Exploiting Structure:Asymptotically-Reduced and Low-Order Models of

Convective and Shear TurbulenceGregory P. Chini

Program in Integrated Applied Mathematics, University of New Hampshire

Keith JulienDepartment of Applied Mathematics, University of Colorado, Boulder

Edgar KnoblochDepartment of Physics, University of California, Berkeley

Charles R. DoeringDepartments of Mathematics and Physics, University of Michigan

Zhexuan Zhang, Ziemowit Malecha, Baole WenDepartment of Mechanical Engineering, University of New Hampshire

GPC acknowledges support from NSF awards CMG 0934827, DMS award 0928098.

February 2nd, 2011

Modeling/Analysis Spectrum for Spatiotemporally Complex(“Turbulent”) Flows

Ad hoc physically motivated models. . .

. . . Formal systematic analysis (“1st principles”?). . .

. . . . . . Rigorous analysis of Navier–Stokes equations

Themes:

• Geophysical applications (ocean surface boundary layer).

• Deterministic models.

• Mechanistic viewpoint: Turbulence structure (anisotropy, localization). . .

“The mind imbued with Newtonian mechanics seeks simple deterministicexplanations of phenomena.” (S. Pope, Turbulent Flows, p. 323)

3 Classes of Systematic Models. . . and 3 Questions

1. Systematic multiscale models of upper ocean turbulence.Question: Reliance on scale separation?

2. Asymptotically-exact reduced PDEs for constrained turbulent flows.Question: Application to wall-bounded shear flow turbulence?

3. Low-order ODE models of thermal convection from upper bound theory.Question: What if upper bounds are poor (or don’t exist!)?

I. Systematic Multiscale Models ofGeophysical Flows

Ocean Turbulence (R. Ferrari)

Approaches

(i) Physically motivated, ad hoc parameterizations (or complete omission!) ofSGS processes in OGCMs, etc.

(ii) Presuming scale separation, systematic (but formal) multiple space andtime scale asymptotic analysis (Klein, Majda, Julien. . . ):

– To mitigate closure problems;

– To provide valuable guidelines regarding the reduced or averaged infothat must be communicated from small scales to large;

– To provide the basis for multiscale computational strategies (e.g. theHeterogeneous Multiscale Method of E & Engquist).

– Successes: Madden–Julien oscillation in atm. dynamics (Majda).

– Illustrative example: upper ocean turbulence.

Turbulence in Ocean Surface BL (R. Weller)

• O(10–100)-m non-hydrostatic, weakly stratified, “fully” 3D turbulence.

Submesoscale Phenomena (B. Fox-Kemper)

• O(10)-km rotationally influenced (if not constrained), stratified, hydrostaticinternal waves (IWs), eddies, fronts, and their associated instabilities.

Master PDEs: Stratified Rotating Craik–Leibovich (CL) Eqns

∂Tu+ (v⊥ ·∇⊥)u−1

Rov =

1

Re

∂2Y

+1

δ2∂2z

u

∂Tv+ (v⊥ ·∇⊥) v+1

Rou = −∂Y p+Us∂Y u+

1

Re

∂2Y

+1

δ2∂2z

v

∂Tw+ (v⊥ ·∇⊥)w = −1

δ2∂zp+

1

δ2Us∂zu+

Γ

δ2b+

1

Re

∂2Y

+1

δ2∂2z

w

∂T b+ (v⊥ ·∇⊥) b =1

Pe

∂2Y

+1

δ2∂2z

b

∇⊥ · v⊥ = 0

Free–Surface BC

1

Reδ∂zu =

u∗U

2

Scaling & Parameters

Scale Submesocale Value BL Value

Horizontal Length L 1–10 km l = h 50-100 mHorizontal Velocity U 0.1 m/s U 0.05–0.1 m/s

Wind Stress u∗ 0.01 m/s u∗ 0.01 m/sBL Depth h 50-100 m h 50-100 m

Stokes Drift Velocity us0 0.1 m/s us0 0.1 m/sBuoyancy Anomaly B = g|∆ρ|/ρ0 0.001 m/s2 B 0.001 m/s2

Vertical Velocity Uh/L <0.01 m/s U 0.05–0.1 m/sAdvection Time L/U 5–10 hr h/U 0.5 hr

