Generalized Stability of Non–Geostrophic Baroclinic Shear Flow

Part I – Large Richardson Number Regime ∗

E. Heifetz and Brian F. FarrellDepartment of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

∗Corresponding author address: Eyal Heifetz, Dept. of Geophysics and Planetary Sciences, Tel–Aviv Univer-sity, Israel. Email: eyalh@cyclone.tau.ac.il

1

Abstract

The generalized stability analysis of baroclinic shear flow is made using the primitive equa-

tions (PE), and with typical synoptic mid–latitudinal values of vertical shear and stratification.

Unbounded constant shear flow and the Eady model are taken as example basic states and

energy is chosen as the reference norm. Comparison is made with results obtained using quasi–

geostrophic (QG) analysis. Because the rotational and the divergent modes are orthogonal the

PE and the QG dynamics give similar results for time scales of a few days. However, the PE

initial growth rate during the first few hours, exceeds the QG growth at all wavelengths, and

attains values as much as two orders of magnitude greater as the wavelength decreases and ap-

proaches unit aspect ratio. This extreme PE growth is due to the direct kinetic energy growth

mechanism which is filtered out by QG.

The PE response to spatially and temporally uncorrellated stochastic forcing reveals a com-

parable amount of variance supported by the rotational and the divergent modes. This strong

response of the divergent modes can be brought into accord with observed gravity wave variance

at large scale by forcing with temporally correllated noise and taking damping proportional to

frequency.

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1 Introduction

Quasi–geostrophy (hereafter QG) provides a theoretical basis for understanding atmospheric dy-

namics at the synoptic scale including the essential nature of cyclone development. However, QG

theory is strictly valid only in the limit of small Rossby number and large Richardson number

(flows strongly constrained by rotation and stratification) and in the mid–latitude atmosphere

these constraints are often quantitatively violated, particularly in association with strong cy-

clogenesis and frontogenesis. In order to secure and extend understanding of synoptic scale

dynamics the effects of relaxing the QG assumptions must be determined. A necessary first step

in exploring the effects of relaxing the QG constraints is to examine nongeostrophic dynamics in

the linear limit. This problem has been studied previously using the method of modal analysis

(e.g. Aranson(1963), Stone(1966), Derome and Dolph(1970), Mak(1977), Bannon(1989), Naka-

mura(1988), Snyder(1994)) but because atmospheric flows are generally highly non-normal a

comprehensive assessment of linear stability of these flows requires using the methods of Gener-

alized Stability Theory (Farrell and Ioannou, 1996) which we use in this study.

Among the effects of nongeostrophic dynamics in the presence of baroclinic shear are modi-

fication of the primarily rotationally constrained Rossby waves which are supported by QG and

in addition the excitation and growth of new waves with primarily ageostrophic balance which

are not supported in QG. Moreover, the separation between these rotational and divergent wave

manifolds is not absolute except in the limit of vanishing shear and the interaction between

primarily rotational and primarily divergent waves in the presence of shear is a central issue in

nongeostrophic dynamics. A related issue is the nature of geostrophic adjustment in the presence

of strong shear where the conceptual separation of gravity waves and Rossby waves as separate

agents in the adjustment process which was so useful in the unsheared case can no longer be

justified.

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Another problem is understanding the observed confinement of synoptic and subsynoptic dis-

turbances in mid–latitudes to a primarily geostrophic balance. Among the explanations which

have been advanced for this phenomena are that the potential for disturbance growth is confined

to the primarily rotational modes and that rapid adjustment continually returns disturbances

with divergent components back to the primarily rotational manifold. We examine these mecha-

nisms below and find that the potential growth of ageostrophic disturbances in fact may greatly

exceed that of primarily geostrophic disturbances and that the primarily nongeostrophic modes

are potentially highly persistent.

We choose the Eady problem as an example to examine non-geostrophic dynamics because of

its simplicity as well as because there are a number of previous works based on the Eady model

so that the properties of this model are familiar (Stone(1966), Farrell(1984), Nakamura(1988),

Bannon(1989)). In part I we analyze the generalized stability properties of non–geostrophic

synoptic scale dynamics, for typical mid–latitude values of mean baroclinic shear, Λ = Uz =

O(10−3s−1), and stratification, N = O(10−2s−1), corresponding to Richardson number Ri =

(NΛ )2 = O(100). The baroclinic primitive equations (PE) are introduced in section 2 and a 3D

Helmholtz decomposition applied to separate geostrophic from non–geostrophic components.

The Eady model is reviewed and its optimal growth explored as a function of spatial scale in

section 3. The response of the system to stochastic forcing is examined in section 4 and we

conclude with a review of our results in section 5.

In part II the generalized stability of non-geostrophic dynamics for intermediate and small

Richardson numbers is examined. Small Richardson number corresponds to a combination of

larger shear and smaller static stability than is typical for midlatitude synoptic scale motions.

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2 Formulation

2.1 The baroclinic primitive equations in unbounded shear

The momentum, thermodynamic and continuity equations linearized about a basic state zonal

wind with constant vertical shear, U = Λz, are:

du

dt= fva − Λw, (1a)

dv

dt= −fua, (1b)

dw

dt= −φz + θ (1c)

dθ

dt= fΛv −N2w, (1d)

uax + vay + wz = 0. (1e)

In (1) the linearized substantial derivative is denoted by ddt ≡ ∂

∂t + U ∂∂x ; the Boussinesq ap-

proximation is made, and a constant value of the Coriolis parameter, f , and a constant mean

stratification are assumed so that N2 = gθ0

θz, where N is the Brunt–Vaisala frequency. The

mean zonal shear is assumed to be in thermal wind balance, fΛ = − gθ0

θy, where θθ0

is the

mean state potential temperature scaled by a representative tropospheric value and g is the

constant gravitational acceleration. The zonal, meridional, and vertical directions are denoted

(x, y, z) respectively, and the total vector velocity perturbation is u = (u, v,w) = ug +ua where

ug = (ug, vg, 0) and ua = (ua, va, wa) are the geostrophic and ageostrophic components. The

geostrophic component of the perturbation velocity is ug = f−1(−φy, φx, 0), where φ is the

geopotential perturbation in log pressure coordinates.

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The QG equations differ from (1) in assuming that the substantial derivative of the horizontal

velocity components on the LHS of 1a and 1b involves only the geostrophic components of the

velocity; by assuming quasi–hydrostatic balance (vanishing of the LHS of 1c); and by neglecting

the advection of the mean wind by the vertical component of perturbation velocity (the term

Λw on the RHS of 1a). As a result of the latter, the QG framework excludes the mechanism of

energy transfer from the mean to the perturbation mediated by Reynolds stress.

2.2 An upper bound on energy growth

In the QG limit the growth of volume integrated total perturbation energy can be shown to be

(e.g. Gill 1982, p.566):

∂

∂t

∫EqgdV =

∂

∂t

∫ 12

[ug

2 + vg2 + (

θg

N)2

]dV = (

f

N)Λ

∫vg(

θg

N)dV, (2)

where for simplicity constant shear and stratification have been assumed. From (2) it can be

seen that total perturbation energy growth is due solely to positive meridional heat flux, which

converts available potential energy (hereafter APE) from the basic state to the perturbations.

By appeal to Schwartz’s inequality we have:

∂

∂t

∫EqgdV ≤ (

f

N)Λ

∫ 12

[vg

2 + (θg

N)2

]dV ≤ (

f

N)Λ

∫EqgdV, (3)

and therefore,1∫

EqgdV

∂∫

EqgdV

∂t≤ (

f

N)Λ. (4)

Thus, the maximum QG energy growth rate cannot exceeds ( fN )Λ, which for typical mid–

latitudinal tropospheric values of f = 10−4s−1; N = 10−2s−1 gives an upper bound on the

energy growth rate of 10−2 times the shear. On the other hand, the total energy growth in PE

can be shown (using 1) to be:

∂

∂t

∫EdV =

∂

∂t

∫ 12

[u2 + v2 + w2 + (

θ

N)2

]dV = Λ

∫ [(f

N)v(

θ

N)− uw

]dV ≤

6

Λ∫ 1

2

[(f

N)(

v2 + (θ

N)2

)+ u2 + w2

]dV ≤ Λ

∫EdV, (5)

and therefore,1∫

EdV

∂∫

EdV

∂t≤ Λ. (6)

Hence, the theoretical upper bound on PE energy growth rate is Nf times the upper bound

on energy growth rate in QG. Inequalities (5) and (6) indicate that the KE shear mechanism

could be, at least in principle, much more effective than the conventional APE mechanism

in transferring energy from the mean flow to the perturbations. In other words, when uw is

comparable to v( θN ), perturbation growth may be dominated by the momentum flux mechanism

which directly reduces the mean shear, rather than by the heat flux mechanism which reduces

the mean shear indirectly (by reducing the temperature gradient that maintains the shear). A

scale analysis presented below suggests that the KE shear term should dominate, producing

larger growth than the baroclinic QG growth, when the perturbation horizontal scale is smaller

than 1000km. This raises the fundamental question of why most growth is at synoptic scale and

observed to be baroclinic in origin.

2.3 The non-geostrophic Eady model

Consider the Eady (1949) model for the mid–latitude troposphere consisting of a mean zonal

wind which increases linearly with height, U(z) = Λz, between two horizontal solid boundaries

on z = 0;H, that model the earth’s surface and the tropopause. As in the unbounded shear

problem above, an f–plane, constant stratification, and the Boussinesq approximation are as-

sumed. Characterizing the Eady model are the following quantities: the tropopause height, H,

which provides the scale for distance in the vertical, the vertical shear, Λ (and thus the scale for

horizontal velocity ΛH), the stratification, N , and the rotational time scale, f−1. A horizontal

scale, L, is not imposed by the model and one could be chosen arbitrarily. Here, we adopt the

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conventional scaling by the Rossby radius of deformation, L = NH/f , (e.g. Nakamura 1988)

1, however perturbation dynamics with larger and smaller scale than the Rossby radius will be

investigated as well.

