arXiv subjects

Emmanuel Dormy

Publications and source records attributed to Emmanuel Dormy.

At least 19 recordsLinked to original sources

arXiv2025

Scaling of strong-field spherical dynamos

Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak-field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading-order effect on the flow in strong-field solutions. We demonstrate the persistence of the strong-field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected $O(1)$ values of strong-field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong-field dynamos in parameter space.

physics.geo-ph↗

arXiv2025

Diffusion-free boundary conditions for the Navier-Stokes equations

We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain $\Omega$ of $\mathbb{R}^d$, $d=2$ or $3$. We show that diffusion-free boundary conditions $$ \Delta u \cdot \tau \vert_{\partial \Omega} = 0, \quad u \cdot n\vert_{\partial \Omega} = 0 \quad \text{ when } d=2, $$ $$ \Delta u \times n\vert_{\partial \Omega} = 0, \quad u \cdot n\vert_{\partial \Omega} = 0 \quad \text{ when } d=3, $$ allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for $d=2$, local for $d=3$). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity $\nu \rightarrow 0$. We establish that the amplitude of the boundary layer flow is in this case of order $\nu$, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions.

math.AP↗

arXiv2025

The cross-over from viscous to inertial lengthscales in rapidly-rotating convection

Convection is the main heat transport mechanism in the Earth's liquid core and is thought to power the dynamo that generates the geomagnetic field. Core convection is strongly constrained by rotation while being turbulent. Given the difficulty in modelling these conditions, some key properties of core convection are still debated, including the dominant energy-carrying lengthscale. Different regimes of rapidly-rotating, unmagnetised, turbulent convection exist depending on the importance of viscous and inertial forces in the dynamics, and hence different theoretical predictions for the dominant flow lengthscale have been proposed. Here we study the transition from viscously-dominated to inertia-dominated regimes using numerical simulations in spherical and planar geometries. We find that the cross-over occurs when the inertial lengthscale approximately equals the viscous lengthscale. This suggests that core convection in the absence of magnetic fields is dominated by the inertial scale, which is hundred times larger than the viscous scale.

physics.flu-dyn↗

arXiv2024

Do irrotational water waves remain irrotational in the limit of a vanishing viscosity?

Theoretical results on water waves almost always start by assuming irrotationality of the flow in order to simplify the formulation. In this work, we investigate the well-foundedness of this hypothesis via numerical simulations of the free-surface Navier-Stokes equations. We show that, in the presence of a non-flat bathymetry, either angular or smooth, a gravity wave of finite amplitude can shed vortex pairs from the bottom boundary layer into the bulk of the flow. As these eddies approach the free surface they modify the shape of the wave. It is found that this perturbation does not vanish as the Reynolds number is increased. The vanishing viscosity limit of water waves is therefore singular when no-slip boundary conditions are enforced on the bottom.

physics.flu-dyn↗

arXiv2024

What can Hurricane SAM (2021) tell us about ocean waves under tropical cyclones?

We investigate the ocean wave field under Hurricane SAM (2021). Whilst measurements of waves under Tropical Cyclones (TCs) are rare, an unusually large number of quality in situ and remote measurements are available in that case. First, we highlight the good consistency between the wave spectra provided by the Surface Waves Investigation and Monitoring (SWIM) instrument onboard the China-France Oceanography Satellite (CFOSAT), the in situ spectra measured by National Data Buoy Center (NDBC) buoys, and a saildrone. The impact of strong rains on SWIM spectra is then further investigated. We show that whereas the rain definitely affects the normalized radar cross section, both the innovative technology (beam rotating scanning geometry) and the post-processing processes applied to retrieve the 2D wave spectra ensure a good quality of the resulting wave spectra, even in heavy rain conditions. On this basis, the satellite, airborne and in situ observations are confronted to the analytical model proposed by Kudryavtsev et al (2015). We show that a trapped wave mechanism may be invoked to explain the large significant wave height observed in the right front quadrant of Hurricane SAM.

physics.ao-ph↗

arXiv2024

Numerical study of a viscous breaking water wave and the limit of vanishing viscosity

We introduce a numerical strategy to study the evolution of 2D water waves in the presence of a plunging jet. The free-surface Navier-Stokes solution is obtained with a finite but small viscosity. We observe the formation of a surface boundary layer where the vorticity is localised. We highlight convergence to the inviscid solution. The effects of dissipation on the development of a singularity at the tip of the wave is also investigated by characterising the vorticity boundary layer appearing near the interface.

physics.flu-dyn↗

arXiv2023

Inviscid Water-Waves and interface modeling

We present a rigorous mathematical analysis of the modeling of inviscid water waves. The free-surface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and Green-Naghdi solitary waves. It is finally tested on a wave breaking problem, for which an odd-even coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering.

