arXiv subjects

Michael Jolly

Publications and source records attributed to Michael Jolly.

5 recordsLinked to original sources

arXiv2022

Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations

We introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a predetermined path at a given speed. We prove that, if the movement is fast enough, then the algorithm perfectly synchronizes with a reference solution. The analysis suggests an informed scheme in which the subdomain moves according to a region where the error is dominant is optimal. Numerical simulations are presented that compare the efficacy of movement that follows a regular pattern, one guided by the dominant error, and one that is random.

math.AP↗

arXiv2021

Continuous Data Assimilation For the 3D Ladyzhenskaya Model: Analysis and Computations

We analyze continuous data assimilation by nudging for the 3D Ladyzhenskaya equations. The analysis provides conditions on the spatial resolution of the observed data that guarantee synchronization to the reference solution associated with the observed, spatially coarse data. This synchronization holds even though it is not known whether the reference solution, with initial data in $L^2$, is unique; a particular reference solution is determined by the observed, coarse data. The efficacy of the algorithm in both 2D and 3D is demonstrated by numerical computations.

math.DS↗

arXiv2020

Three-dimensional shear driven turbulence with noise at the boundary

We consider the incompressible 3D Navier-Stokes equations subject to a shear induced by noisy movement of part of the boundary. The effect of the noise is quantified by upper bounds on the first two moments of the dissipation rate. The expected value estimate is consistent with the Kolmogorov dissipation law, recovering an upper bound as in [15] for the deterministic case. The movement of the boundary is given by an Ornstein-Uhlenbeck process; a potential for over-dissipation is noted if the Ornstein-Uhlenbeck process were replaced by the Wiener process.

math.AP↗

arXiv2016

Energy spectra and passive tracer cascades in turbulent flows

We study the influence of the energy spectrum on the extent of the cascade range of a passive tracer in turbulent flows. The interesting cases are when there are two different spectra over the potential range of the tracer cascade (in 2D when the tracer forcing is in the inverse energy cascade range, and in 3D when the Schmidt number Sc is large). The extent of the tracer cascade range is then limited by the width of the range for the shallower of the two energy spectra. Nevertheless, we show that in dimension $d=2,3$ the tracer cascade range extends (up to a logarithm) to $\kappa_{d\text{D}}^{p}$, where $\kappa_{d\text{D}}$ is the wavenumber beyond which diffusion should dominate and $p$ is arbitrarily close to 1, provided Sc is larger than a certain power (depending on $p$) of the Grashof number. We also derive estimates which suggest that in 2D, for Sc${}\sim1$ a wide tracer cascade can coexist with a significant inverse energy cascade at Grashof numbers large enough to produce a turbulent flow.

math.AP↗

arXiv2013

On whether zero is in the global attractor of the 2D Navier-Stokes equations

The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.

math.AP↗