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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Rossby waves in rotating magnetized fluids Teimuraz Zaqarashvili temury@genao.org Solar Physics Group Abastumani Astrophysical Observatory at Chavchavadze State University (Georgia)

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory The outline Introduction Vorticity and magnetic field Hydrodynamic Rossby waves Magnetic Rossby waves Final remarks

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Why Rossby waves? Hurricane Katrina Hurricane Elena

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory For most of our purposes the fluid can be considered as incompressible. The momentum equation in the frame rotating with constant angular velocity Main equation

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Vorticity A dynamic variable of preeminent importance in rotating fluid dynamics is vorticity For a fluid with uniform rotation the vorticity is

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Vorticity The vorticity of the fluid as observed from an inertial, nonrotating frame is called absolute vorticity The vorticity vector is nondivergent

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School on Space Plasma Physics August 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

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

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

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

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Vorticity The vorticity equation Hydrodynamic case The vorticity equation is analogous to the induction equation for magnetic field

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Vorticity Using the continuity equation we have Hydrodynamic case

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

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Vorticity However, if we have the magnetic field, then in general, the potential vorticity is not conserved But:

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Magnetic field If is a conserved quantity for each fluid element i.e. Then the potential induction is conserved by each fluid element i.e.

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Hydrodynamic Rossby waves Shallow water theory The fundamental parametric condition which characterizes shallow water theory is

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Hydrodynamic Rossby waves Following the fluid motion the potential vorticity 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.

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School on Space Plasma Physics August 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 The beta-plane

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Hydrodynamic Rossby waves The beta-plane The dispersion relation of Rossby waves is

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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)

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Magnetic Rossby waves MHD Shallow water equations Here B and u are horizontal magnetic field and velocity, H is the thickness of the layer, g is the reduced gravity.

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School on Space Plasma Physics August 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 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.

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School on Space Plasma Physics August 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. Rectangular case The magnetic field is supposed to be directed along the x axis. The Fourier analyses with exp(-iωt +k x x +k y y) gives the dispersion relation Zaqarashvili, Oliver and Ballester, 2007

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School on Space Plasma Physics August 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 Poincar é waves with dispersion relation

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School on Space Plasma Physics August 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. For larger wave lengths, this equation has two different solutions.

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 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 and can 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.

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Magnetic Rossby waves Rectangular case, numerical dispersion diagram

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Magnetic Rossby waves We use an unperturbed toroidal magnetic field it has a maximum value at the equator and tends to zero at the poles. The solar magnetic field can be approximated by However, analytical dispersion relation hardly be obtained in this case.

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School on Space Plasma Physics August 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 here μ=cosθ, s is the longitudinal wave number, R is the radius of sphere, V A is the Alfv é n speed.

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School on Space Plasma Physics August 31-September 7, Sozopol Abastumani Astrophysical Observatory Magnetic Rossby waves If then the equation is the associated Legendre differential equation, those typical solutions are associated Legendre polynomials

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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.

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School on Space Plasma Physics August 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!

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School on Space Plasma Physics August 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.

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

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