Gunnar Hornig- Topological Methods in Fluid Dynamics

Introduction

MHD ↔ HD

Topological . . .

Structure of magnetic . . .

Helicity

Non-ideal evolution

Helicity conservation

Summary

Appendix

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Topological Methods inFluid Dynamics

Gunnar HornigTopologische FluiddynamikRuhr-Universitat-Bochum

IBZ, Februar 2002

Collaborators:H. v. Bodecker, J. Kleimann , C. Mayer, E. Tassi, S.V. Titov

Introduction

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Topological . . .


Helicity

Non-ideal evolution


Summary

Appendix

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1. Introduction

Why are we interested in topological features of fluids?

Topological properties do not depend on a (continuous/differentiable) deformationof the object.

Topological properties are an appropriate description for fluids.

Topological properties are often very robust ⇔ Changes of topological propertiesare very violent.

But where and when topological properties change most often depends on non-topological properties.

A topological description requires:

• Characterisation of the topological properties

• Description under which conditions they are conserved or destroyed

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2. MHD ↔ HD

Hydrodynamics:

∂tρ+∇ · (ρv) = 0ρ∂tv + ρv · ∇v = −∇p+ ν4v ν = const., (∇ · v = 0)

Navier-Stokes Eq. ⇒ Equation for the vorticity w = ∇× v:

∂tw −∇× (v ×w) = ∇×(−1/ρ∇p−∇v2/2 + 1/ρν4v

)≈ ∂tw −∇× (v ×w) = 0⇒ Conservation of vorticity (Kelvin’s Theorem)

≈ : Approximation of large Reynolds Number: Re := V0L0/ν and isentropic flow(p = p(ρ))

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Magnetohydrodynamics:

∂t ρ+∇ · (ρv) = 0ρ ∂tv + ρv · ∇v = −∇p+ J×B

E + v ×B = ηJ + other terms (?)∇×E + ∂tB = 0∇ ·B = 0∇×B = µJ

Ohm’s law (?) ⇒ Induction equation:

∂tB−∇× (v ×B) = −∇× (ηJ + other terms)≈ ∂tB−∇× (v ×B) = 0⇒ Conservation of magnetic flux (Alfvens’s Theorem)

≈ : Approximation of large magnetic Reynolds Number: RM := V0L0/η

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3. Topological Conservation Laws

3.1. Flux conservation

∂tB−∇× (v ×B) = 0

⇒∫

C2(t)

B · da = const. for a comoving surface C2,

⇒ Conservation of flux

⇒ Conservation of field lines

⇒ Conservation of null points

⇒ Conservation of knots and linkages of field lines

Transport of fields (ωk) in the flow v(x, t) implies the conservation of an integralover a k-dimensional comoving volume Ck.

∂tωk + Lvω

k = 0

⇒∫

F (Ck)

ωk = const.

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3.2. For physical space (IR3):

∂tω0α + Lvω

0α = 0 ⇔ ∂t α+ v · ∇α = 0

⇒ α(F (x, t), t) = const.

Example: α decribes the color in a mixture of paints

∂tω1A + Lv ω

1A = 0 ⇔ ∂tA +∇(v·A)− v ×∇×A = 0

⇒∫

C1(t)

A·dl = const.

Example: A = ∇α

∂tω2B + Lv ω

2B = 0 ⇔ ∂tB + v·∇B−B·∇v + B ∇·v = 0

⇒∫

C2(t)

B · da = const. ,

Example: B vorticity in hydrodynamics (Kelvins Theorem)

∂tω3ρ + Lv ω

3ρ = 0 ⇔ ∂tρ+∇· (v ρ) = 0

⇒∫

C3(t)

ρ d3x = const.

Example: ρ mass-density in hydrodynamics

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3.3. Conservation laws in space-time

LV ω0α = 0 ⇔ V t∂tα+ V·∇α = 0

⇒∫

C0ω0

α = α = const.

LV ω1A = 0 ⇔

{∂t(V tAt) + V·∇At −A · ∂tV = 0V t∂tA +∇ (V·A)−V ×∇×A−At∇V t = 0

⇒∫

C1ω1

A =∫

C1Atdt−

∫C1

A·dl = const.

