1. Viewing baroclinic initial value problems in terms of Rossby wave components
John Methven, University of Reading
Hylke de Vries, University of Reading
Tom Frame, University of Reading
Brian Hoskins, University of Reading
Eyal Heifetz, Tel Aviv University
Craig Bishop, Naval Research Laboratories, Monterrey
14th Cyclone Workshop, 25 September 2008

2. Motivation
Attractive to break down complex flow into a number of components with predictable evolution.
Examples
Wave-mean flow partition
Distinct coherent vortices
Wave-like disturbances focussed at different levels
Can anticipate what would have happened if basic state or initial disturbances had been different (from physical arguments alone).

3. Obstacles 1. Appropriate definition of background state
2. Theory for Rossby wave evolution
- isolation of Rossby wave components
- non-modal growth
- nonlinear wave breaking
3. Moisture and latent heat release

4. Rossby wave components
… are waves in potential vorticity (PV).
On zonal basic state flows they are advected but also propagate relative to the flow if there is a meridional PV gradient.
Linear baroclinic instability  interaction between two counter-propagating Rossby waves (CRWs).
BUT, the CRW components only describe the evolution of normal modes (the discrete spectrum) [Heifetz et al, 2004].
Is a compact description of evolution possible from general initial conditions (including the continuous spectrum) ?
… first consider instability in terms of CRW structures…

5. Interacting Rossby edge waves
When basic state PV is piecewise uniform with only 2 jumps
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
U=z

6. CRW propagation and interaction

7. Basic States with Continuous PV Gradients
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.

8. e.g., CRW structures in Charney model
Contours show meridional wind
N = northward
S = southward
Upper CRW N S
Colour shading shows QGPV
Red = +ve
Blue = -ve
+/- indicate boundary temperature anomalies + 
Lower CRW S N
Growing NM in Charney model L=1/k=1000 km +  S N

9. Determining CRW structures from NM
PV anomalies of adiabatic NMs arise by meridional displacements of air, η, advecting PV:
CRWs are defined as orthogonal in pseudomomentum and “shear term” appearing in pseudoenergy:
(=0 for i≠j )
where brackets are defined as
Given above defs, PV equation yields evolution equations for CRW amplitudes, a1, a2, and phases, ε1, ε2, as given by Davies and Bishop (1994) for Eady model.
BUT only covers discrete spectrum. What about other PV?

10. Eady model Initial PV dipole
Uniform vertical shear and static stability
f-plane (no interior PV gradient)
Rigid lid

11. Eady model Initial PV dipole
Uniform vertical shear and static stability
f-plane (no interior PV gradient)
Rigid lid
Black/red = +ve boundary PV anomaly
White = -ve boundary PV anomaly

12. Eady model Initial PV dipole
Uniform vertical shear and static stability
f-plane (no interior PV gradient)
Rigid lid
Result at T=12
Red = +ve boundary PV anomaly
White = -ve boundary PV anomaly

13. β-plane with broad tropopause (and no lid)

14. β-plane with broad tropopause (and no lid)

15. β-plane with broad tropopause (and no lid)
residual PV
project

16. β-plane with broad tropopause (and no lid)

17. Why is the residual PV almost passive?
Consider partitioning PV anomalies:
where
Substituting into linearised PV equation gives:
(1)
Meridional wind and parcel displacements are related:
Left side of (1) is 0  qp advected as if passive.
 residual is passive if CRWs describe all of qd for all t

18. PAR PV approximation Passively advected residual PV approximation
“Residual PV” = PV left after subtracting projection onto CRWs.
Approximation is that residual evolves by passive advection only.
Not exact because residual contains part of qd as well as qp.
BUT, partition of q(0) into qd and qp is arbitrary!
However, if qp(0)=q(0), qd (0)=0 is assumed,
found maximum perturbation energy error (for any t ) < 10%.
Can be reduced further by varying initial partition.

19. Solution to initial value problem under PAR PV approximation
Partition q(0) into qd and qp.
Project qd onto CRW structures:
 a1, a2, ε1, ε2 at t=0
Evolution of qp known
qp structure excites CRWs by advection as “forcing” term:
j=aj exp(iεj)
Amplitude and phase
Creation of displacement PV anomalies by winds induced by passive PV

20. Limitations in application to atmosphere
Only 3 Rossby wave components to consider, but…
Theory assumes small amplitude waves (single k)
Partition of large amplitude waves from “background state” affects interpretation of evolution
Diabatic processes:
Create PV anomalies  qp (but need to know history)
Latent heat release alters dynamics
(e.g., saturated ascent with reduced effective N2)
Analysis of nonlinear life cycles
CRW evolution in moist 2-layer model with ascent/descent asymmetry.
Multi-layer models (but sinusoidal waves).

21. Bringing Theory and Observation Closer
PE dynamics -near balance
Eady model
General jet: U(y,z), Qy(y,z)
Unstable NMs
CRW pair
Large amplitude, but simplified situation
Nonlinear life cycle expts
Wave-mean interaction
Simple basic state
Sinusoidal wave
Small waveslope
QG dynamics
Dry
General initial value problems
Continuous spectrum
Atmosphere
RWs on vertical heating gradients
RW breaking
Stationary waves
Simultaneous systems in different stormtracks
Water phases
moisture
Asymmetry of ascent/descent

22. Upper level precursor

24. Relative evolution of the CRW components
π/2
π/2
Angle = U-L phase
r ~ tan-1(U/L)
Eady
t=0
t=0
Charney
(+β, no lid) π π
-π/2
-π/2
π/2
π/2
Green
(+β)
Broad tropopause
(+β, no lid)
t=0
t=0 π π
-π/2
-π/2

25. β-plane without lid (Charney model)

26. β-plane + rigid lid (Green model)

27. β-plane + rigid lid (Green model)