- Email: [email protected]
Chapter 16 Hydrodynamic instability
Chapter 16
Hydrodynamic Instability
Wave propagation is supported by a positive restoring force, which opposes air displacements by driving parcels back toward their undisturbed positions (Chapter 14). Under certain conditions, the sense of the restoring force is reversed. Air displacements are then reinforced by a negative restoring force, one which accelerates parcels away from their undisturbed positions. Instability was encountered earlier in connection with hydrostatic stratification (Chapter 7). If distributions of temperature and moisture violate the conditions for hydrostatic stability, small vertical displacements produce buoyancy forces that accelerate parcels away from their undisturbed positions. Unlike the response under stable stratification, this reaction leads to finite displacements of air. Fully developed convection then drives the stratification toward neutral stability by rearranging air. Two classes of instability are possible. Parcel instability follows from reinforcement of air displacements by a negative restoring force, such as that occurring in the development of convection. Wave instability occurs in the presence of a positive restoring force, but one that amplifies parcel oscillations inside wave motions. Unstable waves amplify by extracting energy from the mean circulation (e.g., from available potential energy associated with baroclinic stratification and vertical shear; Chapter 15). Strong zonal motion that results from the nonuniform distribution of heating and geostrophic equilibrium makes this class of instability the one most relevant to the large-scale circulation. Like parcel instability, it develops to neutralize instability in the mean flow, which it achieves by rearranging air.
16.1
Inertial Instability
The simplest form of large-scale instability relates to the inertial oscillations described in Sec. 12.4. Consider disturbances to a geostrophically balanced zonal flow K on an f-plane. If the disturbances introduce no pressure perturbation, the total motion is governed by the horizontal momentum balance du dt f v - O , (16.1.1)
dv dt + f ( u - ~) = 0, 517
(16.1.2)
518
16
Hydrodynamic Instability
where geostrophic equilibrium has been used to eliminate pressure in favor of ~. Since a parcel's motion satisfies (16.2)
dy dt'
v-
(16.1.1) implies dt = f
(16.3)
"
Integrating from the parcel's initial position Y0 to its displaced position Y0+ Y' gives u(yo + y') - -a(yo) = fy'.
To first order in parcel displacement, this can be expressed d-~y, u(Yo) + -~y - -u(yo) = fY' or
( u - ~)ly0 -
OK) y,
f
- 7y
= O.
Then incorporating (16.1.2) yields d2y ' et---r + f
dK) y'
f - Uy
= o.
(16.4)
If the mean flow is without shear, (16.4) reduces to a description of the inertial oscillations considered previously. In the presence of shear, ' displacements either oscillate, decay exponentially, or grow without bound. The system possesses unstable solutions if the absolute vorticity of the mean flow f + st= f
~y
(16.5)
has sign opposite to the planetary vorticity f. Displacements then amplify exponentially and the zonal flow is inertially unstable. These circumstances make the specific restoring force f ( f + ()y' negative, so the ensuing instability is of the parcel type. Because f + sr usually has the same sign as f, the criterion for inertial instability is tantamount to the absolute vorticity reversing sign somewhere. Inertial instability does not play a major role in the atmosphere. Extratropical motions tend to remain inertially stable, even locally in the presence of synoptic and planetary wave disturbances. However, the criterion for inertial instability is violated more easily near the equator, where f is small. Evidence of inertial instability exists in the tropical stratosphere, where horizontal shear w
16.2
S h e a r Instability
519
flanking the strong zonal jets (Fig. 1.8) can violate the criterion for inertial stability.
16.2
Shear Instability
More relevant to the large-scale circulation is instability associated directly with shear. Shear instability is of the wave type, so it requires a more involved analysis than that applying to an individual parcel. Like the treatment of wave motions, describing shear instability requires the solution of partial differential equations that govern perturbation properties. Closed-form solutions can be found only for very idealized profiles of zonal-mean flow K(y, z). However, an illuminating criterion for instability, due originally to Rayleigh, can be developed under fairly general circumstances.
16.2.1 Necessary Conditions for Instability Consider quasi-geostrophic motion on a beta plane in an atmosphere that extends upward indefinitely and is bounded below and laterally at y - +L by rigid walls. Disturbances to the zonal-mean flow K(y, z) are governed by first-order conservation of potential vorticity (14.75)
DQ' - -
-]- 1)'/~ e --" 0
Dt
(16.6.1)
where D / D t - 3/3t + -~3/3x,
la (fff ,~g,') (2'- v2q/+ po ~ ~P~ ' o~2~ ~e
-
-
-
~
Oy2
13
(f2
PO Oz
(16.6.2)
3~)
- ~ Po -~z
~Q
(16.6.3)
3y'
and z denotes log-pressure height. Requiring vertical motion to vanish at the ground (which is treated as an isobaric surface) gives, via the thermodynamic equation and thermal wind balance, the lower boundary condition Dt \ 3z J
3z 3x
=0
z-0.
(16.6.4)
Physically meaningful solutions must also have bounded column energy, which provides the upper boundary condition finite energy condition
z ~ oo.
(16.6.5)
520
16 Hydrodynamic Instability
At the lateral walls, v' must vanish, so 0' =const. It suffices to prescribe 0' = 0
y - +L.
(16.6.6)
Equations (16.6) define a second-order boundary value problem for the disturbance streamfunction ~'(x, t)--one that is homogeneous. Containing no imposed forcing, (16.6) describes a system that is self-governing or autonomous. Nontrivial solutions (i.e., other than q,' -- 0) exist only for certain eigenfrequencies that enable boundary conditions to be satisfied. Determined by solving the homogeneous boundary value problem for a given zonal flow ~(y, z), those eigenfrequencies are, in general, complex. Consider solutions of the form (16.7)
q,' = ~ ( y , z ) e ik(x-ct),
ici
where 9 and c - Cr + transforms (16.6) into
can
assume complex values. Substituting (16.7)
+ --P0~zz N2P~
- k2
+/~eXIt-- 0.
(16.8.1)
The lower boundary condition (16.6.4)becomes 3~ 3~ (~- c)-* - 0 z - 0. (16.8.2) 0z 3z For c real, (16.8) is singular at a critical line where ~ = c. The singularity disappears if ci :/: O, in which case (16.7) contains an exponential modulation in time. If the flow is stable, wave activity incident on the critical line is absorbed when dissipation is included (Sec. 14.3). The wave field then decays exponentially. If the flow is unstable, wave activity can be produced at the critical line. The wave field can then amplify exponentially. Multiplying the conjugate of (16.8) by 9 and (16.8) by the conjugate of and then subtracting yields
"u Oy---g--
ay 2 j +
~ - -po - - Oz k N 2 P0 -~z
..10(, 0.)] PO Oz
N2 Po~z
(16.9.1)
- 2 i c i I-u - c l fie "-" 0
and a~* _ ~ , a ~ 3z
I~] 2 3~ = 0
3 z + 2ici [-u -- c[ 2 o~z
z - 0.
(16.9.2)
The chain rule allows terms in the first set of square brackets in (16.9.1) to be expressed 02 XI/'* XI/'*o~2a~r o~ [ ~ o~aIt'* _ XI/'*o~aIr] ,I, oy-----7- 07 = k oy oy j
16.2
Shear
Instability
521
Terms in the second set of square brackets can be expressed in a similar manner. Then (16.9.1) can be written
d I ~ d ~ * _ x i , , O ~ 1 1 0 [f~ (O~* ~,d~)l Oz 3y L 3y 3y + ~Po 7z - ~ Po ~ az - 2ic i l_~ _
(16.10)
c[ 213 e -- O.
Integrating over the domain unravels the exterior differentials in (16.10) to give xI, 3xI't'*
'~Y +
L
-
x[t* ~
y=L
-~po Ve~-o~Z
2icifLfo0 ~L
podz
aY y=-L
dy o~Z
I~I*12 {~._ C[2 ~ e d y p o d z
(16.11)
z--O - O.
By (16.6.6), the first integral vanishes. The finite energy condition makes the upper limit inside the second integral also vanish. Then incorporating (16.9.2) for the lower limit yields the identity
/3 e ~P01XP'I2d y d z -
Ci
L
1 ~ - el 2
~
L N 2 I-u -- cl 2 3 z
dy
~--0
= O,
(16.12)
which must be satisfied for ~(y, z) to be a solution of (16.6). Advanced by Charney and Stern (1962), the preceding identity provides "necessary conditions" for instability of the zonal-mean flow K(y, z). If c is complex, (16.7) describes a disturbance whose amplitude varies in time exponentially eik(x-ct) = ekci t . eik(x-Crt),
with the growth/decay r a t e k c i. Without loss of generality, k can be considered positive, so the existence of unstable solutions requires ci > 0. Unstable solutions are then possible only if the quantity inside braces in (16.12) vanishes. If it does not, (16.12) implies ci = 0 and solutions to (16.6) are stable.
