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# A baroclinic instability blog

2011-07-21 Predrag you think we can go baroclinic by AUG meeting time? Would require new code to (1) quotient symmetries and (2) find exact solutions. I, of course, am Spiegel in this configuration (but I can change diapers)

2011-07-21 Annalisa Bracco well… my desktop had to be dismantled and the OS changed, so nothing works any longer (no fft, no compilers, lost all old emails with pine…). I can try to fix the issue next week. If I get the baroclinic code to work, then we can start searching for the symmetries. I'll need help on that though, because it'll be stupid to code it from scratch if has been done already. The baroclinic geometry is similar to Is channelflow. It is a multi-layer channel. Boundary conditions: lid on the top, slip boundary condition on top, stick slip on bottom, periodic only east-west, wall north south (or vice versa)

2011-07-21 Predrag Otherwise a flat parallelepiped? Is it Fourier in the wall north south? Channelflow.org has most of the code that we need in that case. I'll talk to Gibson. Any feeling for how low a low Reynolds number can be, the minimal cell size?

2011-07-21 Annalisa Bracco 2 or 3 layers. I have to run it before I can be precise on low Re solutions. Should not be an issue though. I can reduce it as much as I wish. Cannot change much the geometry without recoding everything because of fft use.

2011-07-21 Predrag starting to read up on matters baroclinic: here are a few snippets clipped from the baroclinic wikipedia.

The baroclinity (sometimes called baroclinicity) of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. The baroclinic vector

`  ∇p × ∇ρ`

which is proportional to sine of the angle between surfaces of constant pressure and surfaces of constant density, is of interest both in compressible and in incompressible (but inhomogenous) fluids.

Baroclinic instability is a fluid dynamical instability of fundamental importance in the atmosphere and in the oceans. In the atmosphere it is the dominant mechanism shaping the cyclones and anticyclones that dominate weather in mid-latitudes. In the ocean it generates a field of mesoscale (100 km or smaller) eddies that play various roles in oceanic dynamics and the transport of tracers. Baroclinic instability is a concept relevant to rapidly rotating, strongly stratified fluids.

Whether a fluid counts as rapidly rotating is determined by the Rossby number, which is a measure of how close the flow is to solid body rotation. More precisely, a flow in solid body rotation has vorticity that is proportional to its angular velocity. The Rossby number is a measure of the departure of the vorticity from that of solid body rotation. The Rossby number must be small for the concept of baroclinic instability to be relevant. When the Rossby number is large, other kinds of instabilities, often referred to as inertial, become more relevant.

The simplest example of a stably stratified flow is an incompressible flow with density decreasing with height. In a compressible gas such as the atmosphere, the relevant measure is the vertical gradient of the entropy, which must increase with height for the flow to be stably stratified. One measures the strength of the stratification by asking how large the vertical shear of the horizontal winds has to be in order to destabilize the flow and produce the classic Kelvin-Helmholtz instability. This measure is the Richardson number. When the Richardson number is large, the stratification is strong enough to prevent this shear instability.

The energy source for baroclinic instability is the potential energy in the environmental flow. As the instability grows, the center of mass of the fluid is lowered. In growing waves in the atmosphere, cold air moving downwards and equatorwards displaces the warmer air moving polewards and upwards. The term “baroclinic” refers to the mechanism by which vorticity is generated. Vorticity is the curl of the velocity field. in general, the evolution of vorticity can be broken into contributions from advection (as vortex tubes move with the flow), stretching and twisting (as vortex tubes are pulled or twisted by the flow) and baroclinic vorticity generation, which occurs whenever there is a density gradient along surfaces of constant pressure. The study of the evolution of these baroclinic instabilities as they grow and then decay is a crucial part of developing theories for the fundamental characteristics of midlatitude weather.