N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Basics of atmospheric dynamics

N. Kampfer

Institute of Applied PhysicsUniversity of Bern

20. / 23. March 2012

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Outline

Introduction

Equation of motioncontinuity equation

Primitive equations

Geostrophic wind

Thermal wind

VorticityCirculationAbsolute vorticity

Rossby waves

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

IntroductionIn an atmosphere with horizontal temperature gradients,density gradients will be generated and as such pressuregradients. These in turn will lead to a circulation of theatmosphere.The circulation of the atmosphere of a planet is a keycomponent of its climate. In case of for example:

I Atmospheric motions carry heat from the tropics to thepole

I winds transport humidity from the oceans to the landI Distribution of most chemical species in the atmosphere

is the result of transport processesI O3 is mainly produced in equatorial regions but is

transported polewardsI H2O enters the middle atmosphere in the tropics and is

moved toward the polesI CFCs are released in industrial areas and generate

ozone-holes in polar regionsI Aerosols are produced in industrial areas and affect

regions far away

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Introduction

Air motions are strongly constrained by:

I Density stratification→ gravitational force resists vertical displacement

I Earth rotation→ Coriolis force is a barrier against meridionaldisplacements

Transport occurs at a variety of spatial and temporal scales.In case of :

I global

I synoptic ≈ 1000km→ e.g. high and low pressure systems

I mesoscale ≈ 10 - 1000km→ e.g.fronts

I small scale→ e.g. planetary boundary layer

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Introduction

In order to describe the dynamical behavior of theatmosphere, we treat it as a fluid→Fundamental equations of fluid mechanics must be used

Circulation of a planet’s atmosphere is governed by threebasic principles:

1. Newton’s law of motion

2. Conservation of energy → first law of thermodynamics

3. Conservation of mass → equation of continuity

plus the equation of state

In using these laws we must consider that we are operatingin a rotating frame of reference!→ we have to consider centrifugal and the Coriolis force

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Coriolis forceExample: Planes heading to Miami from Toronto and Quito

Coriolis effect causes things moving toward the poles to leadthe earth’s rotation because they are headed into regionswhere the earth’s rotational speed is slower⇒ They are deflected to the east

Coriolis effect causes things moving toward the equator tolag the earth’s rotation because they are headed into regionswhere the earth’s rotational speed is faster⇒ They are deflected to the west

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Coordinate systemIt is practical to use a spherical coordinate system with originat the center of the Earth to describe dynamical aspects

The coordinate system is rotating together with the planet

λ longitude in easterly directionφ latitude in northerly directionz geometric altitudeR radius of planet, r = R + z

u = r cosφdλ/dt zonal component of wind velocity ~vmeasured towards the east

v = rdφ/dt meridional component of wind velocity ~vmeasured towards the north

w = dz/dt vertical component of ~v

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Real forcesReal forces entering in the equation of motion are:

1. Pressure gradient force, ~FpWhenever there is a gradient in pressure the resultingforce per mass is given by

~Fp = −1

ρ~∇p

~Fp ⊥ isobars, directed from higher to lower pressureDistance of isobars → measure for the pressure gradientTypical values for pressure gradient for :→ ca. 1mb per 8m in vertical direction and

1mb per 10km in horizontal direction2. Gravity force, ~FG

~FG = −~∇Φ = −~g3. Frictional force, ~FR is nearly proportional to wind speed

~FR = −a~vimportant in planetary boundary layer

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Equation of motion without rotationConsidering all forces: general equation of motion per mass

d~v

dt= ~Fp + ~FR + ~FG

Neglecting friction

d~v

dt= −1

ρ~∇p − ~g

Considering vertical direction only

dw

dt= −g − 1

ρ

∂p

∂z

In case there is no vertical acceleration, i.e. in equilibrium wearrive at

∂p

∂z= −gρ

This is the hydrostatic equilibrium

What means ddt , i.e. the time derivative in a fluid?

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Material derivativeConsider an atmospheric parameter ψ, like p, T or ~v

I ψ is a vector or a scalar field variable

I ψ depends on location and time ψ = ψ(x , y , z , t)

I ψ is measured by fixed instruments

Equations so far however are valid for a moving air parcels

What is the time derivative dψdt of a moving air parcel?

