Teimuraz Zaqarashvili

Rossby waves: an introduction

Teimuraz Zaqarashvili

Institute für Geophysik, Astrophysik und Meteorologie

University of Graz, Austria Abastumani Astrophysical Observatory at Ilia state

University, Georgia

Hurricane Sandy

Courtesy to NASA GSFC

Carl-Gustaf Arvid Rossby

Massachusetts Institute

of Technology

University of Chicago

Woods Hole Oceanographic

Institution

Swedish Meteorological and

Born: December 28, 1898 Stockholm, Sweden

Rossby Waves – as seen by Rossby

Platzman 1968

Rossby and collaborators, 1939

The picture is taken from above the South Pole, shows a number of mid latitude cyclones circling Antarctica.

Palmen 1949

• Rossby waves may play a significant role in large-scale dynamics of Earth (and planetary) core, astrophysical discs, solar/stellar atmospheres/interiors, etc.

• Solar/stellar atmospheres/interiors and planetary cores contain magnetic fields.

• Therefore, the hydrodynamic Rossby wave theory should be modified in the presence of large-scale magnetic fields.

Vorticity and magnetic field

( ) ( ) .211 2 Ω=Ω∂

∂=

∂

∂= r

rrru

rrz ϕω

.uω ×∇=

A dynamic variable of preeminent importance in rotating fluid dynamics is vorticity

For a fluid with uniform rotation, the vorticity is

,ru Ω=ϕ

The vorticity vector is nondivergent

.0=⋅∇ ω

( ) .1t

uuuuΔ+∇−=∇⋅−

∂

∂ν

ρp

Momentum equation in fluid dynamics is

Taking its curl gives the vorticity equation

( ) .t

ωωuωΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇−××∇=

∂

∂ν

ρ

p

Electron and proton momentum equations

)(1

),(1

ieeiiiii

i

ieeieeee

e

cenp

dtd

cenp

dtd

uubuEu

uubuEu

−+⎟⎠

⎞⎜⎝

⎛ ×++−∇=

−−⎟⎠

⎞⎜⎝

⎛ ×+−−∇=

αρ

αρ

bjjbuE ×+=∇+×+ee

eie

ei cenne

penc

11122

α

Ohm’s law is obtained from the electron equation

( )ieeen uuj −−= current density

and substituting into Maxwell equation

tc ∂∂

−=×∇bE 1

( ) .t

bbubΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇+××∇=

∂

∂η

ρ

pemc

one gets the induction equation

jbuE 22

11

e

eie

ei ne

penc

α+∇−×−=

Defining electric field as

The vorticity equation is analogous to the induction equation

( ) ( ) .t

ωuωuωωuωΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇−⋅∇−∇⋅=∇⋅+

∂

∂ν

ρ

p

( )pp∇×∇−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇ ρ

ρρ 2

1

The term proportional to

is called the baroclinic term in the vorticity equation and Biermann battery term in the induction equation.

( ) ( ) .t

bububbubΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇+⋅∇−∇⋅=∇⋅+

∂

∂η

ρ

pemc

If the fluid density is constant, or if the density is a function of only a pressure, then the baroclinic term vanishes

( ) ( ) 01122 =∇×∇=∇×∇

ρρρ

ρρ

ρ ddpp

and the vorticity and the induction equations become neglecting Lorentz force and Hall term

( ) ( ) ,t

ωuωuωωuωΔ+⋅∇−∇⋅=∇⋅+

∂

∂ν

( ) ( ) .t

bububbubΔ+⋅∇−∇⋅=∇⋅+

∂

∂η

Using the continuity equation

,t

uωω⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛

ρρDD

.t

ubb⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛

ρρDD

0t

=⋅∇+ uρρ

DD

the vorticity and the induction equations in the ideal fluid become

λρ

∇⋅=Πω

is conserved by each fluid element.

The vorticity equation easily leads to the statement known as Ertel theorem: If λ is some conserved scalar quantity, then the potential vorticity

The same theorem is valid for the magnetic field as

λρ∇⋅=Π

bm

is conserved by each fluid element.

( ) ( ) .t

ωuωuωωuωΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇−⋅∇−∇⋅=∇⋅+

∂

∂ν

ρ

p

( ) ( ) .t

bububbubΔ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∇×∇+⋅∇−∇⋅=∇⋅+

∂

∂η

ρ

pemc

Sum of vorticity and induction equations gives

( ) ( ) uΩuΩΩuΩBBB

B ⋅∇−∇⋅=∇⋅+∂

∂

t

( )uΩΩB

B ∇⋅=dtd

ωbΩB emc

+= is conserved.

