale of τ =5.20 ms.\nη\nFor the strong-turbulence case, the impellers rotate at 1.21 Hz. The corresponding mean energy\ndissipation rate is ε = 0.075±0.001 m2/s3, which yields a Taylor-scale Reynolds number of Re = 447, a\nλ\nKolmogorov length scale of η =0.061 mm, and a Kolmogorov time scale of τ =3.66 ms.\nη\nWe have clarified these aspects in the revised Supplementary Material.\nThe modifications in the Supplementary Material are listed below:\nThemeasureddissipationrateεisdeterminedfromthedrivingtorqueonthetwoimpellersasε=TΩ/(ρV )\nw\n(Mordant, 1997; Labbé et al., 1996), where T is the sum of the driving torques from the impellers at the\ntop and bottom of the water tank and V = 2.8×10−3 m3 is the volume of the water tank. The driving\nw\ntorques are measured using two strain gauge torque meters mounted on the motors. In our experiments, we\ninvestigatedtwoturbulenceintensities: (1)theweak-turbulencecondition, forwhichtheresultsareshownin\nFig. 2(b–d) of the main text; and (2) the strong-turbulence condition, for which the results are presented in\nSupplementary Fig. 1(b). For the weak-turbulence case, the impellers rotate at 0.83 Hz. The corresponding\nmean energy dissipation rate is ε = 0.037±0.002 m2/s3, which yields a Taylor-scale Reynolds number of\nλ η\nFor the strong-turbulence case, the impellers rotate at 1.21 Hz. The corresponding mean energy dissipation\nrate is ε=0.075±0.001 m2/s3, which yields a Taylor-scale Reynolds number of Re =447, a Kolmogorov\nλ\nlength scale of η =0.061 mm, and a Kolmogorov time scale of τ =3.66 ms.\nη\nWhat is their size relative to the turbulence length scales? Can they be reasonably treated as point particles,\nor must finite-size effects be considered? Quantify diverse sources of polydispersity and provide an estimate\nof the magnetic susceptibility. The roughness appears significant from images, has this been measured or\ncharacterized?\nResponse: Below, we provide the point-by-point explanation.\n- The particles used in our experiments possess three key characteristics that enable the study of their\nrotational motion under the combined influence of turbulence and magnetic interactions with the rotating\nmagnetic field. The particle diameter is in the range between the Kolmogorov scale and the Taylor mi-\ncroscale, with the radius r ≈5η, which ensures interaction with the smallest turbulent flow structures.\n- Regarding the point-particle assumption, the particles cannot be strictly treated as point-like because\ntheir size is larger than the Kolmogorov length scale, η. However, they are still much smaller than the\nTaylor microscale. As such, they reside in an intermediate regime where the dominant interaction is with\ndissipative-scale turbulence, and finite-size effects are limited. Our analysis primarily focuses on the ro-\ntational dynamics of the particle, which are more sensitive to the externally applied magnetic torque and\nlocal velocity gradients than to large-scale finite-size wake effects. Thus, while the point-particle approxi-\nmation is not strictly valid, the qualitative conclusions of our study, and most crucially the conception of\nthe stochastic resonance mechanism, remain robust.\n- The magnetic susceptibility and anisotropy of our particles are statistically characterized using a Super-\nconducting Quantum Interference Device (SQUID) magnetometer manufactured by Quantum Design Inc.\nThe mean volume magnetic susceptibility χ is determined via batch measurements of 20 particles ran-\nV\ndomly oriented in the sample holder, yielding a mean value of |χ | = 0.136±0.046. Magnetic anisotropy\nV\nwas evaluated through single-particle measurements, yielding an average value of ∆χ = 0.011±0.002. A\nmagnetic field of 1 T was first applied to allow the particle to freely rotate and align its magnetic moment\nm with the field B. Once aligned, the hysteresis loop was recorded along this direction to determine the\np\nparallelsusceptibilityχ . Theperpendicularsusceptibilityχ wasthenmeasuredbyapplyingthemagnetic\n∥ ⊥\nfield orthogonal to the aligned magnetic moment. A more detailed description of the particles’ magnetic\nproperties can be found in our experimental paper, referenced as (Wu et al., 2025). Sources of polydis-\npersity include variation in particle diameter, density, and magnetic susceptibility. Based on imaging of\na batch of 50 randomly selected particles, the diameter distribution has a mean of d = 0.762mm with a\nstandard deviation of ±0.066mm, corresponding to approximately 8.7% relative polydispersity. The den-\nsity is ρ =0.208kg/m3 with a variation below 0.014% across the sampled batch. The spread in magnetic\np\nsusceptibility values has already been quantified above. Taken together, these sources of polydispersity are\nmoderateanddonotqualitativelyaffecttherotationaldynamicsand thestochasticresonancebehavior we\nreport.