Using f ,H ,L to nondimensionalize (1) results in:

du

dt= va − Sw, (7a)

dv

dt= −ua (7b)

α2 dw

dt= −φz + θ (7c)

dθ

dt= Sv − w, (7d)

uax + vay + wz = 0, (7e)

where the tilde superscript denotes nondimensional variables; the linearized substantial deriva-

tive is ddt

= ∂∂t

+ zS ∂∂x , the shear number is defined as S = 1/

√Ri = Λ

N , and α = H/L denotes

the aspect ratio. We hereafter approximate (7c) by assuming quasi–hydrostatic balance, θ = φz,

which is valid when α2 << 1. Thus, we can not address perturbation dynamics for disturbances

with horizontal scale equal to or smaller than the associated vertical scale 2.1This is equivalent to setting B = RiRo2 = 1, where B, Ri = (N

Λ)2, Ro = U

fL= ΛH

fL, are the Burger,

Richardson and Rossby numbers, respectively.2Note that the shear growth mechanism discussed in subsection 2.2, that is omitted in QG, does not require

violation of hydrostatic balance. In order for the transfer of mean kinetic energy to the perturbations mediatedby downgradient momentum flux to dominate transfer of mean potential energy to the perturbations mediated bydowngradient heat flux i.e., for |uw| > | f

Nv θ

N|, it is required that the aspect ratio α > (f/N). Hence, in the range

of 1 > α > (f/N) which is equivalent to O(10km) < L < O(1000km), both quasi–hydrostaticity and potentialdominance of the KE shear mechanism simultaneously hold. Also, note that under quasi–hydrostatic balance, itis valid to neglect the square of the vertical velocity in the total energy term. Inequalities (5) and (6) remaincorrect but the total energy term is modified there to E = 1

2[u2 + v2 + ( θ

N)2].

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Eady (1949) suggested transforming (7) into a set of three equations for the three variables:

vertical component of relative vorticity, vertical velocity and the laplacian of the potential tem-

perature. He also showed how (7) can be written as a single, third order equation for the verti-

cal velocity. In this study, however, we choose to decouple the geostrophic and nongeostrophic

contributions. Therefore, we transform (7) into the following set (hereafter we omit the tilde

superscript):d

dtvaz = −2Svx + (wx − uaz), (8a)

d

dtuaz = 2Svy − (wy − vaz), (8b)

d

dt(vx − uy + θz) = S(wy + vaz), (8c)

in which the perturbation geostrophic components satisfy the thermal wind relations, vgz =

θx, ugz = −θy. In the QG limit the LHS of (8a,b) vanish and these equations become the

linearized version of the Q–vector diagnostic equations (Hoskins et. al. 1978) which assume

a perfect balance between mean meridional temperature advection by the geostrophic velocity

and the ageostrophic circulation response. However, in the non–geostrophic dynamic regime, the

imbalance between those two terms forces the tendency of the ageostrophic wind. On the RHS

of the PV equation, (8c), Swy is the solenoidal term that tilts the horizontal mean vorticity

into the vertical, due to meridional differential vertical advection of the mean zonal wind. The

second term, Svaz, alters the thermal component of the PV due to vertical differential advection

of the mean temperature by the ageostrophic velocity. In the QG limit both terms vanish.

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2.4 3D geostrophic Helmholtz decomposition

Applying the 3D Helmholtz decomposition, suggested by Muraki et al. (1999):

v−u

θ

= ∇Φ +∇×

FG0

=

Φx −Gz

Φy + Fz

Φz + Gx − Fy

, (9)

to (8), we obtain

∂

∂t

FGΦ

= A

FGΦ

, (10)

where A is the finite difference propagator matrix (details are given in Appendix A). The benefit

of this representation is that the geostrophic component 3 (vg,−ug, θg) = ∇Φ, is decoupled from

the ageostrophic component, (va,−ua, θa) = ∇×(F,G, 0). Furthermore, for a horizontal Fourier

decomposition of the solution spatial structure:

FGΦ

=

F (z, t)G(z, t)Φ(z, t)

ei(kx+ly), (11)

where k, l are the scaled zonal and meridional wavenumbers, it can be shown that the geostrophic

and the ageostrophic components are orthogonal to each other under the energy norm. Thus,

the volume integrated energy can be decomposed into

E =∫

EdV =∫ 1

2

[u2 + v2 + θ2

]dV =

∫EgdV +

∫EadV ; (12)

where Eg = 12 [ug

2 + vg2 + θg

2]; Ea = 12 [ua

2 + va2 + θa

2]. Hence, this decomposition provides a

convenient partitioning of the energy between geostrophic and the ageostrophic components.

3Here we include θg , which is the part of the temperature that satisfies the thermal wind balance, under theterm “geostrophic”.

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3 Generalized stability analysis

3.1 The energy norm

We choose the total energy inner product to measure perturbation magnitude. Energy provides

a convenient and physically interpretable norm while (potential) enstrophy does not because

the divergent modes, in the absence of shear, have zero PV (see a brief analysis in Appendix

B), which is also true for the QG Eady normal modes. It follows that the enstrophy inner

product does not possess the positive definiteness required to generate a norm. In addition, the

non–normal energetic interaction among modes, which occurs in the presence of shear, plays a

central role in our study which further motivates the choice of the energy norm.

Denote as η the state vector on the LHS of (11), and define the generalized energy coordinates

transformation of η as χ, with the property that the Euclidean inner product of χ is the energy

χ†χ = E (the dagger superscript indicates the Hermitian transpose). It is clear that χ =

M1/2η with M the positive definite Hermitian matrix with the property E = η†Mη. The

nongeostrophic dynamics (10) becomes in the energy coordinate representation:

∂χ

∂t= Dχ, (13)

where D = M1/2AM−1/2 (the energy coordinate transformation is given explicitly in Appendix A).

We study both the inviscid version of the Eady model and the viscid version (Appendix C).

The inviscid problem is singular and the continuous spectrum is composed of singular modes

while with vertical diffusion however small the problem is nonsingular and the singular modes

of the inviscid problem are replaced by a discrete spectrum of analytic modes. Analysis of

optimal growth in the inviscid case, which is due to non–normal interaction among the singular

continuous spectrum and the analytic normal modes, should be verified in the presence of small

diffusion in which case the interaction is among physically realistic discrete analytic modes which

11

have no counterpart in the continuous spectrum of singular modes.

We begin with an investigation of the instantaneous growth of the system, and then continue

with optimal excitation analysis for different intermediate time targets. We end this section by

referring to the optimal excitation for the asymptotic t →∞ limit.

3.2 Instantaneous growth

The maximum instantaneous growth rate of system (13) is equal to the largest eigenvalue of

the matrix (D + D†)/2, and the structure χ producing this growth rate is its associated

eigenvector (Farrell and Ioannou, 1996). The growth rate of χ, scaled by the time scale for

horizontal advection, ( fN )Λ, as a function of the zonal wave number nondimensionalized by

the the horizontal space scale (Nf )H (we consider symmetric horizontal plane waves, k = l),

is shown in Fig.1(a)4. The solid line indicates the most unstable normal mode growth rate

of the classical QG Eady model, while the dashed line shows the maximal instantaneous QG

growth rate. As expected from the QG energy growth rate bound (4), the QG growth rate does

not exceed the scaled value 0.5, i.e., ( fN )Λ/2 (the instantaneous growth rate of χ is half the

instantaneous energy growth rate). In contrast the circled symbols indicate the instantaneous

growth rate obtained from the PE solution which increases dramatically with wavenumber, from

scaled values of 0.5 as k = l → 0. This growth rate is composed of a contribution from the

APE growth term, vθ/2, indicated by the asterix, and a contribution from the KE shear term,

−uw/2, indicated by the plus symbols. We can see that the APE mechanism dominates at

small wavenumbers but as k exceeds 2.25, corresponding to a cyclone with half wavelength of

approximately 1000km, the shear term begins to dominate so that as k reaches 10 nearly all of

the instantaneous growth arises from the KE shear term.

The energy partition between the geostrophic component (indicated by the asterix symbols)4We restrict ourselves to the wave number range 0 < k ≤ 10 in which the hydrostatic approximation is valid.

12

and the ageostrophic component (indicated by the plus symbols) for the instantaneous optimal

structures is shown in Fig.1(b). The geostrophic and the ageostrophic energy components are

the first and the second terms on the RHS of (12). For small wavenumber the partition is even,

however, as the wavenumber increases the ageostrophic component begins to dominate so that

for k > 5 its fraction of the total energy approaches 75%. The dependence of energetics on

wavenumber is shown in Fig.1 for the shear S = 0.1 or equivalently for Ri = 100. Moreover, the

result in Fig.1 is generic and insensitive to the Richardson number (over the range O(10−4) ≤

Ri ≤ O(104)). It is also insensitive to the addition of small amounts of vertical diffusion to the

model in order to regularize the modes.

Contributions to the APE growth term, vθ/2, indicated as before by the asterix symbols,

are shown in Fig.2(a): the pure geostrophic contribution vgθg/2, is indicated by the dashed line,

while the pure ageostrophic contribution vaθa/2, is indicated by the dashed–dot line. The cross

interactions vaθg/2, and vgθa/2, are indicated by the circle and plus symbols, respectively. We

see that at small wavenumbers the large APE growth is dominated by cross interactions in which

the ageostrophic meridional wind advects the geostrophic potential temperature. The structure

of va and θg are shown in zonal–vertical cross–section in Fig.3(a,b), for k = 1 revealing a

remarkable correlation betweeen the va and θg fields. As the wavenumber increases the APE

growth becomes dominated by the ageostrophic heat flux, however the APE growth is by then

far smaller than the shear growth. The shear growth, indicated by plus symbols on Fig.2(b),

is composed of the pure ageostrophic component −uaw/2, indicated by the dashed line, and

by the cross interaction −ugw/2, indicated by the circle symbols. These two components are

of the same order and become equal for large k (not shown here). The (u,w) wind vectors

for k = 10, shown in Fig.3(c), displays the characteristic “leaning against the mean zonal flow”

orientation associated with downgradient momentum transport and perturbation energy growth

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with a shear KE source.

