Generalized linear stability theory is used to calculate the optimal initial conditions that result in transient amplification of the thermohaline circulation (THC) in a zonally-averaged single basin ocean model. The eigenmodes of the tangent linear model verify that the system is asymptotically stable but the nonnormality of the system permits the growth of perturbations for a finite period through the interference of nonorthogonal eigenmodes. It is found that the maximum amplification of the THC anomalies occurs after 6 years with both the thermally driven and salinity driven components playing major roles in the amplification process. The transient amplification of THC anomalies is due to the constructive and destructive interference of a large number of eigenmodes and the evolution over time is determined by how the interference pattern evolves. It is found that five of the most highly nonnormal eigenmodes are critical to the initial cancellation of the salinity and temperature contributions to the THC while 11 oscillating modes with decay timescales ranging from 2 to 6 years are the major contributors at the time of maximum amplification. This analysis demonstrates that the different dynamics of salinity and temperature anomalies allows the dramatic growth of perturbations to the THC on relatively short (interannual to decadal) timescales. In addition the ideas of generalized stability theory are used to calculate the stochastic optimals which are the spatial patterns of stochastic forcing that are most efficient at generating variance growth in the THC. It is found that the optimal stochastic forcing occurs at high latitudes and induces low-frequency THC variability by exciting the salinity-dominated modes of the THC. The first stochastic optimal is found to have its largest projection on the same five highly nonnormal eigenmodes found to be critical to the structure of the optimal initial conditions. The model’s response to stochastic forcing is not controlled by the least damped eigenmodes of the tangent linear model but rather by the linear interference of these highly nonnormal eigenmodes. The process of pseudoresonance suggests that the nonnormal eigenmodes are excited and sustained by stochastically induced perturbations which in turn lead to maximum THC variance. Finally, it was shown that the addition of wind stress did not have a large impact on the nonnormal dynamics of the linearised system. Adding wind allowed the value of the vertical diffusivity to be reduced to achieve the same maximum linearised THC amplitude as was used in the case with no wind stress.

Generalized linear stability theory is used to calculate the optimal initial conditions that result in transient amplification of the thermohaline circulation (THC) in a zonally-averaged single basin ocean model. The eigenmodes of the tangent linear model verify that the system is asymptotically stable but the nonnormality of the system permits the growth of perturbations for a finite period through the interference of nonorthogonal eigenmodes. It is found that the maximum amplification of the THC anomalies occurs after 6 years with both the thermally driven and salinity driven components playing major roles in the amplification process. The transient amplification of THC anomalies is due to the constructive and destructive interference of a large number of eigenmodes and the evolution over time is determined by how the interference pattern evolves. It is found that five of the most highly nonnormal eigenmodes are critical to the initial cancellation of the salinity and temperature contributions to the THC while 11 oscillating modes with decay timescales ranging from 2 to 6 years are the major contributors at the time of maximum amplification. This analysis demonstrates that the different dynamics of salinity and temperature anomalies allows the dramatic growth of perturbations to the THC on relatively short (interannual to decadal) timescales. In addition the ideas of generalized stability theory are used to calculate the stochastic optimals which are the spatial patterns of stochastic forcing that are most efficient at generating variance growth in the THC. It is found that the optimal stochastic forcing occurs at high latitudes and induces low-frequency THC variability by exciting the salinity-dominated modes of the THC. The first stochastic optimal is found to have its largest projection on the same five highly nonnormal eigenmodes found to be critical to the structure of the optimal initial conditions. The model’s response to stochastic forcing is not controlled by the least damped eigenmodes of the tangent linear model but rather by the linear interference of these highly nonnormal eigenmodes. The process of pseudoresonance suggests that the nonnormal eigenmodes are excited and sustained by stochastically induced perturbations which in turn lead to maximum THC variance. Finally, it was shown that the addition of wind stress did not have a large impact on the nonnormal dynamics of the linearised system. Adding wind allowed the value of the vertical diffusivity to be reduced to achieve the same maximum linearised THC amplitude as was used in the case with no wind stress.

Wird der Äquator-Pol-Energietransfer als Wärmediffusion berücksichtigt, so gehen Energiebilanzmodelle in Reaktions-Diffusionsgleichungen über, deren Modellfall die (deterministische) Chafee-Infante-Gleichung darstellt. Ihre Lösung besitzt zwei stabile Zustände und mehrere instabile auf der separierenden Mannigfaltigkeit (Separatrix) der stabilen Anziehungsgebiete. Es wird bewiesen, dass die Lösung auf geeignet verkleinerten Anziehungsgebieten mit Minimalabstand zur Separatrix innerhalb von Zeitskalen relaxiert, die höchstens logarithmisch darin anwachsen. Motiviert durch statistische Belege aus grönländischen Zeitreihen wird diese partielle Differentialgleichung unter Störung mit unendlichdimensionalem, Hilbertraum-wertigen, regulär variierenden Lévy''schen reinen Sprungrauschen mit index alpha und Intensität epsilon untersucht. Ein kanonisches Beispiel dieses Rauschens ist alpha-stabiles Rauschen im Hilbertraum. Durch Erweiterung einer Methode von Imkeller und Pavlyukevich auf stochastische partielle Differentialgleichungen wird unter milden Bedingungen bewiesen, dass im Gegensatz zu Gauß''schem Rauschen die erwarteten Austritts- und übertrittszeiten zwischen Anziehungsgebieten polynomiell mit Ordnung in der inversen Intensität für kleine Rauschintensität anwachsen. In Kapitel 6 wird eine zusätzliche natürliche “Separatrixhypothese” über das Sprungmaß, eingeführt, die eine obere Schranke für die Austrittszeiten aus einer Umgebung der Separatrix impliziert. Dies ermöglicht den Nachweis einer oberen Schranke für die Austrittszeiten, welche gleichmäßig für Anfangsbedingungen in dem ganzen Anziehungsgebiet gilt. Es folgen zwei Lokalisierungsergebnisse. Schließlich wird gezeigt, dass die Lösung metastabiles Verhalten aufweist. Unter der “Separatrixhypothese” wird dies auf ein Ergebnis erweitert, welches gleichmäßig im Raum gilt. / If equator-to-pole energy transfer by heat diffusion is taken into account, Energy Balance Models turn into reaction-diffusion equations, whose prototype is the (deterministic) Chafee-Infante equation. Its solution has two stable states and several unstable ones on the separating manifold (separatrix) of the stable domains of attraction. We show, that on appropriately reduced domains of attraction of a minimal distance to the separatrix the solution relaxes in time scales increasing only logarithmically in it. Motivated by the statistical evidence from Greenland ice core time series, we consider this partial differential equation perturbed by an infinite-dimensional Hilbert space-valued regularly varying (pure jump) Lévy noise of index alpha and intensity epsilon. A proto-type of this noise is alpha-stable noise in the Hilbert space. Extending a method developed by Imkeller and Pavlyukevich to the SPDE setting we prove under mild conditions that in contrast to Gaussian perturbations the expected exit and transition times between the domains of attraction increase polynomially in the inverse intensity. In Chapter 6 we introduce an additional natural separatrix hypothesis on the jump measure that implies an upper bound on the exit time of a neighborhood of the separatrix. This allows to obtain an upper bound for the asymptotic exit time uniform for the initial positions inside the entire domain of attraction. It is followed by two localization results. Finally we prove that the solution exhibits metastable behavior. Under the separatrix hypothesis we can extend this to a result that holds uniformly in space.