Conditional Symmetric Instability (CSI): Conditional Symmetric Instability (CSI): Is it possible for anyone to Is it possible for anyone to really grasp what this is?really grasp what this is?

Paul MarkowskiDepartment of Meteorology, Penn State University

CSI: What is it?CSI: What is it?

• It’s due to a combination of baroclinic instability and centrifugal instability, the latter of which results from an unstable distribution of angular momentum (here angular momentum is defined with respect to the Earth’s axis)

The release of CSI has been used to explain the mesoscale banding of precipitation within the broad precipitation shields of mid-latitude cyclones.

Courtesy of Dave Schultz (NSSL)

http://www.nssl.noaa.gov/csi

Centrifugal instability

radial wind

specific volume

tangential wind

} }

radial PGF centrifugal force

Centrifugal Instability

• It is convenient to introduce angular momentum, XXXXX , which is conserved if XX , i.e., if the pressure field is symmetric about the axis of rotation

• We can then write the momentum equations as

Centrifugal Instability

• Let us displace a ring of fluid in the r direction and evaluate the new force balance at r = ro + r – do the PGF and centrifugal

force balance each other at the new location, and if not, what is the net acceleration on the ring of fluid?

r = ro + r

r = ro

Centrifugal Instability

• We can define a mean state whereby no radial acceleration exists, i.e., the PGF and centrifugal force balance each other:

r = ro + r

r = ro

overbars denote mean (equilibrium) state

Centrifugal Instability

• Subtracting r = ro + r

r = ro

from

yields

Centrifugal Instability

neglect

Evaluate at r = ro + r

Centrifugal Instability

• If M increases with r, the radial acceleration is opposite r and we have stability.

• When M decreases with r, the radial acceleration has the same sign as r and we have instability.

Centrifugal Instability

• Centrifugal instability constrains vortex characteristics—angular momentum must increase with distance from the axis of rotation– e.g., tornadoes,

hurricanes

• Wind field in tornadoes is often approximated as a Rankine vortex (and it’s no coincidence that the Rankine vortex is centrifugally stable):

v 1/r v r

• Wind field in tornadoes is often approximated as a Rankine vortex (and it’s no coincidence that the Rankine vortex is centrifugally stable):

M = constant M r2

M = vr

Angular Momentum on Larger Scales

• Angular momentum is with respect to Earth’s axis

A closer look at angular momentum surfaces…

N S

Mg = 80 m/s

*

• It’s actually geostrophic absolute momentum (Mg) that’s relevant, but Mg surfaces are just angular momentum surfaces; Mg = ug - fy

Mg = 70 m/s

Mg = 60 m/s

A closer look at angular momentum surfaces…

For a stable configuration of angular momentum surfaces, parcels of air tend to return to their original angular momentum surface if perturbed horizontally.

Mg = 80 m/s

Mg = 70 m/s

Mg = 60 m/s

Recall potential temperature surfaces…

For a stable configuration of potential temperature surfaces (potential temperature increases with height), parcels of air tend to return to their original potential temperature surface if perturbed vertically.

N

= 310 K = 305 K

= 300 K

S

But what about this configuration of potential temperature and angular momentum surfaces?

Parcels are stable with respect to both vertical and horizontal displacements, but parcels displaced along certain slanted paths will accelerate away from their initial position—this is called symmetric instability.

N

= 310 K

= 305 K

= 300 K

Mg = 80 m/s

Mg = 70 m/s

Mg = 60 m/s

S

Symmetric Instability• Its release results in so-called

“slantwise convection” – not the same as buoyant

convection in shear, which has slanted trajectories but an acceleration that is directed vertically only (at least the part of the acceleration that results from the instability)

green = acceleration (due to buoyancy)

orange = parcel trajectory

Symmetric Instability• Potential temperature surfaces must be sloped

more steeply than angular momentum surfaces, and the displacement must have a slope between that of the angular momentum and potential temperature surfaces

Conditional Symmetric Instability

• In most operational meteorology applications, we are concerned with the presence of this instability in a saturated atmosphere—thus, we replace potential temperature surfaces with equivalent potential temperature surfaces (e), and the instability becomes known as conditional symmetric instability (CSI).

*

• The slope of e vs Mg

surfaces actually defines PSI, but where the atmosphere is saturated, e and e surfaces are equivalent.

*

CSI

• It can be shown (with not too much additional work) that the “slope criterion” for CSI is met when (Ri/f)(f-dug/dz) < 1, EPV < 0, or when absolute vorticity on a potential temperature surface becomes negative

• CSI is perhaps most likely to arise in the cold sector east of intensifying cyclones where the potential temperature surfaces are steeply sloped (perhaps implying strong isentropic ascent) and the midtropospheric absolute vorticity is small (or the relative vorticity is negative)

Final comments• The terms CSI and slantwise convection do not

have interchangeable meanings– CSI is to slantwise convection as conditional

instability is to buoyant convection

• CSI is notoriously difficult to observe—it often is removed as quickly as it is generated (observations of an atmosphere in which and Mg surfaces are parallel might suggest that slantwise convection has occurred or is occurring)

• There are dynamical processes other than slantwise convection that can result in banded precipitation too!