\n5\n- The particles appear “rough” in optical images, which originates from their manufacturing process. Each\nparticle consists of a lightweight polystyrene core coated with a thin layer of magnetic paint containing\niron powder. During manual spraying, variations in paint distribution, the side of the particle in contact\nwiththeadhesivetape,andminorinconsistenciesinthesprayingangleandamountresultinanon-uniform\ncoating. Thisprocessimpartsthemagneticpropertiesandanisotropynecessaryforparticlerotationunder\nthe applied magnetic field, and also generates surface patterns that visually resemble “roughness”. How-\never, this apparent roughness is primarily due to the contrast between the bright polystyrene core and the\ndarker magnetic coating, rather than actual geometric protrusions. On the macroscopic scale relevant to\nparticle rotational dynamics, the particle surfaces remain closely spherical. Therefore, we expect that this\nsmallroughnessdoesnotsignificantlyincreasetranslationaldragorrotationalresistancebeyondtheeffects\nalready captured by particle size and magnetic torque.\nWe have clarified these aspects in the revised Supplementary Material.\nThe modifications in the main paper are listed below:\nThe particles are designed to probe rotational dynamics under the combined influence of turbulence and\na rotating magnetic field. Their size lies between the Kolmogorov and Taylor microscales, with r ≈ 5η,\nensuring interactions with the smallest turbulent structures while remaining much smaller than the inte-\ngral scale and Taylor microscale. Consequently, they cannot be regarded as point-like; however, finite-size\neffects on the rotation are limited, and the dominant particle–flow interactions occur at dissipative scales.\nThe particles’ magnetic properties were characterized using a Superconducting Quantum Interference De-\nvice (SQUID) magnetometer (Quantum Design Inc.). Batch measurements yielded a mean volume sus-\nceptibility of |χ | = 0.136±0.046, while single-particle measurements revealed an average anisotropy of\nV\n∆χ=0.011±0.002.\nIn optical images as shown in Fig. 1, the particle surfaces may appear “rough.” This appearance originates\nfrom the contrast between the bright Styrofoam core and the darker magnetic coating, rather than actual\ngeometric protrusions. Variations in paint deposition, the side of the particle in contact with the substrate\nduring spraying, and small inconsistencies in the magnetic paint spraying angle result in non-uniform coat-\ning patterns. These patterns enhance optical contrast, creating the visual impression of surface roughness.\nHowever, the actual geometric irregularities are negligible compared to the particle diameter, and the sur-\nfaces remain effectively smooth on the hydrodynamic scale. Therefore, while such patterns may introduce\nminor perturbations to local flow, they do not significantly alter translational drag or rotational resistance\nbeyond the effects already captured by particle size and magnetic torque.\nWhile magnetic field amplitudes are mentioned in figure captions, they should also be clearly specified in\nthe Methods section, along with the typical frequency range used.\nResponse: The rotating planar magnetic field is calibrated using a SENIS F3A magnetic-field-to-voltage\ntransducer equipped with a fully integrated 3D Hall probe. Within the frequency range of 1 to 50 Hz, the\nmaximummagneticfluxdensityremainsstableat1.60±0.02mT.Thefrequencyissettobe20HzinFig.2\nof the main text, and is set to be 10 Hz in Fig. 4 of the main text, which shows the stochastic resonance.\nDetailed characterization of the magnetic field can also be found in our experimental paper (Wu et al.,\n2025).\nThe modifications in the Method section of the main paper are listed below:\nThe rotating planar magnetic field was calibrated using a SENIS F3A magnetic-field-to-voltage transducer\nwith an integrated 3D Hall probe. Across the frequency range of 1–50 Hz, the maximum magnetic flux\ndensity remained stable at 1.60±0.02 mT. The driving frequency is set to be 20 Hz for Fig. 2 and 10 Hz for\nFig. 4, which illustrates the stochastic resonance. Further details of the field characterization are reported\nin our experimental study.\nPlease elaborate on the method used for tracking rotational motion:\nWhat pattern or texture is exploited for angular tracking? Is any post-processing or filtering applied? What\nis the temporal resolution and signal-to-noise ratio on angular velocity. Is it sufficient to eventually hope\nextracting rotational accelerations for future studies?