To summarize, for the Eady model the maximum instantaneous growth rate obtained by the

PE solution always exceeds the maximum growth rate obtained by the QG solution (the dashed

line in Figs.1a,b). For small wavenumbers, the conventional baroclinic APE growth mecha-

nism dominates. However, PE dynamics differs from QG dynamics by including explicitly the

ageostrophic component, and the PE solution uses this ageostrophic wind to advect geostrophic

temperature, achieving a larger growth rate than is attained by advecting the geostrophic tem-

perature with the geostrophic wind alone, as in QG dynamics. At larger wavenumbers PE dy-

namics uses the direct KE shear mechanism, which is filtered out in QG, to achieve instantaneous

growth rate that can reach, for very large k (not shown here), up to two orders of magnitudes

larger than QG growth rate. The question of whether the system is able to maintain such large

growth, or whether it is a short term phenomena, is examined next.

3.3 Intermediate time optimal growth

The solution of (13) can be written as

χ(t) = eDtχ(0) = UΣV†χ(0), (14)

where UΣV† is the singular value decomposition (hereafter SVD) of the matrix propagator

from an initial time zero to target time t, Φ(0, t) = eDt, in which U and V are two unitary

matrices and Σ is a diagonal matrix with positive elements ordered by growth (e.g. Noble

and Daniel, 1988). We identify the optimal initial unit magnitude perturbation χ(0) producing

maximal growth at target time t to be the first column vector of V which is projected by Φ(0, t)

onto the first column unit vector of U with magnitude increase σ = Σ11, where σ denotes the

optimal growth over time t.

We analyse next the optimal growth and the associated structures for the shear number

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S = 0.1, corresponding to Ri = 100, focusing on two characteristic wavenumbers k = 1, 10.

3.3.1 The case S=0.1, k=1

The optimal growth properties are shown in Figs.4(a,b) where the circles indicate the PE solution

and the asterices indicate the QG solution with the same shear number. The maximum growth

σ(t) of a perturbation over the interval of time t is shown in Fig.4(a). The mean growth rate

γ(t), for these target times, satisfying σ(t) = exp (γt), where γ(t) is normalized by fS is shown

in Fig.4(b). We see that although the optimal instantaneous PE growth rate is larger than the

optimal instantaneous QG growth rate, the optimal growth attained for times larger than three

hours in PE is almost identical to the optimal growth attained in the QG anaysis.

In order to understand these results we consider next the dispersion relation and the optimal

growth structures. Recall that in the abscence of shear, the pure Poincare divergent mode

frequencies (Appendix B) satisfy ωr/f = ±[1 + (K/nπ)2]1/2 (where K =√

k2 + l2 is the

horizontal wavenumber) and thus for k = l = 1, ωr/f ∈ ±(1, 1.1). On the other hand, the

pure QG rotational mode’s phase speeds lie within the zonal mean wind range and therefore

ωr/f ∈ ±(0, kS) for these modes. The dispersion relation is shown in Fig.5(a) where the circles

indicate the ω values of the two QG normal modes. The energy partition of the modes is shown

in Fig.5(b) sorted increasing in ωr with the geostropic and the ageostropic components indicated

by gray and black thick lines, respectively. We can see three distinct branches where the left and

the right branches are the nearly nondivergent modes. Their frequency range is almost identical

to that of the Poincare modes with no shear (when the Doppler shift, kS/2 = 0.05, due to

the averaged mean zonal wind is taken into account). The middle branch contains two normal

modes, which are almost identical to the QG normal modes, and also a spectrum of singular

modes, with steering levels within the domain. A contour plot of Ei,j = χ†iχj, at the intersection

of (x = i, y = j), where χi is the normalized generalized coordinate vector of mode i, so that

15

χ†iχi = 1 is shown in Fig.5(c). This contour value is a measure to the energy orthogonality

between modes, and we can see that the divergent modes are almost orthogonal to each other,

and are completely orthogonal to the rotational modes. The latter is not surprising since the

geostrophic and the ageostrophic components are orthogonal under the energy norm (c.f. (12)).

The rotational singular modes and the two normal modes are not orthogonal to each other, as

in QG dynamics (Farrell, 1984).

As was shown by Nakamura (1988), critical lines in the PE dynamics exist if the mode’s

phase speed satisfies

[k(Sz − c) + 1](Sz − c)[k(Sz − c)− 1] = 0. (15)

The rotational modes, excluding the two unstable and decaying normal modes, are all singular

due to vanishing of the quantity in the mid bracket of (15). Since for the left branch c ∈

(−1.05,−0.95), all modes, excluding the very first, are singular due to vanishing of the quantity

in the left bracket. Similarly for the right branch c ∈ (1.05, 1.15), and thus all modes, excluding

the very last, are singular due to vanishing of the quantity in the right brackets.

The result of adding the smallest value of viscosity required to resolve these singular modes

(Re = 108, see Appendix C) is shown in Figs.5(d–f). Since diffusion is proportional to the

second derivative in z, the Poincare waves, as well as the rotational discrete modes, decay in

proportion to the square of their vertical wavenumber (Fig.5d). The energy partition, shown in

Fig.5(e), is robust and so is the orthogonality between the rotational and the divergent modes.

The viscid dynamics produce a slight increase in the amount of non–orthogonality among the

modes, Fig.5(f), and slightly decreases the optimal growth.

We examine next the optimal evolution over a time scale typical of cyclone development.

The evolution of the distribution of fluxes and energy for a two day optimal are shown in

Figs.6(a) and 6(c) respectively. The evolution is almost purely on the geostrophic manifold and

16

the growth is almost completely due to the geostrophic component of meridional heat flux (the

total heat flux matches almost exactly the flux obtained by the QG dynamics). A zonal–vertical

crossection of the initial and final meridional wind for the two day optimal is shown in Figs.

7(a,b). These structures are almost identical to the structures obtained in QG dynamics. We

wish to understand why the two day optimal does not exploit the large instantaneous growth,

that is available to it from the ageostrophic heat flux vaθg. The evolution over 48 hours of the

initial perturbation which has the largest instantaneous growth is shown in Figs.6(b,d). The

ageostrophic heat flux vaθg falls rapidly and becomes negative by 6 hours. Figs.7(c,d) and the

same fields after 3 hours are shown in Figs.7(e,f). While the geostrophic field remains almost

unchanged, the ageostrophic field is tilted strongly by the mean shear, decreasing the correlation

between the two fields by more than 60% over this time interval. Hence, the vaθg mechanism is

not sustained, and in addition even its large initial growth does not project onto the geostrophic

manifold, due to the orthogonality between the geostrophic and the ageostrophic modes (cf.

Figs.5c,f).

3.3.2 The case S=0.1, k=10

The optimal growth properties are shown in Figs.8(a,b) where as in the previous case although

the instantaneous PE growth rate is enormous, the optimal growth for all times (larger than

three hours) is almost identical to the optimal growth obtained in the QG anaysis. The inviscid

dispersion relation for k=10, differs from that obtained for, k=1 (Figs.9a–c). Recall that in the

abscence of shear, for k = l = 10, the pure Poincare modes’ frequencies satisfy ωr/f ∈ ±(1, 4.6)

(Appendix B), where the pure QG rotational mode’s frequencies satisfy ωr/f ∈ ±(0, 1). The

two divergent branches have the non–sheared Poincare’s frequency range, shifted by kS/2 = 0.5.

Although they do not overlap with the rotational modes (on the frequency range of (0 < kS <

1) ), they are approaching. The small instability obtained by the rotational modes is due to

17

the presence of critical layers arising from vanishing of the quantities in the left and the right

brackets of (15) (Nakamura, 1988). Since for the left branch kc ∈ (−4.1,−0.5), only the mode’s

frequencies which overlap with (kSz − 1) ∈ (−1, 0) are singular. Similarly, for the right branch,

where kc ∈ (1.5, 5.1), the singular modes have frequencies which overlap with (kSz +1) ∈ (1, 2).

The singular modes are resolved and the rotational instability vanishes when a small vis-

cosity is added (Fig.9d). The divergent modes are orothogonal to the rotational modes, and

viscosity increases the non–orthogonality in each branch (Figs.9c,f). Energy growth by the two

day optimal is dominated by the geostrophic heat flux (Figs.10a,c), and the meridional wind

structure, is almost identical to that of QG dynamics (Figs.11a,b). The enormous instantaneous

KE shear growth, which is mainly ageostrophic, drops to zero after 3 hours (Figs.10b,d), and

the initial zonal–vertical wind structure that is leaning against the mean shear, is rotated by the

mean shear and becomes untilted after 3 hours (Figs.11c,d). Due to the orthogonality between

the geostrophic and the ageostrophic modes, this enormous initial growth has no projection on

the geostrophic manifold. As in the previous case the viscosity increase slightly the amount of

non–orthogonality among the modes, Fig.9(f), and decreases slightly the optimal growth.

3.4 The asymptotic t →∞ optimal

The optimal excitation for t → ∞ target time is the structure that optimaly excites the most

unstable mode 5. It is the conjugate of the biorthogonal vector of the most unstable mode (which

is the conjugate of the vector that is orthogonal to all other modes except the most unstable

mode), Farrell and Ioannou (1996).

