Eady model, CRW theory & moist baroclinic instability

Eady model, CRW theory &

moist baroclinic instability

Bergen, March 2019

Hylke de Vries ([email protected])

Royal Netherlands Meteorological Institute (KNMI)

Me

¤ MSc (Amsterdam) Th.Physics [quark-gluon plasma]

¤ Phd (Utrecht, 2006) Atmospheric dynamics [idealized models of baroclinic instability & non-modal growth]

¤ Postdoc (Reading 2007-2009): CRWs & diabatic heating

¤ Researcher (KNMI, Netherlands 2009-):

¤ Atmospheric blocking; Future cold spells; snowfall

¤ Sea level rise; KNMI climate scenarios;

¤ Ocean waves; TC in high-res GCMs (‘Future weather’);

¤ NWP model-verification;

¤ High-res reg. climate modeling with Harmonie

2

Outline

¤ Part 1: Intro, Shear flow, Eady model & CRWs

¤ Part 2: Extensions & diabatic effects

3

Intro

Weather is...

¤ “... nothing more than the existence of large- scale wavelike

fluctuations in the

atmospheric circulation, whose occurrence

cannot be predicted, as the tides can be, by a simple almanac of assured recurrence based on past

experience.”

5

Pedlosky (1987, Ch7)

Fluctuations

¤ “The existence of fluctuations ... can be attributed to the

instability of the dynamical state to very small disturbances.”

¤ “Such small disturbances are inevitably present present in any real system, but their effect on stable systems is ephemeral”

¤ “If the flow is unstable... these fluctuations will grow in amplitude, with time and space scales determined by the

dynamics of the interaction of the initial perturbation and the structure of the original flow state.”

6

Pedlosky (1987, Ch7)

Rotation

Temperature contrast

Key elements

Two key elements 1. Basic state

2. Exposure to (small) disturbances

Cautious note

The observed structure of the mean flow is inevitably affected by the presence of the very fluctuations we seek to predict, since non-linear processes give rise to fluxes of heat and momentum with non-zero time-averages ...”

7

Pedlosky (1987, Ch7)

Aim

¤ Various aspects of baroclinic development in sheared flows

¤ Elements

¤ Focus on initial stage / linear regime (almost analytic)

¤ Different geometries => Unbounded shear => Eady model => ...

¤ Counter-propagating Rossby waves

¤ Growth mechanisms

¤ Moisture / Diabatic effects

8

Real world and models, level of understanding

9

MODEL COMPLEXITY

real world too simple?

LEVEL OF UNDERSTANDING

Instability of shear flows

Setting the scene – basic state

¤ Atmospheric zonal flow U

₀

(y,z)

¤ Can be unstable to small perturbations

¤ Horizontal and vertical shear of U

₀

important

dU₀/dy => barotropic dU₀/dz => baroclinic

Basic state / mean flow

Annual mean zonal wind (source: ECMWF-atlas)

WEST surface EAST

Further approximations

“Make things simpler but ... not too simple!”

Einstein

¤ Small Rossby Number: U/f

₀

L < 1

¤ Large scales, L~1000km, H~10km, U~30m/s

¤ Linear quasi-geostrophic dynamics

¤ Mid-Latitude f

₀

- or b -Plane (f=f

₀

+ b y)

¤ Inviscid (or Ekman), no bottom topography

Quasi-geostrophic dynamics (linearized)

PV conservation Thermodynamic

Boundary conditions

¤ At rigid lids (surface/top)

¤ zero vertical velocities, w=0

¤ [extension: Ekman pumping]

¤ At a ‘tropopause’-interface: material surface and continuous streamfunction

¤ Unbounded model: at infinite height:

¤ zero perturbation amplitude, y=0

Qy=0: ‘passive’ PV-advection

¤ If mean PV-gradient is zero: existing PV is passively advected by the mean flow

¤ What makes the basic-state PV-gradient become zero?

¤ Each term vanishes individually (e.g. f-plane, linear shear, constant density and N**2) [Eady 1949]

¤ Terms vanish only in combination (e.g. beta-plane but special basic- state wind U or buoyancy frequency) [Lindzen 1994]

1

Model ‘hierarchy’ for today

¤ Stratified flow – no shear

¤ Adding vertical wind shear

¤ Adding a lower boundary

¤ Adding a lid/tropopause

¤ Add diabatic heating

¤ ....

¤ ...

¤ ...

Q

Model 1: constant stratified flow

Unbounded flow on f-plane with constant U, N^2 and density.

U z

¤ Try wavelike disturbances

¤ Homogeneous PV, q=0

¤ Solving QGPV equation: inverting the Laplace equation

¤ Boundary conditions:

¤ f(z) = constant as z tends to minus and plus infinity

¤ Lower BC: B = 0, Upper BC: A = 0.