16.2.2
Barotropic and Baroclinic Instability
Requiring (16.12) to be satisfied with for instability:
ci
>
0 provides two alternative criteria
522
16 Hydrodynamic Instability
1. If 3-~13z vanishes at the lower boundary, so does the temperature gradient by thermal wind balance. Then /3e - 3Q/3y must reverse sign somewhere in the interior. Since/3 e isnormally positive, a region of negative potential vorticity gradient, oQ/3y < 0, is identified as an unstable region of the mean flow. 2. If/3 e > 0 throughout the interior, 3-u/3z must be positive somewhere on the lower boundary. By thermal wind balance, this implies the existence of an equatorward temperature gradient at the surface. Other combinations are also possible, but these are the ones most relevant to the atmosphere. Neither represents a "sufficient condition" for instability. Satisfying criterion (1) or (2) does not ensure the existence of unstable solutions. Criterion (1) defines a necessary condition for free-field instability (e.g., instability for which boundaries do not play an essential role). From (16.6.3), the mean gradient of potential vorticity can reverse sign through strong horizontal curvature of the mean flow or through strong (density-weighted) vertical curvature of the mean flow. It is customary to distinguish these contributions to eQ/3y. If the necessary condition for instability is met through horizontal shear, amplifying disturbances are referred to as barotropic instability. If it is met through vertical shear (which is proportional to the horizontal temperature gradient and the departure from barotropic stratification), amplifying disturbances are referred to as baroclinic instability. Realistic conditions often lead to criterion (1) being satisfied by both contributions, in which case amplifying disturbances are combined barotropic-baroclinic instability. In the absence of rotation, criterion (1) reduces to Rayleigh's (1880) necessary condition for instability of one-dimensional shear flow (Problem 16.12). Criterion (1) is then equivalent to requiring the mean flow profile to possess an inflection point. Since /3 is everywhere positive, rotation is stabilizing. It provides a positive restoring force that inhibits instability and supports stable wave propagation. Recall that (16.8) is singular at a critical line K = Cr if ci = 0. Exponential amplification removes the singularity by making ci > 0. When boundaries do not play an essential role, amplifyin__g solutions usually possess a critical line inside the unstable region where ~Q/~y < 0 (see, e.g., Dickinson, 1973). Rather than serving as a localized sink of wave activity, as its does under conditions of stable wave propagation (eQ/dy > 0), the critical line then functions as a localized source of wave activity. Wave activity flux then diverges out of the critical line, where it is produced by a conversion from the mean flow. Alternatively, incident wave activity that encounters the critical line is "overreflected": More radiates away than is incident on the unstable region. Criterion (2) describes instability that is produced through the direct involvement of the lower boundary. This criterion applies to baroclinic instability because it requires a temperature gradient at the surface and hence
16.3 The Eady Problem
523
baroclinic stratification. Since air must move parallel to it, the boundary can then drive motion across mean isotherms, which transfers heat meridionally (e.g., in sloping convection). By weakening the temperature gradient, eddy heat transfer drives the mean thermal structure toward barotropic stratification and releases available potential energy (Sec. 15.1), which in turn is converted to eddy kinetic energy. This situation underlies the development of extratropical cyclones. Temperature gradients introduced by the nonuniform distribution of heating make the stratification baroclinic and produce available potential energy, on which baroclinic instability feeds.
16.3
The Eady Problem
The simplest model of baroclinic instability is that of Eady (1949). Consider disturbances to a mean flow that is invariant in y, bounded above and below by rigid walls at z = 0, H on an f plane, and within the Boussinesq approximation (Sec. 12.5). A uniform meridional temperature gradient is imposed, which, by thermal wind balance, corresponds to constant vertical shear (Fig. 16.1) - Az
A - const.
(16.13)
3Q/o~y n
Under these circumstances, vanishes identically in the interior, so instability can follow solely from the temperature gradient along the boundaries. Disturbances to this system are governed by the perturbation potential vorticity equation in log-pressure coordinates
D (v2~t'-4-f2 32-----~u)--O
Dt
N 2 3z 2
(16.14.1)
'
=Az
Figure 16.1 Geometry and mean zonal flow in the Eady problem of baroclinic instability.
524
Hydrodynamic Instability
16
with the boundary conditions
D (3q/) Dt ~
z=O,H.
o~3q~'=O
o~z 3x
(16.14.2)
Considering solutions of the form ~' = xIt(z) cos(ly)e ik(x-ct)
(16:15)
reduces (16.14) to the one-dimensional boundary value problem d2~
-- a2a~ t = O,
(16.16.1)
dz 2
(-~ -
d~ c)--~z - A ~ - 0
z = O, H ,
(16.16.2)
where
a-
N
~lkhl
(16.16.3)
with [kh]2 = k 2 - + - / 2 , is a weighted horizontal wavenumber. 1 Solutions of (16.16.1) are of the form = A cosh (az) + B sinh (az). Substituting (16.17) into the boundary geneous system of two equations for solutions exist only if the determinant the dispersion relation for Eady modes c
= A2H2
(16.17)
conditions (16.16.2) leads to a homothe coefficients A and B. Nontrivial of that system vanishes, which yields (Problem 16.14)
coth (all) 1 aH + (all) 2 .
(16.18)
If the right-hand side of (16.18) is positive, c is real and the system is stable. If it is negative, unstable solutions exist. Because c - (AH/2) must then be imaginary, AH
Cr = --~-.
(16.19)
Thus, amplifying disturbances have phase speeds equal to the mean flow at the middle of the layer. Unstable disturbances are advected eastward by the mean flow with its speed at the the steering level: z = HI2. While influenced by rotation, baroclinic waves are not Rossby waves in a strict sense because, as the Eady model demonstrates, they can exist in the absence of/3. The quantity k 2 ( c - AH/2) 2 reflects the square of the complex frequency and is plotted as a function of a H in Fig. 16.2. For a greater than a critical
1(
)
1Considering structure of the form cos(/y) = 5 eily+e-ily implicitly presumes that disturbances are trapped meridionally (e.g., by rigid walls at y = •
16.3
525
The Eady Problem
1.0
0.8
0.6
tat) v o
~:~
0.4
T-X
G6
o.2
0.0
~c
-0.2
-0.4
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
(zH
Figure 16.2 Frequency squared of Eady modes as a function of scaled horizontal wavenumber. Instability occurs for wavenumbers smaller than the cutoff c~c.
value ac ~ 2.4, (c - AH/2) 2 > 0. Thus, the system possesses a "shortwave cutoff' for instability, amplifying solutions existing only for smaller a (larger scales). Wavenumbers oL < a~ are all unstable: (c - AH/2) 2 < 0. A maximum growth r a t e k c i is achieved at a H ~ 1.6 for l - 0. This value of a H also maximizes the growth rate for waves of fixed aspect ratio l / k , which have smaller kci. For square waves (k = l) and values representative of the troposphere, a H ~- 1.6 predicts a wavenumber of order 5, which is typical of extratropical cyclones. The maximum growth rate is proportional to A and therefore to the meridional temperature gradient. Representative values give an e-folding time for amplification of a couple of days, consistent with the observed development of extratropical cyclones. Insight into how instability is achieved follows from the structure of the most unstable disturbance. The lower boundary condition (16.16.2) implies
526
16
Hydrodynamic Instability
the following relationship between the coefficients in (16.17): B A
=
A
(16.20)
c~c'
which yields the structure shown in Fig. 16.3 for k - l. The fastest growing Eady mode is characterized by a westward tilt with height that places geopotential anomalies about 90 ~ out of phase between the lower and upper boundaries. This can be inferred from the eddy meridional velocity v' = i k q / , which is contoured (solid/dashed lines) in Fig. 16.3a on a vertical section passing zonally through the center of the disturbance (y - 0). Eddy geopotential and motion maximize at z = 0 and H in the form of two edge waves that are sandwiched between the upper and lower boundaries. In addition to driving air across mean isotherms, those boundaries trap instability, which allows the disturbance to amplify. Each edge wave decays vertically with the Rossby height scale H R - a-1. In the limit Ikhl --+ c ~ , H R is short enough for the two edge waves to be isolated from one another. Because neither has vertical phase tilt, hydrostatic balance implies a temperature perturbation that is in phase with the geopotential perturbation (Problem 16.15). The perturbation velocity v' is then everywhere in quadrature with 0', so the eddy heat flux averaged over a wavelength v' 0' vanishes. Consequently, eddy motion does not alter the baroclinic stratification of the mean state and no available potential energy is released. It is for this reason that short wavelengths in the Eady problem are neutral. For a < a c, H R is long enough for the two edge waves to influence one another. Phase-shifted, they produce the westward tilt apparent in Fig. 16.3a. The temperature perturbation is then no longer in phase with the geopotential perturbation. Isentropes (dotted lines in Fig. 16.3a) actually tilt slightly eastward with height to make 0' positively correlated with v" Cold air moves equatorward (out of the page) and warm air moves poleward (into the page). 2 Both contribute positively to v' 0' to produce a poleward eddy heat flux, which releases available potential energy. The mode's amplification is thus directly related to its westward tilt with height. The streamfunction in a horizontal section at z - H / 2 (Fig. 16.3b) and in a meridional section at k x - 37r/2 (Fig. 16.4) illustrates that polewardmoving air ascends, whereas, half a wavelength away, equatorward-moving air descends. Characteristic of sloping convection, this is the same sense meridional motion would have were it to move along zonal-mean isentropic surfaces in Fig. 16.4 (dotted lines), which slope upward toward the pole. However, meridional displacements of the fastest growing Eady mode have a slope at midlevel which is only about half that of mean isentropic surfaces (Problem 16.18). Eddy motion directed across those surfaces advects high 0 poleward and low 0 equatorward in adjacent branches of a cell, as is evident from isentropes at midlevel (Fig. 16.3b). This asymmetric motion produces a net heat 2In fact, v' and 0' are perfectly in phase at the steering level: z =
HI2.
16.3
(a)
v'
v ',,_.,..~ ~-'~:
. . . . ......
-.........