In the case for the inviscid Orr–Somerfeld eq’s this adjoint structure is seen to be concentrated

near the critical level, as shown by Drazin and Reid (1982, section 21). There it is exactly5Here, by “most unstable mode” we refer to the dominant mode which could be the neutral, or the least

damped mode.

18

proportional to χ(t = ∞)/[U (z) − c], where χ(t = ∞) is the most unstable normal mode with

the phase speed c. Although the functional form of the adjoint mode in our problem is not as

simply expressed it can be shown that the adjoint structure in this case is also concentrated in

the region of the critical layer. Therefore, if the “most unstable mode” is neutral so that c is

real, and also if the critical level lies within the domain, then the optimal excitation should be

concentrated at the critical level. If the instability is small, that is the imaginary part of c is

small, we should still expect to find the initial optimal to be concentrated at the vicinity of the

steering level.

The meridional wind zonal–vertical crossections of the initial optimal for time infinity, and of

the most unstable normal mode are shown in Figs.12(a,b) with S = 0.1 and k = 1. Structures

for the same case except in the presence of small viscosity are shown in Figs.12(c,d). The

structures are almost purely geostrophic and are almost identical to the structures obtained

from QG dynamics, where the initial optimal is found to be maximized at the mid–tropospheric

steering level and the most unstable mode is found to be maximized at the boundaries. The

viscid structures are different from the inviscid ones by being slightly more smooth in vertical

structure.

The meridional wind structures where S = 0.1 and k = 10 is shown in Figs.13(a,b). As

can be seen in Fig.9(a), the two most unstable rotational modes are equally unstable, with

critical layers located within the domain, either near the ground or near the tropopause. The

former mode is chosen for display in Fig.13(b) . Although, the presence of explicit ageostrophic

dynamics in the PE framework destabilize the neutral QG Eady modes , their structure remains

almost purely geostrophic, gaining a slight tilt. Since the instability of this mode is small, the

optimal initial excitation is highly concentrated in the vicinity of the critical layer (Fig.13(a)).

It is interesting to note that in the presence of small viscosity all modes becomes damped,

19

(Fig.9(c)), but the least damped modes (almost neutral) are now the two divergent modes with

the smallest and largest frequencies (with steering levels outside the flow), whose structures

are almost identical to the pure Poincare modes with vertical wavenumber ±1. It can be seen

from Fig.9(f) that these two modes are almost orthogonal to all other modes, and therefore

their biorthogonal vectors are nearly identical with the modes themselves as can be seen in

Figs.13(c,d), where the mode with positive wavenumber has been chosen for display.

The above analysis of the optimal linear excitation for target time infinity is a limiting case

shown in part because of its simplicity in that the optimal way to excite the most unstable mode

is its biorthogonal conjugate. In order to assess the relevance of this limit we need to find the

minimal target time required for the optimal excitation to converge to the asymptotic t → ∞

optimal. For the inviscid S = 0.1, k = 1, case this minimal target time is found to be 15 days,

and about 20 days for the viscid case. When S = 0.1, k = 10, the minimal times are 60 and 30

days for the inviscid and the viscid cases, respectively (The large difference between these last

is due to the fact that the least damped divergent mode in the presence of viscosity is entierly

different from the most unstable rotational mode obtained by the inviscid dynamics). As the

characteristic time scale for extra–tropical cyclone development is of the order of a day or two,

these limiting cases are unlikely to be observed.

3.5 Summary

To summarize the large Richardson numbers dynamics, we can see a clear separation between

modes that are almost purely divergent and modes which are almost purely rotational 6. We6This partition can be understood from (A.1). When S = 0.1, then if the divergent part (F, G), is at the order

of one, (A.1c) indicates that the rotational part Φ is of the order of 0.1. Its interaction with the divergent modes,(A.1a,b), is then of the order of 0.01. Similarly, if Φ is of order one, (A.1a,b) shows that (F, G) are of order 0.1and their interaction with Φ,(A.1c), is of order 0.01. In fact, not only are the rotational modes almost identical tothe QG modes (that is the solution of (A.1c) when F = G = 0), but also the divergent modes are almost identicalto the solution obtained from (A.1a,b) when Φ is set to zero. Thus, for small S the divergent and the rotationalcomponents are practically decoupled.

20

conclude that for typical time scale of cyclone development (12 – 48 hrs), QG dynamics provides

an accurate approximation to PE dynamics. The potentially highly effective growth mechanisms,

which are filtered out by the QG, are important only for the first few hours of evolution, but

irrelevant to the synoptic time scale. This is because these mechanisms are not sustained and

because, due to the orthogonality between the balanced and the unbalanced manifolds, these

unbalanced modes do not project onto the “balanced QG slow manifold dynamics”, and therefore

cannot deposite their initial large growth into the rotational balanced modes (For intermidiate

and small Richardson numbers, the interaction can be very different, as is shown in part II of

this paper.).

Although including explicit divergent dynamics has little efect on the development of bal-

anced motions, still the first couple of hours of extreme growth produced by the PE dynamics

and the existence of modal divergent structures in the shear flow, together suggest that the diver-

gent modes might respond strongly to uncorrellated forcing that arises from nonlinear interaction

among waves in the flow and from processes such as latent heat release. Since the mid–latitudinal

troposphere statistical equilibrium can be understood as an asymptotically stable linear dynam-

ical system exposed to stochastic forcing (Farrell and Ioannou, 1995) it is important to examine

the response of the PE to noise.

21

4 Response to stochastic forcing

Adding a forcing vector f(t) to (13) yields,

∂χ

∂t= Dχ + f , (16)

with solution in the frequency domain (Farrell and Ioannou, 1996):

χ(ω) = R(ω)f (ω), (17)

in which apears the resolvent:

R(ω) = (iωI−D)−1, (18)

where the hat superscript denotes the Fourier transform,

χ(ω) =12π

∫ ∞

−∞χ(t)e−iωtdt, (19)

and I is the identity matrix.

Using the SVD of the resolvent, R = UΣV†, and assuming for a given ω a spacially uncor-

related forcing with unit variance, i.e., 〈fi, f∗j 〉 = δij , the covariance matrix of χ can be written

as :

C(ω) = χχ† = RR† = UΣ2U†. (20)

Hence, the column vectors of U are the EOFs of χ(ω) ordered by their weight (which is given

by σ2i /trace(Σ

2), where σ2i is the ith diagonal element of Σ2). The ith EOF vector results from

a forcing equals to the ith column vector of V which can be found alternatively from SVD of

the Hermitian matrix B(ω) = R†R = VΣ2V†.

When a spectrum of frequencies, ω ∈ (ω1, ω2), is excited by spacially and temporally uncor-

related forcing with unit variance, i.e., 〈fi(ωm), f∗j (ωn)〉 = δijδ(ωm−ωn)/2π, then the EOFs and

22

the stochastic optimals are obtained by the SVD of the matricies

C =12π

∫ ω2

ω1

R(ω)R†(ω)dω = UΣ2CU

†; B =12π

∫ ω2

ω1

R†(ω)R(ω)dω = VΣ2BV

†, (21)

and the ensemble response variance is given by

〈χ2〉 =12π

∫ ω2

ω1

F (ω)dω (22)

where F (ω) = trace[R†(ω)R(ω)] = trace[R(ω)R†(ω)]. Note that unless the matrix D is normal

ΣC 6= ΣB, and therefore in general a stochastic optimal is not uniquelly related to an EOF.

When all frequencies are excited (white noise), that is ω ∈ (ω1 = −∞, ω2 = ∞), C and B can be

directly obtained from the Lyapunov equations (Farrell and Ioannou, 1996):

DC + CD† = −I; D†B + BD = −I. (23)

In the presence of equal amounts of Rayleigh damping and Newtonian cooling, corresponding

together to an e–folding decaying rate of 2.5 days, all modes decay and therefore F (ω) is bounded.

F (ω) for the two wavenumbers with this damping is shown in Fig.14(a,b) with the thick gray line

indicates the PE response. For the k = l = 1 case which corresponds to horizontal wavelength of

λ ≈ 4400km, each branch of the divergent modes maintains almost the same amount of variance

as the rotational modes, Fig.14(a). The stochastic optimals and the EOFs calculated exclusively

for the rotational spectrum, by using (21), are found to maintain ∼ 35% of the rotational

perturbation energy. Their structures, which are almost identical to the corresponding structures

obtained from the QG analysis, are shown in Fig.15(a,b). Moreover, Fig.15(c,d) shows that the

first stochastic optimal and first EOF, when calculated exclusively for each of the divergent

branches, are almost identical to each other and almost identical as well to the structures of the

first and the last analytic modes which are shown in the dispersion relations in Fig.(5). As is

shown in Fig.5(c,f), these modes are almost orthogonal to all other modes and therefore their

conjugate biorthogonals are almost identical to the modes themselves.

23

There is no evidence in synoptic scale observations for divergent modes with variance compa-

rable to that of the rotational modes (and with structure as in Fig.15(d), where cyclonic motion

circulates high pressure), and indeed this variance is filtered out by QG, as indicated by the

thin solid line in Fig.14. In order to explore further maintenance of the divergent variance, we

study modifications of the model. Adding a sponge layer above, in order to account for the

stratosphere as a sink for the inertio–gravity waves was found to have little effect on the diver-

gent wave variance since these waves have a very small vertical group velocity, and therefore are

not strongly absorbed by the sponge. Adding drag within the boundary layer to account for

increased eddy mixing was also found not to affect dramatically the structure of F (ω).

Since observations of gravity waves show a power law behaviour, e.g. Peixoto and Oort

(1992), it follows that the forcing, which arises in part from nonlinear interaction among the

eddies, should also obey a power law, and therefore should be assumed red rather than white.

