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Baroclinic Instability Basic question: what causes storms (large scale turbulence) in the atmosphere? A simplified picture of atmospheric general circulation

In time scales longer than days, incoming solar radiation is zonally symmetric, but varies with latitude

Flux per unit area ~ cos! (ignore the seasonality), or Fs = Fso + Fs 'cos! The radiative equilibrium is balanced between the incoming solar radiation and outgoing long wave radiation:

(1! a)Fs = "Te4

2

where a is albedo, Te is some kind of mean atmospheric temperature . We divide Te into a constant and lat-dependent part: Te = Teo + Te ' , and let (1! a)Fso = "Teo

4 . Thus, Te

4 = (Teo + Te ')4 ! Teo

4 + 4Teo3Te '

è (1! a)Fs 'cos" = #Teo3Te '

è Te ' = T̂e cos!, where T̂e "(1# a)Fs '$Teo

3 is the pole-to-equator radiative equilibrium

temperature difference. Since now !Te / !y " 0, u " 0 based on thermal wind relation. In spherical geometry, using log p ( z* ! "H ln(p / ps ) ) vertical coordinate, the zonal momentum equation becomes

2!u sin" +u2 tan"

a= #

1a$%$"

!"

! ln p= #RT upon dividing through by -H and using definition of z* ==> !"

!z *= RT / H .

But H = RT0 / g , therefore, !"!z *

= gT /T0

Taking the vertical derivative of the momentum equation and sub from !"!z *

= gT /T0

into it (and neglecting the metric term), we then have

2!asin" #ue#z *

= $gT0

#Te#"

= $gT0

#T̂e cos"#"

= +gT̂eT0sin"

Therefore,

ue =g2!a

T̂eT0z * + ueo

The radiative equilibrium wind is in a form of solid body rotation at angular velocity

!̂ =g2!a

T̂eT0

.

3

So, why don’t we see this hypothetical zonally symmetric, storm-free state, i.e., u = ue, v = 0, T = Te in the real atmosphere? Instead, the zonally and time averaged general circulation is like pictured below: (note that in the figure below the prime denotes the stationary wave component, or time mean zonal asymmetric part; and * indicates transient wave component).

Zonal wind

4

1. Why do we not see radiative convective equilibrium? 2. How do we understand the structure and magnitude of the observed midlatitude

storm system?

Meridional wind

5

Eddy heat flux

6

Eddy momentum flux

7

Hydrodynamic instability A zonal mean flow field is said to be hydrodynamically unstable if a small disturbance introduced into the flow grows exponentially, drawing energy from the basic flow.

A. parcel instability: static instability, convective overturning, inertial instability (Section 7.5.1), Symmetric instability (Section 9.3). B. wave instability: barotropic instability, baroclinic instability Barotropic instability is a wave instability associated with horizontal shear in a jet-like mean-flow field, it grows by extracting kinetic energy from the mean-flow field. Baroclinic instability, however, is associated with the vertical shear of the mean flow, grows by converting potential energy associated with the mean horizontal temperature gradient that must exist to provide thermal wind balance for the vertical wind shear in the basic mean flow.

Baroclinic Instability Baroclinic instability theory provides a linear perspective on the existence of weather, its origin, growth rate, frequency and structure. Normal Mode Baroclinic Instability (Chapter 8.2) We first focus on the simplest possible model that can incorporate baroclinic processes—the 2-layer model (Phillips model).

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The QG vorticity equation for midlatitude beta-plane is applied at the 250- and 750-hPa levels, designated by 1 and 3 in Fig.8.2, whereas the hydrostatic thermodynmic energy equation is applied at 500-hPa level, designated by 2 in Fig.8.2.

Dg

Dt!2" + # $"

$x= f0

$%$p

(8.2)

Dg

Dt!"!p

#$%

&'(+)f0* = 0 (8.3)

Using finite difference approach,

!"!p

#$%

&'( 1

)"2 *"0

+ p, !"

