Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2. Pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy.A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Download Full-text

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Journal of Fluid Mechanics ◽

10.1017/s0022112087002209 ◽

1987 ◽

Vol 181 (-1) ◽

pp. 527 ◽

Cited By ~ 142

Author(s):

M. E. Mcintyre ◽

T. G. Shepherd

Keyword(s):

Shear Flows ◽

Hamiltonian Structure ◽

Finite Amplitude ◽

Local Conservation ◽

Stability Theorems ◽

Parallel Shear Flows ◽

Conservation Theorem

Download Full-text

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory

Journal of Fluid Mechanics ◽

10.1017/s0022112095002424 ◽

1995 ◽

Vol 290 ◽

pp. 67-104 ◽

Cited By ~ 10

Author(s):

Paul J. Kushner ◽

Theodore G. Shepherd

Keyword(s):

Conservation Laws ◽

Potential Vorticity ◽

Large Scale ◽

Equations Of Motion ◽

Lower Boundary ◽

Wave Activity ◽

Eddy Correlation ◽

Mean Flow ◽

Local Conservation ◽

Stability Theorems

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Download Full-text

Deformed Sine-Gordon Models, Solitons and Anomalous Charges

10.5772/intechopen.95432 ◽

2021 ◽

Author(s):

Harold Blas ◽

Hector F. Callisaya ◽

João P.R. Campos ◽

Bibiano M. Cerna ◽

Carlos Reyes

Keyword(s):

Conservation Laws ◽

Infinite Number ◽

Soliton Solutions ◽

Linear Formulation ◽

Reflection Symmetry ◽

Local Conservation ◽

Time Symmetry ◽

Non Local ◽

Sine Gordon Model ◽

Special Space

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential V q w = 64 q 2 tan 2 w 2 1 − sin w 2 q 2 , q ∈ R , which contains the usual SG potential V 2 w = 2 1 − cos 2 w .

Download Full-text

Numerical preservation of multiple local conservation laws

Applied Mathematics and Computation ◽

10.1016/j.amc.2021.126203 ◽

2021 ◽

Vol 403 ◽

pp. 126203

Author(s):

Gianluca Frasca-Caccia ◽

Peter E. Hydon

Keyword(s):

Conservation Laws ◽

Local Conservation

Download Full-text

Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽

10.3390/sym13061077 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1077

Author(s):

Yarema A. Prykarpatskyy

Keyword(s):

Conservation Laws ◽

Hamiltonian Structure ◽

Necessary Condition ◽

Type Representation ◽

Infinite Hierarchy ◽

Polynomial Coefficients ◽

Kdv Type Equations ◽

Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

Download Full-text