The dashed–dot lines in Fig.14 show the frequency response to red noise with an ω−2 dependence

showing that the divergent branch variance is suppressed. Nonlinear eddies interaction can be

regarded not only as a mechanism that supplies forcing to the system, by constantly exciting

all frequencies, but also as a mechanism which impose damping in all frequencies (Farrell and

Ioannou, 1995). The Rayleigh damping that is being added to the system represents partly

physical processes such as the Ekman damping and Newtonian cooling, and partly parametrizes

this nonlinear damping. For Poincare waves Ekman damping is not effective (since convergence

into low pressure is actually reinforcing the anti–cyclonic negative vorticity), and therefore it is

likely that in the troposphere most of the damping arises from nonlinear processes. The self–

similarity property of the power law spectrum suggests self–similarity of the eddy decorrelation

time as a function of frequency. The autocorrelation for an eddy χ, with a frequency ω0, satisifies

∫ ∞

0χ†(ω0, t)χ(ω0, t + τ)dt ∼ Re{exp [(iω0 − r)τ ]}, (24)

24

where r is the damping coefficient, then choosing r to be linearly proportion to the frequency

preserves the self–similarity in the sense that the e folding time of the autocorrelation remains

proportional to the eddy period. The dashed lines in Fig.14 show the effect of such a damping,

and the solid–dot lines show the response where both red noise forcing and linear damping are

applied. The red noise primarily attenuates the variance leaving the frequency dependence of

the response while the frequency dependent linear damping primarily smooths it.

When k = l = 10 (λ ≈ 440km), Fig.14(b), the response of the divergent motion to white

noise corresponds to almost 45% of the total variance (the thick gray line), and it is strongly

suppressed when the noise is made red (the dashed–dot line). The sharp peaks in the divergent

variance which are associated with the analytic divergent modes (Fig.9) are smoothed by linear

damping (dashed line) and in combination (the solid–dot line) these reduce the divergent variance

to approximately the level obtained by QG anaysis (the solid line). For the rotational spectrum

the first stochastic optimal and the first EOF are almost identical to the corresponding structures

obtained from the QG analysis (fig.16(a,b)). Although the divergent variance response to white

noise is distributed along all of the modes, still the sharpest peaks of the first and the last

analytic modes dominate the spectrum. Therefore, the first stochastic optimal and the first

EOF of the divergent branches, are mainly affected by those two modes (Fig.16(c,d)).

25

5 Discussion

This work is part of a generalized stability analysis of baroclinic shear flow using the primitive

equations. In this part we focused on the synoptic mid–latitude regime, which is characterized

by Richardson number of order 100. This synoptic regime is mainly geostrophically balanced

and its dynamics adequately described by the relatively simple quasi–geostrophic equations.

However, it is not obvious why QG provides such a good approximation to the dynamics at

high Richardson numbers. Observations and PE models indicate that synoptic eddies can be

highly non geostrophic, and theory shows that in their first few hours of evolution optimally

growing perturbations experience a much larger growth than is predicted by QG. It is never-

theless true that in their mature stage, these eddies attain a primarily geostrophic balance and

their asymptoptic growth rate is in general agreement with the QG prediction.

The QG framework assumes a typical perturbation spatial scale of O(1000km) and a typical

time scale of days. Therefore, the QG framework cannot address the more basic question of why

most of the eddies of synoptic scale have such spatial and temporal scales. In PE analyses we

can free the spatial scaling while using the Coriolis parameter to set the time scale. We can then

explore eddies with all spatial scales (provided only that they remain quasi–hydrostatic).

Total energy growth in a baroclinic shear flow arises from the available potential energy source

tapped by positive meridional heat flux which acts to reduce the mean temperature gradient

and the kinetic energy source tapped by downward zonal momentum flux which acts directly to

reduce the vertical shear. The latter mechanism can produce growth rate N/f = O(102) times

larger than the former, however this kinetic energy growth mechanism is filtered out by QG since

for large eddies (of scale O(1000km)) the vertical velocity is too weak by a factor of Ro2 and

therefore the KE growth mechanism is still smaller than the APE growth mechanism. However

for eddies with scale of a few hundred kilometers the potential KE growth rate can be larger

26

by an order of magnitude than the APE growth rate and on approaching the quasi–hydrostatic

limit (aspect ratio 1), this KE growth rate can be 100 times larger than the QG growth. This

result begs the question of why eddies have the scales observed, and why smaller eddies do not

take advantage of this KE mechanism.

One explanation of why motions on the synoptic scale are mainly geostrophically balanced

is the Rossby adjustment process which continuously readjusts perturbations towards geostro-

phy by radiating the ageostophic residual portion as Poincare inertio–gravity waves. Since the

frequency of Poincare waves is larger than the frequency of the rotational geostrophic Rossby

waves, this adjustment is approximated by QG to be instantaneous and the Poincare waves are

filtered out.

The Rossby adjustment mechanism is commonly presented for the case of zero shear and it

can be shown that in that case the Poincare and the Rosby waves are orthogonal. However, in

the presence of shear, there is no a-priori guarantee that these manifolds will remain orthogonal.

Thus there is no guarantee that perturbations can not attain a large transient growth, filtered

in QG, which is due to non–normal interaction between these modes.

In order to examine these issues we explored the generalized stabilty of the baroclinic PE

in the large Richardson number regime using the Eady model basic state in both inviscid and

viscid versions. We applied a 3D Helmholtz decomposiiton in order to decouple the divergent

components from the rotational ones and took the total hydrostatic energy as the norm. We

showed representative results for eddies with scales of the order of 1000km and 100km.

The instantaneous growth rate in PE was found to increase with wave number and to exceed

the QG instantaneous growth. For cyclones with scales smaller than 1000km the KE growth

mechanism dominates and reaches an order of magnitude larger than that obtained in QG for

the mesoscale eddies. For cyclones larger than 1000km the APE growth domniates, however the

27

reason that the PE instantaneous growth was still found to be larger than the QG is because the

PE dynamics use the ageostrophic wind to advect of the geostrophic potential temperature7.

However, this large initial growth is not sustained for more than few hours after which time

the PE optimal growth becomes almost identical to the QG optimal growth and the evolved

structures become almost purely geostrophic. The reason is that for large Richardson numbers,

the divergent unbalanced manifold and the rotational balanced manifold are nearly orthogonal.

Therefore, the large initial ageostrophic growth cannot be projected on and affect the evolution

at the slow manifold. Hence, the enormous instantaneous KE growth rate for mesoscale eddies

does not result in sustained growth since structures where the (u,w) wind field is leaning against

the vertical mean shear become untilted after times of order three hours.

Orthogonality between the divergent and the rotational modes is the key to understand why

the QG provides such a good approximation to large Richardson number dynamics. However,

it will be shown in Part II that as the shear increases and/or the static stability decreases, the

orthogonality is broken and the divergent and the rotational parts can interact.

The response of the PE dynamics to white noise forcing was somewhat surprising. For the

large eddies (order of 1000km), the divergent eastward and westward propagating manifolds

maintain variance comparable to that maintained by the rotational manifold. For the mesoscale

eddies (order of 100km) we found that the divergent manifolds maintain 45% of the total variance.

There is no observational support for such strong variance in divergent large eddies. On the other

hand, for mesoscale eddies, it has been shown by Koshyk et al. (1999), that GCM high resolution

simulation matched aircraft observations which indicate that the variance of mesoscale eddies

is approximately equipartitioned between the rotational and the divergent contribution. These7In QG both the meridional wind and the potential temperature are derivatives of the geostrophic streamfunc-

tion and therefore the maximization of the correlation between the two depends on the structure of the geostrophicstreamfunction. The PE, on the other hand, use the ageostrophic part as an additional degree of freedom withwhich to maximize the meridional heat flux by correlating the ageostrophic wind with the geostrophic temperature.

28

results are in apparant agreement with ours, although it is hard to determine the Richardson

number for the observed variance.

In order to examine the mechanisms supporting divergent variance we first included in the

Eady model a sponge layer above (in order to model the stratosphere as a sink for gravity waves)

and a drag within the boundary layer, to model boundary layer dissipation. However, because

the divergent modes have small vertical group velocity localized damping in the vertical did not

affect them strongly.

On the other hand, replacing the white noise by red, which is in agreement with the power

law frequency spectrum that is observed for gravity waves strongly suppressed the divergent

variance. Also the damping which is due to nonlinearity can be parametrized to be linearly

proportional to the frequency. This makes the eddy decorrelation time proportional to the eddy

period and this self–similarity is also in agreement with the observed power law spectrum. Such a

parameterized damping tends to smooth the divergent variance producing variance distribution

that agrees with observed distrubutions.

Acknowledgments. Discussions with Prof. Petros J. Ioannou have been highly fruitfull.

29

Appendices

A. Energy norm transformation

The matrix A, in (10), can be obtained by substituting the vector η ≡ (F,G,Φ) into (8)

resulting in:d

dt

[(Fx + Gy)y −∇2G

]= −2S(Φx −Gz)x +∇2F, (A.1a)

d

dt

[−(Fx + Gy)x +∇2F

]= −2S(Φx −Gz)y +∇2G, (A.1b)

d

dt∇2Φ = S

[2(Fx + Gy)y −∇2G

], (A.1c)

where ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 . The horizontal derivatives are assumed to act on the expansion in

Fourier modes of η in (11), and the first and the second vertical derivatives are approximated

by central finite differences.

