MeteorologyVoyager flyby

Lecture 7b, AST111

Topics

❑Thermal gradients in an atmosphere, the wet and dry adiabatic lapse rates.

❑The formation of clouds and the partial pressure of the precipitant

❑Precipitation, terminal velocity ❑Global winds caused by temperature

gradients. ❑Fluid equations and the Coriolis force

The temperature profile• Hydrostatic equilibrium • When the atmosphere is convective, convective

motions are very efficient at transferring heat. • The temperature profile is set by the condition that

heat is instantly convected away. • When a parcel of gas moves vertically, no heat is

exchanged with its surroundings. “Adiabatic” • The temperature profile derived then is

dP gdz

ρ= −

dT

dz= �� � 1

�

µg

kdry lapse rate

temperature drops with increasing height

Clouds tend to form at particular a particular height

The cloud forms when the vapor pressure of the cloud is equal to the saturation vapor pressure.

This height depends on T,P, and the volume filling factor (partial pressure) of the molecule

If you have a formula for the saturation vapor pressure (Clausius Clapeyron eqn) then you can predict the height that clouds are formed (from the abundance of the molecular which sets to the volume filling factor)

The “wet adiabatic lapse rate”

• When clouds form, energy is gained from the latent heat of crystallization.

• This then affects the thermal temperature gradient. • If you assume 100% humidity (vapor continues to

crystallize out more and more as the height increase, but always maintaining 100% humidity) then you can predict what is called the “wet adiabatic lapse rate”

cV dT = �PdV � Lsdws

The “wet adiabatic lapse rate” (continued)

adiabatic, but with addition of heat from release of latent heat from a phase change of moisture

Ls latent heat, energy per unit mass cV specific heat, energy per unit mass per degree dws amount of vapor mass condensing per unit mass ws = grams of vapor per gram of air

cP dT =dP

⇢� Lsdws

using relation between cV, cP and ideal gas law

The wet adiabatic lapse rate

Latent heat, Specific heat amount of vapor condensing per unit mass

/

/

Wet adiabatic Lapse rate

P s s

s P

s

P s s

P s s

P s s

dPc dT L dw

L cdw

dP c dT L dwdP g dzc dT gdz L dwdT gdz c L dw dz

ρ

ρρ

= −

= += −= − −

= −+

In the limit Ls ! 0 we recover the dry adiabatic lapse rate. Dry lapse rate is larger than the wet one Water lapse rate is larger than the Ice one as latent heat is larger for ice

cP dT =dP

⇢� Lsdws

Ls latent heat, cP specific heat dws amount of vapor condensing per unit massdP = �g⇢dz hydrostatic

equilibrium

wet adiabatic lapse rate

cP dT = �gdz � Lsdws

Lapse rates

Cloud base

Graphed like dz/dT rather than the other way around. High lapse (dry) means shallow slope here

Cooler or hotter?

Ice

Liquid

Temperature

Alti

tude

dry

wet

ice

For water: dry lapse rate 9.8 oC/km wet lapse rate 5oC/km

Lapse rates

Cloud base

hotter

Ice

Liquid

Temperature

Alti

tude

dry

wet

ice

base

Mountain top

Clouds on JupiterGalileo Probe -The Net Flux Radiometer (NFR) detected clouds by observing sky brightness oscillations as the probe rotated. NFR detected a cloud layer near the 0.6 bar pressure level. Composed of ammonia ice particles based on its location. -The Nephelometer (NEP) directly detected clouds by observing scattered laser light passing between the body of the probe and a deployed 0.1 meter arm. NEP found one tenuous cloud structure with a base at the 1.6 bar pressure level and faint traces of cloud particles at a wide range of altitudes. No other cloud layers were found.

The simplest explanation for the contradiction between the NEP and NFR experiments is that the upper cloud layer near 0.6 bars is patchy and the Probe went through a clear area. Credit NASA

Precipitation• Droplets and ice crystals fall under the influence of gravity • Drag resists free fall • The terminal velocity 𝑣terminal determines whether a drop is likely to

continue to fall

FDrag =CD

2A⇢airv

2terminal

Drag force for drop with area A=𝜋Rdrop2 Rdrop is the radius of the drop

• Speculation that Titan has seas of hydrocarbons based on clouds observed (note Huygen’s probe saw a dry surface).

