The variable link between PNA and NAO in observations and in multi-century CGCM simulations

The link between the Pacific/North American pattern (PNA) and the North Atlantic Oscillation (NAO) is investigated in reanalysis data (NCEP, ERA40) and multi-century CGCM runs for present day climate using three versions of the ECHAM model. PNA and NAO patterns and indices are determined via rotated principal component analysis on monthly mean 500 hPa geopotential height fields using the varimax criteria. On average, the multi-century CGCM simulations show a significant anti-correlation between PNA and NAO. Further, multi-decadal periods with significantly enhanced (high anti-correlation, active phase) or weakened (low correlations, inactive phase) coupling are found in all CGCMs. In the simulated active phases, the storm track activity near Newfoundland has a stronger link with the PNA variability than during the inactive phases. On average, the reanalysis datasets show no significant anti-correlation between PNA and NAO indices, but during the sub-period 1973–1994 a significant anti-correlation is detected, suggesting that the present climate could correspond to an inactive period as detected in the CGCMs. An analysis of possible physical mechanisms suggests that the link between the patterns is established by the baroclinic waves forming the North Atlantic storm track. The geopotential height anomalies associated with negative PNA phases induce an increased advection of warm and moist air from the Gulf of Mexico and cold air from Canada. Both types of advection contribute to increase baroclinicity over eastern North America and also to increase the low level latent heat content of the warm air masses. Thus, growth conditions for eddies at the entrance of the North Atlantic storm track are enhanced. Considering the average temporal development during winter for the CGCM, results show an enhanced Newfoundland storm track maximum in the early winter for negative PNA, followed by a downstream enhancement of the Atlantic storm track in the subsequent months. In active (passive) phases, this seasonal development is enhanced (suppressed). As the storm track over the central and eastern Atlantic is closely related to the NAO variability, this development can be explained by the shift of the NAO index to more positive values.

European storminess and associated circulation weather types: future changes deduced from a multi-model ensemble of GCM simulations

A range of possible changes in the frequency and characteristics of European wind storms under future climate conditions was investigated on the basis of a multi-model ensemble of 9 coupled global climate model (GCM) simulations for the 20th and 21st centuries following the IPCC SRES A1B scenario. A multi-model approach allowed an estimation of the (un)certainties of the climate change signals. General changes in large-scale atmospheric flow were analysed, the occurrence of wind storms was quantified, and atmospheric features associated with wind storm events were considered. Identified storm days were investigated according to atmospheric circulation, associated pressure patterns, cyclone tracks and wind speed patterns. Validation against reanalysis data revealed that the GCMs are in general capable of realistically reproducing characteristics of European circulation weather types (CWTs) and wind storms. Results are given with respect to frequency of occurrence, storm-associated flow conditions, cyclone tracks and specific wind speed patterns. Under anthropogenic climate change conditions (SRES A1B scenario), increased frequency of westerly flow during winter is detected over the central European investigation area. In the ensemble mean, the number of detected wind storm days increases between 19 and 33% for 2 different measures of storminess, only 1 GCM revealed less storm days. The increased number of storm days detected in most models is disproportionately high compared to the related CWT changes. The mean intensity of cyclones associated with storm days in the ensemble mean increases by about 10 (±10)% in the Eastern Atlantic, near the British Isles and in the North Sea. Accordingly, wind speeds associated with storm events increase significantly by about 5 (±5)% over large parts of central Europe, mainly on days with westerly flow. The basic conclusions of this work remain valid if different ensemble contructions are considered, leaving out an outlier model or including multiple runs of one particular model.

Statistical uncertainty of changes in winter storms over the North Atlantic and Europe in an ensemble of transient climate simulations

Winter storms are among the most important natural hazards affecting Europe. We quantify changes in storm frequency and intensity over the North Atlantic and Europe under future climate scenarios in terms of return periods (RPs) considering uncertainties due to both sampling and methodology. RPs of North Atlantic storms' minimum central pressure (CP) and maximum vorticity (VOR) remain unchanged by 2100 for both the A1B and A2 scenarios compared to the present climate. Whereas shortened RPs for VOR of all intensities are detected for the area between British Isles/North-Sea/western Europe as early as 2040. However, the changes in storm VOR RP may be unrealistically large: a present day 50 (20) year event becomes approximately a 9 (5.5) year event in both A1B and A2 scenarios by 2100. The detected shortened RPs of storms implies a higher risk of occurrence of damaging wind events over Europe.