Dynamic Pressure ρ0U2 10 Pa ρ0U

2 10 Pa

Parameters: δ = h/L Ro = U/fL Γ = Bh/U2 (Re, Pe) = UL/(ν,κ)

Asymptotic Analysis I: Distinguished Limit

1. Perform asymptotic analysis by exploiting smallness of δ ≈ 0.01− 0.1.

2. Consider distinguished limit in which δ → 0 with:

Ro = O(1); Γ = O(1)

(Re, Pe) = δ−2(R, P ), where (R, P ) = O(1)

(u∗/U)2 = δτ , where τ = O(1)

3. Introduce fast space and time scales: y = Y/δ and t = T/δ.

4. Multiscale differentiation: ∂Y → ∂Y + δ−1

∂y; ∂T → ∂T + δ−1

∂t.

Asymptotic Analysis II: Multiscale Asymptotic Expansions

5. Decompose fields into fast-(y,t) mean plus fluctuation:

u(Y, z, T ) ∼ u(Y, z, T ) + u(y, Y, z, t, T )

v(Y, z, T ) ∼ v(Y, z, T ) + v(y, Y, z, t, T )

w(Y, z, T ) ∼ w(Y, z, T ) +1

δw(y, Y, z, t, T )

p(Y, z, T ) ∼ p(Y, z, T ) + p(y, Y, z, t, T )

b(Y, z, T ) ∼ b(Y, z, T ) + b(y, Y, z, t, T )

f(Y, z, T ) =lim

L→∞T → ∞

1

4T L

T

−T

L

−Lf(y, Y, z, t, T ) dydt

Mean Equations: Submesoscale Dynamics

DT u+∂Y

vu+δ

−1∂z

wu−Ro

−1v = R

−1∂2z u

DT v+∂Y

vv+δ

−1∂z

wv+Ro

−1u = −∂Y p+Us∂Y u+R

−1∂2z v

0 = −∂zp+b+Us∂zu− ∂z

ww

DT b+∂Y

vb+δ

−1∂z

wb

= P−1

∂2z b

∂Y v + ∂zw = 0

Free-Surface BC: R−1

∂zu = τ

where DT ≡ ∂T + v∂Y + w∂z.

Fluctuation Equations: Langmuir Turbulence

Dtu

+w

∂zu = δR

−1∂2y+∂

2z

u

Dtv

+w

∂zv = −∂yp

+Us∂yu

+δR

−1∂2y+∂

2z

v

Dtw

− ∂z

ww

= −∂zp+Us∂zu

+b

+δR

−1∂2y+∂

2z

w

Dtb+w

∂zb = δP

−1∂2y+∂

2z

b

∂yv+ ∂zw

= 0

Free-Surface BC: R−1

∂zu = 0

where Dt≡ ∂t + (v + v

)∂y + w∂z.

CommentsX

Z

Y

!+"!

!

Y

y

• Explicit identification of dominant multiscale coupling terms.

• Pattern-forming geophysical BLs: LC, buoyancy-driven convection, Ekmanrolls, even uni-directional shear flows?

• Opportunity to borrow ideas, techniques from pattern formation theory:Phase-diffusion/mean-drift equations via nonlinear WKBJ analysis.

• Informs design of multiscale numerical algorithms (e.g. HMM).

Discussion Topic # 1: Scale Separation?

• How small does δ have to be in practice? Dynamic assessment?

• How wide does micro-scale domain have to be for stable averages/fluxes?

• “There may very well be clean scale separations even if they are not visiblein spectral decompositions” (R. Klein, Annu. Rev. Fluid. Mech. 2010).

• Extension of homogenization and related multiscale asymptotic techniquesto problems lacking (spectral) scale separation (T. Hou). . . ?

Discussion Topic # 1: Scale Separation? (Cont’d)

Adaptation of T. Hou Strategy

Consider passive scalar advection for Pe 1.

∂TC + (V ·∇)C =1

Pe∇2

C

C(X,Y,0) = C0(X,Y )

C(X,Y ), L–periodic in 0 ≤ X ≤ L and 0 ≤ Y ≤ L.

1. Re-partition C0, C, and V into formal 2-scale representation; e.g.:

C0(X) = C0(X) + C0(X,X/ε), where ε = ∆X/L = ∆Y/L.

