Fast waves and incompressible models · Herv e Guillard Universit e C^ote d’Azur, Inria, CNRS,...

$: Fast waves and incompressible models · Herv e Guillard Universit e C^ote d’Azur, Inria, CNRS, LJAD, France Herv e GuillardANU, Canberra{August 2017- 1. IntroductionReduced MHDAerodynamicsConvergence$
Introduction Reduced MHD Aerodynamics Convergence proof

Fast waves and incompressible models

Herve Guillard

Universite Cote d’Azur, Inria, CNRS, LJAD, France

Herve Guillard ANU, Canberra–August 2017- 1


MHD MODEL (here ideal)

∂

∂tρ+∇ · (ρu) = 0

∂

∂tρu +∇ · (ρu ⊗ u) +∇p = J × B

∂

∂tp + u · ∇p + γp∇ · u = 0

∂

∂tB +∇× B × u = 0



MHD waves

Hyperbolic system with 3 different types of waves (+ material or entropy waves) Ifn is the direction of propagation of the wave

Fast Magnetosonic waves : λF = u.n ± CF

C 2F =

1

2(V 2

t + v2A +

√(V 2

t + v2A)2 − 4V 2

t C2A)

Alfven waves : λF = u.n ± CA C 2A = (B.n)2/ρ

Slow Magnetosonic waves : λS = u.n ± CS

C 2S =

1

2(V 2

t + v2A −

√(V 2

t + v2A)2 − 4V 2

t C2A)

v2A = |B|2/ρ vA : Alfven speedV 2t = γp/ρ Vt : acoustic speed



Transverse MHD waves

propagation speed depends on the direction w r to the magnetic field.

If n · B = 0 (transverse waves) :

Alfven waves : λF = 0

Slow Magnetosonic waves : λS = 0

Fast Magnetosonic waves : λF = ±CF with C 2F = V 2

t + v2A

only the Fast Magnetosonic waves survive !



Reduced MHD modelsFeng Liu & al : Nonlinear MHD simulations of Quiescent H-mode pedestal in DIII-D andimplications for ITER

Jorek Code (https://www.jorek.eu/)

∂tρ+∇ · ρv = Sρ

ρ(∂tT + v · ∇T ) + (γ − 1)p∇ · v =Diss + ST

∂tψ + R[U , ψ] + F0∂φU = R2Diss

eφ · ∇ × [ρ(∂tv+v · ∇v) +∇p = J × B + Diss]

B = F0∇φ+∇ψ ×∇φ

u = R2∇U ×∇φ

3D he-lical structure of the densityperturbation for the saturatedn=1-10 modes at separatrix forthe JOREK simulation of DIII-DTokamak.



Goal of this work

To give a mathematically rigorous proof that full compressible MHD solutions→ Reduced MHD

Summary

Abstract formulation of the problemApplication to MHDApplication in aerodynamics (convergence toward the incompressible limit)The idea of the mathematical proof



Singular limit of hyperbolic PDEs

Let W ∈ IRN solution of the hyperbolic system with a large operator∂tW +

∑j [Aj(W , ε) +

1

εCj ]∂xjW = 0

W (0, x , ε) = W 0(x , ε)

What is the behavior of the solutions when ε→ 0 ?

Let n be a arbitrary direction, then some eigenvalues of∑

j nj(Aj +1

εCj) are of

the form ak +1

εck → ±∞ while the others (kernel of

∑j njCj) are simply ak

What is the behavior of the solutions when Slow and Fast waves co-exist ?



Singular limit of hyperbolic PDEs : Slow limit

∂tW +∑j

Aj(W , ε)∂xjW +1

ε

∑j

Cj∂xjW = 0

LW =∑

j Cj∂xjW has to be O(ε)

Look for the solution as W = W 0 + εW 1 with LW 0 = 0, obtain :

∂tW 0 +∑j

Aj(W , ε)∂xjW 0 + LW 1 = O(ε)

and the solutions converge to W 0 defined by :LW 0 = 0∂tW 0 + P

∑j Aj(W 0, 0)∂xjW = 0

P projection on the kernel of L



But we forget the fast waves...

