arXiv subjects

Philip S. Marcus

Publications and source records attributed to Philip S. Marcus.

15 recordsLinked to original sources

arXiv2026

Perturbation analysis of triadic resonance in columnar vortices: selection rules and the roles of external forcing and critical layers

The remarkable robustness of columnar vortices suggests the existence of fundamental constraints that prevent spontaneous disintegration. In this work, we investigate the weakly nonlinear stability of such flows, demonstrating that the triadic resonance of wave modes is governed by a set of hydrodynamic ``selection rules''. By employing a multi-scale perturbation analysis, we prove that resonant interactions between smooth neutral modes, specifically regular Kelvin waves and discrete critical layer modes with passive singularities, are strictly conservative and confined to the Manley--Rowe relations. Using wave pseudoenergy within a large-$k$ WKBJ framework, we show that these rules topologically prohibit intrinsic instability, analogous to the forbidden transitions of quantum mechanics. Consequently, the breakdown of a columnar vortex requires a specific symmetry-breaking mechanism to overcome this barrier. We identify two distinct pathways: (1) \textit{Parametric instability}, a limiting case where one mode is maintained externally. By generalising beyond the specific spatial and temporal assumptions of classical studies (e.g., elliptical instability) and leveraging a tuning method based on non-degenerate perturbation theory, our framework admits arbitrary driving frequencies and identifies new instability configurations involving discrete critical layer modes. (2) \textit{Active critical layers}, where an embedded wave-mean resonance enables the direct, non-conservative extraction of mean-flow energy. These findings provide theoretical guidance for flow control, suggesting that aircraft wake vortex mitigation requires either tuned external forcing or the excitation of critical layers (e.g., via thermal stratification) to trigger the forbidden transitions.

physics.flu-dyn↗

arXiv2026

A semi-analytical pseudo-spectral method for 3D Boussinesq equations of rotating, stratified flows in unbounded cylindrical domains

We present a pseudo-spectral method for solving the three-dimensional Boussinesq equations in unbounded cylindrical domains, specifically tailored for rotating, stably stratified flows subject to strong azimuthal shear. To effectively capture the global geometry without sacrificing spectral accuracy, the spatial discretization employs Fourier expansions in the azimuthal and axial directions alongside mapped associated Legendre polynomials in the radial direction. This basis spans the semi-infinite domain while analytically resolving the coordinate singularity at the origin. While this spectral framework ensures high spatial fidelity, the temporal integration of these rotating shear flows presents a formidable computational challenge due to the numerical stiffness driven by fast restorative wave forces and rapid background advection. To circumvent this, we develop an exponential time differencing (ETD) scheme that analytically integrates the fully coupled linear operator, including the radially dependent advective cross terms. By encoding the physical resonance characteristics and stability limits of the background flow directly into the integration operators, the proposed ETD formulation removes the numerical stability constraints imposed by the background shear and stratification. This permits integration time steps scaled by the slow macroscopic evolution of the physical instabilities rather than the fast background kinematics, offering significant performance gains over standard mixed implicit-explicit schemes. The method's accuracy and stability are validated through the precise conservation of energy and angular momentum, establishing a robust framework for simulating instabilities in astrophysical and geophysical vortices.

physics.flu-dyn↗

arXiv2025

Transient growth of a wake vortex and its initiation via inertial particles

The transient dynamics of a wake vortex, modelled as a strong swirling $q$-vortex, are investigated with a focus on optimal transient growth driven by continuous eigenmodes associated with continuous spectra. The pivotal contribution of viscous critical-layer eigenmodes (Lee & Marcus, J. Fluid Mech., vol. 967) amongst the entire eigenmode families to optimal perturbations is numerically confirmed, utilising a spectral collocation method for a radially unbounded domain that ensures correct analyticity and far-field behaviour. The consistency of the numerical method across different sensitivity tests supports the reliability of the results and provides flexibility for tuning. Both axisymmetric and helical perturbations with axial wavenumbers of order unity or less are examined through linearised theory and non-linear simulations, yielding results that align with existing literature on energy growth curves and optimal perturbation structures. The initiation process of transient growth is also explored, highlighting its practical relevance. Inspired by ice crystals in contrails, the backward influence of inertial particles on the vortex flow, particularly through particle drag, is emphasised. In the pursuit of optimal transient growth, particles are initially distributed at the periphery of the vortex core to disturb the flow. Two-way coupled vortex-particle simulations reveal clear evidence of optimal transient growth during ongoing vortex-particle interactions, reinforcing the robustness and significance of transient growth in the original non-linear vortex system over finite time periods.

