Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.

Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection

It is known that the empirical orthogonal function method is unable to detect possible nonlinear structure in climate data. Here, isometric feature mapping (Isomap), as a tool for nonlinear dimensionality reduction, is applied to 1958–2001 ERA-40 sea-level pressure anomalies to study nonlinearity of the Asian summer monsoon intraseasonal variability. Using the leading two Isomap time series, the probability density function is shown to be bimodal. A two-dimensional bivariate Gaussian mixture model is then applied to identify the monsoon phases, the obtained regimes representing enhanced and suppressed phases, respectively. The relationship with the large-scale seasonal mean monsoon indicates that the frequency of monsoon regime occurrence is significantly perturbed in agreement with conceptual ideas, with preference for enhanced convection on intraseasonal time scales during large-scale strong monsoons. Trend analysis suggests a shift in concentration of monsoon convection, with less emphasis on South Asia and more on the East China Sea.

The troposphere-to-stratosphere transition in kinetic energy spectra and nonlinear spectral fluxes as seen in ECMWF analyses

Global horizontal wavenumber kinetic energy spectra and spectral fluxes of rotational kinetic energy and enstrophy are computed for a range of vertical levels using a T799 ECMWF operational analysis. Above 250 hPa, the kinetic energy spectra exhibit a distinct break between steep and shallow spectral ranges, reminiscent of dual power-law spectra seen in aircraft data and high-resolution general circulation models. The break separates a large-scale ‘‘balanced’’ regime in which rotational flow strongly dominates divergent flow and a mesoscale ‘‘unbalanced’’ regime where divergent energy is comparable to or larger than rotational energy. Between 230 and 100 hPa, the spectral break shifts to larger scales (from n 5 60 to n 5 20, where n is spherical harmonic index) as the balanced component of the flow preferentially decays. The location of the break remains fairly stable throughout the stratosphere. The spectral break in the analysis occurs at somewhat larger scales than the break seen in aircraft data. Nonlinear spectral fluxes defined for the rotational component of the flow maximize between about 300 and 200 hPa. Large-scale turbulence thus centers on the extratropical tropopause region, within which there are two distinct mechanisms of upscale energy transfer: eddy–eddy interactions sourcing the transient energy peak in synoptic scales, and zonal mean–eddy interactions forcing the zonal flow. A well-defined downscale enstrophy flux is clearly evident at these altitudes. In the stratosphere, the transient energy peak moves to planetary scales and zonal mean–eddy interactions become dominant.

Extensivity of two-dimensional turbulence

This study is concerned with how the attractor dimension of the two-dimensional Navier–Stokes equations depends on characteristic length scales, including the system integral length scale, the forcing length scale, and the dissipation length scale. Upper bounds on the attractor dimension derived by Constantin, Foias and Temam are analysed. It is shown that the optimal attractor-dimension estimate grows linearly with the domain area (suggestive of extensive chaos), for a sufficiently large domain, if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed. For sufficiently small domain area, a slightly “super-extensive” estimate becomes optimal. In the extensive regime, the attractor-dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined, on physical grounds, in terms of the average rate of shear. This dissipation length scale (which is not necessarily the scale at which the energy or enstrophy dissipation takes place) can be identified with the dimension correlation length scale, the square of which is interpreted, according to the concept of extensive chaos, as the area of a subsystem with one degree of freedom. Furthermore, these length scales can be identified with a “minimum length scale” of the flow, which is rigorously deduced from the concept of determining nodes.

The correlation of ULF waves and auroral intensity before, during and after substorm expansion phase onset

We present case studies of the evolution of magnetic wave amplitudes and auroral intensity through the late growth phase and the expansion phase of the substorm cycle. We present strong evidence that substorm-related auroral enhancements are clearly and demonstrably linked to ULF wave amplitudes observed at the same location. In most cases, we find that the highest correlations are observed when the magnetometer time series is advanced in time, indicating that the ULF wave amplitudes start to grow before measured auroral intensities, though interestingly this is not always the case. Further we discuss the four possible reasons that may be able to explain both the timing and the high correlations between these two phenomena, including: a simple coincidence, an artifact of instrumental effects, the response of the ionosphere to magnetic waves and auroral particle precipitation, and finally that ULF waves and auroral particle precipitation are physically linked. We discount coincidence and instrumental effects since in the studies presented here they are unlikely or in general will contribute negligible effects, and we find that the ionospheric response to waves and precipitation can explain some, but not all of the results contained within this paper. Specifically, ionospheric response to substorm waves and auroral precipitation cannot explain that the result that previous studies have shown, that onset of ULF wave activity and the onset of auroral particle precipitation occur at the same time and in the same location. This leaves the possibility that ULF waves and auroral particles are physically linked.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Lateral boundary contributions to wave-activity invariants and nonlinear stability theorems for balanced dynamics

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear. This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models. Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

On nonlinear symmetric stability and the nonlinear saturation of symmetric instability

A nonlinear symmetric stability theorem is derived in the context of the f-plane Boussinesq equations, recovering an earlier result of Xu within a more general framework. The theorem applies to symmetric disturbances to a baroclinic basic flow, the disturbances having arbitrary structure and magnitude. The criteria for nonlinear stability are virtually identical to those for linear stability. As in Xu, the nonlinear stability theorem can be used to obtain rigorous upper bounds on the saturation amplitude of symmetric instabilities. In a simple example, the bounds are found to compare favorably with heuristic parcel-based estimates in both the hydrostatic and non-hydrostatic limits.

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Nonlinear stability of continuously stratified quasi-geostrophic flow

Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-geostrophic flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-geostrophic model and the modified quasi-geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

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.

Nonlinear stability of Eady's model

A nonlinear stability theorem is established for Eady's model of baroclinic flow. In particular, the Eady basic state is shown to be nonlinearly stable (for arbitrary shear) provided (Δz)/(Δy) > 2(5)^1/2f/(πN),where Δz is the height of the domain, Δy the channel width, f the Coriolis parameter, and N the buoyancy frequency. When this criterion is satisfied, explicit bounds can be derived on the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids, which are expressed in terms of the initial disturbance fields. The disturbances are completely general (with nonzero potential vorticity) and are not assumed to be of small amplitude. The results may be regarded as an extension of Arnol'd's second nonlinear stability theorem to continuously stratified quasigeostrophic baroclinic flow.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

Nonlinear saturation of baroclinic instability: part III: bounds on the energy

Rigorous upper bounds are derived on the saturation amplitude of baroclinic instability in the two-layer model. The bounds apply to the eddy energy and are obtained by appealing to a finite amplitude conservation law for the disturbance pseudoenergy. These bounds are to be distinguished from those derived in Part I of this study, which employed a pseudomomentum conservation law and provided bounds on the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity. Bounds on the eddy energy are worked out for a general class of unstable westerly jets. In the special case of the Phillips model of baroclinic instability, and in the limit of infinitesimal initial eddy amplitude, the bound states that the eddy energy cannot exceed ϵβ2/6F where ϵ = (U − Ucrit)/Ucrit is the relative supercriticality. This bound captures the essential dynamical scalings (i.e., the dependence on ϵ, β, and F) of the saturation amplitudes predicted by weakly nonlinear theory, as well as exhibiting remarkable quantitative agreement with those predictions, and is also consistent with heuristic baroclinic adjustment estimates.