Geophysical Fluid Dynamics

We propose an infrastructure based on the Open64 compiler for analyzing, modeling and optimizing MPI and/or OpenMP applications. The framework consists of four main parts: a compiler, microbenchmarks, a user interface and a runtime library. The compiler generates the application signature containing a portable representation of the application structure that may influence program performance. Microbenchmarks are needed to capture the system profile, including MPI latency and OpenMP overhead. The user interface, based on Eclipse, is used to drive code transformation, such as OpenMP code generation. And lastly, our runtime library can be used to balance the MPI workload, thus reducing load imbalance. In this paper we show that our framework can analyze and model MPI and/or OpenMP applications. We also demonstrate that it can be used for program understanding of large scale complex applications.

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We will investigate, from the angle of partial differential equation analysis, some oscillatory geophysical fluid dynamics in three dimensions: Navier-Stokes equations in a fast rotating, spherical shell, and Magnetohydrodynamics subject to strong Coriolis and Lorentz forces. Upon averaging their oscillatory solutions in time, interesting patterns such as zonal flows can emerge. More rigorously, we will prove that, when the restoring forces are strong enough, time-averaged solutions stay close to the null spaces of the wave operators, whereas the solutions themselves can be arbitrarily far away from these subspaces.

This paper considers the existence of extreme solutions of periodic boundary value problem for the second-order impulsive functional differential equations. We prove some new maximum principles. This allows us to introduce a new definition of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid. Meanwhile, we extend previous results.

The re-analysis programmes of numerical weather prediction (NWP) centres provide global, comprehensive descriptions of the atmosphere and of the Earth surface over long periods of time. The high realism of their representation of key NWP parameters, like temperature and winds, implies some realism for less emblematic parameters, such as cloud cover, but the degree of this realism needs to be documentated. This study aims to evaluate the high clouds over open oceans in the ECMWF 15-yr and 45-yr re-analyses. The assessment is based on a new 23-yr climatology of monthly frequencies of high cloud occurrence retrieved from the infrared radiances measured by operational polar satellites. It is complemented by data from the International Satellite Cloud Climatology Project. It is shown that the 45-yr ECMWF re-analysis dramatically improves on the previous 15-yr re-analysis for the realism of seasonal and interannual variations in high clouds, despite remaining systematic errors. More than ...

The framework of invariant parameterization is extended to higher-order closure schemes. We also define, for the first time, generalized invariant parameterization schemes, where symmetries of the corresponding original model are preserved as equivalence transformations of related classes of closed system of differential equations. As a particular problem, we consider invariant parameterization schemes for geostrophic eddies in a barotropic ocean. Here the initial model is the barotropic vorticity equation, which is equivalent to the system of incompressible inviscid two-dimensional Euler equations on a midlatitude beta-plane. The maximal Lie invariance algebra of this model is infinite-dimensional, and we intend to preserve it in the course of invariant parameterization, at least partially. The parameterizations proposed for the eddy vorticity flux and the energy flux are of order one and a half since we explicitly consider the equation for the turbulent kinetic energy in the closu...

Theoretical analyses and laboratory experiments have been performed on the stability of a flow generated by the differential cyclonic corotation of a flat, rigid disk in a uniformly rotating, linearly stratified fluid contained within a cylindrical tank. The undisturbed fluid is stably stratified with salt ͑Schmidt number Ϸ 670͒ and the ͑vertical͒ axes of rotation of the disk and the fluid container are coincident. The theoretical analysis shows that when the interior flow satisfies gradient wind balance ͑or, alternatively, thermal wind balance͒, it is destabilized by the action of viscosity. In the experiments, the manifestation of the viscous overturning instability is seen to be the formation of steplike internal microstructures in the density field, observed as regularly spaced, curved ring-shaped sheets with associated localized sharp, vertical density gradients. A stability analysis of the flow shows that the instability criterion is dependent on local values of the vertical and radial gradients of zonal velocity and the background density field. These quantities are measured in the experiments using a combination of horizontal-plane particle image velocimetry and an array of traversing microconductivity probes. The stability criterion based on this linear analysis predicts that the interior of the fluid is unstable. Using the ӷ 1 condition, simple asymptotic expressions for the maximum growth rate and associated wave number have been derived from the cubic dispersion relation. The theoretically predicted length scales and e-folding times associated with the fastest growing modes are found to give excellent agreement with the corresponding values obtained from the laboratory experimental data.

