Scalar and photon masses are equal
The equality of the radial extension with the Rossby radius for the tropical cyclone Florin Spineanu and Madalina Vlad National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
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The asymptotic states of the atmospheric vortices are determined by universal extremum-action constraints acting on the vorticity distribution Large scale flows are dominated by the ideal vorticity dynamics. The thermodynamic processes provide the energy necessary to cover the friction loss. This is relatively small compared with the two energies: heat flow and mechanical motion.
Content • The 2D discrete systems and the field theory formalism • 2D water • planetary atmosphere (2D quasi-geostrophic) – tropical cyclone; relationships vθmax , Rmax , rvθmax – Rmax ∼ Rossby radius • absolute minimum: annular vortices • Related subjects: Concentration of vorticity; Contour Dynamics; statistics of turbulence; etc.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
Scalar and photon masses are equal
Main idea : The conservation laws are not able to identify the natural states. The action can provide solutions. Quasi-coherent structures are observed in • fluids (in oceans and in laboratory experiments) • plasma (confined in strong magnetic field) • planetary atmosphere (2D quasi-geostrophic) • non-neutral plasma (crystals of vortices)
There are common features suggesting to develop models based on the self-organization of the vorticity field. The fluids evolve at relaxation precisely to a subset of stationary states. Decoupling between thermodynamic and mechanics (vorticity dynamics) with the latter becoming dominant
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Coherent structures in fluids and plasmas (numerical 1)
∂∇2 ⊥ ψ +−∇ ψ × n · ∇⊥ ∇2 ⊥ ⊥ψ = 0 ∂t
D. Montgomery, W.H. Matthaeus, D. Martinez,
S.
Phys.
Oughton, Fluids
A4
(1992) 3. Numerical simulations of the Euler equation.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Equivalence with discrete models We will try to write Lagrangians not directly for fluids and plasmas but for equivalent discrete models.
An equivalent discrete model for the Euler equation drki ∂ = εij j dt ∂rk
N
ωn G (rk − rn ) , i, j = 1, 2 , k = 1, N
(1)
n=1,n=k
the Green function of the Laplacian |r − r | 1 ln G r, r ≈ − 2π L
(2)
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
An equivalent discrete model for the plasma in strong magnetic field and for the planetary atmosphere The equations of motion for the vortex ωk at (xk , yk ) under the effect of the others are dxk dt dyk −2πωk dt
−2πωk
where W = π
N N
i=1 j=1 ωi ωj K0 i=j
= =
∂W ∂yk ∂W − ∂xk
(m |ri − rj |)
Physical model → point-like vortices → field theory. The field theory is defined by the Lagrangian L = Lmatter + LCS + Linteraction
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
The water Lagrangian 2D Euler fluid: Non-Abelian SU (2), Chern-Simons, 4th order L
=
2 (3) −εμνρ T r ∂μ Aν Aρ + Aμ Aν Aρ + 3
1 1 2 † † † +iT r Ψ D0 Ψ − T r (Di Ψ) Di Ψ + T r Ψ , Ψ 2 4
where Dμ Ψ = ∂μ Ψ + [Aμ , Ψ] The equations of motion are 1 2 1
† iD0 Ψ = − D Ψ − Ψ, Ψ , Ψ 2 2
(4)
i Fμν = − εμνρ J ρ 2
(5)
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
Scalar and photon masses are equal
The Hamiltonian density is
1 2 1 H = T r (Di Ψ)† (Di Ψ) − T r Ψ† , Ψ 2 4
8
(6)
Using the notation D± ≡ D1 ± iD2
1 † † † † T r (Di Ψ) (Di Ψ) = T r (D− Ψ) (D− Ψ) + T r Ψ Ψ, Ψ , Ψ 2 † 1 Then the energy density is H = 2 T r (D− Ψ) (D− Ψ) ≥ 0 and the Bogomol’nyi inequality is saturated at self-duality D− Ψ = 0
† ∂+ A− − ∂− A+ + [A+ , A− ] = Ψ, Ψ
(7) (8)
We obtain the equation Δψ + γ sinh (βψ) = 0, which is known to be the eq. describing the asymptotic states of the 2D ideal fluid.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
Scalar and photon masses are equal
The Lagrangian of 2D plasma in strong magnetic field: Non-Abelian SU (2), Chern-Simons, 6th order • gauge field, with “potential” Aμ , (μ = 0, 1, 2 for (t, x, y)) described by the Chern-Simons Lagrangean; • matter (“Higgs” or “scalar”) field φ described by the covariant kinematic Lagrangean (i.e. covariant derivatives, implementing the minimal coupling of the gauge and matter fields) † • matter-field self-interaction given by a potential V φ, φ with 6th power of φ; • the matter and gauge fields belong to the adjoint representation of the algebra SU (2)
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
L
=
2 −κεμνρ tr ∂μ Aν Aρ + Aμ Aν Aρ 3
−tr (Dμ φ)† (Dμ φ) † −V φ, φ
(9)
Sixth order potential
† 1 † 2 † 2 tr φ, φ φ φ . , φ − v φ, φ , φ − v V φ, φ† = 4κ2 (10) The Euler Lagrange equations are Dμ Dμ φ =
∂V ∂φ†
−κενμρ Fμρ = iJ ν
(11) (12)
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
The energy can be written as a sum of squares. The self-duality eqs. D− φ
=
F+−
=
0
1 2 † † ± 2 v φ − φ, φ , φ , φ κ
(13)
The algebraic ansatz : in the Chevalley basis φ = φ1 E+ + φ2 E− and A+ = aH, A− = −a∗ H. This simplest form of the equation governing the stationary states
Δψ +
1 sinh ψ (cosh ψ − 1) = 0 2
The ’mass of the photon’ is m =
v2 κ
=
1 ρs
where κ ≡ cs and v 2 ≡ f0 .
