Atmospheric vortices
A field theoretical model of stationary atmospheric vortices Florin Spineanu and Madalina Vlad National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Objective: to develop a field-theoretical model that can provide a description of the 2D fluids close to stationarity. The theory is relevant for the classical debate on the relation between (1) large-scale organized, quasi-coherent flows, and (2) ”structures” (solitons, vortices, etc.)
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 – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Main idea : there exist preferred states of the system 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. It is found that besides conservation there is also action
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Coherent structures in fluids and plasmas (reality)
Toroidal surfaces of vorticity
Nice tornado vortex.
Vortex ring emitted by the volcano Etna.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Coherent structures in fluids and plasmas (numerical 1)
∂∇2 ⊥ ψ + −∇ ψ × n b · ∇⊥ ∇2 ⊥ ⊥ψ = 0 ∂t
Numerical simulations of the Euler equation.
D. Montgomery, W.H. Matthaeus, D. Martinez, S. Oughton, Phys. Fluids A4 (1992) 3.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Coherent structures in fluids and plasmas (numerical 2)
2 1 − ∇⊥
∂φ ∂t
∂φ −κ
∂y
b · ∇⊥ ∇2 − −∇⊥ φ × n ⊥φ = 0
Khukharin and Orszag, Carry-le-Rouet, 2000. Numerical simulations of the Charney-Hasegawa-Mima equation.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Coherent structures in fluids and plasmas (numerical 3)
R. Kinney, J.C. McWilliams, T. Tajima Phys. Plasmas 2 (1995) 3623.
Numerical simulations of the MHD equations.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Compare the two approaches Conservation eqs. ∂n ∂t mn
∂ ∂t
3 2
n
∂ ∂t
+ ∇· (nv)
=
Lagrangian
L xµ , φν , ∂ρ φν
0
→
S =
=
0
Z
dxdtL
!
v
=
−∇p − ∇ · π + F
∂xµ δ
!
T
=
−∇ · q − p (∇ · v) − π : ∇v + Q
Valid for : a single system. Just give the initial state.
+v·∇ +v·∇
∂
Valid for : coffee, ocean, sun.
δL
δL − ν ∂φ δφν ∂xµ
Lagrangians are preferable. But, how to find a Lagrangian ? See Phys.Rev.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
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 X
n=1,n6=k
ωn G (rk − rn ) , i, j = 1, 2 , k = 1, N
the Green function of the Laplacian ′ 1 |r − r | G r, r′ ≈ − ln 2π L
(1)
(2)
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
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
∂W ∂yk ∂W = − ∂xk =
where N X N X W =π ωi ωj K0 (m |ri − rj |) i=1 j=1 i6=j
Physical model → point-like vortices → field theory.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Field theoretical version of the discrete-vortex models From physical variables : ψ, v, ω to a new description: matter, field, interaction The continuum version of the point-like vortices is a field theory. The field theory is defined by the Lagrangian L = Lmatter + LCS + Linteraction
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The water Lagrangian 2D Euler fluid: Non-Abelian SU (2), Chern-Simons, 4th order L
where
=
2 −εµνρ T r ∂µ Aν Aρ + Aµ Aν Aρ + (3) 3 h 1 1 i2 † † † +iT r Ψ D0 Ψ − T r (Di Ψ) Di Ψ + T r Ψ , Ψ 2 4 Dµ Ψ = ∂µ Ψ + [Aµ , Ψ]
The equations of motion are i i 1 2 1 hh † iD0 Ψ = − D Ψ − Ψ, Ψ , Ψ 2 2 i Fµν = − εµνρ J ρ 2
(4) (5)
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The Hamiltonian density is h 1 i2 1 H = T r (Di Ψ)† (Di Ψ) − T r Ψ† , Ψ 2 4
(6)
Using the notation D± ≡ D1 ± iD2 † † T r (Di Ψ) (Di Ψ) = T r (D− Ψ) (D− Ψ) + hh i i 1 † † Tr Ψ Ψ, Ψ , Ψ 2
Then the energy density is
1 † H = T r (D− Ψ) (D− Ψ) ≥ 0 2
(7)
D− Ψ = 0
(8)
and the Bogomol’nyi inequality is saturated at self-duality
h
∂+ A− − ∂− A+ + [A+ , A− ] = Ψ, Ψ
†
i
(9)
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The static solutions of the self-duality equations The algebraic ansatz: [E+ , E− ] = H
(10)
[H, E± ] = ±2E±
taking
tr (E+ E− ) = 1 2 tr H = 2 ψ = ψ1 E+ + ψ2 E−
(11)
and Ax
=
Ay
=
1 (a − a∗ ) H 2 1 (a + a∗ ) H 2i
(12)
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The gauge field tensor F+− = (−∂+ a∗ − ∂− a) H and from the first self-duality equation ∂ψ1 ∂ψ1 −i − 2ψ1 a∗ = 0 ∂x ∂y
(13)
∂ψ2 ∂ψ2 −i + 2ψ2 a∗ = 0 ∂x ∂y
(14)
†
and their complex conjugate from (D− ψ) = 0. 2
Notation : ρ1 ≡ |ψ1 | , ρ2 ≡ |ψ2 |
2
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
∆ ln (ρ1 ρ2 ) = 0
(15)
∆ ln ρ1 + 2(ρ1 − ρ−1 1 ) = 0
(16)
∆ψ + γ sinh (βψ) = 0.