\nResponse: The magnetic particles are made of Styrofoam cores coated with purple magnetic paint. The\noriginal white color of the Styrofoam combined with the purple magnetic paint forms the non-uniform\nsurfacepattern. Thisrandompatternservesasthefeatureforangulartracking,asthealgorithmcomputes\nrotationalmotionbyanalyzingthechangesinthetextureacrossframes. Thedistributionofmagneticpaint\nis not easily controllable due to the manual fabrication process: we characterize the particle magnetism\nstatistically. To reduce image noise, a Gaussian filter is applied to each frame before tracking. The kernel\nsize is set to 3 pixels in our experiments, which was found sufficient to smooth minor fluctuations with-\nout affecting the rotational features. A more detailed description and validation of the rotational tracking\nmethodologyarereportedinourexperimentfocusedworkunderthesectionof“IV.IMAGEPROCESSING\nAND PARTICLE ROTATION TRACKING” (Wu et al., 2025). The frame rate of our recordings is 3000\nfps, corresponding to a temporal resolution of 0.33 ms. In validation experiments conducted to implement\nthis rotation tracking algorithm on magnetic particles (Wu et al., 2025), the signal-to-noise ratio (SNR)\nof the measured angular velocity (normalized by the reference angular velocity of the magnetic field) was\nevaluated using the definition SNR = 20log (µ/σ), where µ and σ denote the mean and standard de-\n10\nviation of the normalized angular velocity, respectively. The SNR was found to range from 16.0 to 37.0\nas the reference angular velocity increased. For rotation frequencies above 20 Hz, the SNR exceeded 30,\nindicatingthatrotationalaccelerationscanbereliablyextractedwhenparticlesrotateatfrequencieshigher\nthan 20 Hz. In turbulent flows, particle rotation can be more irregular than in controlled conditions; while\nourcurrentresultsdemonstratethatrotationalvelocitiesandaccelerationscanbereliablymeasuredunder\nwell-characterized rotations, further investigation is needed to fully assess the method performance espe-\ncially in highly turbulent environments.\nThe authors rely on Stokes rotational drag. Given the aspect ratio 1.2, have they estimated the role of\ninertial torques?\nThese can be non-negligible even for weakly anisotropic, sub-Kolmogorov particles.\nResponse: First, we clarify that the particles in the simulation is modeled as a sphere. To enable these\nspherical particles experience a magnetic torque in a rotating magnetic field, we assume that the particles\npossesses magnetic properties, specifically anisotropic susceptibilities χ and χ . These susceptibilities are\n∥ ⊥\nset to be identical to those of a prolate spheroid with an aspect ratio 1.2, since χ and χ are uniquely\n∥ ⊥\ndetermined by the aspect ratio of the particle (Osborn, 1945). This specific aspect ratio is chosen based\non experimental estimations of the magnetic particle properties, aiming to closely mimic the behavior of\nparticles observed in experiments for a direct one-to-one comparison between simulation and experimental\nresults.\nSecondly, we agree that a careful assessment of inertial effects is crucial for the model validity. Our\nrotational dynamics model relies on the overdamped condition, which assumes that inertial torques are\nnegligible compared to viscous and magnetic torques. To justify this assumption, we now estimate the\nparticlerotationalresponsetime(τ )andcomparedittothecharacteristictimescaleofthefluidvorticity\nrot\n(τ ).\nη\nBased on our experimental parameters:\n• Particle diameter (mean): d=0.762 mm (radius r =0.381 mm)\n• Particle density (mean): ρ =208 kg/m3\np\n• Fluid viscosity (water at room temp): µ≈1.0×10−3 Pa·s\n• Characteristic timescale of fluid vorticity: τ =7.56 ms\nη\nForasphericalparticle,therotationalresponsetimecanbeestimatedasτ = I ,whereI = 2V ρ r2\nrot ξr 5 p p\nis the particle moment of inertia, and ξ =8πµr3 is the rotational drag coefficient. So τ = ρpr2.\nr rot 15µ\nSubstituting the values:\n208 kg/m3×(0.381×10−3 m)2\nτ =\nrot 15×1.0×10−3 Pa·s\n=2.01×10−3 s=2.01 ms.\nNow, comparing the two timescales:\n• Particle rotational response time: τ =2.01 ms.\nrot\n• Fluid vorticity timescale: τ =7.56 ms.\nη\nWeobservethatτ isapproximately2.01/7.56=0.266oraboutone-quarterofτ ,indicatingthatthe\nrot η\nparticle rotational dynamics are significantly faster than the characteristic changes in the fluid vorticity.\nanoverdampedregimeareasonablefirstapproximation. Furthermore, manynovelphenomenaofmagnetic\nparticles have been found in an overdamped system, e.g., colloidal chains, microscopic rods, flexible fila-\nments, or dynamical re-orientations (Melle et al., 2002; Qian et al., 2008; Belovs and C¯ebers, 2006; Dhar\net al., 2007; Tierno et al., 2009).\nOfcoursethemagneticparticlesusedinourexperimentsmaydisplayaslightanisotropy,thederivation\nof τ for an equivalent sphere provides a strong first-order estimate. For weakly anisotropic particles and\nrot\nin situations where the particle response time is considerably shorter than the characteristic fluid scales,\ninertial torques are often considered negligible. In such cases, the particle angular response is largely\ndominated by drag torque rather than inertial ones. Moreover, the good agreement observed between\nour experimental results and numerical simulations provides strong direct evidence that inertial effects are\nnegligible in the regimes studied.