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

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-B\'{e}nard 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.Comment: 12 page

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.Comment: 17 pages, 5 figure

Northern Hemisphere midlatitude cyclone variability in GCM simulations with different ocean representations

Abstract : The impact of different ocean models or sea surface temperature (SST) and sea-ice concentrations on cyclone tracks in the Northern Hemisphere midlatitudes is determined within a hierarchy of model simulations. A reference simulation with the coupled atmosphere ocean circulation model ECHAM/HOPE is compared with simulations using ECHAM and three simplified ocean and sea-ice representations: (1) a variable depth mixed layer (ML) ocean, (2) forcing by varying SST and sea-ice, and (3) with climatological SST and sea-ice; the latter two are from the coupled ECHAM/HOPE integration. The reference simulation reproduces the observed cyclone tracks. The cyclones are tracked automatically by a standard routine and the variability of individual cyclone trajectories within the storm tracks is determined by a cluster approach. In the forced simulation with varying SST, the geographical distribution and the statistics of the cyclones are not altered compared to the coupled reference simulation. In the ML- and the climatological simulation, deviations of the mean cyclone distribution are found which occur mainly in the North Pacific, and can partially be traced back to missing El Niño/Southern Oscillation (ENSO) variability. The climatological experiment is superior to the ML-experiment. The variability of the individual cyclone trajectories, as determined by the cluster analysis, reveals the same types and frequencies of propagation directions for all four representations of the lower boundary. The largest discrepancies for the cluster occupations are found for the climatological and the ML-simulatio