2. Introduce phase variable and perform Lagrangian (not Eulerian) averaging:

∂TΘ+ (V ·∇)Θ = 0

Θ(X,0) = X

Discussion Topic # 1: Scale Separation? (Cont’d)

C0 =

|k|>1/2ε

C0(k)expi2π

Lk ·X

=

1εk(s)+k(l)

C0

1εk(s)+k(l)

exp

i2π

Lk(l) ·X

exp

i2π

Lk(s) ·

X

ε

=

k(s) =0

C0s

k(s),X

exp

i2π

Lk(s) ·

X

ε

= C0

X,X

ε

where: C0s

k(s),X

=

k(l)C0

1εk(s)+k(l)

exp

i2π

Lk(l) ·X

⇒ C= C

X,Θ

ε, T,

T

ε

II. Asymptotically-Reduced PDEs forConstrained Turbulent Flows

Reduced Models of Constrained Flows:Old Idea. . .

• “Large-scale” geophysical and astrophysical flows constrained by rotation,stratification, and/or magnetic fields.

• Strong constraint imposed by dominant force on flow drives near two-dimensionalization, reduced mode coupling in certain directions.

• Departures from 2D dynamics remain fundamentally important, and flowsremain strongly nonlinear.

• Reduced models obtained by linking emergence of strongly anisotropicflow structures to imposed constraint and exploiting asymptotic disparity inlength and time scales.

• Example: Quasi-Geostrophic equations in oceanic/atmospheric flows.

Reduced Models of Constrained Flows:Modern Developments

Extensions to constrained but non-hydrostatic flows:

• Julien, Knobloch have greatly extended these techniques to:

(i) rapidly rotating thermal convection;(ii) thermal convection in a strong magnetic field;(iii) the magnetorotational instability (in accretion disks).

• These flows are both constrained and centrifugally unstable.

Are extensions to shear flows possible?

• Anisotropic Langmuir turbulence. . . (Tandon & Leibovich, JPO 1995,Chini et al. GAFD 2009)

• Plane Couette flow (PCF) turbulence? (Waleffe, Nagata, Busse. . . )

Anisotropic LC Dynamics

–Szeri (1996) –Marmorino et al. (2005) –McWilliams et al. (1997)

Isotropically Scaled CL Equations

• Consider full 3D, isotropically-scaled CL equations, where two parameters

R∗ ≡ u∗H/νe, Lat =u∗/us0 replace single parameter La ≡ LatR

−3/2∗ :

Du

Dt= −∇p +

1

La2t

(Us × ω) +1

R∗∇2u

• Two turbulence regimes:

Shear flow turbulence regime: Lat 1 with R∗ 1.Langmuir turbulence regime: Lat = O(0.1) with La 1.

• Motivates consideration of formal limit Lat → 0 with R∗ or La fixed.

Implications of Leading-Order Balance

• Upon rescaling the pressure, the leading-order balance is:

∇P = Us ×Ω ⇒ Us ·∇Ω = Ω ·∇Us

• From this balance, the following deductions can be made:

∂xP = 0

Us∂xΩx = ΩzUs(z)

Us∂xΩy = 0

Us∂xΩz = 0

• 2D dynamics: Ωx =0, ∂xΩx=0, u–fluctuations (v, w)–fluctuations.

Anisotropic Velocity Scalings

• Employ anisotropic velocity scales to capture nonlinear, spatially anisotropicreduced dynamics:

Lx = H, (Ly, Lz) = H, T = H/V

U = u∗R∗, (V,W) =Uus0, P = ρV2

• In essence, perturbing off of strictly 2D [∂(·)/∂x = 0] problem.

• Identify U/W = Lat(Lat/La)1/3 ≡ ε 1

(cf. Tejada-Martinez & Grosch JFM 2007, Teixeira & Belcher JFM 2002).

Rescaled CL Equations in Strong CL Vortex-Force Limit

∂tu+ εu∂xu+ (v⊥ ·∇⊥)u = −ε−1

∂xP +La

∂2x +∇2

⊥u

∂tv⊥ + εu∂xv⊥ + (v⊥ ·∇⊥)v⊥ = −∇⊥P +La

∂2x +∇2

⊥v⊥

+ Us

∇⊥u− ε

−1∂xv⊥

ε ∂xu+∇⊥ · v⊥ = 0

• Wind stress BC: ∂zu = 1 along z = 0,−1.