Fgravity

= mdrop

g =4⇡

3⇢drop

R3drop

g

In the low Reynolds number limit the drag coefficient is the inverse of the Reynolds number, Re

CD

⇠ Re�1 =⌫air

Rdrop

vterminal

mdrop is the drop mass 𝜌drop is the drop mass density g is gravitational acceleration

𝜈air is fluid viscosity

Equate forces and solve vterminal

⇠2gR2

drop

9⌫air

⇢drop

⇢air

Formation of drops and crystals

• Direct condensation as molecules collide. • Seed dust particles or charged particles can

be important sites for condensation, particularly for crystal formation.

• Possible role of cosmic rays. • After a seed is formed, the growth rate is

determined through collisions with other molecules and particles.

Winds forced by Solar Heating

Differential solar heating causes pressure differentials which cause motions in the atmosphere ❑Hadley circulation ❑Thermal tidal winds ❑Condensation flows Coriolis force breaks up large scale patterns causing trade/zonal winds.

Hadley circulation• When the planet’s rotation axis

is perpendicular to the ecliptic, the equator receives more sunlight than the poles.

• Hot air rises and flows toward regions with lower pressures.

• This air then cools, drops and flows back to the equator at low altitudes.

• Venus, for example, has one Hadley cell per hemisphere.

Atmospheric circulation (contd.)

From Sun

Equator

N. pole

Warm air rising

Cool air sinking

Atmospheric circulation (contd.)❑ Halley’s idea doesn’t explain the direction of the trade

winds, which tend to be easterly between the tropics. ❑ The N-S flow must be modified, since the planet and

atmosphere are rotating about the axis, at higher speeds closer to the equator, this speed given by:

2cos

dayrRv π

λ⊕= (λ = latitude)

At the equator (λ = 0) Earth’s rotational speed is 0.46 km/s, but at λ = ±45º, it’s down to 0.33km/s, and at ±75º, 0.12 km/s, decreasing to zero at the poles.

Atmospheric circulation (contd.)

❑ The Earth rotates counterclockwise, as viewed from the North. As warm air is pushed north or south, it finds itself moving into slower air, and each bubble of it moves out somewhat ahead of the normally-rotating airmass.

❑ Thus the warm flow turns toward the rotation direction as it goes. Underneath, the cooler return flow goes the other way.

Direction of rotation

Warm (high-altitude) flow

Cool (surface) flow

Atmospheric circulation (contd.)

❑ Between cooling and turning, the warm high-altitude flow only makes it to latitude ±30º before sinking, driving surface flow back the way the warm flow came. This circulation pattern is called a Hadley cell.

❑ Taking advantage of the extreme difference between rotational speed and solar heating from the poles to latitudes just below, another Hadley cell, usually called the polar cell to distinguish it from the equatorial one, works the same way between latitudes ±60º and the poles.

❑ In between (i.e. in the temperate zones) a circulation pattern is driven by the Hadley cells that has the opposite sense of these two, counter to rotation. This one’s called the Ferrel cell.

Hadley cell circulation on Earth

When the planet is rotating, the overall equator to polar flow breaks up.

Surface easterlies

Surface Westerlies

Hadley cells and Coriolis force

Hadley was correct about the effects of the Earth’s rotation on circulation. About 100 years later, it was described theoretically in accurate detail by Gaspard Coriolis. The Coriolis force, like the more-familiar centrifugal force, is due to an accelerating reference frame. It’s given by

( )Coriolis 2m= ×F v ω ω = angular velocity of reference frame v = velocity within reference frame

The Coriolis force is fictitious: it’s an artifact of the acceleration of one’s reference frame

21

Atmospheric circulation (contd.)

❑ The faster a planet rotates, the more bands of alternating Hadley and Ferrel cells are obtained in the atmosphere.

❑ By the same token, a slow enough rotator would have only one, ideal-looking, Hadley cell per hemisphere, stretching all the way from the equator to the poles.