The vorticity of the fluid as observed from an inertial, nonrotating frame is called absolute vorticity

Ωωω 2+=a

and this quantity is conserved during the fluid motion on a rotating sphere.

Rossby waves are produced from the conservation of absolute vorticity.

The vorticity of rotating sphere (e.g. Earth) is maximal at poles and tends to zero at the equator.

• As an air parcel moves northward or southward over different latitudes, it experiences change in Earth vorticity.

• In order to conserve the absolute vorticity, the air has to rotate to produce relative vorticity.

• The rotation due to the relative vorticity bring the air back to where it was.

Momentum equation in rotating frame (with angular velocity )

Rossby waves (hydrodynamics)

( ) ,21t

guΩuuu+×−∇−=∇⋅+

∂

∂ pρ

where is the Coriolis force. uΩ×− ρ2

Ω

Ratio of convective and Coriolis terms is called a Rossby number

Ω=LU

Ro

When then the rotation effects are significant. 1Ro <<

is co-latitude, is longitude, g is the acceleration, H is the layer thickness, is the surface elevation, is the angular frequency.

In 18th century, Laplace formulated his “tidal” equations

( ) .0sinsin

,cos2

,sin

cos2

=⎥⎦

⎤⎢⎣

⎡

∂

∂+

∂

∂+

∂

∂

∂

∂−=Ω−

∂

∂

∂

∂−=Ω+

∂

∂

ϕθ

θθ

θθ

ϕθθ

ϕθ

ϕθ

θϕ

uu

RH

th

hRgu

tu

hRgu

tu

θ ϕ

h Ω

( ) ( ) .0

,

,

=∂

∂+

∂

∂+

∂

∂

∂

∂−=+

∂

∂∂

∂−=−

∂

∂

yx

xy

yx

Huy

Huxt

hyhgfu

tu

xhgfu

tu

ϑsin2Ω=f is the Coriolis parameter.

Rectangular coordinates:

θϑ −= 090 is the latitude.

gHc = is the surface gravity speed.

.012

2

2

22

2

2

2 =∂

∂

∂

∂−⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂−⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂

∂

∂

∂

xu

yf

uyx

ftct

yy

These equations can be cast into one equation

If one neglects the surface elevation or 0≈h 1<<Hh

.02

2

2

2

=∂

∂

∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂

∂

∂

xu

yf

uyxt

yy

This approximation eliminates surface gravity (or Poincare) waves and induces small change in Rossby wave dispersion relation.

At this point came up Rossby with his plane approximation −β

When spatial scales of considered process is less than sphere radius then one can expand the Coriolis parameter at a given latitude as

,0 yff β+=

.cos2

constRy

f=

Ω=

∂

∂= ϑβ

.02

2

2

2

=∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂

∂

∂

xu

uyxt

yy β

Fourier analysis of the form leads to the dispersion relation of Rossby (or planetary) waves

)~exp( yikxikti yx ++− ω

.~22yx

x

kkk+

−=β

ω

Rossby waves always propagate in the opposite direction of rotation!

Phase speed

.~xkβ

ω −=

Long wavelength waves propagate faster!

.~

22yxx

ph kkkv

+−==

βω

For purely toroidal propagation

Group speed ( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

−−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

∂

∂= 222222

22 2,

~,

~

yx

yx

yx

xy

yx kk

kk

kk

kkkk

ββωω

gv

If there is a constant zonal flow then the phase speed can be written as (Rossby 1939)

Long wavelength waves propagate westward and short wavelength waves propagate eastward!

It appears that the waves become stationary when

β=1.6 10-11 m-1 s-1 at mid-latitudes.

The ratio of Rossby wave and Earth rotation periods is around 6!

The observed Rossby wave period on the Earth is 4-6 days (Yanai and Maruyama 1966, Wallace 1973, Madden 1979).

For the wavelength of 10000 km, one gets the period of 5.6 days.

Phase speed - 20 m/s.

What does a Rossby wave look like? Recall that ψ is proportional to

the geopotential, or the pressure in the ocean. So a sinusoidal wave is a

sequence of high and low pressure anomalies. An example is shown in

Fig. (13). This wave has the structure:

ψ = cos(x− ωt)sin(y) (122)

(which also is a solution to the wave equation, as you can confirm). This

appears to be a grid of high and low pressure regions.

x

y

Rossby wave, t=0

0 2 4 6 8 10 120

2

4

6

8

10

12

x

t

Hovmuller diagram, y=4.5

0 2 4 6 8 10 120

2

4

6

8

10

12

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 13: A Rossby wave, with ψ = cos(x − ωt)sin(y). The red corresponds to highpressure regions and the blue to low. The lower panel shows a “Hovmuller” diagram ofthe phases at y = 4.5 as a function of time.