Changing Northern Hemisphere storm tracks in an ensemble of IPCC climate change simulations

Winter storm-track activity over the Northern Hemisphere and its changes in a greenhouse gas scenario (the Special Report on Emission Scenarios A1B forcing) are computed from an ensemble of 23 single runs from 16 coupled global climate models (CGCMs). All models reproduce the general structures of the observed climatological storm-track pattern under present-day forcing conditions. Ensemble mean changes resulting from anthropogenic forcing include an increase of baroclinic wave activity over the eastern North Atlantic, amounting to 5%–8% by the end of the twenty-first century. Enhanced activity is also found over the Asian continent and over the North Pacific near the Aleutian Islands. At high latitudes and over parts of the subtropics, activity is reduced. Variations of the individual models around the ensemble average signal are not small, with a median of the pattern correlation near r = 0.5. There is, however, no evidence for a link between deviations in present-day climatology and deviations with respect to climate change.

Changing European storm loss potentials under modified climate conditions according to ensemble simulations of the ECHAM5/MPI-OM1 GMC

A simple storm loss model is applied to an ensemble of ECHAM5/MPI-OM1 GCM simulations in order to estimate changes of insured loss potentials over Europe in the 21st century. Losses are computed based on the daily maximum wind speed for each grid point. The calibration of the loss model is performed using wind data from the ERA40-Reanalysis and German loss data. The obtained annual losses for the present climate conditions (20C, three realisations) reproduce the statistical features of the historical insurance loss data for Germany. The climate change experiments correspond to the SRES-Scenarios A1B and A2, and for each of them three realisations are considered. On average, insured loss potentials increase for all analysed European regions at the end of the 21st century. Changes are largest for Germany and France, and lowest for Portugal/Spain. Additionally, the spread between the single realisations is large, ranging e.g. for Germany from −4% to +43% in terms of mean annual loss. Moreover, almost all simulations show an increasing interannual variability of storm damage. This assessment is even more pronounced if no adaptation of building structure to climate change is considered. The increased loss potentials are linked with enhanced values for the high percentiles of surface wind maxima over Western and Central Europe, which in turn are associated with an enhanced number and increased intensity of extreme cyclones over the British Isles and the North Sea.

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.

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.

Nonlinear saturation of baroclinic instability: part II: continuously stratified fluid

Rigorous upper bounds are derived that limit the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow in a continuously stratified, quasi-geostrophic, semi-infinite fluid. Bounds are obtained bath on the depth-integrated eddy potential enstrophy and on the eddy available potential energy (APE) at the ground. The method used to derive the bounds is essentially analogous to that used in Part I of this study for the two-layer model: it relies on the existence of a nonlinear Liapunov (normed) stability theorem, which is a finite-amplitude generalization of the Charney-Stern theorem. As in Part I, the bounds are valid both for conservative (unforced, inviscid) flow, as well as for forced-dissipative flow when the dissipation is proportional to the potential vorticity in the interior, and to the potential temperature at the ground. The character of the results depends on the dimensionless external parameter γ = f02ξ/β0N2H, where ξ is the maximum vertical shear of the zonal wind, H is the density scale height, and the other symbols have their usual meaning. When γ ≫ 1, corresponding to “deep” unstable modes (vertical scale ≈H), the bound on the eddy potential enstrophy is just the total potential enstrophy in the system; but when γ≪1, corresponding to ‘shallow’ unstable modes (vertical scale ≈γH), the eddy potential enstrophy can be bounded well below the total amount available in the system. In neither case can the bound on the eddy APE prevent a complete neutralization of the surface temperature gradient which is in accord with numerical experience. For the special case of the Charney model of baroclinic instability, and in the limit of infinitesimal initial eddy disturbance amplitude, the bound states that the dimensionless eddy potential enstrophy cannot exceed (γ + 1)2/24&gamma2h when γ ≥ 1, or 1/6;&gammah when γ ≤ 1; here h = HN/f0L is the dimensionless scale height and L is the width of the channel. These bounds are very similar to (though of course generally larger than) ad hoc estimates based on baroclinic-adjustment arguments. The possibility of using these kinds of bounds for eddy-amplitude closure in a transient-eddy parameterization scheme is also discussed.