Dissipation and Spectrum Peter Constantinpeople.cs.uchicago.edu/~const/disspetalk.pdf · E(l)dl 5E...

Dissipation and Spectrum

Peter Constantin

1

References• P. Constantin and Ch. Doering, Heat transfer in convective turbulence,

Nonlinearity 9 (1996) 1049-1060.

• P. Constantin, The Littlewood-Paley spectrum in 2D turbulence, Theor.Comp. Fluid Dyn.9 (1997), 183-189.

• P. Constantin and Ch. Doering, Infinite Prandtl number convection,J. Stat. Phys., 94 (1999), 159-172.

• P. Constantin, C. Hallstrom and V. Putkaradze, Heat transport inrotating convection, Physica D 125, (1999),27.

• P. Constantin, Variational Bounds in Turbulent Convection, in Con-temporary Mathematics 238 (1999), G-Q. Chen and E. Di Benedettoand Chen Edtrs., AMS, Providence R.I, 77-88. 5-284.

• Ch. Doering and P. Constantin, On upper bounds for infinite Prandtlnumber convection with or without rotation, Journal Math. Phys., 42(2001), 784-795.

• P. Constantin, C. Hallstrom and V. Putkaradze, Logarithmic Boundsfor Infinite Prandtl Number Rotating Convection Journal Math. Phys.,42 (2001), 773-783.

• P. Constantin, Q. Nie and S. Tanveer, Bounds for second order struc-ture functions and energy spectrum in turbulence Phys. Fluids 11(1999), 2251-2256.

• P. Constantin, Energy spectrum of quasi-geostrophic turbulence, Phys.Rev. Lett. 89 (October 2002) 18, 184501.

2

Navier-Stokes Equations

∂u

∂t+ u · ∇u +∇p = ν∆u + f

∇ · u = 0.

f = deterministic, time independent. DomainD ⊂ R3. No slip boundary conditions condi-tions.

Energy Dissipation

ε = ν〈|∇u(x, t)|2〉.

〈. . .〉 is space-time average.Long time average

Mu = MT (u) =1

T

T∫0u(·, t)dt

Notation:

F 2 =1

|D|∫D|f |2dx

L−1 = F−1‖∇f‖L∞(dx)

U 2 = lim supT→∞

1

|D|MT

∫D|u|2dx = 〈|u|2〉

3

ε ∼ U 3/L

Upper bound:

ε ≤ U 3

L+√ε√νU

L

This implies, of course,

ε ≤ 2U 3

L+ ν

U 2

L.

and, if one does not like 2, one can use thequadratic equation...

ε ≤ U 3

L+νU 2

4L2+

√νU

2L

√√√√√√νU 2

L2+

4U 3

L

Proof: (Foias) WLOG ∇ · f = 0.

fi = ∂jM(ujui) + ∂iMp− ν∆Mu + Mut

because Mf = f . Take scalar product with f .∫D|f |2dx = −

∫D

(∂jfi)M(ujui)dx

+ν∫D∇f ·∇M(u)dx+

1

T(∫Df ·(u(·, T )−u(·, 0))dx.

4

It follows that:

F ≤ U 2

L+

ν

F |D|∫D|∇f | |∇Mu| dx + O(

1

T)

But|∇Mu|2 ≤M |∇u|2

(very sad...), so anyway,

F ≤ U 2

L+√ν√ε/L + O(

1

T).

Butε ≤ FU.

Lower Bound ?

5

Littlewood-PaleyDecomposition

Nonnegative, nonincreasing, radially symmetricfunction

φ(0)(k) = φ(0)(k)

φ(0)(k) = 1, ∀k ≤ 5

8k0,

φ(0)(k) = 0, ∀k ≥ 3

4k0.

The positive number k0 is the wavenumber unit.

φ(n)(k) = φ(2−nk), ψ(0)(k) = φ(1)(k)−φ(0)(k)

ψ(n)(k) = ψ(0)(2−nk), n ∈ Z.

ψ(n)(k) = 1, ∀k ∈ 2n[3

4k0,

5

4k0],

ψ(n)(k) = 0, ∀k /∈ 2n[5

8k0,

3

2k0].

