The dynamics of a low-order coupled ocean-atmosphere model

A system of ve ordinary dierential equations is studied which combines the Lorenz model for the atmosphere and a box model for the ocean The behaviour of this system is studied as a function of the coupling parameters For most parameter values the dynamics of the atmosphere model is dominant Stable equilibria are found as well as periodic solutions and chaotic attractors For a range of parameter values competing attractors exist The KaplanYorke dimension and the correlation dimension of the chaotic attractor are numerically calculated and compared to the values found in the uncoupled Lorenz model The correlation dimension diers much less than te KaplanYorke dimension indicating that there is little variability in the ocean model In the transition from periodic behaviour to chaos intermittency is observed This is explained by means of bifurcation analysis The intermittent behaviour occurs near a NeimarkSacker bifurcation at which a periodic solution loses its stability The average length of a periodic interval in the intermittent regime l is studied as a function of the bifurcation parameter Near the bifurcation point it shows a power law scaling It diverges as l where and is the distance from the bifurcation point in reasonable agreement with the results of Pomeau and Manneville Commun Math Phys The intermittent behaviour persists beyond the point where the unstable periodic solution disappears in a saddle node bifurcation The length of the periodic intervals is governed by the time scale of the ocean component Thus in this regime the ocean model has a considerable inuence on the dynamics of the coupled syste

Dynamics of singular vectors in the semi-infinite eady model: nonzero ß but zero mean PV gradient

A nonmodal approach based on the potential vorticity (PV) perspective is used to compute the singular vector (SV) that optimizes the growth of kinetic energy at the surface for the β-plane Eady model without an upper rigid lid. The basic-state buoyancy frequency and zonal wind profile are chosen such that the basic-state PV gradient is zero. If the f-plane approximation is made, the SV growth at the surface is dominated by resonance, resulting from the advection of basic-state potential temperature (PT) by the interior PV anomalies. This resonance generates a PT anomaly at the surface. The PV unshielding and PV-PT unshielding contribute less to the final kinetic energy at the surface. The general conclusion of the present paper is that surface cyclogenesis (of the 48-h SV) is stronger if p is included. Three cases have been considered. In the first case, the vertical shear of the basic state is modified in order to retain the zero basic-state PV gradient. The increased shear enhances SV growth significantly first because of a lowering of the resonant level (enhanced resonance), and second because of a more rapid PV unshielding process. Resonance is the most important contribution at optimization time. In the second case, the buoyancy frequency of the basic state is modified. The surface cyclogenesis is stronger than in the absence of β but less strong than if the shear is modified. It is shown that the effect of the modified buoyancy frequency profile is that PV unshielding occurs more efficiently. The contribution from resonance to the SV growth remains almost the same. Finally, the SV is calculated for a more realistic buoyancy frequency profile based on observations. In this experiment the increased value of the surface buoyancy frequency reduces the SV growth significantly as compared to the case in which the surface buoyancy frequency takes a standard value. All growth mechanisms are affected by this change in the surface buoyancy frequency

Interpretation of discrete and continuum modes in a two-layer Eady model

The upper rigid lid of the conventional Eady model for baroclinic instability is replaced by a more realistic stratosphere with an increased buoyancy frequency and a different shear of the zonal wind. Previously reported results of a normal-mode stability analysis are re-interpreted using the concept of interacting surface and tropopause PV anomalies, called PV building blocks (PVBs). In this perspective, which directly relates to the counter-propagating Rossby wave formalism, the appearance of both the short-wave and the long-wave cut-off becomes physically transparent. New results include a discussion of the continuum modes in terms of interacting PVBs. Continuum modes are modal solutions to the inviscid linearised equations, specified by non-zero PV at one interior level (as well as non-zero PV at the surface and the tropopause). If the stratospheric zonal wind decreases with height, the continuum modes cause resonances at multiple (even stratospheric) levels. These resonant continuum modes may play an important role in the explanation of disturbance growth from initial conditions in which the discrete normal modes are neutral

Resonance in optimal perturbation evolution. Part I. Two-layer eady model

A detailed investigation has been performed of the role of the different growth mechanisms (resonance, potential vorticity unshielding, and normal-mode baroclinic instability) in the evolution of optimal perturbations constructed for a two-layer Eady model and a kinetic energy norm. The two-layer Eady model is obtained by replacing the conventional upper rigid lid by a simple but realistic stratosphere. To make an unambiguous discussion possible, generally applicable techniques have been developed. At the heart of these techniques lies a description of the linear dynamics in terms of a variable number of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically localized sheets of potential vorticity. If the optimal perturbation is composed of only one PVB, the rapid surface cyclogenesis can be attributed to the growth of the surface PVB (the edge wave), which is excited by the tropospheric PVB via a linear resonance effect. If the optimal perturbation is constructed using multiple PVBs, this simple picture is modified only in the sense that PV unshielding dominates the surface amplification for a short time after initialization. The unshielding mechanism rapidly creates large streamfunction values at the surface, as a result of which the resonance effect is much stronger. A similar resonance effect between the tropospheric PVBs and the tropopause PVB acts negatively on the surface streamfunction amplification. The influence of the stratosphere to the surface development is negligible. In all cases reported here, the growth due to traditional normal-mode baroclinic instability contributes either negative or only little to the surface development up to the optimization time of two days. It takes at least four days for the flow to become fully dominated by normal-mode growth, thereby confirming that finite-time optimal perturbation growth differs in many aspects fundamentally from asymptotic normalmode baroclinic instability

