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DYNAMICAL OCEANOGRAPHY
Summary
Necessary conditions for instability of a zonal flow can be determined
through a linear stability analysis. If the flow is unstable, then there
will be an interval of wavenumbers that will grow exponentially. After
some time, the amplitudes of the perturbations become so large that
they will influence the flow and nonlinear interactions of the different
modes can no longer be neglected. Only a nonlinear theory can then
provide an answer to how the perturbations develop and how they will
modify the basic state.
A necessary condition for barotropic instability of a zonal flow with
velocity U ∗ (y ∗ ) is that there is sign change in the quantity U ′′
∗ − β 0 .
Note that this is not a sufficient condition: when U ′′
∗ −β 0 changes sign,
the flow may still be stable.
The barotropic instability mechanism can be considered as an amplification of Rossby waves through the background horizontal shear.
Whenever there is vertical shear in a rotating stratified flow, baroclinic
instability will occur and perturbations over a band of wavenumbers
are amplified. During the instability process, potential energy from
the background density field is converted into kinetic energy of the
disturbances.
The β-effect is a stabilizing effect in baroclinic instability as it modifies the potential vorticity of the background state.
DYNAMICAL OCEANOGRAPHY
Summary
Necessary conditions for instability of a zonal flow can be determined
through a linear stability analysis. If the flow is unstable, then there
will be an interval of wavenumbers that will grow exponentially. After
some time, the amplitudes of the perturbations become so large that
they will influence the flow and nonlinear interactions of the different
modes can no longer be neglected. Only a nonlinear theory can then
provide an answer to how the perturbations develop and how they will
modify the basic state.
A necessary condition for barotropic instability of a zonal flow with
velocity U ∗ (y ∗ ) is that there is sign change in the quantity U ′′
∗ − β 0 .
Note that this is not a sufficient condition: when U ′′
∗ −β 0 changes sign,
the flow may still be stable.
The barotropic instability mechanism can be considered as an amplification of Rossby waves through the background horizontal shear.
Whenever there is vertical shear in a rotating stratified flow, baroclinic
instability will occur and perturbations over a band of wavenumbers
are amplified. During the instability process, potential energy from
the background density field is converted into kinetic energy of the
disturbances.
The β-effect is a stabilizing effect in baroclinic instability as it modifies the potential vorticity of the background state.