Buku ini menyajikan pengantar yang sistematis dan mendalam mengenai dinamika fluida geofisika (Geophysical Fluid Dynamics, GFD), bidang ilmu yang mempelajari pergerakan fluida di atmosfer dan lautan dalam skala besar. Fokus utama pembahasan mencakup dua faktor penting yang membedakan GFD dari mekanika fluida klasik, yaitu pengaruh rotasi Bumi dan stratifikasi densitas fluida. Materi disusun secara bertahap, dimulai dari konsep dasar seperti persamaan Navier-Stokes dalam kerangka rotasional, hingga fenomena yang lebih kompleks seperti gelombang Rossby, penyesuaian geostrofik, dan prinsip kekekalan vortisitas potensial.

Buku ini menjelaskan berbagai fenomena fisik yang terjadi di atmosfer dan lautan, termasuk pergerakan fluida berskala besar yang dipengaruhi oleh rotasi Bumi dan gaya pemulih gravitasi. Pembaca akan diajak memahami bagaimana keseimbangan geostrofik terbentuk, bagaimana gelombang bergerak dalam sistem perairan dangkal, serta bagaimana sirkulasi dan ketidakstabilan berkembang dalam aliran fluida yang kompleks. Semua pembahasan dikemas dengan pendekatan yang mudah dipahami namun tetap memuat kedalaman ilmiah.

Dirancang sebagai sumber belajar sekunder bagi mahasiswa, peneliti, dan peminat ilmu meteorologi dan oseanografi, buku ini memberikan penjelasan mendalam tentang prinsip-prinsip dasar yang membentuk dinamika fluida di Bumi. Dengan materi yang tersusun secara runtut, buku ini diharapkan dapat memperkaya pemahaman pembaca tentang fenomena fisis yang mendasari dinamika atmosfer dan lautan, serta interaksi kompleks di antara keduanya.

The context of this work is Automatic Differentiation (AD). Fundamentally, AD transforms a program that computes a mathematical function into a new program that computes its derivatives. Being automatic, AD can spare a large amount of development effort. AD is increasingly used in Scientific Computing, but some problems still delay its widespread use. In this thesis we propose elements of solution for two of these problems. The first problem is non-differentiability of most real-life programs for some particular inputs. AD tools often overlook this problem. However users need a strong degree of confidence in the derivatives they obtain, to use them e.g. in optimization engines. Non-differentiability in programs may come from various causes, from the mathematical model to the actual implementation choices. Practically, non-differentiability in programs is embodied by the presence of control. Rather than studying notions of extended derivatives, our approach is to describe the domain in the input space for which the function actually remains differentiable. We describe several alternatives to find this domain of validity and we evaluate their complexity. We formally study one method to compute the complete domain, but we discard it because of its cost. Alternatively, we propose a simpler directional method that we have implemented and validated on several examples. The second problem is efficiency of the reverse mode of AD, which computes gradients and is therefore of high interest. Reverse AD programs use the intermediate values from the original program in the reverse order, and this has a cost, whatever the strategy used. Available strategies all rely on a combination of storing and recomputing intermediate values, thus costing memory space and execution time. The central tactic, called "checkpointing", trades duplicate execution of complete segments of the code for memory space. In this work, we formalize the static data-flow behavior of reverse AD programs, taking checkpointing into account. Based on these formal results, we contribute two improvements to reverse AD strategies. First, the data-flow analysis lets us reduce the number of stored values to a minimum, and we give elements of proof of minimality. Second, we obtain indications on the code segments that are most appropriate for checkpointing. To experiment on checkpointing schemes, we extend the data-flow analyses and the reverse mode in our AD tool. We show the benefits on several large Scientific Computing codes.

Rotating grid turbulence experiments have been carried out in a stably stratified fluid for relatively large Reynolds numbers (mesh Reynolds numbers up to 18 000). Under the combined effects of rotation and stratification the flow degenerates into quasihorizontal motions. This regime is investigated using a scanning imaging velocimetry technique which provides time-resolved velocity fields in a volume. The most obvious effect of rotation is the inhibition of the kinetic energy decay, in agreement with the quasi-geostrophic model which predicts the absence of a direct energy cascade, as found in two-dimensional turbulence. In the regime of small Froude and Rossby numbers, the dynamics is found to be non-dissipative and associated with a symmetric and highly intermittent vertical vorticity field, that displays k -3 h energy spectra. For higher Rossby numbers, fundamental differences with the quasi-geostrophic model are found. A significant decay of kinetic energy, which does not depend on the stratification, is observed. Moreover, in this regime, although both cyclones and anticyclones are initially produced, the intense vortices are only cyclones. For late times the flow consists of an assembly of coherent interacting structures. Under the influence of both rotation and stratification, they take the form of lens-like eddies with aspect ratio proportional to f/N.

We explain the emergence of organized structures in two-dimensional turbulent flows by a theory of equilibrium statistical mechanics. This theory takes into account all the known constants of the motion for the Euler equations. The microscopic states are all the possible vorticity fields, while a macroscopic state is defined as a probability distribution of vorticity at each point of the domain, which describes in a statistical sense the fine-scale vorticity fluctuations. The organized structure appears as a state of maximal entropy, with the constraints of all the constants of the motion. The vorticity field obtained as the local average of this optimal macrostate is a steady solution of the Euler equation. The variational problem provides an explicit relationship between stream function and vorticity, which characterizes this steady state. Inertial structures in geophysical fluid dynamics can be predicted, using a generalization of the theory to potential vorticity.

The variational principle of Hamilton is applied to develop an analytical formulation to describe the volume viscosity in fluids. The procedure described here differs from those used in the past in that a dissipative process is represented by the chemical affinity and progress variable ͑sometimes called "order parameter"͒ of a reacting species. These state variables appear in the variational integral in two places: first, in the expression for the internal energy, and second, in a subsidiary condition accounting for the conservation of the reacting species. As a result of the variational procedure, two dissipative terms appear in the Navier-Stokes equation. The first is the traditional volume viscosity term, proportional to the dilatational component of velocity; the second term is proportional to the material time derivative of the pressure gradient. Values of the respective volume viscosity coefficients are determined by applying the resulting volume-viscous Navier-Stokes equation to the case of acoustical propagation and then comparing expressions for the dispersion and absorption of sound. The formulation includes the special case of equilibration of the translational degrees of freedom. As examples, values are tabulated for dry and humid air, argon, and sea water.

The increase of computational capabilities led recent studies to implement very high-resolution simulations that gave access to new scale interaction processes, particularly those associated with the transfer of energy from the oceanic mesoscales to smaller scales through an interior route to dissipation, which is still underexplored. In this context, we study spindown simulations of a mesoscale interior vortex, unstable to a mixed baroclinic–barotropic instability. Even though the global energy is almost conserved, some energy is transferred down to dissipation scales during the development of instabilities. However, in our parameter regime, there is no substantial forward energy cascade sustained by unbalanced dynamics. Rather than exploring the physical parameter range, we clarify numerical discretization issues that can be detrimental to the physical solutions and our interpretation of finescale dynamics. Special care is given to determining the effective resolution of the diffe...

We use a parameterized convection model to investigate the effects of deep water cycling on the thermal evolution of an Earth-like planet. The model incorporates two water reservoirs, a surface and an interior mantle reservoir. Exchange between the two is calculated using a mantle convection parameterization that allows for temperature-and water-dependent mantle viscosity together with internally self-consistent degassing and regassing parameterizations. The balance between degassing and regassing depends on the average spreading rate of tectonic plates, the amount of water partitioned into melt, the thickness of a mantle melt zone, and of a hydrated layer at the top of subducting plates. Degassing scales with melt zone thickness such that an early period of extensive melting would create a drier and more viscous mantle, shifting the solidus line in a direction that would reduce the melt zone thickness and the rate of mantle heat loss. Coupling a hydrated zone thickness-dependent regassing factor to the model, to mimic water delivery to the mantle via a serpentinized layer, allows for the potential of a reversing point where the overall water flow direction switches from degassing to regassing as the mantle cools. The water effect on viscosity creates a negative feedback that tends to regulate the final amount of water in the mantle so it is not strongly dependent on the initial amount of planetary water. The final amount of water in the surface reservoir is then determined by this feedback effect together with the initial water budget of the entire planet. This implies that if the initial water budget of a planet can be estimated, from planetary formation models, then the volume of surface water can be used to estimate the volume of water in the mantle of an Earth-like planet. Applying this methodology to the Earth leads to predictions for water concentration in the Earth's mantle that are in line with geochemical and petrological constraints.

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.

Geostrophic turbulence is a flow regime attained by turbulent, rotating, stably stratified fluids in near-geostrophic balance. When a small-scale forcing is present, flows in this regime may develop an inverse energy cascade. Geostrophic turbulence has been used in geophysical fluid dynamics as a relatively simple model of the large-scale planetary and terrestrial circulations. When the meridional variation of the Coriolis parameter (or a β-effect) is taken into account, the horizontal flow symmetry breaks down giving rise to the emergence of jet flows. In a certain parameter range, a new flow regime comes to life. Its main characteristics include strongly anisotropic kinetic energy spectrum and slowly evolving systems of alternating zonal jets. This regime is a subset of geostrophic turbulence and has been coined zonostrophic turbulence; it can develop both on a β-plane and on the surface of a rotating sphere. This regime was first discovered in computer simulations but later revealed in the laboratory experiments, in the deep terrestrial oceans, and on solar giant planets where it is believed to be the primary physical mechanism responsible for the generation and maintenance of the stable systems of alternating zonal jets. The hallmarks of zonostrophic turbulence are the anisotropic inverse energy cascade and complicated interaction between turbulence and Rossby-Haurwitz waves. Addressing the goals of the conference 'Turbulent Mixing and Beyond' that took place in August 2007 in Trieste, Italy, this paper exposes the regime of zonostrophic turbulence to a wide scientific community, provides a survey of this regime, elaborates its main characteristics, offers novel approaches to describe and understand this phenomenon, and discusses its applicability as a model of the large-scale planetary and terrestrial circulations.

The performance of 18 coupled Chemistry Climate Models (CCMs) in the Tropical Tropopause Layer (TTL) is evaluated using qualitative and quantitative diagnostics. Trends in tropopause quantities in the tropics and the extratropical Upper Troposphere and Lower Stratosphere (UTLS) are analyzed. A quantitative grading methodology for evaluating CCMs is extended to include variability and used to develop four different grades for tropical tropopause temperature and pressure, water vapor and ozone. Four of the 18 models and the multi‐model mean meet quantitative and qualitative standards for reproducing key processes in the TTL. Several diagnostics are performed on a subset of the models analyzing the Tropopause Inversion Layer (TIL), Lagrangian cold point and TTL transit time. Historical decreases in tropical tropopause pressure and decreases in water vapor are simulated, lending confidence to future projections. The models simulate continued decreases in tropopause pressure in the 21st ...

This work evaluates the performance of two radiation parameterization schemes in 30-day clear-sky runs of the Eta model over a region in Southeast Brazil. Two versions of the Eta model are compared: a version using the radiation scheme developed by the Geophysical Fluid Dynamics Laboratory (GFDL) and a recently developed version using the Rapid Radiative Transfer Model for GCM (RRTMG). These simulations are compared against CMSAF satellite data and surface station data. The simulation using RRTMG produced downward surface shortwave radiation fluxes closer to observations and reduced the systematic positive bias of the Eta simulation using the GFDL scheme. The 2-m temperature negative bias found in the Eta-GFDL simulations is reduced in the Eta-RRTMG simulations, which results from a larger net total radiation in the Eta-RRTMG simulations. The new version has better accuracy than the Eta using the GFDL scheme for most of the evaluated variables, particularly for clear-sky conditions.

