Eady growth rate of baroclinic instability

How fast baroclinic instability grows: the rate at which the available potential energy stored in tilted density surfaces is converted into eddies. High where a sheared, weakly stratified flow can overturn, which on this shelf means the shelf-break front and the edges of warm-core rings — the places eddies are actually generated.

Usage

eady_growth_rate(
  env_dat,
  shallow = c("UO", "VO"),
  deep,
  depths,
  stratification = "N2",
  per = c("day", "second"),
  name = NULL
)

Arguments

env_dat: an sf POINT object with one row per location and time step, as datamatch's access functions return
shallow: length-2 character vector naming the eastward and northward velocity columns at the shallower level, in m/s
deep: the same for the deeper level
depths: the two depths in metres, shallower first
stratification: name of an \(N^2\) column, in s^-2, as produced by buoyancy_frequency()
per: "day" (the default) or "second"
name: name for the new column

Value

env_dat with a growth rate column, in day^-1 or s^-1

Details

Computed in the maximum-growth-rate form of Lindzen and Farrell (1980), \(\sigma = 0.31\,|f|\,|\partial U/\partial z| / N\), for the instability described by Eady (1949).

Eady is a person, not a spelling of "eddy". Eric Eady set out the model in 1949. The collision is unlucky, because the rate named after him is precisely a predictor of where eddies form, so the two words look interchangeable beside each other and are not.

What it needs

Velocities at two depths and a stratification spanning the same layer. From Copernicus that means two calls at different depth ranges joined as columns; from HYCOM it means UO/VO beside UO_BOTTOM/VO_BOTTOM, which arrive together. See buoyancy_frequency(), which produces the stratification and explains how to choose the levels.

What it is and is not

An index of where and when instability is favoured, not a prediction of growth. The formula assumes quasi-geostrophic scaling and uniform stratification through the layer, neither of which holds exactly on a shelf, and the 0.31 is the maximum over wavenumber rather than the rate of any particular disturbance. Read it as a comparative field.

It is undefined where the column is not stably stratified, since \(N\) is then not a frequency. Those cells come back NA rather than as a very large rate, which is what dividing by a vanishing \(N\) would otherwise produce and which would look like intense instability exactly where the assumption has failed.

References

Eady ET (1949). Long waves and cyclone waves. Tellus 1(3), 33-52. doi:10.3402/tellusa.v1i3.8507

Lindzen RS, Farrell B (1980). A simple approximate result for the maximum growth rate of baroclinic instabilities. Journal of the Atmospheric Sciences 37(7), 1648-1654. doi:10.1175/1520-0469(1980)037<1648:ASARFT>2.0.CO;2

Examples

if (FALSE) { # \dontrun{
env <- buoyancy_frequency(env, "rho_surface", "rho_deep",
                          depths = c(0.494, 92.326))
env <- eady_growth_rate(env,
                        shallow = c("UO", "VO"),
                        deep = c("UO_deep", "VO_deep"),
                        depths = c(0.494, 92.326))
} # }