1 = φ(n)(k) +∞∑m=n

ψm(k), ∀n ∈ Z.

I = S(n) +∞∑m=n

∆m, ∀n ∈ Z.

6

Littlewood-Paley operators S(m) and ∆n : mul-tiplication, in Fourier representation by φ(m)(k)and, respectively by ψ(n)(k).

For mean-zero functions F that decay at in-finity, S(m)F → 0 as m→ −∞.The LP decomposition is:

F =∞∑

n=−∞∆nF

∆nF =∫R3 Ψ(n)(y)(δyF )dy = F(n).

Ψ(n)(y) = (2π)−3∫eiy·ξψ(n)(ξ)dξ

Ψ(n) = ψ(n)

and(δyF )(x) = F (x− y)− F (x).

∆n is a weighted sum of finite difference opera-tors at scale 2−nk−1

0 in physical space. For eachfixed k > 0 at most three ∆n do not vanish intheir Fourier representation at k:

∆nF (k) 6= 0⇒ n ∈ Ik = [−1, 1] + log2

kk0

).

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LP Evolution for NSE

u(x, t) =∞∑

n=−∞u(n)(x, t)

where

u(n)(x, t) =∫R3 Ψ(n) (y)u(x−y, t)dy = (∆nu)(x, t)

Forcef (x) =

∞∑n=−∞

f(n)(x)

Navier-Stokes Littlewood-Paley components equa-tion:

(∂t + u · ∇ − ν∆)u(n) +∇p(n) = W(n) + f(n)

where p(n) = ∆np, are the Littlewood-Paleycomponents of the pressure and

W(n)(x, t) =∫R3 Ψ(n) (y) ∂yj (δy(uj)(x, t)δy(u)(x, t)) dy.

8

2D NS

(∂t + u · ∇ − ν∆)ω = f

with

u(x, t) =1

2π

∫ y⊥|y|2

ω(x− y, t)dy

LP vorticity 2DNS

(∂t + u · ∇ − ν∆)ω(n) = f(n) + W(n),

whereW(n)(x, t) =

−∫(∂yj

(Ψ(n)(y)

))(δyuj)(x, t)(δyω)(x, t)dy.

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Surface QG

(∂t + u · ∇ + υΛ) θ = f

u = ∇⊥Λ−1θ

Λ = (−∆)12

LP SQG

∂tθ(n) + u · ∇θ(n) + υΛθ(n) = W(n) + f(n)

with

W(n) =∫

Ψ(n)(y)∇y · (δy(u)δyθ)dy

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Littlewood-Paley spectrum

Energy spectrum:

E(k) =∫l=k〈|u(l, t)|2〉dS(l)

∫ ∞0 E(k)dk =

⟨|u|2

⟩

LP Spectrum:

ELP (k) =1

k

∑n∈Jk〈|u(n)(k, t)|2〉

with

Jk = [−2, 2] + log2

kk0

.

This represents the mean-square average ofthe components u(n) associated to the scale k.Note:

u(k) =∑n∈Ik

u(n)(k)

with

Ik = [−1, 1] + log2

kk0

.

11

The Littlewood-Paley spectrum is closely re-lated to a shell average of the traditional energyspectrum. If l is a wave number whose magni-tude l is comparable to k, k2 ≤ l ≤ 2k, then

u(l, t) =∑

n∈Jku(n)(l, t)

because Il ⊂ Jk. It follows that

1

k

∫ 2kk2E(l)dl ≤ 5ELP (k).

Viceversa, because the functions ψ(n), n ∈ Jkare non-negative, bounded by 1, and supportedin [ 5

32k, 6k] one has also

ELP (k) ≤ 5

k

∫ 6k5k32E(l)dl

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2DNS Spectrum

2D Kraichnan spectrum:

E(k) = C 〈η〉23 k−3.

η = ν|∇ω|2 is the rate of dissipation of enstro-phy.

Theorem 1 Consider forcing with spectrumlocated in k ≤ kf . For any 0 < a < 1 suchthat kf ≤ akd there exists a constant Ca suchthat the Litlewood-Paley energy spectrum ofsolutions of two dimensional forced Navier-Stokes equations obeys the bound

ELP (k) ≤ Cak−3

for k ∈ [akd, kd]. Consequently the tradi-tional spectrum obeys

1

k

2k∫k2

E(l)dl ≤ Cak−3.