\nTherefore, based on this quantitative comparison of timescales, we maintain that the overdamped\nmodel provides a robust framework for analyzing the rotational dynamics in this study. We acknowledge\nthatforcaseswithsignificantlylargerτ orspecifichighlydynamicflowconditions,theinclusionofinertial\nrot\ntorques would indeed be necessary, but this is beyond the scope of the present research.\nWe have clarified this careful assessment of inertial effect to justify the overdamped model validity in\nthe revised manuscript.\nThe modifications in the SI of the paper are listed below:\nTo justify the validity of the overdamped model, we carefully assessed inertial effects. For a spherical\nparticle, the rotational response time can be estimated as τ = I , where I = 2V ρ r2 is the particle\nrot ξr 5 p p\nmoment of inertia, and ξ = 8πµr3 is the rotational drag coefficient. Substituting these into the equation\nr\nfor τ simplifies to τ = ρpr2. Using our experimental parameters of r = 0.381 mm, ρ = 208 kg/m3,\nrot rot 15µ p p\nµ≈1.0×10−3 Pa·s, and τ =7.56 ms. The rotational response time is τ =2.01 ms. We observe that\nη rot\nτ is approximately 2.01/7.56 = 0.266 or about one-quarter of τ , indicating that the particle rotational\nrot η\ndynamicsaresignificantlyfasterthanthecharacteristicchangesintheflowvorticity. Therefore,theparticle\ncan respond quickly to the flow vorticity, making the assumption of an overdamped regime a reasonable\nfirst approximation. Note that our experimental frequencies are chosen to ensure the particle can follow\nthe driving field within the overdamped regime, i.e., the frequency of the magnetic field ω is slower than\nH\n1/τ ≈2000 Hz, so that the particle rotational motion will remain in the overdamped regime.\nrot\nHow appropriate is the point-particle model used in simulations, considering the actual particle size in ex-\nperiments?\nResponse: Thepoint-particleapproximationoffersaleading-orderdescriptionofparticle–fluidinteraction\nby neglecting finite-size hydrodynamic effects, and it has been widely adopted in studies of particle-laden\nflows and particle–field interactions (Maxey and Riley, 1983; Toschi and Bodenschatz, 2009).\nIn our system, the particle radius is comparable to the Kolmogorov length scale of the turbulence\n(r ∼ η), placing the particle near the lower end of the inertial range. Under such conditions, the velocity\ngradients across the particle remain small, allowing the flow to be treated as locallylinear. As a result, the\nrotational dynamics of the particle are primarily governed by the local vorticity rather than by finite-size-\ninduced flow disturbances. This provides a physically sound justification for employing the point-particle\nmodel in our numerical simulations.\nFurthermore, the dominant driving mechanism for particle rotation in our system includes a well-\ncharacterized external magnetic torque, which acts directly on the particle magnetic moment rather than\nthrough hydrodynamic interactions. This reduces the sensitivity of the results to finite-size corrections.\nImportantly, the suspension used in the experiments is in the dilute regime, with a particle volume\nfractionofΦ =0.17%. Suchalowconcentrationresultsinthefactthattheaverageinter-particledistance\nV\nis much greater than the particle diameter and thus minimizes particle-particle interactions, allowing us to\nneglect hydrodynamic interactions between particles. This also justifies the one-way coupling situation in\nour system (Balachandar and Eaton, 2010), where the flow affects particle motion, but the back-reaction\nofparticlesontheflowfieldisneglected, whichfurthersimplifiestheproblemtoapoint-particleapproach.\nFinally,wehavevalidatedthesimulationresultsagainstexperimentalmeasurementsofparticleangular\nvelocity, and observed good agreement within the bounds of experimental uncertainty. This empirical\nconsistency supports the qualitative and semi-quantitative validity of the point-particle approximation in\nour system.\nWe have comment on this in the revised manuscript.\nThe modifications in the Supplementary Information are listed below:\nDespite neglecting finite-size hydrodynamic effects, the point-particle approximation offers a leading-\norderdescriptionofparticle–fluidinteractionandiswidelyadoptedinstudiesofparticle-ladenflows(Maxey\nand Riley, 1983; Toschi and Bodenschatz, 2009). In our system, the particle radius is comparable to the\nKolmogorov length scale (r ∼ η), ensuring locally linear flow gradients that primarily govern rotational\ndynamics via local vorticity. This, coupled with a dominant external magnetic torque and a dilute suspen-\nsion(volumefractionofΦ =0.17%thatminimizesparticle-particleinteractions(BalachandarandEaton,\nV\n2010)), provides a robust physical justification for our model. Crucially, the good agreement between our\nsimulations and experimental measurements of particle angular velocity further validates the point-particle\n11\napproximation for this system.