Boundary conditions are incorporated in A by noting that the vertical velocity follows from

continuity, (7e), which requires wz = (Fx + Gy)z, and that rigid horizontal boundary conditions

satisfy w(z = 0) = w(z = 1) = 0, which in turn requires:

kF (0) + lG(0) = kF (1) + lG(1), (A.2)

assuming harmonic solutions in x and y. We choose to set

F (0) = G(0) = 0; kF (1) + lG(1) = 0, (A.3)

since this choice eliminates a zero energy subspace, as discussed below. Evaluated on the bound-

aries the thermodynamic equation (7d) is:

d

dtΦz = S(Φx −Gz), on z = 0, 1. (A.4)

30

The transformation to generalized coordinates can be obtained by first writing (9) in matrix

form,

ζ =1√2

v−u

θ

=

1√2

0 − ∂∂z

∂∂x

∂∂z 0 ∂

∂y

− ∂∂y

∂∂x

∂∂z

FGΦ

= Bη, (A.5)

so that

E = ζ†ζ = η†(B†B)η = η†Mη = χ†χ, (A.6)

from which the generalized coordinate χ = M1/2η, satisfying (13) is identified.

Transformation (A.5) has a null space consisting of the three vectors

η′ =

−ilikK

eKz ;

−ilik

−K

e−Kz ;

1l/k0

ei(kx+ly), (A.7)

(K2 = k2 + l2) so that

ζ = B(η + η′) = Bη. (A.8)

These η′ also satisfy the boundary conditions (A.2), and thus these three degree of freedoms

represent arbitrariness in the Helmholtz decomposition of the wind and the temperature fields

into geostrophic and ageostrophic components. However, boundary conditions (A.3) filter these

solutions.

The η is computed on n vertical levels, including the boundaries, and has a length of (3n−4),

given the aformentioned boundary conditions. ζ is computed on the mid levels and thus has a

length of (3n− 3). The matrix B has the size of (3n− 3)× (3n− 4), and thus M and M1/2 are

squared matrices of (3n−4)× (3n−4), where (3n−4) is the length of the generalized coordinate

vector χ. All matrices are full ranked. The results shown in this paper corresponds to n = 51

levels, but were computed for n = 151 as well, with almost no differences.

31

B. Poincare waves in the Eady channel

The Poincare waves, which are the inertio–gravity waves in the absence of shear, satisfies the

vertical wind eq.,∂2

∂t2wzz +∇2w = 0, (B.1)

(∇2 is the 3D Laplacian) which is the non–dimensional version of eq.(8.4.11), P.258 in Gill

(1982), and is also the degenerate homogeneous version of the complete w eq. in the abscence

of shear, e.g. Phillips (1964). As pointed out by Muraki et al. (1999), the solution can be

obtained as well by solving eqs.(A.1a,b) with zero geostrophic component Φ, and with zero

shear S. The horizontal boundaries of the Eady channel force w to vanish there and thus a

wavelike solution of the form w = w(z)ei(kx+ly−ωt), should satisfy w(z) ∝ sin[± K√ω2−1

z], with

the dispersion relation ω = ±√K2 + (nπ)2/nπ, where n is an integer. This solution differs

from the unbounded free wave solution, by being untilted and by having a descritized vertical

wavenumber. The vertical structure of the other fields can be obtained by solving eqs.(A.1a,b)

with the boundary conditions of (A.3).

The Poincare waves are characterized by having zero PV and by cyclonic motion that circles

high pressure. Fig.17 shows an example of such a wave (for n = 3, and l = 0) that is propagating

toward the east. In Fig.17(a) we can see that the isentropes contours form the streamlines for

the (u,w) field, so that downdraft (which cause warming for positive stratification N) is alaways

to the east of positive θ, while updraft (which cause cooling) is alaways to the east of negative

θ. Hence, this mechanism makes the temperature anomaly field to propagate to the east. The

positive vorticity anomaly, Fig.17(b), (which coalesence with positive pressure anomaly) is due

to the cyclonic circulation caused by the meridional wind, shown in Fig.17(c), and it is loctaed

where the streching term ∂θ∂z , is negative. This can be shown to sum the PV anomaly to zero.

Figs.17(a,b) indicate that regions of horizontal convergence are always to the east of positive

32

vorticity anomalies, making the vorticity field to propagate to the east as well. The same picture,

but with a miror image of the (u,w) vectors field would cause the wave to propagate to the west.

This is the wave structure for the negative ω solution.

In the Rossby adjustment mechanism, Blueman(1971) these Poincare waves are responsible

to radiate away the ageostrophic componenets of the leading order. Since these waves carry

zero PV they do not alter the remaining PV magnitude but redistribute it to be geostrophic.

In the QG framework the Poincare waves are filtered out and the PV is assumed to be purely

geostrophic. The latter is equivalent to assume inifinite propagation speed of the Poincare waves

and thus an instanteneous Rossby adjustment mechanism.

C. The PE Eady Viscid problem

Adding vertical diffusion to the dimensional momentum and thermodynamic equations (1(a,b,d))

gives:du

dt= fva − Λw − ∂

∂z(νm

∂u

∂z), (C.1a)

dv

dt= −fua − ∂

∂z(νm

∂v

∂z), (C.1b)

dθ

dt= fΛv −N2w − ∂

∂z(νθ

∂θ

∂z), (C.1c)

where νm, νθ are the momemtum and thermal diffusivity coefficients, which we assume for sim-

plicity to be constant and equal: νm = νθ = ν = const. Thus, to the RHS of the nondimensional

equation (7(a,b,d)) we add the terms −ν ∂2

∂z2 (u, v, θ), respectively, where ν ≡ S(Nf )2Re−1, and

Re ≡ UL/ν is the Reynolds number.

The smallest value for ν to resolve the wall boundary layer, for vertical resolution of δz =

O(100m), was found numerically to satislfy O(kRe) = 108. This corresponds to ν = O(10−1)m2s−1

if we take shear S = 0.1 and k = 1. This value of ν is larger than is realistic for diffusion in

33

the troposphere. Furthermore, since the value of ν required to resolve the modes is proportional

to the shear and wavenumber, the viscosity required becomes even less realistic as S and k

increase. Since the optimal evolution eventually converges to the most unstable mode (which in

the presence of viscosity becomes the least damped mode), the optimal growth obtained in the

viscid problem can be very different than that obtained in the inviscid problem, for large times

and large shears.

The viscid problem introduces an additional second derivative with height which requires

an additional boundary condition. In order to prevent the boundaries from being a source of

momentum, we require (in additional to the vanishing of the vertical velocity) the free slip

condition, ∂u∂z = ∂v

∂z = 0, on the boundaries. Similarly, to prevent the boundaries from being a

thermal source we set ∂θ∂z = 0 there as well. In the QG limit the momentum free slip condition

is equivalent to vanishing θy, due to the thermal wind balance, and this boundary condition is

in agreement with the Bretherton (1966) alternative description of the Eady basic state, which

requires the zonal mean flow to satisfy U z = −θy = 0 on the boundaries. Since everywhere else

within the domain the mean shear is a non zero constant, in the limit these boundary conditions

form delta functions in U zz with opposite signs just above the lower boundary and just below

the upper boundary. U zz = −qy where qy, is the mean PV gradient. Solving the QG PV

equation, with this basic state and boundary conditions θ(z = 0; 1) = 0, is equivalent to solving

the PV equation within the domain and the thermodynamic equations on the boundaries in the

conventional Eady model. Although we solve the PE , we would like to obtain the QG limit for

large Richardson numbers and thus we adopt Bretherton description and solve (A.1) under the

approximated Bretherton basic state,

U =

12 [(z2/zB) + zB ]z12 [(1 − z)2/(1− zT ) + (zT + 1)]

; U z =

zzB

1(1− z)/(1 − zT )

; U zz =

1/zB

0−1/(1 − zT )

for0 ≤ z < zB

zB < z < zT

zT < z ≤ 1(C.2)

34

Shown in Fig.18, where zB and (1 − zT ) indicate the width of the lower and upper parabolic

boundary layers which are taken to have vertical extent 0.1 for a vertical resolution of 50 layers.

As the numbers of layers increase zB and (1−zT ) can be taken to be smaller and in the continuous

limit Uzz forms the boundary delta functions with opposite signs. When viscosity is small, the

phase speeds, growth rates and structures of the analytic PE normal modes, that are obtained

from using this basic state, approximate the analytic PE normal modes, obtained using the

inviscid Eady model.

35

REFERENCES

Aranson, G., 1963: The Stability of Nongeostrophic Perturbations in a Baroclinic Zonal Flow.

Tellus, 15, 205 – 211.

Bannon, P.R., 1989: Linear Baroclinic Instability with the Geostrophic Momentum Approxi-

mation. J. Atmos. Sci., 46, 402 – 409.

Blumen, W., 1972: Geostrophic Adjustment. Rev. Geophys. Space. Phys., 10, 485 – 528.

Bretherton, F.P., 1966: Baroclinic instability and the short wave cut–off in terms of potential

vorticity. Quart. J. Roy. Meteor. Soc., 92, 335 – 345.

Derome, J., and C.L. Dolph, 1970: Three–Dimensional Non–Geostrophic Disturbances in a

Baroclinic Zonal Flow. Geo. Fluid. Dyn., 1, 91 – 122.

Drazin, P.G, and W.H. Reid 1981: Hydrodynamic Stability. Cambridge University Press, 525

pp.

Eady, E.T., 1949: Long Waves and Cyclone Waves. Tellus, 1, 33 – 52.

Farrell, B.F., 1984: Modal and non–modal baroclinic waves. J. Atmos. Sci., 41, 668 – 673.

Farrell, B.F., and P.J. Ioannou 1995: Stochastic Dynamics of the Mid–latitude Atmospheric

Jet. J. Atmos. Sci., 52, 1642 – 1656.

Farrell, B.F., and P.J. Ioannou 1996: Generalized Stability Theory. Part I:Autonomous Oper-

ators. J. Atmos. Sci., 53, 2025 – 2040.

36

Gill, A.E., 1982: Atmosphere – Ocean Dynamics, Academic Press, 662pp.

Hoskins, B.J., I. Darghici and H.C. Davies 1978: A New Look at the ω–Equation. Quart. J.

Roy. Met. Soc., 104, 31 – 38.