13

‖ω(n)‖L2(dx) ≤ C2−nk−20 ‖∇ω(n)‖L2(dx)

‖δyu‖L∞(dx) ≤ |y|‖∇u‖L∞(dx)

‖δyω‖L2(dx) ≤ |y|‖∇ω‖L2(dx)

∫|y|

∣∣∣∣∇y

{Ψ(m) (y)

}∣∣∣∣ dy ≤ C

‖W(n)(·, t)‖L2(dx) ≤ Γ(t)2−nk−10

Γ(t) = ‖∇u(·, t)‖L∞(dx)‖∇ω(·, t)‖L2(dx)

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3D NS Spectrum

Kolmogorov spectrum:

E(k) = C 〈ε〉23 k−

53 .

ε = ν{〈|∇u|3〉

}23

kd = ν−34(ε)

14 .

Theorem 2 Consider forcing with spectrumlocated in k ≤ kf . Assume solutions of thethree dimensional Navier-Stokes equation sat-isfy ε < ∞. For any 0 < a < 1 such thatkf ≤ akd there exists a constant Ca such that

ELP (k) ≤ Cak−5

3

holds for k ∈ [akd, kd]. Consequently

1

k

2k∫k2

E(l)dl ≤ Cak−5

3 .

Nie, Tanveer, C: Taylor microscale Reynoldsnumbers up to Rλ = 155: inertial range withE(k) ∼ C ε2/3 k−5/3 with constant C.

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SQG Inverse cascade

Swinney experiment:

E(k) ∼ k−2, k ≤ kf .

Theorem 3 Consider forcing with spectrumlocated in k ≥ kf . Then

ELP (k) ≤ Ck−2

holds for solutions of SQG, for all k < kf .The constant has units of length per timesquared and is given by

C = c0E12υ−3〈|f (r, t)|2〉.

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Ideas of proof

Basic balance

υk〈∣∣∣∣θ(n)(k, t)

∣∣∣∣2〉 = 〈W(n)(r, t)θ(n)(r, t)〉.for k < kf , n ∈ Ik. The inverse cascade re-gion will be described by wave numbers smallerthan the minimal injection wave number kf .The inverse cascade region corresponds thus, inthe Littlewood-Paley decomposition, to indicesn > −∞ that satisfy 2n+1k0 < kf . We shownow that the right hand side of the equation isbounded above, uniformly for all such n > −∞.

〈W(n)(r, t)θ(n)(r, t)〉 =

−∫∇yΨ(n)(y)〈δy(u)(r, t)δy(θ)(r, t)θ(n)(r, t)〉dy.

The term θ(n) is bounded pointwise by apply-ing the Fourier inversion formula and Schwartzinequality:

|θ(n)(r, t)| ≤ cψ2nE(t)12

where c2ψ = (2π)−2 ∫ |ψ(0)(k)|2dk and

E(t) =∫|θ(r, t)|2dr =

∫|u(r, t)|2dr

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is proportional to k0 and depends on the choiceof the Littlewood-Paley template ψ(0) only throughthe non-dimensional positive absolute constantc0. The number

η = 〈η(t)〉

is related to the long time dissipation. It canbe bound in terms of the forcing term using theSQG

1

2

d

dtE(t) + υη(t) =

∫f (r, t)θ(r, t)dr.

Time average:

η ≤ υ−2k−1f 〈|f (r, t)|2〉

This bound diverges for very large scale forcing,i.e. when kf → 0. Nevertheless, because of thepresence of the coefficient 2n+1 and the fact that2n+1k0 ≤ kf in the inverse cascade region, thetotal bound on the spectrum does not divergeas kf → 0:

|〈W(n)(r, t)θ(n)(r, t)〉| ≤ c0υ−2E

12〈|f (r, t)|2〉.

19

Higher order structure functionsTraditional assumptions of scaling:

〈|δyu|m〉 ∼ Um

|y|L

ζm

UL/ν → ∞. Kolmogorov ’41: ζm = m3 . Lan-

dau : Intermittency. Kraichnan: passsive scalaranomalous scaling.

