CSI

1. CSI Bluestein, H.B., 1993: Synoptic dynamic meteorology in mid-latitudes, Vol II, observation and theory of weather systems. pp. 339-350 and 545-561 Holton, J.R., 1979: An introduction to dynamic meteorology, 2nd ed., pp 49-51 and 215-16 Emanuel, K., 1983: The lagrangian parcel dynamics of moist symmetric instability, J. Atmos. Sci., 40, 2368-2376. Emanuel, K., 1983: On assessing local conditional symmetric instability from atmospheric soundings. Mon. Wea. Rev., 111, 2016-2033. Schultz, D. M., and P. N. Schumacher, 1999: The use and misuse of conditional symmetric instability. Mon. Wea. Rev., 127, 2709-2732.

2. Goal: Derive a dynamical mechanism to explain commonly observed precipitation bands that occur over warm fronts and within the trowal region of extratropical cyclones

3. Assume base state density is a function of height and and perturbation is given by Instabilites: Convective instability: Body force is gravity, buoyancy acts opposite gravity and parcels accelerate vertically vertical momentum equation (1) (2) Assume base state pressure is a function of height and and perturbation is given by (3) (4) Base state is in hydrostatic balance Put (2), (3), (4) into (1), approximate , approximate do some algebra and get: Vertical accelerations result from imbalances between the vertical perturbation pressure gradient force and buoyancy

4. real atmosphere H L Assume for for rising parcel, environmental pressure instantly adjusts to parcel movement (atmosphere is everywhere hydrostatic (p = 0) v = virtual potential temperature A parcel’s stability can be determined by displacing it vertically a small distance z, assuming that the environmental virtual potential temperature at z is , and realizing that the parcel virtual potential temperature will be conserved From this equation, we obtain the criteria for gravitational stability in an unsaturated environment: stable neutral unstable

5. Condensation makes the stability problem considerably more complicated. In the interest of time, I will state the stability criteria for moist adiabatic vertical ascent (see Holton p.333, Bluestein’s books or other books for details): Absolute instability Conditional Instability (CI) (parcel) Potential Instability (PI) (layer) Saturation equivalent potential temperature Definitions: Equivalent potential temperature

6. Inertial instability: Body force is centrifugal acceleration due to Coriolis effect, parcels accelerate horizontally horizontal momentum equations Assume a base state flow that is geostrophic (2) (1) Assume a parcel moving at geostrophic base state velocity is displaced across stream Parcel conserves its absolute angular momentum (3) (4) Geostrophic wind at location y + y Put (3) and (4) into (1) Equation governing inertial instability

7. = absolute geostrophic vorticity If the absolute vorticity is negative, a parcel of air when displaced in a geostrophically balanced flow will accelerate away from its initial position To understand inertial instability Consider this simple example  Inertially stable 300 mb height field in the vicinity of a jetstream COR > PGF COR= PGF COR= PGF COR> PGF Inertially unstable  +8   For a parcel displaced south of jet axis, f is positive while is positive. If f exceeds the geostrophic shear , is negative and parcel will accelerate away from its original position. For a parcel displaced north of jet axis, f is positive while is negative. Therefore is negative and parcel will return to its original position.

8. Instability summary In an atmosphere characterized by a hydrostatic and geostrophic base state: Horizontal displacement Vertical displacement Conditional Instability Inertial Instability Or if we define the absolute geostrophic momentum as so that Horizontal displacement Vertical displacement Conditional Instability Inertial Instability Momentum equations What happens if a parcel of air is displaced slantwise in an atmosphere that is inertially and convectively stable?

9. Starting point • Let’s assume: • We have an east-west oriented front with cold air to the north. • The base state flow in the vicinity of the front is in hydrostatic and geostrophic balance • No variations occur along the front in the x (east-west) direction • We consider the stability of a tube of air located parallel to the x axis (east-west oriented tube)

10. RECALL THIS FROM KEYSER AND SHAPIRO PAPER m surfaces and q surfaces in a barotropic and baroclinic environment m2 m3 m4 m5 m1 m only a function of f along y direction q5 q4 p q3 q2 q1 y x Barotropic Atmosphere (no temperature gradient) m2 m3 m4 m5 m1 Because of temperature gradient geostrophic wind increases with height And m surfaces tilt since m = ug + fy q5 q4 q3 q2 p q1 y x Baroclinic Atmosphere (temperature gradient)

