• INTRODUCTION
  • MODEL
  • STRATEGY
  • RESULTS
  • THEORY
  • CONCLUSIONS
  • REFERENCES

    Linear stability analysis

    By adopting a PV perspective we are able to couch the following stability problem and successfully predict the growth rate and scales of the eddies which ultimately break up the fluid column.

    Defining potential vorticity Q as

    where

    In a convectively mixed chimney Q approaches zero, whilst during the subsequent chimney collapse DQ/Dt=0.

    Thus we can:

  • initially envisage the chimney as a column of zero PV fluid.

  • interpret the cold upper surface as equivalent to a Bretherton sheet.

    • For a two layer baroclinic vortex like that illustrated in Fig.(1a) above, the growth rate, Omega_i, of an unstable wave of mode number m can be shown to be

    where

    and

    and where h_s and h_i refer to the depths of the upper and lower layers respectively such that h_s + h_i = h, the total depth of the chimney. Bessels functions I_m,K_m have argument r/lambda where lambda is related to the deformation radius

    but depends on delta; Decreasing the ratio of the upper to the lower layer depth decreases the deformation radius so that the effective radius in the Bessels functions arguments becomes

    Thus lowering delta decreases the deformation radius, thereby INCREASING the mode number of the fastest growing mode and REDUUCING the growth rate of the unstable mode. Assuming

    we are able to predict for m and the growth rate, as for example in the following figure for growth rate versus mode number as a prediction for the parameters of the reference experiment .