❑ This is the situation of Venus, the slowest rotator among the planets. • Most of the air circulation on Venus is north-south: by

far the simplest atmospheric structure of the planets. • This is why the surface temperature on Venus is so

uniform –as hot at the poles as at the equator, and as hot at night as during the day.

Thermal tides• If there is a large day/night temperature difference, then air can flow

from the hot day side to the night side. “Thermal tidal winds.” • There is a return flow at low altitudes. • The temperature differential determines whether such a flow occurs.

It needs to be large. • The temperature change depends on the total mass of the

atmosphere. • The fractional change is small for planets with large atmospheres

like Jupiter and Venus. • Mars, however has a relatively tenuous atmosphere and so a strong

thermal tide. • The Earth and Venus have thermal tides only in the thermosphere

(high up).

Condensation flows

• Seasonal and orbital changes can cause ices to sublime and condense.

• Mars’s CO2 summer polar cap sublimes increasing the CO2 content of the atmosphere. This gas condenses falling onto the opposite pole.

• On Triton and Pluto, nitrogen and methane condense onto the surface as they get further from the Sun.

Fluid equationsTo describe a fluid we track the velocity and density everywhere Mass flux 𝜌v

The rate mass leaves a region depends on the gradient of the mass flux

If this is zero there is the same amount of mass coming in one side as leaving the other side and the density remains constant

@

@x

(⇢v)

There is a change in the mass density locally@

@x

(⇢v) 6= 0

@⇢

@t

= � @

@x

(⇢v) In one dimension

Conservation of mass@⇢

@t

= � @

@x

(⇢vx

)� @

@y

(⇢vy

)� @

@z

(⇢vz

)

@⇢

@t+r · (⇢v) = 0

r · v = 0

An incompressible fluid had constant density

The mass density in a blob depends on the mass flux moving in the x,y,z directions

Advective Derivatives

f(x, y, z, t)

Df

Dt

=@f

@t

+@f

@x

dx

dt

+@f

@y

dy

dt

+@f

@z

dz

dt

Df

Dt=

@f

@t+ v ·rf

v

Conservation of massconservation of mass using mass flux

Gaus’ law

expand advective derive@⇢

@t+ v ·r⇢ = �⇢r · v

Z

V

D⇢

DtdV = �

Z

V⇢r · vdV

Z

V

D⇢

DtdV = �

Z

S⇢v · dA

D⇢

Dt= �⇢r · v

@⇢

@t+r · (⇢v) = 0

reorder and you get the conservation law

Lagrangian vs Eulerian viewpoints

D⇢

Dt= �⇢r · v

Using the advective derivative Lagrangian viewpoint, moving with the fluid

@⇢

@t+r · (⇢v) = 0 Eulerian viewpoint

External reference

Same conservation law

Conservation of Momentum

In steady state, consistent with hydrostatic equilibrium

⇢Dv

Dt= �rP + ⇢g equivalent to F=ma

Conservation of moment Without viscosity or gravity gives “Euler’s equation” With viscosity “Navier-Stokes equation”

Dv

Dt=

@v

@t+ (v ·r)v = �1

⇢rP + ⌫r2v

Pressure: = force per unit area = momentum flux through a surface

Rotating frame and Coriolis force

Because the earth is rotating, a particle on an apparentlystraight path is actually on a curved oneand so feels an acceleration. This additional acceleration is the sum of the Coriolis force and the

2

centrifugal forces. Euler's equation becomes:

12 ( )rot rotD PDt

ω ωρ

= − × − ∇ + +v z v g r

Coriolis force Centrifugal term

Pressure term

Because the Earth is rotating a particle that looks like it is moving on a straight path may actually be on a curved one and so feel an acceleration The extra acceleration is the sum of the Coriolis and centrifugal forces

Euler’s equation becomes

ωrot spin rate of Earth

Geostrophic balance2

2

12 ( )

Assume steady state, 0, assume is small.

Take 2-dimensional limit (shallow approximation),12 ( )

Coriolis force is exactly balanced by pressure gradi

rot rot

rot

rot

DP

DtDDt

P

ω ωρ

ω

ωρ

= − × − ∇ + +

=

− × = ∇

v z v g r

v r

z v

ent. is perpendicular to isobars.

is perpendicular to flows along isobars.