LV ω2AB = 0 ⇔

{∂t(V tA) +∇ (V·A)−V ×∇×A− ∂tV ×B = 0V t∂tB−∇× (V ×B) + V ∇·B +∇V t ×A = 0

⇒∫

C2ω2

AB =∫

C2B·da−

∫C2

A·dl dt = const.

LV ω3A = 0 ⇔

{V t∂tA

t +∇·(VAt)−A·∇V t = 0∂t(V tA)−∇× (V ×A) + V ∇·A−At∂tV = 0

⇒∫

C3ω3

A =∫

C3AtdV −

∫C3

A·da dt = const.

LV ω4ρ = 0 ⇔ ∂t(ρV t) +∇·(ρV) = 0

⇒∫

C4ω4

ρ =∫

C4ρ dV (4) = const.

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3.4. Conservation in IRn

∂sω0α + LV ω

0α = 0

⇔ ∂sα+ V·∇α = 0

⇒∫

C0ω0

α = α = const.

...

...∂sω

nρ + LV ω

nρ = 0

⇔ ∂sρ+∇·(ρV) = 0

⇒∫

Cn

ω4ρ =

∫Cn

ρ dV (n) = const.

Example: V Hamilton flow on IR(2m), ω(2k) = dq1 ∧ dp1 ∧ ... ∧ dqk ∧ dpk yieldsPoincare integral invariants

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4. Structure of magnetic fields

Long range forces:

• Gravitation

• Electromagnetism

– Electric fields (shielded)

– Magnetic fields

Magnetic fields are found in many astrophysical objects:planets, stars, white dwarfs, pulsars, interstellar clouds, galaxy, intergalactic field

Are these all dipolar fields?

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The magnetic field of the Earth’s core according to a numerical simulation by G.Glazmeier.

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X-ray image of the Sun, Yohkoh satellite

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Helicity

Non-ideal evolution


Summary

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Introduction

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Even simple magnetic configurations contain linked and knotted field lines.

A field line forming a simpletorus-knot.

A field line linking the centralfield line of the torus.

Can we quantify the knottedness or linkage of flux in a magnetic field?

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5. Helicity

5.1. Total Helicity

The (total) magnetic helicity is defined as

H(B) :=∫

V

A ·B︸︷︷︸hel. density

d3x for B · n|∂V = 0 ,

where A is the vector potential for the magnetic field B, which is tangent to theboundary ∂V .In terms of the magnetic field only:

H(B) =14π

∫ ∫B(x′)× x− x′

|x− x′|3·B(x) d3x′ d3x

which shows that the helicity is of 2nd order in the magnetic field.

The total helicity measures the mutuallinkage of flux in the volume.

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5.2. Cross Helicity

The helicity integral can be considered as a special case of the mutual or crosshelicity integral:

H(B1,B2) :=∫

V

A1 ·B2 d3x =∫

V

A2 ·B1 d3x for B1 ·n|∂V = B2 ·n|∂V = 0

For B = B1 + B2 we have the relation

H(B1 + B2)︸︷︷︸total helicty

= H(B1) +H(B2)︸︷︷︸self helicity of comp.

+2 H(B1,B2)︸︷︷︸cross helicity

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5.3. Evolution of helicity

The homogeneous Maxwell’s equations yield a balance equation for the helicitydensity:

∂

∂tA ·B︸︷︷︸

hel. density

+∇ · (ΦB + E×A︸︷︷︸hel. current

) = −2E ·B︸︷︷︸hel. source

Remarks:

• There is no freedom to add certain terms either to the current or to the sourcesince the covariant formulation uniquely determines the helicity current.

• The helicity density and the helicity current are not gauge invariant, but thesource term is gauge invariant.

Integrating over a volume yields an expression for the evolution of the total helicity

d

dt

∫V

A ·B d3x = −2∫

V

E ·B d3x ,

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5.4. Topology and free energy of a magnetic field

Energy of a magnetic field:

Em(B) :=18π

∫B2d3x

Free energy of a magnetic field:

EF := Em(B)− Em(B0)

where B0 is the vacuum field satisfying the same boundary conditions.

