School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory...

School on Space Plasma PhysicsAugust 31-September 7, Sozopol

Abastumani Astrophysical Observatory

Rossby waves in rotating magnetized fluids

Teimuraz Zaqarashvili

[email protected]

Solar Physics GroupAbastumani Astrophysical Observatory at Chavchavadze State University (Georgia)



The outline

IntroductionVorticity and magnetic fieldHydrodynamic Rossby wavesMagnetic Rossby wavesFinal remarks



Why Rossby waves?

Large-scale dynamics of the Earth’s atmosphere and oceans are governed by Rossby waves.

Rossby waves arise due to the latitudinal variation of Coriolis parameter.

Cyclon/Anticyclon and Hurricanes are due to the Rossby waves.



Why Rossby waves?

Hurricane Katrina Hurricane Elena



Why Rossby waves?

The Rossby waves can play major role in large-scale dynamics of stellar atmospheres and interiors.

However, the stellar interiors contain magnetic fields.

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



BBVV

)(

4

12

p

dt

d

For most of our purposes the fluid can be considered as incompressible.

0 V

The momentum equation in the frame rotating with constant angular velocity

Main equation



Vorticity

2)( r

V

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

For a fluid with uniform rotation the vorticity is



Vorticity

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

2aThe vorticity vector is nondivergent

0 a



Vorticity

A vortex line or vortex filament is a line in the fluid which at each point is parallel to the vorticity vector.

A vortex tube is formed by the surface consisting of the vortex filaments, which pass through a closed curve



Magnetic field

.0B

The magnetic field vector is nondivergent

A magnetic filed line or magnetic field filament is a line in the fluid which at each point is parallel to the magnetic field vector.

A magnetic tube is formed by the surface consisting of the magnetic field lines, which pass through a closed curve.



Vorticity

dr,VdAn CA

aa

dA.n aa

The absolute strength or flux of a vortex tube is

The circulation of the velocity around the closed contour C is

where n is the outward normal to the surface element dA.

Hydrodynamic case



Vorticity

.0dt

d

a

If fluid is barotropic on C then absolute circulation is conserved following the motion

Kelvin theorem

Hydrodynamic case

If the surfaces of the constant pressure and the constant density coincide i.e. the state of fluid is termed barotropic.,0 p



Vorticity

F

dt

d2

p

aa VV ωωω

The vorticity equation

Hydrodynamic case

The vorticity equation is analogous to the induction equation for magnetic field

BBBB 2

dt

d VV



Vorticity

The rate of change of the relative vorticity is equal to the sum of

1). the production of vorticity by baroclinicity,

2). the diffusive effects of friction,

3). the vortex-tube stretching, which alters the vorticity parallel to the filament by convergence of the filaments,

4). the vortex tilting by the variation, along the filament, of the velocity component perpendicular to the filaments.



Vorticity

Using the continuity equation we have

.1F

Vdt

d3

paa

Hydrodynamic case



Vorticity

0dt

d

2

Ertel theorem:

Hydrodynamic case

Consider some scalar which is a conserved quantity for each fluid element i.e.

Then if the friction force is negligible and either the fluid is barotropic then the potential vorticity

is conserved by each fluid element i.e.

.0dt

d



Vorticity

However, if we have the magnetic field, then

in general, the potential vorticity is not conserved

!0dt

d

But:



Magnetic field

0dt

d

BB

If is a conserved quantity for each fluid element i.e.

Then the potential induction

is conserved by each fluid element i.e.

.0dt

d

B



Hydrodynamic Rossby waves

Earth vorticity is a function of latitude.

The higher the latitude, the greater the vorticity.

Earth vorticity is zero at the equator.

Earth (planetary) vorticity




Air which rotates in the direction of Earth’s rotation is said to exhibit positive vorticity.

Air which spins oppositely exhibit negative vorticity.

Relative vorticity




Rossby waves are produced from the conservation of absolute vorticity.

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.

Vorticity and Rossby wave




The density is considered to be constant and uniform.

Total body force g is directed antiparallel to the vertical axis.

The fluid is assumed inviscid.

The parameter D characterizes the average depth of the layer as well as vertical scale of the motion.

Similarly there exists a characteristic horizontal length scale for the motion L.

Shallow water theory





The fundamental parametric condition which characterizes shallow water

theory is

.1L

D




Following the fluid motion the potential vorticity

H

fs


is conserved. Here ς is z-component of relative vorticity and

is the Coriolis parameter.

If H increases the absolute vorticity must decrease to keep potential vorticity constant.

sin2f




If the scale of the motion is sufficiently small in north-south direction then locally flat Cartesian system can be used in which the only effect of the Earth’s sphericity is the variation of the Coriolis parameter with latitude

.cos2

,sin2

,

00

00

0

R

f

yff

The beta-plane




The beta-plane

The dispersion relation of Rossby waves is

.22yx

x

kk

k




When wave motions are at the scale of the radius, then

the spherical coordinate system (r,θ,φ) should be considered.

In this case the dispersion relation of Rossby waves is

Spherical coordinates

where n and s are poloidal and toroidal wave numbers.

,)1(

2 0 snn



Magnetic Rossby waves

In MHD shallow water theory the basic principles of HD is retained, but in addition nearly horizontal magnetic field exists within the thin layer.