!p#$%

&'( 3

)" 4 *"2

+ pwhere + p = 500 hPa

(8.4)

The resulting vorticity equations at level 1 and 3 are

!!t"2# 1 + V1 $" "2# 1( ) + % !# 1

!x=f0&2

' p (8.5)

!!t"2# 3 + V3 $" "2# 3( ) + % !# 3

!x= &

f0'2

( p (8.6)

where we used the fact that !0 = 0 and assumed that ! 4 = 0 . We next write the thermodynamic equation at level 2. Here we must evaluate !" / !p using the difference formula !" / !p( )2 # " 3 $" 1( ) /% p The result is

!!t

" 1 #" 3( ) = #V2 $% " 1 #" 3( ) + &' pf0

(2 (8.7)

(8.5)-(8.7) is a closed set of prediction equations in the variables ! 1,! 3, and"2 . Linear perturbation Analysis We assume that the streamfunctions consist of basic state parts that depend linearly on y alone, plus perturbations that depend on x and t. Thus, we let

! 1 = "U1y +! 1

' (x,t)! 3 = "U3y +! 3

' (x,t)#2 =#2

' (x,t) (8.8)

Substituting into (8.5-7) yields the perturbation equations

!!t

+U1!!x

"#$

%&'!2( '

1

!x2+ ) !( '

1

!x=f0*

'2

+ p (8.9)

!!t

+U3!!x

"#$

%&'!2( '

3

!x2+ ) !( '

3

!x= *

f0+'2

, p (8.10)

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!!t

+Um!!x

"#$

%&'( '1 )(

'3( ) )UT

!!x

( '1 +(

'3( ) = *+ p

f0, '2 (8.11)

where we have linearly interpolated to express V2 in terms of ! 1 and ! 3 , and have defined Um ! U1 +U3( ) / 2, UT ! U1 "U3( ) / 2 They are, respectively, the vertically averaged mean zonal wind and the mean thermal wind. The dynamical properties of the system are better expressed if adding (8.9) to (8.10) to eliminate ! '

2 , and subtracting (8.9) from (8.10) and combining with (8.11) to eliminate ! '2 . As a result, we have, respectively,

!!t

+Um!!x

"#$

%&'!2( m

!x2+ ) !( m

!x+UT

!!x

!2( T

!x2"#$

%&'= 0 (8.15)

!!t

+Um!!x

"#$

%&'

!2( T

!x2) 2* 2( T

"#$

%&'+ + !( T

!x+UT

!!x

!2( m

!x2+ 2* 2( m

"#$

%&'= 0 (8.16)

where !2 " f02 / [# ($ p)2 ] (!"1 is a sort of Rossby radius of deformation, the typical

tropospheric value of !"1 for midlatitude is about 4300 km), and we have defined ! m " ! '

1 +!'3( ) / 2, ! T " ! '

1 #!'3( ) / 2 (8.14)

Equations (8.15) and (8.16) govern the evolution of the barotropic (vertically averaged) and baroclinic (thermal) perturbation vorticity, respectively. It is through the UT term the barotropic and the baroclinic components of the flow interact with each other. Again we seek for wave-like solution ! m = Aeik (x"ct ), ! T = Beik (x"ct ) (8.17) and substitute into (8.15-16). From (8.15): ik c !Um( )k2 + "#$ %&A ! ik 3UTB = 0 (8.18) From (8.16): ik c !Um( ) k2 + 2"2( ) + #$% &'B ! ikUT k2 ! 2"2( )A = 0 (8.19) Because this set is homogeneous, no trivial solutions will exist only if the determinant of the coefficients of A and B is zero. Thus the phase speed c must satisfy the condition

(c !Um )k

2 + "

!UT (k2 ! 2#2 )

!k2UT

(c !Um )(k2 + 2#2 ) + "

= 0 (8.20)

Remind that

a11 a12b11 b12

= a11b12 ! a12b11

10

So writing (8.20) out, we have

c !Um( )2 k2 k2 + 2"2( ) + 2 c !Um( )# k2 + "2( ) +

# 2 +U 2T k

2 2"2 ! k2( )$% &' = 0 (8.20)

which is the dispersion equation that is quadratic in c !Um( ) . Solve for c !Um( ) :

c =Um !" k2 + #2( )k2 k2 + 2#2( ) ± $

1/2

where $ %" 2# 4

k 4 k2 + 2#2( )2!U 2

T2 2#2 ! k2( )k2 + 2#2( )