physics.flu-dyn↗

arXiv2024

Stable 3-dimensional Vortex Families Consistent with Jovian Observations Including the Great Red Spot

Detailed observations of the velocities of Jovian vortices exist at only one height in the atmosphere, so their vertical structures are poorly understood. This motivates this study that computes stable 3-dimensional, long-lived planetary vortices that satisfy the equations of motion. We solve the anelastic equations with a high-resolution pseudo-spectral method using the observed Jovian atmospheric temperatures and zonal flow. We examine several families of vortices and find that {\it constant-vorticity} vortices, which have nearly-uniform vorticity as a function of height and horizontal areas that go to zero at their tops and bottoms, converge to stable vortices that look like the Great Red Spot (GRS) and other Jovian anticyclones. In contrast, the {\it constant-area} vortices proposed in previous studies, which have nearly-uniform areas as a function of height and vertical vorticities that go to zero at their tops and bottoms, are far from equilibrium, break apart, and converge to {\it constant-vorticity} vortices. Our late-time vortices show unexpected properties. Vortices that are initially non-hollow become hollow (i.e., have local minima of vertical vorticity at their centers), which is a feature of the GRS that cannot be explained with 2-dimensional simulations. The central axes of the final vortices align with the planetary-spin axis even if they initially align with the local direction of gravity. We present scaling laws for how vortex properties change with the Rossby number and other non-dimensional parameters. We analytically prove that the horizontal mid-plane of a stable vortex must lie at a height above the top of the convective zone.

physics.flu-dyn↗

arXiv2023

Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions

A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier-Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling $q$-vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (2011). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the $Re^{-1/3}$ scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.

physics.flu-dyn↗

arXiv2023

Airfoil Optimization using Design-by-Morphing

We present Design-by-Morphing (DbM), a novel design methodology applicable to creating a search space for topology optimization of 2D airfoils. Most design techniques impose geometric constraints and sometimes designers' bias on the design space itself, thus restricting the novelty of the designs created, and only allowing for small local changes. We show that DbM methodology does not impose any such restrictions on the design space and allows for extrapolation from the search space, thus granting truly radical and large search space with a few design parameters. In comparison to other shape design methodologies, we apply DbM to create a search space for 2D airfoils. We optimize this airfoil shape design space for maximizing the lift-over-drag ratio, $CLD_{max}$, and stall angle tolerance, $Δα$. Using a bi-objective genetic algorithm to optimize the DbM space, it is found that we create a Pareto-front of radical airfoils exhibiting remarkable properties for both objectives.

math.GT↗

arXiv2022

Optimization of the Shape of a Hydrokinetic Turbine's Draft Tube and Hub Assembly Using Design-by-Morphing with Bayesian Optimization

Finding the optimal design of a hydrodynamic or aerodynamic surface is often impossible due to the expense of evaluating the cost functions (say, with computational fluid dynamics) needed to determine the performances of the flows that the surface controls. In addition, inherent limitations of the design space itself due to imposed geometric constraints, conventional parameterization methods, and user bias can restrict {\it all} of the designs within a chosen design space regardless of whether traditional optimization methods or newer, data-driven design algorithms with machine learning are used to search the design space. We present a 2-pronged attack to address these difficulties: we propose (1) a methodology to create the design space using morphing that we call {\it Design-by-Morphing} (DbM); and (2) an optimization algorithm to search that space that uses a novel Bayesian Optimization (BO) strategy that we call {\it Mixed variable, Multi-Objective Bayesian Optimization} (MixMOBO). We apply this shape optimization strategy to maximize the power output of a hydrokinetic turbine. Applying these two strategies in tandem, we demonstrate that we can create a novel, geometrically-unconstrained, design space of a draft tube and hub shape and then optimize them simultaneously with a {\it minimum} number of cost function calls. Our framework is versatile and can be applied to the shape optimization of a variety of fluid problems.