11. y This surface represents a surface where a parcel of air Rising slantwise would be in equilibrium shape of surface depends on moisture distribution in environment Weak shear z Strong shear Absolute geostrophic momentum surface X S N Consider a tube at X that is displaced to A At A, the tube’s v is less that its environment At A, the tube’s m is greater than its environment Tube will accelerate downward and southward…. Return to its original position STABLE TO SLANTWISE DISPLACEMENT

12. y This surface represents a surface where a parcel of air Rising slantwise would be in equilibrium shape of surface depends on moisture distribution in environment Weak shear z Strong shear Absolute geostrophic momentum surface X S N Consider a tube at B that is displaced to C At C, the tube’s v is greater that its environment At A, the tube’s m is less than its environment Tube will accelerate upward and northward…. Accelerate to D UNSTABLE TO SLANTWISE DISPLACEMENT

13. Requirements for convection (slantwise or vertical) Instability Moisture Lift

14. Evaluating Moist Symmetric Instability Three different methods 1. Cross sectional analysis 1. Flow must be quasi-two dimensional on a scale of u0/f where u0 is the speed of the upper level jet (e.g. 50 m s -1/10 -4 s -1 = 500 km) 2. Cross section must be normal to geostrophic shear vector (parallel to mean isotherms) in the layer where the instability is suspected to be present 3. Air either must be saturated, or a lifting mechanism (e.g. ageostrophic circulation associated with frontogenesis) must be present to bring the layer to saturation. 4. Air must not be conditionally (or potentially) unstable, or inertially unstable. If either condition is true, the vertical or horizontal instability will dominate.

15. Two approaches depending on the nature of the lifting: is it expected that a layer will be lifted to saturation or a parcel? LAYER: Potential Symmetric Instability PARCEL: Conditional Symmetric Instability On cross section plot (superimpose on RHw or RHi to determine saturation) On cross section (superimpose on RHw or RHi to determine saturation) Slantwise instability evaluation z z z Stable Neutral Unstable RH = 100% RH = 100% RH = 100% y y y

16. Evaluating Moist Symmetric Instability 2. Evaluation of (saturation) equivalent geostrophic potential vorticity Determining if CSI possible is equivalent to determining if the saturation equivalent geostrophic potential vorticity is negative Determining if PSI possible is equivalent to determining if the equivalent geostrophic potential vorticity is negative Note that using the MPVg criteria does not differentiate between regions of CI/PI and CSI/PSI An independent assessment of CI must be done to isolate regions of CSI/PSI

17. Evaluating Moist Symmetric Instability 3. Evaluation of slantwise convective available potential energy (SCAPE) using single soundings Governing equations for displaced tube Potential energy for reversible lifting of tube Emanuel (1983, MWR, p.2018-19) shows that the maximum potential energy available to a parcel ascending slantwise in an environment characterized by CSI occurs when the parcel ascends along an Mg surface. SCAPE for this ascent is The susceptibility of the atmosphere to slantwise convection can be assessed by reversibly lifting a (2-D) parcel along a surface of constant Mg and comparing its virtual temperature (or v) to that of its environment

18. 3 Dec 82 00 UTC Surface 3 Dec 82 00 UTC 500 mb 3 Dec 82 12 UTC Surface 3 Dec 82 12 UTC 500 mb Meteorological conditions at the surface and 500 mb on 3 Dec 1982

19. Satellite images showing storm system – winter frontal squall line with trailing stratiform region

20. Cross section approximately normal to geostrophic shear showing 12Z during more “stratiform” period 00Z during strong upright convection Conditionally unstable: dominant mode will be upright convection Neutral to slantwise convection : implies that slantwise convective adjustment may have occurred

21. Stable to upright convection Neutral to slantwise convection Dots take into account centrifugal potential energy To compare to M surface Moist Adiabat for parcel lifted from 690 mb Sounding along M = 40 T Td Neutral to slantwise convection M = 70 M = 40 Sounding along M = 70

22. Nature of banding Vertical velocity in model simulation: solid = upward, dashed = downward Frontogenetic forcing in the presence of small negative EPVg Frontogenetic forcing in the presence of large negative EPVg As EPVg is reduced from positive values toward 0, the single updraft becomes narrow and more intense. For more widespread and larger negative EPVg the preferred mode becomes multiple bands