\nComment 7: Neglecting Rotation-Translation Coupling\nTheassumptionofdecoupledtranslationalandrotationaldynamicsmaynotholdforfinite-sizeoranisotropic\nparticles. Could the authors assess the possible impact of these couplings?\nResponse: The particles used in our experiments/simulations are sufficiently small and nearly spheri-\ncal. Moreover,theirmassdistributionisapproximatelyuniform. Thesefactorsminimizetheoffsetbetween\nthe center of mass and the hydrodynamic center (center of resistance), reducing the torque generated by\ntranslational motion.\nThis is consistent with previous research efforts. For example, it has been found experimentally\n(Zimmermann et al., 2011) that the linear and angular velocities and accelerations reveal an intermittency\nphenomenafortheparticleswhosesizeliesintheinertialrange,closetotheintegralscaleoftheunderlying\nturbulentflow. Thisisinherentlyafinitesizeeffect; Butforparticleswithverysmalldiameters(compared\nto the Kolmogorov length scale) the translational and rotational dynamics are not coupled.\nEven if weak coupling exists, we note that the main results of our study – such as the emergence\nof stochastic resonance and the phase lag behavior – are primarily determined by the balance between\nexternal torque and rotational viscous drag. Therefore, any minor translation-rotation coupling would not\nqualitatively alter our conclusions and the revealed mechanisms.\nWhileweassumedecoupledtranslationalandrotationaldynamicsbasedonparticlepropertiesandsta-\ntisticalverification,weagreethatitisworthwhiletoconsiderhowsuchcoupling,ifpresent,mightinfluence\nrotational dynamics of the magnetic particles. For particles with non-uniform mass distribution or shape\nasymmetry, translational motion through a shear or turbulent flow can generate additional hydrodynamic\ntorquesduetoamismatchbetweenthecenterofmass(CM)andthecenterofresistance(CR).Thistorque\neffectively acts as an additional noise or deterministic bias on the rotational motion.\nBelow we estimate the magnitude of this effect in our system.\nWeconsideraparticlewithradiusr =0.381 mmandapossibleCM–CRoffsetδ ∼0.1r =0.0381 mm.\nThe translational velocity at the Kolmogorov scale is approximately\nη 8.7×10−5 m\nv ≈v ≈ = ≈0.0115 m/s. (1)\nη τ 7.56×10−3 s\nη\nThe torque induced by translational drag is then estimated as\nT ∼δ·6πµr·v, (2)\ntrans\nwith dynamic viscosity µ=1.0×10−3 Pa·s. Substituting values yields\nT ∼0.0381×10−3·6π·1.0×10−3·0.381×10−3·0.0115≈3.14×10−13 Nm. (3)\ntrans\nT =mB, (4)\nmag\nwhere B =1.6 mT is the magnetic field strength, and m∼10−8 A·m2 is the typical magnetic moment of\nthe particle. This gives\nT ∼10−8·1.6×10−3 =1.6×10−11 Nm. (5)\nmag\nTherefore, the ratio is\nT 3.14×10−13\ntrans ≈ ≈0.0196≈2%, (6)\nT 1.6×10−11\nmag\nindicating that even under conservative assumptions, the translation-rotation coupling induced torque is\nlessthan2%ofthemagnetictorque. Ofcoursespecificestimationsmaychangebasedondifferentparameter\nsettings of the magnetic field, particle properties and the background turbulence.\nNotethatinoursystemthephaselagbetweentheparticlerotationandtheappliedmagnetictorqueis\na key observable for identifying stochastic resonance. If translational motion were to induce non-negligible\ntorques: (i) The effective torque acting on the particle would become a combination of magnetic torque,\nviscous drag, and translationally-induced torque. (ii) This additional component could either amplify or\nsuppress the rotational response, depending on its sign and correlation with the magnetic forcing. (iii) As\naresult, thephaselagmightrepresentasmallshift, andthelocationofthestochasticresonancepeak(i.e.,\nthe optimal magnetic frequency where lag is maximized) could move or broaden.\nHowever, because these translation-rotation coupling induced torques are uncorrelated with the exter-\nnallyappliedrotationalmagneticfieldandaretypicallynoisy,theireffectisexpectedtoincreaserotational\nfluctuationsbutnotgeneratecoherentresonancebehavior. Therefore,theirpresencemayreducethesharp-\nness of the stochastic resonance peak, but not eliminate it.\nCrucially, the emergence of stochastic resonance in our system arises from the interplay between\nexternal periodic forcing and intrinsic rotational noise. If translation-rotation coupling were to increase\nthe noise level slightly, it would shift the resonance condition, but the resonance mechanism itself remains\nintact. This means that our key conclusion – that maximal phase lag emerges when magnetic forcing\nbalances turbulent fluctuations – remains qualitatively valid.\nTo address this concern more explicitly, we have added a paragraph discussing these potential effects\nin the revised manuscript.