The whole wave in this case is propagating westward. Thus if we take a

cut at a certain latitude, here y = 4.5, and plot ψ(x, 4.5, t), we get the plot

54

Courtesy to Lacasce

The red corresponds to high pressure regions and the blue to low. The lower panel shows a “Hovmuller” diagram of the phases at y = 4.5 as a function of time.

Fourier analysis with leads to the equation

( ) ,0~~

21

1 2

22 =⎥

⎦

⎤⎢⎣

⎡ Ω−

−−

∂

∂−

∂

∂θωµµ

µµ

umm

where , m is the toroidal wave number and

)~exp( ϕω imti +−

θµ cos=

When the spatial scale of considered process is longer than the sphere radius then one should use the spherical coordinates

( ) .0sin

,sin1

cos2

,cos2

=∂

∂+

∂

∂

∂

∂−=Ω+

∂

∂∂

∂−=Ω−

∂

∂

ϕθ

θ

ϕθρθ

θθ

ϕθ

θϕ

ϕθ

uu

pR

utu

pRgu

tu

.sin~θθ θuu =

( )1~2

+=Ω

− nnmω

If

then the equation is associated Legendre differential equation, those typical solutions are associated Legendre polynomials

( ),cos~ θθmnPu =

where n-m is a number of nodes along the latitude.

It defines the dispersion relation for spherical Rossby waves (Haurwitz 1940, Longuet-Higgins 1968, Papaloizou and Pringle 1978)

( ).

12~+

Ω−=

nnm

ω

.)1(

2~

+

Ω−==

nnmvph

ω

First consideration of magnetic field effects on Rossby waves was done by Hide (1966).

Rossby waves (magnetohydrodynamics)

He considered incompressible 2D MHD approximation with uniform 2D magnetic field.

Then Gilman (2000) wrote MHD shallow water equations for nearly horizontal magnetic field

u

uBBuB

zug-BBuuu

HH

H

⋅−∇=∂

∂

∇⋅+∇⋅−=∂

∂

×+∇∇⋅+∇⋅−=∂

∂

t

t

ft

Here B and u are horizontal magnetic field and velocity, H is the thickness of the layer, g is the reduced gravity.

The divergence-free condition for magnetic fields is now written as

( ) 0HB =⋅∇

This states simply that at every point the magnetic flux associated with the horizontal magnetic field, which are independent with height, is conserved.

The total magnetic field is made up of horizontal fields independent of the vertical together with a small vertical field that is, like the vertical velocity, a linear function of height, being zero at the bottom and maximum at the top.

Linear equations in x-y plane

xB is the unperturbed horizontal magnetic field.

0

4

4

0 =⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

+∂∂

∂

∂=

∂

∂∂∂

=∂∂

∂∂

−∂

∂=+

∂

∂

∂∂

−∂∂

=−∂∂

yu

xuH

th

xu

Btb

xuB

tb

yhg

xbBfu

tu

xhg

xbBfu

tu

yx

yx

y

xx

x

yxx

y

xxy

x

πρ

πρ

( ) .022222220

22

20

222

20

2

2

2

=⎥⎦

⎤⎢⎣

⎡

∂∂

−−

−−−−+

∂

∂y

Ax

x

Ax

Axx

y uyf

vkk

vkCf

Cvk

kCy

uω

ωωωω

,0 yff β+=

( )[ ] ( )[ ] .02 2220

222220

22220

20

224 =+++−+++− yxAxAxxyxAx kkCvkvkkCkkCfvk βωωω

.cos2

0

0 ΘΩ

=R

β

At a given latitude we can expand the Coriolis parameter as

Away from the equator then we get 0fy <<β

( ) .022222220

22

20

222

20

2

2

2

=⎥⎦

⎤⎢⎣

⎡

−−

−−−−+

∂

∂y

Ax

x

Ax

Axx

y uvk

kvkC

fCvkk

Cyu

ωωβ

ωωω

This equation gives the dispersion relation (Zaqarashvili et al. 2007)

is surface gravity speed. 00 gHC =

πρ4x

ABv = is the Alfvén speed.

Low frequency branch (magnetic Rossby waves):

02222

2 =−+

+ xAyx

x kvkk

kω

βω

For this equation has two different branches

( )2220

20

2yx kkCf ++=ω

0CvA <<

Higher frequency branch (Poincaré waves):

Hide (1966) considered only x-y plane and obtained:

( ) .0sincos~~ 2222

2 =+−+

+ ϑϑωβ

ω lkvlk

kA

The dispersion relation has two solutions: fast (high-frequency) and slow (low-frequency) magnetic Rossby waves.

The magnetic field splits the ordinary HD Rossby waves into fast and slow magnetic Rossby modes!