Fractional structure functions

For arbitrary m ≥ 1 and 0 ≤ r ≤ 1 definefractional structure functions

sm;r(u) = supx0

lim supT→∞

1

T

T∫0

1

|Bρ|∫Bρ

|δyu(x, t)|mdxr

dt.

Assume: scaling down to the dissipation scale,and spatial ergodicity. Then:

ζ6m

6m≥ ζ4m

4m− 1

m.

Rules out ζm ∼ mp for 0 < p < 1.

20

For the proof:

Ψ = |δyu|m = |q|m

From PDE

ν

4

1− 1

m

|∇Ψ|2 ≤ I + II + III

I = ∂yj((δyuj)Ψ

2),

II = 2mq · g|q|2m−2.

(with gi = δyf − ∂xi(δyp))

III = − (∂t + u · ∇ − ν∆) Ψ2

Use local Sobolev inequality, keep track of non-local terms and m, evaluate at dissipative cut-off.

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Bounds for Bulk HeatTransport

Boussinesq Rayleigh Benard

∂u

∂t+ u · ∇u +∇p = σ∆u + σRaeT,

∇ · u = 0∂T

∂t+ u · ∇T = ∆T.

Box of height 1 and lateral side L. T = 1 atthe bottom boundary and T = 0 at top.⟨

|∇T |2⟩

= N.⟨|∇u|2

⟩= Ra(N − 1).

Theorem 4 There exists an absolute con-stant C, independent of Rayleigh number Ra,aspect ratio L and Prandtl number σ suchthat

N ≤ 1 + C√Ra

holds for all solutions of the Boussinesq equa-tions.

Howard, Busse; Doering, C.

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Rotating Infinite PrandtlNumber Convection

(∂t + u · ∇)T = ∆T

−∆u− E−1v + px = 0−∆v + E−1u + py = 0−∆w + pz = RT.

∇ · u = 0

B. C.: ((u, v, w), p, T ) periodic in x and y withperiod L; u, v, and w vanish for z = 0, 1, T = 0at z = 1, T = 1 at z = 0.

N =⟨‖∇T‖2

⟩

Theorem 5 There exist absolute constantsc1, ..., c4 so that the Nusselt number for ro-tating infinite Prandtl-number convection isbounded by

N − 1 ≤ minc1R

25 ; (c2E

2 + c3E)R2;

c4R1/3(E−1 + log+R)

23

.Hallstrom, Putkaradze, Doering, C.

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Infinite Prandtl Number Equa-tions

Active scalar equation

(∂t + u · ∇)T = ∆T (1)

equations of state:

−∆u + px = 0, (2)

together with

−∆v + py = 0 (3)

and−∆w + pz = RT. (4)

R represents the Rayleigh number. The velocityis divergence-free

ux + vy + wz = 0. (5)

The horizontal independent variables (x, y)belong to a basic square Q ⊂ R2 of side L.Sometimes we will drop the distinction betweenx and y and denote both horizontal variables x.

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The vertical variable z belongs to the interval[0, 1]. The non-negative variable t representstime. The boundary conditions are as follows:all functions ((u, v, w), p, T ) are periodic in xand y with period L; u, v, and w vanish forz = 0, 1, and the temperature obeys T = 0 atz = 1, T = 1 at z = 0. We write

‖f‖2 =1

L2

∫ 10

∫Q |f (x, y, z)|2dzdxdy

for the (normalized) L2 norm on the whole do-main. We denote by ∆D the Laplacian withperiodic-Dirichlet boundary conditions. We willdenote by ∆h the Laplacian in the horizontal di-rections x and y. We will use < · · · > for longtime average:

〈f〉 = lim supt→∞

1

t

∫ t0 f (s)ds.

We will denote horizontal averages by an over-bar:

f (·, z) =1

L2

∫Q f (x, y, z)dxdy.

25

We will also use the notation for scalar product

(f, g) =1

L2

∫ 10

∫Q(fg)(x, y, z)dxdydz.

