arXiv subjects

Richard Blender

Publications and source records attributed to Richard Blender.

11 recordsLinked to original sources

arXiv2023

Madden-Julian Oscillation described as a Nonlinear Burgers Kink in the Meridional Vorticity Equation

A dynamic equation for a large scale convective event in the tropical atmosphere similar to the Madden--Julian Oscillation (MJO) is suggested based on the meridional vorticity equation with buoyancy parametrized by Convective Available Potential Energy (CAPE). The propagation is determined by the nonlinear Burgers equation with a stationary solution describing a kink moving towards the moisture source. In this conceptual model, the propagation speed depends on the asymmetry in the zonal surface winds which is observed in the boundary layer in the vicinity of an MJO event and attributed to Rossby and Kelvin waves. Furthermore, the model predicts convection at the equator in the east of the MJO which is not correlated with Kelvin and Rossby waves.

physics.ao-ph↗

arXiv2020

Wave Triad with Forcings as a Nambu System

The dynamics of an ideal wave triad with real amplitudes has a well-known Nambu representation with energy and enstrophy as conservation laws. Here we derive Nambu representations for systems with constant forcings. These equations have been applied to triads of Rossby-Haurwitz waves in the atmosphere where they are forced with orography. The conservation laws are based on relations for the unforced amplitudes and a Hamiltonian given by the total energy plus terms involving the unforced amplitudes. The forcing of the unstable wavenumber causes a recharge cycle.

physics.ao-ph↗

arXiv2018

Fluctuation Analysis of the Atmospheric Energy Cycle

The atmosphere gains available potential energy by solar radiation and dissipates kinetic energy mainly in the atmospheric boundary layer. We analyze the fluctuations of the global mean energy cycle defined by Lorenz (1955) in a simulation with a simplified hydrostatic model. The energy current densities are well approximated by the generalized Gumbel distribution (Bramwell, Holdsworth and Pinton, 1998) and the Generalized Extreme Value (GEV) distribution. In an attempt to assess the fluctuation relation of Evans, Cohen, and Morriss (1993) we define entropy production by the injected power and use the GEV location parameter as a reference state. The fluctuation ratio reveals a linear behavior in a finite range.

physics.ao-ph↗

arXiv2017

Construction of Hamiltonian and Nambu forms for the shallow water equations

A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations, using the conservation for energy and potential enstrophy, is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws (CLs) for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators and result in the sum of two Nambu brackets, one for the vortical flow and one for the wave-mean flow interaction, and a Poisson bracket representing the interaction between divergence and geostrophic imbalance. The advantage of this approach is that the Hamiltonian and Nambu forms can be here written in a coordinate independent form.

math-ph↗

arXiv2017

Representation of Viscous Dissipation in 2D Fluid Dynamics as a Symplectic Process and its Metriplectic Representation

Dissipation can be represented in Hamiltonian mechanics in an extended phase space as a symplectic process. The method uses an auxiliary variable which represents the excitation of unresolved dynamics and a Hamiltonian for the interaction between the resolved dynamics and the auxiliary variable. This method is applied to viscous dissipation (including hyper-viscosity) in a two-dimensional fluid, for which the dynamics is non-canonical. We derive a metriplectic representation and suggest a measure for the entropy of the system.

physics.flu-dyn↗

arXiv2015

Hydrodynamic Nambu Brackets derived by Geometric Constraints

A geometric approach to derive the Nambu brackets for ideal two-dimensional (2D) hydrodynamics is suggested. The derivation is based on two-forms with vanishing integrals in a periodic domain, and with resulting dynamics constrained by an orthogonality condition. As a result, 2D hydrodynamics with vorticity as dynamic variable emerges as a generic model, with conservation laws which can be interpreted as enstrophy and energy functionals. Generalized forms like surface quasi-geostrophy and fractional Poisson equations for the stream-function are also included as results from the derivation. The formalism is extended to a hydrodynamic system coupled to a second degree of freedom, with the Rayleigh-Bénard convection as an example. This system is reformulated in terms of constitutive conservation laws with two additive brackets which represent individual processes: a first representing inviscid 2D hydrodynamics, and a second representing the coupling between hydrodynamics and thermodynamics. The results can be used for the formulation of conservative numerical algorithms that can be employed, for example, for the study of fronts and singularities.

physics.flu-dyn↗

arXiv2014

Instability of Planetary Flows using Riemann Curvature: a Numerical Study

The instability of ideal non-divergent zonal flows on the sphere is determined numerically by the instability criterion of Arnol'd (1966) for the sectional curvature. Zonal flows are unstable for all perturbations besides for a small set which are in approximate resonance. The sectional curvature scales with $m/\ell$ for large total and zonal wave numbers $\ell$ and $m$ of the perturbations. The planetary rotation is stable and the presence of rotation reduces the instability of perturbations.

physics.flu-dyn↗

arXiv2014

Mathematical and Physical Ideas for Climate Science

The climate is a forced and dissipative nonlinear system featuring non-trivial dynamics of a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics, and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the non-equilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate and characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on non-equilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

physics.ao-ph↗

arXiv2013

Towards a Fluctuation Theorem in an Atmospheric Circulation Model

An investigation of the distribution of finite time trajectory divergence is performed on an Atmospheric Global Circulation Model. The distribution of the largest local Lyapunov exponent shows a significant probability for negative values over time spans up to 10 days. This effect is present for resolutions up to wave numbers l=42 (~ 250km). The probability for a negative local largest Lyapunov exponent decreases over time, similarly to the predictions of the Fluctuation Theorem for entropy production. The model used is hydrostatic with variable numbers of vertical levels and different horizontal resolutions.

physics.ao-ph↗

arXiv2013

Avalanches, Breathers and Flow Reversal in a Continuous Lorenz-96 Model

For the discrete model suggested by Lorenz in 1996 a one-dimensional long wave approximation with nonlinear excitation and diffusion is derived. The model is energy conserving but non-Hamiltonian. In a low order truncation weak external forcing of the zonal mean flow induces avalanche-like breather solutions which cause reversal of the mean flow by a wave-mean flow interaction. The mechanism is an outburst-recharge process similar to avalanches in a sand pile model.

nlin.AO↗

arXiv2012

Nambu representation of an extended Lorenz model with viscous heating

We consider the Nambu and Hamiltonian representations of Rayleigh-Benard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec=0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivative of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.

physics.flu-dyn↗