∂tW +∑j

Aj(W , ε)∂xjW +1

ε

∑j

Cj∂xjW = 0

Let us do the simple change of variable : t = ετ :

1

ε∂τW +

∑j

Aj(W , ε)∂xjW +1

ε

∑j

Cj∂xjW = 0

and when ε→ 0 the limiting form becomes :

∂τW +∑j

Cj∂xjW = 0

Solution are fast waves moving at velocity1

ε



Need to take into account the interaction between fast andslow waves

Fast transverse waves are absent from RMHD system

“An important feature of RMHD systems is that they eliminate the fast compressional Alfvenwave” (Strauss, Reduced MHD in nearly potential magnetic fields, Journal of Plasma Physics1997; 57(1):8387)

However the full MHD system is non linear and contains quadratic terms :

Q(U,U) = (v · ∇)v or (B · ∇)B or (B · ∇)v , · · ·But the fast wave do exist :

UMHD = USlow wave + UFast wave

thusQ(U,U) = Q(USlow,USlow) +Q(USlow,UFast) +Q(UFast ,USlow)

+Q(UFast,UFast)

How do we know that the term : Q(UFast ,UFast ) is not important for the dynamics of thesystem ?



Putting the MHD model into the abstract formulation

Following Strauss, Physics of Fluids, 19, p 134, 1976 : Assumptions

1 Slab geometry

2 Large dominant magnetic field B⊥/Bz = ε << 1

3 Low plasma β : β = p/B20 ∼ ε2

4 Constant density



Scaling assumptions

Large Dominant Magnetic field Bz = B0(1 + εBz) (1.2)

Large Dominant Magnetic field B⊥ = B0εB⊥ (1.1)

Pressure : Low β assumption p = ρ0v2A(ε2q) (1.3)

Velocity : small transverse velocities u⊥ = εvAv⊥ (1.4)

Velocity : very small parallel velocities uz = ε2vAv z (1.5)

Time : low frequencies t =a

εvAτ (1.6)



Reduced MHD scaling

Plug this into the full compressible MHD system :

D

Dtu +∇(p + B2/2)− (B.∇)B = 0 (2.1)

D

DtB − (B.∇)u + B∇.u = 0 (2.2)

1

γp

D

Dtp +∇.u = 0 (2.3)



Scaled full MHD equations

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥(B2/2 + q)− ∂zB⊥ +

1

ε∇⊥Bz = O(ε) (3.2)

∂

∂τBz + (v⊥ · ∇⊥)Bz + Bz∇⊥ · v⊥ +

1

ε∇⊥ · v⊥ = O(ε) (3.1)

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ + B⊥∇⊥ · v⊥ − ∂zv⊥ = O(ε) (3.3)

1

γp(∂

∂τp + v⊥ · ∇⊥p) +∇⊥.u = 0

Indeed of the form :

∂tW +∑j

Aj (W , ε)∂xj W +1

εLW = 0 with LW =

0 ∇ 0 0∇· 0 0 00 0 0 00 0 0 0

W



Slow limit of MHD

Look for a solution into the kernel of L and obtain :“Almost 2D” Incompressible MHD

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥π − ∂zB⊥ = 0 (4.1)

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ − ∂zv⊥ = 0 (4.2)

∇⊥ · v⊥ = ∇⊥ · B⊥ = 0 (4.3)

Formulation of RMHD model in term of stream functions :

∇⊥ · v⊥ = ∇⊥ · B⊥ = 0⇒ ∃ 2 scalar functions ψ,U such that :

v⊥ = z×∇⊥U B⊥ = z×∇⊥ψ

1 Plug these expression in eqs (4)

2 apply the operator z · ∇⊥× to (4.1) (vorticity form of the momentum equation)



Slow limit of MHD in term of stream functions

Two equation model of Strauss, Physics of Fluids, 19, p 134, 1976variables : U : electric potential ψ : magnetic flux

∂ω

∂t+ [ω,U]− [ψ, J]− ∂

∂zJ = 0

∂ψ

∂t+ [U , ψ]− ∂U

∂z= 0

ω = ∇2⊥U J = ∇2

⊥ψ

[f , g ] = z.∇⊥f ×∇⊥g



Reduced MHD - physical interpretation

In the presence of a large dominant magnetic field, the dynamic can be described by

2D incompressible MHD in the transverse direction and

Alfven waves propagating in the direction of the dominant magnetic field.

