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

Born: December 28, 1898 Stockholm, Sweden

Massachusetts Institute of Technology

University of Chicago

Woods Hole Oceanographic Institution

Swedish Meteorological and Rossby Waves – as seen by Rossby

Platzman 1968 Rossby and collaborators, 1939

The picture is taken from above the Palmen 1949 South Pole, shows a number of mid latitude cyclones circling Antarctica. • 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 A dynamic variable of preeminent importance in rotating fluid dynamics is vorticity ω = ∇×u. The vorticity vector is nondivergent ∇ ⋅ω = 0.

For a fluid with uniform rotation, u ϕ = Ω r , the vorticity is

1 ∂ 1 ∂ 2 ωz = (ruϕ )= (r Ω)= 2Ω. r ∂r r ∂r Momentum equation in fluid dynamics is ∂u 1 − (u ⋅∇)u = − ∇p +νΔu. ∂t ρ

Taking its curl gives the vorticity equation

∂ω ⎛ ∇p ⎞ = ∇×(u×ω)− ∇×⎜ ⎟ +νΔω. ∂t ⎝ ρ ⎠ Electron and proton momentum equations

due ⎛ 1 ⎞ ρe = −∇pe − ene ⎜E + ue ×b⎟ −αei (ue − ui ), dt ⎝ c ⎠

dui ⎛ 1 ⎞ ρi = −∇pi + eni ⎜E + ui ×b⎟ +αei (ue − ui ) dt ⎝ c ⎠ Ohm’s law is obtained from the electron equation

1 1 αei 1 E + ui ×b + ∇pe = 2 2 j+ j×b c ene e ne cene

j = −ene (ue − ui ) current density Defining electric field as

1 1 αei E = − ui ×b − ∇pe + 2 2 j c ene e ne and substituting into Maxwell equation 1 ∂b ∇×E = − c ∂t one gets the induction equation

∂b mc ⎛ ∇p ⎞ = ∇×(u×b)+ ∇×⎜ ⎟ +ηΔb. ∂t e ⎝ ρ ⎠ ∂ω ⎛ ∇p ⎞ + (u ⋅∇)ω = (ω⋅∇)u − ω∇ ⋅u − ∇×⎜ ⎟ +νΔω. ∂t ⎝ ρ ⎠

The vorticity equation is analogous to the induction equation

∂b mc ⎛ ∇p ⎞ + (u ⋅∇)b = (b ⋅∇)u − b∇ ⋅u + ∇×⎜ ⎟ +ηΔb. ∂t e ⎝ ρ ⎠

The term proportional to

⎛ ∇p ⎞ 1 p ∇×⎜ ⎟ = − 2 (∇ρ ×∇ ) ⎝ ρ ⎠ ρ is called the baroclinic term in the vorticity equation and Biermann battery term in the induction equation. If the fluid density is constant, or if the density is a function of only a pressure, then the baroclinic term vanishes

1 1 dp (∇ρ ×∇p)= (∇ρ ×∇ρ) = 0 ρ 2 ρ 2 dρ and the vorticity and the induction equations become neglecting Lorentz force and Hall term ∂ω + (u ⋅∇)ω = (ω⋅∇)u − ω∇ ⋅u +νΔω, ∂t

∂b + (u ⋅∇)b = (b ⋅∇)u − b∇ ⋅u +ηΔb. ∂t Using the continuity equation

Dρ + ρ∇ ⋅u = 0 Dt the vorticity and the induction equations in the ideal fluid become

D ⎛ ω ⎞ ⎛ ω ⎞ ⎜ ⎟ = ⎜ ⋅∇⎟u, Dt ⎝ ρ ⎠ ⎝ ρ ⎠

D ⎛ b ⎞ ⎛ b ⎞ ⎜ ⎟ = ⎜ ⋅∇⎟u. Dt ⎝ ρ ⎠ ⎝ ρ ⎠ The vorticity equation easily leads to the statement known as Ertel theorem: If λ is some conserved scalar quantity, then the potential vorticity ω Π = ⋅∇λ ρ is conserved by each fluid element. The same theorem is valid for the magnetic field as b Π = ⋅∇λ m ρ is conserved by each fluid element. ∂ω ⎛ ∇p ⎞ + (u ⋅∇)ω = (ω⋅∇)u − ω∇ ⋅u − ∇×⎜ ⎟ +νΔω. ∂t ⎝ ρ ⎠

