Lecture 10 Static Stability. General Concept An equilibrium state can be stable or unstable Stable...

Lecture 10Lecture 10

Static StabilityStatic Stability

General ConceptGeneral Concept

An equilibrium state can be stable or An equilibrium state can be stable or unstableunstableStable equilibrium: A displacement induces a Stable equilibrium: A displacement induces a restoring force restoring force i.e., system tends to move back to its original i.e., system tends to move back to its original

statestate

Unstable equilibrium: A displacement Unstable equilibrium: A displacement induces a force that tends to drive the induces a force that tends to drive the system even further away from its original system even further away from its original statestate

A More Realistic ScenarioA More Realistic Scenario

Equilibrium

Small DisplacementSmall Displacement

Small DisplacementSmall Displacement

Stable.

Large DisplacementLarge Displacement

Large DisplacementLarge Displacement

Unstable.

Idea of Previous SlidesIdea of Previous Slides

There may be a critical displacement There may be a critical displacement magnitudemagnitude

displacement < critical displacement < critical stable stable

displacement > critical displacement > critical unstableunstable

(More about this shortly)(More about this shortly)

Atmospheric StabilityAtmospheric Stability

Unsaturated AirUnsaturated Air

Consider a vertical parcel displacement, Consider a vertical parcel displacement, zz Assume displacement is (dry) adiabaticAssume displacement is (dry) adiabatic

Change in parcel temperature = -Change in parcel temperature = -d d zz

Denote lapse rate of environment by Denote lapse rate of environment by

z

Parcel

T = T0

T=T0 - dz

Environment

T = T0

T = T0 - z

Temp of displaced parcel temp of environment

Two CasesTwo Cases

Parcel temp. > environment temp. Parcel temp. > environment temp. parcel less dense than environmentparcel less dense than environment

parcel is buoyantparcel is buoyant

Parcel temp. < environment temp.Parcel temp. < environment temp. parcel denser than environment parcel denser than environment

parcel is negatively buoyantparcel is negatively buoyant

Lapse RatesLapse Rates

TTparcelparcel = T = T00 - - ddzz

TTenvenv = T = T00 - - zz

TTparcelparcel > T > Tenvenv if T if T00 - - ddz > Tz > T00 - - zz

> > dd

TTparcelparcel < T < Tenvenv if T if T00 - - ddz < Tz < T00 - - zz

< < dd

StabilityStability

> > d d resultant force is positive resultant force is positive parcel acceleration is upward (away from parcel acceleration is upward (away from

original position)original position) equilibrium is unstableequilibrium is unstable

< < d d resultant force is negative resultant force is negative parcel acceleration is downward (toward parcel acceleration is downward (toward

original position)original position) equilibrium is stableequilibrium is stable

Graphical DepictionGraphical Depiction

Temperature

z

Temp of rising parcel

Stable lapse rate

Unstable lapse rate

Saturated AirSaturated Air

Recall: Vertically displaced parcel Recall: Vertically displaced parcel cools/warms at smaller ratecools/warms at smaller rate Call this the moist-adiabatic rate, Call this the moist-adiabatic rate, mm

Previous analysis same with Previous analysis same with dd replaced replaced

by by mm

Equilibrium stable if Equilibrium stable if < < mm

Equilibrium unstable if Equilibrium unstable if > > mm

General ResultGeneral Result

Suppose we don’t know whether a layer of Suppose we don’t know whether a layer of the atmosphere is saturated or notthe atmosphere is saturated or not

> > dd > > mm equilibrium is unstable, equilibrium is unstable,

regardlessregardless Equilibrium is Equilibrium is absolutely unstableabsolutely unstable

< < mm < < dd equilibrium is stable, equilibrium is stable,

regardlessregardless EquilibriumEquilibrium is is absolutely stableabsolutely stable

ContinuedContinued

Suppose Suppose mm < < < < dd

Layer is stable if unsaturated, but unstable Layer is stable if unsaturated, but unstable if saturatedif saturated

Equilibrium is Equilibrium is conditionallyconditionally unstableunstable

md

Absolutely unstable

Absolutely stable

Conditionally unstable

ApplicationApplication

If a layer is unstable and clouds form, they If a layer is unstable and clouds form, they will likely be cumuliformwill likely be cumuliform

If a layer is stable and clouds form, they If a layer is stable and clouds form, they will likely be stratiformwill likely be stratiform

Example: Mid-Level CloudsExample: Mid-Level Clouds

Suppose that clouds form in the middle Suppose that clouds form in the middle tropospheretroposphere

