Instability, Cyclogenesis, and Anticyclogenesis

The concept of instability !  When pressure systems intensify, the increased

pressure gradient increases wind speeds

!  Two ways this happens in the atmosphere:

!  Conversion of available potential energy to kinetic energy through the lifting of warm air and sinking of cold air Baroclinic instability

!  Conversion of kinetic energy of the mean flow into the kinetic energy of the growing disturbance Barotropic instability

Barotropic instability !  [1.94] !  Discovered by Kuo in

the late 1940’s !  A necessary condition

for barotropic instability in a zonal current is where the gradient in absolute vorticity vanishes

!  This condition is often met near jets where ug and the background flow ! go to zero and the wave’s ! can grow

( . f ..,. C+ I

o nlit ,c lonic . hear s ide

of jet

y

- . ." Q"hcyclon,c - Sh OOf . ,,'"

'--- -.

Baroclinic instability !  Consider a simple

example of a coin sitting on a table

!  Standing on its side, a coin has available potential energy given by its mass times gravity times the height z0 of the center of mass (above right) minus the potential energy of the coin sitting on its side (below right)

z0

Baroclinic instability (continued) !  “Tipping” the coin over

will convert available potential energy to kinetic energy

!  The coin still has some potential energy, but it is unavailable to the system

!  The earth’s atmosphere is more complicated, because it is compressible and the earth rotates

z0

Sanders analytic model !  We will use an analytic model developed by Fred

Sanders in 1971 to understand the baroclinic instability process on cyclones in the real atmosphere

!  To the board!

---

L y

4"-'1',---- 0.0 0

0.0 0

/---...... / ,

6'

/ '°0

\0 0 I<D L I -10.2 /\

/ '-.... /'0 "-

,,,, ,,,, x

o

ell 10.2 '0 0

0

L L 0 L L-"2 -4" 4" "2 Figure 1.27 Isopleths of 1000-mb height at intervals of 6 dam in Sanders' analytic model. <1>0 = 1020 m2 S-2, and A = L/4. The abscissa is the x direction; the ordinate is the y direction.

is an infinite checkerboard of warm and cold perturbation centers - [t cos(2.7l'/ L)x cos(2.7l' / L)y] superimposed upon a field of uniform meridional temperature gradient (-ay). The x direction points toward the east, and the y direction points toward the north. The parameter L is the (isotropic) horizontal wavelength, and A is the phase lag of the lOOO-mb temperature field

y .L....

0 f 0 ci N

L"'4" x L-"2 L"'4" o L L

4" "2

\ 0 0 N

J

Figure 1.28 Isotherms at 1000 mb in °C (solid lines) in Sanders' analytic model. a = 1.0 x 10-5 °C m-1

, f = 10°C, and L = 3500 km. The abscissa is the x direction; the ordinate is the y direction.

[1.30]

500 hPa height at intervals of 6 dm in Sanders’ analytic model. (" = 0.722, !m/g = 5555 m)

y h 4

552.

H 576.

...r

56".

/'

0

_h X4 l l 0 l -2 -'4 '4 2

Figure 1.30 500-mb height at intervals of 6 dam (solid lines) in Sanders' analytic model. The value of ll' is 0.722, and «11m (500 mb)/g =5555 m. The abscissa is the x direction; the ordinate is the y direction.

!  Derivations on board – see text

/ /

/I. ....

. ., /

L (103 km)

Figure 'l.96 Deepening rate (a<l>/at) of the 1000-mb cyclone as a function of wavelength (L) and meridional temperature gradient (ordinate) for selected values of the vorticity-stability parameter and for f =1°C. The isopleths are labeled in units of 10- 3 m2 S-3 . The dashed, solid, and dotted lines are for values of the vorticity-stability parameter as in Fig. 1.39. The longer dashed lines are loci of the wavelength of maximum deepening rate (from Sanders, 1971). (Courtesy of the American Meteorological Society)

Sanders’ analytic model: height tendency

!