(17)
We then have
The water we drink is self-dual
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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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 – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
L
=
2 −κεµνρ tr ∂µ Aν Aρ + Aµ Aν Aρ 3 h i † µ −tr (D φ) (Dµ φ) † −V φ, φ
(18)
Sixth order potential hh i i † hh i i 1 † 2 † 2 V φ, φ† = tr φ, φ , φ − v φ φ, φ , φ − v φ . 4κ2 (19) The Euler Lagrange equations are Dµ Dµ φ =
∂V ∂φ†
−κενµρ Fµρ = iJ ν
(20) (21)
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The energy can be written as a sum of squares. The self-duality eqs. D− φ
=
F+−
=
0
hh i i i 1 h 2 † † ± 2 v φ − φ, φ , φ , φ κ
(22)
The algebraic ansatz : in the Chevalley basis
The fields
[E+ , E− ]
=
H
[H, E± ]
=
±2E±
tr (E+ E− ) 2 tr H
=
1
=
2
(23)
φ = φ1 E+ + φ2 E− A+ = aH, A− = −a∗ H
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
This simplest form of the equation governing the stationary states 1 ∆ψ + sinh ψ (cosh ψ − 1) = 0 2 The ’mass of the photon’ is v2 1 m= = κ ρs κ
≡ cs
v2
≡ Ωci
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Absolute minimum of the action-functional for atmosphere The Abelian dominance In certain cases the model collapses to an Abelian structure, where (φ, Aµ ) are complex scalar functions 1 µνρ 2 µ ∗ L = (D φ) (Dµ φ) + κε Aµ Fνρ − V |φ| 4 where
and
∂φ Dµ φ = + ieAµ φ ∂xµ
2
V |φ|
2 e2 2 2 = 2 |φ| |φ| − v 2 κ
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The SELF-DUALITY The energy is transformed similar to the Bogomolnyi form Z h 2 2 E = d r |(Dx ± iDy ) φ|
# 2 2 2 κ −1 e ∂ |φ| 2 ∗ 2 + φ B ± φ |φ| − v + 2e κ ∂t Z 1 ±ev 2 Φ + dl · J 2 r=∞
Restrict to the states 1. static (∂/∂t ≡ 0); 2. the current goes to zero at infinity such that the last integral is zero.
Then the energy consists of a sum of squared terms plus an additional term that has a topological nature, proportional with the total magnetic flux through the area.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
∆ψ = exp (ψ) [exp (ψ) − 1] At infinity (|φ| ≃ v) the covariant derivative term goes to 0 Dk φ → 0 at r → ∞ ∂k φ + ieAk φ → 0 Z Z dl · ∇ ln (φ) = i d (phase of φ) = 2πin
(24)
r=∞
The flux is
Z
2π n e The magnetic flux is discrete, integer multiple of a physical quantity. Φ=
d2 r (∇ × A) =
The equation has annular-type vortex solutions; radial scale is ρ, Rossby radius
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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Review of the field-theoretical method We have started from fluid models in 2D, for which discrete models are available. We have provided a field theoretical formulation of the continuum limit of the discrete models. This has allowed to obtain FT equations of motion, energy, currents, etc. and also have shown that the extrema of the action are obtained at self-duality. We have derived differential equations governing the stationary states at self-duality which correspond to the lowest action states of the fluid/plasma. We have studied these equations in relation with concrete applications of major interest for fluid and plasma dynamics. There are numerous confirmations of the validity of this approach and very interesting new aspects have been revealed. Much remains to be done.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Various applications Theroertical (line) vs. experimental (o) vorticity ω(r)
6
3
x 10
2.5
−1
vorticity ω(r) (s )
2
1.5
1
0.5
0
−0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
r (m)
Figure 1: The atmospheric vortex, the plasma vortex, the flows in tokamak,the crystal of vortices in non-neutral plasma.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The tropical cyclone The tangential component of the velocity, vθ, center is (0,0)
0.35 0.3 0.25
vθ
0.2 0.15 0.1 0.05 −0.5
0 −0.05 −0.5
0 −0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.5 y
x
Figure 2: The tangential component of the velocity, vθ (x, y)
The exact solution is a smooth vortex with the morphology of the tropical cyclone.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The tropical cyclone , comparisons The tangential component of the velocity, v , center is (0,0)
vθ
θ
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 3: The solution and the image from a satelite.