This research explores the application of Finite Element Analysis (FEA) software in simulating metal forming processes, a crucial technique in modern manufacturing and engineering. FEA models the complex behavior of metals under various forming conditions, such as stamping, forging, and extrusion, by discretizing the workpiece and tooling into finite elements. This method allows for detailed analysis of stresses, strains, and deformations, helping optimize process parameters and tooling design, thereby improving product quality and reducing costs. Despite challenges like accurately modeling material properties and computational demands, FEA remains a powerful tool for enhancing understanding of material behavior, minimizing waste, and supporting sustainable manufacturing practices.

We analyse the exchange of energy for an axisymmetric gravity current, released instantaneously from a lock, propagating over a horizontal boundary at high Reynolds number. The study is relevant to flow in either a wedge or a full circular geometry. Attention is focused on effects due to a linear stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is fair agreement between these two approaches for the energy changes of the dense fluid (the current). The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The total energy of the axisymmetric current decays considerably faster with distance of propagation than for the two-dimensional counterpart.

Our goal is to study the e ffects of atmospheric density stratification on the formation and longevity of aircraft trailing vortices. Toward s this end, we study a series of model two-dimensional problems, two of which are presented here. Th studies are summarised here from [1, 2, 3]. The first is on the dynamics of a single vortex lo cated at a density stratified layer. Due to the presence of the vortex, the density interfa c rolls up into a spiral. Centrifugal acceleration is shown to play a major role in reducing the life-span of the vortex. Due to the misalignment of the acceleration vector with the normal to the spiral interface, vorticity is created along the density interface, resulting in a spira l Kelvin-Helmholtz instability. At every interface where heavier fluid is closer to the vortex th an lighter, the flow is also unstable due to a centrifugal Rayleigh-Taylor instability. In th e second problem, centrifugal e ff cts are neglected, i.e., the Boussinesq approximation is...

We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the ‘stable stratification’ of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh–Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood numberAtshow growth rates scaling asAt1/3close to the critical radius, andAt1/2further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vo...

In the framework of background currents, we examine a dynamically consistent model of a periodic flow over an isolated submerged obstacle of Gaussian shape. Chaotic advection of passive scalars in the ocean is studied from the viewpoint of the dynamic system theory. Much attention is given to the nonlinear resonance role. The relationship between the rotation frequency of an unperturbed fluid particle and the appearance and disappearance of a nonlinear resonance is established. So, the frequency gives us the opportunity to explain the trajectories' chaotization via the Chirikov criteria. This simple estimation is verified by numerical calculations. A simplified model is used for more exact reasoning. The distance between nonlinear resonance and its width was established for large action, and accurate estimation of chaotic region boundary was calculated. Also, numerically and analytically, there are studied chaotization dependencies from some other flow parameters.

Effects of stratification on the dissolution of a vertical ice-face: effect of Rayleigh number Bishakhdatta Gayen, Mainak Mondal and Ross W. Griffiths Research School of Earth Science, The Australian National University, Canberra, Australia Numerical simulation is performed to investigate the effect of ambient stratification on a vertical ice face dissolving into cold and saline water. The three coupled interface equations are used, along with the Boussinesq and non-hydrostatic governing equations of motion and equation of state for seawater, to solve for interface temperature, salinity and melt rate. The main focus is on the rate of dissolving of ice at ambient water temperatures between 0 ◦ C and 4 ◦ C and salinity around 35 psu and the dependence on stratification (as characterizes many sites around Antarctica). In order to examine the process and scaling we use very strong stratification and vertical scales much less than those in the oceans. Figure 1: (a) The formation of doubl...

This thesis is organised as follows. In Chapter 2 checkpointing techniques for double sweep simulations are introduced. Here, reversal schedules are classified and defined. Moreover, notations, definitions, and some basic results are introduced. They will be needed for the further development of nested checkpointing later on. Chapter 3 gives a survey of numerical methods for separated boundary value problems. Here, it is shown how these methods can be applied to the solu

A foil-based secondary emission monitor (SEM) was developed for beam profile measurements at the Ultra-low energy Storage Ring (USR) that will be installed at the future Facility for Low-energy Antiproton and Ion Research (FLAIR) in Darmstadt, Germany. Simulations of the optimised design of the monitor are presented. Furthermore, its usability for the low energy antiproton (p) beam profile monitoring is discussed.

