Transcript Great Storms” Hitting Europe: Future Perspectives

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
Baroclinic instability theory
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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?
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Parallel flow with shear.
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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.
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In continuous system, just 2 Rossby waves exist if vorticity (PV) is
piecewise uniform with only 2 jumps.
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
Basic States with Continuous PV Gradients
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What happens when the positive PV gradient is not
concentrated at a lid but is non-zero throughout interior
(e.g., the Charney model).
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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.
A Pair of Waves Associated with Instability
How can a pair of interacting CRWs be identified?
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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).
Meridional wind
PV
Upper CRW
M=7 lon-sig
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+
Lower CRW
+
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+
Example:
Fastest
growing NM
on realistic
zonal jet Z1
Conclusions so far
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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).
Problems in Application to the Atmosphere (I)
1. 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).
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)
No barotropic shear
A/C turning
Cyclonic shear
cyclonic turning
Seclusion of
warm air
Secondary
cyclogenesis
Occlusion of
warm sector
Phase difference between upper and
lower CRWs from linear theory