cs.CG↗

arXiv2022

Bayesian Optimization For Multi-Objective Mixed-Variable Problems

Optimizing multiple, non-preferential objectives for mixed-variable, expensive black-box problems is important in many areas of engineering and science. The expensive, noisy, black-box nature of these problems makes them ideal candidates for Bayesian optimization (BO). Mixed-variable and multi-objective problems, however, are a challenge due to BO's underlying smooth Gaussian process surrogate model. Current multi-objective BO algorithms cannot deal with mixed-variable problems. We present MixMOBO, the first mixed-variable, multi-objective Bayesian optimization framework for such problems. Using MixMOBO, optimal Pareto-fronts for multi-objective, mixed-variable design spaces can be found efficiently while ensuring diverse solutions. The method is sufficiently flexible to incorporate different kernels and acquisition functions, including those that were developed for mixed-variable or multi-objective problems by other authors. We also present HedgeMO, a modified Hedge strategy that uses a portfolio of acquisition functions for multi-objective problems. We present a new acquisition function, SMC. Our results show that MixMOBO performs well against other mixed-variable algorithms on synthetic problems. We apply MixMOBO to the real-world design of an architected material and show that our optimal design, which was experimentally fabricated and validated, has a normalized strain energy density $10^4$ times greater than existing structures.

cs.LG↗

arXiv2017

Stability of 3D Gaussian vortices in an unbounded, rotating, vertically-stratified, Boussinesq flow: Linear analysis

The linear stability of three-dimensional (3D) vortices in rotating, stratified flows has been studied by analyzing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely-used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby number ($-0.5 < Ro < 0.5$) and Burger number ($0.02 < Bu < 2.3$) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro-Bu$. We have found neutrally-stable vortices only over a small region of the $Ro-Bu$ parameter space: cyclones with $Ro \sim 0.02-0.05$ and $Bu \sim 0.85-0.95$. However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g., $Ro<0$ and $0.5 \lesssim Bu \lesssim 1.3$) is slower than $50$ turn-around times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to $0<Ro<0.1$ and $0.5 \lesssim Bu \lesssim 1.3$. While most calculations have been done for $f/\bar{N}=0.1$ (where $f$ and $\bar{N}$ are the Coriolis and background Brunt-Väisälä frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing $f/\bar{N}$ to the more ocean-relevant value of $0.01$. The results of this paper provide a steppingstone to study the more complicated problems of the stability of geophysical (e.g., those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).

physics.ao-ph↗

arXiv2016

Zombie Vortex Instability. II. Thresholds to Trigger Instability and the Properties of Zombie Turbulence in the Dead Zones of Protoplanetary Disks

In Zombie Vortex Instability (ZVI), perturbations excite critical layers in stratified, rotating shear flow (as in protoplanetary disks), causing them to generate vortex layers, which roll-up into anticyclonic zombie vortices and cyclonic vortex sheets. The process is self-sustaining as zombie vortices perturb new critical layers, spawning a next generation of zombie vortices. Here, we focus on two issues: the minimum threshold of perturbations that trigger self-sustaining vortex generation, and the properties of the late-time zombie turbulence on large and small scales. The critical parameter that determines whether ZVI is triggered is the magnitude of the vorticity on the small scales (and not velocity), the minimum Rossby number needed for instability is $Ro_{crit}\sim0.2$ for $β\equiv N/Ω= 2$, where $N$ is the Brunt-Väisälä frequency. While the threshold is set by vorticity, it is useful to infer a criterion on the Mach number, for Kolmogorov noise, the critical Mach number scales with Reynolds number: $Ma_{crit}\sim Ro_{crit}Re^{-1/2}$. In protoplanetary disks, this is $Ma_{crit}\sim10^{-6}$. On large scales, zombie turbulence is characterized by anticyclones and cyclonic sheets with typical Rossby number $\sim$0.3. The spacing of the cyclonic sheets and anticyclones appears to have a "memory" of the spacing of the critical layers. On the small scales, zombie turbulence has no memory of the initial conditions and has a Kolmogorov-like energy spectrum. While our earlier work was in the limit of uniform stratification, we have demonstrated that ZVI works for non-uniform Brunt-Väisälä frequency profiles that may be found in protoplanetary disks.

astro-ph.SR↗

arXiv2013

Self-Replicating Three-Dimensional Vortices in Neutrally-Stable Stratified Rotating Shear Flows

A previously unknown instability creates space-filling lattices of 3D vortices in linearly-stable, rotating, stratified shear flows. The instability starts from an easily-excited critical layer. The layer intensifies by drawing energy from the background shear and rolls-up into vortices that excite new critical layers and vortices. The vortices self-similarly replicate to create lattices of turbulent vortices. The vortices persist for all time. This self-replication occurs in stratified Couette flows and in the dead zones of protoplanetary disks where it can de-stabilize Keplerian flows.

astro-ph.EP↗

arXiv2012

The Universal Aspect Ratio of Vortices in Rotating Stratified Flows: Theory and Simulation

