Equations Of Geophysical Fluid Dynamics Research Articles

Exact solutions for geophysical flows with discontinuous variable density and forcing terms in spherical coordinates

ABSTRACT We present here exact solutions to the equations of geophysical fluid dynamics that depict inviscid flows moving in the azimuthal direction on a circular path, around the globe, and which admit a velocity profile below the surface and along it. These features render this model suitable for the description of the Antarctic circumpolar current (ACC). The governing equations we work with–taken to be the Euler equations written in spherical coordinates–also incorporate forcing terms which are generally regarded as means that ensure the general balance of the ACC. Our approach allows for a variable density (depending on the depth and latitude) of discontinuous type which divides the water domain into two layers. Thus, the discontinuity gives rise to an interface. The velocity in both layers and the pressure in the lower layer are determined explicitly, while the pressure in the upper layer depends on the free surface and the interface. Functional analytical techniques render (uniquely) the surface and interface-defining functions in an implicit way. We conclude our discussion by deriving relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations. A regularity result concerning the interface is also derived.

Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates

AbstractThis paper is concerned with some analytical aspects pertaining to the Antarctic Circumpolar Current. We use spherical coordinates in a rotating frame to derive a new exact and partially explicit solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible azimuthal flow with a discontinuous density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows, that is, flows that are observed in an averaged sense in the ocean. The discontinuous density triggers the appearance of an interface that plays the role of an internal wave. Although the velocity and the pressure are determined explicitly, we use functional analytical techniques that uniquely render the surface and interface defining functions in an implicit way as soon as a small enough pressure is applied on the free surface. Additionally, we consider a particular example, where the interface can be determined explicitly. We conclude our discussion by setting out monotonicity relations between the surface pressure and its distortion that concur with the physical expectations. A regularity result concerning the interface is also derived.

Explicit and exact solutions concerning the Antarctic Circumpolar Current with variable density in spherical coordinates

We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows, that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g., of f- and β-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely azimuthal stratified flow with a free surface that—thanks to the inclusion of forcing terms and the consideration of the Earth’s geometry via spherical coordinates—makes it suitable for describing the Antarctic Circumpolar Current and enables an in-depth analysis of the structure of this flow. In line with the latter aspect, we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface. We conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations.

A steady stratified purely azimuthal flow representing the Antarctic Circumpolar Current

We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.

Time discrete approximation of weak solutions to stochastic equations of geophysical fluid dynamics and applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Ito or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.

Compatible finite element spaces for geophysical fluid dynamics

Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.

A new hierarchically-structured n-dimensional covariant form of rotating equations of geophysical fluid dynamics

We introduce n-dimensional equations of geophysical fluid dynamics (GFD) valid on rotating n-dimensional manifolds. Moreover, by using straight and twisted differential forms and an auxiliary velocity field, we introduce hierarchically-structured equations of (geophysical) fluid dynamics in which the equations are split into metric-free and metric-dependent parts. For these sets of equations we provide representations in local coordinate charts and we show that they conserve potential vorticity and that Kelvin’s circulation theorem holds. As such general n-dimensional formulations do not exist in vector calculus, we provide for both covariant and vector-invariant equations a representation on a rotating coordinate frame in an Euclidean space and compare these representations. This study reveals, among others, that the prognostic variables, described by straight and twisted differential forms, are independent of both metric and orientation. This makes them perfect descriptors of the fluid’s quantities of interest, as they assign, analogously to physical measurement devices, real valued numbers to finite distances, areas, or volumes. This is not the case for prognostic variables described by (vector) proxy fields, as they depend on metric and orientation. The new structuring reveals also important geometrical features of the equations of GFD. For instance, the dimensioned differential forms display the geometric nature of the fluid’s characteristics, while the equations’ structuring illustrates how the metric-free momentum and continuity equations geometrically interact and how they are connected by the metric-dependent Hodge star and Riemannian lift. Besides this geometric insight, this structuring has also practical benefits: since equations containing only topological structure are less complicated to implement and can be integrated exactly, our results may contribute to more efficient and accurate discretizations.

The utility of Earth system Models of Intermediate Complexity (EMICs)

AbstractIntermediate‐complexity models are models which describe the dynamics of the atmosphere and/or ocean in less detail than conventional General Circulation Models (GCMs). At the same time, they go beyond the approach taken by atmospheric Energy Balance Models (EBMs) or ocean box models by using sophisticated parameterizations of the unresolved flow or by explicitly resolving the equations of geophysical fluid dynamics albeit at coarse spatial resolution. Being computationally fast, intermediate‐complexity models have the capability to treat slow climate variations. Hence, they often include components of the climate system that are associated with long‐term feedbacks like ice sheets, vegetation and biogeochemical cycles. Here again they differ from conventional GCM‐type models that feature only atmosphere and ocean/sea‐ice components. Many different approaches exist in building such a reduced model, resulting in a ‘spectrum of Earth system Models of Intermediate Complexity closing the gap between EBMs and GCMs’. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Climate Models and Modeling > Earth System Models

Lagrangian Analysis and Prediction of Coastal Ocean Dynamics

The basic governing equations of geophysical fluid dynamics have been known for many years, but how meaningful these equations are to our understanding of coastal and open oceans depends crucially on the accuracy of constitutive equations that relate the internal and external forces acting on these bodies of water. Field measurements, whether Eulerian or Lagrangian, have played and continue to play a central role in two important aspects of geophysical fluid dynamics. First, the data collected often point to the presence of various stable and coherent structures, eddies, or streams, whose study naturally becomes the focus of analytical and computational efforts. Second, the availability of good data serves to calibrate how the response of bodies of water to forces should be modeled. This book provides a substantial collection of well‐ written articles on how Lagrangian data are collected, analyzed, and eventually assimilated into models.

