The Eady problem is modified to allow for a non-zero ß parameter but retaining a zero meridional gradient of potential vorticity. Two different basic states are examined for which analytical solutions of the linearized quasi-geostrophic potential vorticity equation were obtained. As has to be expected in addition to the short wave cutoff to instability a non-zero ß parameter implies a long wave cutoff, too. In both cases the solutions turn out to converge towards the classical Eady solution if ß-70. lt is found that the qualitative structure of the phase speed diagramm and also the qualitative shape of the vertical structure of the unstable solutions turned out to be rather insensitive to the specific settings of the basic state. / Das klassische Eadyproblem wird auf die ß-Ebene verlagert, wobei der Grundstrom so modifiziert wird, daß der meridionalen Gradienten der potentiellen Vorticity nach wie vor verschwindet. Zwei verschiedene Grundzustände werden untersucht, für die analytische Lösungen der linearisierten potentiellen Vorticitygleichung erhalten werden konnten. Wie man erwarten konnte, bedingt die Einführung eines von Null verschiedenen ß-Parameters das Auftreten einer Langwelleninstabilitätsgrenze zusätzlich zu der Kurzwelleninstabilitätsgrenze des klassischen Problems. In beiden Fällen konnte weiterhin gezeigt werden, daß die Lösungen für ß-70 gegen die klassischen Eadylösungen konvergieren. Ferner stellte sich heraus, daß die qualitative Struktur des Phasengeschwindigkeitsdiagramms und die qualitative Gestalt der Vertikalstruktur der instabilen Lösungen relativ unempfindlich gegenüber der genauen funktionalen Darstellung des Grundzustandes sind.

As pointed out by Farrell, a normalmode analysis alone may be not enough for a convicing investigation of baroclinic stability. In some models growth rates can be achieved large enough to enable nonlinear growth also in parameter ranges of neutral normalmodes. According to Farrell one has also to consider that structures, which achieve optimal growth for a given, fixed time interval (i.e. the singular vectors). Fischer (1998) investigated this problem for the classical Eady-model with q' = 0 - a case which can be treated analytically. In this paper we want to give a short overview of an investigation of singular vectors in Eady-models with β ≠ 0 and q' = 0. Our aim was to understand the influence of β ≠ 0 on optimal growth. Qualitative differences to Fischer’s results are only found at small wavenumbers below the longwave cutoff. The most remarkable difference beyond the longwave cutoff is the fact, that the singular vectors of the model with β ≠ 0 grow faster in the upper half of the fluid than in the lower half for the considered basic flows. The growth rates for parameter ranges of neutral normalmodes are too small to enable nonlinear growth effects in meteorologically relevant times. For long timescales we find, that the cutoffs must be understood more as a smooth transition to instability.

The Eady problem is modified to allow for a non-zero ß parameter but retaining a zero meridional gradient of potential vorticity. Two different basic states are examined for which analytical solutions of the linearized quasi-geostrophic potential vorticity equation were obtained. As has to be expected in addition to the short wave cutoff to instability a non-zero ß parameter implies a long wave cutoff, too. In both cases the solutions turn out to converge towards the classical Eady solution if ß-70. lt is found that the qualitative structure of the phase speed diagramm and also the qualitative shape of the vertical structure of the unstable solutions turned out to be rather insensitive to the specific settings of the basic state. / Das klassische Eadyproblem wird auf die ß-Ebene verlagert, wobei der Grundstrom so modifiziert wird, daß der meridionalen Gradienten der potentiellen Vorticity nach wie vor verschwindet. Zwei verschiedene Grundzustände werden untersucht, für die analytische Lösungen der linearisierten potentiellen Vorticitygleichung erhalten werden konnten. Wie man erwarten konnte, bedingt die Einführung eines von Null verschiedenen ß-Parameters das Auftreten einer Langwelleninstabilitätsgrenze zusätzlich zu der Kurzwelleninstabilitätsgrenze des klassischen Problems. In beiden Fällen konnte weiterhin gezeigt werden, daß die Lösungen für ß-70 gegen die klassischen Eadylösungen konvergieren. Ferner stellte sich heraus, daß die qualitative Struktur des Phasengeschwindigkeitsdiagramms und die qualitative Gestalt der Vertikalstruktur der instabilen Lösungen relativ unempfindlich gegenüber der genauen funktionalen Darstellung des Grundzustandes sind.

As pointed out by Farrell, a normalmode analysis alone may be not enough for a convicing investigation of baroclinic stability. In some models growth rates can be achieved large enough to enable nonlinear growth also in parameter ranges of neutral normalmodes. According to Farrell one has also to consider that structures, which achieve optimal growth for a given, fixed time interval (i.e. the singular vectors). Fischer (1998) investigated this problem for the classical Eady-model with q'' = 0 - a case which can be treated analytically. In this paper we want to give a short overview of an investigation of singular vectors in Eady-models with β ≠ 0 and q'' = 0. Our aim was to understand the influence of β ≠ 0 on optimal growth. Qualitative differences to Fischer’s results are only found at small wavenumbers below the longwave cutoff. The most remarkable difference beyond the longwave cutoff is the fact, that the singular vectors of the model with β ≠ 0 grow faster in the upper half of the fluid than in the lower half for the considered basic flows. The growth rates for parameter ranges of neutral normalmodes are too small to enable nonlinear growth effects in meteorologically relevant times. For long timescales we find, that the cutoffs must be understood more as a smooth transition to instability.

Ocean submesoscales are the subject of increasing focus in the oceanographic literature; with instrumentation now more capable of observing them in situ and numerical models now able to reach the resolution required to more fully capture them. Submesoscales are typified by horizontal spatial scales of O(1 − 10) km, vertical scales O(100) m and time-scales of O(1) day and are known to be associated with regions of high vertical velocity and vorticity. Occurring most commonly at density fronts at the ocean surface they can control mixed layer restratification and provide an important control on fluxes between the atmosphere and the deep ocean. This thesis sets out to better understand the fundamental physical processes underpinning submesoscale instabilities using a number of idealised process models. Linear stability analysis complemented by non-linear, high-resolution simulations will be used initially to explore the ways in which submesoscale instabilities in the mixed layer may compete and interact with one another. In particular, we will investigate the way in which symmetric and ageostrophic baroclinic instabilities interact when simultaneously present in a flow, with focus on the growth rates and energetic pathways of previously unexplored dynamic instabilities that arise in this paradigm; three-dimensional, mixed symmetric-baroclinic instabilities. Further, these non-linear simulations will allow us to investigate the transition to dissipative scales that can occur in the classical Eady model via a multitude of small-scale secondary instabilities that result from primary submesoscale instabilities. Finally, observational data, taken aboard the SMILES project cruise to the Southern Ocean, helps to motivate the consideration of a new dynamical paradigm; the Eady model with superimposed high amplitude barotropic jet. Non-linear simulations investigate the extent to which the addition of such a jet is capable of damping submesoscale growth. The causes of this damping are then investigated using linear analysis. With this approach eventually demonstrated as being unable to fully explain growth rate reductions, we introduce a new framework combining potential vorticity mixing by submesoscale instabilities with geostrophic adjustment, which relaxes the flow back to a geostrophic balanced state. This framework will help to explain, conceptually, how non-linear eddies control the linear stability of the flow.