Koshyk, J.N., K. Hamilton and J.D. Mahlman 1999: Simulation of the k−5/3 mesoscale spectral

regime in the GFDL SKYHI general circulat ion model. Geophys. Rev. Lett., 26, 843 –

846.

Mankin, M., 1977: On the Nongeostrophic Baroclinic Instability Problem J. Atmos. Sci., 34,

991 – 1002.

Muraki, D.J., C. Snyders and R. Rotunno 1999: The Next–Order Corrections to QUasi-

geostrophic Theory. J. Atmos. Sci., 56, 1547 – 1560.

Nakamura, N., 1988: Scale Selection of Baroclinic Instability–Effects of Stratification and Non-

geostrophy. J. Atmos. Sci., 45, 3253 – 3267.

Noble, B., and J. Daniel 1988: Applied Linear Algebra. Prentice Hall, 521 pp.

Peixoto, J.P, and A.H. Oort 1992: Physics of Climate. American Institute of Physics, 520 pp.

Phillips, N.A., 1964: An Overlooked Aspect of the Baroclinic Stability Problem. Tellus, 16,

268 – 270.

Snyder, C., 1994: Stability of Steady Fronts with Uniform Potential Vorticity. J. Atmos. Sci.,

52, 724 – 736.

Stone, P.H, 1966: Om Non–Geostrophic Baroclinic Stability J. Atmos. Sci., 23, 390 – 400.

37

FIGURE CAPTIONS

Figure 1. (a) Circled symbols indicate the maximal instantaneous growth rate, scaled by

( fN )Λ, that is obtained from PE dynamics as a function of symmetric horizontal plane

wavenumbers, k = l ∈ (0, 10), where k is scaled by (Nf )H. The APE term contribution to

the growth, vθ/2, is indicated by the asterix, and the KE shear term contribution, −uw/2,

is indicated by the plus symbols. The maximal instantaneous growth rate, obtained from

the QG dynamics, is indicated by the dashed line and the most unstable QG normal modes

growth rate is indicated by the solid line. (b) Energy partition between the geostrophic

and the ageostrophic component, indicated by the asterix and the plus symbols, respec-

tively, for the optimal instantaneous structures, as a function of k = l ∈ (0, 10).

Figure 2. (a) Contributions to the instantaneous APE growth term. The total vθ/2, is indi-

cated by the asterix symbols, as in Fig.1(a). The ageostrophic contribution vaθa/2, is

indicated by the dashed–dot line, where the cross interactions vaθg/2, and vgθa/2, are

indicated by the circle and plus symbols, respectively. (b) Contributions to the instanta-

neous KE growth term. The total −uw/2 is indicated by the plus symbols, as in Fig.1(a).

The pure ageostrophic contribution −uaw/2, is indicated by the dashed line and the cross

interaction contribution −ugw/2, is indicated by the circle symbols.

Figure 3. Zonal–vertical crossection for : (a) the instantaneous ageostrophic meridional wind

va, for wavenumber k = 1. (b) the geostrophic potential temperature component θg, for

k = 1. (c) the (u,w) wind vector field for k = 10.

38

Figure 4. (a) The optimal growth σ(t), over consecutive time interval t, for k = 1, and shear

number S = 0.1. (b) The effective growth rate γ(t) = ln σ(t), scaled by ( fN )Λ, along

the time interval t. For both figs.(a,b) the PE growth is indicated by circles, and the QG

growth is indicated by asterices.

Figure 5. (a) The inviscid PE dispersion relation for S = 0.1, k = 1. ωr, ωi are scaled by

the Coriolis parameter f . The two circles indicates growing and decaying mormal modes,

obtained from the QG dynamics. (b) Energy partition of the modes, sorted increasing

in ωr. The geostropic and the ageostropic components indicated by gray and black thick

lines, respectively. (c) A contour plot which measure the orthogonality between modes.

It shows contours of Ei,j = χ†iχj , at the intersection of (x = i, y = j), where χi is the

normalized generalized coordinate vector of mode i, so that χ†iχi = 1. (d,e,f) The same

as (a,b,c) but for viscid dynamics where the Reynolds number Re = 10(8).

Figure 6. (a) Evolution of the fluxes, scaled by ( fN )Λ and by the total energy perturbation,

which contribute to the energy growth of the two days optimal perturbation, for S = 0.1,

k = 1. Circled symbols indicate the total growth rate, where asterix and plus symbols

indicate the normalized heat flux vθ, and normalized momentum flux −uw, respectively.

The dashed line indicate the total growth rate obtained from the QG dynamics. (b) The

same as in (a), but for the optimal instantaneous perturbation. (c) Energy partition of the

two days optimal perturbation. The geostropic and the ageostropic components indicated

by a plus and an asterix respectively. (d) Same as in (c), but for the optimal instantaneous

perturbation.

Figure 7. Zonal–vertical crossections for S = 0.1, k = 1, of : (a) the initial, and (b) the final

meridional wind v, for the two days optimal. (c) The ageostrophic meridional wind va,

39

and (d), the geostrophic potential temperature θg, of the initial optimal instantaneous.

Their 3 hours evolution are shown in (e) and (f), respectibvely.

Figure 8. Same as in figs.4(a,b), but for the case S = 0.1, k = 10.

Figure 9. Same as in figs.5(a–f), but for the case S = 0.1, k = 10.

Figure 10. Same as in figs.6(a–d), but for the case S = 0.1, k = 10.

Figure 11. Zonal–vertical crossections for S = 0.1, k = 10, of : (a) the initial, and (b) the

final meridional wind v, for the two days optimal. (c) The initial, and (d) the 3 hours

evolution of the (u,w) vector field of the optimal instantaneous.

Figure 12. Zonal–vertical crossections of the meridional wind v, for S = 0.1, k = 1, of : (a)

the inviscid initial optimal, and (b) the inviscid final optimal for target time infinity. (c)

and (d) are the smae as (a) and (b), but when viscosity is added.

Figure 13. The same as in figs.12(a–d) but for the S = 0.1, k = 10 case.

Figure 14. Frequency response to: white noise forcing (thick gray line), red noise forcing with

power spectrum of ω−2 (dashed–dot line), white noise with damping that increases linearly

with frequency (dashed line), red noise and linear damping (solid–dot line), white noise in

the QG framework (solid line) for: (a) S = 0.1, k = l = 1, (b) S = 0.1, k = l = 10.

Figure 15. S = 0.1, k = l = 1: (a) The first stochastic optimal and (b) the first EOF when

computed exclusively for the rotational branch. (c) The first stochastic optimal and (d)

the first EOF when computed exclusively for each of the divergent branch.

Figure 16. The same as in figs.15(a–d) but for the S = 0.1, k = l = 10 case.

40

Figure 17. zonal–vertical crossection of eastward propagating Poincare wave: (a) temperature

contours lines with the (u,w) vector field, (b) vorticity contours which are in phase with

the pressure contours, (c) meridional wind contours. Positive and negative values are

indicated by solid and dashed lines, respectively.

Figure 18. The “Bretherton viscid basic state profile”, as described by (C.2):

(a) U , (b) U z, (c) U zz. zB = zT = 0.1, where the vertical increment resolution is dz = 0.02.

41

0 2 4 6 8 100

0.5

1

1.5

2

2.5

k=l

grow

th r

ate

Instantaneous growth

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

k=l

ener

gy

Energy partition

Figure 1: (a) Circled symbols indicate the maximal instantaneous growth rate, scaled by ( fN )Λ,

that is obtained from PE dynamics as a function of symmetric horizontal plane wavenumbers,k = l ∈ (0, 10), where k is scaled by (N

f )H. The APE term contribution to the growth, vθ/2, isindicated by the asterix, and the KE shear term contribution, −uw/2, is indicated by the plussymbols. The maximal instantaneous growth rate, obtained from the QG dynamics, is indicatedby the dashed line and the most unstable QG normal modes growth rate is indicated by the solidline. (b) Energy partition between the geostrophic and the ageostrophic component, indicatedby the asterix and the plus symbols, respectively, for the optimal instantaneous structures, as afunction of k = l ∈ (0, 10).

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

k=l

grow

th r

ate

Instantaneous APE growth

0 2 4 6 8 100

0.5

1

1.5

2

2.5

k=l

grow

th r

ate

Instantaneous kinetic growth

Figure 2: (a) Contributions to the instantaneous APE growth term. The total vθ/2, is indicatedby the asterix symbols, as in Fig.1(a). The ageostrophic contribution vaθa/2, is indicated by thedashed–dot line, where the cross interactions vaθg/2, and vgθa/2, are indicated by the circle andplus symbols, respectively. (b) Contributions to the instantaneous KE growth term. The total−uw/2 is indicated by the plus symbols, as in Fig.1(a). The pure ageostrophic contribution−uaw/2, is indicated by the dashed line and the cross interaction contribution −ugw/2, isindicated by the circle symbols.

−0.3 −0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1(u,w) for k=10

x

z

−2 0 2

0.2

0.4

0.6

0.8

va for k=1

x

z

−2 0 2

0.2

0.4

0.6

0.8

θg for k=1

x

z

Figure 3: Zonal–vertical crossection for : (a) the instantaneous ageostrophic meridional windva, for wavenumber k = 1. (b) the geostrophic potential temperature component θg, for k = 1.(c) the (u,w) wind vector field for k = 10.

0 20 40 60 801

1.5

2

2.5S=0.1, k=l=1

t (hours)

σ(t)

0 20 40 60 800.3

0.35

0.4

0.45

0.5

0.55S=0.1, k=l=1

t (hours)

γ(t)

Figure 4: (a) The optimal growth σ(t), over consecutive time interval t, for k = 1, and shearnumber S = 0.1. (b) The effective growth rate γ(t) = ln σ(t), scaled by ( f

N )Λ, along the timeinterval t. For both figs.(a,b) the PE growth is indicated by circles, and the QG growth isindicated by asterices.