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥π − ∂zB⊥

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ − ∂zv⊥ = 0

∇⊥ · v⊥ = ∇⊥ · B⊥ = 0



Aerodynamics : incompressible limit and acoustic

ε = Mach Number (ratio Fluid Velocity/speed ofsound).

r pressure deviation1

(1 + εr)[∂r

∂t+ u.∇r ] +

1

εdiv u = 0

(1 + εr)[∂u∂t

+ (u.∇)u] +1

ε∇r = 0



Aerodynamics : The slow limit

1

(1 + εr)[∂r

∂t+ u.∇r ] +

1

ε∇ · u = 0

(1 + εr)[∂u∂t

+ (u.∇)u] +1

ε∇r = 0

Look for the solution in the kernel of the large operator :

∇r = 0,∇ · u = 0

and let ε→ 0 we get the incompressible Euler equation∇ · u = 0

∂u∂t

+ (u.∇)u +∇r1 = 0



Aerodynamics : The fast limit

1

(1 + εr)[∂r

∂t+ u.∇r ] +

1

ε∇ · u = 0

(1 + εr)[∂u∂t

+ (u.∇)u] +1

ε∇r = 0

take t = ετ and let ε→ 0 we get the Acoustic equations∂r

∂τ+∇ · u = 0

∂u∂t

+∇r = 0



Superposition incompressible + acoustic

same situation than previously :MHD Fast magnetosonic waves ↔ Aero Acoustic waves

Does Acoustic phenomena modify incompressible flows ?



Acoustic stirring

Not only a jet can generate sound but also sound can generate a jet! 1

Figure: Reproduced from : V. Botton, , T. Cambonie, B. Moudjed, S. Miralles, D. Henryand H. Ben Hadid : How to drive a square flow in a liquid: acoustic stirring, 7thInternational Conference on Computational Methods for Coupled Problems in Scienceand Engineering, June 2017

1S. J., Lighthill, Acoustic streaming, J. Sound Vibr. 61, pp. 391418 (1978)



Convergence proof : Some notations

The variables : V ε = (Bεz , vε⊥,Bε⊥)t

The equations : ∂tVε + H(V ε,V ε) +

1

εLV ε = O(ε)

H(V ,V ) is a non-linear operator (at most quadratic)

H(V ,V ) =

(v⊥ · ∇⊥)Bz + Bz∇⊥ · v⊥(v⊥ · ∇⊥)v − (B⊥ · ∇⊥)B⊥ +∇⊥(B2/2 + q)− ∂zB⊥(v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ + B⊥∇⊥ · v⊥ − ∂zv⊥

LV is the constant coefficient linear operator

LV =

∇ · v⊥∇Bz0



The proof strategyS. Schochet, E. Grenier, P.L.Lions-N.Masmoudi, B. Desjardins...

1 Introduce a filtered variable Vε

= FV ε to remove the oscillations

2 Prove that the filtered variable Vε → V

0satisfying some equation

∂tV0 + H(V 0,V 0) = 0 with H time-independent.

3 Prove that the original variable V ε → F−1V0

4 Since F−1V0 → PV

0where P is the L2 projection on the kernel of L

Result

V ε → V = PV0

and V satisfies :∂tV + PH(V 0,V 0) = 0



The wave operator L

LV =

∇ · v⊥∇Bz0

L2(Ω)× (L2(Ω))2 = KerL⊕ ImLKerL = (Bz , v);Bz = cte,∇⊥ · v = 0ImL = (Bz , v);

∫Bz = 0,∃Φv = ∇⊥Φ

Spectrum of L on ImLLet ψk , k ≥ 1 the eigenvectors of the Laplace operator

−∆ψk = λ2kψk λk > 0

then the eigenvectors of L are :

Φ±k =

ψk

±∇ψk

iλk

with LΦ±k = ±iλkΦ±k



The solution operator L of the wave equation

Let L(t) be the semi-group (L(t), t ∈ IR) defined by

L(t) = exp(−Lt) (6)

In other words

V (t, x) = L(t)V 0(x) means that∂V

∂t+ LV = 0 with V (t = 0, x) = V 0(x)

Using the expression of the spectrum of L we can have an explicit representation of the solutionoperator L(t) : Let P be the L2 projection on KerL

on the velocity component Lv (t)V

if V − PV =∑k,±

a±k Φ±k then Lv (t)V = πv⊥ +∑k,±±a±k e±iλk t

∇ψk

iλk

a−k = (a+k )∗ conjugate (real functions)