\
--',.
........
~ -__,,___,__,_,__,_,_,__ - - y - , - ; - , - - c - < - ~ k - \ - N - \_- & ~ /
./
L
-- - , - , - - , - - , - , - - - , - - - - ,
527
0
--.
k .... < -_. . . . . . . . . . . --'~,"":-"_ ........... ~----r--~--~--~--~-.." k,_.-. ..... . .' ""
The Eady Problem
L
~
\
\
""--
..... 2:.
--~--Z:I-I
0
1-7-~-_c-_"~.:Q~-1 ,
". . . . . . . . . . . . . . . . . . . . .
""
"\'" r
- .............
7:t
...
- ......... " . . . . . . . . . . . . .
--~___
:_.,.;.:I
-V.-~--~"
-_ _-\ ~ - -
"'"
rt
"
2rt
kx (b)
V'
O
- .........
/1:
---~T . . . . . . :,. . . .
2
---L:-S-L'--'~" - -_...~Z__-". . . . . . . . .
- - - ~/-.- - r . . ./. . ,"" ~ ." . . . ='--:."-"_ - - . - ~ , . . . . ._".~------<--m--'--" --'-"-<--',"- ....
V-f- .:'-- "- - ,";, ~, .-:..~-:-----~-~S.".--.-:~--~-~ --~--
-~-l-.'--l--L-+--~ ," _.-----:-=" .:...--~-<-'-.-~--C-_Y:
1 - - r - r - c T ~ z . ~ - - ' : " : . . . . . . . "~.-~--~--'~'i"~-[-~ + + ~ - r r ' ; " l . - , - " ' ; ' ; . . :->-~'. :~- ~--~-'; ~: Z ~:L: r-P -~-r-r~-T~s -~-'-/"" . . . . . . . L - @ - ~ ' T l'_[.i.~-~, I-I" -v 1-r 1"1-1: I . ~ - - r ' " :
~-P-~l-~'-'~] ' -l---e'" -'- 4-~" "~"
-
.......
- .....
- ....
: : L . J~-'-~7, Z - ~ ,-v.
i .-~-~-7 ~ ~j t-~
---T"
-~-
-
"l-
-
"~" + t r ~a_~._c.~- ~.~,. . . . . .~ . . . . ~-, r ~.,.~_~_~. -1" . - v . - r - ' - ,, x__C-~,--:--~-':~'~'7 ~ / ~ L
-r- - [ - -~'.,I._
....
x----,'"
V_---N--~-
t
. . . .
. . . . . .
"~---~'"
~
9-~- - - - ' ~ . - - " ~ ' ~
. . . . . . . .
~, . . . .
,--->"
3.._:.=....=--'-
i---~--~-~:-"--"---
..--
~--..----~.~ ""
',,,.-~ . . . .
_ _ _-
-=---
_
/ _-~--- . . . .
.,.,, . , ~ , , o t
":--='-'~'~':"
... ~ --"----Z.->" --
/
.
-.:-,:.-.
l
I
-~- 7"- 7 -"
I ....
Z--4---
/ ...-. ~ _ _ _ ~ , , .
/ .....
_ > _ . , ~ _ _ _ - ~ - - -'='- ,-.-- -...-l~-'-- . . . . . . .
_ m
2
0
rt
2rt
kx Figure 16.3 Structure of the fastest growing square Eady mode (k = I). (a) Vertical section in the zonal plane at y = 0 of eddy meridional velocity v' (solid/dashed lines) and isentropic surfaces 0 = const (dotted lines). Potential temperature increases upward. | marks equatorward motion (v' < 0) and @ marks poleward motion (v' > 0). (b) Horizontal section at z = H/2 of eddy streamlines (solid/dashed lines) and isentropes (dotted lines). Potential temperature increases equatorward.
528
16
Hydrodynamic Instability m
..........
0
H
.
.
..... "-" Z
..o..- ....
..,
...,.---'.~
-
....
...,,
o~.,,,oO~
..m,,..--_:~"
.,,.---~'" ...-
. . . . . --=.'"
....
-.,,
--
.
..
......
".~
o~ .,,
"."
_
~
~
_..,
._~.o~
.... ~
_.~,,~ _..,
._.,,
__.~.~o--.r.:-~
~
_.r~....,,,~r,"...-~
_.-.~,.~
~ . . . . .=.z,'"_..,
....
~
~
..-.po..,.~~176
~
.-.~.~176176
...r:,...,,.~~
~ ..7~..o..,~.~ ~
.._~_......,,..~'....._~
.-..,,
.,,.,,-ooT.~
.,.-~-~
....,,
_.,..-:.T,
~
.~,.--:.~" _..
_...,,~
..rT~....~.z.~ ~ ~
~
~
.,_..~'T.....~
..,..,~'~ ~
~
...,, ..~,~176
.-...=.,~-'-~.:-7,
_
.
..
_ ....----
..~.o--~" .,,
--4,
.....
~ .... -~.... .
...., . _ , . - - = . : - " _
_..~~176
~
~
-
. _
..., . . . . , , - ' "
...,o-"
-
0
-~
o ly
7
F i g u r e 16.4 Vertical section in the meridional plane at kx = 37r/2 of m e a n isentropic surfaces ( d o t t e d lines) and motion for the fastest growing square E a d y mode. Potential t e m p e r a t u r e increases u p w a r d and e q u a t o r w a r d ( c o m p a r e Fig. 12.5).
flux poleward, which releases available potential energy by driving the thermal structure toward barotropic stratification. Poleward heat flux acts to eliminate the horizontal temperature gradient and shallow the slope of isentropic surfaces, which in turn reduces the zonal-mean available potential energy. Extratropical cyclones have qualitatively similar structure during their development. Figure 16.5 shows distributions of 700-mb height and temperature for an amplifying cyclone situated off the coast of Africa on March 2, 1984. This disturbance is the precursor to the cyclone apparent in Figs. 1.15 and 1.24 two days later. During amplification, the system tilts westward, which transfers heat poleward and releases available potential energy--analogous to an unstable Eady mode with c~ < a c. Eddy heat flux tends to maximize near 700 mb, which typifies the steering level of observed cyclones. This is lower than the steering level predicted by the Eady model. However, an unbounded model treated by Charney (1947) reproduces the observed steering level, while retaining the essential ingredients captured by Eady's solution. Figure 16.5 contains the characteristic signature of sloping convection: A tongue of warm air is drawn poleward ahead of the closed low, while cold air is advected equatorward behind it. Those bodies of air have disparate histories, which are reflected in contemporaneous infrared (IR) and water vapor imagery (Figs. 16.6a and b). A tongue of high cloud cover and moisture that extends northwestward from the African coast defines the w a r m s e c t o r ahead of the cyclone. A complementary tongue of cloud-free conditions and low moisture is being drawn equatorward behind it. Sharp gradients separating those bodies
16.4
529
Nonlinear Considerations
March 2, 1984
""-..,
,,
......,..:
. . .~i.. ..}::.-i-~, ll.... . . . . .". ,
H3183.o ..:.
,
:.. "._.....
::-
_. _
/
i
...
?... -... . -.. "......
.~..
~ .
'" ..................... ~"" ......... .. ...... -::..:
.. - . . . . .
-..................... E,~........~!..............
.
:...
~:
....
"..
~
,
~
',
..
.
.
.~.
Figure 16.5 700-mb height (solid lines) and selected isotherms (dashed lines) on March 2, 1984. A surface frontal analysis is superposed.
of air mark warm and cold fronts at the surface, which are superposed in Fig. 16.5 and delineate the warm sector. Moisture inside the warm sector can be traced back in water vapor imagery (e.g., Fig. 16.6d) to its source: tropical convection over the Amazon basin (compare Fig. 1.24). Air advancing behind the cold front undercuts the warm sector in sloping convection, lifting moist air to produce the extensive cloud shield in Fig. 16.6a.
16.4
Nonlinear Considerations
While capturing the development of extratropical cyclones, Eady's solution provides only a hint of their behavior at maturity. Conversion of available
530
16
Hydrodynamic Instability
Water Vapor
Infrared (a) Mar 2 1500 (
.............
......................
(b) M a - " 1500 (
(c) Ma-"
(d) Ma-"
0300 (
0300 (
(e) Ms 1200 ~
(f) Mar 1200 (
Figure 16.6 Infrared and water vapor imagery from Meteosat between March 2 and March 4, 1984, that reveal the evolution of an extratropical cyclone off the northwest coast of Africa. The warm sector ahead of the cyclone is marked by a tongue of high cloud cover and moisture, which slopes northwestward on March 2. That air mass subsequently overturns near the juncture of cold and warm fronts delineating it to form an occlusion, in which cold and warm air are entrained and mixed horizontally (compare Fig. 9.21).