Minimum free energy of a magnetic field:

min(EF ) := Em(B∗)− Em(B0)

where B∗ is the lowest energy state accessible from B by an ideal relaxation.

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Every kind of linkage or knottedness contributes to a lower bound for the freeenergy of a magnetic field. E.g. the total magnetic helicity yields the inequality

EF ≥ min(EF (B)) ≥ C H(B)

([Arnold 1986], [Freedman 1988], [Freedman & He 1991a], [Freedman & He 1991b],[Berger 1993])

⇒ There is a need for higher order invariants and new energy limits which measuremore complex linkage or knottedness.

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6. Non-ideal evolution

∂tB−∇× (v ×B) = −∇×N (1)N : Non-ideal term, e.g. N = ηJ

If B ·N = 0 we can rewrite (1)

⇒ ∂tB−∇× (w ×B) = 0

with w := v −∆v = v − B×NB2

• Existence of a smooth w ⇒ slippage solution (See example of J. Kleimann)

• Existence of a smooth w with exception of points where B = 0 but N 6= 0⇒ Reconnection-like processes

– B has a X-line ⇒ classical (2d) reconnection

– B has a 3d null point ⇒ null-point reconnection ?

• B ·N 6= 0 ⇒ no w exists.

– Localized N ⇒ 3d reconnection

– N 6= 0 globally ⇒ global dissipation (very slow for astrophysical plas-mas)

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A simple reconnection process

Movie

• An evolution of field lines integrated from two cross-sections (black) comovingwith the fluid is shown. The cross-section are located outside the non-idealregion (N 6= 0 only in a neighborhood of the z-axis) and initially belong tothe same magnetic flux tube.

• The flux tube is transported with velocity w, which outside the non-idealregion coincides with the plasma velocity v.

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• The reconnection occurs along a line, the reconnection line. This is the linealong which the virtual transport velocity w is singular (here the z-axis ofthe box).

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6.1. 3-D Reconnection

B 6= 0 ⇒ No flux conserving flow w exists!

Movie

• An evolution of field lines integrated from two cross-sections (black) comovingwith the fluid is shown. The cross-section are located outside the non-idealregion (N 6= 0 only in a neighborhood of the z-axis) and initially belong tothe same magnetic flux tube.

• There exists no transport velocity w as in the 2-d case. Thus the flux tubesplits as soon as it enters the non-ideal region. The strands of the flux tube flip

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around each other and finally merge again. Note that there exist a covariantformulation where this type of reconnection is defined by a singularity of a4-vector flow W 4 analogously to the singularity of the 3-velocity w in 2-dreconnection.

• The final result are two flux tubes which have non-vanishing twist (not shownin the figure) and non-vanishing total helicity although the initial flux tubehad zero total helicity. ⇒ 3-d reconnection in general produces helicity.

How much helicity is created in 3-d reconnection?

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7. Helicity conservation

7.1. General considerations

For a non-ideal evolution E + v × B = N the change of the total helicity in amagnetically closed volume is given by

d

dt

∫V

A ·B d3x = −2∫

V

N ·B d3x,

if N vanishes on the boundary. The total helicity is strictly conserved for idealMHD or more general for N ·B = 0, e.g. in case of 2-d reconnection.

The total helicity is approximately conserved on the time scale of energy dissipationfor a resistive plasma [Berger 1984].∣∣∣∣∆HH

∣∣∣∣ ≤√∆tτd

with τd = L2/η and L =∣∣∣∣AB

∣∣∣∣ =∣∣∣∣A ·BB2

∣∣∣∣⇒ If ∆t� τd then ∆H/H � 1.

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7.2. Production of helicity in reconnection

Astrophysical plasmas differ from many technical plasmas in the size of the regionsVdiss, where the dissipation dominates the evolution compared to the volume V ofmagnetic flux connected to Vdiss.

Especially d� L for astrophysical plasnas, where d is the diameter of Vdiss and Lof V .