HD Rossby waves 22

~yx

x

kkk+

−=β

ω

No rotation: 222~xAkv=ω Alfvén waves

No magnetic field:

Low-frequency solution corresponds to slow magnetic Rossby waves ( )

.222

βω yxAx kkvk +≈

High-frequency solution corresponds to fast magnetic Rossby waves.

Zaqarashvili et al. 2007, A&A

Solid lines: fast and slow modes

Triangles: HD Rossby waves

Dashed lines: Alfvén waves

0sin

0

0sinsin

0cos42

4sincos2

0cos42

4sincos2

0

0

002

000

000

=∂

∂+

∂

∂

=∂

∂−

∂

∂

=∂

∂+

∂

∂+

∂

∂

=−∂

∂−

∂

∂+Ω−

∂

∂

=+∂

∂−

∂

∂+Ω−

∂

∂

θθ

ϕ

ϕθθθ

θπρϕπρϕ

θθ

θπρϕπρθ

θθ

θϕ

ϕθ

ϕθ

θϕ

θϕ

ϕϕ

ϕθ

uRB

tb

uRB

tb

uRHu

RH

th

bR

BbR

BhRgu

tu

bR

BbR

BhRgum

tu

0H is the thickness of the layer.

0sin),0,,0( BBBB θφφ ==

Linear MHD equations in rotating frame (spherical coordinates)

where µ=cosθ and m is the toroidal wave number.

( )1R

2R22222

2220 +=

−+Ω nnVmVmm

A

A

ωω

If

then the equation is associated Legendre differential equation, those typical solutions are associated Legendre polynomials

( ).cos~ ϑϑmnPu =

For Fourier analysis with leads to )~exp( ϕω imti +−

( ) .0~~R

2~R21

1 2222

222

2

22 =⎥

⎦

⎤⎢⎣

⎡

−

+Ω−

−−

∂

∂−

∂

∂θω

ω

µµµ

µu

vmvmmm

A

A

0→h

It defines the dispersion relation for spherical magnetic Rossby waves

(Zaqarashvili et al. 2007)

The magnetic field causes the splitting of ordinary HD mode into the fast and slow magnetic Rossby waves.

( )( )( )

.0112

RB

12

220

220

0

2

0

=++−

Ω+

Ω++⎟⎟

⎠

⎞⎜⎜⎝

⎛

Ω nnnnm

nnm

µρωω

In nonmagnetic case it transforms into HD Rossby wave solution

( ).

12 0

+Ω

−=nnm

ω

For slow magnetic Rossby waves

( ).2

12R

B22

0

20

0+−

ΩΩ−=

nnmµρ

ω

Period of particular harmonics depend on the magnetic field strength.

Zaqarashvili et al. 2007

Solid line: slow mode Dashed line: fast mode Dotted line: HD Rossby waves

The solution is presented in terms of spheroidal wave functions

For Fourier analysis with leads to the complicated second order equation, which for the magnetic field profile was solved analytically.

0≠h )~exp( ϕω imti +−

0cossin),0,,0( BBBB θθφφ ==

14

0

22

<<Ω

=gHR

ε1) (strongly stable stratification)

( )ϑεϑ cos,~nmSu =

and the dispersion relation of magnetic Rossby waves is (Zaqarashvili et al. 2009)

( ) ( ).0

11

R4B

12

220

220

0

2

0

=+Ω

+Ω+

+⎟⎟⎠

⎞⎜⎜⎝

⎛

Ω nnm

nnm

πρ

ωω

Hence, the dispersion relation of magnetic Rossby waves depends on the magnetic field structure.

In this case the governing equation is transformed into Weber equation, which has the solution in terms of Hermite polynomials.

14

0

22

>>Ω

=gHR

ε1) (weakly stable stratification)

The dispersion relation for magnetic Rossby waves is (Zaqarashvili et al. 2009)

( ) ( ).0

121

R4B

122

220

220

0

2

0

=+Ω

+Ω+

+⎟⎟⎠

⎞⎜⎜⎝

⎛

Ω ενπρ

ω

εν

ω mm

The solution is concentrated near the equator and hence it describes equatorially trapped waves.

➢ Rossby waves arise due to the conservation of absolute vorticity and govern the large scale dynamics on rotating spheres.

➢ Horizontal magnetic field splits HD Rossby waves into fast and slow modes.

➢ The magnetic field and differential rotation may lead to the instability of magnetic Rossby waves in solar/stellar interiors and in astrophysical discs.

➢ Rossby waves can be important to explain solar/stellar activity variations.

➢ Observed and theoretical periods can be used to probe the dynamo

layers of the Sun and solar-like stars.

Final remarks