Multiscale Interactions in Geophysical Fluids

The dynamics of the atmosphere and ocean involves a broad range of spatial and temporal scales, many of which emerge through complex nonlinear mechanisms from forcings at very different scales. This poses major challenges for the numerical prediction of the weather, ocean state and climate: many processes have scales that are too small to be resolved yet they play an essential role in determining large-scale features. This workshop examined how modern mathematical methods – ranging from multiscale asymptotics to adaptive numerical methods and stochastic modelling – can be applied to represent the large-scale impact of these small-scale processes and improve both deterministic and probabilistic predictions

2004 program of study : tides

The summer of 2004 saw the GFD program tackle “Tides”. Myrl Hendershott (Scripps Institution of Oceanography) gave a fabulous introduction to the subject in the first week of the course, laying the foundations from astronomy and classical geophysical fluid dynamics. In the second week, Chris Garrett (University of Victoria) admirably followed up with recent developments on the subject, including the recent observations from satellite altimetry, their implications to mixing and circulation, and even a memorable lecture on the noble theme of how we might solve the world's energy crisis. The principal lectures proved unusually popular this summer, and the seminar room at Walsh often overflowed in the first two weeks. Following on from the lectures, the seminar schedule of the summer covered in greater detail the oceanographic issues with which researchers are actively grappling. We also heard about related problems regarding atmospheric, planetary and stellar tides, together with the usual mix of topics on GFD in general. The summer once again featured a lecture for the general public in the Woods Hole area. Carl Wunsch delivered a very well received lecture entitled “Climate Change Stories”, in which he gave an impression of how scientists generally believe our climate is currently changing, whilst simultaneously urging caution against some of the more outrageous and exaggerated claims. The lecture was held at Lilly Auditorium, thanks to the hospitality of the Marine Biology Laboratory. The reception following the lecture was enjoyed by all. Neil Balmforth and Stefan Llewellyn Smith acted as Co-Directors for the summer. Janet Fields, Jeanne Fleming and Penny Foster provided the administrative backbone to the Program, both during the summer and throughout the year beforehand. As always, we were grateful to the Woods Hole Oceanographic Institution for the use of Walsh Cottage, and Keith Bradley's solid service could not be overlooked. Shilpa Ghadge and Shreyas Mandre are to be thanked for their part in comforting the fellows, developing the summer's proceedings volume (available on the GFD web site) and for running the computer network.Funding was provided by the Office of Naval Research under Contract No. N00014-04-1-0157 and the National Science Foundation under Grant No. OCE-0325296

Conservation Properties of the Hamiltonian Particle-Mesh method for the Quasi-Geostrophic Equations on a sphere

The Hamiltonian particle-mesh (HPM) method is used to solve the Quasi-Geostrophic model generalized to a sphere, using the Spherepack modeling package to solve the Helmholtz equation on a colatitude-longitude grid with spherical harmonics. The predicted energy conservation of a Poisson system is shown to be approximately retained and statistical mean-eld theory is veried

Atmospheric Circulation of Exoplanets

We survey the basic principles of atmospheric dynamics relevant to explaining existing and future observations of exoplanets, both gas giant and terrestrial. Given the paucity of data on exoplanet atmospheres, our approach is to emphasize fundamental principles and insights gained from Solar-System studies that are likely to be generalizable to exoplanets. We begin by presenting the hierarchy of basic equations used in atmospheric dynamics, including the Navier-Stokes, primitive, shallow-water, and two-dimensional nondivergent models. We then survey key concepts in atmospheric dynamics, including the importance of planetary rotation, the concept of balance, and scaling arguments to show how turbulent interactions generally produce large-scale east-west banding on rotating planets. We next turn to issues specific to giant planets, including their expected interior and atmospheric thermal structures, the implications for their wind patterns, and mechanisms to pump their east-west jets. Hot Jupiter atmospheric dynamics are given particular attention, as these close-in planets have been the subject of most of the concrete developments in the study of exoplanetary atmospheres. We then turn to the basic elements of circulation on terrestrial planets as inferred from Solar-System studies, including Hadley cells, jet streams, processes that govern the large-scale horizontal temperature contrasts, and climate, and we discuss how these insights may apply to terrestrial exoplanets. Although exoplanets surely possess a greater diversity of circulation regimes than seen on the planets in our Solar System, our guiding philosophy is that the multi-decade study of Solar-System planets reviewed here provides a foundation upon which our understanding of more exotic exoplanetary meteorology must build.Comment: In EXOPLANETS, edited by S. Seager, to be published in the Spring of 2010 in the Space Science Series of the University of Arizona Press (Tucson, AZ) (refereed; accepted for publication

1989 summer study program in Geophysical Fluid Dynamics : general circulation of the oceans

The success of this summer's Geophysical Fluid Dynamics Program owes much to Myrl Hendershott's excellent and engaging survey of the Oceans General Circulation, including recent developments In the Theory of Recirculation Gyres and Thermocline Ventilation.Funding was provided by the National Science Foundation through Grant Number OCE-89-0101

The Dynamics of Geophysical and Astrophysical Turbulence

Turbulence is ubiquitous within geophysical and astrophysical fluid flows. Its interaction with physical ingredients such as rotation and stratification gives rise to spectacular dynamics, including the layering of material properties, which in turn influence the transport and distribution of heat, momentum and tracers. We consider the effects of rotation and stratification individually in the study of two different problems of scientific interest, using idealised models which retain only the essential ingredients. Our first problem investigates the role of rotation in geophysical flows. We consider a barotropic, stochastically-forced turbulent flow on a beta-plane, which is well known to exhibit the spontaneous formation and equilibration of persistent zonal jets. The equilibrated jets are not steady and the focus here is on their time variability, which is of interest both because of its relevance to the behaviour of naturally occurring jet streams and for the insights it provides into the dynamical mechanisms operating in these systems. We compare the behaviour of a nonlinear (NL) system to a quasilinear (QL) model in which eddy-eddy interactions are neglected. Both systems reveal a rich zoology of dynamics, nevertheless, key differences exist. The NL model admits the formation of systematically migrating jets, a phenomenon that has not been previously identified. Jets migrate north or south with a speed of translation that is a function of the Rhines scale and the frictional damping rate, occasionally changing their direction of migration. The QL model does not exhibit jet migration, but a generalised quasilinear (GQL) model, in which certain eddy-eddy interactions are systematically restored, does, demonstrating that long waves, generated by such interactions, play a key dynamical role. The importance of these waves, in addition to the role of random fluctuations, is affirmed using a statistical formulation in which the flow statistics are solved for directly. Our second problem considers the interaction of a stable density stratification with a background velocity distribution, which can develop into stratified turbulence. Geophysical flows, in which the diffusivities of momentum and heat are commensurate, are often very strongly stratified, nevertheless, turbulence still occurs. Density layering is key to understanding the properties of this `layered anisotropic stratified turbulence' (LAST) regime. On the other hand, astrophysical flows are typically characterised by strong thermal diffusion, inhibiting the formation of density layers. This suggests that LAST dynamics cannot occur, raising the interesting question of whether analogous or fundamentally different regimes exist in the limit of strong thermal diffusion. This thesis addresses this question for the case of a vertically stratified, horizontally-forced Kolmogorov flow. Using linear stability theory, we show that three-dimensional perturbations of the horizontal shear are always unstable in the limit of strong stratification and strong thermal diffusion, causing the flow to develop vertical layers, and hence vertical shear, in the velocity field, thereby allowing vertical shear instabilities to develop. The subsequent nonlinear evolution and transition to turbulence is studied numerically using direct numerical simulations, where four distinct dynamical regimes emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws which explain the empirical observations.Murray Edwards College Cambridge Philosophical Societ

Scaling Baroclinic Eddy Fluxes: Vortices and Energy Balance

The eddy heat flux generated by the statistically equilibrated baroclinic instability of a uniform, horizontal temperature gradient is studied using a two-mode f-plane quasigeostrophic model. An overview of the dependence of the eddy diffusivity D on the bottom friction κ, the deformation radius λ, the vertical variation of the large-scale flow U, and the domain size L is provided by numerical simulations at 70 different values of the two nondimensional control parameters κλ/U and L/λ. Strong, axisymmetric, well-separated baroclinic vortices dominate both the barotropic vorticity and the temperature fields. The core radius of a single vortex is significantly larger than λ but smaller than the eddy mixing length ℓ_mix. On the other hand, the typical vortex separation is comparable to ℓ_mix. Anticyclonic vortices are hot, and cyclonic vortices are cold. The motion of a single vortex is due to barotropic advection by other distant vortices, and the eddy heat flux is due to the systematic migration of hot anticyclones northward and cold cyclones southward. These features can be explained by scaling arguments and an analysis of the statistically steady energy balance. These arguments result in a relation between D and ℓ_mix. Earlier scaling theories based on coupled Kolmogorovian cascades do not account for these coherent structures and are shown to be unreliable. All of the major properties of this dilute vortex gas are exponentially sensitive to the strength of the bottom drag. As the bottom drag decreases, both the vortex cores and the vortex separation become larger. Provided that ℓ_mix remains significantly smaller than the domain size, then local mixing length arguments are applicable, and our main empirical result is ℓ_mix ≈ 4λ exp(0.3U/κλ)

Variational integrators for the rotating shallow water equations

The numerical simulation of the Earth’s atmosphere plays an important role in developing our understanding of climate change. The atmosphere and ocean can be seen as a shallow fluid on the globe; here, we use the shallow water equations as a first step to approximate these geophysical flows. Then, the numerical model can only be accurate if it has good conservation properties, e.g. without conserving mass the simulation can not be physical. Obtaining such a numerical model can be achieved using numerical variational integration. Here, we have derived a numerical variational integrator for the rotating shallow water equations on the sphere using the Euler–Poincaré framework. First, the continuous Lagrangian is discretized; then, the numerical scheme is obtained by computing the discrete variational principle. The conservational properties and accuracy of the model are verified with standard test cases. However, in order to obtain more realistic simulations, the shallow water equations need to include physical parametrizations. Thus, we introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. Then, benchmarks are carried out to demonstrate that the spatial part of the stochastic scheme preserves the total energy. The proposed random model better captures the structure of a large-scale flow than a comparable deterministic model. Furthermore, to be able to carry out long term simulations we extend the discrete Euler–Poincaré framework with a selective decay. The selective decay dissipates an otherwise conserved quantity while conserving energy. We apply the new framework to the shallow water equations to dissipate the potential enstrophy. Then, we carry out standard benchmarks to demonstrate the conservation properties. We show that the selective decay resolves more small scales compared to a standard dissipation