P∇×

→z v vv

Dv

Dt= 0 Steady state small

2-dimensional limit (shallow approximation)

Coriolis force balanced by pressure gradient

∇P perpendicular to isobars! (definition of a gradient) z𝗑v is perpendicular to v so v flows along isobars

�2!rot

(z⇥ v) =rP

⇢

!2rot

r

Geostrophic flowsExamples of Geostrophic flows: • Trades winds on Earth • Earth’s stratospheric jet streams • Zonal winds on Giant planets.

�2!rot

(z⇥ v) =rP

⇢

Cyclostrophic balance

When the pressure is balanced by centrifugal force rather than Coriolis force, the we speak of “cyclostrophic balance”

Examples of cyclostrophic balance: • Titan and Venus global patterns. • Equatorial flows • Dust devils, tornados.

212 ( )rot rotD

PDt

ω ωρ

= − × − ∇ + +v z v g r

Clouds on Venus

Venus, because it is a slow rotator has one Hadley cell and a polar cyclostrophic flow pattern. The clouds show structure in UV images. Credit UCAR, UV Mariner 10 image of Venus.

3.8 micron images of Venus’s polar vortex. From ESAs Thermal Imaging SpectrometerShowing temperature of cloud tops with darker being hotter and at lower altitude

Venus’s polar vortex

Jupiter

The clouds consist of • Ammonia (top deck, bright white) • Water (bottom deck, dull white) • Ammonium Hydrosulfide (middle, red/

brown) Wet air rises, dry air sinks. The different colors in Jupiter represent

different “humidity” gas.

Zonal flows on Jupiter

Jupiter Near IR Optical

Infrared bright regions correspond to dry regions that are dark in optical images. Dry air sinks so these are downward flows.

Cassini flyby

Storms

Warm humid air rises. As releases water, heats further, and rises further. This makes a low pressure core. Hurricanes.

Coriolus + centrifual force

Pressure force

anticyclone

cyclone hurricane

The Red Spot

• Red spot on Jupiter is an anti-cyclone. • Differential rotation causes vortices or eddies. • The merging of two eddies produces a lower energy

state. • So eddies can increase in size. • Large eddies can be long lasting. • Non-linear 2D turbulence problem. • Both cyclones and anti-cyclones are on Jupiter, but

with different colors

The dark spot on NeptuneNeptune's blue color is largely the result of absorption of red light by methane in the atmosphere but there is some additional as-yet-unidentified chromophore which gives the clouds their rich blue tint. At the time of the Voyager encounter, Neptune's most prominent feature was the Great Dark Spot (left) in the southern hemisphere. It was about half the size as Jupiter's Red Spot. Neptune's winds blew the Great Dark Spot westward at 300 meters/second

Features on NeptuneVoyager 2 also saw a smaller dark spot in the southern hemisphere and a small irregular white cloud that zips around Neptune every 16 hours or so now known as "The Scooter" (right). It may be a plume rising from lower in the atmosphere but its true nature remains a mystery. Recent HST observations suggest that the dark spot has dimmed or disappeared

Uranus

Uranus is nearly featureless. The picture on the right uses false colors and contrast enhancement to bring out subtle details in the polar region of Uranus. Images obtained through ultraviolet, violet, and orange filters were respectively converted to blue, green, and red colors to produce the picture. In this false-color picture, Uranus reveals a dark polar hood surrounded by a series of progressively lighter concentric bands. One possible explanation is that a brownish haze or smog, concentrated over the pole, is arranged into bands by zonal motions of the upper atmosphere.

Exoplanet atmospheres

Winds on exoplanets

Louden and Wheatley ApJL 2015 HD 189733b

Summary

• Using hydrostatic equilibrium, the equation of state, and the adiabatic index to predict the vertical temperature gradient.

• Dry/wet adiabatic lapse rates. • Precipitation and formation of clouds • How fluid equations determine geostrophic and

cyclostrophic flows. • Hadley cells, Thermal tides, Condensation flows.