16.4
Nonlinear Considerations
531
potential energy into eddy kinetic energy enables a baroclinic system to intensify and eventually attain finite amplitude. Finite horizontal displacements then invalidate the linear description on which Eady's model is based and which predicts exponential amplification to continue indefinitely--analogous to vertical displacements under hydrostatically unstable conditions (Sec. 7.3). Second-order effects then modify the zonal-mean state, which in turn limits subsequent amplification of the baroclinic system. As a baroclinic disturbance amplifies, horizontal displacements (e.g., Fig. 16.3b) become increasingly exaggerated. Warm tropical air is eventually folded north of cold polar air, as is revealed by the distribution of potential vorticity on March 2, 1984 (Fig. 12.10). Deformations experienced by those air masses steepen potential temperature gradients separating warm and cold air (e.g., Fig. 12.4), which intensifies the accompanying fronts. Continued advection leads to warm and cold air eventually encircling the low, with the warm sector overturning at the junction of the warm and cold fronts. (Cloud cover in Fig. 9.21 provides a textbook example.) The cyclone in Fig. 16.5 then occludes, with warm and cold fronts overlapping in the center of the system. The occlusion actually develops when the surface trough (not shown) separates from the junction of warm and cold fronts and deepens farther back into the cold air mass (see, e.g., Wallace and Hobbs, 1977). This structure marks the mature stage of the cyclone's life cycle because the surface trough is then positioned beneath the upper-level trough, which eliminates the system's westward tilt and hence its release of available potential energy--now analogous to a neutral Eady mode with c~ > c~C. Cold and warm air drawn into the occlusion are then wound together and mixed horizontally. Figure 16.6 shows sequences of IR and water vapor imagery while the disturbance in Fig. 16.5 matures. At 1500 GMT on March 2 (Figs. 16.6a and b), the cold front is approaching the warm front near their junction, with the warm sector clearly defined. Twelve hours later, the 700-mb trough has deepened (not shown). The warm sector in IR and water vapor imagery (Figs. 16.6c and d) has then been sheared to the northwest, where cold dry air is being entrained with warm moist air in the occlusion that has formed. This process culminates in cold and warm air at the occlusion winding up into a spiralbsimilar to behavior in a cylindrical annulus at high rotation (compare Fig. 15.7f). By 1200 GMT on March 3 (Figs. 16.6e and f), air inside the occlusion has wound up into a vortex, which is seen to separate from the remaining warm sector to its south and east. Interleaving bands of cold and warm air that are apparent in both cloud cover and moisture symbolize efficient horizontal mixing. One day later (Figs. 1.15 and 1.23), that mass of air has become nearly homogeneous. The 700-mb trough has then weakened and high cloud cover that developed earlier through sloping convection is dissipating. Only a broad spiral of equatorward-moving air remains, drawn cyclonically around the now-diffuse anomaly of potential vorticity, in which cold and warm air have been mixed.
532
16 Hydrodynamic Instability
Since 0 (more generally, 0e) is conserved for individual air parcels, horizontal mixing makes the potential temperature uniform and restores the thermal structure to barotropic stratification. Baroclinic instability that was present initially has then been neutralized and no more potential energy is available for conversion to eddy kinetic energy. This process is a direct counterpart of vertical mixing by convection, which neutralizes hydrostatic instability (Sec. 7.3). Strong modification of the mean state by unstable eddies contrasts sharply with wave propagation under stable conditions. Parcel displacements are then bounded, so the mean state remains largely unaffected outside regions of dissipation, where wave activity is absorbed (Chapter 14). The preceding treatment applies formally to a zonally symmetric mean state. In practice, the results also lend insight into isolated regions of instability. Amplified planetary waves in the Northern Hemisphere reinforce zonal-mean vertical shear to produce a broken storm track that is marked by localized jets east of Asia and North America (refer to Figs. 15.8a and 9.38). Strongly baroclinic, the north Atlantic and north Pacific storm tracks are preferred sites for cyclone development. Weaker planetary waves in the Southern Hemisphere leave stratification there nearly uniform in longitude to produce a continuous storm track. This enables cyclones there to assume a more regular distribution than is observed in the Northern Hemisphere, one that occasionally resembles baroclinic modes in a rotating annulus (see Figs. 2.10 and 15.7e). The criteria for instability also have implications important to planetary waves that have attained large amplitude. The situation is analogous to the breaking of gravity waves when isentropic surfaces are overturned (Fig. 14.25). High 0 is then folded underneath low 0 to make ~O/3z < 0 and render the region hydrostatically unstable. Convective mixing that ensues absorbs organized wave motion and neutralizes the instability by driving ~O/3z to zero. Planetary waves of sufficient amplitude overturn the distribution of the conserved property Q (Fig. 14.26). High potential vorticity folded poleward of low potential vorticity then reverses ~Q/~y locally to render the motion dynamically unstable. At that point, the planetary wave field breaks. Smallscale eddies that develop in the region of instability mix Q horizontally, which absorbs organized wave motion and neutralizes the instability by driving ~Q/~y to zero. Suggested Reading
Atmospheric Science: An Introductory Survey (1977) by Wallace and Hobbs contains a synoptic analysis of the life cycle of extratropical cyclones.
Atmosphere-Ocean Dynamics (1982) by Gill includes a nice comparison between Charney's (1947) model of baroclinic instability and the Eady model.
An Introduction to Dynamic Meteorology (1992) by Holton provides a complete description of baroclinic extratropical disturbances, their energetics, and
Problems
533
aspects surrounding their prediction. It also discusses the process of frontal formation.
Middle Atmosphere Dynamics (1987) by Andrews et aL discusses inertial instability in the tropical stratosphere and includes a description of planetary wave breaking. Problems 16.1. Derive equations (16.1). 16.2. Carry out the Lagrangian integration of (16.3) to relate an air parcel's meridional displacement to its change of zonal velocity. 16.3. Provide an expression for the specific restoring force inside an inertially unstable layer. 16.4. (a) At what latitudes is inertial instability favored? (b) Identify those regions of the zonal-mean flow in Fig. 1.8 that would be most susceptible to inertial instability. 16.5. Consider westerly flow that corresponds to an angular velocity A(z) = -~/(a cos ~b), which varies with height but not with latitude. (a) Within the framework of quasi-geostrophic motion and the Boussinesq approximation, what sign of curvature must the velocity profile A(z) have for shear instability to develop? (b) At what latitudes is shear instability favored most if A(z) = E(z)-2~, with E < < 17 16.6. Obtain the lower boundary condition (16.6.4). 16.7. A wave packet approaches its critical line with phase speed Cr equal to the real part of c in (16.8.1). Describe the wave packet's evolution in the presence of weak dissipation if, under inviscid adiabatic conditions, (a) c in (16.8.1) is real and (b) c in (16.8.1) is complex. 16.8. Obtain equations (16.9) and (16.10). 16.9. Recover the necessary condition (16.12). 16.10. Derive a necessary condition for instability of a quasi-geostrophic zonal flow that is unbounded above and bounded below by a rigid surface of constant height. 16.11. Derive a necessary condition for instability of a quasi-geostrophic zonal flow that is bounded vertically at z = 0 and H by rigid walls. 16.12. (a) Show that, in the absence of rotation, criterion (1) in Sec. 16.2.2 recovers Rayleigh's condition for instability: a barotropic flow must possess an inflection point in its interior. (b) Discuss the influence rotation has on shear instability. 16.13. Why do midlatitude cyclones intensify during winter? 16.14. Obtain the dispersion relation (16.18) for Eady modes.
534
16 Hydrodynamic Instability
16.15. Demonstrate that the limiting structure of Eady modes for H ~ oo has v' in quadrature with 0'. 16.16. Derive an expression for the shortwave cutoff ac for instability in the Eady problem. 16.17. (a) Show that the maximum growth rate of Eady modes is achieved for l = 0. (b) Express the growth rate of the fastest growing square Eady mode (k = l) in terms of that of the fastest growing Eady mode (l = 0). 16.18. Show that, for the fastest growing Eady mode, the maximum slope of motion in the meridional plane is only about half the slope of mean isentropic surfaces. 16.19. Calculate the e-folding times of square Eady modes (k = l) for an f plane at 45 ~ N 2 - 1 0 - 4 S-2, vertical shear of 3 m s -1 km -1, a rigid lid at 10 km, and for zonal wavenumbers 1-8. 16.20. Within the framework of an initial value problem, describe the structure and evolution predicted by the Eady model if the flow in Problem 15.19 is initialized with random structure having a broad wavenumber spectrum (a) under inviscid adiabatic conditions, (b) in the presence of linear dissipation 3 with a timescale equal to the shortest e-folding time of zonal wavenumber 2 under inviscid adiabatic conditions and (c) in the presence of linear dissipation with a timescale equal to the shortest e-folding time of zonal wavenumber 4 under inviscid adiabatic conditions. 16.21. Meridional components of the eddy momentum flux u'v' and group velocity %y vanish for Eady modes. Why? 16.22. Show that the eddy heat flux v'0' is positive and independent of height for Eady modes. 16.23. Use the Eady problem to explain (a) why midlatitude cyclones develop and track along the jet stream, (b) how baroclinic instability would be altered if the jet were displaced equatorward and (c) where the storm tracks should be positioned in relation to the time-mean motion in Fig. 1.9. 16.24. Precipitation inside cyclones often assumes a banded structure. Discuss this feature in relation to the evolution in Fig. 16.6 and the corresponding distribution of potential vorticity Qg. 16.25. Consider an idealized atmosphere in which the circulation is described by the barotropic zonal flow K(y) = - U cos(24~), 3Rayleigh friction and Newtonian cooling of the same timescale.
Problems
535
where U = 0.55~a. (a) Characterize the inertial stability of this flow as a function of latitude. (b) Describe the zonal-mean circulation toward which inertial instability will drive the flow. 16.26. Discuss the relationship of barotropic and baroclinic instability to stratification (Sec. 12.2.1). 16.27. Unlike extratropical cyclones, tropical cyclones are driven by latent heat release. Suppose a typhoon drifts poleward from the ITCZ. (a) Describe its evolution as it migrates over colder sea surface temperatures. (b) What structure of midlatitude westerlies will allow it to sustain itself?