The Nusselt number is

N = 1 + 〈(w, T )〉 . (6)

One can prove using the equation (1) and theboundary conditions that

N =⟨‖∇T‖2

⟩(7)

and using the equations of state (2 - 4) that⟨‖∇u‖2

⟩= R(N − 1). (8)

This defines a Nusselt number that depends onthe choice of initial data; we take the supre-mum of all these numbers. The system hasglobal smooth solutions for arbitrary smoothinitial data. The solutions exist for all timeand approach a finite dimensional set of func-tions. If we think in terms of this dynamicalsystem picture then the Nusselt number repre-sents the maximal long time average distance

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from the origin on trajectories. Because all in-variant measures can be computed using tra-jectories the Nusselt number is also the maxi-mal expected dissipation, when one maximizesamong all invariant measures.

Bounding the heat flux

We take a function τ (z) that satisfies τ (0) = 1,τ (1) = 0, and write T = τ + θ(x, y, z, t). Therole of τ is that of a convenient background;there is no implied smallness of θ, but of courseθ obeys the same homogeneous boundary con-ditions as the velocity. The equation obeyed byθ is

(∂t + u · ∇ −∆) θ = −τ ′′ − wτ ′ (9)

where we used τ ′ = dτdz . We are interested in the

function b(z, t) defined by

b(z, t) =1

L2

∫Qw(·, z)T (·, z)dx.

27

Its average is related to the Nusselt number:

N − 1 =⟨∫ 1

0 b(z)dz⟩.

Note thatT − T = θ − θ

Also note that from the boundary conditionsand incompressibility

w(z, t) = 0

and therefore

b(z, t) =1

L2

∫Qw(·, z)θ(·, z)dx.

From the equation (9) it follows that

N =⟨−2

∫ 10 τ′(z)b(z)dz − ‖∇θ‖2

⟩+∫ 10 (τ ′(z))

2dz.

(10)Now we are in a position to explain the varia-tional method and some previous results. Con-sider a choice of the background τ that is “ad-missible” in the sense that⟨

−2∫ 10 τ′(z)b(z)dz − ‖∇θ‖2

⟩≤ 0

28

holds for all functions θ. Then of course

N ≤∫ 10 (τ ′(z))

2dz.

The set of admissible backgrounds is not empty,convex and closed in the H1 topology. Thebackground method, as originally applied, isthen to seek the admissible background thatachieves the minimum

∫10 (τ ′(z))2 dz. Such an

approach would predictN ≤ cR12 for this active

scalar, just as in the case of the full Boussinesqsystem. One can do better. Let us write

b(z, t) =1

L2

∫Q

∫ z0

∫ z10 wzz(x, z2, t)θ(x, z)dxdzdz2dz1.

(11)It follows that

|b(z, t)| ≤ z2 (1 + ‖τ‖L∞) ‖wzz‖L∞(dz;L1(dx)).(12)

Now we use two a priori bounds. First, onecan prove using (9) and (8) that there exists apositive constant C∆ such that⟨‖∆θ‖2

⟩≤ C∆

{RN +

∫ 10

[(τ ′′(z))

2+ Rz(τ ′(z))2

]dz

}(13)

29

holds. Secondly, one has the basic logarithmicbound

‖wzz‖L∞ ≤ CR(1+‖τ‖L∞)[1+log+(R‖∆θ‖)]2.(14)

Using (14) together with (13) in (12) one de-duces from (10)

N ≤∫ 10 (τ ′(z))2dz+CR(1+‖τ‖L∞)2

[∫ 10 z

2|τ ′|dz]

[1 + log+

{RN +

∫ 10

[(τ ′′(z))

2+ Rz(τ ′(z))2

]dz

}](15)

Choosing τ to be a smooth approximation ofτ (z) = 1−z

δ for 0 ≤ z ≤ δ and τ = 0 for z ≥ δand optimizing in δ one obtains

Theorem 6 There exists a constant C0 suchthat the Nusselt number for the infinite Prandtlnumber equation is bounded by

N ≤ N0(R)

where

N0(R) = 1 + C0R1/3(1 + log+R)

23

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Dissipation and Spectrum Peter Constantinpeople.cs.uchicago.edu/~const/disspetalk.pdf · E(l)dl 5E...

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