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Mixed baroclinicbarotropic instability with oceanic applications Wright, Daniel Gordon
Abstract
A brief introduction to the general subject of baroclinicbarotropic instability is given in chapter I followed by a discussion of the work done in the following chapters. In chapter II a threelayer model is derived to study the stability of largescale oceanic zonal flows over topography to quasigeostrophic 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 crosschannel 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 twolayer 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 baroclinicbarotropic instability in a channel is studied using two and threelayer models. The equations appropriate to the twolayer 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. (1cos 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 threelayer model, whose governing equations were derived in chapter II,the mean velocities are chosen in the form U₁= U₀(1cos 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 baroclinicbarotropic 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 longchannel wavenumber. For each model, it is found that below the short wave cutoff 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 threelayer model although it is also present in the twolayer 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 cutoff for pure barotropic instability, only the former of these two types of instabilities persists (i.e. the baroclinic instability). Qualitative results for the threelayer 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 threelayer 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 lowfrequency quasigeostrophic disturbances. Recent satellite infrared imagery and hydrographic maps show eddies in the deep ocean just beyond the continental shelf in the northeast Pacific. The wavelength of these patterns is about 100 km and the eddies are aligned in the northsouth direction paralleling the continental slope region. A modification of the threelayer 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.
Item Metadata
Title 
Mixed baroclinicbarotropic instability with oceanic applications

Creator  
Publisher 
University of British Columbia

Date Issued 
1978

Description 
A brief introduction to the general subject of baroclinicbarotropic instability is given in chapter I followed by a discussion of the work done in the following chapters.
In chapter II a threelayer model is derived to study the stability of largescale oceanic zonal flows over topography to quasigeostrophic 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 crosschannel 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 twolayer 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 baroclinicbarotropic instability in a
channel is studied using two and threelayer models. The equations
appropriate to the twolayer 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.
(1cos 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 threelayer model, whose
governing equations were derived in chapter II,the mean velocities are
chosen in the form U₁= U₀(1cos 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 baroclinicbarotropic 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 longchannel wavenumber. For each model, it is found that below the short wave cutoff 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 threelayer model although it is also present in the twolayer 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 cutoff for pure barotropic instability, only the former of these two types of instabilities persists (i.e. the baroclinic instability).
Qualitative results for the threelayer 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 threelayer 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 lowfrequency quasigeostrophic disturbances.
Recent satellite infrared imagery and hydrographic maps show eddies in the deep ocean just beyond the continental shelf in the northeast Pacific. The wavelength of these patterns is about 100 km and the eddies are aligned in the northsouth direction paralleling the continental slope region. A modification of the threelayer 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.

Genre  
Type  
Language 
eng

Date Available 
20100318

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080244

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.