→ total derivative

dψ

dt=∂ψ

∂x

dx

dt+∂ψ

∂y

dy

dt+∂ψ

∂z

dz

dt+∂ψ

∂t

dt

dt

I partial derivatives (e.g. ∂ψ∂y ) are valid for the field ψ

I derivatives like dydt are valid for particles

I dxdt = u, dy

dt = v , dzdt = w are the components of the

velocity ~v of the particle

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Eulerian and Lagrangeian point of view

dψ

dt=

∂ψ

∂x

dx

dt+∂ψ

∂y

dy

dt+∂ψ

∂z

dz

dt+∂ψ

∂t

dt

dt

= u∂ψ

∂x+ v

∂ψ

∂y+ w

∂ψ

∂z+∂ψ

∂t

=∂ψ

∂t+ ~v · ~∇ψ

Total change of ψ with time dψdt , the so called material

derivative, is built up of

I change of time at fixed position, i.e. local: ∂ψ∂t

I advective change ~v · ~∇ψ→ Eulerian point of view→ Lagrangeian point of view

Keep in memory: ddt = ∂

∂t + ~v · ~∇

For the velocity ~v this leads to d~vdt = ∂~v

∂t + ~v · ~∇~v

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Apparent forces in a rotating reference systemThe centrifugal force is showing up in a rotating referenceframe→ its effect is

”absorbed“ in the gravitational force

An object moving with velocity ~v in a plane perpendicular tothe axis of rotation experiences an apparent force, calledCoriolis force, ~FC .

~FC = 2~v × ~ωThe angular velocity is for : ω = Ω = 7.3 · 10−5 sec−1

The Coriolis force in horizontal direction can be written as

~FCh =

(d~vhdt

)C

= −f ~k × ~vh

~k is a unit vector in z-direction where f is calledCoriolis-Parameter

f = 2Ω sinϕ

f is equal to zero at the equator and has a maximum at thepoles f = ±1.46 · 10−4sec−1

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Apparent forcesThe Coriolis force per mass for zonal velocity, u, andgeographic latitude ϕ is

dv

dt= −2Ωu sinϕ = −fu

and analogue for meridional direction

du

dt= 2Ωv sinϕ = fv

I The Coriolis force is acting normal to the direction ofmotion→ to the right in the northern hemisphere→ to the left in the southern hemisphere

I As the force is acting normal to the direction of motionno work is performed

I The force vanishes at the equator and is maximum atthe poles

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Equation of motion with rotation

Considering all forces: general equation of motion per mass

d~v

dt= ~Fp + ~Fc + ~FR + ~FG

resp.d~v

dt= −1

ρ~∇p − 2~Ω× ~v − a~v + ~g

We remember

d~v

dt=∂~v

∂t+(~v · ~∇

)~v

→ nonlinear in ~v → difficult to forecast atmospheric state

In atmospheric dynamics usually spherical coordinates areused what makes look the equations more complicated

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Equation of motion in spherical coordinates

The whole set in spherical coordinates without friction

du

dt=

uv

rtanϕ− uw

r+ 2Ω sinϕv − 2Ω cosϕw − 1

ρr cosϕ

∂p

∂λ

dv

dt= −u2

rtanϕ− uw

r− 2Ω sinϕu − 1

ρr

∂p

∂ϕ

dw

dt=

u2 + v2

r+ 2Ω cosϕu − g − 1

ρ

∂p

∂r

In the Lagrangian frame we have to take care of the total

derivative

d

dt=

∂

∂t+

u

r cosϕ

∂

∂λ+

v

r

∂

∂ϕ+ w

∂

∂z

In order to simplify, terms of minor importance areneglected, depending on case of investigation

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Continuity equationProperty flux

A fluid can transport substances or a property ψ like mass,density, momentum etc. → Flux of property ψ is ~vψExamples:

I mass flux (per volume): ~vρ

I heat flux (per mass): ~vcVT

I momentum flux (per volume): ~vρ~v

Consider ψ-flux in a volume:flux divergence → more ψ flows out than influx convergence → more ψ flux flows in than out