math-ph↗

arXiv2022

Solenoidal force balances in numerical dynamos

Numerical simulations of the geodynamo (and other planetary dynamos) have made significant progress in recent years. As computing power has advanced, some new models claim to be ever more appropriate for understanding Earth's core dynamics. One measure of the success of such models is the ability to replicate the expected balance between forces operating within the core; Coriolis and Lorentz forces are predicted to be most important. The picture is complicated for an incompressible flow by the existence of the pressure gradient force which renders the gradient parts of all other forces dynamically unimportant. This can confuse the situation, especially when the scale dependence of forces are considered. In this work we investigate force balances through the alternative approach of eliminating gradient parts of each force to form `solenoidal force balances'. We perform a lengthscale dependent analysis for several spherical simulations and find that removal of gradient parts offers an alternative picture of the force balance compared to looking at traditional forces alone. Solenoidal force balances provide some agreement with the results of previous studies but also significant differences. They offer a cleaner overall picture of the dynamics and introduce differences at smaller scales. This has implications for geodynamo models purporting to have reached Earth-like regimes: in order to achieve a meaningful comparison of forces, only the solenoidal part of forces should be considered.

physics.geo-ph↗

arXiv2021

Weak branch and multimodal convection in rapidly rotating spheres at low Prandtl number

The focus of this study is to investigate primary and secondary bifurcations to weakly nonlinear flows (weak branch) in convective rotating spheres in a regime where only strongly nonlinear oscillatory sub- and super-critical flows (strong branch) were previously found in [E. J. Kaplan, N. Schaeffer, J. Vidal, and P. Cardin, Phys. Rev. Lett. 119, 094501 (2017)]. The relevant regime corresponds to low Prandtl and Ekman numbers, indicating a predominance of Coriolis forces and thermal diffusion in the system. We provide the bifurcation diagrams for rotating waves (RWs) computed by means of continuation methods and the corresponding stability analysis of these periodic flows to detect secondary bifurcations giving rise to quasiperiodic modulated rotating waves (MRWs). Additional direct numerical simulations (DNS) are performed for the analysis of these quasiperiodic flows for which Poincaré sections and kinetic energy spectra are presented. The diffusion time scales are investigated as well. Our study reveals very large initial transients (more than 30 diffusion time units) for the nonlinear saturation of solutions on the weak branch, either RWs or MRWs, when DNS are employed. In addition, we demonstrate that MRWs have multimodal nature involving resonant triads. The modes can be located in the bulk of the fluid or attached to the outer sphere and exhibit multicellular structures. The different resonant modes forming the nonlinear quasiperiodic flows can be predicted with the stability analysis of RWs, close to the Hopf bifurcation point, by analyzing the leading unstable Floquet eigenmode.

physics.flu-dyn↗

arXiv2020

The vortex method for 2D ideal flows in exterior domains

The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices and that the total vorticity around the obstacle is conserved. In this work, we provide a rigorous justification of this method for any smooth exterior domain, one of the main mathematical difficulties being that the Biot--Savart kernel defines a singular integral operator when restricted to a curve (here, the boundary of the domain). We also introduce an alternative method---the fluid charge method---which, as we argue, is better conditioned and therefore leads to significant numerical improvements.

math.AP↗

arXiv2018

Formation of eyes in large-scale cyclonic vortices

We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow, swirling vortex combined with a poloidal flow in the r-z plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested by Oruba et al 2017, an eye forms at the centre of the vortex which is reminiscent of that seen in a tropical cyclone and is characterised by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with V=(αg β)^{1/2}H, where g is gravity, αthe expansion coefficient, βthe background temperature gradient, and H is the depth of the domain. We also show that the two most important parameters controlling the flow are Re=VH/νand Ro=V/ΩH, where Ωis the background rotation rate and νthe viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on Re, upper and lower bounds on Ro, and an upper bound on Ekman number.

physics.flu-dyn↗

arXiv2017

Three Branches of Dynamo Action

In addition to the weak-dipolar state and to the fluctuating-multipolar state, widely discussed in the literature, a third regime has been identified in Dormy (2016). It corresponds to a strong-dipolar branch which appears to approach, in a numerically affordable regime, the magnetostrophic limit relevant to the dynamics of the Earth's core. We discuss the transitions between these states and point to the relevance of this strong-dipolar state to Geodynamo modelling.

physics.geo-ph↗

arXiv2017

The vortex method for 2D ideal flows in the exterior of a disk

The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices. In this work, we provide a rigorous justification for this method in exterior domains. One of the main mathematical difficulties being that the Biot-Savart kernel defines a singular integral operator when restricted to a curve. For simplicity and clarity, we only treat the case of the unit disk in the plane approximated by a uniformly distributed mesh of point vortices. The complete and general version of our work is available in [arXiv:1707.01458].