(8.21)

Two special cases Case 1: UT = 0 thus the basic state thermal wind vanishes and the mean flow is barotropic. And the solution (8.21) becomes c1 =Um ! "k!2 (8.22) c2 =Um ! "(k2 + 2#2 )!1 (8.23) (8.22) is simply the dispersion relationship for a barotropic Rossby wave with no y-dependence discussed in Section 7.7. Sub (8.22) for c into (8.18) and (8.19) we see that in this case B = 0 so that the perturbation is barotropic in structure. The expression (8.23) however, may be interpreted as the phase speed for an internal baroclinic wave. In this case there is an extra factor 2!2 appears in the denominator and there is vertical motion associated with the Rossby wave so that static stability modifies the wave speed. It is interesting to note that the wave fields in level 1 and 3 are 180 degree out of phase so that the perturbation is baroclinic, although the basic state is barotropic. Furthermore, the !2

' field is 1/4 cycle out of phase with the 250hPa geopotential field, with the maximum upward motion occurring west of the 250-hPa trough. (Homework: show this and interpret it in terms of physical processes) Case 2: ! = 0 .

c =Um ±UTk2 ! 2"2

k2 + 2"2#$%

&'(

1/2

(8.24)

For waves with zonal wave number satisfying k2 < 2!2 , (8.24) has an imaginary part. Thus, all waves longer than the critical wavelength Lc = 2! / " , which is approximately 3000 km for typical tropospheric conditions, will grow exponentially with time. From the definition of ! , we notice that the critical wavelength for baroclinic instability increases with static stability. The role of static stability in stabilizing the shorter waves can be understood qualitatively as follows: For a sinusoidal perturbation, the relative

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vorticity, and hence the differential vorticity advection (last term on the lfs of 8.16), increases with the square of wave number. However, as shown in Chapter 6, a secondary vertical circulation is required in order to maintain hydrostatic temperature changes and geostrophic vorticity changes in the presence of differential vorticity advection. Thus, for a geopotential perturbation of fixed amplitude, the relative strength of the accompanying vertical circulation must increase sharply as the wavelength of the disturbance decreases. Because static stability tends to resist vertical displacement, the shortest wavelengths will thus be stabilized. The short wave cutoff for baroclinic instability could also be interpreted as the consequence of the inability of forming the phase-lock between the counter-propagating waves at levels 1 and 3 when the vertical influence of PV disturbance is confined with the scale of !"1 . It is also of interest to note that for the case of ! = 0 , the critical wavelength for instability does not depend on the thermal wind, whereas the growth rate does, it increases linearly with the mean thermal wind. Also note that there is not long-wave cutoff for the case of ! = 0 . General case Neutral curve, which connects all values of Um and k for which ! = 0 so that the flow is marginally stable. From (8.21), the condition ! = 0 implies that

! 2" 4

k 4 k2 + 2"2( ) =U2T 2"

2 # k2( ) (8.26)

Solving for k 4 / 2! 4 yields k 4 / (2! 4 ) = 1± 1" # 2 / 4! 4UT

2( )$% &'1/2

The neutral curve shown in Fig.8.3 separates the unstable region of the UT -k plane from the stable region. It is clear that the inclusion of ! effect serves to stabilize the flow for now unstable roots exist only for UT > ! / (2"2 ) . This is the Phillips criterion for baroclinic instability of a 2-layer model (Phillips, 1954). In addition, the minimum value of UT required for unstable growth depends strongly on k. Thus, the ! effect strongly stabilizes the long wave end of the wave spectrum---long wave cutoff. Considering long waves, i.e., k

2 ! 2! 2 , (8.26) can be written as ! 2" 4 =U 2

T k4 4" 4 # k 4( ) $U 2

T k4 4" 4 (8.26”)

Thus, the neutral curve at the small wave number side may be approximated as UT = ! / 2k2 (8.26”’) As such, for a given wave number near the long wave cutoff, wind shear UT > ! / 2k2 for the instability to occur.

