Contents
My research is on the stability of axisymmetric
zonal
flows in the equatorial
stratosphere.
To be in equilibrium the velocity and temperature
fields must be such that the Coriolis, gravitational, and
pressure-gradient forces balance each other. This condition is known
as thermal wind balance. The stability of the equilibrium depends on the signs
of the temperature and velocity gradients.
An unstable zonal flow may be interpreted physically as either
convectively or inertially unstable, or a combination of
the two, depending on whether the instability is primarily due to the
temperature distribution, or the velocity distribution. If potential temperature decreases with
height, then a small parcel of air displaced upwards would be less
dense than its surroundings and continue to rise. If the instability
extends over a finite region, it results in overturning cells called
convection which redistribute the air to flatten the potential
temperature gradient.
If the velocity is such that absolute angular momentum
increases with latitude away from the equator, then pressure must
increase away from the equator to balance the Coriolis force, and
fluid displaced northward would have a lower velocity than the
surrounding fluid and be pushed further north by the pressure
gradient. The equilibrium is said to be inertially unstable
because the apparent force that pushes the fluid away from equilibrium
is equivalent to the fluid continuing its motion in the
absolute nonrotating reference frame. If the unstable
condition exists over a large latitude range, then the equivalent of
convective cells, called Taylor Vortices, redistributes the
fluid to flatten the angular momentum gradient.
The following table summarizes the two instability mechanisms
INERTIAL INSTABILITY
Convection occurs when the potential temperature (theta) decreases
with height. Rising fluid expands and cools as the pressure of the
surrounding air decreases. It will be warmer (cooler) than the
surrounding air if its potential temperature is greater (less) than
that of the surrounding air.
COLD
WARM
Schematic of convection cells: Air with low potential temperature
lying over air with high potential temperature is unstable. If it is
disturbed slightly, the warm fluid will rise and the cold fluid will
sink. The process organizes into cells.
CONVECTIVE INSTABILITY
A cicular flow is inertially unstable if angular momentum decreases
with distance from the centre. The speed of a ring of fluid increases
(decreases) if its radius is reduced (increased) in the same way as a
figure skater spins faster when she brings her arms close to her body,
but its angular momentum does not change.
Schematic of Taylor Vortices: The inner ring of fluid has greater
angular momentum than the outer ring. The pressure at the radius of
the outer ring keeps the fluid moving in a circle. If the fluid in
the inner ring is displaced outwards, it will have a greater velocity,
hence a greater centrifugal tendency, than the surrounding fluid, and
the pressure field will no longer be able to keep it from moving
further out. If disturbed, the inner fluid will move outward and the
outer fluid will move inward (while still rotating around the axis).
The Taylor Vortex state is features the fluid moving in a helical
pattern about the axis of rotation.
A famous example of inertial instability occurs in flow between
coaxial cylinders (Taylor-Couette flow). If the inner cylinder
rotates fast enough, then the flow can become inertially unstable.
The instability manifests itself in the form of vertical rolls
(Taylor vortices) superposed on the tangential flow.
A similar phenomenon can occur in zonal (East-West) flows in the
equatorial ocean and middle atmosphere. Since the radius of the fluid
rings must increase towards the equator (to negotiate the spherical
Earth), the system can become inertially unstable if the absolute
angular momentum (i.e. including the contribution of the rotating
Earth) is anywhere increasing with latitude. In this case,
instability would result in the formation of vertical rolls in the
meridional (North-South) direction.
Comparison of inertial instability in Taylor-Couette experiment
and in earth's atmosphere. Note that the angular momentum gradient is
radial in the Taylor-Couette experiment, but latitudinal in the
atmosphere. This is because in the atmosphere, gravity and the pressure
gradient force are approximately balanced in the radial direction so
motion is almost entirely horizontal. As such, vertical gradients of
angular momentum are less important than latitudinal gradients.
It is felt that the hydrostatic system might be particularly poor at
representing equatorial inertial instability. Firstly, when using the
hydrostatic system in a spherical domain, we are forced to consider
the entire atmosphere (or ocean) as if it were at the same altitude
(the shallow atmosphere approximation). This makes it
insensitive to radial angular momentum gradients. Secondly, and more
significantly, the component of the Earth's rotation vector in the
horizontal plane is absent from the hydrostatic system. One of the
signatures of inertial instability is the appearance of zonal jets
that develop due to the Coriolis force (the apparent sideways
force on moving objects viewed from a rotating reference frame) acting
on the overturning fluid. Without the full Earth rotation vector,
these jets would not develop in a realistic way (in numerical
simulations, for example).
I am loosely affiliated with the Modelling of
Global Chemistry for Climate initiative, and presented a short
talk at the 9th Annual GCC Workshop (November, 2001) here at the
University of Toronto.
Watch this space for progress, if any!
Last updated April 27th, 2004