Shear Instability Viewed as Interaction between Counter-propagating Waves

1. Shear Instability Viewed as Interaction between Counter-propagating Waves John Methven, University of Reading Eyal Heifetz, Tel Aviv University Brian Hoskins, University of Reading Craig Bishop, Naval Research Laboratories, Monterey

2. Baroclinic instability theory • Attempts to describe the growth of synoptic scale weather systems. • Early successes using the Charney (1947), Eady (1949) and Phillips (1954) models - very simple basic states - perturbations described by linearised quasigeostrophic eqns • Mechanism of growth in 2-layer (Phillips) model was explained in terms of Counter-propagating Rossby Waves (CRWs) by Bretherton (1966).

3. A Real Extratropical Weather System

4. Bringing Theory and Observation Closer • Baroclinic instability theory is insufficiently developed to predict weather system development. • We have been approaching from both ends: • Simplify atmospheric situation, but retain full nonlinear dynamic equations and solve numerically (e.g., baroclinic wave life cycles). • Explore generalisation of instability theory to more complete dynamic equations (e.g., PEs on sphere) and situations (e.g., realistic jets). • Focus is on developing theory that can give quantitative predictions for nonlinear life cycles, with new diagnostic framework that can also be applied to atmospheric analyses.

5. Idealised Baroclinic Wave Life Cycle Potential temperature at ground Potential vorticity on 300K potential temperature (isentropic) surface

6. Idealised Baroclinic Wave Life Cycle Potential temperature at ground Potential vorticity on 300K potential temperature (isentropic) surface

7. CRW propagation and interaction

8. When Does This Picture Apply? • Parallel flow with shear. • In two layer model, the 2 waves can be Rossby or gravity waves. Necessary criteria for instability: • Waves propagate in opposite directions (have opposite signed pseudomomentum), • Wave on more +ve basic state flow has –ve propagation speed so that phase speeds of 2 waves without interaction are similar. • In continuous system, just 2 Rossby waves exist if vorticity (PV) is piecewise uniform with only 2 jumps.

9. Interacting Rossby edge waves Rayleigh Model Horizontal shear, no vertical variation  barotropic instability Eady Model Vertical shear in thermal wind balance with cross-stream temperature gradient and no cross-stream variation in flow  baroclinic instability

10. Basic States with Continuous PV Gradients • What happens when the positive PV gradient is not concentrated at a lid but is non-zero throughout interior (e.g., the Charney model). • Cross-stream advection by surface temperature wave can create PV perturbations at any height  no longer just 2 waves. • Two parts to solution of linear dynamics: Discrete spectrum (normal) modes with distributed PV structure + continuous spectrum modes, each consisting of a PV -function at given height and associated flow perturbation.

11. A Pair of Waves Associated with Instability How can a pair of interacting CRWs be identified? • Superposing any growing normal mode (NM) and its decaying complex conjugate results an untilted PV structure. • Cross-stream wind (v) induced by such PV will also be untilted, as in CRW schematic. • Seek 2 CRWs whose phase and amplitude evolution equations have the same form as those for the Eady (or 2-layer) models. • Decomposition achieved by requiring the CRWs to be orthogonal in pseudomomentum and pseudoenergy (globally conserved properties of disturbed component of flow).

12. Meridional wind PV M=7 lon-sig Upper CRW +  Lower CRW +   + Example: Fastest growing NM on realistic zonal jet Z1

13. Conclusions so far • The CRW perspective applies to linear disturbances on any parallel jet. • Although only an alternative basis to NMs, the CRW structures enable new insights into growing baroclinic wave properties (e.g., up-gradient momentum fluxes). • The CRW propagation and interaction mechanism is robust at large amplitude, explaining why some of the predictions of linear theory apply even during wave breaking (e.g., phase difference maintained).

14. Problems in Application to the Atmosphere (I) • Identification of relevant background state • Atmosphere never passes through zonally symmetric state. • Modified Lagrangian Mean state •  find mass and circulation within PV contours in isentropic layers and re-arrange adiabatically to be zonally symmetric. • Advantages: retains strong PV gradients and background state is steady solution of equations (when adiabatic and frictionless). Also pseudomomentum conservation law extends to nonlinear evolution if waves are defined relative to the MLM state (Haynes, 1988). • Collaboration with Paul Berrisford (CGAM).

15. Problems in Application to the Atmosphere (II) • Transient Growth from Finite Perturbations • Relevant to cyclogenesis, but partly described by the continuous spectrum rather than CRW interaction. • Exploring excitation of CRWs by PV -functions. • Collaboration with Eyal Heifetz (Tel Aviv) and Brian Hoskins (Met). • Nonlinear Effects, Especially Rossby Wave Breaking • Examine atmosphere using modified Lagrangian mean framework. • Collaboration with Brian Hoskins (Meteorology).

16. The End

17. Phase difference between trough of upper wave and crest of lower wave (both +ve PV) No barotropic shear Cyclonic shear A/C turning cyclonic turning Seclusion of warm air Secondary cyclogenesis Occlusion of warm sector Phase difference between upper and lower CRWs from linear theory