Oceananigans.jl is a fast and friendly software package for the numerical simulation of incompressible, stratified, rotating fluid flows on CPUs and GPUs. Oceananigans.jl is fast and flexible enough for research yet simple enough for students and first-time programmers. Oceananigans.jl is being developed as part of the Climate Modeling Alliance project for the simulation of small-scale ocean physics at high-resolution that affect the evolution of Earth's climate.

Lyapunov vectors are natural generalizations of normal modes for linear disturbances to aperiodic deterministic flows and offer insights into the physical mechanisms of aperiodic flow and the maintenance of chaos. Most standard techniques for computing Lyapunov vectors produce results which are norm-dependent and lack invariance under the linearized flow (except for the leading Lyapunov vector) and these features can make computation and physical interpretation problematic. An efficient, norm-independent method for constructing the n most rapidly growing Lyapunov vectors from n − 1 leading forward and n leading backward asymptotic singular vectors is proposed. The Lyapunov vectors so constructed are invariant under the linearized flow in the sense that, once computed at one time, they are defined, in principle, for all time through the tangent linear propagator. An analogous method allows the construction of the n most rapidly decaying Lyapunov vectors from n decaying forward and n − 1 decaying backward singular vectors. This method is demonstrated using two low-order geophysical models.

These lecture notes discuss various representations and approximations of ideal shallow water dynamics in a rotating frame. These rotating shallow water (RSW) equations possess a slow + fast decomposition in which they reduce approximately to quasigeostrophic (QG) motion (conservation of energy and potential vorticity) plus nearly decoupled equations for gravity waves in an asymptotic expansion in small Rossby number, 1. The solution properties of the RSW equations are discussed, and some alternative representations of the RSW equations which highlight the slow + fast interactions are given. At the conclusion, the RSW equations are re-derived as Euler–Poincaré equations from Hamilton’s principle in the Eulerian fluid representation.

In this paper, we consider two-phase drift flux model and give a full symmetry group classification of the governing quasilinear system of partial differential equations (PDEs). An exact solution is obtained by mapping the governing system to an analogous system of PDEs, where the flow variable exhibits space-time dependence. The drift flux model, which is a strictly hyperbolic system, is shown to have a linearly degenerated characteristic field, therefore, the associated wave is a contact discontinuity. The transport equations for the evolution of the contact and weak discontinuity waves are studied explicitly and some intriguing observations on the behavior of these elementary waves are obtained.

It is well established that the Coriolis force deflects wind and water currents. However, its influence on groundwater flow is neglected. Earth's rotation causes inertia circles in groundwater that create vortices ending up in different local pressure zones, similar to the high and low pressures in air. High pressure zones in groundwater induce, under certain conditions, a vertical flow to ground surface. This could be the missing link where hydrostatic pressure cannot explain springs in deserts, mountains and on islands in the sea. Here, simulations on the Coriolis force acting on groundwater flow are presented.

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.

The time-dependent meandering in a thin baroclinic jet over bottom topography is discussed in the quasi-geostrophic approximation. The motion of the axis of the jet is taken to be vertically coherent and the axis itself is defined as inextensible. The motion is examined from a frame of reference moving with the axis but fixed at an arbitrary longitude in terms of an open ocean spatial initial value problem. The velocities of the axis and of the jet are quasi-geostrophic, and vorticity conservation for the first non-geostrophic components constrains the evolution of the axis and gives a path equation. The spatial linearized stability problem is studied and the jet is found to be baroclinically unstable to path disturbances of sufficiently high frequency which amplify downstream. An exact solution is obtained to the nonlinear path equation over a flat bottom with no ß-effect. The evolution of the path of these unstable meanders is such that the path closes itself and forms rings (at which point the analysis breaks down). It is proposed that the baroclinic jet processes studied here are fundamental to the dynamics of Gulf Stream meandering and isolated eddy production.

