PRECIPITATION PROCESSES AT FRONTS

POSSIBLE CONDITIONS PRESENT AT FRONT

1. Air ahead of the front is stable to all forms of instability

Forcing mechanism for vertical motion: ageostrophic circulation associated with frontogenesis

2. Air is potentially or conditionally unstable

Lifting of air to saturation by ageostrophic circulation triggers convection

3. Air is stable to potentially or conditionally unstable to upright displacement but unstable to slantwise displacement

Lifting of air to saturation by ageostrophic circulation triggers slantwiseconvection

Frontogenesis

As we have learned from the previous analysis of the SE equation:

Frontogenesis leads to a direct circulation

Warm (moist) air rises on the warm side of the front, leading to widespread clouds and precipitation

HOW DO INSTABILITES MODULATE THIS PRECIPITATION?

Instabilites:

Convective instability: Body force is gravity, buoyancy acts opposite gravity and parcels accelerate vertically

gz

p

dt

dw

1

vertical momentum equation

tzyxz ,,, Assume base state density is a function of height and and perturbation is given by

z

tzyxpzpp ,,,Assume base state pressure is a function of height and and perturbation is given by

zpp

gz

p

Base state is in hydrostatic balance

(1)

(2)

(3)

(4)

Put (2), (3), (4) into (1), approximate , approximate do some algebra and get:

111

dz

pd

dz

pd

1

gz

p

dt

dw

1Vertical accelerations result from imbalances between the vertical perturbation pressure gradient force and buoyancy

gz

p

dt

dw

1 Assume for for rising parcel, environmental pressure instantly adjusts to parcel movement (atmosphere is everywhere hydrostatic (p = 0)

H

L

realatmosphere

gggdt

dw

v

vv

zdz

d vvv 0

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 0vv z

zdz

dgz

dz

dg

dt

dw v

v

vvv

v

00

From this equation, we obtain the criteria for gravitational stability in an unsaturated environment:

0dz

d vstable 0

dz

d vneutral 0

dz

d vunstable

v = virtual potential temperature

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):

0*

dz

d e

0dz

d vAbsoluteinstability

ConditionalInstability (CI)

(parcel)

PotentialInstability (PI)

(layer)0

dz

d e

Definitions:

Tc

qL

p

vsve exp*

LCLp

vsve Tc

qLexp

Saturation equivalent potential temperature

Equivalent potential temperature

Example of Potential Instability

Stable sounding to parcel ascent

Lift layer between 1 and 1.25 kmone km in altitude

AIR DESTABILIZES

Synoptic environment conducive to the development of potential instability

Inertial instability: Body force is centrifugal acceleration due to Coriolis effect, parcels accelerate horizontally

fuydt

dv

yfug

1Assume a base state flow that is geostrophic

fvxdt

du

horizontal momentum equations

0x

uufdt

dvg fv

dt

du

Assume a parcel moving at geostrophic base state velocity is displaced across stream

yfyuyyu g 00Parcel conserves its absolute angular momentum

yy

uyuyyu g

gg

00 Geostrophic wind at location y + y

(1) (2)

(3)

(4)

Put (3) and (4) into (1)

yy

uff

dt

dv g

Equation governing inertial instability

yy

uff

dt

dv g

gagg fy

uf

= absolute geostrophic vorticity

If the absolute vorticity is negative, a parcel of air when displaced in a geostrophically balancedflow will accelerate away from its initial position

To understand inertial instabilityConsider this simple example

+8

300 mb heightfield in the vicinity

of a jetstream

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.

y

ug

dt

dv

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.

y

ug

COR= PGF COR= PGF

COR > PGF

COR> PGF

Inertially stable

Inertially unstable

y

ug

dt

dv

Instability summary

In an atmosphere characterized by a hydrostatic and geostrophic base state:

0*

dz

d eConditionalInstability

InertialInstability

0 fg

Vertical displacement Horizontal displacement

Or if we define the absolute geostrophic momentum as fyum g

so that ffy

u

y

mg

g

0*

dz

d eConditionalInstability

InertialInstability

0y

m

Vertical displacement Horizontal displacement

What happens if a parcel of air is displaced slantwise in an atmosphere that is inertially and convectively stable?

vvv

g

dt

dw

Momentumequations

gmmfdt

dv

Let’s assume:

1) We have an east-west oriented front with cold air to the north.