Conclusion

No zero-PV solution for this geometry/basic flow

Solutions with zero interior PV

¤ Try delta-function PV-anomaly

¤ Solving Laplace equation with delta-function source term

¤ Exercise: Solve for the streamfunction:

¤ Applying the BCs => B=0, C=0.

Solutions with non-zero interior PV

Solution: Green’s function

for the unbounded domain known as KRW (Kernel Rossby Wave) (Heifetz)

Solutions with non-zero interior PV

¤ Properties of the solutions

¤ Vortices floating with mean flow U

¤ phase-speed: c=U independent of wavelength

¤ Vert.velocity: w=0 Why? no meridional tilt of isentropes

¤ un-tilted structures: v ~ -ikq (v and q pi/2 out of phase)

thermodynamic eq’n

dashed = mer.

wind at z=0 full = PV at z=0

PV-invertibility / KRW-structure

¤ Positive PV: warm anomaly above, cold below, cyclonic winds

¤ If you know the PV, you know the flow & theta (not w...)

+

warm

cold

theta(x,z)

v(x,z)

from Thorpe (1985)

A real upper-level PV feature

Model 2. Add wind shear

¤ Still away from rigid boundaries

¤ Still Q/dy=0 (PV passive)

¤ Still no mode instability

¤ Continuous spectrum in terms of KRWs (Kernel Rossby Waves) New: Transient energy growth and decay via the Orr-mechanism (PV- unshielding)

UNBOUNDED

west east

L

_t

N_t²

+

warm cold

+

warm

cold

¤ Adding constant shear to the basic-state zonal wind

¤ Only difference is in the thermodynamic equation

¤ The vertical velocity of the KRW is proportional to v and the sign of shear

¤ Northward rising motion

Adding shear => adding w

+ (u,v)

w KRW gets ‘dressed’

¤ Vertical velocity field: isentropic up-glide and down-glide

¤ isentropic motion: motion along lines of constant theta

¤ Equal amounts of heat are transported northward (by positive v’) as southward (by negative v’). Therefore there is ZERO NET heat transport by KRW. => Need multiple for transient growth

Adding shear

Q Q-D Q+D

N Z Y-Z-view

U

U

Q Q-D Q+D X-Z-view

trajectories

Non-modal/transient development

¤ 2 KRWs at distance d and pi out of phase initially

¤ Contours: meridional wind

d=1 d=0.2

Exercise: derive an expression for the amplitude of the streamfunction at z=0 as a function of time

PV evolution f-plane

T<0

kx z

T=0

T>0

No PV growth Temporal growth and decay of of streamfunction 1 RW component

Orr(1907), β-plane: Boyd(1983) Badger & Hoskins (2001)

GrowthDecay

“Venetian” blind

Linearity of the KRW-solution

¤ KRW solution. Compute the non-linear term (Jacobian) in the QGPV-equation

¤ The KRW is linear (and neutral) in isolation despite the shear of the basic-state wind

¤ Main cause for this is that the structure is an un-tilted pancake

¤ Non-linearity arises solely through interactions bonus

Boundaries introduce boundary PV gradients, along which PV

anomalies (‘edge waves’) can propagate [Bretherton1966].

Qy(z=0) = -Uz/N**2

¤ Still too simple for exponential growth on f-plane (necessary conditions not satisfied)

¤ Transient growth via

¤ Orr (PV-unshielding)

¤ Resonance (excitation of edge wave by the interior PV)

Model 3. Add a lower boundary

RIGID / EKMAN

west east

L

_t

N_t²

warm

+

warm

cold Semi-infinite Eady model

Example: Surface development triggered by a mid-tropospheric PV anomaly

Contours: meridional wind

- +

+ - +

-

‘Optimal’ perturbation (with beta)

NoteAmplitude constantly rescaled Zonal jet in a channel,

unbounded from above

Mechanisms in time: ORR => Resonance => Exponential growth

Edge waves

¤ Eastward propagating neutral solution with no interior PV

¤ At steering level (U(z)=c) w is at maximum (derive yourself) bonus

Edge waves

¤ The edge wave is a neutral solution despite the fact that there is available potential energy

¤ Because it is untilted in the vertical

¤ Meridional heat transport will be nonzero if phase-lines tilt with height

¤ Westward tilt / growth: positive net mer. heat transport

¤ Eastward tilt / decay: negative net meridional heat transport

¤ How to get development/growth/tilt in the perturbation??