∂b mc ⎛ ∇p ⎞ + (u ⋅∇)b = (b ⋅∇)u − b∇ ⋅u + ∇×⎜ ⎟ +ηΔb. ∂t e ⎝ ρ ⎠

Sum of vorticity and induction equations gives

∂ΩB + (u ⋅∇)ΩB = (ΩB ⋅∇)u − ΩB∇ ⋅u ∂t

dΩB = (ΩB ⋅∇)u dt mc Ω = b + ω is conserved. B e The vorticity of the fluid as observed from an inertial, nonrotating frame is called absolute vorticity

ωa = ω + 2Ω

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

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

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.

Rossby waves (hydrodynamics) Momentum equation in rotating frame (with angular velocity Ω ) ∂u 1 + (u ⋅∇)u = − ∇p − 2Ω×u + g, ∂t ρ where − 2 ρ Ω × u is the Coriolis force.

Ratio of convective and Coriolis terms is called a Rossby number U Ro = LΩ

When R o < < 1 then the rotation effects are significant. In 18th century, Laplace formulated his “tidal” equations

∂u g ∂h ϕ + 2Ωcosθu = − , ∂t θ Rsinθ ∂ϕ ∂u g ∂h θ − 2Ωcosθu = − , ∂t ϕ R ∂θ

∂h H ⎡ ∂ ∂uϕ ⎤ + ⎢ (sinθuθ )+ ⎥ = 0. ∂t Rsinθ ⎣∂θ ∂ϕ ⎦

θ is co-latitude, ϕ is longitude, g is the acceleration, H is the layer thickness, h is the surface elevation, Ω is the angular frequency. Rectangular coordinates:

∂u ∂h x − fu = −g , ∂t y ∂x

∂u y ∂h + fu = −g , ∂t x ∂y ∂h ∂ ∂ + (Hux )+ (Hu y )= 0. ∂t ∂x ∂y f = 2Ωsinϑ is the Coriolis parameter.

ϑ = 900 −θ is the latitude. These equations can be cast into one equation

2 2 2 ∂ ⎡ 1 ⎛ ∂ ⎞ ⎛ ∂ ∂ ⎞⎤ ∂f ∂u y ⎜ f 2 ⎟ ⎜ ⎟ u 0. ⎢ 2 ⎜ 2 + ⎟ − ⎜ 2 + 2 ⎟⎥ y − = ∂t ⎣c ⎝ ∂t ⎠ ⎝ ∂x ∂y ⎠⎦ ∂y ∂x c = gH is the surface gravity speed. h If one neglects the surface elevation h ≈ 0 or <<1 H 2 2 ∂ ⎛ ∂ ∂ ⎞ ∂f ∂u y ⎜ ⎟u 0. ⎜ 2 + 2 ⎟ y + = ∂t ⎝ ∂x ∂y ⎠ ∂y ∂x

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 β − plane approximation

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

f = f0 + βy,

∂f 2Ω β = = cosϑ = const. ∂y R

2 2 ∂ ⎛ ∂ ∂ ⎞ ∂u y ⎜ ⎟u 0. ⎜ 2 + 2 ⎟ y + β = ∂t ⎝ ∂x ∂y ⎠ ∂x ~ Fourier analysis of the form e x p ( − i ω t + i k x x + i k y y ) leads to the dispersion relation of Rossby (or planetary) waves

~ βkx ω = − 2 2 . kx + k y Rossby waves always propagate in the opposite direction of rotation! β For purely toroidal propagation ω~ = − . kx ω~ β Phase speed v ph = = − 2 2 . kx kx + k y Long wavelength waves propagate faster!

~ ~ ⎛ 2 2 ⎞ ⎛ ∂ω ∂ω ⎞ k y − kx 2kxk y Group speed v = ⎜ , ⎟ = ⎜− β , β ⎟ g ⎜ ⎟ 2 2 2 2 2 2 ∂kx ∂k y ⎜ k k k k ⎟ ⎝ ⎠ ⎝ ( x + y ) ( x + y ) ⎠ If there is a constant zonal flow then the phase speed can be written as (Rossby 1939)

It appears that the waves become stationary when

Long wavelength waves propagate westward and short wavelength waves propagate eastward! β=1.6 10-11 m-1 s-1 at mid-latitudes.

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

Phase speed - 20 m/s.

