arXiv subjects

Artur Prugger

Publications and source records attributed to Artur Prugger.

3 recordsLinked to original sources

arXiv2023

Rotating shallow water equations with bottom drag: bifurcations and growth due to kinetic energy backscatter

The rotating shallow water equations with f-plane approximation and nonlinear bottom drag are a prototypical model for mid-latitude geophysical flow that experience energy loss through simple topography. Motivated by numerical schemes for large-scale geophysical flow, we consider this model on the whole space with horizontal kinetic energy backscatter source terms built from negative viscosity and stabilizing hyperviscosity with constant parameters. We study its interplay with linear and non-smooth quadratic bottom drag through the existence of coherent flows. Our results highlight that backscatter can have undesired amplification and selection effects, generating obstacles to energy distribution. We find that decreasing linear bottom drag destabilizes the trivial flow and generates nonlinear flows that can be associated with geostrophic equilibria (GE) and inertia-gravity waves (IGWs). The IGWs are periodic travelling waves, while the GE are stationary and can be studied by a plane wave reduction. We show that for isotropic backscatter both bifurcate simultaneously and supercritically, while for anisotropic backscatter the primary bifurcation are GE. In all cases presence of non-smooth quadratic bottom drag implies unusual scaling laws. For the rigorous bifurcation analysis by Lyapunov-Schmidt reduction care has to be taken due to this lack of smoothness and since the hyperviscous terms yield a lack of spectral gap at large wave numbers. For purely smooth bottom drag, we identify a class of explicit such flows that behave linearly within the nonlinear equations: amplitudes can be steady and arbitrary, or grow exponentially and unboundedly. We illustrate the results by numerical computations and present extended branches in parameter space.

physics.flu-dyn↗

arXiv2021

Geophysical fluid models with simple energy backscatter: explicit flows and unbounded exponential growth

Motivated by numerical schemes for large scale geophysical flow, we consider the rotating shallow water and Boussinesq equations on the whole space with horizontal kinetic energy backscatter source terms built from negative viscosity and stabilising hyperviscosity with constant parameters. We study the impact of this energy input through various explicit flows, which are simultaneously solving the nonlinear equations and the linear equations that arise upon dropping the transport nonlinearity, i.e. the linearisation in the zero state. These include barotropic, parallel and Kolmogorov flows as well as monochromatic inertia gravity waves. With focus on stable stratification we find that the backscatter generates numerous solutions of this type that grow exponentially and unboundedly, also with vertical structure. This signifies the possibility of undesired energy concentration into specific modes due to the backscatter. Families of steady state flows of this type arise as well and superposition principles in the nonlinear equations provide explicit sufficient conditions for instability of some of these. For certain steady barotropic flows of this type we provide numerical evidence of eigenmodes whose growth rates are proportional to the amplitude factor of the flow. For all other arising steady solutions we prove this is not possible.

physics.flu-dyn↗

arXiv2020

Explicit superposed and forced plane wave generalized Beltrami flows

We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier-Stokes equations on $\mathbb{R}^n$, as well as the rotating Boussinesq equations on $\mathbb{R}^3$. We cast these solutions are superpositions of certain linear plane waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave vectors and flow directions. For $n\leq 3$ these are explicit examples for generalized Beltrami flows. We show that forcing terms of corresponding plane wave type yield explicit solutions by linear variation of constants. We work in Eulerian coordinates and distinguish the two situations of vanishing and of gradient nonlinear terms, where the nonlinear terms modify the pressure. The methods, that we introduce to find explicit solutions in some nonlinear fluid models, can also be used in other equations with material derivative. Our approach offers another view on known explicit solutions of different fluid models from a plane wave perspective, and provides transparent nonlinear interactions between different flow components.

math.AP↗