Have more than one PV anomaly [or: add the upper rigid lid]

integrate over one wavelength

bonus

Positive boundary PV’

Negative boundary PV’

Bretherton / boundary PV

W

C = +

Thorpe 1985 HMR 1985

bonus

Hoskins (1997)

bonus

3. Finally: ‘classic’ Eady model

¤ Two edge waves: may interact to form an exponentially growing normal mode

[Davies&Bishop1994]

¤ Linear stability analysis (DIY) RIGID / tropopause

RIGID / EKMAN

west east

L

_t

N_t² cold

warm

+

^warm

cold

73.3 Barnclinic Insrabilrty : The Eady Problem 557

part c, and imaginary part ci, i.e.,

c = c,

+

^{iCi, ( I 3.3.8)}

the amplitudes A and B vary as does (13.3.4), i.e., as

cos ly exp(ik(x - c,t)) exp(kcit). (1 3.3.9) If ci is positive, disturbances grow spontaneously, whereas ci < 0 corresponds to a decaying disturbance. Formula (13.3.7) shows that both types of disturbance exist, but the growing one will soon dominate because of its exponentially growing ampli- tude. In fact, there will be a wavenumber selection in favor of the fastest-growing disturbance. Figure 13.3 shows the growth rate ⁰

=

kci as a function of wavenumber

(k, I). It is small for long waves because (13.3.7) gives

a

=

^{kci x} 3-'I2kHdUldz,, ( 1 3.3.10) whereas ( 1 3.3.5) shows that growth occurs only when

H tanh(H/H,) c H , , i.e., H < 1.19978, or N,K,H < 1.1997.f (13.3.11)

by ( I 3.2.4). Maximum growth is achieved when

1 = 0 and H ⁼ 0.8031HR, i.e., N,kH ⁼ 0.8031f, (13.3.12) the maximum value being given by

omnx 3 0.3098( f I N , ) d U l d z ,

.

^(13.3.13)

Fig. 13.3. Growth rate u of an Eady wave as a function of wavenumber (k, I). Contours are shown in units of (//N,dUldz,, Values are zero on k = 0 and when the magnitude K" of the wavenumber equals 1.1997. The maximum value 0.3098(1/N,) dUldz, is achieved when I = 0 and N,HK,/~ = 0.8031. For fixed ratio kll, the maximum is at the same value of K,. The maximum for a fixed I (corresponding to a baroclinic zone of fixed width) is at a value of k that decreases as I increases (longer unstable waves for narrower baroclinic zones).

4. Replace rigid top by tropopause

¤ Basic-state wind:

¤ Piecewise constant shear L_t/s of zonal wind

¤ Basic-state stratification

¤ Piecewise constant buoyancy frequency N_t/s² (N_t²=10^-4s^-2)

¤ tropopause: material surface

N

_t²

N

_s²

WEST surface (rigid lid) EAST

Stratosphere (unbounded)

Tropopause (interface)

L _t L _s

Troposphere

y/L

Normal modes

¤ Analytic solutions

¤ Growing-decaying pair or neutral

¤ Wedge of instability: (ultra)-long and short wavelengths are neutral

growth neutrality

neutrality

Smooth numerical model

=> Next questions:

Understand the growth

• Why short-wave cut-off?

• Why long-wave cut-off?

• Why cut-off at strong positive stratospheric shear?

strat shear

wave number

Normal modes (2)

¤ Realistic features:

¤ Baroclinic structure, tilt increases with height.

Westward tilt: ‘lean’ against the shear of troposphere;

¤ Zonal propagation rate and growth rate.

¤ Quasi-unrealistic features:

¤ Amplifies at a constant rate at all levels.

¤ Structure is time-independent.

N S

Meridional wind X

Z

Understanding [C]RW interaction

¤ Quasi geostrophic PV

¤ Material conservation

¤ Invertibility & smoothing

¤ dQ/dy>0: westward Rossby wave propagation

41

dQ/dy>0 high PV

low PV

v

_own

v

_own

+

y

-

x (west to east) north

south

- +

Understanding [C]RW interaction

¤ Quasi geostrophic PV

¤ Material conservation

¤ Invertibility & smoothing

¤ dQ/dy<0: eastward Rossby wave propagation

dQ/dy<0

high PV

low PV y

x (west to east)

v

_own

v

_own

+ -

north

south

+ -

+ -+

-

42

Counter-propagating Rossby waves

¤ Normal-mode growth: constructive interaction between surface and tropopause ‘edge’ waves = CRWs in this case

43

west surface east

+ warm

cold

+ warm

tropopause home base wave 2

home base wave 1 U(z)

Interaction

44

p q1 0

-p/2 p/2

Qy>0

Qy<0

q2

q2

q2

q2

q2 q2

+

+

q

₂

q

₁

phase-lock angle

HELP ROSSBY PROPAG OF EACH OTHER HINDER

ROSSBY PROPAG OF EACH OTHER

AMPLIFY EACH OTHER

DECAY EACH OTHER They hinder each

others counter- propagation &

help each other grow

U=c

dQ/dy>0

dQ/dy<0

Necessary conditions

¤ Charney-Stern (1962)

¤ dQ/dy changes sign

=> Required to get CRW mutual growth

¤ Fjørtoft (1951)

¤ U and dQ/dy positively correlated Þ Required to get CRW phase-locking

45

If necessary conditions are

satisfied, this does not mean

all modes grow!