math.AP↗

arXiv2016

Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

A comprehensive study of the classical linear spin-down of a constant density viscous fluid (kinematic viscosity ν) rotating rapidly (angular velocity Ω) inside an axisymmetric cylindrical container (radius L, height H) with rigid boundaries, that follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=ν/H^2Ω\ll 1$, was provided by Greenspan & Howard (1963). $E^{1/2}$-Ekman layers form quickly triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}Ω^{-1})$ time-scale. On the longer lateral viscous diffusion time-scale $O(L^2/ν)$, the QG-flow responds to the $E^{1/3}$-side-wall shear-layers. In our variant the side-wall and top boundaries are stress-free; a setup motivated by the study of isolated atmospheric structures, such as tropical cyclones, or tornadoes. Relative to the unbounded plane layer case, spin-down is reduced (enhanced) by the presence of a slippery (rigid) side-wall. This is evinced by the QG-angular velocity, ω*, evolution on the O(L^2/ν) time-scale: Spatially, ω* increases (decreases) outwards from the axis for a slippery (rigid) side-wall; temporally, the long-time ($\gg L^2/ν)$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery side-wall case, the $E^{1/2} \times E^{1/2}$ corner region that forms at the side-wall intersection with the rigid base is responsible for a $\ln E$ singularity within the $E^{1/3}$-layer causing our asymptotics to apply only at values of E far smaller than can be reached by our Direct Numerical Simulation (DNS) of the entire spin-down process. Instead, we solve the $E^{1/3}$-boundary-layer equations for given E numerically. Our hybrid asymptotic-numerical approach yields results in excellent agreement with our DNS.

physics.flu-dyn↗

arXiv2016

A systematic method to enforce conservativity on semi-Lagrangian schemes

Semi-Lagrangian schemes have proven to be very efficient to model advection problems. However most semi-Lagrangian schemes are not conservative. Here, a systematic method is introduced in order to enforce the conservative property on a semi-Lagrangian advection scheme. This method is shown to generate conservative schemes with the same linear stability range and the same order of accuracy as the initial advection scheme from which they are derived. We used a criterion based on the column-balance property of the schemes to assess their conservativity property. We show that this approach can be used with large CFL numbers and third order schemes.

physics.flu-dyn↗

arXiv2016

Strong-Field Spherical Dynamos

Numerical models of the geodynamo are usually classified in two categories: those denominated dipolar modes, observed when the inertial term is small enough, and multipolar fluctuating dynamos, for stronger forcing. We show that a third dynamo branch corresponding to a dominant force balance between the Coriolis force and the Lorentz force can be produced numerically. This force balance is usually referred to as the strong-field limit. This solution co-exists with the often described viscous branch. Direct numerical simulations exhibit a transition from a weak-field dynamo branch, in which viscous effects set the dominant length scale, and the strong-field branch in which viscous and inertial effects are largely negligible. These results indicate that a distinguished limit needs to be sought to produce numerical models relevant to the geodynamo and that the usual approach of minimizing the magnetic Prandtl number (ratio of the fluid kinematic viscosity to its magnetic diffusivity) at a given Ekman number is misleading.

physics.geo-ph↗

arXiv2016

The Equatorial Ekman Layer

The steady incompressible viscous flow in the wide gap between spheres rotating about a common axis at slightly different rates (small Ekman number E) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the limit E<<1, was undertaken by Stewartson (J. Fluid Mech. 1966, vol. 26, pp. 131-144). The mainstream flow, exterior to the E^{1/2} Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder C, is geostrophic. Stewartson identified a complicated nested layer structure on C, which comprises relatively thick quasi-geostrophic E^{2/7} (inside C) and E^{1/4} (outside C) layers. They embed a thinner E^{1/3} ageostrophic shear layer (on C), which merges with the inner sphere Ekman layer to form the E^{2/5} Equatorial Ekman layer of axial length E^{1/5}. Under appropriate scaling, this $E^{2/5}$--layer problem may be formulated, correct to leading order, independent of E. Accordingly, the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial co-ordinate z) solution. We present a numerical solution, which uses a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting z^{-1} as a small parameter we extend Stewartson's similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

physics.flu-dyn↗

arXiv2016

Multi-stage high order semi-Lagrangian schemes for incompressible flows in Cartesian geometries

Efficient transport algorithms are essential to the numerical resolution of incompressible fluid flow problems. Semi-Lagrangian methods are widely used in grid based methods to achieve this aim. The accuracy of the interpolation strategy then determines the properties of the scheme. We introduce a simple multi-stage procedure which can easily be used to increase the order of accuracy of a code based on multi-linear interpolations. This approach is an extension of a corrective algorithm introduced by Dupont \& Liu (2003, 2007). This multi-stage procedure can be easily implemented in existing parallel codes using a domain decomposition strategy, as the communications pattern is identical to that of the multi-linear scheme. We show how a combination of a forward and backward error correction can provide a third-order accurate scheme, thus significantly reducing diffusive effects while retaining a non-dispersive leading error term.

physics.comp-ph↗