• x-invariance at leading-order: ∂xP = ∂xv = ∂xw = 0 and ∇⊥ · v⊥ = 0.

Multiple Scale Expansion

1. Limit process: ε → 0 with La fixed.

2. Introduce slow x scale: X ≡ εx so that ∂x → ∂x + ε∂X .

3. Expand fields:

u(x, y, z, t) = u0(x,X, y, z, t) + εu1(x,X, y, z, t) + . . .

v⊥(x, y, z, t) = v0⊥(X, y, z, t) + εv1⊥(x,X, y, z, t) + . . .

P (x, y, z, t) = P0(X, y, z, t) + εP1(x,X, y, z, t) + . . .

4. Substitute into PDEs, collect terms of like order and average over fast x.

5. Obtain closed set of equations for u0 ≡ U(X, y, z, t), v0⊥ ≡ V⊥(X, y, z, t)and P0 ≡ Π(X, y, z, t).

Reduced PDEs

• Define:D

⊥t (·) ≡ ∂t(·)+(V⊥ ·∇⊥)(·) ≡ ∂t(·)+J[(·),ψ],

where J[(·),ψ] = ∂zψ∂y(·)− ∂yψ∂z(·).

• Reduced dynamics governed by:

D⊥t U = −∂XΠ+La∇2

⊥U

D⊥t Ω+Us(z)∂XΩ = U

s(z)(∂XV − ∂yU)+La∇2

⊥Ω

∇2⊥Π = 2J[∂yψ, ∂zψ]+∇⊥ · (Us(z)∇⊥U)+U

s(z)∂X(∂yψ)

∇2⊥ψ = −Ω, V⊥ ≡ ∇⊥ × ψı

• Fast x averaged BCs along z = 0,−1: ∂zU = 1, Ω = 0, ψ = 0.

• Advection by U and stretching of Ω are subdominant processes.

Pseudospectral Numerical Simulations of Reduced PDEs

300 400 500 600 700 800 9000.128

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Surface Tracer Evolution: Windrows and Y-Junctions

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Comments

• Derived reduced PDEs for anisotropic turbulent Langmuir circulation instrong vortex-force limit.

• Reduced PDEs capture dominant linear and secondary instabilities.

• Reduced PDEs offer several analytical and computational advantages:

1. Filter fine x-scale variability ⇒ larger ∆x and possibly ∆t.

2. Vortex stretching is sub-dominant process ⇒ facilitates analysis (e.g.homogenization theory [T. Hou]).

3. Limiting procedure suppresses “self-sustaining process” proposed byWaleffe for wall-bounded shear-flow turbulence.

4. Methodology being applied to PCF turbulence. . .

Discussion Topic # 2: Extension to Wall-Bounded ShearFlow Turbulence?

Low Reynolds Number Plane Couette Flow (PCF) Turbulence

U = Uw, (V,W) = 1Re

Uw – J. Gibson

Reduction of NSE for Lower-Branch PCF “Turbulence”

Averaged Equations

∂T u0 + (y+u0) ∂Xu0 + (v1⊥ ·∇⊥) u0 + v1 =1

R∇2

⊥u0

∂T v1⊥ + ∂X [(y+u0) v1⊥] +∇⊥ ·v1⊥v1⊥+v1⊥v

1⊥

= −∇⊥p2 +

1

R∇2

⊥v1⊥∂Xu0 +∇⊥ · v1⊥ = 0

where R ≡ Re = O(1) is the reduced Reynolds number.

Fluctuation Equations

∂tu1 + (y+u0) ∂xu

1 +

v1⊥ ·∇⊥

u0 + v

1 = −∂xp

1

∂tv1⊥ + (y+u0) ∂xv

1⊥ = −∇⊥p

1

∂xu1 +∇⊥ · v1⊥ = 0

• See recent paper by Hall & Sherwin (JFM 2010).• SGS modeling of “outer–inner” BL coupling, spatial dynamics studies?

III. Low-Order (ODE) Models fromUpper Bound Theory

Motivation

• Low-dimensional (ODE) descriptions of nonlinear fluid phenomena requiredin increasing number of applications (flow control, parameter estimation,fine-scale dynamics in multiscale algorithms).