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

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

Rossby wave, t=0 0.8 12

0.6

10 0.4

8 0.2

y 0 6

−0.2 4 The red corresponds to −0.4

2 high pressure regions and −0.6

0 −0.8 the blue to low. The 0 2 4 6 8 10 12 x lower panel shows a “Hovmuller” diagram of Hovmuller diagram, y=4.5 1 the phases at y = 4.5 as a 12 0.8 function of time. 10 0.6

0.4 8

0.2

t 6 0

4 −0.2

−0.4 2 −0.6 Courtesy to Lacasce

0 −0.8 0 2 4 6 8 10 12 x

Figure 13: A Rossby wave, with ψ = cos(x ωt)sin(y).Theredcorrespondstohigh 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.

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

∂u g ∂p θ − 2Ωcosθu = − , ∂t ϕ R ∂θ ∂u 1 ∂p ϕ + 2Ωcosθu = − , ∂t θ ρRsinθ ∂ϕ ∂ ∂u (sinθu )+ ϕ = 0. ∂θ θ ∂ϕ

Fourier analysis with e x p ( − i ω ~ t + i m ϕ ) leads to the equation

⎡ ∂ ∂ m2 2mΩ⎤ 1 2 u~ 0, ⎢ ( − µ ) − 2 − ~ ⎥ θ = ⎣∂µ ∂µ 1− µ ω ⎦ ~ where µ = c o s θ , m is the toroidal wave number and uθ = sinθuθ . 2mΩ If − = n(n +1) ω~ then the equation is associated Legendre differential equation, those typical solutions are associated Legendre polynomials

~ m uθ = Pn (cosθ ), 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) 2mΩ ω~ = − . n(n +1) ω~ 2Ω v = = − . ph m n(n +1) Rossby waves (magnetohydrodynamics) First consideration of magnetic field effects on Rossby waves was done by Hide (1966). 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 = −u⋅∇u+B⋅∇B- g∇H + fu× z ∂t ∂B = −u⋅∇B+B⋅∇u ∂t ∂H = −∇⋅Hu ∂t 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

∇ ⋅(HB)= 0

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 ∂u B ∂b ∂h x − fu = x x − g ∂t y 4πρ ∂x ∂x

∂uy B ∂by ∂h + fu = x − g ∂t x 4πρ ∂x ∂y ∂b ∂u x = B x ∂t x ∂x ∂b ∂u y = B y ∂t x ∂x

∂h ⎛ ∂u ∂uy ⎞ H x 0 + 0 ⎜ + ⎟ = ∂t ⎝ ∂x ∂y ⎠

Bx is the unperturbed horizontal magnetic field. 2 2 2 2 2 2 ∂ uy ⎡ω k v ω f k ω ∂f ⎤ k 2 x A x u 0. 2 + ⎢ 2 − x − 2 − 2 2 2 2 − 2 2 2 ⎥ y = ∂y ⎣C0 C0 C0 (ω − kx vA ) ω − kx vA ∂y ⎦

C0 = g H 0 is surface gravity speed.

Bx vA = is the Alfvén speed. 4πρ At a given latitude we can expand the Coriolis parameter as

2Ω0 f = f0 + βy, β = cosΘ. R0 Away from the equator β y < < f 0 then we get 2 2 2 2 2 2 ∂ uy ⎡ω k v ω f k ωβ ⎤ k 2 x A x u 0. 2 + ⎢ 2 − x − 2 − 2 2 2 2 − 2 2 2 ⎥ y = ∂y ⎣C0 C0 C0 (ω − kx vA ) ω − kx vA ⎦ This equation gives the dispersion relation (Zaqarashvili et al. 2007)

4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ω − [2kx vA + f0 + C0 (kx + k y )]ω − C0 kxβω + kx vA[kx vA + C0 (kx + k y )]= 0. For v A < < C 0 this equation has two different branches Higher frequency branch (Poincaré waves):

2 2 2 2 2 ω = f0 + C0 (kx + k y ) Low frequency branch (magnetic Rossby waves):

2 kxβ 2 2 ω + 2 2 ω − vAkx = 0 kx + k y

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

~ 2 kβ ~ 2 2 ω + ω − vA (k cosϑ + l sinϑ) = 0. k 2 + l 2 The dispersion relation has two solutions: fast (high-frequency) and slow (low-frequency) magnetic Rossby waves.

