1. Boundary Layer Meteorology Lecture 10: Role of the boundary layer in large-scale dynamics Review of baroclinic instability theory Ekman layer damping time-scale Boundary conditions for baroclinic neutrality calculations derived from Ekman theory Literature review on Ekman layer’s influence on the stability of the real troposphere.

2. Baroclinic instability Rossby waves; gradient of potential vorticity as restoring force; PV anomalies and their implied circulations (see Holton’s text book). Instability: Eady model and counter propagating Rossby waves. Waves can propagate on the top and bottom boundaries only. These waves take the shape of PV anomalies that exist only at the boundaries.

3. Baroclinic instability, cont’d Although the PV anomalies in the Eady model are confined to the surface (because there’s no PV gradient in the model troposphere), the winds associated with these anomalies are not; they decay exponentially upward from the lower boundary and downward from the top. If the bottom and top are not too far apart, and if waves are not too short, these winds can strengthen the anomalies. (Lindzen 1994 is not a bad reference for this stuff, see me for others).

4. Baroclinic Instability, cont’d. We typically try to solve instability problems like the Eady model by assuming a time and space dependence like: exp[ik(x-ct)] where c = c r + ic im where c r is the phase speed, and c im is the growth rate. How should we take surface friction into account? One way is to subtract some friction decay rate from c im.

5. Ekman Layer Damping Time From Pedlosky (1987), Geophysical Fluid Mechanics:

6. Ekman Layer Damping Time, cont’d. This damping time scale could be applied to the growth rate of eddies, following calculation of the growth rate using the Eady or Charney model But a more sophisticated approach is to use the Ekman layer theory to impose a boundary condition on the vertical velocity.

7. Boundary conds. derived from Ekman Theory Starting from mass conservation and incompressibility:

8. Literature review of Ekman layer’s influence on tropospheric stability Williams, G.P. and J.B. Robinson, 1974: Generalized Eady Waves with Ekman Pumping. J. Atmos. Sci., 31:1768-1776. Card, P.A., and A. Barcilon, 1982: The Charney problem with a lower Ekman layer. J. Atmos. Sci., 39:2128-2137. Farrell, B., 1985: Transient growth of damped baroclinic waves. J. Atmos. Sci., 42: 2718-2727. Lin, S.-J., and R.T. Pierrehumbert, 1988: Does Ekman friction suppress baroclinic instability? J. Atmos. Sci., 45:2920-2933. Grotjahn, R., M. Chen, J. Tribbia, 1994: Linear instability with Ekman and interior friction. Part I: Quasigeostrophic eigenanalysis. J. Atmos. Sci., 52:754-763. Grotjahn, R., R. Pederson, J. Tribbia, 1994: Linear instability with Ekman and interior friction. Part II: Initial value analysis. J. Atmos. Sci., 52:764-777.