2) The base state flow in the vicinity of the front is in hydrostatic and geostrophic balance

3) No variations occur along the front in the x (east-west) direction

4) We consider the stability of a tube of air located parallel to the x axis (east-west oriented tube)

Starting point

1

2

3

4

5

y

m1 m2 m3 m4 m5

x

p

Barotropic Atmosphere (no temperature gradient)

m surfaces and surfaces in a barotropic and baroclinic environment

1

2

3

4

5

y

m1 m2 m3 m4 m5

x

p

Baroclinic Atmosphere (temperature gradient)

Because of temperature gradientgeostrophic wind increases with heightAnd m surfaces tilt since m = ug + fy

m only a function of f along y direction

yN S

z

Strong shear

Weak shear

This surface represents a surface where a parcel of airRising slantwise would be in equilibrium shapeof surface depends on moisture distribution in environment

vv

Absolute geostrophicmomentum surface

Consider a tube at X that is displaced to A

X

At A, the tube’s v is less that its environmentAt A, the tube’s m is greater than its environment

Tube will accelerate downward and southward…. Return to its original position

vvv

g

dt

dw

gmmfdt

dv

STABLE TO SLANTWISE DISPLACEMENT

yN S

z

Strong shear

Weak shear

This surface represents a surface where a parcel of airRising slantwise would be in equilibrium shapeof surface depends on moisture distribution in environment

vv

Absolute geostrophicmomentum surface

Consider a tube at B that is displaced to C

X

At C, the tube’s v is greater that its environmentAt A, the tube’s m is less than its environment

Tube will accelerate upward and northward…. Accelerate to D

vvv

g

dt

dw

gmmfdt

dv

UNSTABLE TO SLANTWISE DISPLACEMENT

Requirements for convection (slantwise or vertical)

Instability

LiftMoisture

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.

RH = 100%RH = 100%

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)

ge Mvs ge Mvs*

e ee 2y

z

gM

z z

y y

gg MM 2

e ee 2

gM

gg MM 2

Stable Neutral Unstable

e ee 2

gM

g

g

M

M

2*: ee orred gMblue :

Slantwise instability evaluation

RH = 100%

iw RHorRHgreen :

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

0*, egsg kfvgMPV

0***

pf

y

u

x

v

xp

v

yp

ueggegeg

Note that using the MPVg criteria does not differentiate between regions of CI/PI and CSI/PSIAn independent assessment of CI must be done to isolate regions of CSI/PSI

Determining if PSI possible is equivalent to determining if the

equivalent geostrophic potential vorticity is negative

0 egg kfvgMPV

0

pf

y

u

x

v

xp

v

yp

ueggegeg

Evaluating Moist Symmetric Instability

3. Evaluation of slantwise convective available potential energy (SCAPE) using single soundings

vvv

g

dt

dw

gmmfdt

dv Governing equations for displaced tube

Potential energy for reversible lifting of tube 2

1

ldkg

immfPE vvv

g

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

2

1,M

vvv

ldkg

SCAPE

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 itsvirtual temperature (or v) to that of its environment

3 Dec 8200 UTCSurface

3 Dec 8212 UTCSurface

3 Dec 8200 UTC500 mb

3 Dec 8212 UTC500 mb

Meteorological conditions at the surface and 500 mb on 3 Dec 1982

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

Cross section approximately normal to geostrophic shear showing ge Mand*

00Z during strong upright convection 12Z during more “stratiform” period

Neutral to slantwise convection : implies that slantwise convective adjustment may have occurred

Conditionally unstable: dominant mode will be upright convection

TTd

Moist Adiabat for parcel lifted from 690 mb

Sounding alongM = 40

M = 40

M = 70

Sounding alongM = 70

Stable to uprightconvection

Neutral to slantwiseconvection

Neutral to slantwiseconvection

Dots take into account centrifugal potential energyTo compare to M surface

Nature of banding

Frontogenetic forcing in the presence of small positive EPVg

Frontogenetic forcing in the presence of large negative EPVg

Vertical velocity in model simulation: solid = upward, dashed = downward

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