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
Scalar and photon masses are equal
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The exact solution has the morphology of the tropical cyclone
v
θ
The tangential component of the velocity, vθ, center is (0,0)
0 0.5 0.4
0.5
0.3 0.2 0.1 0 0
−0.1 −0.2 −0.3 y
−0.4 −0.5
x
−0.5
Figure 1: The solution and the image from a satelite. The solution reproduces the eye radius, the radial extension and the vorticity magnitude.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Scaling relationships between main parameters of the tropical cyclone eye-wall radius, maximum tangential wind, maximum radial extension
90 80
0.25
0.2 max
60 /R
50
0.15 θ max
2
vmax and (e /2)*[α e
1/L
−1]
70
r
v
40
0.1
θ
30 20
0.05 10 0 0
1
max vθ (L)
2
e2 2
3 L
α exp
4
5
√ 2 Rmax
0 0
6
−1
0.5
1
1.5
2 L
2.5
3
3.5
4
rvmax Rmax 1 θ 1 − exp − = Rmax 4 2
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Few remarkable hurricanes Table 1: Comparison between calculated and respectively observed magnitudes of the maximum tangential wind for four cases of tropical cyclones Name
Input (obs) phys Rmax
(km)
rvmax θ
Rmax
Calculated L
Observed
Rossby ρR
(vθmax )
(vθmax )
(km)
(m/s)
(m/s)
Andrew
120
0.1
0.72
117.85
64.31
68
Katrina
300
0.111
0.83
212
88.6
77.8
Rita
350
0.125
0.98
252.47
77.5
77.8
Diana
160
0.1125
0.845
133.81
56.86
55
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Profile of the azimuthal wind velocity vθ (r) 40 35 30
vθ(r) (m/s)
25 20 15 10 5 0 0
2
4
6 r (m)
8
10
12 4
x 10
Comparison between the Holland’s empirical model for vθ (continuous line) and our result (dotted line).
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Relationship between the maximum radial extension of the atmospheric vortex and the Rossby radius The field theoretical formalism defines two fields: • the scalar (”matter”) field φ; its particle is the Higgs scalar; the mass of the particle is mH . The mass mH is the inverse of the characteristic range of spatial decay of the field φ solution, i.e. the inverse of the radius of characteristic decay of the vortex flow mH =
1 Rmax
• the gauge field Aμ whose particle is the ”photon”. For atmosphere the photon has a mass, mphoton , which is the inverse of the characteristic range of spatial decay of the interaction. The latter is Rossby radius mphoton =
1 RRossby
The field theory finds: mH = mphoton , which means Rmax ∼ RRossby . F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Rmax = RRossby is confirmed by the scaling relationships derived from the max asymptotic equation rvθmax , vθ , Rmax Using the relationship between the ratio of the radius where the velocity becomes maximum and the radius of maximum vortex extension Rmax , rvθmax Rmax 1 1 − exp − = Rmax 4 2 √ it results, with Rmax = 2L, √ rvθmax L − √ = ln 1 − 2 2 L 2 √
rvmax
An approximation is possible if 2 2 θL ≡ ε 1 .Then the equation √ rvmax √ L √ becomes − 2 −2 2 θL or L 2 rvθmax . All length are normalised to the Rossby radius, ρR , and since
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
L=
√ Rphys max / 2 ρR
and rvθmax =
phys rv
θ max ρR
, the equation becoms
ρR
1 8
phys 2 Rmax rvphys θ max
Then the Rossby radius for a particular atmospheric vortex can be estimated on the basis of the knowledge of the maximum radius of the vortex and of the radius of maximum tangential wind. Since very often the ratio of these two quantities is in the range phys Rmax ∼ 8...10 phys rvθ max
we obtain that phys ρR (1 · · · 1.2) × Rmax
This very rough estimation is only useful to suggest that the adimensional L , the essential parameter of the problem, is of the order 1.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Scalar and photon masses are equal
Radius of max.velocity versus the max.velocity from Equation 1
Returning to the first law, 2 1 e α exp vθmax = √ max − 1 2 2 rv θ
radius of the maximum velocity (nondimensional)
0.9
0.8
0.7
0.6
0.5
or
0.4
0.3 Rita +
0.2
Katrina +
0.1
0
0
2
4
6
8
10
rvθmax =
α
Andrew +
12
1 4 ln 1
14
maximum azimuthal velocity vmax (nondimensional) θ
16
18
20
1 2 max v e2 θ
+1
2
where α ∼ 0.97.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
Scalar and photon masses are equal
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Conclusion The field theoretical formalism provides interesting results: • identifies preferred states as extrema of an action functional • derives explicit differential equations for these states • allows to investigate neighboring states and reveals the existence of cuasi-degenerate directions and multiple minima of the action in the function space • reveals the universal nature of the extrema, as self-dual states • practical applications: tropical cyclone, tornadoes, crystals of atmospheric vortices, vorticity concentration, etc. It invites to study the natural extension of the theory.
F. Spineanu – 19th Conf. Atmospheric and Oceanic Fluid Dynamics –