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Meanwhile, the flow is always stable for waves shorter than the critical wavelength Lc = 2! / " ---short wave cutoff. Shortwave cutoff always exists irrespective of the existence of the ! effect, although the curve of the cutoff is somewhat modified. This long wave stabilization associated with the ! effect is caused by the rapid westward propagation of long Rossby waves, which occurs only when the ! effect is included in the model. It can be shown that baroclinic instability waves always propagate at a speed that lies between maximum and minimum mean zonal wind speeds. Thus, for the 2-layer model in the usual midlatitude case where U1 >U3 > 0 , the real part of the phase speed satisfies the inequality U3 < cr <U1 for unstable waves. In a continuous atmosphere, this would imply that there must be a level where U = cr . Such a level is called a critical level by theoreticians and a steering level by synopticians. For long waves and weak basic state wind shear, the solution given by (8.21) will have cr <U3 , therefore there is no steering level, and unstable growth cannot occur. The above argument about the long wave cutoff is also in conformity with the view of counter-propagating Rossby edge waves. The long wave cutoff criterion is exactly the condition for the formation of the phase-lock of two counter propagating Rossby waves in levels 1 and 3: The phase speed of the free Rossby waves in level 1 and 3 are

c1 =U1 !"1k2 , and c3 =U3 !

"3k2 , respectively, where !1 =

"q1"y

= ! + #2 (U1 $U3) and

!3 ="q3"y

= ! # $2 (U1 #U3) . Phase-lock implies c1 = c3 , so that the two edge waves

always see each other at the same phase lag and they can interact and reinforce each other. The critical condition for the instability at the long wave cutoff side coincides with the phase-lock condition when the low level beta becomes neutral (the same condition for unstable root to exist for phase speed):!3=0, or ! " 2# 2UT . Using both conditions c1 = c3 and ! <= 2" 2UT , we may derive

U1 !U3 " 2UT =#1 !#3k2

= 4$2UT

k2>= #

k2

And indeed, UT =!2k2 demarcates the boundary between the stable and unstable regimes

in the UT -k space and k2 > !

2UT is required for unstable growth to occur. For a given

vertical wind shear, β effect stabilizes the small wave number disturbance---a long wave cutoff. For a given β, on the other hand the stronger wind shear is required for unstable growth of longer waves.

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Differentiating (8.26) with respect to k and setting dUT / dk = 0 we find that the minimum value of UT for which unstable waves may exist occurs when k2 = 2!2 . This wave number corresponds to the wave of maximum growth. Wave numbers of observed disturbances should be close to the wave number of maximum instability. Under the normal conditions of static stability the wavelength of max instability is about 4000 km, which is close to the average wavelength for midlatitude synoptic systems. Therefore, the observed behavior of midlatitude synoptic systems is consistent with the hypothesis that such systems can originate from infinitesimal perturbations of a baroclinically unstable basic flow.

Please read Section 8.2.2 for the structure of the baroclinically unstable disturbances.

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Charney-Stern-Pedlosky Criterion for Instability Log-pressure coordinates (8.4.1 of Holton)

z* = !H ln(p / ps ) (8.4) where is a standard reference pressure and H is a standard scale height, H ! RTs / g , with Ts a global average temperature. For an atmosphere with a realistic temperature profile, z* is only approximately equivalent to the height, but in the troposphere the difference is usually quite small. The vertical velocity in this system is w* = Dz* / dt The hydrostatic equation now becomes

!"!z*

= RT /H (8.43)

And continuity equation

!u!x

+ !v!y

+ 1"0

!("0w*)

!z*= 0 (8.44)

The first law of thermodynamics can be expressed as

!!t

+V i"#$%

&'(!)!z*

+w*N 2 = *JH (8.45)

where

N 2 ! (R /H )("T / "z* + #T /H )

