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Atmospheric Oscillation: Linear Perturbation Theory The QG PV equation derived from Chapter 6 is still a nonlinear model and remains difficult to solve analytically and precludes any simple interpretation of the physical processes it produces. The QG PV can be solved numerically though and the numerical model based on the QG PV equation has been extensively used to understand the mid-latitude dynamics and geostrophic turbulence. If we wish to gain physical insight into the fundamental nature of the atmospheric motions, it is useful to employ simplified models in which certain processes are omitted and compare the results with those of more complete models. Although the linear theory is not an accurate representation for the observed mid latitude weather and climate system, one can still gain much insight by treating the system as wave disturbances riding on the time mean, zonal mean flow (which is a function of altitude and latitude, x, t-independent). The theoretical framework for this system is called linear perturbation theory. This is also a framework from which one can form basic concepts of wave dynamics and wave-mean flow interactions. In the perturbation method, all field variables are divided into two parts: a basic state portion independent of time and longitude, and a perturbation portion, which is the local deviation of the field from the basic state. Fro example, u = u (y, z) + u '(x, y, z,t) and the inertial acceleration term can be written as

u!u!x

= (u + u ') !!x(u + u ') = u !u '

!x+ u ' !u '

!x

The basic assumptions of perturbation theory are that the basic state variables must themselves satisfy the governing equations when the perturbations are set to zero, and that the perturbation fields must be small enough so that all terms that involve products of the perturbations can be neglected. The latter requirement means for the example above:

u!u '!x! u ' !u '

!x

7.7.1 Free barotropic Rossby waves The dispersion relationship for barotropic Rossby waves may be derived formally by finding wave-type solutions of the linearized barotropic vorticity equation. For a midlatitude beta-plane the barotropic vorticity equation (4.27) has the form

!!t

+ u!!x

+ v!!y

"#$

%&'( + )v = 0 (7.89)

We now assume the motion consists of a constant basic state zonal velocity plus a small horizontal perturbation:

u = u + u ', v = v ', ! = "v '/ "x # "u '/ "y = ! '

u = constant, v = 0, ! = 0

We define a perturbation streamfunction ! ' according to

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u ' = !"# '/ "y, v ' = "# '/ "x from which ! ' = "2# ' . The perturbation form of (7.89) is then

!!t

+ u!!x

"#$

%&'(2) + * !) '

!x= 0 (7.90)

where as usual we have neglected terms involving the products of perturbation quantities. We see a solution of the form ! ' = Re " exp(i(kx + ly #$t))[ ] Here k and l are wave numbers in the zonal and meridional directions. Substituting ! ' for in (7.90):

(!"i + iku )(!k2 ! l2 ) + ik# = 0

$

(!" + ku )(!k2 ! l2 ) + k# = 0

which may be immediately solved for ! : ! = uk " k / (k2 + l2 ) (7.91) Recalling that c = ! / k , we have the dispersion relation:

c = u !"

k2 + l2 (7.92)

Thus, the Rossby wave zonal phase propagation is always westward relative to the mean zonal flow. Furthermore, the Rossby wave phase speed depends inversely on the square of the horizontal wavenumber. Therefore, Rossby waves are dispersive waves whose (westward) phase speeds increase rapidly with increasing wavelength. This result is consistent with the discussion in Section 6.2.2 in which we showed that the advection of planetary vorticity, which tends to make disturbances retrogress, increasingly dominates over relative vorticity advection as the wavelength increases.

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For longer wavelengths the westward Rossby wave phase speed may be large enough to balance the eastward advection by the mean zonal wind so that the resulting disturbance is stationary wrt the ground. From (7.92) it is clear that the free Rossby wave becomes stationary when

ks2 =

!u" l2 (7.93)

Note that l cannot be too large (or the meridional scale of the disturbance cannot be too small) to have a stationary Rossby wave. For the case l = 0 (uniform in y-direction) and relaxing the requirement of stationarity,

! = kc = ku " # / k

Cgx =$!$k

= u + # / k2

We have

phase speed: cx = u ! " / k2

group speed: cgx = u + " / k2

See graphics below for the illustration of the relationship between phase speed, group velocity and the zonal wind as they vary with wave number k.