Hydrodynamic Instability
Wave propagation is supported by a positive restoring force, which opposes air displacements by driving parcels back toward their undisturbed positions (Chapter 14). Under certain conditions, the sense of the restoring force is reversed. Air displacements are then reinforced by a negative restoring force, one which accelerates parcels away from their undisturbed positions. Instability was encountered earlier in connection with hydrostatic stratification (Chapter 7). If distributions of temperature and moisture violate the conditions for hydrostatic stability, small vertical displacements produce buoyancy forces that accelerate parcels away from their undisturbed positions. Unlike the response under stable stratification, this reaction leads to finite displacements of air. Fully developed convection then drives the stratification toward neutral stability by rearranging air. Two classes of instability are possible. Parcel instability follows from reinforcement of air displacements by a negative restoring force, such as that occurring in the development of convection. Wave instability occurs in the presence of a positive restoring force, but one that amplifies parcel oscillations inside wave motions. Unstable waves amplify by extracting energy from the mean circulation (e.g., from available potential energy associated with baroclinic stratification and vertical shear; Chapter 15). Strong zonal motion that results from the nonuniform distribution of heating and geostrophic equilibrium makes this class of instability the one most relevant to the large-scale circulation. Like parcel instability, it develops to neutralize instability in the mean flow, which it achieves by rearranging air.
16.1
Inertial Instability
The simplest form of large-scale instability relates to the inertial oscillations described in Sec. 12.4. Consider disturbances to a geostrophically balanced zonal flow K on an f-plane. If the disturbances introduce no pressure perturbation, the total motion is governed by the horizontal momentum balance du dt f v - O , (16.1.1)
dv dt + f ( u - ~) = 0, 517
(16.1.2)
518
16
Hydrodynamic Instability
where geostrophic equilibrium has been used to eliminate pressure in favor of ~. Since a parcel's motion satisfies (16.2)
dy dt'
v-
(16.1.1) implies dt = f
(16.3)
"
Integrating from the parcel's initial position Y0 to its displaced position Y0+ Y' gives u(yo + y') - -a(yo) = fy'.
To first order in parcel displacement, this can be expressed d-~y, u(Yo) + -~y - -u(yo) = fY' or
( u - ~)ly0 -
OK) y,
f
- 7y
= O.
Then incorporating (16.1.2) yields d2y ' et---r + f
dK) y'
f - Uy
= o.
(16.4)
If the mean flow is without shear, (16.4) reduces to a description of the inertial oscillations considered previously. In the presence of shear, ' displacements either oscillate, decay exponentially, or grow without bound. The system possesses unstable solutions if the absolute vorticity of the mean flow f + st= f
~y
(16.5)
has sign opposite to the planetary vorticity f. Displacements then amplify exponentially and the zonal flow is inertially unstable. These circumstances make the specific restoring force f ( f + ()y' negative, so the ensuing instability is of the parcel type. Because f + sr usually has the same sign as f, the criterion for inertial instability is tantamount to the absolute vorticity reversing sign somewhere. Inertial instability does not play a major role in the atmosphere. Extratropical motions tend to remain inertially stable, even locally in the presence of synoptic and planetary wave disturbances. However, the criterion for inertial instability is violated more easily near the equator, where f is small. Evidence of inertial instability exists in the tropical stratosphere, where horizontal shear w
16.2
S h e a r Instability
519
flanking the strong zonal jets (Fig. 1.8) can violate the criterion for inertial stability.
16.2
Shear Instability
More relevant to the large-scale circulation is instability associated directly with shear. Shear instability is of the wave type, so it requires a more involved analysis than that applying to an individual parcel. Like the treatment of wave motions, describing shear instability requires the solution of partial differential equations that govern perturbation properties. Closed-form solutions can be found only for very idealized profiles of zonal-mean flow K(y, z). However, an illuminating criterion for instability, due originally to Rayleigh, can be developed under fairly general circumstances.
16.2.1 Necessary Conditions for Instability Consider quasi-geostrophic motion on a beta plane in an atmosphere that extends upward indefinitely and is bounded below and laterally at y - +L by rigid walls. Disturbances to the zonal-mean flow K(y, z) are governed by first-order conservation of potential vorticity (14.75)
DQ' - -
-]- 1)'/~ e --" 0
Dt
(16.6.1)
where D / D t - 3/3t + -~3/3x,
la (fff ,~g,') (2'- v2q/+ po ~ ~P~ ' o~2~ ~e
-
-
-
~
Oy2
13
(f2
PO Oz
(16.6.2)
3~)
- ~ Po -~z
~Q
(16.6.3)
3y'
and z denotes log-pressure height. Requiring vertical motion to vanish at the ground (which is treated as an isobaric surface) gives, via the thermodynamic equation and thermal wind balance, the lower boundary condition Dt \ 3z J
3z 3x
=0
z-0.
(16.6.4)
Physically meaningful solutions must also have bounded column energy, which provides the upper boundary condition finite energy condition
z ~ oo.
(16.6.5)
520
16 Hydrodynamic Instability
At the lateral walls, v' must vanish, so 0' =const. It suffices to prescribe 0' = 0
y - +L.
(16.6.6)
Equations (16.6) define a second-order boundary value problem for the disturbance streamfunction ~'(x, t)--one that is homogeneous. Containing no imposed forcing, (16.6) describes a system that is self-governing or autonomous. Nontrivial solutions (i.e., other than q,' -- 0) exist only for certain eigenfrequencies that enable boundary conditions to be satisfied. Determined by solving the homogeneous boundary value problem for a given zonal flow ~(y, z), those eigenfrequencies are, in general, complex. Consider solutions of the form (16.7)
q,' = ~ ( y , z ) e ik(x-ct),
ici
where 9 and c - Cr + transforms (16.6) into
can
assume complex values. Substituting (16.7)
+ --P0~zz N2P~
- k2
+/~eXIt-- 0.
(16.8.1)
The lower boundary condition (16.6.4)becomes 3~ 3~ (~- c)-* - 0 z - 0. (16.8.2) 0z 3z For c real, (16.8) is singular at a critical line where ~ = c. The singularity disappears if ci :/: O, in which case (16.7) contains an exponential modulation in time. If the flow is stable, wave activity incident on the critical line is absorbed when dissipation is included (Sec. 14.3). The wave field then decays exponentially. If the flow is unstable, wave activity can be produced at the critical line. The wave field can then amplify exponentially. Multiplying the conjugate of (16.8) by 9 and (16.8) by the conjugate of and then subtracting yields
"u Oy---g--
ay 2 j +
~ - -po - - Oz k N 2 P0 -~z
..10(, 0.)] PO Oz
N2 Po~z
(16.9.1)
- 2 i c i I-u - c l fie "-" 0
and a~* _ ~ , a ~ 3z
I~] 2 3~ = 0
3 z + 2ici [-u -- c[ 2 o~z
z - 0.
(16.9.2)
The chain rule allows terms in the first set of square brackets in (16.9.1) to be expressed 02 XI/'* XI/'*o~2a~r o~ [ ~ o~aIt'* _ XI/'*o~aIr] ,I, oy-----7- 07 = k oy oy j
16.2
Shear
Instability
521
Terms in the second set of square brackets can be expressed in a similar manner. Then (16.9.1) can be written
d I ~ d ~ * _ x i , , O ~ 1 1 0 [f~ (O~* ~,d~)l Oz 3y L 3y 3y + ~Po 7z - ~ Po ~ az - 2ic i l_~ _
(16.10)
c[ 213 e -- O.
Integrating over the domain unravels the exterior differentials in (16.10) to give xI, 3xI't'*
'~Y +
L
-
x[t* ~
y=L
-~po Ve~-o~Z
2icifLfo0 ~L
podz
aY y=-L
dy o~Z
I~I*12 {~._ C[2 ~ e d y p o d z
(16.11)
z--O - O.
By (16.6.6), the first integral vanishes. The finite energy condition makes the upper limit inside the second integral also vanish. Then incorporating (16.9.2) for the lower limit yields the identity
/3 e ~P01XP'I2d y d z -
Ci
L
1 ~ - el 2
~
L N 2 I-u -- cl 2 3 z
dy
~--0
= O,
(16.12)
which must be satisfied for ~(y, z) to be a solution of (16.6). Advanced by Charney and Stern (1962), the preceding identity provides "necessary conditions" for instability of the zonal-mean flow K(y, z). If c is complex, (16.7) describes a disturbance whose amplitude varies in time exponentially eik(x-ct) = ekci t . eik(x-Crt),
with the growth/decay r a t e k c i. Without loss of generality, k can be considered positive, so the existence of unstable solutions requires ci > 0. Unstable solutions are then possible only if the quantity inside braces in (16.12) vanishes. If it does not, (16.12) implies ci = 0 and solutions to (16.6) are stable.
