Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\nu$ and density diffusivity $\kappa$. The initial diffusive error function density distribution varies continuously so that $\rho \in [\bar{\rho} - (1/2)\rho_0, \bar{\rho} + (1/2) \rho_0]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite P\'eclet number $Pe= U_m h /\kappa=500$ and Prandtl number $Pr=\nu/\kappa=1$, and a range of bulk Richardson numbers $Ri_b= g \rho_0 h /(\bar{\rho} U^2) \in [0,1]$ where $U_m$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing results of Foures, Caulfield \& Schmid (2014) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a `mix-norm' as first introduced by Mathew, Mezic \& Petzold (2005), further discussed by Thi eault (2012) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kinetic-energy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm-minimising optimal perturbations can still trigger substantial mixing for $Ri_b \lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into `available potential energy', which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs

On the equatorial Ekman layer

This is the author accepted manuscript. The final version is available from Cambridge University Press via the DOI in this record.The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) 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 small Ekman number limit E≪1, was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the E1/2 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 quasigeostrophic E2/7- (inside C) and E1/4E1/4- (outside C) layers. They embed a thinner ageostrophic E1/3 shear layer (on C), which merges with the inner sphere Ekman layer to form the E2/5-equatorial Ekman layer of axial length E1/5. Under appropriate scaling, this E2/5-layer problem may be formulated, correct to leading order, independent of E. Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate z) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using 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.F.M. and E.D. have been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01. F.M. acknowledges a PhD mobility grant from Institut de Physique du Globe de Paris. A.M.S. visited ENS, Paris (19–25 October 2014), while F.M. and E.D. visited the School of Mathematics and Statistics, Newcastle University (respectively, 7–25 September 2015 and 25–30 November 2015); the authors wish to thank their respective host institutions for their hospitality and support

Tayler-Spruit dynamos in simulated radiative stellar layers

The Tayler-Spruit dynamo mechanism has been proposed two decades ago as a plausible mechanism to transport angular momentum in radiative stellar layers. Direct numerical simulations are still needed to understand its trigger conditions and the saturation mechanisms. The present study follows up on (Petitdemange et al. 2023), where we reported the first numerical simulations of a Tayler-Spruit dynamo cycle. Here we extend the explored parameter space to assess in particular the influence of stratification on the dynamo solutions. We also present numerical verification of theoretical assumptions made in (Spruit 2002), which are instrumental in deriving the classical prescription for angular momentum transport implemented in stellar evolution codes. A simplified radiative layer is modeled numerically by considering the dynamics of a stably-stratified, differentially rotating, magnetized fluid in a spherical shell. Our simulations display a diversity of magnetic field topologies and amplitudes depending on the flow parameters, including hemispherical solutions. The Tayler-Spruit dynamos reported here are found to satisfy magnetostrophic equilibrium and achieve efficient turbulent transport of angular momentum, following Spruit's heuristic prediction

Dynamo generated by the centrifugal instability

International audienceWe present a scenario for magnetic field amplification where an electrically conducting fluid is confined in a differentially rotating, spherical shell with thin aspect ratio. When the angular momentum sufficiently decreases outwards, a hydrodynamic instability develops in the equatorial region, characterized by pairs of counter-rotating toroidal vortices similar to those observed in cylindrical Couette flow. These spherical Taylor-Couette vortices generate a subcritical dynamo magnetic field dominated by nonaxisymmetric components. We show that the critical magnetic Reynolds number seems to reach a constant value at large Reynolds number and that the global rotation can strongly decrease the dynamo onset. Our numerical results are understood within the framework of a simple dynamical system, and we propose a low-dimensional model for subcritical dynamo bifurcations. Implications for both laboratory dynamos and astrophysical magnetic fields are finally discussed

Data supporting "Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers"

Data supporting publication. Contains codes to create data, initial conditions and figure files, as well as original paper and explanatory "note". Optimization.m : Main routine for optimization (use param.m to specify the flow parameters, mixing norms, resolution etc.) Diagnostic_diag.m : Main routine for diagnostic (forward integration in time only). Use param.m again. The *_diag.m routines are called for diagnostic only. Flow_measures.m and Flow_measures_diag.m contain the definitions of all the diagnostic variables. Their timeseries are stored in norms_timeseries.mat. All the figures have been produced with the data stored in norms_timeseries.mat and the flow snapshots. Omega0.mat is produced by Diagnostic_diag.m and contains the initial vorticity field for the perturbation

Spin-down by dynamo action in simulated radiative stellar layers

International audienceThe evolution of a star is influenced by its internal rotation dynamics through transport and mixing mechanisms, which are poorly understood. Magnetic fields can play a role in transporting angular momentum and chemical elements, but the origin of magnetism in radiative stellar layers is unclear. Using global numerical simulations, we identify a subcritical transition from laminar flow to turbulence caused by the generation of a magnetic dynamo. Our results have many properties of the theoretically proposed Tayler-Spruit dynamo mechanism, which strongly enhances transport of angular momentum in radiative zones. The dynamo generates deep toroidal fields that are screened by the stellar outer layers. This mechanism could produce strong magnetic fields inside radiative stars without an observable field on their surface

Systematic Route to Subcritical Dynamo Branches

International audienceWe demonstrate that the nonlinear optimisation of a finite-amplitude disturbance over a freely evolving and possibly even turbulent flow, can successfully identify subcritical dynamo branches as well as the structure and amplitude of their critical perturbations. As this approach does not require prior knowledge of the magnetic field amplification mechanisms, it opens a new avenue for systematically probing subcritical dynamo flows