"#"t

L2$ %, 0, 1000 hPa&

' ( )

= K13P13 $ K14P14( )sin 2*%L

Physical effect: vertical motion accompanying vertical motion due to advection of relative vorticity by the thermal wind Related to L, static stability ", and meridional temperature gradient a

!

K13 =f0Ra ˆ T L4"T0#

!

K14 =f0

2 ˆ T "L3

4 2#( )3T0$

Physical effect: vertical motion accompanying advection of the earth’s vorticity (a.k.a. beta effect) by the thermal wind Related to L3, static stability ", and meridional temperature gradient a.

Analysis !  If there were no beta effect, troughs of all

wavelengths would deepen ! The wavelength that this happens is called the long

wave cutoff. !  At short wavelengths, advection of geostrostropic

vorticity is dominant – but Q-G assumptions get violated. ! Sanders’ model shows deepening rate going to 0 –

short wave cutoff !  The most rapid intensification occurs at

wavelengths between 1500 and 3000 km

Analysis, cont. !  Since K13 is proportional to a and inversely

proportional to ", baroclinic instability increases with meridional temperature gradient, and decreases with static stability ! This makes sense since the thermal wind is

proportional to a, and the magnitude of vertical motion predicted by the omega equation is inversely proportional to static stability

! The magnitude of the vertical circulation that lifts/sinks warm/cold air (converting APE to KE) is proportional to a and 1/"#

!  Seasonal effects?

Baroclinic instability and the flow pattern !  Sanders’ model is limited for many reasons

! Why? ! Must be careful in interpreting its solution

!  It has been noted since the 1930s that the shape of the troughs have something to do with their likelihood to produce a cyclone ! Bjerknes (1954) and Polster (1960) identified that

the majority of cyclones intensify under diffluent troughs

! Only a few occur under confluent troughs !  Why?

Diffluent Confluent

Tilting troughs !  Waves in the westerlies whose axes slope in the

direction of the flow with latitude are called positively tilted. ! Associated with poleward fluxes of angular

momentum !  Negatively tilted troughs slope against the

direction of the flow with latitude in the westerlies. ! Associated with equatorward fluxes of angular

momentum, also removing momentum from westerly jet and growing system due to barotropic instability

Why do waves tilt? !  Consider the Rossby wave phase

dispersion relation:

!  What happens when ! U increases with latitude? ! U decreases with latitude?

!  Fractured waves?

!

c =U " #

2$L

% &

' (

2

Fractured trough

Cont.

Cont.

Bombs !  Explosive cyclogenesis occurs most frequently

over the ocean during the cold season.

Definition: Pressure fall of 1 mb/h for 24 hours (termed a bergeron)

Vertical cross section through a bomb

Seclusion

Polar low/instant occlusion

Dryline-front intersection low Thermal low

Subtropical high

Idealized structures of baroclinic waves in the westerlies !  Based on surface obs in Europe, Bjerknes,

Bergeron, and Solberg of the Bergen “Norwegian School” formulated the Polar-Front Theory: cyclones form on fronts

!  Now, it is generally accepted that surface cyclones often*** intensify when DPVA downstream of an upper level trough encounters a frontal zone at the surface (Palmen and Newton 1955)

***not always!

East coast lows

Other types of cyclones !  Colorado lows !  Alps lows

!  Flow over topography, subsidence generates cyclonic low level relative vorticity -> not sufficient to generate a propagating cyclone

!  Short wave aloft triggers cyclogenesis !  Low initially moves southeasterly (higher terrain to the

right combined with warm air advection to the east) !  Low then moves more easterly apart from topography,

usually towards Chicago in North America.

Homework !  Create a Miller-type Diagram consisting of the key

parameters conducive for severe weather for the period 18Z-06Z tomorrow covering the region outlined on the SPC risk map

!  Write a SPC-style mesoscale discussion which textually describes the ingredients regarding the potential for severe weather during this period. Use the convective forecasting handbook.

!  Submit via COMPASS as a PPT file, due tomorrow at 21Z