Resonance in optimal perturbation evolution. Part II. Effects of a nonzero mean PV gradient

Optimal perturbations are constructed for a two-layer -plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without . The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution. The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the f plane only the PVB at the surface and at the tropopause can be amplified. In the presence of , however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer f-plane model, the inclusion of slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface. If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time

The dynamics of a low-order coupled ocean-atmosphere model

A system of ve ordinary dierential equations is studied which combines the Lorenz model for the atmosphere and a box model for the ocean The behaviour of this system is studied as a function of the coupling parameters For most parameter values the dynamics of the atmosphere model is dominant Stable equilibria are found as well as periodic solutions and chaotic attractors For a range of parameter values competing attractors exist The KaplanYorke dimension and the correlation dimension of the chaotic attractor are numerically calculated and compared to the values found in the uncoupled Lorenz model The correlation dimension diers much less than te KaplanYorke dimension indicating that there is little variability in the ocean model In the transition from periodic behaviour to chaos intermittency is observed This is explained by means of bifurcation analysis The intermittent behaviour occurs near a NeimarkSacker bifurcation at which a periodic solution loses its stability The average length of a periodic interval in the intermittent regime l is studied as a function of the bifurcation parameter Near the bifurcation point it shows a power law scaling It diverges as l where and is the distance from the bifurcation point in reasonable agreement with the results of Pomeau and Manneville Commun Math Phys The intermittent behaviour persists beyond the point where the unstable periodic solution disappears in a saddle node bifurcation The length of the periodic intervals is governed by the time scale of the ocean component Thus in this regime the ocean model has a considerable inuence on the dynamics of the coupled syste

Stratospheric impact on the troposphere deduced from potential vorticity inversion in relation to the Arctic Oscillation

The zonal mean state of the atmosphere in the Northern Hemisphere in winter is determined by the temperature at the Earth’s surface and by two potential vorticity (PV) anomalies (defined as that part of the PV field that induces a wind field) centred over the North Pole: one in the upper troposphere/lower stratosphere (UTLS), extending to the Subtropics, and the other over the polar cap in the lower to middle stratosphere. Isentropic PV inversion demonstrates that the UTLS PV anomaly induces themain part of the zonalmean wind in the troposphere, including the subtropical jet stream, while the stratospheric PV anomaly induces the polar night stratospheric jet. The stratospheric PV anomaly has a greater amplitude and extends further downwards if the Arctic Oscillation (AO) index is positive. Also, the UTLS PV anomaly has a slightly larger amplitude if the AO index is positive, but the meridional PV gradient in the Subtropics that is associated with this anomaly is greatest when the AO index is negative, resulting in a stronger subtropical jet when the AO index is negative. PV inversion translates the UTLS PV anomaly into a wind anomaly and a static stability anomaly. The resulting differences in the vertical wind shear and in the Brunt–V¨ais¨al¨a frequency between the two AO phases show a larger baroclinicity in the extratropics when the AO index is positive. This explains why more extratropical cyclones are observed when the AO index is positiv

Interpretation of discrete and continuum modes in a two-layer Eady model

The upper rigid lid of the conventional Eady model for baroclinic instability is replaced by a more realistic stratosphere with an increased buoyancy frequency and a different shear of the zonal wind. Previously reported results of a normal-mode stability analysis are re-interpreted using the concept of interacting surface and tropopause PV anomalies, called PV building blocks (PVBs). In this perspective, which directly relates to the counter-propagating Rossby wave formalism, the appearance of both the short-wave and the long-wave cut-off becomes physically transparent. New results include a discussion of the continuum modes in terms of interacting PVBs. Continuum modes are modal solutions to the inviscid linearised equations, specified by non-zero PV at one interior level (as well as non-zero PV at the surface and the tropopause). If the stratospheric zonal wind decreases with height, the continuum modes cause resonances at multiple (even stratospheric) levels. These resonant continuum modes may play an important role in the explanation of disturbance growth from initial conditions in which the discrete normal modes are neutral

Chaos en Orde

Verschillende wetenschapsgebieden kunnen veel van elkaar leren. Zo af en toe steken ergens een omvattende theorie, onderzoeksthema, model of concept de kop op, die met succes toegepast kunnen worden op andere terreinen. Dan vindt er een vruchtbare kruisbestuiving plaats tussen onderzoeksgebieden die op zich weinig met elkaar gemeen hebben. De chaostheorie is een voorbeeld van een theorie die uitnodigt tot een blik over de eigen disciplinegrenzen heen. In eerste instantie is de chaostheorie ontwikkeld binnen de exacte wetenschappen om niet-lineaire dynamische systemen te beschrijven en te analyseren. Daarna werden de concepten en modellen van de theorie in diverse zeer uiteenlopende wetenschapsterreinen gehanteerd