Turbulence of near-inertial waves in the continuously stratified fluid

Abstract By using the normal form of continuously stratified “primitive” equations of geophysical fluid dynamics with density (in the ocean), or potential temperature (in the atmosphere) playing the role of the vertical coordinate, we decouple vortex and wave motions in the system, introduce normal variables, and derive the effective Hamiltonian for waves with frequencies close to the inertial frequency (near-inertial waves, NIW). We then apply the weak turbulence approach to the random-phase ensembles of these waves. We show how the anisotropic scale-invariance of NIW may be exploited in order to obtain the stationary power-law spectra. The non-decay anisotropic scale-invariant dispersion laws of the NIW-type were not studied previously in the weak-turbulence literature.

Hadley and Rossby regimes in a simple model of convection of a rotating fluid

This paper analyzes the properties of solutions to the equations describing the motion of a stratified fluid in the class of velocity and temperature fields linear in coordinates. For an ideal fluid, these equations, on the one hand, are exact for the corresponding hydrodynamic problem and, on the other hand, are identical to the equations of motion for a heavy top. In a conservative case, the equations of motion of a top share common solutions with the equations of geophysical fluid dynamics and reproduce motions similar to those existing in the theory of the large-scale atmospheric circulation. This study considers the effects of viscosity and heat conduction in the fluid, which are, in a sense, similar to the effect of friction in the case of a top. The influence of deflections of the vectors of gravity and external rotation from their standard directions for a plane-parallel atmosphere is also considered. The regimes of motions that are described by the starting equations and approximations commonly used to model the atmospheric general circulation (the quasi-geostrophic approximation) are analyzed. It is shown that these equations correctly describe the Hadley and Rossby circulation regimes and transitions between them that are observed in numerical and laboratory experiments. Particular attention is given to the consistency between different regimes of the exact equations and their quasi-geostrophic approximations, which is manifested for small Rossby numbers and is generally absent for large Rossby numbers. The asymptotic behaviors of the curves of transition between the Hadley and Rossby regimes under the conditions of breaking the external symmetry of flows are obtained. These asymptotics explain the corresponding transition boundaries for the regimes observed in the known experiments in annuluses.

Mathematical Justification of the Hydrostatic Approximation in the Primitive Equations of Geophysical Fluid Dynamics

Geophysical fluids all exhibit a common feature: their aspect ratio (depth to hori- zontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanog- raphy, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier-Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.

Magnetohydrodynamic “Shallow Water” Equations for the Solar Tachocline

We argue that the classical shallow water equations of geophysical fluid dynamics should be useful for studying the global dynamics of the solar tachocline and demonstrate the existence of an MHD analog that would allow taking into account the strong toroidal magnetic field likely to be present there.

On the transformation of discontinuity waves from spatial to pressure coordinates for the primitive system of geophysical fluid dynamics

We give a complete analysis of the transformation of discontinuity waves from spatial coordinates to pressure coordinates for the primitive equations of geophysical fluid dynamics. These waves are moving surfaces in physical space across which some or all of the fluid variables are discontinuous, and therefore correspond to regions of steep gradient in solutions of related models which incorporate frictional forces or viscosity. Discontinuity waves consequently represent the most easily discernable and frequently the most interesting and important features of the flow. Our results show that the primitive equations in either coordinate system inherit from the underlying Euler equations all the physically correct shear waves; that the acoustic waves are completely eliminated in pressure coordinates but not in spatial coordinates; and that unphysical, anomolous waves occur in both coordinate systems.

Nonlinear inertia-gravity internal wave in stratified fluid

Based on the fundamental equations of geophysical fluid dynamics and on the consequence of vertical density stratification, travelling wave coordinates are used in this work to study the geometric topological structures of nonlinear permanent wave in phase plane. Rigorous mathematical mechanics demonstrate that the solution of permanent solitary wave does not exist. Hamilton functions and “action-angle” variables are used to express the travelling wave system in the simplest form and the analytic solution of the nonlinear inertia-gravity internal wave is obtained.

Some splitting methods for equations of geophysical fluid dynamics

In this paper, equations of atmospheric and oceanic dynamics are reduced to a kind of evolutionary equation in operator form, based on which a conclusion that the separability of motion stages is relative is made and an issue that the tractional splitting methods established on the physical separability of the fast stage and the slow stage neglect the interaction between the two stages to some extent is shown. Also, three splitting patterns are summed up from the splitting methods in common use so that a comparison between them is carried out. The comparison shows that only the improved splitting pattern (ISP) can be in second order and keep the interaction well. Finally, the applications of some splitting methods on numerical simulations of typhoon tracks made clear that ISP owns the best effect and can save more than 80% CPU time.