Fundamentals Of Geophysical Fluid Dynamics Part 6 ppt

By analogy,since the final term in ˜qQG,1has no counterpart in ˜qQG,0, the two modal˜QG,m can be said to have an identical definition in terms of ˜ψm if thebarotropic deformation radius

or for continuous height modes,

1H

Z H 0

dz Gp(z) Gq(z) = δp,q , (5.32)

with δp,q = 1 if p = q, and δp,q = 0 if p 6= q (i.e., δ is a discretedelta function) This is a mathematically desirable property for a set ofvertical basis functions because it assures that the inverse transformationfor (5.30) is well defined as

dz ψ(z)Gm(z) (5.34)

The physical motivation for making this transformation comes frommeasurements of large-scale atmospheric and oceanic flows that showthat most of the energy is associated with only a few of the gravestvertical modes (i.e., ones with the smallest m values and correspondinglylargest vertical scales) So it is more efficient to analyze the behavior

of ˜ψm(x, y, t) for a few m values than of ψ(x, y, z, t) at all z values withsignificant energy A more theoretical motivation is that the verticalmodes can be chosen — as explained in the rest of this section — soeach mode has a independent (i.e., decoupled from other modes) lineardynamics analogous to a single fluid layer (barotropic or shallow-water)

In general a full dynamical decoupling between the vertical modes cannot

be achieved, but it can be done for some important behaviors, e.g., theRossby wave propagation in Sec 5.2.1

For specificity, consider the 2-layer quasigeostrophic equations (N =2) to illustrate how the Gmare calculated The two vertical modes arereferred to as barotropic (m = 0) and baroclinic (m = 1) (For a N -layermodel, each mode with m≥ 1 is referred to as the mthbaroclinic mode.)

To achieve the linear-dynamical decoupling between layers, it is sufficient

to ”diagonalize” the relationship between the potential vorticity andstreamfunction That is, determine the 2x2 matrix Gm(n) such thateach modal potential vorticity contribution (apart from the planetaryvorticity term), i.e.,

˜QG,m− βy = H1 Σ2n=1Hn(qQG,n− βy) Gm(n) ,

depends only on its own modal streamfunction field,

˜

ψm = 1

H Σ

2 n=1HnψnGm(n) ,and not on any other ˜ψm 0 with m0 6= m This is accomplished by thefollowing choice:

of height, while the baroclinic mode reverses its sign with height andhas a larger amplitude in the thinner layer Both modes are normalized

quanti-˜QG,0 = βy +∇2ψ˜0

˜QG,1 = βy +∇2ψ˜1−R12ψ˜1 . (5.38)These relations exhibit the desired decoupling among the modal stream-function fields Here the quantity,

R21 = g0H1H2

defines the deformation radius for the baroclinic mode, R1 By analogy,since the final term in ˜qQG,1has no counterpart in ˜qQG,0, the two modal

˜QG,m can be said to have an identical definition in terms of ˜ψm if thebarotropic deformation radius is defined to be

The form of (5.38) is the same as the quasigeostrophic potential vorticityfor barotropic and shallow-water fluids, (3.28) and (4.113), with thecorresponding deformation radii, R =∞ and R =√gH/f0, respectively.This procedure for deriving the vertical modes, Gm, can be expressed

in matrix notation for arbitrary N The layer potential vorticity andstreamfunction vectors,

qQG = {qQG,n; n = 1, , N} and ψ =ψ {ψn; n = 1, , N} ,are related by (5.27) re-expressed as

qQG = P ψψ + Iβy ψ (5.41)Here I is the identity vector (i.e., equal to one for every element), and P

is the matrix operator that represents the contribution of ψψψ derivatives

op-ψ

ψ = G˜ψ ,ψ ψψ = G˜ −1ψ ,ψ (5.43)with analogous expressions relating qQG− Iβy and ˜qQG− Iβy Thematrix G is related to the functions in (5.29) by Gnm= Gm(n) Thus,

