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Mixed baroclinic-barotropic instability with oceanic applications

University of British Columbia

A brief introduction to the general subject of baroclinic-barotropic instability is given in chapter I followed by a discussion of the work done in the following chapters. In chapter II a three-layer model is derived to study the stability of large-scale oceanic zonal flows over topography to quasi-geostrophic wave perturbations. The mean density profile employed has upper and lower layers of constant densities p*₁ and p*₃ respectively (p*₁<p*₃) and a middle layer whose density varies linearly from p*₁ to p*₃. The model includes vertical and horizontal shear of the zonal flow in a channel as well as the effects of β (the variation with latitude of twice the local vertical component of the earth's rotation) and cross-channel variations in topography. In chapter II the effects of density stratification, vertical curvature in the mean velocity profile, β, constant slope topography and layer thicknesses H[sub i] (i=l,2,3) are studied. The following general results with regard to the stability of the flow are found: (1) curvature in the mean velocity profile has a very strong destabilizing influence (2) density stratification stabilizes (3) the β -effect stabilizes (4) topography stabilizes one of two possible classes of instability (a bottom intensified instability) and (5) increasing either H₁, or H₃ relative to H₂ stabilizes. Finally, the model is compared with two-layer models and results clearly indicate the importance of having at least three layers when curvature of the mean velocity profile is present or when H₂ is significant. In chapter III, mixed baroclinic-barotropic instability in a channel is studied using two- and three-layer models. The equations appropriate to the two-layer model used have been derived previously by Pedlosky (1964a). This model consists of two homogeneous layers of fluid with upper and lower layers of densities p*₁ and p*₂ respectively p*₁ > p*₂ and the corresponding mean velocities are taken as U₁= U₀. (1-cos π(y+1)), U₂ = εU₁ (ε=constant) . The choice of a cosine jet allows the possibility of barotropic instability (Pedlosky, 1964b) while the possibility of baroclinic instability is introduced by considering values of ε other than 1. In the study of the three-layer model, whose governing equations were derived in chapter II,the mean velocities are chosen in the form U₁= U₀(1-cos π(y+1), U₂ = εU₁ and U₃ = 0 and to simplify the interpretation of results, the effects of $ and topography are neglected. Again the study of mixed baroclinic-barotropic instability is studied by varying ε. The study of pure baroclinic or pure barotropic instability in either model is justified for the cases (L/r[sub i])² « 1 or (L/r[sub i])² » 1, respectively (L is the horizontal length scale of the mean currents and r[sub i] is a typical internal (Rossby) radius of deformation for the system). For the case (L/r[sub i])² ~ 1 it is found that the properties of the most unstable waves vary with the long-channel wavenumber. For each model, it is found that below the short wave cut-off for pure barotropic instability there are generally two types of instabilities: (1) a baroclinic instability which generally loses kinetic energy to the mean currents through the mechanism of barotropic instability and (2) a "barotropic instability" which in some cases extracts the majority of its energy from the available potential energy of the mean state. The latter type of instability is most apparent in the study of the three-layer model although it is also present in the two-layer case. It is a very interesting case since its structure is largely dictated by the mechanism of barotropic instability even when its energy source is that of a baroclinic instability. Beyond the short wave cut-off for pure barotropic instability, only the former of these two types of instabilities persists (i.e. the baroclinic instability). Qualitative results for the three-layer model are also derived in chapter III (section 3). The energy equation is discussed, bounds on phase speeds and growth rates of unstable waves are derived and the. condition for marginally stable waves with phase speed within the range of the mean currents is presented. Chapter IV is concerned with oceanic applications. Low frequency motions (≤0.25 cpd) have recently been observed in Juan de Fuca Strait. The three-layer model developed in chapter II is used to show that at least part of this activity may be due to an instability (baroclinic) of the mean current to low-frequency quasi-geostrophic disturbances. Recent satellite infrared imagery and hydrographic maps show eddies in the deep ocean just beyond the continental shelf in the north-east Pacific. The wavelength of these patterns is about 100 km and the eddies are aligned in the north-south direction paralleling the continental slope region. A modification of the three-layer model derived in chapter II is used to study the stability of the current system in this area. It is found that for typical vertical and horizontal shears associated with this current system (which consists of a weak flow to the south at shallow depths, a stronger poleward flow at intermediate depths and a relatively quiescent region below), the most unstable waves have properties in agreement with observations.

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2010-03-18

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http://hdl.handle.net/2429/22071

Mathematics

University of British Columbia

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