Not all waves are unstable

¤ Complex c=c

_r

+ic

_i

¤ short-wave cutoff

¤ long-wave cutoff

c11

c22

horizontal wave number K

Im{c}

Re{c}

two-layer Eady dispersion relation

46

(c11+c22)/2

Summary – part 1

¤ Growth rate:

¤ Mechanism: Constructive interaction between CRW-L and CRW- U

¤ Zonal propagation:

¤ c = Re[c_B+c_T]/2 => mean of the speed that the two CRWs would have had in isolation.

¤ Cut-off wave numbers:

¤ Short waves: too less interaction to phase-lock.

¤ Long waves: difference in PV-gradient causes different speeds, no phase- lock.

¤ All normal modes stable for large positive stratospheric shear:

dQ/dy(d)<0 => Charney-Stern (1962) condition violated.

¤ Follow-up questions:

¤ Rigid structure of growing normal mode realistic?

¤ How to generalize the theory to include interior PV (diabatic sources)?

Part 2: Generalizations

General basic states & diabatic heating

“Didn’t we all know this

since Hoskins et al. (1985)?”

Well… maybe yes, but what about…

§ More complex models?

§ Complex initial conditions?

§ Continuous spectrum and non-modal growth?

§ Effects of moisture?

49

Generalization 1: more complex models

¤ Counter-propagating Rossby waves (CRWs)

¤ Construction / interpretation possible whenever gNM has been identified (Heifetz etal. 2004, Methven etal. 2005)

¤ Vertically un-tilted and self-neutral

¤ PV perspective - action at a distance

¤ Hope / Expectation

¤ Certain aspects remain valid at ‘finite’ amplitude

¤ Based on wave-activity arguments

=> Charney model

¤ Beta-plane, constant shear, unbounded from above

¤ Simplest model not described by two edge waves

home bases ____ = PV - - - - = - PSI

Heifetz etal (2004) part ii Similar CRW approach possible

for barotropic shear flows, mixed barotropic,baroclinic models etc

=> Primitive equations on the sphere

1402 J. METHVENet al.

a) b)

c) d)

Figure 1. The basic state Z1. (a)θ (contour interval 5 K up to 350 K) and zonal angular velocityU =u/(acosφ) (contour interval 4 deg day⁻¹) in σ-coordinates. The thick line marks the 2 PVU (=2×10⁻⁶ K kg⁻¹m²s⁻¹) tropopause. (b) U in isentropic coordinates. The heavy line marks the ground. (c) The Ertel potential vorticity (PV) (contour interval 1 PVU). (d) The isentropic PV-gradient measure Q_y (contour interval 1×10⁻¹¹ m⁻¹s⁻¹ up to 17). In all panels, positive contours are solid, the zero contour is dashed and negative contours are dotted.

In order to perform the CRW analysis it is necessary to interpolate the basic-state fields onto isentropic surfaces using linear vertical interpolation from the σ-coordinates of the primitive-equation spectral model of Hoskins and Simmons (1975). For the CRW calculations the model has spectral truncation T42 and there are 15 levels located at σ =0.018, 0.060, 0.106, 0.152, 0.197, 0.241, 0.287, 0.338, 0.400, 0.477, 0.569, 0.674, 0.784, 0.887, 0.967. The fields are interpolated onto isentropic surfaces with a spacing of 2 K and a maximum value θ_K =400 K. The zonal angular velocity, u/(acosφ), and PV are shown in isentropic coordinates in Figs. 1(b) and (c). Note that the isentropic coordinates stretch the top of the model domain, since isentropic density, r, falls with height from about 300 kg m⁻²K⁻¹ at the surface to about 120 kg m⁻²K⁻¹ at 2 PVU^∗, and then rapidly across the tropopause zone to 30 kg m⁻²K⁻¹ at about 6 PVU. The isentropic PV gradients are calculated using centred differencing of the interpolated PV field and are set to zero at the grid points closest to the lower boundary on each θ-surface, giving rise to the step-like nature of the field close to the lower boundary. Figure 1(d) shows the PV gradient term that appears in the definition of wave- activity density, r P_y cosφ. Although it has a clear maximum along the tropopause, the PV-gradient measure is also significant in the mid-latitude troposphere and stratosphere

∗ Potential vorticity unit: 1 PVU=10⁻⁶ K kg⁻¹m²s⁻¹.