The solution reproduces the eye radius, the radial extension and the vorticity magnitude.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Scaling relationships between main parameters of the tropical cyclone eye-wall radius, maximum tangential wind, maximum radial extension
90 80
0.25 vmax and (e2/2)*[α e1/L−1] θ
70
0.2 max
60
0.15 θ max
/R
50
r
v
40
0.1
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 – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
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 ρg
(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 – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
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 – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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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 phi solution, i.e. the inverse of the radius of characteristic decay of the vortex flow; • 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.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
32
The field theory finds: mH = mphoton , which means Rmax ∼ RRossby . Actually we should expect roughly Rmax ∼ 1.2RRossby
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The crystals of atmospheric vortices The vorticity ω(x,y) resulting from the solution ψ(x,y)
The solution streamfunction ψ(x,y)
1
7
0
6 5
−1
4
−2
3
−3
2
−4
1
−5
0
−6
−1 0.5
−7 0.5
0.5 0
0.5 0
0
y
−0.5
−0.5
x
0
y
−0.5
−0.5 x
Figure 4: The crystals of atmospheric vortices, L = 1.
Similar with crystal-type solutions obtained in numerical simulation 1 . 1
Kossin and Schubert, JAS 58 (2001) 2196
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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Vortex crystals in non-neutral plasma
Comparison of our vortex solution with experiment.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
The concentration of vorticity
Tornado vortex.
Concentration of vorticity in fluids.
What makes the vorticity which is initially spread in the volume to get concentrated into such narrow, almost string-like, vortices? The answer may be that lower energy states are accessed in this way. A hint comes from the shape of the action functional around solution.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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Peaked profiles have lower energy
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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Numerical solution starting with sech4/3
Figure 5: Three intervals on the (peaking factor, amplitude) parameter space. Very weak variation of the error functional along the path (line of minimum error relative to the exact solution).
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
Radial integration
4
3.5
3
2.5
2
1.5
1 5
0.5 4 0 0
Figure 6: The functional error
R
3 2
4
6
8
10
12
14
2
d2 r(ω + nl)2 .
String of quasi-solutions.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
Atmospheric vortices
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Along the string of quasi-solutions the vortices are more and more concentrated
Figure 7: Green points: smooth, but progressively more peaked vortices; red: quasi-singular vortices. The energies Ef inal and the vorticities Ωf inal are only slightly different. We conclude that the system can drift along this path, under the action of even a small external drive.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
What the field-theoretical formulation can tell us about vorticity concentration?
The current of point-like vortices h µ † i µ † µ J = −i φ , D φ − (D φ) , φ
∂ψ ∂ (2iχ) ∂ −i (ρ1 + ρ2 ) − i (ρ1 − ρ2 ) ∂y ∂x ∂x ∂ψ ∂ (2iχ) ∂ Jy ≃ −i (ρ1 + ρ2 ) − i (ρ1 − ρ2 ) ∂x ∂y ∂y
Jx ≃ −
(25) (26)
(χ is the phase of φ1 ). The second term is, for the Euler case Jx(2)
∂ (−2ω) = −i ∂x
Can-we say we have the pinch of vorticity?
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
What about the pressure from this terms? For the Euler equation, ω + sinh(ψ) = 0 vxω
∼
vyω
∼
∂ (ρ1 − ρ2 ) ∂x ∂ (ρ1 − ρ2 ) ∂y
1 (v · ∇) v ∼ − ∇p ρ0 with the pressure " 2 # 2 1 ∂ω 1 ∂ω + p∼− 2 ∂x 2 ∂y ω
ω
This contribution to the pressure is negative, as expected.
F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
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Atmospheric vortices
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 The FT motel still has to be examined: It needs a clear mapping: ”formalism” - ”physics” It needs better investigation of the equations of motion It invites to study the natural extension of the theory. F. Spineanu – 17th Conf. Atmospheric and Oceanic Fluid Dynamics –