Note that unlike the static stability parameter, Sp , in the isobaric form of the thermodynamic equation (3.6), the parameter N 2 varies only weakly with height in the troposphere; it can be assumed to be constant without serious error. This is a major advantage of the log-pressure formulation. The quasi-geostrophic potential vorticity equation (6.24) has the same form as in the isobaric system, but with q defined as

q ! "2# + f + 1$0

%%z*

&$0%#%z*

'()

*+, (8.46)

Rayleigh Theorem The linearized form of the QG PV equation (6.24) can be expressed in a log-p coordinates as

!!t

+ u!!x

"#$

%&'q '+ !q

!y!( '!x

= 0 (8.47)

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where

q ' ! "2# '+ 1$0

%%z *

&$0%# '%z *

'()

*+,

(8.48)

and

!q!y

= " #!2u!y2

#1$0

!!z *

%$0!u!z *

&'(

)*+

(8.49)

Assuming (i) the flow is adiabatic at the lower and upper boundary pressure surfaces;

!!t

+ u!!x

"#$

%&'!( '!z *

)!( '!x

!u!z *

= 0 (8.50)

(ii) vertical motion vanishes at these boundaries; (iii) v component of wind vanishes at sidewall boundaries; (iv) rigid lid upper boundary condition, or alternatively streamfunction ! ' remains finite as z*!" . Inserting into the set of equations a single wave solution as follows ! '(x, y, z,t) = Re "(y, z)exp ik(x # ct)[ ]{ } (8.52) and after some manipulation, one derive the constraint for the instability:

ci!q!y

"0 #2

u $ c 2dydz*$ % !u

!z*"0 #

2

u $ c 2z*=0

dy$L

+L

&0

'

&$L

+L

&(

)**

+

,--= 0 (8.61)

(8.61) has important implications for the stability of quasi-geostrophic perturbations. For unstable mode, ci must be nonzero, and thus the quantity in the square brackets must vanish. For this to be true, !u / !z * and !q / !y in the whole domain must satisfy certain constraints:

(a) If !u / !z* = 0 at z* = 0 . The first integral must vanish for instability to occur.

This can occur only if !q / !y changes sign within the domain (i.e., !q / !y =0 somewhere). This is referred to as the Rayleigh necessary condition for instability.

(b) If !q / !y > 0 everywhere, then it is necessary that !u / !z* > 0 somewhere at the lower boundary for ci > 0 .

(c) If !u / !z* < 0 everywhere at z* = 0 , then it is necessary that !q / !y < 0 somewhere for instability to occur. Thus, there is an asymmetry between westerly and easterly shear at the lower boundary, with the former more favorable for baroclinic instability.

My own thoughts: All the 3 conditions above can be generalized to be a single necessary condition for baroclinic instability: PV changes sign in the whole domain (including

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boundaries) of the problem, if we interpret !u / !z* > 0 as a source of negative PV

gradient at z*= 0 via !!"2u"z*2

!"!"z*

"u"z*

terms.

The normal baroclinic instability in midlatitudes is associated with mean flows in which !q / !y > 0 and !u / !z* > 0 at the ground. Hence, a mean meridional temperature gradient at the ground is essential for the existence of such instability. Baroclinic instability can also be excited at the tropopause due to the rapid decrease of ! with height if there is a sufficiently strong easterly mean wind shear to cause a local reversal in the mean potential vorticity gradient. The Eady Problem The Eady problem deals with the baroclinic instability in a continuous atmosphere governed by equations (8.47)-(8.50). For further simplification, we make the following assumptions:

(i) Basic state density is constant (Boussinesq approximation) (ii) f-plane geometry (! = 0 ) (iii) !u / !z* = " = constant (iv) Rigid lid at z* = 0 and H.

Despite the zero mean PV in the domain, the Eady model satisfies the necessary conditions for instability discussed above because vertical shear of the basic state mean flow at the upper boundary provides an additional term in (8.61) that is equal and opposite to the lower boundary integral.