16.2.2
Barotropic and Baroclinic Instability
Requiring (16.12) to be satisfied with for instability:
ci
>
0 provides two alternative criteria
522
16 Hydrodynamic Instability
1. If 3-~13z vanishes at the lower boundary, so does the temperature gradient by thermal wind balance. Then /3e - 3Q/3y must reverse sign somewhere in the interior. Since/3 e isnormally positive, a region of negative potential vorticity gradient, oQ/3y < 0, is identified as an unstable region of the mean flow. 2. If/3 e > 0 throughout the interior, 3-u/3z must be positive somewhere on the lower boundary. By thermal wind balance, this implies the existence of an equatorward temperature gradient at the surface. Other combinations are also possible, but these are the ones most relevant to the atmosphere. Neither represents a "sufficient condition" for instability. Satisfying criterion (1) or (2) does not ensure the existence of unstable solutions. Criterion (1) defines a necessary condition for free-field instability (e.g., instability for which boundaries do not play an essential role). From (16.6.3), the mean gradient of potential vorticity can reverse sign through strong horizontal curvature of the mean flow or through strong (density-weighted) vertical curvature of the mean flow. It is customary to distinguish these contributions to eQ/3y. If the necessary condition for instability is met through horizontal shear, amplifying disturbances are referred to as barotropic instability. If it is met through vertical shear (which is proportional to the horizontal temperature gradient and the departure from barotropic stratification), amplifying disturbances are referred to as baroclinic instability. Realistic conditions often lead to criterion (1) being satisfied by both contributions, in which case amplifying disturbances are combined barotropic-baroclinic instability. In the absence of rotation, criterion (1) reduces to Rayleigh's (1880) necessary condition for instability of one-dimensional shear flow (Problem 16.12). Criterion (1) is then equivalent to requiring the mean flow profile to possess an inflection point. Since /3 is everywhere positive, rotation is stabilizing. It provides a positive restoring force that inhibits instability and supports stable wave propagation. Recall that (16.8) is singular at a critical line K = Cr if ci = 0. Exponential amplification removes the singularity by making ci > 0. When boundaries do not play an essential role, amplifyin__g solutions usually possess a critical line inside the unstable region where ~Q/~y < 0 (see, e.g., Dickinson, 1973). Rather than serving as a localized sink of wave activity, as its does under conditions of stable wave propagation (eQ/dy > 0), the critical line then functions as a localized source of wave activity. Wave activity flux then diverges out of the critical line, where it is produced by a conversion from the mean flow. Alternatively, incident wave activity that encounters the critical line is "overreflected": More radiates away than is incident on the unstable region. Criterion (2) describes instability that is produced through the direct involvement of the lower boundary. This criterion applies to baroclinic instability because it requires a temperature gradient at the surface and hence
16.3 The Eady Problem
523
baroclinic stratification. Since air must move parallel to it, the boundary can then drive motion across mean isotherms, which transfers heat meridionally (e.g., in sloping convection). By weakening the temperature gradient, eddy heat transfer drives the mean thermal structure toward barotropic stratification and releases available potential energy (Sec. 15.1), which in turn is converted to eddy kinetic energy. This situation underlies the development of extratropical cyclones. Temperature gradients introduced by the nonuniform distribution of heating make the stratification baroclinic and produce available potential energy, on which baroclinic instability feeds.
16.3
The Eady Problem
The simplest model of baroclinic instability is that of Eady (1949). Consider disturbances to a mean flow that is invariant in y, bounded above and below by rigid walls at z = 0, H on an f plane, and within the Boussinesq approximation (Sec. 12.5). A uniform meridional temperature gradient is imposed, which, by thermal wind balance, corresponds to constant vertical shear (Fig. 16.1) - Az
A - const.
(16.13)
3Q/o~y n
Under these circumstances, vanishes identically in the interior, so instability can follow solely from the temperature gradient along the boundaries. Disturbances to this system are governed by the perturbation potential vorticity equation in log-pressure coordinates
D (v2~t'-4-f2 32-----~u)--O
Dt
N 2 3z 2
(16.14.1)
'
=Az
Figure 16.1 Geometry and mean zonal flow in the Eady problem of baroclinic instability.
524
Hydrodynamic Instability
16
with the boundary conditions
D (3q/) Dt ~
z=O,H.
o~3q~'=O
o~z 3x
(16.14.2)
Considering solutions of the form ~' = xIt(z) cos(ly)e ik(x-ct)
(16:15)
reduces (16.14) to the one-dimensional boundary value problem d2~
-- a2a~ t = O,
(16.16.1)
dz 2
(-~ -
d~ c)--~z - A ~ - 0
z = O, H ,
(16.16.2)
where
a-
N
~lkhl
(16.16.3)
with [kh]2 = k 2 - + - / 2 , is a weighted horizontal wavenumber. 1 Solutions of (16.16.1) are of the form = A cosh (az) + B sinh (az). Substituting (16.17) into the boundary geneous system of two equations for solutions exist only if the determinant the dispersion relation for Eady modes c
= A2H2
(16.17)
conditions (16.16.2) leads to a homothe coefficients A and B. Nontrivial of that system vanishes, which yields (Problem 16.14)
coth (all) 1 aH + (all) 2 .
(16.18)
If the right-hand side of (16.18) is positive, c is real and the system is stable. If it is negative, unstable solutions exist. Because c - (AH/2) must then be imaginary, AH
Cr = --~-.
(16.19)
Thus, amplifying disturbances have phase speeds equal to the mean flow at the middle of the layer. Unstable disturbances are advected eastward by the mean flow with its speed at the the steering level: z = HI2. While influenced by rotation, baroclinic waves are not Rossby waves in a strict sense because, as the Eady model demonstrates, they can exist in the absence of/3. The quantity k 2 ( c - AH/2) 2 reflects the square of the complex frequency and is plotted as a function of a H in Fig. 16.2. For a greater than a critical
1(
)
1Considering structure of the form cos(/y) = 5 eily+e-ily implicitly presumes that disturbances are trapped meridionally (e.g., by rigid walls at y = •
16.3
525
The Eady Problem
1.0
0.8
0.6
tat) v o
~:~
0.4
T-X
G6
o.2
0.0
~c
-0.2
-0.4
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
(zH
Figure 16.2 Frequency squared of Eady modes as a function of scaled horizontal wavenumber. Instability occurs for wavenumbers smaller than the cutoff c~c.
value ac ~ 2.4, (c - AH/2) 2 > 0. Thus, the system possesses a "shortwave cutoff' for instability, amplifying solutions existing only for smaller a (larger scales). Wavenumbers oL < a~ are all unstable: (c - AH/2) 2 < 0. A maximum growth r a t e k c i is achieved at a H ~ 1.6 for l - 0. This value of a H also maximizes the growth rate for waves of fixed aspect ratio l / k , which have smaller kci. For square waves (k = l) and values representative of the troposphere, a H ~- 1.6 predicts a wavenumber of order 5, which is typical of extratropical cyclones. The maximum growth rate is proportional to A and therefore to the meridional temperature gradient. Representative values give an e-folding time for amplification of a couple of days, consistent with the observed development of extratropical cyclones. Insight into how instability is achieved follows from the structure of the most unstable disturbance. The lower boundary condition (16.16.2) implies
526
16
Hydrodynamic Instability
the following relationship between the coefficients in (16.17): B A
=
A
(16.20)
c~c'
which yields the structure shown in Fig. 16.3 for k - l. The fastest growing Eady mode is characterized by a westward tilt with height that places geopotential anomalies about 90 ~ out of phase between the lower and upper boundaries. This can be inferred from the eddy meridional velocity v' = i k q / , which is contoured (solid/dashed lines) in Fig. 16.3a on a vertical section passing zonally through the center of the disturbance (y - 0). Eddy geopotential and motion maximize at z = 0 and H in the form of two edge waves that are sandwiched between the upper and lower boundaries. In addition to driving air across mean isotherms, those boundaries trap instability, which allows the disturbance to amplify. Each edge wave decays vertically with the Rossby height scale H R - a-1. In the limit Ikhl --+ c ~ , H R is short enough for the two edge waves to be isolated from one another. Because neither has vertical phase tilt, hydrostatic balance implies a temperature perturbation that is in phase with the geopotential perturbation (Problem 16.15). The perturbation velocity v' is then everywhere in quadrature with 0', so the eddy heat flux averaged over a wavelength v' 0' vanishes. Consequently, eddy motion does not alter the baroclinic stratification of the mean state and no available potential energy is released. It is for this reason that short wavelengths in the Eady problem are neutral. For a < a c, H R is long enough for the two edge waves to influence one another. Phase-shifted, they produce the westward tilt apparent in Fig. 16.3a. The temperature perturbation is then no longer in phase with the geopotential perturbation. Isentropes (dotted lines in Fig. 16.3a) actually tilt slightly eastward with height to make 0' positively correlated with v" Cold air moves equatorward (out of the page) and warm air moves poleward (into the page). 2 Both contribute positively to v' 0' to produce a poleward eddy heat flux, which releases available potential energy. The mode's amplification is thus directly related to its westward tilt with height. The streamfunction in a horizontal section at z - H / 2 (Fig. 16.3b) and in a meridional section at k x - 37r/2 (Fig. 16.4) illustrates that polewardmoving air ascends, whereas, half a wavelength away, equatorward-moving air descends. Characteristic of sloping convection, this is the same sense meridional motion would have were it to move along zonal-mean isentropic surfaces in Fig. 16.4 (dotted lines), which slope upward toward the pole. However, meridional displacements of the fastest growing Eady mode have a slope at midlevel which is only about half that of mean isentropic surfaces (Problem 16.18). Eddy motion directed across those surfaces advects high 0 poleward and low 0 equatorward in adjacent branches of a cell, as is evident from isentropes at midlevel (Fig. 16.3b). This asymmetric motion produces a net heat 2In fact, v' and 0' are perfectly in phase at the steering level: z =
HI2.
16.3
(a)
v'
v ',,_.,..~ ~-'~:
. . . . ......
-.........
\
--',.
........