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The physical intuition of Rossby waves

Dh

Dt(! + f ) = 0 f = f0 + "y

Suppose we deflect parcel from equilibrium latitude with the absolute vorticity conserved, we must have !" = #! f = #$!y !" < 0 for northward displacement, flow curves to right (counter clockwise).!" > 0 for southward displacement, flow curves to left (clockwise)

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Fig.7.14 Perturbation vorticity field and induced velocity field (dashed arrows) for a meridionally displaced chain of parcels. The speed of propagation can be computed as follows: Letting

δy = asin(k(x − ct )) then

v = D(!y)Dt

= "kcacos[k(x " ct)] and relative vorticity

! = "v"x

= k2casin[k(x # ct)]

Substitution for δ y andζ in (7.87) then yields

k2casin[k(x − ct )] = −β a sin[k(x − ct )] or

c = ! "k2

If u is a function of y, ie., u = u (y), then !" = #($ # uyy )! y , so if (! " uyy ) changes sign, !" can increase as δy increases, thus leading to instability---barotropic instability.

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Dispersion and Group Velocity For general propagating waves, frequency ! generally depends on the wave number of the perturbation as well as the physical properties of the medium. Thus, because c = ! / k , the phase speed also depends on the wave number except in the special case where ! " k . For waves in which the phase speed varies with k, the various sinusoidal components of a disturbance originating at a given location are at later time found in different places (thus the shape of the disturbances varies with time), that is, they are dispersed. Such waves are referred to as dispersive, and the formula that relates ! and k is called a dispersion relationship. In nondispersive waves, such as the acoustic waves, a spatially localized disturbance consisting of a number of Fourier wave components (a wave group) will preserve its shape as it propagates in space at the phase speed of the wave. For dispersive waves, however, the shape of a wave group will not remain constant as the group propagates. Because the individual Fourier components of a wave group may either reinforce or cancel each other, depending on the relative phases of the components, the energy of the group will be concentrated in limited region as illustrated in Fig.7.3. Furthermore, the group generally broadens in the course of time, that is, the energy is dispersed because the group velocity is function of wave number.

A non-dispersive wave packet.

A dispersive wave packet.

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Fig.7.3

An expression for the group velocity, which is the velocity at which the energy propagates, can be derived as follows: we use notation for wave: !(x,t) = exp(ik(x " ct)) [real part] ! is constant when x ! ct = constant = "0 or x = !0 + ct Or: !(x,t) = exp[i(kx "#t))] [real part] Then we consider the superposition of two horizontally propagating waves of equal amplitude but slightly different wavelengths with wave numbers and frequencies differing by 2!k and 2!" , respectively. The total disturbance is thus !(x,t) = exp i[(k + "k)x # ($ + "$)t]{ } + exp i[(k # "k)x # ($ # "$)t]{ } Rearranging terms and applying Euler formula gives

! = ei(" kx#"$t ) + e# i(" kx#"$t )%& '(e

i(kx#$t )

= 2cos("kx # "$t) ei(kx#$t ) (7.5)

The disturbance (7.5) is the product of a high-frequency carrier wave of wavelength 2! / k whose phase speed is the average for the two Fourier components, and a low-frequency envelope of wavelength 2! /"k that travels at the speed !" /!k . Thus for waves with continuous spectrum, in the limit as !k" 0 , the horizontal velocity of the envelope, or group velocity, is

cgx =!"!k

Similarly for 3D, ! ~ ei(kx+ ly+nz"#t )

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cgx =!"!k, cgx =

!"!l, cgx =

!"!n

This result applies generally to arbitrary wave envelopes provided that the wavelength of the wave group, 2! /"k , is large compared to the wavelength of the dominant component, 2! / k . For Rossby wave case discussed in Section 7.7.1: Dispersion relationship for linear Rossby wave riding on a constant mean wind is

c = u !"

k2 + l2 (7.92)

Unlike the phase speed, which is always westward relative to the mean flow, the zonal group velocity for a Rossby wave may be either eastward or westward relative to the mean flow, depending on the ratio of the zonal and meridional wave numbers (see Problem 7.20). Stationary Rossby modes (c=0) have zonal group velocities that are eastward relative to the ground. Synoptic-scale Rossby wave also tends to have eastward zonal group velocity relative to the ground. For these waves, the envelopes (representing the group velocity) move eastward faster than the crests and troughs (representing the phase). As indicated in Fig7.4b, this implies that new disturbances tend to develop downstream of existing disturbances, which is an important consideration for forecasts.