Methven et al. (2005) part III

1406 J. METHVENet al.

a) b)

e) c)

f) d)

Figure 4. Zonal structure at the latitude of maximum surfaceθy(46^◦N) in meridional wind (left-hand panels) and PV (right-hand panels) form=7, shown inσ-coordinates for (a) and (b) the upper CRW, (c) and (d) the

lower CRW, and (e) and (f) the fastest-growing NM. Contour intervals are the same as in Fig. 3.

structure with identically zero displacement at the ground (and, therefore, positive- definite pseudomomentum). The PV of the upper CRW obtained by this method closely resembles Fig. 4(b). The lower CRW would be defined as the untilted structure with zero displacement at the level of the upper CRW’s tropospheric PV maximum. As a test, two home-bases were used for the upper CRW corresponding to the model level above and below its mid-tropospheric PV maximum at 46^◦N (σ =0.569, 0.674). The lower CRWs obtained were very similar to the orthogonal CRW in Fig. 4(d) but withP1=0 lines displaced upwards and downwards, respectively. The phase-locked angle also spanned

1406 J. METHVENet al.

a) b)

e) c)

f) d)

Figure 4. Zonal structure at the latitude of maximum surfaceθy(46^◦N) in meridional wind (left-hand panels) and PV (right-hand panels) form=7, shown inσ-coordinates for (a) and (b) the upper CRW, (c) and (d) the

lower CRW, and (e) and (f) the fastest-growing NM. Contour intervals are the same as in Fig. 3.

structure with identically zero displacement at the ground (and, therefore, positive- definite pseudomomentum). The PV of the upper CRW obtained by this method closely resembles Fig. 4(b). The lower CRW would be defined as the untilted structure with zero displacement at the level of the upper CRW’s tropospheric PV maximum. As a test, two home-bases were used for the upper CRW corresponding to the model level above and below its mid-tropospheric PV maximum at 46^◦N (σ=0.569, 0.674). The lower CRWs obtained were very similar to the orthogonal CRW in Fig. 4(d) but with P1=0 lines displaced upwards and downwards, respectively. The phase-locked angle also spanned

1406 J. METHVENet al.

a) b)

e) c)

f) d)

Figure 4. Zonal structure at the latitude of maximum surfaceθy(46^◦N) in meridional wind (left-hand panels) and PV (right-hand panels) form=7, shown inσ-coordinates for (a) and (b) the upper CRW, (c) and (d) the

lower CRW, and (e) and (f) the fastest-growing NM. Contour intervals are the same as in Fig. 3.

structure with identically zero displacement at the ground (and, therefore, positive- definite pseudomomentum). The PV of the upper CRW obtained by this method closely resembles Fig. 4(b). The lower CRW would be defined as the untilted structure with zero displacement at the level of the upper CRW’s tropospheric PV maximum. As a test, two home-bases were used for the upper CRW corresponding to the model level above and below its mid-tropospheric PV maximum at 46^◦N (σ=0.569, 0.674). The lower CRWs obtained were very similar to the orthogonal CRW in Fig. 4(d) but withP₁=0 lines displaced upwards and downwards, respectively. The phase-locked angle also spanned

v PV

CRW-U

CRW-L

LINEAR AMPLIFICATION OF MARGINALLY NEUTRAL BAROCLINIC WAVES 1081

at which CRW-2 attains its PV maximum. This level is called the home base of CRW-2, and for obvious reasons, the surface is called the home base of CRW-1.

The lower and upper CRW are labelled by subscripts 1 and 2, respectively: q

_1,2

(z) denote the PV structures and v

_1,2

(z) the meridional winds. We limit ourselves to initial conditions formed by CRWs only (or, equivalently, to gNM–dNM superpositions only). Any perturbation PV state at time t can then be written as q (z, t) = α

₁

(t)q

₁

(z) + α

₂

(t)q

₂

(z). The complex factor α

_i

is further expressible in a real amplitude a

_i

(t) and phase ϵ

_i

(t) of the CRW

α

_1,2

(t ) ≡ a

_1,2

(t )e

^iϵ1,2(t)

. (2) The linear evolution of this two-wave system is described by:

! α ˙

₁

˙ α

₂

"

= − ik A

! α

₁

α

₂

"

# $% &

≡α

, A =

! c

₁₁

c

₁₂

c

₂₁

c

₂₂

"

. (3)

The coefficients c

_ij

depend on the model geometry and on the properties of the perturbation, such as the zonal wavenumber k. The diagonal elements c

₁₁

and c

₂₂

of A determine the phase speeds of the lower and upper CRW in absence of interaction, respectively, and the off-diagonal elements describe their inter- action (Heifetz et al., 2004a). If the home-base method is used for constructing the CRWs, the c

_ij

are given by

c

_ij

= u ¯

_i

δ

_ij

− γ

_ij

k , γ

_ij

=

' v

_j

q

_i

∂ q ¯

∂ y

( )) ) ) )

z_i

, (4)

where z

_1,2

mark the home bases of the two CRWs. The final step is to re-normalise the structure of the CRWs such that the interaction coefficients become equal and opposite:

c

₁₂

= − c

₂₁

≡ σ

k > 0. (5)

Note that the coefficients c

_ij

and their parameter dependence change if the orthogonality method were used instead of the home-base method, or if linearized primitive-equation dynamics were used. However, the structure of the two-wave equations, as written in (3) does not change. Therefore, the results obtained in the remainder of this section, as well as in Section 3 are valid in general.