~ -__,,___,__,_,__,_,_,__ - - y - , - ; - , - - c - < - ~ k - \ - N - \_- & ~ /
./
L
-- - , - , - - , - - , - , - - - , - - - - ,
527
0
--.
k .... < -_. . . . . . . . . . . --'~,"":-"_ ........... ~----r--~--~--~--~-.." k,_.-. ..... . .' ""
The Eady Problem
L
~
\
\
""--
..... 2:.
--~--Z:I-I
0
1-7-~-_c-_"~.:Q~-1 ,
". . . . . . . . . . . . . . . . . . . . .
""
"\'" r
- .............
7:t
...
- ......... " . . . . . . . . . . . . .
--~___
:_.,.;.:I
-V.-~--~"
-_ _-\ ~ - -
"'"
rt
"
2rt
kx (b)
V'
O
- .........
/1:
---~T . . . . . . :,. . . .
2
---L:-S-L'--'~" - -_...~Z__-". . . . . . . . .
- - - ~/-.- - r . . ./. . ,"" ~ ." . . . ='--:."-"_ - - . - ~ , . . . . ._".~------<--m--'--" --'-"-<--',"- ....
V-f- .:'-- "- - ,";, ~, .-:..~-:-----~-~S.".--.-:~--~-~ --~--
-~-l-.'--l--L-+--~ ," _.-----:-=" .:...--~-<-'-.-~--C-_Y:
1 - - r - r - c T ~ z . ~ - - ' : " : . . . . . . . "~.-~--~--'~'i"~-[-~ + + ~ - r r ' ; " l . - , - " ' ; ' ; . . :->-~'. :~- ~--~-'; ~: Z ~:L: r-P -~-r-r~-T~s -~-'-/"" . . . . . . . L - @ - ~ ' T l'_[.i.~-~, I-I" -v 1-r 1"1-1: I . ~ - - r ' " :
~-P-~l-~'-'~] ' -l---e'" -'- 4-~" "~"
-
.......
- .....
- ....
: : L . J~-'-~7, Z - ~ ,-v.
i .-~-~-7 ~ ~j t-~
---T"
-~-
-
"l-
-
"~" + t r ~a_~._c.~- ~.~,. . . . . .~ . . . . ~-, r ~.,.~_~_~. -1" . - v . - r - ' - ,, x__C-~,--:--~-':~'~'7 ~ / ~ L
-r- - [ - -~'.,I._
....
x----,'"
V_---N--~-
t
. . . .
. . . . . .
"~---~'"
~
9-~- - - - ' ~ . - - " ~ ' ~
. . . . . . . .
~, . . . .
,--->"
3.._:.=....=--'-
i---~--~-~:-"--"---
..--
~--..----~.~ ""
',,,.-~ . . . .
_ _ _-
-=---
_
/ _-~--- . . . .
.,.,, . , ~ , , o t
":--='-'~'~':"
... ~ --"----Z.->" --
/
.
-.:-,:.-.
l
I
-~- 7"- 7 -"
I ....
Z--4---
/ ...-. ~ _ _ _ ~ , , .
/ .....
_ > _ . , ~ _ _ _ - ~ - - -'='- ,-.-- -...-l~-'-- . . . . . . .
_ m
2
0
rt
2rt
kx Figure 16.3 Structure of the fastest growing square Eady mode (k = I). (a) Vertical section in the zonal plane at y = 0 of eddy meridional velocity v' (solid/dashed lines) and isentropic surfaces 0 = const (dotted lines). Potential temperature increases upward. | marks equatorward motion (v' < 0) and @ marks poleward motion (v' > 0). (b) Horizontal section at z = H/2 of eddy streamlines (solid/dashed lines) and isentropes (dotted lines). Potential temperature increases equatorward.
528
16
Hydrodynamic Instability m
..........
0
H
.
.
..... "-" Z
..o..- ....
..,
...,.---'.~
-
....
...,,
o~.,,,oO~
..m,,..--_:~"
.,,.---~'" ...-
. . . . . --=.'"
....
-.,,
--
.
..
......
".~
o~ .,,
"."
_
~
~
_..,
._~.o~
.... ~
_.~,,~ _..,
._.,,
__.~.~o--.r.:-~
~
_.r~....,,,~r,"...-~
_.-.~,.~
~ . . . . .=.z,'"_..,
....
~
~
..-.po..,.~~176
~
.-.~.~176176
...r:,...,,.~~
~ ..7~..o..,~.~ ~
.._~_......,,..~'....._~
.-..,,
.,,.,,-ooT.~
.,.-~-~
....,,
_.,..-:.T,
~
.~,.--:.~" _..
_...,,~
..rT~....~.z.~ ~ ~
~
~
.,_..~'T.....~
..,..,~'~ ~
~
...,, ..~,~176
.-...=.,~-'-~.:-7,
_
.
..
_ ....----
..~.o--~" .,,
--4,
.....
~ .... -~.... .
...., . _ , . - - = . : - " _
_..~~176
~
~
-
. _
..., . . . . , , - ' "
...,o-"
-
0
-~
o ly
7
F i g u r e 16.4 Vertical section in the meridional plane at kx = 37r/2 of m e a n isentropic surfaces ( d o t t e d lines) and motion for the fastest growing square E a d y mode. Potential t e m p e r a t u r e increases u p w a r d and e q u a t o r w a r d ( c o m p a r e Fig. 12.5).
flux poleward, which releases available potential energy by driving the thermal structure toward barotropic stratification. Poleward heat flux acts to eliminate the horizontal temperature gradient and shallow the slope of isentropic surfaces, which in turn reduces the zonal-mean available potential energy. Extratropical cyclones have qualitatively similar structure during their development. Figure 16.5 shows distributions of 700-mb height and temperature for an amplifying cyclone situated off the coast of Africa on March 2, 1984. This disturbance is the precursor to the cyclone apparent in Figs. 1.15 and 1.24 two days later. During amplification, the system tilts westward, which transfers heat poleward and releases available potential energy--analogous to an unstable Eady mode with c~ < a c. Eddy heat flux tends to maximize near 700 mb, which typifies the steering level of observed cyclones. This is lower than the steering level predicted by the Eady model. However, an unbounded model treated by Charney (1947) reproduces the observed steering level, while retaining the essential ingredients captured by Eady's solution. Figure 16.5 contains the characteristic signature of sloping convection: A tongue of warm air is drawn poleward ahead of the closed low, while cold air is advected equatorward behind it. Those bodies of air have disparate histories, which are reflected in contemporaneous infrared (IR) and water vapor imagery (Figs. 16.6a and b). A tongue of high cloud cover and moisture that extends northwestward from the African coast defines the w a r m s e c t o r ahead of the cyclone. A complementary tongue of cloud-free conditions and low moisture is being drawn equatorward behind it. Sharp gradients separating those bodies
16.4
529
Nonlinear Considerations
March 2, 1984
""-..,
,,
......,..:
. . .~i.. ..}::.-i-~, ll.... . . . . .". ,
H3183.o ..:.
,
:.. "._.....
::-
_. _
/
i
...
?... -... . -.. "......
.~..
~ .
'" ..................... ~"" ......... .. ...... -::..:
.. - . . . . .
-..................... E,~........~!..............
.
:...
~:
....
"..
~
,
~
',
..
.
.
.~.
Figure 16.5 700-mb height (solid lines) and selected isotherms (dashed lines) on March 2, 1984. A surface frontal analysis is superposed.
of air mark warm and cold fronts at the surface, which are superposed in Fig. 16.5 and delineate the warm sector. Moisture inside the warm sector can be traced back in water vapor imagery (e.g., Fig. 16.6d) to its source: tropical convection over the Amazon basin (compare Fig. 1.24). Air advancing behind the cold front undercuts the warm sector in sloping convection, lifting moist air to produce the extensive cloud shield in Fig. 16.6a.
16.4
Nonlinear Considerations
While capturing the development of extratropical cyclones, Eady's solution provides only a hint of their behavior at maturity. Conversion of available
530
16
Hydrodynamic Instability
Water Vapor
Infrared (a) Mar 2 1500 (
.............
......................
(b) M a - " 1500 (
(c) Ma-"
(d) Ma-"
0300 (
0300 (
(e) Ms 1200 ~
(f) Mar 1200 (
Figure 16.6 Infrared and water vapor imagery from Meteosat between March 2 and March 4, 1984, that reveal the evolution of an extratropical cyclone off the northwest coast of Africa. The warm sector ahead of the cyclone is marked by a tongue of high cloud cover and moisture, which slopes northwestward on March 2. That air mass subsequently overturns near the juncture of cold and warm fronts delineating it to form an occlusion, in which cold and warm air are entrained and mixed horizontally (compare Fig. 9.21).