\nThe modifications in the SI of the paper are listed below:\nIn the model of particle angular dynamics, we neglect rotation–translation coupling by assuming that\nthe particle center of mass and hydrodynamic center are approximately aligned. This assumption is justi-\nfied by the nearly spherical shape, uniform mass distribution, and small size of the particles (compared to\nthe Kolmogorov scale), whichcollectively minimizetranslation-inducedhydrodynamic torque. Consequently,\nthe hydrodynamic torque induced by translational motion is expected to be negligible. Even if weak coupling\nexists, its effect can be interpreted as an additional source of rotational noise that is uncorrelated with the\n13\nexternal magnetic field. This may slightly modify the amplitude of angular fluctuations or slightly shift the\nresonance frequency, but it does not alter the underlying mechanism of stochastic resonance. Since the\nemergence of stochastic resonance in our system fundamentally arises from the balance between external\nmagnetic forcing and intrinsic rotational noise, the presence of small translation–rotation coupling does not\nqualitatively affect our main conclusions.\nThe overdamped limit is central to the analysis. The authors should discuss its validity more thoroughly:\n-Underwhatconditions(e.g. magneticfieldstrength,frequency,particlesize/moentinertia/magnetization)\nis the approximation justified?\n- Have they evaluated rotational acceleration statistics to confirm the absence of inertial effects beyond first\nmoments?\nResponse: We now provide a more detailed justification for the conditions under which the rotational\ninertia of the particle can be neglected, both from dimensional analysis and experimental evidence.\nWe first discuss the conditions that justify the overdamped approximation for particle rotational dy-\nnamics.\nThevalidityofthe overdampedapproximation forrotationaldynamicsdepends ontherelative impor-\ntance between inertial effects and viscous (drag) effects. Specifically, it is justified when the characteristic\ntime scale for rotational inertia is significantly smaller than the characteristic time scale of the driving\nforces or the fluid rotational motion. For a light spherical particle, the rotational response time (τ )\nrot\nquantifies how quickly a particle rotation adapts to changes in the surrounding fluid or applied torques.\nAs previously derived in the reply to comment 5, τ can be estimated as τ = ρpr2. Based on the\nrot rot 15µ\nexperimental parameters, we have τ = 2.01 ms, which is smaller than the characteristic time scale of\nrot\nthe fluid vorticity, τ = 7.56 ms. This indicates that the particle can respond significantly faster than the\nη\ncharacteristic changes in the fluid rotational fields. This rapid response allows us to neglect the rotational\ninertialterm(angularacceleration)intherotationalequationofmotion,simplifyingittoabalancebetween\napplied torques (magnetic) and viscous drag.\nNow we discuss the specific conditions of several related parameters. With respect to particle size\n(r) and moment of inertia (I), ss shown by the formula, τ is directly proportional to ρ r2 (which is\nrot p\nproportional to the moment of inertia I). Smaller particles with lower inertia will naturally exhibit a\nshorter τ , further justifying the overdamped limit. Our chosen particle size lies within a regime where\nrot\nthisconditionholdsrelativetotheturbulenttimescales. Whilethemagnetictorqueisasignificantdriving\nmechanism, the overdamped approximation primarily concerns the particle response time to any applied\ntorque (magnetic or hydrodynamic). As long as the frequency of the magnetic field and the resulting\nparticle rotation are slower than or comparable to 1/τ , the particle rotational motion will remain in the\nrot\noverdamped regime. If the magnetic field were to oscillate at frequencies significantly higher than 1/τ ,\nrot\nthen inertial effects would become relevant as the particle would struggle to instantaneously follow the\nrapidly changing magnetic field. Our experimental frequencies are chosen to ensure the particle can follow\nthe driving field within the overdamped regime.\nThe observed good agreement between our numerical simulations (which is built on the overdamped\nassumption, i.e., neglecting the angular acceleration term Iω˙ ) and the experimental measurements of\np\n15\nparticle angular velocity provides strong direct evidence. This suggests that rotational acceleration is\nindeed negligible on average, and its fluctuating component does not dominate the dynamics to the extent\nthat it invalidates the overdamped model. The relatively small ratio of τ /τ further supports this by\nrot η\nindicating that angular velocity fluctuations occur on time scales where the particle can rapidly adjust,\nminimizing the build-up of significant inertial terms.\nIn summary, the combined analytical estimation of τ relative to the flow characteristic time scales,\nrot\nthe physical parameter settings of our system, and the strong empirical validation from experimental\ncomparison collectively justify the use of the overdamped approximation for particle rotation in our study.\nWe have clarified this in the revised manuscript.