τhel =∣∣∣∣∫

VA ·B d3x∫

VE ·Bd3x

∣∣∣∣ ∼ B V L

E Vrecwith L =

∣∣∣∣AB∣∣∣∣ ,

τdiss =∣∣∣∣∫

VB2/(8π)d3x∫V

E · Jd3x

∣∣∣∣ ∼ B V d

E Vdisswith d =

∣∣∣∣BJ∣∣∣∣Vdiss

⇒ τhel

τdiss∼ L

d� 1

In plasmas with high magnetic Reynolds numbers the helicity production in a singlereconnection event is small compared to the potential helicity of the field(e.g. acorresponding constant-α force-free field ([Hornig 1999]).

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⇒ How is the helicity created which we observe on the solar surface ?

A principal method:

⇒ ⇒

An initially untwisted flux tube with vanishing total helicity is twisted and ...

⇒ ⇒

...reconnected into two flux tubes with negative and positive total helicity.

Reconnection does not produce helicity but separates helicity!

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7.3. Creation of helicity in the Sun

In the interior of the sun, the equator rotates faster than the poles. Differentialrotation provides a strong source of helicity injection (from [Berger 2000]).

Helicity Transfer into the sun from magnetogram and solar rotation data Helicitytransfer into the southern interior (predominantly positive curve)and northern inte-rior (predominantly negative curve). The units are 1040 Mx2 /day. The differencesin magnitude between the two curves go up to 5 x 1042 Mx2 /day

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8. Summary

• Astrophysical magnetic fields often have a complex topological structure.

• Topological invariants are important in the dynamics of fluids, e.g. the mag-netic helicity is a better invariant then magnetic energy.

Thus we need:

• A better description of the complexity of field structures, i.e. higher ordertopological invariants.

⇒ C. Mayer, H.v. Bodecker

• Additional knowledge where and under which condition the topological struc-ture of fields changes, i.e. about reconnection and related processes.

⇒ S.V. Titov, E. Tassi, J. Kleimann

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References

[Arnold 1986] Arnold, V.I., 1986,The asymptotic Hopf invariant and its applica-tion, Sel. Math. Sov., 5, 327

[Berger 1996] Berger, M.A.,1996 Inverse cascades in a periodic domain , Astro-physical Letters & Communications, 34, 225 (1996).

[Berger 1984] Berger, M.A., 1984, Rigorous new limits on magnetic helicity dissi-pation in the solar corona, Geophys. Astrophys. Fluid Dynamics, 30, 79

[Berger 2000] Berger, M.A., Ruzmaikin, A.,Rate of helicity production by solarrotation, J. Geophysical Research 105, 10481-10490 (2000)

[Hornig & Rastatter, 1997] Hornig, G., and L. Rastatter, The role of Helicity inthe Reconnection Process, Adv. Space Res. 19, 1789, 1997a.

[Hornig 2000] Hornig, G., The Geometry of Reconnection, in An Introduction tothe Geometry and Topology of Fluid Flows, Kluwer 2001 http://www.tp4.ruhr-uni-bochum.de/∼gh/publiste/gtf.ps.gz.

[Hornig 1999] Hornig, G., 1999. In: Brown, M.R., Canfield, R.C., Pevtsov,A.A.(eds.), Helicity in Space and Laboratory Plasmas, Geophysical Monographs,AGU, 157

[Freedman 1988] Freedman, M.H., 1988, Journal of Fluid Mechanics, 194, 549

[Freedman & He 1991a] Freedman, M.H., He, Z-X., 1991, Topology, 30, 283

[Freedman & He 1991b] Freedman, M.H., He, Z-X., 1991, Annals of Mathematics,134, 189

http://www.tp4.ruhr-uni-bochum.de/~gh/publiste/gtf.ps.gz

http://www.tp4.ruhr-uni-bochum.de/~gh/publiste/gtf.ps.gz

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[Berger 1993] Berger, M.A., 1993, Physical Review Letters, 70, 705

[Moffatt 1969] Moffatt H K 1969,Journal of Fluid Mechanics 35 117

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9. Appendix

9.1. Covariant helicity

The helicity density together with the helicity current form the helicity 4-vector

hα = εαβγδAαFβγ = (A ·B,ΦB + E×A)

and the balance equation reads

εαβγδ∂δAαFβγ = FαδFβγ

or in differential forms:d(A ∧ F ) = F ∧ F

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9.2. Helicity in a periodic domain

A box⊂ IR3 with periodic boundary conditions on all sides is topologically a 3-torusand therefore different from any domain in IR3. Especially it allows for magneticfields which do not have a vector potential (see [Berger 1996]).