2.2. Normal modes and exponential growth

Following Heifetz and Methven (2005) the matrix A is rewritten as

A =

! c ¯ +

_2k^{µ σ}_k

−

^σ_k

c ¯ −

_2k^µ

"

, (6)

where ¯ c ≡

¹₂

(c

₁₁

+ c

₂₂

) is the mean of the self-propagation speeds of the two CRWs and µ ≡ − k(c

₂₂

− c

₁₁

) is proportional to their difference. The NMs are obtained as the eigenvectors of

A . The eigenvalues are c

_±

= c ¯ ± σ

k

√ ' , ' ≡ ρ

²

− 1, (7)

where ρ ≡ µ/(2σ ). Unstable waves (' < 0) propagate zonally with speed ¯ c, have CRW amplitude ratio χ ≡ a

₂

/a

₁

= 1. The CRW phase-difference ϵ ≡ ϵ

₂

− ϵ

₁

= ϵ

₊

in the gNM is given by cos (ϵ

₊

) = − ρ (0 ≤ ϵ

₊

≤ π ) and the gNM growth rate is k Im(c ) = σ ˆ ≡ σ sin(ϵ

₊

).

3. Marginally neutral waves

3.1. The cut-off condition

If an instability cut-off wavenumber k

_c

is found in the dispersion diagram, it satisfies ' = 0 following (7), further implying that ρ = ρ

_c

where

ρ

_c

≡ µ

_c

2σ

_c

= ± 1, (8)

where µ

_c

≡ µ |

^k=k_c

and σ

_c

≡ σ |

^k=k_c

. The factor ρ

_c

is introduced to simplify notation and arises because µ can be either positive or negative at an instability cut-off. The physical interpretation here is that, at the cut-off wavenumber, either the upper ( − ) or the lower ( + ) CRW propagates fastest in the zonal (eastward) direction. In the Eady problem, for instance, the short-wave cut- off is obtained when ρ

_c

= − 1 (i.e. µ < 0 and c

₂₂

> c

₁₁

).

From (7), it also follows that if ' = 0, there is only one eigen- value c = c ¯ with multiplicity 2. There is also only one eigenvec- tor, which has the form (α

₁

, α

₂

) = ( − ρ

_c

, 1). This eigenvector is the mNM, in which the CRWs phase-lock either with zero phase-difference (if ρ

_c

= − 1) or with a phase-difference equal to π (if ρ

_c

= 1).

Clearly, the gNM and dNM have become identical. There- fore, at this wavenumber, the NMs—or, more precisely, the single NM—cannot describe correctly the evolution from ini- tial conditions with arbitrary upper and lower CRW.

¹

What then happens with such initial conditions in this limit?

As already announced in Section 1, it turns out that such initial conditions trigger a resonance-like phenomenon as a result of which the amplitude of the mNM increases linearly with time. Mathematically, the behaviour of the system under these circumstances is well understood (Boyce and DiPrima, 2003):

the inclusion of a secular term proportional to t is needed if the eigenvalue has multiplicity 2, and there is only one associated eigenvector. The physics of the resonance can be understood as follows. The mNM itself is a neutral solution of the equations.

Any initial condition differing from this mNM can use part of its wind field to excite the mNM. Since there is no feedback from the mNM to the initial condition, the growth is linear and, in principle, unbounded (as long as friction is ignored). The above

1

Since the CRWs require the existence of the gNM, their construction can be difficult in the limit of vanishing growth rate.

Tellus 60A (2008), 5

Generalization 2: initial conditions

¤ 2-wave system (Davies&Bishop1994,…)

¤ Understand (non)-modal evolution of arbitrary superpositions of growing and decaying NM: e.g. upper-level precursor

¤ 2 coupled non-linear ode’s for amplitude-ratio and phase- difference

self-propagation interaction

complex amplitudes crw-L and crw-U

Example

⁽Eady with stratosphere)

¤ vertical: scaled CRW amplitude ratio (R=1 <=> gNM)

¤ horizontal: CRW phase difference

un-stable

(un+stable fixed point)

0 pi

-pi marginal

(one saddle point) 1

0 pi

-pi

1

long waves

0 pi

-pi 1

short waves

R=inf

0

stable

(two stable centers)

R=inf

0

short-wave cutoff

De Vries & Ehrendorfer 2008

Generalization 3: continuous spectrum

¤ Required for evolution of general initial conditions

¤ Peculiar structure & neutral <= previously ignored

¤ Important for singular vector / non-modal growth (Farrell …)

¤ Resonating continuum modes ⁽Chang1992, V&Opsteegh05, Jenkner&Ehrendorfer2006)

KRW

≠ ^{A + B}

+

56

time

time

KRW spirals clockwise to origin

RED: crw-u x crw-l BLACK: rem x crw-l

..As if just three waves

¤ Radius: amplitude ratio of v

¤ Angle: phase-difference

¤ thin: 3-wave system (Dirren

& Davies 2004, extended in de Vries et al 2009)

¤ Simple evolution

¤ KRW excites upper and lower CRW coherently

¤ CRWs then approach phase- lock from a hindering config.

