Shear Instability Viewed as Interaction between Counter-propagating Waves

John Methven, University of ReadingEyal Heifetz, Tel Aviv UniversityBrian Hoskins, University of ReadingCraig Bishop, Naval Research Laboratories, Monterey

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).

A Real Extratropical Weather System

Bringing Theory and Observation Closer

Baroclinic instability theory is insufficiently developed to predict weather system development.

We have been approaching from both ends:

1. Simplify atmospheric situation, but retain full nonlinear dynamic equations and solve numerically (e.g., baroclinic wave life cycles).

2. 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.

Idealised Baroclinic Wave Life Cycle

Potential temperature at ground

Potential vorticity on 300K potential temperature (isentropic) surface

Idealised Baroclinic Wave Life Cycle

Potential temperature at ground

Potential vorticity on 300K potential temperature (isentropic) surface

CRW propagation and interaction

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:

1. Waves propagate in opposite directions (have opposite signed pseudomomentum),

2. 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.

Interacting Rossby edge waves

Rayleigh ModelRayleigh ModelHorizontal shear, no vertical variation barotropic instability

Eady ModelEady ModelVertical shear in thermal

wind balance with cross-stream temperature gradient and no cross-stream variation in flow

baroclinic instability

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 spectrumDiscrete spectrum (normal) modes with distributed PV structure + continuous spectrumcontinuous spectrum modes, each consisting of a PV -function at given height and associated flow perturbation.

A Pair of Waves Associated with Instability

How can a pair of interacting CRWs be identified?How can a pair of interacting CRWs be identified? Superposing any growing normal mode (NM) and its

decaying complex conjugate results an untilted 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 Seek 2 CRWs whose phase and amplitude evolution equations have the same form as those for the Eady (or 2-equations have the same form as those for the Eady (or 2-layer) models.layer) models.

Decomposition achieved by requiring the CRWs to be orthogonal in pseudomomentum and pseudoenergy (globally conserved properties of disturbed component of flow).

M=7 lon-sig

Example:

Fastest growing NM on realistic zonal jet Z1

Meridional wind PV

Upper CRW

Lower CRW

++

+

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 (e.g., up-gradient momentum fluxes)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)(e.g., phase difference maintained).

Problems in Application to the Atmosphere (I)

1. Identification of relevant background state

Atmosphere never passes through zonally symmetric state.

Modified Lagrangian Mean 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 Also pseudomomentum conservation law extends to nonlinear evolution if waves conservation law extends to nonlinear evolution if waves are defined relative to the MLM state (Haynes, 1988).are defined relative to the MLM state (Haynes, 1988).

Collaboration with Paul Berrisford (CGAM).

Problems in Application to the Atmosphere (II)

2. 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).

3. Nonlinear Effects, Especially Rossby Wave Breaking

Examine atmosphere using modified Lagrangian mean framework.

Collaboration with Brian Hoskins (Meteorology).

The End

Phase difference between trough of upper wave and crest of lower wave (both +ve PV)

A/C turning

Occlusion of warm sector

Seclusion of warm air

cyclonic turning

Secondary cyclogenesis

Phase difference between upper and lower CRWs from linear theory

No barotropic shear

Cyclonic shear