The density is considered to be constant and medium is incompressible.

Horizontal velocity and magnetic field are independent of the vertical coordinate.

Then the MHD analog to classical HD shallow water equations are:

MHD Shallow water equations (Gilman 2000)




MHD Shallow water equations

u

uBBuB

zug-BBuuu

HH

H

t

t

ft

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




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

MHD Shallow water equations

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.




If wave lengths are less than the radius of sphere than the local Cartesian frame (x,y,z) can be considered.

02 2220

222220

22220

20

224 yxAxAxxyxAx kkCvkvkkCkkCfvk

Rectangular case

The magnetic field is supposed to be directed along the x axis.

The Fourier analyses with exp(-iωt +kxx +kyy) gives the dispersion relation

gHC 0

Zaqarashvili, Oliver and Ballester, 2007




In the case of small Alfven speed i.e.

Rectangular case

which is the case in the stellar interior, the high-frequency branch

contains Poincaré waves with dispersion relation

0CvA

2220

20

2yx kkCf




The lower frequency branch yields the dispersion relation

Rectangular case

This equation describes the magnetic Rossby waves.

02222

2

xAyx

x kvkk

k

For larger wave lengths, this equation has two different solutions.




Higher frequency solution corresponds to the HD Rossby waves i.e.

Rectangular case

And the lower frequency solution yields

222yxAx kkvk

22yx

x

kk

k




Hence, the horizontal magnetic field causes the splitting of ordinary large-scale Rossby waves propagating in opposite directions.

Rectangular case

The high frequency mode has the properties of HD Rossby wave andcan be called as fast magnetic Rossby mode.

But, additionally, a lower frequency mode arises whose frequency is significantly smaller than that of Rossby and Alfven waves at the same spatial scales.

Due to the small frequency it can be called slow magnetic Rossby mode.




The phase speed of the mode in the x direction depends on both Alfven speed and the β parameter

Rectangular case

The phase speed is different from Alfven and Rossby phase speeds, which again indicates different nature of this wave mode.

222yxA

xph

kkv

kv




Rectangular case, numerical dispersion diagram




We consider magnetized fluid on a sphere rotating with angular velocity Ω. We used spherical coordinate system system (r,θ,φ), where r is the distance from the center, θ is the co-latitude and φ is the longitude.

r

Zaqarashvili, Oliver and Ballester, 2007




We use an unperturbed toroidal magnetic field

,sinBB 0

it has a maximum value at the equator and tends to zero at the poles.

The solar magnetic field can be approximated by

,cossinBB 0 However, analytical dispersion relation hardly be obtained in this case.




After Fourier analysis by expi(-ωt +sφ), the equation governing

the dynamics of linear magnetic Rossby waves is

0R

2R2

11

2222

222

2

22

u

Vs

Vsss

A

A

here μ=cosθ, s is the longitudinal wave number, R is the radius of

sphere, VA is the Alfvén speed.




1R

2R22222

222

nnVs

Vss

A

A

If

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

cossnPu




This defines the dispersion relation for spherical magnetic Rossby waves

In nonmagnetic case this equation reduces to the HD Rossby wave

solution.

The magnetic field causes the splitting of ordinary HD mode into

the fast and slow magnetic Rossby waves.

.0

1

12

R4

B

1

222

220

2

nn

nns

nn

s




The dispersion relations for lower order harmonics of fast and slow

magnetic Rossby waves are

.

2

12

R

,2

12

R1

2

2

2

2

2

nnsV

nnsV

nn

s

As

Af




The dependence of wave frequency on the poloidal wave number n. The continuous and dashed lines are the solutions for slow and fast magnetic Rossby waves, respectively. The dotted line is the solution for HD Rossby wave solution.




Frequencies of then s=1, n=2 harmonics of fast (dashed) and slow (continuous) magnetic Rossby waves vs the ratio of the Alfven speed to the rotation rate.



Applications: solar tachocline

Helioseismic observations suggest that the radiative zone rotates uniformly with both latitude and radius.

The convection zone has strong differential rotation with latitudes and almost uniform rotation with radius.




There is a thin transition layer at the base of convection zone called tachocline.

The tachocline is very important for magnetic field generation and storage; and also for the transport of angular momentum.

The tachocline contains large-scale toroidal magnetic field, therefore magnetic Rossby wave theory can be successfully applied here.




However, the differential rotation of the tachocline may lead to the instability of magnetic Rossby waves.

The instability may lead to the periodic emergence of magnetic flux towards the surface.

The magnetic Rossby waves can be of importance for the explanation of intermediate term periodicities in the solar activity!



Final remarks

Horizontal, large scale magnetic field on the rotating sphere causes the splitting of ordinary Rossby waves into fast and slow magnetic Rossby waves.

Fast magnetic Rossby waves are similar to HD Rossby waves.

Slow magnetic Rossby waves are new wave modes and their frequencies are very low compared to the rotation period.



Future study

Instability of magnetic Rossby waves due to the differential rotation.

Tachoclines of solar and solar like stars.

Interiors of giant planets.

Magnetized galactic and accretion discs.

School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory...

Documents

Transcript of School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory...