Vertical structure of baroclinic annulus waves

Baroclinic instability, as its name implies, is a three-dimensional process that entails the transport of heat both upwards and horizontally, `down' the zonal mean horizontal temperature gradient. This means that the baroclinic waves resulting from this instability must have a quite complicated vertical and horizontal structure.

Pressure varies ~sinusoidally in azimuth/longitude, so that the pressure gradient force can support radial geostrophic flow to transport heat horizontally.
Temperature is related to dp/dz (via hydrostatic balance) so, in order to result in heat transport, pressure must also vary with height. In practice this usually results in a `westward' phase-tilt with height of the pressure field.
Temperature must also then vary ~sinusoidally in azimuth/longitude, and in the vertical with a phase tilt that is consistent with p.
Vertical velocity will also vary ~sinusoidally within the wave, and will develop a pattern that leads to a positive correlation between w and T. This is necessary to ensure upward heat transport - necessary to release buoyant potential energy. However, w must also be consistent with the geostrophically balanced horizontal flow, e.g. through the quasi-geostrophic omega equation.

This kind of 3D multivariate pattern can be seen, for example, in the structure of a growing Eady wave. The figure below shows a schematic illustration of the phase lines in the zonal and vertical directions at fixed latitude for the most rapidly growing unstable Eady mode (Eady 1949). This clearly shows the westward phase tilt with height of p and a generally eastward phase tilt for T. The phase lines for w are consistent with upward vertical velocity occurring mostly where the temperature part of the wave is relatively warm, and vice versa for the downward branch of the wave.

diagram representing cross-sections in longitude and height in a growing Eady wave

This kind of structure is evident in real baroclinic waves, such as in the rotating annulus experiments in the laboratory. The figure below shows the structure of the pressure, temperature and vertical velocity fields as contours in azimuth and height at mid-channel in a numerical simulation of m=3 fully developed flow in a thermally-driven rotating annulus. In this case the detailed structure of the wave in p, T and w is somewhat different to the Eady wave, but still shows similar characteristics.

diagram representing cross-sections in azimuth and height in a fully developed baroclinic wave

p still exhibits a `westward' phase tilt with height, though a somewhat different amplitude structure from the Eady wave.
Temperature follows dp/dz fairly closely, with a generally eastward phase tilt with height, though with more complicated structure near the Ekman boundary layers.
Vertical velocity looks rather different from the Eady wave, though still shows the required correlation with T to achieve upward heat transport.
It is surprising how closely these simulations of fully developed baroclinic waves resemble the structure of growing Eady waves, given that the latter are solutions to a highly idealised, LINEAR model of baroclinic instability.

These features can also be seen in actual visualisations of baroclinic wave flows in the laboratory. The images below show some examples of streak visualisations at different heights within fully-developed baroclinic wave flows under various conditions.

The properties of the annulus used for the streak images below are as follows:

Inner radius a = 2.5cm
Outer radius b = 8.0cm
Height d = 14.0cm
Rigid lid
Vertical axis of rotation
Rotation rate (Omega) = 0.05-4 rad/s

The following parameters are the same for each experiment:

Density (1.050 g/cm^3)
Time interval between level switching (1 min)
Streak length (1 s)
Streaks change colour from orange at the leading end to blue at the trailing end.

Constant temperature gradient: DT = 4 K; Flow at 5 different height levels