\nThe modifications in the SI are listed below which is combined with that in the reply to\ncomment 5:\nTo justify the validity of the overdamped model, we carefully assessed inertial effects. For a spherical\nparticle, the rotational response time can be estimated as τ = I , where I = 2V ρ r2 is the particle\nrot ξr 5 p p\nmoment of inertia, and ξ = 8πµr3 is the rotational drag coefficient. Substituting these into the equation\nr\nfor τ simplifies to τ = ρpr2 . Using our experimental parameters of r = 0.381 mm, ρ = 208 kg/m3,\nrot rot 15µ p p\nµ≈1.0×10−3 Pa·s, and τ =7.56 ms. The rotational response time is τ ≈2.01 ms. We observe that\nη rot\nτ is approximately 2.01/7.56 ≈ 0.266 or about one-quarter of τ , indicating that the particle rotational\nrot η\ndynamics are significantly faster than the characteristic changes in the flow vorticity. Therefore, the parti-\ncle can respond quickly to the flow vorticity, making the assumption of an overdamped regime a reasonable\nfirst approximation. Note that our experimental frequencies are chosen to ensure the particle can follow\nthe driving field within the overdamped regime, i.e., the frequency of the magnetic field ω is slower than\nH\n1/τ ≈2000Hz, so that the particle rotational motion will remain in the overdamped regime.\nrot\nWhile the Von Kármán setup is a standard turbulence generator, it is not isotropic or homogeneous, par-\nticularly due to its central core with non-zero mean helicity. Could this large-scale anisotropy influence the\nobserved symmetry breaking or other phenomena? A discussion on this point would improve the interpre-\ntation of results and their generalizability.\nResponse: The Von Kármán-type flow is indeed not perfectly homogeneous or isotropic but it features\nan intense turbulent region at its center and is homogeneous in the radial directions, i.e., parallel to the\ntwo rotating disks. The axial velocity component typically exhibits stronger intermittency than its radial\ncounterparts. As a consequence, for example, the velocity autocorrelation function decays more rapidly in\nthe axial direction. Though it is true that the large-scale helicity and inhomogeneity may influence global\nflow organization and intermittency in the axial direction, many studies have shown that a fully devel-\noped turbulent flow can be generated in the central region of the container, and the small-scale turbulent\nstatistics approach local isotropy, with velocity gradients and dissipation-dominated dynamics resembling\nthose of homogeneous and isotropic turbulence (Mordant et al., 2004; Volk et al., 2008; Zocchi et al., 1994;\nVoth et al., 2002; Mordant et al., 2001; La Porta et al., 2001; Volk et al., 2011). This region, measuring\napproximately 20 mm × 20 mm × 20 mm, serves as the primary domain for measurements.\nIn our experiments, the particles are of size r ∼ 5η, such that their rotational dynamics are primar-\nily coupled to the dissipative scales of turbulence. These scales are much less sensitive to the large-scale\nanisotropy or mean flow structures, and are instead governed by nearly isotropic velocity gradients, es-\npecially in the chosen measurement domain. The observed stochastic resonance and symmetry-breaking\nbehaviorsthusoriginatefromtheinterplaybetweenparticle-scaleturbulentfluctuationsandtheexternally\napplied magnetic field, rather than from the large-scale anisotropy of the Von Kármán-type flow.\nWe have now added a discussion of these limitations to the Method section in the revised manuscript\nto clarify the generalizability of our results.\nThe modifications in the revised SI are listed below:\nWhile the Von Kármán-type turbulent flow is not perfectly homogeneous and isotropic, a fully developed\nturbulent flow can be generated in the central region of the container with small-scale statistics close to local\nisotropy (Mordant et al., 2004; Volk et al., 2008; Zocchi et al., 1994; Voth et al., 2002; Mordant et al.,\n2001; La Porta et al., 2001; Volk et al., 2011). This region, measuring approximately 20 mm × 20 mm\n× 20 mm, serves as the primary domain for measurements. Since our small particles (r ∼ 5η) primarily\ninteract with dissipative-scale velocity gradients in this central region, the observed stochastic resonance and\nsymmetry breaking are governed by particle-scale turbulence and magnetic forcing rather than large-scale\nflow anisotropy.\nTheauthorsthanktherevieweroncemoreforthemanydetailedcommentsonthepaper,whichhelped\nto improve the manuscript. We have implemented all the suggestions in the revised paper. We hope that\nwith these changes and additions, the paper can now be given a full recommendation for publication in\nNature Communications.\nReferences\nBalachandar, S. and Eaton, J. K. (2010). Turbulent dispersed multiphase flow. Annual review of fluid\nmechanics, 42(1):111–133.\nBelovs, M. and C¯ebers, A. (2006). Dynamic fluctuations of dipolar semiflexible filaments. Physical Review\nE—Statistical, Nonlinear, and Soft Matter Physics, 73(2):021507.