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divergence convergenve

(uψ)2−(uψ)1

∆x > 0 (uψ)2−(uψ)1

∆x < 0

How is ψ in the volume affected? What is dψ/dt

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Continuity equationConservation of mass

Consider densityflux ~vρ(x , y , z) through a volume dx , dy , dz

Mass-flux in y -direction: vρ(y + dy) = vρ(y) + ∂vρ∂y dy

The flux through area dxdz in the cube is vρdxdy

The flux out of the cube is accordingly(vρ+ ∂vρ

∂y dy)dxdz

→ net flux:(vρ+ ∂vρ

∂y dy)dxdz − vρdxdy = ∂vρ

∂y dydxdz

In three dimensions → by analogy for the net mass flux:(∂uρ

∂x+∂vρ

∂y+∂wρ

∂z

)dxdydz

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Continuity equation ctd.What happens to the mass in case of a divergence, i.e. moreflows out than in?Mass is conserved → mass will thus decrease in time.

∂m

∂t=∂ρV

∂t=∂ρ

∂tdxdydz

⇒ decrease in mass is equal to negative of flux divergence

∂ρ

∂tdxdydz = −

(∂uρ

∂x+∂vρ

∂y+∂wρ

∂z

)dxdydz

Divide by volume to obtain continuity equation

∂ρ

∂t= −

(∂uρ

∂x+∂vρ

∂y+∂wρ

∂z

)= −~∇ · ~vρ

Note ∂ρ∂t = −~∇ · ~vρ = −~v · ~∇ρ− ρ~∇ · ~v

∂ρ

∂t+ ~v · ~∇ρ =

dρ

dt= −ρ~∇ · ~v

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Primitive equations

The whole set of equations used to describe atmosphericdynamics is called primitive equations

d~v

dt= −1

ρ~∇p − 2~Ω× ~v − a~v + ~g

the gas lawp = ρRT

the first law of thermodynamics

dT

dt=

1

cpρ

dp

dt+

Q

cp

and the continuity equation

dρ

dt+ ρ~∇ · ~v = 0

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Geostrophic windConsider equation of motion in horizontal direction

d~vhdt

= ~Fp + ~Fch + ~FR

= −1

ρ~∇hp − f ~k × ~vh − a~vh

Above approx. 1 km friction can be neglectedAir parcel starts to move due to pressure gradient → evokesCoriolis force → deviation of track to the right → equilibrium

f ~k × ~vh = −1

ρ~∇hp

Vector multiplication with ~k on the left allows to solve for ~vh⇒ Velocity is called geostrophic wind ~vg

~vg =1

ρ · f~k × ~∇hp

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Geostrophic wind

,

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Geostrophic wind

~vg =1

ρ · f~k × ~∇hp where f = 2Ω sinϕ

vg =1

f ρ

∂p

∂xug = − 1

f ρ

∂p

∂y

Note:

I Geostrophic wind is parallel to isobars

I On northern hemisphere → low pressure system on theleft

I Wind around low pressure system in the same directionas Earth rotation

I The denser the isobars the higher the wind speed

I As geostrophic wind is parallel to isobars (normal togradient) → pressure imbalance can not be changed

I When friction is present → subgeostrophic wind →pressure imbalance can be changed

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Geostrophic wind

Geopotential height at 500 hPa in decameters and winds

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Rossby waves qualitativelyExample of geopotential wave field at 30 mb30 mb Geopotential Height

H L

H HL L

Dec. 28, 1997 90

60 High-LowWave 1patternL

atit

ud

e

30

0180 90 0 90

Longitude

Nov. 18, 199790

60

Lat

itu

de

30

Wave 2pattern

0180 90 0 90

Longitude

As we move from east to west, we observe high-low structures.A single high-low structure is a wave 1 pattern, while 2 high-lowstructures are a wave 2 pattern. Winds tend to follow along a lineof constant geopotential. The units used here are geopotentialkilometers.