We derive a relationship for the vortex aspect ratio $α$ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt-Vaisala frequencies within the vortex $N_c$ and in the background fluid outside the vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the vortex: $α^2 = Ro(1+Ro) f^2/(N_c^2-\bar{N}^2)$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for $α$ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have $N_c^2 > \bar{N}^2$; weak anticyclones (with $|Ro| < 1$) must have $N_c^2 < \bar{N}^2; and strong anticyclones must have $N_c^2 > \bar{N}^2$. We verify our relation for $α$ with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localised region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically-computed vortices validate our relationship for $α$, and generally they differ significantly from the values obtained from the much-cited conjecture that $α= f/\bar{N}$ in quasi-geostrophic vortices.

physics.flu-dyn↗

arXiv2012

The Universal Aspect Ratio of Vortices in Rotating Stratifi?ed Flows: Experiments and Observations

We validate a new law for the aspect ratio $α= H/L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter f and buoyancy (or Brunt-Vaisala) frequency $\bar{N}$ of the background flow, but also on the buoyancy frequency $N_c$ within the vortex and on the Rossby number $Ro$ of the vortex such that $α= f \sqrt{[Ro (1 + Ro)/(N_c^2- \bar{N}^2)]}$. This law for $α$ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly-stratified salt water. The vortices are far from the top and bottom boundaries of the tank, so there is no Ekman circulation. In one set of experiments, the vortices viscously decay, but as they do, they continue to obey our law for $α$, which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection after they form, so they evolve more slowly and have larger |Ro|, but they also obey our law for $α$. The law for $α$ is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter's Great Red Spot and Oval BA. The relationship for $α$ is derived and examined numerically in a companion paper by Hassanzadeh et al. (2012).

physics.flu-dyn↗

arXiv2007

On the interaction of Jupiter's Great Red Spot and zonal jet streams

In this paper, Jupiter's Great Red Spot (GRS) is used to determine properties of the Jovian atmosphere that cannot otherwise be found. These properties include the potential vorticity of the GRS and its neighboring jet streams, the shear imposed on the GRS by the jet streams, and the vertical entropy gradient (i.e., Rossby deformation radius). The cloud cover of the GRS, which is often used to define the GRS's area and aspect ratio, is found to differ significantly from the region of the GRS's potential vorticity anomaly. The westward-going jet stream to the north of the GRS and the eastward-going jet stream to its south are each found to have a large potential vorticity ``jump''. The jumps have opposite sign and as a consequence of their interaction with the GRS, the shear imposed on the GRS is reduced. The east-west to north-south aspect ratio of the GRS's potential vorticity anomaly depends on the ratio of the imposed shear to the strength of the anomaly. The aspect ratio is found to be $\approx$2:1, but without the opposing jumps it would be much greater. The GRS's high-speed collar and quiescent interior require that the potential vorticity in the interior be approximately half that in the collar. No other persistent geophysical vortex has a significant minimum of potential vorticity in its interior and laboratory vortices with such a minimum are unstable.

physics.flu-dyn↗

arXiv2005

A 3D Spectral Anelastic Hydrodynamic Code for Shearing, Stratified Flows

We have developed a three-dimensional (3D) spectral hydrodynamic code to study vortex dynamics in rotating, shearing, stratified systems (e.g. the atmosphere of gas giant planets, protoplanetary disks around newly forming protostars). The time-independent background state is stably stratified in the vertical direction and has a unidirectional linear shear flow aligned with one horizontal axis. Superposed on this background state is an unsteady, subsonic flow that is evolved with the Euler equations subject to the anelastic approximation to filter acoustic phenomena. A Fourier-Fourier basis in a set of quasi-Lagrangian coordinates that advect with the background shear is used for spectral expansions in the two horizontal directions. For the vertical direction, two different sets of basis functions have been implemented: (1) Chebyshev polynomials on a truncated, finite domain, and (2) rational Chebyshev functions on an infinite domain. Use of this latter set is equivalent to transforming the infinite domain to a finite one with a cotangent mapping, and using cosine and sine expansions in the mapped coordinate. The nonlinear advection terms are time integrated explicitly, whereas the Coriolis force, buoyancy terms, and pressure/enthalpy gradient are integrated semi-implicitly. We show that internal gravity waves can be damped by adding new terms to the Euler equations. The code exhibits excellent parallel performance with the Message Passing Interface (MPI). As a demonstration of the code, we simulate the merger of two 3D vortices in the midplane of a protoplanetary disk.

astro-ph↗