There are two ways to ensure the existence of a vector potential in such a domain:

• Use a periodic A in your numerical integration and derive B from A or ...

• Make sure that the magnetic flux through every side of the box vanishes.

An example of what can happen in a periodic domain: Consider a magnetic fieldwith a constant component in z-direction and an evolution of x-y components asshown below (from [Berger 1996]).

The helicity of the field changes its sign from a) to f)! Note that the reconnectionprocess involved can be chosen such that it produces an arbitrary small change ofhelicity.

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9.3. Equivalence of boundary conditions

It is obvious that A× n|∂V = 0. can always be achieved by gauge transformationsif B = 0 outside of an arbitrary integration volume V . Under the less restrictiveassumption B · n|∂V = 0, where V is a simply connected volume, this requirementcan still be fulfilled, as we will see now: Let us assume that A has a non-vanishingcomponent A tangential to the surface ∂V . We can express A as a one-form αdefined only on ∂V . Then the assumption B · n|∂V = (∇ × A) · n = 0 writtenin differential forms reads dα = 0 on ∂V . From V being simply connected itfollows that ∂V has the same homotopy type as the two sphere S2, but since thecohomology vector space H1(S2; IR) = 0, all closed one-forms are exact. Thereforethere exists a scalar function ψ on ∂V such that α = dψ. This in turn implies thata gauge exists such that A|∂V = 0 and thus A× n|∂V = 0

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9.4. Time dependent gauge of the vector potential

We use a gauge {Φ→ Φ = Φ− ∂tΨA→ A = A +∇Ψ,

defined bydΨdt

= Φ−W·A

such thatΦ− v · A = 0 .

Thus A is transported in the plasma flow v as a 1-form:

∂tA +∇ (v·A)− v ×∇×A = 0

in terms of a Lie-derivative for the 1-form A

⇔ ∂tA+ LvA = 0

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9.5. Covariant description

If the effect of the non-ideal term ∇×N is limited to isolated regions, embeddedin an ideal environment (∇×N = 0) then the non-ideal flow v can be mapped toan ideal 4-velocity W (4)= (W 0,W 1,W 2,W 3)= (W 0,W) satisfying

εαβγδ∂αWνF νβ = 0

⇔{∂0(W 0E + W ×B) +∇(E·W) = 0W 0∂0B−∇× (W ×B)−∇W 0 ×E = 0

⇔ LWω2F = 0

⇒∫

C

Fµνdxµdxν = const. (2)

where Fαβ is the electromagnetic field tensor ([Hornig 2000].

Special solutions:W 0E + W ×B = 0

W/W 0 =∂x∂s/∂ct

∂s= w

• The 4-velocity W 4 vanishes at the reconnection line (w is singular in thiscase) .

• The covariant transport of the electromagnetic field tensor generally doesnot imply the conservation of magnetic flux. Exception: W 0 is constant orE ·B = 0.

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9.6. Interpretation

For systems of (untwisted) flux tubes the to-tal magnetic helicity can be expressed as asum over the mutual linking of flux tubes[Moffatt 1969]:

H(B) = 2∑i<j

lk(Ti, Tj)ΦiΦj ,

where lk(Ti, Tj) is the linking number of thetube Ti and Tj with magnetic fluxes Φi andΦj .

c)

This interpretation was generalized by [Arnold 1986] for the generic case wherefield lines are not closed using asymptotic linking numbers.

Note: Twist is a linkage of sub-flux tubes:

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Gunnar Hornig- Topological Methods in Fluid Dynamics