\nDhar, P., Swayne, C. D., Fischer, T. M., Kline, T., and Sen, A. (2007). Orientations of overdamped\nmagnetic nanorod-gyroscopes. Nano letters, 7(4):1010–1012.\nLa Porta, A., Voth, G. A., Crawford, A. M., Alexander, J., and Bodenschatz, E. (2001). Fluid particle\naccelerations in fully developed turbulence. Nature, 409(6823):1017–1019.\nLabbé, R., Pinton, J.-F., and Fauve, S. (1996). Study of the von kármán flow between coaxial corotating\ndisks. Phys. Fluids, 8(4):914–922.\nMaxey, M. R. and Riley, J. J. (1983). Equation of motion for a small rigid sphere in a nonuniform flow.\nPhysics of Fluids, 26(4):883–889.\nMelle, S., Calderón, O. G., Rubio, M. A., and Fuller, G. G. (2002). Chain rotational dynamics in mr\nsuspensions. International Journal of Modern Physics B, 16(17n18):2293–2299.\nMordant, N. (1997). Characterization of turbulence in a closed flow. J. Phys. II France"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I have now carefully read the responses of the authors to the points I raised in my previous review, as well as the new version of their manuscript. They both clearly address all the remarks I pointed and substantially improve in my opinion the clarity and robustness of the study.\nMy recommendation at this point is to accept the article for publication in Nature Communications.", "reviewer_2": "The author considered most of my suggestions, including adding experimental data, changing the title, and focusing the scope of the study on stochastic resonance, which is the main result, rather than the phase-locked or back-and-forth regimes already explored. Moreover, the authors have added more experimental details, but some comments are still lacking.\nThe revised manuscript has been improved and is now ready for publication in Nature Communications, provided the authors address the points outlined below.\ni) In contrast to the response given to my Comment 2, the new experimental points performed do not appear in Fig. 3(m) as in Fig. R1 for Reviewer 2. I think it is just an omission.\nii) The authors still claim that this concept has strong analogies with magnetometry techniques used in biomedical applications, but do not give any references or explain how they can scale up this technique.\niii) The Stokes number asked in my Comment 5 is still not given.\niv) The sentence, answering my Comment 11, “to this end, magnetic particles are designed to be light and of a size comparable to the Kolmogorov scale, allowing them to preferentially enter and interact with vortex filaments” is an important statement that should be clearly emphasized in the manuscript.\nv) The Comment 14 is still not answered: Is it in Nature Communications’ policies to publish the dedicated instrumentation and methodology of this manuscript in another journal – see new Ref. [60] (experimental) and Ref. [54] (numerics) of the same authors’ group? Furthermore, the reliance on arXiv publications posted after the first round of review raises concerns about the transparency of the revision process.\nMinor:\nIn the Theoretical model Section, the sentence “and details can be found later” should be removed.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Reply to Reviewer 2\nSynopsis\nTheauthorconsideredmostofmysuggestions, includingaddingexperimentaldata,\nchanging the title, and focusing the scope of the study on stochastic resonance, which\nis the main result, rather than the phase-locked or back-and-forth regimes already\nexplored. Moreover, the authors have added more experimental details, but some\ncomments are still lacking.\nThe revised manuscript has been improved and is now ready for publication in\nNature Communications, provided the authors address the points outlined below.\nReply\nWe thank the Reviewer for their positive assessment of our work. Below we address\nthe feedback of the Reviewer 2 point-by-point. The resulting changes to the text are\nmarked in blue in the manuscript.\nComment i\nIn contrast to the response given to my Comment 2, the new experimental points\nperformed do not appear in Fig. 3(m) as in Fig. R1 for Reviewer 2. I think it is just\nan omission.\nReply\nWethanktheReviewerforremindingus. Wehavenowaddedthenewexperimental\ndata in Fig. 3(m) in the revised manuscript.\nThe updated Fig.3 in the main text is now as follows:\n1\n5 111...000\n0\n−5 000...000\n−−−−1111....0000\n111...000\n5\n−\n0\n5\n−−11..00\n00..00\n00..00 100\n11..00−−−−1111....0000\n111...000\n22x10-1\n000...000 10−1 100\n−−−−1111....0000\n111...000\n5\n0 −−11..00 00..00\n−5 00..00 11..00−−11..00 10−2 10−1 100\n5 111...000 !ŠŽ„‡ (cid:127)€(cid:129)‚€ƒ„…†‡ $Š % (cid:141) ‡ #‡\n0 ƒ"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The author considered all my suggestions.\nThe revised manuscript is now ready for publication in Nature Communications.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

Decision letter
[Editorial Note: See attachment]