˜

qQG = G−1P G˜ψ + ˜Iψ 0βy = 

I∇2− G−1SG ˜ψ + ˜Iβy ,ψ (5.44)using G−1G = I

Therefore, the goal of eliminating cross-modal coupling in (5.44) isaccomplished by making G−1SG a diagonal matrix, i.e., by choosingthe vertical modes, G = Gm(n), as eigenmodes of S with correspondingeigenvalues, R−2

m ≥ 0, such that

for the diagonal matrix, R−2 = δn,mR−2 As in (5.39)-(5.40), Rm is

called the deformation radius for the mtheigenmode From (5.27), S isdefined by

S11 = f

2

g0 1.5H1

, S12 = − f2

g0 1.5H1, S1n = 0, n > 2

S21 = − f2

g0 1.5H2, S22 = f

2

H2

 1

g0 1.5

+ 1

g0 2.5

,

S23 = − f2

g0 2.5H2, S2n = 0, n > 3

S21 = − f2

g0

IH2, S22 = f

ddz

 f2

N2

dGdz



Vertical boundary conditions are required to make this a well posedboundary-eigenvalue problem for Gm(z) and Rm From (5.20)-(5.22) thevertically continuous formula for the quasigeostrophic vertical velocityis

Fig 5.3 Dynamically determined vertical modes for a continuously stratified

0, 1, 2

WhenN2(z) > 0 at all heights, the eigenvalues from (5.48) and (5.50)are countably infinite in number, positive in sign, and ordered by mag-nitude: R0> R1> R2> > 0 The eigenmodes satisfy the orthonor-mality condition (5.32) Fig 5.3 illustrates the shapes of the Gm(z)for the first few m with a stratification profile, N (z), that is upward-intensified For m = 0 (barotropic mode), G0(z) = 1, corresponding to

R0 =∞ For m ≥ 1 (baroclinic modes), Gm(z) has precisely m crossings in z, so larger m corresponds to smaller vertical scales andsmaller deformation radii, Rm Note that the discrete modes in (5.35)for N = 2 have the same structure as in Fig 5.3, except for having afinite truncation level, M = N− 1 (The relation, H1> H2, in (5.35) isanalogous to an upward-intensifiedN (z) profile.)

zero-5.2 Baroclinic InstabilityThe 2-layer quasigeostrophic model is now used to examine the stabilityproblem for a mean zonal current with vertical shear (Fig 5.4) This isthe simplest flow configuration exhibiting baroclinic instability (cf., the3D baroclinic instability in exercise #8 of this chapter) Even though

u 2 = − U

z

Fig 5.4 Mean zonal baroclinic flow in a 2-layer fluid

the Shallow-Water Equations (Chap 4) contain some of the combinedeffects of rotation and stratification, they do so incompletely compared

to fully 3D dynamics and, in particular, do not admit baroclinic bility because they cannot represent vertical shear

insta-In this analysis, for simplicity, assume that H1 = H2 = H/2; hencethe baroclinic deformation radius (5.39) is

g0

IH 12f .This choice is a conventional idealization for the stratification in themid-latitude troposphere, whose mean stability profile,N (z), is approx-imately constant in z above the planetary boundary layer (Chap 6)and below the tropopause Further assume that there is no horizontalshear (thereby precluding any barotropic instability) and no barotropic

component to the mean flow:

+H2

north-is similar to the mid-latitude, northern-hemnorth-isphere atmosphere, withstronger westerly winds aloft (Fig 5.1) and warmer air to the south.Note that (5.51)-(5.52) is a conservative stationary state; i.e., ∂t = 0

in (5.7) if Fn = 0 The qQG,n are functions only of y, as are the ψn

So they are functionals of each other Therefore, J[ψn, qQG,n] = 0,and ∂tqQG,n= 0 The fluctuation dynamics are linearized around thisstationary state Define

ψn = ψn+ ψ0n

qQG,n = qQG,n+ q0QG,n, (5.53)and insert these into (5.13)-(5.14), neglecting purely mean terms, per-turbation nonlinear terms (assuming weak perturbations), and non-conservative terms:

∂q0 QG,n

∂t + un

∂q0 QG,n

∂x + v

0 n

∂x + v

0 2



β−RU2



5.2.1 Unstable ModesOne can expect there to be normal-mode solutions in the form of



K2Ψ2−2R12(Ψ1− Ψ2)

+



β−RU2



Ψ2 = 0 (5.57)for C = ω/k and K2= k2+ `2 Redefine the variables by transformingthe layer amplitudes into vertical modal amplitudes by (5.36):

˜

Ψ0 ≡ 12(Ψ1+ Ψ2)

˜

Ψ1 ≡ 12(Ψ1− Ψ2) (5.58)These are the barotropic and baroclinic vertical modes, respectively Thelinear combinations of layer coefficients are the vertical eigenfunctionsassociated with R0 = ∞ and R1 = R from (5.39) Now take the sumand difference of the equations in (5.57) and substitute (5.58) to obtainthe following modal amplitude equations:

˜

Ψ0 is the barotropic vertical modal amplitude, and this relation is tical to the dispersion relation for barotropic Rossby waves with an infi-nite deformation radius (Sec 3.1.2) The second equation in (5.59) with

is the same as the dispersion relation for baroclinic Rossby waves withfinite deformation radius, R (Sec 4.7).

When U 6= 0, (5.59) has non-trivial modal amplitudes, ˜Ψ0 and ˜Ψ1,only if the determinant for their second-order system of linear algebraicequations vanishes, viz.,

[CK2+ β] [C(K2+ R−2) + β] − U2K2[K2

− R−2] = 0 (5.62)This is the general dispersion relation for this normal-mode problem

To understand the implications of (5.62) with U 6= 0, first considerthe case of β = 0 Then the dispersion relation can be rewritten as

e−ikCt = ek Imag[C]t This behavior is a baroclinic instability for a mean flow with shear only

in the vertical direction

For U, β 6= 0, the analogous condition for C having a nonzero nary part is when the discriminant of the quadratic dispersion relation(5.62) is negative, i.e.,P < 0 for

imagi-P ≡ β2(2K2+ R−2)2 − 4(β2K2− U2K4(K2− R−2)) (K2+ R−2)

= β2R−4+ 4U2K4(K4− R−4) (5.64)Note that β tends to stabilize the flow because it acts to makeP morepositive and thus reduces the magnitude of Imag [C] whenP is negative.Also note that in both (5.63) and (5.64) the instability is equally strongfor either sign of U (i.e., eastward or westward vertical shear)

The smallest value forP(K) occurs when

0 = ∂P

∂K4 = 4U2(K4

− R−4) + 4U2K4 , (5.65)or

At this K value, the value forP is

P = β2R−4− U2R−8 (5.67)

Therefore, a necessary condition for instability is

Further analysis ofP(K) shows other conditions for instability:

• KR < 1 is necessary (and it is also sufficient when β = 0)

WhenP < 0, the solution to (5.62) is

C = − β(2K

2+ R−2)2K2(K2+ R−2)± i

√

−P2K2(K2+ R−2) . (5.69)Thus the zonal phase propagation for unstable modes (i.e., the real part

of C) is to the west From (5.69),

−Kβ2 < Real [C] < −K2+ Rβ −2 (5.70)The unstable-mode phase speed lies in between the barotropic and baro-clinic Rossby wave speeds in (5.60)-(5.61) This result is demonstrated

by substituting the first term in (5.69) for Real [C] and factoring−β/K2from all three expressions in (5.70) These steps yield

1 ≥ 1 + µ/21 + µ ≥ 1 + µ1 (5.71)for µ = (KR)−2 These inequalities are obviously true for all µ≥ 0

U = 2

βR2(1−K4R4)−1/2

Fig 5.5 Regime diagram for baroclinic instability The solid line indicates themarginal stability curve as a function of the mean vertical shear amplitude,

the marginal stability curve for β = 0

5.2.2 Upshear Phase TiltFrom (5.59),

eiθΨ˜0 , (5.72)where θ is the phase angle for (C + βK−2)/U in the complex plane.Since

Real



C + βK−2U



= Imag [C]

Fig 5.6 Modal and layer phase relations for the perturbation streamfunction,

addition: in each column the modal curves in the top two rows are addedtogether to obtain the respective layer curves in the bottom row

for growing modes (with Real [−ikC] > 0, i.e., Imag [C] > 0), then

0 < θ < π/2 in westerly wind shear (U > 0) As shown in Fig 5.6this implies that ˜ψ1 has its pattern shifted to the west relative to ˜ψ0,

by an amount less than a quarter wavelength A graphical addition andsubtraction of ˜ψ1 and ˜ψ0 according to (5.58) is shown in Fig 5.6 Itindicates that the layer ψ1 has its pattern shifted to the west relative

to ψ2, by an amount less than a half wavelength Therefore, layer disturbances are shifted to the west relative to lower-layer ones;i.e., they are tilted upstream with respect to the mean shear direction(Fig 5.7) This feature is usually evident on weather maps during theamplifying phase for mid-latitude cyclonic synoptic storms and is oftenused as a synoptic analyst’s rule of thumb

upper-5.2.3 Eddy Heat FluxNow calculate the poleward eddy heat flux, v0T0 (disregarding the con-version factor, ρocp, between temperature and heat; Sec 2.1.2) Theheat flux is analogous to a Reynolds stress (Sec 3.4) as a contributor

to the dynamical balance relations for the equilibrium state, except it

ψ ’ (x,z)u(z)

xz

+ +

in baroclinic instability for a continuously stratified fluid

appears in the mean heat equation rather than the mean momentumequation Here v0= ∂xψ0, and the temperature fluctuation is associatedwith the interfacial displacement as in (5.9),