Step 1 : Equation satisfied by the filtered variable Vε

∂tVε + H(V ε,V ε) +

1

εLV ε = O(ε)

introduce the filtered variable Vε

= L(−t/ε)V ε

withL(t) = exp(−Lt)

From the definition of L, we deduce that

∂Vε

∂t=

Lε

Vε

+ L(−t/ε)∂V ε

∂t

=Lε

Vε − L(−t/ε)H(L(t/ε)V

ε,L(t/ε)V

ε)− L(−t/ε)

LεL(t/ε)V

ε+O(ε)

= −L(−t/ε)H(L(t/ε)Vε,L(t/ε)V

ε) +O(ε)

since L(t/ε) and L commute.

Initial data : Vε(t = 0) = V ε(t = 0) since L(0) is the identity



Limit Equation

Step 2 : Limit Equation for the filtered variable Vε

∂Vε

∂t+ L(−t/ε)H(L(t/ε)V

ε,L(t/ε)V

ε) = O(ε)

V0

= limε→0 Vε

∂V0

∂t+H(V

0, V

0) = 0

where H(V0, V

0) = limε→0 L(−t/ε)H(L(t/ε)V

0,L(t/ε)V

0) is a time-independent operator

whose expression can be computed explicitly (see next slides)

Step 3 : Go back to the unfiltered variable V ε

V ε → L(t/ε)V0



Limit for the original variable

But we haveL(t/ε)V

0→ PV

0

sinceL(t/ε)V

0= PV

0+∑k,±

±a±k e±iλk t/εΦ±k PV

0

Final result : weak limit of V ε = PV0

that satisfies

∂PV0

∂t+ PH(V

0, V

0) = 0



Explicit form of the limit equation for PV0

example : computation of the quadratic term (v⊥ · ∇)v⊥ = (v⊥)j∂jv⊥

(Lv (t/ε)QV · ∇)Lv (t/ε)QV =

∑k

(a+k e

iλk t/ε − a−k e−iλk t/ε)∇ψk

iλkj∂j

∑l (a

+l e

iλl t/ε − a−l e−iλl t/ε)∇ψl

iλl =∑

k,l[−a+

k a+l e

i(λk+λl )t/ε − a−k a−l e−i(λk+λl )t/ε]1

λkλl(∇ψk )j∂j (∇ψl )

+∑k,l

[a−k a+l e

i(λl−λk )t/ε + a+k a−l e i(λk−λl )t/ε]

1

λkλl(∇ψk )j∂j (∇ψl )

limε→0 (distribution) of all the terms is 0 except when k = l and we get :

(Lv (t/ε)QV ·∇)Lv (t/ε)QV →∑k

[a−k a+k + a+

k a−k ]

1

λ2k

(∇ψk )j∂j (∇ψk ) =∑k

|a+k |

2

λ2k

∇(|∇ψk |2/2)

On the average (weak limit) fast k-waves interact with l-waves only if k = l and the result is agradient

no interaction between fast waves and slow dynamics



Summary

Weak limit of the solutions of full MHD system :

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥(B2/2 + q)− ∂zB⊥ +

1

ε∇⊥Bz = O(ε)

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ + B⊥∇⊥ · v⊥ − ∂zv⊥ = O(ε)

are the solutions of the “incompressible” reduced MHD system

∂

∂τv⊥ + (v⊥ · ∇⊥)v⊥ − (B⊥ · ∇⊥)B⊥ +∇⊥π − ∂zB⊥ = 0

∂

∂τB⊥ + (v⊥ · ∇⊥)B⊥ − (B⊥ · ∇⊥)v⊥ − ∂zv⊥ = 0

∇⊥ · v⊥ = ∇⊥ · B⊥ = 0

no interaction between fast waves and slow dynamics



Comments and perspectives

In previous theory ε = Poloidal Field /Toroidal Field

equivalent small aspect ratio a/R assumption (since q is of order 1)

not so small is modern tokamaks (∼ 1/3 for ITER)

but simulations with RHMD still give “reasonable” results · · ·- is there in modern tokamks an hidden small ε that can justify the use of RMHD ?

- ⇒ need of detailed comparisons between RMHD and full MHD on large aspectratio machines (e.g Pamela Stanislas Jorek team on MAST)


Fast waves and incompressible models · Herv e Guillard Universit e C^ote d’Azur, Inria, CNRS,...

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