The time-dependent meandering in a thin baroclinic jet over bottom topography is discussed in the quasi-geostrophic approximation. The motion of the axis of the jet is taken to be vertically coherent and the axis itself is defined as inextensible. The motion is examined from a frame of reference moving with the axis but fixed at an arbitrary longitude in terms of an open ocean spatial initial value problem. The velocities of the axis and of the jet are quasi-geostrophic, and vorticity conservation for the first non-geostrophic components constrains the evolution of the axis and gives a path equation. The spatial linearized stability problem is studied and the jet is found to be baroclinically unstable to path disturbances of sufficiently high frequency which amplify downstream. An exact solution is obtained to the nonlinear path equation over a flat bottom with no ß-effect. The evolution of the path of these unstable meanders is such that the path closes itself and forms rings (at which point the analysis breaks down). It is proposed that the baroclinic jet processes studied here are fundamental to the dynamics of Gulf Stream meandering and isolated eddy production.

We identify effective carriers of Sargassum in the Caribbean Sea and describe a mechanism for coastal choking. Revealed from satellite altimetry, the carriers of Sargassum are mesoscale eddies (vortices of 50-km radius or larger) with coherent material (i.e., fluid) boundaries. These are observer-independent-unlike eddy boundaries identified with instantaneously closed streamlines of the altimetric sea-surface height field-and furthermore harbor finite-time attractors for networks of elastically connected finite-size buoyant or "inertial" particles dragged by ocean currents and winds,

During explosive volcanic eruptions, a mixture of pyroclasts and volcanic gases is released from the vent. When the mixture loses its upward momentum before the density of the mixture becomes lower than the atmospheric density, the mixture forms a pyroclastic flow. Dynamics of pyroclastic flows can be approximated by that of an inviscid gravity current. The dynamics of inviscid gravity currents are controlled by an inertial-buoyancy balance on the front (e.g., Benjamin, 1968); we refer to this condition as ”the front condition”. The front condition, and hence, the dynamics of the inviscid gravity currents strongly depends on the density ratio of the current ( ρc) to the ambient ( ρa) (e.g., Ungarish, 2009). When ρc/ρa∼1, the current is characterized by a high front, whereas a front height does not develop when ρc/ρaÀ1. In pyroclastic flows, because density ratio ρc/ρa varies spatially and temporally, the dynamics of pyroclastic flows becomes complicated; the basic features of pyrocl...

The breaking of detailed balance in fluids through Coriolis forces or odd-viscous stresses has profound effects on the dynamics of surface waves. Here we explore both weakly and strongly non-linear waves in a three-dimensional fluid with vertical odd viscosity. Our model describes the free surface of a shallow fluid composed of nearly vertical vortex filaments, which all stand perpendicular to the surface. We find that the odd viscosity in this configuration induces previously unexplored non-linear effects in shallow-water waves, arising from both stresses on the surface and stress gradients in the bulk. By assuming weak nonlinearity, we find reduced equations including Korteweg-de Vries (KdV), Ostrovsky, and Kadomtsev-Petviashvilli (KP) equations with modified coefficients. At sufficiently large odd viscosity, the dispersion changes sign, allowing for compact two-dimensional solitary waves. We show that odd viscosity and surface tension have the same effect on the free surface, but distinct signatures in the fluid flow. Our results describe the collective dynamics of many-vortex systems, which can also occur in oceanic and atmospheric geophysics.

The primitive equation (PE) model can be further simplified while retaining its most essential properties. The simplification is based on the already emphasised disparity of horizontal and vertical scales of synoptic motions in the atmosphere, and meso-scales motions in the ocean, and allows us to obtain a conceptual GFD model, which is the main subject of this book: the rotating shallow-water model (RSW). It permits, at the same time, physically transparent analysis and interpretation and efficient numerical tools for simulations of fundamental dynamical phenomena. We show here how the RSW model(s) can be obtained from PE by vertical averaging. In order to stress the generality of the approach, we apply it to the equations of motion without supposing incompressibility. The same procedure holds for PE for incompressible fluid with obvious simplifications. It can be found, with some modifications, in Chapters 15 and 16.