from electronic textbook about ozone

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Thermal windFrom geostrophic wind equation together with ideal gas law

fv =1

ρ

∂p

∂x=

RT

p

∂p

∂x= RT

∂ ln p

∂x

From hydrostatic balance

− g

RT=∂ ln p

∂z

Cross differentiation and neglecting vertical variations in Twe get

∂v

∂z≈ g

fT

∂T

∂x

∂u

∂z≈ − g

fT

∂T

∂y

These are the thermal wind equationsThey give relations between horizontal temperaturegradients and vertical gradients of the horizontal wind whenboth geostrophic and hydrostatic balance apply

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Thermal wind

hr

il-r

vv tl (r5

\\

I NJ

\\

l

tr v

oa<

from Dutton: Dynamics of atmospheric motion

T decrease in y

∂u

∂z≈ − g

fT

∂T

∂y

Latitudinal temperature gradient causes an increase in thelatitudinal pressure gradient→ geostrophic wind speed increases with height→ thermal wind is parallel to isotherms

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Thermal windExample: The zonally averaged zonal wind in the lower andmiddle atmosphere of the Earth is close to a thermal wind

January 1979-98 Temperature

-90 -60 -30 0 30 60 90Latitude

0

0 00

00

20

20

40

-40-20

-20

0

0

0

0

20

20

40

280300

220

220

220

220

240

240

260

260

280

240

240

260

260

200

Summer Winter

Polar Vortex

J

JJ

Lowestmost stratosphereLowestmost stratosphere

Tropics

1000

100

10

1

Pre

ssu

re (

hP

a)

0

8

16

24

32

40

48

Hei

gh

t (k

m)

from electronic textbook about ozone

Clearly visible are jet streams and polar night jet JJets are linked with strong gradients in T

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

VorticityIn addition to the primitive equations also equationsdescribing vorticty in a fluid field are of importance

Vorticity, ζ, in a horizontal flow is the vertical component ofthe rotation of the velocity field

ζ = ~∇z × ~v =∂vy∂x− ∂vx∂y

=∂v

∂x− ∂u

∂y

I Vorticity is a measure of the local (not global) rotationor spin of the flow at any point in the flow

I Vorticity therefore is a field-parameterI Vorticity shows up in two cases:

I in flows that are bendingI in straight motion with shear

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

CirculationIn order to determine the direction of rotation in a flow,circulation is used. Circulation is defined as

Z =

∮S~v · d~s

Example: Assume border S of area A is a”rope “. Tangential

components of velocities will act on this rope. If the resultingtangential velocity is not compensated → rope will rotate.

I Z > 0→ counter clock wise rotationI Z < 0→ clock wise rotation

In contrast to vorticity that is defined at each point in aflow, circulation is valid for an area A and its border s.

A relation between ζ and Z is given by the law of Stokes:

Z =

∮S~v · d~s =

∫∫~∇n × ~vdA =

∫∫AζdA

Circulation is a measure of the mean vorticity of an area A

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Some casesSimplest case: v = ω · r → Circulation is then

Z =

∮~v · d~s = v · 2πr = 2πr2ω

and vorticity by division of the area element

ζ =Z

r2π= 2 · ω

In this case vorticity is just two times the angular velocity

In general

v(r+dr)

=v(r)*$.otv(r)

, '.z \ - - 'z\rr2

- t l ) t '

.z ' --4z/

/.- clg

ζ =dZ

dA=

v(r)

r+∂v

∂r

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Vorticity of Jupiter’s red spotimages of the Galileo satellite

Choi et al. (Icarus, 188, 35-46, 2007)

Radius of X is about R=70000 kmFrom image: spot radius ≈ 6 deg→ r ≈ 7400 kmTangential velocity is about v=150 m/s⇒ ζ = Z

r2π= 2 · ω = 2 v

r = ... = 4 · 10−5s−1

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Absolute vorticitySo far we have discussed the local vorticity, ζ. In an inertialreference system we have to add the part stemming from theplanetary rotation, i.e. ω = 2Ω sinϕ = f

→ The absolute vorticity, η, is thus

η = ζ + f

How is η of a two dimensional flow changing with time?