Unstable Unstable altocumulus altocumulus

Stable Stable altostratus altostratus

AltocumulusAltocumulus

AltostratusAltostratus

Deep ConvectionDeep Convection

Previous discussion not sufficient to Previous discussion not sufficient to explain thunderstorm developmentexplain thunderstorm development

Thunderstorms start in lower atmosphere, Thunderstorms start in lower atmosphere, but extend high into the tropospherebut extend high into the troposphere

Physics Review: EnergyPhysics Review: Energy

h

Object at height h


h

Remove support: Object falls


Let z(t) = height a time t

z(t)

It Can Be Shown …It Can Be Shown …

mghmgzvm 2

2

1

kinetic energy potential energy

(v = speed)

As object falls, potential energy is converted to kinetic energy.

Available Potential EnergyAvailable Potential Energy

Object may have potential energy, but it Object may have potential energy, but it may not be dynamically possible to may not be dynamically possible to release itrelease it

h

Technically, PE = mgh, but lower energy state is inaccessible.

The energy is unavailable.

Energy BarriersEnergy Barriers

h

a

b

To get from a to b, energy must be supplied to surmount the barrier.

hb

Energy needed: mghb


h

a

b

Now, ball can roll down hill.


h

a

b

Amount of PE converted to KE: mg(h + hb)

Net release of energy: mg(h + hb) – mghb = mgh

hb

CAPE, CINCAPE, CIN

CAPE: Convective Available Potential CAPE: Convective Available Potential Energy Energy (Positive area)(Positive area)

CIN: Convective InhibitionCIN: Convective Inhibition (Negative area at bottom of sounding)(Negative area at bottom of sounding)

Negative area

Positive area

Dry adiabat

Saturated adiabat

LCL

Sounding

CAPE, CINCAPE, CIN

CIN is the energy barrierCIN is the energy barrier

CAPE is the energy that is potentially CAPE is the energy that is potentially available available ifif the energy barrier can be the energy barrier can be surmountedsurmounted

Isolated Severe ThunderstormsIsolated Severe Thunderstorms

Suppose CIN and CAPE are largeSuppose CIN and CAPE are large

Consider a population of incipient Consider a population of incipient thunderstormsthunderstorms

Few of these storms will surmount the Few of these storms will surmount the energy barrier, however …energy barrier, however …

Those that do will have a lot of energy Those that do will have a lot of energy available.available.

Sudden Outbreaks of Severe Sudden Outbreaks of Severe WeatherWeather

Start with a high energy barrier (large CIN)


Now, suppose energy barrier decreases.


Disturbances that previously couldn’t overcome the barrier now can.

If CAPE is large, storms could be severe.

Level of Free Convection (LFC)Level of Free Convection (LFC)

Level of Free Convection (LFC):

When a parcel ceases to be colder and denser than surrounding air (environment), and instead becomes positively buoyant.

On a thermo diagram, this occurs when the moist adiabat being followed by the parcel crosses from the cold side of the environmental profile to the warm side.

The level at which this crossover occurs is the LFC.

Equilibrium Level (EL)Equilibrium Level (EL)

Thermals will continue to rise until their temperature matches that of the environment. The level at which this occurs is called the equilibrium level (EL).

Also called level of neutral buoyancy. At that point, parcels may overshoot a little, becoming colder than environment and ultimately falling back to their EL.

Convective Inhibition (CIN)Convective Inhibition (CIN)

dzzfCINLFCZ

B )(0

dzzT

zTzTgCIN

LFCz

)(

)()(

0

gzT

zTzTzfB

)(

)()()(

Buoyant force is negative by definition, minus sign in front of integral.

T(z) temp of rising parcel, T’(z) temp of environment at same level.

fB = Buoyancy force

Convective Inhibition (CIN)Convective Inhibition (CIN)

pdpTpTRCINLFC

P

d ln)()(0

Invoking hydrostatic approximation, and ideal gas law:

CAPECAPE

pdpTpTRCAPEEL

LFC

d ln)()(

pdpTpTRCINLFC

P

d ln)()(0

Once the parcel has overcome the energy barrier, CIN, and reached its LFC, CAPE is the energy that may be released by resulting buoyant ascent.

CIN represents the energy barrier to initiation of free convection. CAPE is the maximum possible energy that can be released after CIN has been overcome.

[J /kg]

[J /kg]

Lecture 10 Static Stability. General Concept An equilibrium state can be stable or unstable Stable...

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Transcript of Lecture 10 Static Stability. General Concept An equilibrium state can be stable or unstable Stable...