School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Rossby waves in rotating magnetized fluids Teimuraz Zaqarashvili [email protected]Solar Physics Group Abastumani Astrophysical Observatory at Chavchavadze State University (Georgia)
50
Embed
School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Rossby waves in rotating magnetized fluids Teimuraz.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Solar Physics GroupAbastumani Astrophysical Observatory at Chavchavadze State University (Georgia)
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
The outline
IntroductionVorticity and magnetic fieldHydrodynamic Rossby wavesMagnetic Rossby wavesFinal remarks
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Why Rossby waves?
Hurricane Katrina Hurricane Elena
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Vorticity
The vorticity of the fluid as observed from an inertial, nonrotating frame is called absolute vorticity
2aThe vorticity vector is nondivergent
0 a
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Vorticity
Using the continuity equation we have
.1F
Vdt
d3
paa
Hydrodynamic case
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Vorticity
However, if we have the magnetic field, then
in general, the potential vorticity is not conserved
!0dt
d
But:
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
Shallow water theory
The fundamental parametric condition which characterizes shallow water
theory is
.1L
D
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
Following the fluid motion the potential vorticity
H
fs
Shallow water theory
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
The beta-plane
The dispersion relation of Rossby waves is
.22yx
x
kk
k
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Hydrodynamic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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)
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
Rectangular case, numerical dispersion diagram
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Magnetic Rossby waves
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Applications: solar tachocline
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Applications: solar tachocline
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!
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
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.
School on Space Plasma PhysicsAugust 31-September 7, Sozopol
Abastumani Astrophysical Observatory
Future study
Instability of magnetic Rossby waves due to the differential rotation.