The group velocity for waves described by (7.92) is

Cg !"#"k, "#"l

$%&

'()= u +

*(k2 + l2 )K 4 , 2*kl

K 4

$%&

'()

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Fig 3.15.2 of Pedlosky GFD book—dispersion relation for a given meridional wave number l. See Chapter 3 of Pedlosky’s GFD book for more comprehensive treatment of the Rossby

waves. ! = " #kk2 + l2 + F

, where F = f 2L2

gH

Stationary Rossby Waves For stationary Rossby waves, u = ! / Ks

2

Cgx =2!k2

Ks4 > 0; Cgy =

2uklKs2 > 0

So the group velocity is till eastward even though the phase speed is 0. For the easterly wind blowing across mountain ranges, unless u is strong enough ( u > ! / k2 & u < 0 ), the Rossby wave energy excited by the easterlies will be confined to the east of the wave source and won’t propagate over the mountains. In other words, downstream of the mountain will not be influenced by the stationary topographically forced Rossby waves. This is in consistence with the description of the mountain wave influence from the Ertel PV perspective.

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Meridional propagation of Rossby waves require l2 > 0 . Otherwise, if l2 < 0 , which means, when either u < 0 (easterlies) or u > ! / k2 (very strong westerlies), since l2 = ! / u " k2 , then l = i l . Therefore,

! " eily ! e# l y , which means the waves are merdionally confined. This is the reason why easterlies and strong westerlies act as barrier for the meridional Rossby wave propagation. From

Cgx =2!k2

Ks4 > 0; Cgy =

2uklKs2 > 0 ,

or

Cgx =2uk2

Ks2 > 0; Cgy =

2uklKs2 > 0 -----------(10.68)

one can immediately finds that the group velocity for stationary Rossby waves is perpendicular to the wave crests. And the magnitude of group velocity is given by cg = 2u cos! The angle ! is the one at which the group velocity is oriented to the x-axis (see Fig 10.14), and is also the angle between lines of constant phase and the y-axis. The group velocity of stationary wave always has an eastward zonal component and a northward (with positive l) or southward (with negative l) meridional component depending on whether l is positive or negative.

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Because energy propagates at the group velocity, (10.68) indicates that the stationary wave excited by a localized topography should consist of two wave trains, one with l >0 extending eastward and northward and the other with l < 0 extending eastward and southward. An example computed using spherical geometry is given in Fig. 10.15. Although the positions of individual troughs and ridges remain fixed for stationary wave, the wave trains in this exmple do not decay in time, as the effects of dissipation are counteracted by energy propagation from the source at the Rossby wave group velocity.

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For the climatological stationary wave distribution in the atmosphere the excitation comes from a number of sources, both topographic and thermal, distributed around the globe. Thus, it is not easy to trace out distinct paths of wave propagation. Nevertheless, detailed calculations using spherical geometry suggest that two-dimensional barotropic Rossby wave propagation provides a reasonable first approximation for the observed departure of the extratropical time-mean flow from zonal asymmetry. Rossby wave excited by isolated orographic features also play a significant role in the momentum budget. Letting the amplitude coefficient ! be real in (7.90), the merdional momentum flux can be expressed as u 'v ' = !("# '/ "x)("# '/ "y) = !$2kl / 2 Therefore, if u > 0 ,

cgy > 0 implies u 'v ' < 0

cgy < 0 implies u 'v ' > 0

Thus, westerly momentum converges into the wave source region (where the energy flux is divergent). In other words, wave group velocity directs to the opposite direction of the wave momentum flux. This eddy momentum flux convergence is necessary to balance the momentum lost to the surface through the pressure torque mechanism discussed in 10.3. 7.7.2 Forced Topograhic Rossby Waves In the atmosphere, forced stationary Rossby waves are of primary importance for understanding the planetary scale circulation pattern. Such modes may be forced by longitudinally dependent diabatic heating patterns, or by flow over topography. Of particular importance for the NH extratropical circulation are stationary Rossby waves forced by flow over the Rockies and Himalayas.

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As the simplest possible dynamical model of topographic Rossby waves, we use the barotropic PV equation for a homogeneous fluid of variable depth (4.26). We assume the upper boundary is at a fixed height H, and the lower boundary is at the variable height hT (x, y) where hT ! H . We also using quasi-geostrophic scaling so that ! ! f , i.e., small Rossby number. We can than approximate (4.26) by

H!!t

+ V "#$%&

'()(* + f ) = + f0

DhTDt

(7.94)

Linearizing and applying the midlatitude beta-plane approximation yields

!!t

+ u!!x

"#$

%&'( 'g + )v'g = *

f0Hu!hT!x

(7.95)

We now examine solutions of (7.95) for the special case of a sinusoidal lower boundary with the form: hT (x, y) = Re h0 exp(ikx)[ ]cos ly (7.96) and represent the geostrophic wind and vorticity by the perturbation streamfunction ! (x, y) = Re ! 0 exp(ikx)[ ]cos ly (7.97) Then (7.95) has a steady-state solution ! 0 = f0h0 / H K 2 " Ks

2( )#$ %& (7.98) (show it, it could be a problem for final, note that Ks

2 = ! / u ) The streamfunction is either exactly in phase (ridges over mountains) or exactly out of phase (troughs over the mountains) with the topography, depending on the sign of K 2 ! Ks

2 . Thus the situation is more complex (depending on the scale of the mountains) than the qualitative description in the discussion of streamline deflection in potential-vorticity conserving flows crossing mountain ranges in Section 4.3.

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15

The solution (7.98) has the unrealistic characteristic that when the wave number exactly equals the critical wave number Ks the amplitude goes to infinity. From (7.93) it is clear that this singularity occurs at the zonal wind speed for which the free Rossby mode becomes stationary. Thus, it may be thought of as a resonant response of the barotropic system. Charney and Eliassen (1949) removed the resonant singularity by including a boundary drag in the form of Ekman pumping, which for the barotropic vorticity equation is simply a linear damping of the relative vorticity. The vorticity equation thus takes the form

!!t

+ u!!x

"#$

%&'( 'g + )v'g + r(

'g = *

f0Hu!hT!x

(7.99)

For steady flow, (7.99) has a solution with complex amplitude ! 0 = f0h0 / H K 2 " Ks

2 " i#( )$% &' (7.100)

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where ! " rK 2 (ku )#1 . Thus, boundary layer drag shift the phase of the response and removes the singularity at resonance. Now the amplitude is

! 0 = ! 0! 0*( ) = f 20h

20

H 2(K 2 " K 2s " i#)(K

2 " K 2s + i#)

=f 20h

20

H 2 K 2 " K 2s( )2 + # 2( )

where ! 0* is the complex conjugate of ! 0 . ! 0 reaches maximum at the minimum of

(K 2 ! Ks2 ) + " 2 :

!!k

K 2 " Ks2( )2 + # 2$

%&' = 0

2 K 2 " Ks2( )2K = 0

K = 0 or K = Ks

Thus, ! 0 is maximum when the wavenumber of the topography (as well as the wave solution) is Ks ! " / u . For the maximum response,

! 0 = "f0h0H (i#)

=if0h0#H

So, friction eliminates the singularity, but shift the phase of the response. Now ! 0 ! sin kx and thus the trough in the streamfunciton occurs 1/4 cycle east of the mountain crest, in approximate agreement with observations. For this case, the mountain ranges induce a net poleward mass/heat flux: v 'hT ' > 0 . It can also be shown that this can be interpreted as a mountain torque or form drag exerted by the lower topography on the fluid above. By use of a Fourier expansion (7.99) can be solved for realistic distributions of topography. The results for an x-dependence of hT given by a smoothed version of the earth’s topography at 45°N, a meridional wave number corresponding to a latitudinal half-wavelength of 35°, ! e = 5 days, u=17ms

"1, f0 = 10"4 s"1, and H = 8 km are shown in

Figure 7.15. Despite its simplicity, the Charney-Eliassen model does a remarkable job of reproducing the observed 500-hPa stationary wave pattern in Northern Hemisphere midlatitudes.

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End of Lecture 10. Homework: 7.1, 7.2, 7.19, 7.20. Note that in 7.20, you may assume the background wind is always eastward.