Use general equation of motion in horizontal form

d~vhdt

=∂~vhdt

+ (~vh · ~∇h)~vh = −1

ρ~∇hp − f ~k × ~vh

Separate in horizontal components

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y− fv = −1

ρ

∂p

∂x∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ fu = −1

ρ

∂p

∂y

Build ∂/∂y of first - ∂/∂x of second and rearrange terms

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Vorticity equationWe obtain the vorticity equation

∂ζ

∂t+ u

∂ζ

∂x+ v

∂ζ

∂y+ v

∂f

∂y= 0

As ∂f /∂t = ∂f /∂x = 0, we can write

∂

∂t(ζ + f ) + u

∂

∂x(ζ + f ) + v

∂

∂y(ζ + f ) = 0

Expressed with the absolute vorticity η, we get

∂η

∂t+ ~vh~∇hη =

dη

dt= 0

I η is conserved in a horizontal flow without divergenceI For the relative vorticity

dζ

dt= −v ∂f

∂y= −vβ

Relative vorticity will change as soon there is ameridional velocity component

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Conservation of total vorticity

To leading order, absolute vorticity η = ζ + f is constant:the relative vorticity ζ simply being exchanged with theplanetary vorticity f→ Planetary waves set up

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Rossby wavesConsider the vorticity equation in two dimensions withoutdivergence

dη

dt=∂η

∂t+ ~vh~∇hη = 0

that can be expressed as

∂ζ

∂t+u

∂ζ

∂x+v

∂ζ

∂y+vβ = 0 where β =

∂f

∂y=

2Ω cosφ

r

The vorticity equation is a non linear partial differentialequation and analytically not solvable→ In order to find a solution we linearize (as often inhydrodynamics):

u = u + u′ where u′ u

v = v + v ′ where v ′ v

ζ = ζ + ζ ′ where ζ ′ ζ

¯(...) are mean values and (...)′ are disturbancesWe neglect products of disturbances as they are small

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Rossby wavesThis will lead to a linearized vorticity equation

∂ζ ′

∂t+ u

∂ζ ′

∂x+ u′

∂ζ

∂x+ v

∂ζ ′

∂y+ v ′

∂ζ

∂y+ v ′β = 0

Assume a constant flow in zonal and none in meridionaldirection → u = u0 =const., v = 0 → ζ = ∂v

∂x −∂u∂y = 0

This simplifies the linearized vorticity equation

∂ζ ′

∂t+ u0

∂ζ ′

∂x+ v ′β = 0

As disturbance is only in v , i.e. u′ = 0 → ζ ′ = ∂v ′/∂x .

∂2v ′

∂t∂x+ u0

∂2v ′

∂x2+ v ′β = 0

Ansatz: v ′ only depends on x , no variation in y direction

v ′(x , t) = v0 cos

(2π

L(x − ct)

)v0: constant amplitude, L: wavelength and c : phase speed

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Rossby wavesThis leads to[

4π2

L2c − 4π2

L2u0 + β

]v0 cos

(2π

L(x − ct)

)= 0

Expression in [ ] brackets must be zeroWe finally obtain the famous Rossby equation

c = u0 −βL2

4π2

I Phase speed depends on wavelength → dispersionI Stationary wave for c = 0. Wave does not move

Lstat = 2π

√u0

β

I With u0 ≈ 15m/s and β ≈ 6.3 · 10−11m−1s−1

Lstat ≈ 6300 km

I These waves are called long waves or Rossby waves

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Rossby waves examples

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

Atmospheric physics - an interdisciplinary scienceI Thermodynamics

I hydrostatic equilibrium, lapse ratesI water and its transformations

I RadiationI InsolationI Absorption, emission, scattering → heating, coolingI Spectroscopy, radiative transfer

I Chapman layerI Radiative forcing

I ChemistryI Thermodynamics of chemical reactionsI Chapman model of O3 formationI aerosols, heterogeneous reactions → ozone hole

I DynamicsI Temp. differences→ pressure differences→ fluid motionI Earth is rotating → Coriolis force

I Geostrophic wind, thermal wind

I waves (Rossby, gravity), circulation, vorticity, PV

N.Kampfer

Atmosphericdynamics

Introduction

Equation of motion

continuity equation

Primitive equations

Geostrophic wind

Thermal wind

Vorticity

Circulation

Absolute vorticity

Rossby waves

This is the end

... if you think physics of planetary atmospheres is interestingyou might like to contact me for a master thesis→ room A101