−2 0 2−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

ωr

ωi

k=l=1, S=0.1, Re=1.e+8

20 40 60 80 100120140

20

40

60

80

100

120

140

mode #

mod

e #

NM orthogonality

0 50 100 1500

0.2

0.4

0.6

0.8

1

mode #

NM energy partition

−2 0 2−0.03

−0.02

−0.01

0

0.01

0.02

0.03

ωr

ωi

k=l=1, S=0.1

20 40 60 80 100120140

20

40

60

80

100

120

140

mode #

mod

e #

NM orthogonality

0 50 100 1500

0.2

0.4

0.6

0.8

1

mode #

NM energy partition

Figure 5: (a) The inviscid PE dispersion relation for S = 0.1, k = 1. ωr, ωi are scaledby the Coriolis parameter f . The two circles indicates growing and decaying mormal modes,obtained from the QG dynamics. (b) Energy partition of the modes, sorted increasing inωr. The geostropic and the ageostropic components indicated by gray and black thick lines,respectively. (c) A contour plot which measure the orthogonality between modes. It showscontours of Ei,j = χ†iχj, at the intersection of (x = i, y = j), where χi is the normalizedgeneralized coordinate vector of mode i, so that χ†iχi = 1. (d,e,f) The same as (a,b,c) but forviscid dynamics where the Reynolds number Re = 10(8).

0 10 20 30 40 500

0.1

0.2

0.3

0.4

grow

th r

ate

t (hours)

optimal 48 hours fluxes, k=1, S=0.1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

ener

gy

t (hours)

optimal 48 hours energy partition

0 10 20 30 40 50−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

grow

th r

ate

t (hours)

optimal instantaneous, k=1, S=0.1

0 10 20 30 40 500.4

0.45

0.5

0.55

0.6

0.65

ener

gy

t (hours)

optimal instantaneous energy partition

Figure 6: (a) Evolution of the fluxes, scaled by ( fN )Λ and by the total energy perturbation,

which contribute to the energy growth of the two days optimal perturbation, for S = 0.1,k = 1. Circled symbols indicate the total growth rate, where asterix and plus symbols indicatethe normalized heat flux vθ, and normalized momentum flux −uw, respectively. The dashedline indicate the total growth rate obtained from the QG dynamics. (b) The same as in (a),but for the optimal instantaneous perturbation. (c) Energy partition of the two days optimalperturbation. The geostropic and the ageostropic components indicated by a plus and an asterixrespectively. (d) Same as in (c), but for the optimal instantaneous perturbation.

−2 0 2

0.2

0.4

0.6

0.8

instantaneous Va , t=0, k=1, S=0.1

z

−2 0 2

0.2

0.4

0.6

0.8

instantaneous θg , t=0, k=1, S=0.1

z

−2 0 2

0.2

0.4

0.6

0.8

instantaneous Va , t=3h, k=1, S=0.1

x

z

−2 0 2

0.2

0.4

0.6

0.8

instantaneous θg , t=3h, k=1, S=0.1

x

z

−2 0 2

0.2

0.4

0.6

0.8

optimal V, t=0, k=1, S=0.1

z

−2 0 2

0.2

0.4

0.6

0.8

optimal V, t=48h, k=1, S=0.1

z

Figure 7: Zonal–vertical crossections for S = 0.1, k = 1, of : (a) the initial, and (b) the finalmeridional wind v, for the two days optimal. (c) The ageostrophic meridional wind va, and (d),the geostrophic potential temperature θg, of the initial optimal instantaneous. Their 3 hoursevolution are shown in (e) and (f), respectibvely.

0 20 40 60 801

1.5

2

2.5S=0.1, k=l=10

t (hours)

σ(t)

0 20 40 60 800

0.5

1

1.5

2

2.5S=0.1, k=l=10

t (hours)

γ(t)

Figure 8: Same as in figs.4(a,b), but for the case S = 0.1, k = 10.

−5 0 5 10−1

−0.8

−0.6

−0.4

−0.2

0

ωr

ωi

k=l=10, S=0.1, Re=1.e+7

20 40 60 80 100120140

20

40

60

80

100

120

140

mode #

mod

e #

NM orthogonality

0 50 100 1500

0.2

0.4

0.6

0.8

1

mode #

NM energy partition

−5 0 5 10−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

ωr

ωi

k=l=10, S=0.1

20 40 60 80 100120140

20

40

60

80

100

120

140

mode #

mod

e #

NM orthogonality

0 50 100 1500

0.2

0.4

0.6

0.8

1

mode #

NM energy partition

Figure 9: Same as in figs.5(a–f), but for the case S = 0.1, k = 10.

0 10 20 30 40 50−0.1

0

0.1

0.2

0.3

0.4

grow

th r

ate

t (hours)

optimal 48 hours fluxes, k=10, S=0.1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

ener

gy

t (hours)

optimal 48 hours energy partition

0 10 20 30 40 50−2

−1

0

1

2

3

grow

th r

ate

t (hours)

optimal instantaneous, k=10, S=0.1

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

ener

gy

t (hours)

optimal instantaneous energy partition

Figure 10: Same as in figs.6(a–d), but for the case S = 0.1, k = 10.

−0.2 0 0.2

0.2

0.4

0.6

0.8

optimal v, t=0, k=10, S=0.1

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

optimal v, t=48h, k=10, S=0.1

x

z

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

0.8

1instantaneous (u,w) , t=0, k=10, S=0.1

x

z

−0.4 −0.2 0 0.2 0.40

0.2

0.4

0.6

0.8

1instantaneous (u,w) , t=3h, k=10, S=0.1

x

z

Figure 11: Zonal–vertical crossections for S = 0.1, k = 10, of : (a) the initial, and (b) the finalmeridional wind v, for the two days optimal. (c) The initial, and (d) the 3 hours evolution ofthe (u,w) vector field of the optimal instantaneous.

−2 0 2

0.2

0.4

0.6

0.8

t → ∞, initial optimal v, k=1, S=0.1

x

z

−2 0 2

0.2

0.4

0.6

0.8

most unstable NM v, k=1, S=0.1

x

z

−2 0 2

0.2

0.4

0.6

0.8

t → ∞, initial optimal v, k=1, S=0.1, Re = 108

x

z

−2 0 2

0.2

0.4

0.6

0.8

most unstable NM v, k=1, S=0.1, Re = 108

x

z

Figure 12: Zonal–vertical crossections of the meridional wind v, for S = 0.1, k = 1, of : (a) theinviscid initial optimal, and (b) the inviscid final optimal for target time infinity. (c) and (d)are the smae as (a) and (b), but when viscosity is added.

−0.2 0 0.2

0.2

0.4

0.6

0.8

t → ∞, initial optimal v, k=10, S=0.1

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

most unstable NM v, k=10, S=0.1

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

t → ∞, initial optimal v, k=10, S=0.1, Re = 107

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

least damped NM v, k=10, S=0.1, Re = 107

x

z

Figure 13: The same as in figs.12(a–d) but for the S = 0.1, k = 10 case.

−8 −6 −4 −2 0 2 4 6 810

−2

100

102

104

ω

F(ω

)

stochastic response, k=l=10, Re=107

−3 −2 −1 0 1 2 310

0

101

102

103

104

ω

F(ω

)

stochastic response, k=l=1, Re=108

Figure 14: Frequency response to: white noise forcing (thick gray line), red noise forcing withpower spectrum of ω−2 (dashed–dot line), white noise with damping that increases linearly withfrequency (dashed line), red noise and linear damping (solid–dot line), white noise in the QGframework (solid line) for: (a) S = 0.1, k = l = 1, (b) S = 0.1, k = l = 10.

−2 0 2

0.2

0.4

0.6

0.8

Rotational stochastic optimal (35% of <E>), k=l=1

x

z

−2 0 2

0.2

0.4

0.6

0.8

Rotational First EOF (35% of <E>), r=0.1, Re=108

x

z

−2 0 2

0.2

0.4

0.6

0.8

Divergent stochastic optimal (21% of <E>), k=l=1

x

z

−2 0 2

0.2

0.4

0.6

0.8

Divergent First EOF (21% of <E>), r=0.1, Re=108

x

z

Figure 15: S = 0.1, k = l = 1: (a) The first stochastic optimal and (b) the first EOF whencomputed exclusively for the rotational branch. (c) The first stochastic optimal and (d) thefirst EOF when computed exclusively for each of the divergent branch.

−0.2 0 0.2

0.2

0.4

0.6

0.8

Rotational stochastic optimal (19% of <E>), k=l=10

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

Rotational First EOF (25% of <E>), r=0.1, Re=107

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

Divergent stochastic optimal (19% of <E>), k=l=10

x

z

−0.2 0 0.2

0.2

0.4

0.6

0.8

Divergent First EOF (19% of <E>), r=0.1, Re=107

x

z

Figure 16: The same as in figs.15(a–d) but for the S = 0.1, k = l = 10 case.

−3 −2 −1 0 1 2 3

0.2

0.4

0.6

0.8

v contours

x

z

−3 −2 −1 0 1 2 3

0.2

0.4

0.6

0.8

vorticity or pressure contours

z

−3 −2 −1 0 1 2 3

0.2

0.4

0.6

0.8

θ contours with (u,w) wind vectors

z

Figure 17: zonal–vertical crossection of eastward propagating Poincare wave: (a) temperaturecontours lines with the (u,w) vector field, (b) vorticity contours which are in phase with thepressure contours, (c) meridional wind contours. Positive and negative values are indicated bysolid and dashed lines, respectively.

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1U

z

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

z

Uz

−10 0 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Uzz

z

Figure 18: The “Bretherton viscid basic state profile”, as described by (C.2):(a) U , (b) Uz, (c) U zz. zB = zT = 0.1, where the vertical increment resolution is dz = 0.02.