T0 = b0

αg =

1αg

∂φ0

∂z =

fαg

∂ψ0

∂z =

2fαgH(ψ

1in theupper and lower layers, respectively, as in (5.37) Therefore the modalheat fluxes are

˜0

1T0 ≡ 2πk

Z 2π 0

dx ˜v01T0

∝

Z 2π

dx sin[kx + θ] cos[kx + θ] = 0

0T0 ≡ 2πk

Z 2π 0

dx ˜v00T0

∝2πk

Z 2π 0

dx sin[kx + θ] cos[kx] = sin[θ]

2 (5.77)with positive proportionality constants Since each v0

n has a positivecontribution from ˜v00, the interfacial heat flux, v0T0, is proportional to

˜0

0T0, and it is therefore positive, v0T0 > 0 The sign of v0T0 is directlyrelated to the range of values for θ, i.e., to the upshear vertical phasetilt (Sec 5.2.2)

5.2.4 Effects on the Mean FlowThe nonzero eddy heat flux for baroclinic instability implies there is

an eddy–mean interaction A mean energy balance is derived similarly

to the energy conservation relation (5.16) by manipulation of the meanmomentum and thickness equations The result has the following form

in the present context:

en-it from the mean energy when the eddy flux, v0η0, has the opposite sign

to the mean gradient, dyη Since η is proportional to T in a layeredmodel, this kind of conversion occurs when v0T0 > 0 and dyT < 0 (asshown in Sec 5.2.3)

The eddy–mean interaction cannot be fully analyzed in the spatiallyhomogeneous formulation of this section, implicit in the horizontallyperiodic eigenmodes (5.56) It is the divergence of the eddy heat fluxthat causes changes in the mean temperature gradient,

∂T

∂t = −∂y∂ v0T0 ,and the divergence is zero in a homogeneous flow Thus, a more completeinterpretation of the role of eddies in the general circulation requires an

extension to inhomogeneous flows, such as the tropospheric westerly jetthat has its maximum speed at a middle latitude,≈ 45o.

The poleward heat flux in baroclinic instability tends to weaken themean state by transporting warm air fluctuations into the region on thepoleward side of the jet with its associated mean-state cold air (n.b.,Fig 5.1) Equation (5.78) shows that the mean circulation loses energy

as the unstable fluctuations grow in amplitude: the mean meridionaltemperature gradient (hence the mean geostrophic shear) is diminished

by the eddy heat flux, and part of the mean available potential ergy associated with the meridional temperature gradient is convertedinto eddy energy The mid-latitude atmospheric climate is established

en-as a balance between the acceleration of the westerly Jet Stream byEquator-to-pole differential radiative heating and the limitation of thejet’s vertical shear strength by the unstable eddies that transport heatbetween the Equatorial heating and polar cooling zones

A similar interpretation can be made for the zonally directed tic Circumpolar Current (ACC) in the ocean (Fig 6.11) In the wind-driven ACC, the more natural dynamical characterization is in terms ofthe mean momentum balance rather than the mean heat balance, al-though these two balances must be closely related because of thermalwind balance A mean eastward wind stress beneath the westerly windsdrives a surface-intensified, eastward mean current that is baroclinicallyunstable and generates eddies that transfer momentum vertically Thiseddy momentum transfer has to be balanced against a bottom turbulentdrag and/or topographic form stress (a pressure force against the solidbottom topography; Sec 5.3.3) The eddies also transport heat south-ward (poleward in the Southern Hemisphere), balanced by the advectiveheat flux caused by the mean, ageostrophic, secondary circulation in themeridional (y, z) plane, such that there is no net heat flux by their com-bined effects

Antarc-In these descriptions for the baroclinically unstable westerly windsand ACC, notice two important ideas about the dynamical maintenance

of a mean zonal flow:

• An equivalence between horizontal heat flux and vertical momentumflux for quasigeostrophic flows The latter process is referred to asisopycnal form stress It is analogous to topographic form stress ex-cept that the relevant material surface is an isopycnal in the fluidinterior instead of the solid bottom Isopycnal form stress is not thevertical Reynolds stress, < u0w0 >, which is much weaker than the

upper-5.2.3 Eddy Heat FluxNow calculate the poleward eddy heat... therefore positive, v0T0 > The sign of v0T0 is directlyrelated to the range of values for θ, i.e., to the upshear vertical phasetilt (Sec