16.4
Nonlinear Considerations
531
potential energy into eddy kinetic energy enables a baroclinic system to intensify and eventually attain finite amplitude. Finite horizontal displacements then invalidate the linear description on which Eady's model is based and which predicts exponential amplification to continue indefinitely--analogous to vertical displacements under hydrostatically unstable conditions (Sec. 7.3). Second-order effects then modify the zonal-mean state, which in turn limits subsequent amplification of the baroclinic system. As a baroclinic disturbance amplifies, horizontal displacements (e.g., Fig. 16.3b) become increasingly exaggerated. Warm tropical air is eventually folded north of cold polar air, as is revealed by the distribution of potential vorticity on March 2, 1984 (Fig. 12.10). Deformations experienced by those air masses steepen potential temperature gradients separating warm and cold air (e.g., Fig. 12.4), which intensifies the accompanying fronts. Continued advection leads to warm and cold air eventually encircling the low, with the warm sector overturning at the junction of the warm and cold fronts. (Cloud cover in Fig. 9.21 provides a textbook example.) The cyclone in Fig. 16.5 then occludes, with warm and cold fronts overlapping in the center of the system. The occlusion actually develops when the surface trough (not shown) separates from the junction of warm and cold fronts and deepens farther back into the cold air mass (see, e.g., Wallace and Hobbs, 1977). This structure marks the mature stage of the cyclone's life cycle because the surface trough is then positioned beneath the upper-level trough, which eliminates the system's westward tilt and hence its release of available potential energy--now analogous to a neutral Eady mode with c~ > c~C. Cold and warm air drawn into the occlusion are then wound together and mixed horizontally. Figure 16.6 shows sequences of IR and water vapor imagery while the disturbance in Fig. 16.5 matures. At 1500 GMT on March 2 (Figs. 16.6a and b), the cold front is approaching the warm front near their junction, with the warm sector clearly defined. Twelve hours later, the 700-mb trough has deepened (not shown). The warm sector in IR and water vapor imagery (Figs. 16.6c and d) has then been sheared to the northwest, where cold dry air is being entrained with warm moist air in the occlusion that has formed. This process culminates in cold and warm air at the occlusion winding up into a spiralbsimilar to behavior in a cylindrical annulus at high rotation (compare Fig. 15.7f). By 1200 GMT on March 3 (Figs. 16.6e and f), air inside the occlusion has wound up into a vortex, which is seen to separate from the remaining warm sector to its south and east. Interleaving bands of cold and warm air that are apparent in both cloud cover and moisture symbolize efficient horizontal mixing. One day later (Figs. 1.15 and 1.23), that mass of air has become nearly homogeneous. The 700-mb trough has then weakened and high cloud cover that developed earlier through sloping convection is dissipating. Only a broad spiral of equatorward-moving air remains, drawn cyclonically around the now-diffuse anomaly of potential vorticity, in which cold and warm air have been mixed.
532
16 Hydrodynamic Instability
Since 0 (more generally, 0e) is conserved for individual air parcels, horizontal mixing makes the potential temperature uniform and restores the thermal structure to barotropic stratification. Baroclinic instability that was present initially has then been neutralized and no more potential energy is available for conversion to eddy kinetic energy. This process is a direct counterpart of vertical mixing by convection, which neutralizes hydrostatic instability (Sec. 7.3). Strong modification of the mean state by unstable eddies contrasts sharply with wave propagation under stable conditions. Parcel displacements are then bounded, so the mean state remains largely unaffected outside regions of dissipation, where wave activity is absorbed (Chapter 14). The preceding treatment applies formally to a zonally symmetric mean state. In practice, the results also lend insight into isolated regions of instability. Amplified planetary waves in the Northern Hemisphere reinforce zonal-mean vertical shear to produce a broken storm track that is marked by localized jets east of Asia and North America (refer to Figs. 15.8a and 9.38). Strongly baroclinic, the north Atlantic and north Pacific storm tracks are preferred sites for cyclone development. Weaker planetary waves in the Southern Hemisphere leave stratification there nearly uniform in longitude to produce a continuous storm track. This enables cyclones there to assume a more regular distribution than is observed in the Northern Hemisphere, one that occasionally resembles baroclinic modes in a rotating annulus (see Figs. 2.10 and 15.7e). The criteria for instability also have implications important to planetary waves that have attained large amplitude. The situation is analogous to the breaking of gravity waves when isentropic surfaces are overturned (Fig. 14.25). High 0 is then folded underneath low 0 to make ~O/3z < 0 and render the region hydrostatically unstable. Convective mixing that ensues absorbs organized wave motion and neutralizes the instability by driving ~O/3z to zero. Planetary waves of sufficient amplitude overturn the distribution of the conserved property Q (Fig. 14.26). High potential vorticity folded poleward of low potential vorticity then reverses ~Q/~y locally to render the motion dynamically unstable. At that point, the planetary wave field breaks. Smallscale eddies that develop in the region of instability mix Q horizontally, which absorbs organized wave motion and neutralizes the instability by driving ~Q/~y to zero. Suggested Reading
Atmospheric Science: An Introductory Survey (1977) by Wallace and Hobbs contains a synoptic analysis of the life cycle of extratropical cyclones.
Atmosphere-Ocean Dynamics (1982) by Gill includes a nice comparison between Charney's (1947) model of baroclinic instability and the Eady model.
An Introduction to Dynamic Meteorology (1992) by Holton provides a complete description of baroclinic extratropical disturbances, their energetics, and
Problems
533
aspects surrounding their prediction. It also discusses the process of frontal formation.
Middle Atmosphere Dynamics (1987) by Andrews et aL discusses inertial instability in the tropical stratosphere and includes a description of planetary wave breaking. Problems 16.1. Derive equations (16.1). 16.2. Carry out the Lagrangian integration of (16.3) to relate an air parcel's meridional displacement to its change of zonal velocity. 16.3. Provide an expression for the specific restoring force inside an inertially unstable layer. 16.4. (a) At what latitudes is inertial instability favored? (b) Identify those regions of the zonal-mean flow in Fig. 1.8 that would be most susceptible to inertial instability. 16.5. Consider westerly flow that corresponds to an angular velocity A(z) = -~/(a cos ~b), which varies with height but not with latitude. (a) Within the framework of quasi-geostrophic motion and the Boussinesq approximation, what sign of curvature must the velocity profile A(z) have for shear instability to develop? (b) At what latitudes is shear instability favored most if A(z) = E(z)-2~, with E < < 17 16.6. Obtain the lower boundary condition (16.6.4). 16.7. A wave packet approaches its critical line with phase speed Cr equal to the real part of c in (16.8.1). Describe the wave packet's evolution in the presence of weak dissipation if, under inviscid adiabatic conditions, (a) c in (16.8.1) is real and (b) c in (16.8.1) is complex. 16.8. Obtain equations (16.9) and (16.10). 16.9. Recover the necessary condition (16.12). 16.10. Derive a necessary condition for instability of a quasi-geostrophic zonal flow that is unbounded above and bounded below by a rigid surface of constant height. 16.11. Derive a necessary condition for instability of a quasi-geostrophic zonal flow that is bounded vertically at z = 0 and H by rigid walls. 16.12. (a) Show that, in the absence of rotation, criterion (1) in Sec. 16.2.2 recovers Rayleigh's condition for instability: a barotropic flow must possess an inflection point in its interior. (b) Discuss the influence rotation has on shear instability. 16.13. Why do midlatitude cyclones intensify during winter? 16.14. Obtain the dispersion relation (16.18) for Eady modes.
534
16 Hydrodynamic Instability
16.15. Demonstrate that the limiting structure of Eady modes for H ~ oo has v' in quadrature with 0'. 16.16. Derive an expression for the shortwave cutoff ac for instability in the Eady problem. 16.17. (a) Show that the maximum growth rate of Eady modes is achieved for l = 0. (b) Express the growth rate of the fastest growing square Eady mode (k = l) in terms of that of the fastest growing Eady mode (l = 0). 16.18. Show that, for the fastest growing Eady mode, the maximum slope of motion in the meridional plane is only about half the slope of mean isentropic surfaces. 16.19. Calculate the e-folding times of square Eady modes (k = l) for an f plane at 45 ~ N 2 - 1 0 - 4 S-2, vertical shear of 3 m s -1 km -1, a rigid lid at 10 km, and for zonal wavenumbers 1-8. 16.20. Within the framework of an initial value problem, describe the structure and evolution predicted by the Eady model if the flow in Problem 15.19 is initialized with random structure having a broad wavenumber spectrum (a) under inviscid adiabatic conditions, (b) in the presence of linear dissipation 3 with a timescale equal to the shortest e-folding time of zonal wavenumber 2 under inviscid adiabatic conditions and (c) in the presence of linear dissipation with a timescale equal to the shortest e-folding time of zonal wavenumber 4 under inviscid adiabatic conditions. 16.21. Meridional components of the eddy momentum flux u'v' and group velocity %y vanish for Eady modes. Why? 16.22. Show that the eddy heat flux v'0' is positive and independent of height for Eady modes. 16.23. Use the Eady problem to explain (a) why midlatitude cyclones develop and track along the jet stream, (b) how baroclinic instability would be altered if the jet were displaced equatorward and (c) where the storm tracks should be positioned in relation to the time-mean motion in Fig. 1.9. 16.24. Precipitation inside cyclones often assumes a banded structure. Discuss this feature in relation to the evolution in Fig. 16.6 and the corresponding distribution of potential vorticity Qg. 16.25. Consider an idealized atmosphere in which the circulation is described by the barotropic zonal flow K(y) = - U cos(24~), 3Rayleigh friction and Newtonian cooling of the same timescale.
Problems
535
where U = 0.55~a. (a) Characterize the inertial stability of this flow as a function of latitude. (b) Describe the zonal-mean circulation toward which inertial instability will drive the flow. 16.26. Discuss the relationship of barotropic and baroclinic instability to stratification (Sec. 12.2.1). 16.27. Unlike extratropical cyclones, tropical cyclones are driven by latent heat release. Suppose a typhoon drifts poleward from the ITCZ. (a) Describe its evolution as it migrates over colder sea surface temperatures. (b) What structure of midlatitude westerlies will allow it to sustain itself?