Shear Instability Viewed as Interaction between Counterpropagating Waves

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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

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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).

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A Real Extratropical Weather System
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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.

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Idealised Baroclinic Wave Life Cycle
Potential temperature at ground
Potential vorticity on 300K potential temperature
(isentropic) surface
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Idealised Baroclinic Wave Life Cycle
Potential temperature at ground
Potential vorticity on 300K potential temperature
(isentropic) surface
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CRW propagation and interaction
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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.

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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
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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.

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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).

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M7 lon-sig
Meridional wind
PV
Upper CRW

?
Lower CRW

?

?
Example Fastest growing NM on realistic zonal
jet Z1
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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).

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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).

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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).

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The End
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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