¤ Follows PV-thinking

¤ Almost f-plane like

the pseudomomentum bracket (B1)h_i(›!q /›y) 3 (8) and making use of the orthogonality relations to get

_

a5La1F(t), L5!ik c11 !g₁₂/k

!g₂₁/k c₂₂

! "

, (15) wherea5(a₁,a₂)^Tanda_j5a_jexp(ie_j) denote the CRW amplitudes and phases. The components of the ‘‘forc- ing’’ F(t) are given by

F^j(t)5 !i

h_j, ›q!

›yy_p(t)

# $

h_j, ›q!

›yh_j

# $ ,

where {,} is defined in (B1) andy_pis the meridional wind associated with q _p; that is, y_p(z,t) 5 ikÐ

G(z,z9)q_p (z9,t)dz9, with q_p(z,t) 5 q _p(z, 0) exp [!ik!u(z)t]. Note that (15) is a generalization of (13) and could be written in similar form by separating it into real and imaginary components. The formal solution of (15) is

a(t)5P(t, 0)a(0)1 ðt

0

P(t9, 0)F(t9)dt9, (16) where P(t,t₀) 5 exp[L(t 2 t₀)] is the propagator. In general (16) cannot be integrated analytically and a numerical method must be employed. Nevertheless, the phase-space trajectories of the CRWs have a predict- able shape (as illustrated for instance in Fig. 4), and on the basis of the Rossby wave properties the qualitative features of the general initial value problem can be anticipated in many cases without explicit calculation.

b. Partitioning the initial perturbation PV

The single KRW used as an initial condition in section 4 was not able to propagate relative to the basic flow, making it reasonable to assign all initial PV to the pas- sive component:q_p(0)5q(0). However, whenq!_y 6¼0, it is not generally possible to distinguish the PV anomalies that are associated with meridional displacements of basic-state PV contours q _d, and those that are not, q_p. However, the partitioning influences the phase-space trajectories deduced for the CRWs, the relative contri- butions of different growth mechanisms, and the accur- acy of the PAR-PV approximation, as seen in section 4.

The approach taken here is to vary the partition between q_d and q_p systematically and seek the initial partition that minimizes the error made by the PAR-PV approx- imation.

One extreme is to assign the complete IC toq _p. This would be exact if the initial perturbation exists where

!

q_y 5 0 and is also appropriate when the IC is given by a single KRW, as in section 4. In this case the initial CRW

amplitudes must be zero. The opposite extreme is to project the full initial perturbation onto the CRWs (as with modal decomposition), which can only be associated with displacement PV. Intermediate ICs are represented by a single tuning parameter s 2 [0,1], which varies the amount of projection of the IC onto the CRWs:

q_d(0)5s[A₁q ₁1A₂q₂],

where A_1,2 are the projection coefficients of q (0) [NB:

not q _d(0)] onto the CRWs. Above it is implicitly as- sumed that all the PV that is not partitioned into the CRWs at initial time contributes to the passive PV component so that q_p(0) 5 q (0) 2 q _d(0). Here s 5 0 implies q _p(0) 5 q (0) and s 5 1 implies subtracting the entire CRW projections fromq (0). Note that the s 5 1 assumption does not imply the same CRW evolution as obtained by modal decomposition; under the PAR-PV approximation, q _p(0) 5 q (0) 2 q_crws(0) still forces the CRWs following (15), whereas the CRWs in a modal decomposition follow the two-wave equations. The optimal partitioning of the initial condition is given by the value of s for which the error in the subsequent perturbation solution made by the PAR-PV approxi- mation is a minimum. The schematic in Fig. 6 summa- rizes the essential aspects of the PAR-PV approximation.

Another, alternative partition is to vary the initial CRW amplitudes and phases such that |q _r(0)|² is mini- mum, where |#| indicates some norm (for instance, total energy). Although it is certainly possible to devise more sophisticated methods for determining an ‘‘optimal’’

partitioning,⁴ these in general require a prior estimate of the full system’s evolution.

6. Testing the PAR-PV approximation

In this section the PAR-PV approximation is tested for more general ICs. As a measure of the accuracy of the PAR-PV approximation, we calculate perturbation total energy

E5 !1

2[fc,q g1fq,cg] (17) of the reduced model and compare it with the result for the full model (3). The PAR-PV approximation gives exact results in the long-time limit when the gNM emerges as dominant. However, at intermediate times it deviates from the full model and the maximum energy contained in the neglected component q_n is measured.

4One such method is to choose the initial CRW projection co- efficients such that they minimize the error measure over a finite time window, somewhat similar to a four-dimensional variational data assimilation (4D-VAR) approach.