!q!y

> 0

!q!y

< 0 z*= 0

z*= H

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With the assumptions above, the linearized QG PV equations can be expressed as

!!t

+ u!!x

"#$

%&'

(2) '+ * !2) '

!z *2"#$

%&'= 0 , (8.62)

and

!!t

+ u!!x

"#$

%&'!( '!z *

)!( '!x

!u!z *

+ *)1w* = 0 , (8.63)

respectively, where again ! "f02

N 2 .

Letting

! '(x, y, z*,t) = "(z*)cos lyexp ik(x # ct)[ ]u (z*) = $z *

(8.64)

and sub into (8.62) we find that the vertical structure is given by the solution to a 2nd order differential equation d 2! / dz *2 "# 2! = 0 (8.65) where ! 2 = (k2 + l2 ) / " . A similar substitution into (8.63) yields the boundary conditions (!z *"c)d# / dz *"#! = 0, at z* = 0, H (8.66) The general solution of (8.65) can be written in the form !(z*) = Asinh"z *+Bcosh"z * (8.67) Sub from (8.67) into the boundary conditions (8.66) for z* = 0, H yields a set of two linear homogeneous equations in the amplitude coefficient A and B:

! c"A ! B# = 0

"(#H ! c)(Acosh"H + Bsinh"H ) ! #(Asinh"H + Bcosh"H ) = 0

As in the two-layer model a nontrivial solution exists only if the determinant of the coefficients of A and B vanishes. This leads to

c =!H2

±!H2

1" 4 cosh#H#H sinh#H

+4

# 2H 2$%&

'()

1/2

(8.68)

Thus

ci ! 0, if 1" 4 cosh#H#H sinh#H

+4

# 2H 2$%&

'()< 0

and the flow is then baroclinically unstable. When the quantity in square brackets in (8.68) is equal to zero, the flow is deemed to be neutral stable. This condition occurs for ! = ! c , where ! 2

cH2 / 4 "! cH (tanh! cH )

"1 +1 = 0 (8.69) (8.69) can be factored to yield

! cH2

" tanh ! cH2

#$%

&'(

)*+

,-.! cH2

" coth ! cH2

#$%

&'(

)*+

,-.= 0 (8.70)

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Thus the critical value of ! is given by ! cH2

= coth(! cH2) , which implies ! cH " 2.4 .

Hence instability requires ! <! c , or

(k2 + l2 ) / ! < 2.4

H"#$

%&'2

i.e., (k2 + l2 ) < 5.76 f02

N 2H 2

"#$

%&'= 5.76 / LR

2

where LR ! NH / f0 is the Rossby radius of deformation for a continuously stratified fluid [compare !"1 defined just below (8.16)]. For waves with equal zonal and meridional wave numbers, the wavelength of maximum growth rate turns out to be Lm = 2 2!LR / (H"m ) # 5500 km where !m is the value of ! for which the growth rate is maximum. The growth rates of the instabilities are given by the imaginary part of c in (8.68). And the maximum growth rate can be determined by differentiation with wave number. The result is kci ! " E = 0.31 f0Ri

#1/2 (*) where Ri ! N

2 / "2 is the Richardson number. And ! E is the well-known Eady growth rate. Sub this value of !m into the solution for the vertical structure of the streamfunctions (8.67) and using the lower boundary condition to express the coefficient B in terms of A, we can determine the vertical structure of the most unstable mode. As shown in Fig.8.10, trough and ridge axes slope westward with height, in agreement with the requirements for extraction of available potential energy from the mean flow. The axes of the warmest and coldest air, however, tilt eastward with height, a result that could not be determined from the two-layer model where temperature was given at a single level. Furthermore, Fig.8.10 show that east of the upper-level trough axis, where the perturbation merdional velocity is positive, the vertical velocity is also positive. Thus, parcel motion is poleward and upward in the region where ! ' > 0 . Conversely, west of the upper-level trough axis parcel motion is equatorward and downward where ! ' < 0 . Both cases are thus consistent with the energy converting parcel trajectory slopes shown in Fig. 8.8.

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20

(you can reason for the shortwave cutoff from the trajectory slope above)

Process of lowering the center of mass