Zonal flows

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The instability of zonal flows to small disturbances underpins our understanding of the scales of motion and theirpredictability in planetary atmospheres and oceans. Numerous papers have explored aspects of this problem since thepioneering work of Charney (1947) and Eady (1949). The quasi-geostrophic formulation is only valid for small Rossbynumbers and large Richardson numbers and requires the stratification of the basic state to be independent of latitude.Formulations with more general validity have until recently been restricted to laterally uniform flows neglecting the Earth'scurvature.Recently, Bell (2017) has described a simple but novel method for studying stability of any zonal flow that is in thermal windbalance: the linearised primitive equations are reduced to a partial differential equation involving only a single variable, thepressure perturbation; the equation is linear in the pressure perturbation, but nonlinear in the complex phase speed whichsets the growth rate of the instability. The method can be used to study the stability of jets on a sphere in which thestratification of the basic state varies with height and latitude, the motions can be non-hydrostatic and the fluid can becompressible. Instabilities associated with different types of critical layers and symmetric and Kelvin Helmholz instabilitiescan be studied. Integral constraints on the instabilities can also be derived.We will investigate the instability of stratified zonal jets near the equator where the Rossby number is large and the Earth'scurvature of fundamental importance. These flows exhibit a number of instabilities that lead, for example, to the formationof tropical instability waves, the heat transport by which is of fundamental importance in setting ocean structure and heatuptake with widespread impact on tropical climate. Understanding the stability properties of these equatorial flows istherefore crucial for understanding the response of tropical climate to anthropogenic forcing.The dependence of growth rates on the profile of the mean state will be investigated in a hierarchy of progressively morecomplex zonal flows. The heat and momentum transports by the unstable waves will be calculated and compared withthose of tropical instability waves in NEMO ocean model simulations. Three classes of instabilities within the near-surfaceocean mixed layer will be studied: symmetric, baroclinic and Langmuir instabilities using the basic state of Blumen (1979)The student will diagnose the instabilities in ocean circulation models, explore numerical solutions for a hierarchy of zonalflows, consider theoretical aspects such as integral constraints on the instabilities and neutral modes which mark stabilitytransitions, and calculate the non-linear or weakly nonlinear development of the instabilities. Preliminary investigations ofnumerical solutions indicate that the growth rates of some instabilities are overestimated unless very fine vertical gridspacing is used: the student will also investigate the vertical resolution that is required for ocean circulation models to avoidmisrepresenting the different classes of instabilities.