APRIL2009 D E V R I E S E T A L . 873

the pseudomomentum bracket (B1)h_i(›q!/›y)3 (8) and making use of the orthogonality relations to get

_

a5La1F(t), L5!ik c11 !g₁₂/k

!g₂₁/k c₂₂

! "

, (15) wherea5(a1,a2)^Tandaj5ajexp(iej) denote the CRW amplitudes and phases. The components of the ‘‘forc- ing’’F(t) are given by

Fj(t)5 !i

h_j, ›!q

›yyp(t)

# $

h_j, ›!q

›yh_j

# $ ,

where {,} is defined in (B1) andypis the meridional wind associated with q_p; that is, y_p(z,t)5 ikÐ

G(z,z9)q_p (z9,t)dz9, with q_p(z,t)5 q_p(z, 0) exp [!iku(z)t]. Note! that (15) is a generalization of (13) and could be written in similar form by separating it into real and imaginary components. The formal solution of (15) is

a(t)5P(t, 0)a(0)1 ðt

0

P(t9, 0)F(t9)dt9, (16) where P(t,t0) 5 exp[L(t 2 t0)] is the propagator. In general (16) cannot be integrated analytically and a numerical method must be employed. Nevertheless, the phase-space trajectories of the CRWs have a predict- able shape (as illustrated for instance in Fig. 4), and on the basis of the Rossby wave properties the qualitative features of the general initial value problem can be anticipated in many cases without explicit calculation.

b. Partitioning the initial perturbation PV

The single KRW used as an initial condition in section 4 was not able to propagate relative to the basic flow, making it reasonable to assign all initial PV to the pas- sive component:q_p(0)5q(0). However, whenq!_y 6¼0, it is not generally possible to distinguish the PV anomalies that are associated with meridional displacements of basic-state PV contours qd, and those that are not, qp. However, the partitioning influences the phase-space trajectories deduced for the CRWs, the relative contri- butions of different growth mechanisms, and the accur- acy of the PAR-PV approximation, as seen in section 4.

The approach taken here is to vary the partition between qd and qp systematically and seek the initial partition that minimizes the error made by the PAR-PV approx- imation.

One extreme is to assign the complete IC toqp. This would be exact if the initial perturbation exists where q!_y 50 and is also appropriate when the IC is given by a single KRW, as in section 4. In this case the initial CRW

amplitudes must be zero. The opposite extreme is to project the full initial perturbation onto the CRWs (as with modal decomposition), which can only be associated with displacement PV. Intermediate ICs are represented by a single tuning parameter s 2 [0,1], which varies the amount of projection of the IC onto the CRWs:

q_d(0)5s[A1q₁1A2q₂],

where A1,2 are the projection coefficients of q(0) [NB:

not qd(0)] onto the CRWs. Above it is implicitly as- sumed that all the PV that is not partitioned into the CRWs at initial time contributes to the passive PV component so that qp(0) 5 q(0) 2 qd(0). Here s 5 0 implies qp(0) 5q(0) and s5 1 implies subtracting the entire CRW projections fromq(0). Note that thes51 assumption does not imply the same CRW evolution as obtained by modal decomposition; under the PAR-PV approximation, qp(0) 5q(0) 2qcrws(0) still forces the CRWs following (15), whereas the CRWs in a modal decomposition follow the two-wave equations. The optimal partitioning of the initial condition is given by the value of s for which the error in the subsequent perturbation solution made by the PAR-PV approxi- mation is a minimum. The schematic in Fig. 6 summa- rizes the essential aspects of the PAR-PV approximation.

Another, alternative partition is to vary the initial CRW amplitudes and phases such that |q_r(0)|²is mini- mum, where |#| indicates some norm (for instance, total energy). Although it is certainly possible to devise more sophisticated methods for determining an ‘‘optimal’’

partitioning,⁴ these in general require a prior estimate of the full system’s evolution.

6. Testing the PAR-PV approximation

In this section the PAR-PV approximation is tested for more general ICs. As a measure of the accuracy of the PAR-PV approximation, we calculate perturbation total energy

E5 !1

2[fc,qg1fq,cg] (17) of the reduced model and compare it with the result for the full model (3). The PAR-PV approximation gives exact results in the long-time limit when the gNM emerges as dominant. However, at intermediate times it deviates from the full model and the maximum energy contained in the neglected componentqnis measured.

4One such method is to choose the initial CRW projection co- efficients such that they minimize the error measure over a finite time window, somewhat similar to a four-dimensional variational data assimilation (4D-VAR) approach.

APRIL2009 D E V R I E S E T A L . 873

Growth mechanisms (f-plane)

57

Resonance - one-way

passive x active => lin. growth Interaction - two-way

active x active => exp. growth

Orr-mechanism - no-way

passive => dep. on structure

both energy

and enstrophy

growth

energyonly growth

Also hold approx on beta plane and for more complex basic states, with CRWS replacing the Eady edge waves

Similar for more complex IC and states: beta-plane with broad

tropopause

T=13

CRW-U

CRW-L FULL

Residual

PAR-PV framework

(HdV, Methven,Hoskins)