Abstract:
The Rossby wave instability in astrophysical disks is as a potentially important mechanism for driving angular momentum transport in disks. We aim to understand this instability in an approximate three-dimensional disk model environment which we assume to be a single homentropic annular layer we analyze using disk shallow-water theory. We consider the normal mode stability analysis of two kinds of radial profiles of the mean potential vorticity: The first type is a single step and the second kind is a symmetrical step of finite width describing either a localized depression or peak of the mean potential vorticity. For single potential vorticity steps we find there is no instability. There is no instability when the symmetric step is a localized peak. However, the Rossby wave instability occurs when the symmetrical step profile is a depression, which, in turn, corresponds to localized peaks in the mean enthalpy profile. This is in qualitative agreement with previous two-dimensional investigations of the instability. For all potential vorticity depressions, instability occurs for regions narrower than some maximum radial length scale. We interpret the instability as resulting from the interaction of at least two Rossby edgewaves. We identify the Rossby wave instability in the restricted three-dimensional framework of disk shallow water theory. Additional examinations of generalized barotropic flows are needed. Viewing disk vortical instabilities from the conceptual perspective of interacting edgewaves can be useful.

Abstract:
Turbulence and angular momentum transport in accretion disks remains a topic of debate. With the realization that dead zones are robust features of protoplanetary disks, the search for hydrodynamical sources of turbulence continues. A possible source is the baroclinic instability (BI), which has been shown to exist in unmagnetized non-barotropic disks. We present shearing box simulations of baroclinicly unstable, magnetized, 3D disks, in order to assess the interplay between the BI and other instabilities, namely the magneto-rotational instability (MRI) and the magneto-elliptical instability. We find that the vortices generated and sustained by the baroclinic instability in the purely hydrodynamical regime do not survive when magnetic fields are included. The MRI by far supersedes the BI in growth rate and strength at saturation. The resulting turbulence is virtually identical to an MRI-only scenario. We measured the intrinsic vorticity profile of the vortex, finding little radial variation in the vortex core. Nevertheless, the core is disrupted by an MHD instability, which we identify with the magneto-elliptic instability. This instability has nearly the same range of unstable wavelengths as the MRI, but has higher growth rates. In fact, we identify the MRI as a limiting case of the magneto-elliptic instability, when the vortex aspect ratio tends to infinity (pure shear flow). We conclude that vortex excitation and self-sustenance by the baroclinic instability in protoplanetary disks is viable only in low ionization, i.e., the dead zone. Our results are thus in accordance with the layered accretion paradigm. A baroclinicly unstable dead zone should be characterized by the presence of large-scale vortices whose cores are elliptically unstable, yet sustained by the baroclinic feedback. As magnetic fields destroy the vortices and the MRI outweighs the BI, the active layers are unmodified.

Abstract:
The three-dimensional nonlinear quasi-geostrophic potential vorticity equation is reduced to a linear form in the stream function in spherical coordinates for the permanent wave solutions consisting of zonal wavenumbers from 0 ton andr n vertical components with a given degreen. This equation is solved by treating the coefficient of the Coriolis parameter square in the equation as the eigenvalue both for sinusoidal and hyperbolic variations in vertical direction. It is found that these solutions can represent the observed long term flow patterns at the surface and aloft over the globe closely. In addition, the sinusoidal vertical solutions with large eigenvalueG are trapped in low latitude, and the scales of these trapped modes are longer than 10 deg. lat. even for the top layer of the ocean and hence they are much larger than that given by the equatorialΒ-plane solutions. Therefore such baroclinic disturbances in the ocean can easily interact with those in the atmosphere. Solutions of the shallow water potential vorticity equation are treated in a similar manner but with the effective depthH= RT/g taken as limited within a small range for the atmosphere. The propagation of the flow energy of the wave packet consisting of more than one degree is found to be along the great circle around the globe both for barotropic and for baroclinic flows in the atmosphere.

Abstract:
This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-(3+1)-dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding (2+1)-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given.

Abstract:
In the context of 1958-1997 NCFP NCAR re-analyses, the South Asia high (SAH) was divided into two components, barotropic and haroclinic, the former based on mass weighed vertical integration and the latter on the difference between the measured circulation and the barotropic component counterpart, whereupon the barotropic and baroclinic circulation conversion features were addressed of the research SAH during its seasonal variation. Evidence suggests that i) in summer (winter). the SAH is a thermal (dynamical) system. with dominant baroclinicity (barotropicity). either of the components accounting for approximately 70% of the total contribution; ii) as time progresses from winter to summer, accompanied by the barotropic SAH evolving into its baroclinic analog, the SAH is moving under the" thermal guidance" of its baroclinic component circulation, suggesting that the component circulation precedes the system itself in variation. iii) the reversal happens when it goes from summer to winter, with the SAH displacement under the" dynamic steering" of its barotropic component circulation.

Abstract:
In strictly speaking, all the natural phenomena on the earth should be treated under rotating coordinate. The existence of baroclinic nonequivalent barotropic laminar solution for rotating fluids is still open though the laminar solutions for the irrotational fluid had been well studied. In this letter, all the possible equivalent barotropic (EB) laminar solution are firstly explored and all the possible baroclinic non-EB elliptic circulations and hyperbolic laminar modes are discovered. The baroclinic EB circulations (including the vortex streets and hurricane like vortices) possess rich structures because either the arbitrary solutions of arbitrary nonlinear Poison equations can be used or an arbitrary two-dimensional stream function is revealed. The discovery of the baroclinic non-EB modes disproves a known conjecture. The results may be broadly applied in atmospheric and oceanic dynamics, plasma physics, astrophysics and so on.

Abstract:
Lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation. The Lie symmetries of the barotropic vorticity equations on the $f$- and $\beta$-planes, as well as on the sphere in rotating and rest reference frames, are determined. A symmetry background for reducing the rotating reference frame to the rest frame is presented. The one- and two-dimensional inequivalent subalgebras of the Lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations. This provides large classes of exact solutions, which include both Rossby and Rossby--Haurwitz waves as special cases. We also discuss the possibility of partial invariance for the $\beta$-plane equation, thereby further extending the family of its exact solutions. This is done in a more systematic and complete way than previously available in literature.

Abstract:
The complete point symmetry group of the barotropic vorticity equation on the $\beta$-plane is computed using the direct method supplemented with two different techniques. The first technique is based on the preservation of any megaideal of the maximal Lie invariance algebra of a differential equation by the push-forwards of point symmetries of the same equation. The second technique involves a priori knowledge on normalization properties of a class of differential equations containing the equation under consideration. Both of these techniques are briefly outlined.

Abstract:
Some theoretical concepts about the frictionless dynamics of propagation of the barotropic tide over two-dimensional continental slopes of arbitrary shape are developed. A numerical procedure which generalizes the exact solution obtained over a rectilinear sloping topography is given. This technique can be applied to compute the harmonic components of the barotropic tide everywhere over sloping bottom contours of any shape. It permits in particular the avoidance of discontinuities at the boundaries of rectilinear-continental-slope profiles. The barotropic tidal results are used afterwards to calculate the barotropic forcing for the generation of internal tides. Numerical experiments are performed to study the interaction between the tide and some typical sloping topographies. A three-layered model is used for this purpose. Results are compared with those previously obtained over a rectilinear continental slope.

Abstract:
Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, $F$ and $\beta$, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case $F\ne 0$ there exists a well-defined point transformation to set $\beta = 0$. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination $F\ne 0$ and $\beta = 0$. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case $\beta \ne 0$.