High-frequency solution corresponds to fast magnetic Rossby waves.

Low-frequency solution corresponds to slow magnetic Rossby waves 2 2 2 kxvA (kx + k y ) ω ≈ . β

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

k β No magnetic field: ~ x HD Rossby waves ω = − 2 2 kx + k y ~ 2 2 2 No rotation: ω = vAkx Alfvén waves Solid lines: fast and slow modes Triangles: HD Rossby waves Dashed lines: Alfvén waves Zaqarashvili et al. 2007, A&A Linear MHD equations in rotating frame (spherical coordinates)

B = (0, Bφ ,0), Bφ = sinθB0 ∂u g ∂h B ∂b B θ − 2mΩ cosθu + sinθ − 0 ϕ + 2 0 cosθb = 0 ∂t 0 ϕ R ∂θ 4πρR ∂ϕ 4πρR ϕ ∂u g ∂h B ∂b B ϕ − 2Ω cosθu + sinθ − 0 ϕ − 2 0 cosθb = 0 ∂t 0 θ R ∂ϕ 4πρR ∂ϕ 4πρR θ ∂h H ∂u H ∂u sin 2 θ + 0 sinθ θ + 0 ϕ = 0 ∂t R ∂θ R ∂ϕ ∂b B ∂u θ − 0 ϕ = 0 ∂t R ∂ϕ ∂b B ∂u ϕ + 0 sinθ θ = 0 ∂t R ∂θ

H0 is the thickness of the layer. For h → 0 Fourier analysis with e x p ( − i ω ~ t + i m ϕ ) leads to ⎡ ∂ ∂ m2 2mΩR 2ω~ + 2m2v2 ⎤ 1 2 A u~ 0. ⎢ ( − µ ) − 2 − 2 ~ 2 2 2 ⎥ θ = ⎣∂µ ∂µ 1− µ R ω − m vA ⎦ where µ=cosθ and m is the toroidal wave number.

2mΩ R 2ω + 2m2V 2 If 0 A 2 2 2 2 = n(n +1) R ω − m VA then the equation is associated Legendre differential equation, those typical solutions are associated Legendre polynomials

~ m uϑ = Pn (cosϑ). It defines the dispersion relation for spherical magnetic Rossby waves (Zaqarashvili et al. 2007) 2 2 2 ⎛ ω ⎞ 2m ω B0m 2 − n(n +1) ⎜ ⎟ 0. ⎜ ⎟ + + 2 2 = ⎝ Ω0 ⎠ n(n +1)Ω0 µρΩ0R n(n +1) The magnetic field causes the splitting of ordinary HD mode into the fast and slow magnetic Rossby waves. In nonmagnetic case it transforms into HD Rossby wave solution 2mΩ ω = − 0 . n(n +1) For slow magnetic Rossby waves 2 B0 2 − n(n +1) ω = −mΩ0 2 2 . µρΩ0R 2 Period of particular harmonics depend on the magnetic field strength. Solid line: slow mode Dashed line: fast mode Dotted line: HD Rossby waves

Zaqarashvili et al. 2007 ~ For h ≠ 0 Fourier analysis with e x p ( − i ω t + i m ϕ ) leads to the complicated second order equation, which for the magnetic field profile B = ( 0 , B φ , 0 ) , B φ = s i n θ c o s θ B 0 was solved analytically.

4Ω2 R2 1) ε = < < 1 (strongly stable stratification) gH 0 The solution is presented in terms of spheroidal wave functions ~ uϑ = Snm (ε,cosϑ) and the dispersion relation of magnetic Rossby waves is (Zaqarashvili et al. 2009) 2 ⎛ ω ⎞ 2m ω B2m2 1 ⎜ ⎟ 0 0. ⎜ ⎟ + + 2 2 = ⎝ Ω0 ⎠ n(n +1)Ω0 4πρΩ0R n(n +1) Hence, the dispersion relation of magnetic Rossby waves depends on the magnetic field structure. 4Ω2 R2 1) ε = > > 1 (weakly stable stratification) gH 0

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

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

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

2 ⎛ ω ⎞ 2m ω B2m2 1 ⎜ ⎟ 0 0. ⎜ ⎟ + + 2 2 = ⎝ Ω0 ⎠ (2ν +1) ε Ω0 4πρΩ0R (2ν +1) ε Final remarks

➢ 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.