• Popular existing approaches fall into two broad categories:

1. Fully predictive models generated by (e.g.) Galerkin projection (GP)onto orthogonal basis functions (e.g. spectral expansions), but thesemodes generally not well adapted to reduced description of highly non-linear systems.

2. Proper Orthogonal Decomposition (POD) based approaches are oftenbetter suited for the low order description of highly nonlinear dynamics,but the POD modes are empirical.

Key Heuristic of New Approach

• Assertion: Eigenfunctions from energy stability theory provide a suitablea priori basis for reduced description of highly nonlinear dynamics. . . ifenergy stability is enforced about a suitable nonlinear base state (e.g.temperature profile) rather than the “conduction” solution.

• Idea is that, on attractor, dynamical system adjusts so that fluctuations aremarginally stable (Malkus 1954, Howard 1963).

• Poje & Lumley (1995,1997) closest in spirit to the methodology described,but they employed a semi-empirical mean profile and sought the fastestgrowing energy modes (≈ coherent structures).

Case Study: (2D) Porous Medium Convection

PDEs ∂tT +u ·∇T = ∇2T

∇ · u = 0

u+∇P = RaT ez

BCs z = 0,1 : T = 1,0; w = 0

x periodic

• The Nusselt number Nu is

Nu ≡ 1+ wT =|∇T |2

= 1+Ra

−1|u|2

≥ 1,

where · denotes a space–time average.

Algorithm

1. Following Doering & Constantin (1998), Otero et al. (2004), decompose:

T (x, z, t) ≡ τ(z) + θ(x, z, t), where τ(0) = 1 and τ(1) = 0.

2. Substituting yields dynamical evolution of θ:

∂tθ + u ·∇θ = ∇2θ − τ

w + τ

= −S(θ) − A(θ) + τ,

where: w = L(θ); S(θ) =1

2

τL(θ)+L(τ θ)

−∇2

θ.

3. Motivated by (rigorous) upper bound analysis, determine τ(z) by requiringΛ(S) ≥ 0 (“energy stability”) and minimizing

10 τ

(z)2dz (heat flux, Nu).

Nusselt–Rayleigh Scaling: Upper Bound Theory vs. DNS80

101

102

103

104

100

101

102

103

PSfrag replacements

Ra

Nu

Figure 4.6: Bounds and data for porous medium convection. The solidline shows the lower envelope of the numerical bound and the rigor-ous bound shown in figure 4.4. The square boxes show the data from[Graham and Steen, 1994]. The data indicated by asterisks is from a di-rect numerical simulation, for the two-dimensional problem, carried outby Hans Johnston. The box indicates the range of Elder’s 1967 exper-imental data where Nu = .025Ra ± 10% for 100 < Ra < 5000, see[Elder, 1967].

θ and the vertical component of velocity w. One then maximizes this functional over

all the relevant fields. By restricting the set over which the optimization takes place

to those fields that have one mode, two modes, etc., one obtains the one-alpha bound,

two-alpha bound, etc. Because this partial optimization will, in general, not give the

absolute maximum one seeks, the corresponding ‘bounds’ serve as lower estimates

above which the true maximum must lie.

Computation of Basis Functions

• A priori basis functions obtained by solving constrained, non-local eigen-value problem using Chebyshev spectral collocation method:

θ(x, z) =∞

n=−∞

∞

m=0Θmn(z)einkx, etc.

• 1. Guess initial background profile τ(z). 2. Solve e-value problem. 3.Repeat, subject to constraint λ0 ≥ 0, adjusting τ(z) to minimize upperbound functional

J [τ(z);Ra, k] ≡ 1

0τ(z)2dz

* For a given domain size (L) and Rayleigh number Ra, optimization codemust sample all admissible modes to ensure spectral constraint satisfiedfor all relevant horizontal wavenumbers.

Background Profile, Spectra, Basis Functions

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Steady Single Roll-Pair Solution at Ra=100T(x,z) Contours: DNS (upper) vs. 6-Mode Model (lower)

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Discussion Item #3: Challenges of New Low-Order ModelingApproach

• Ultimately, aim to construct low-order models of “turbulent” porous mediumconvection (e.g. at Ra = O(5000)) – with goal of reproducing turbulentstatistics. May be able to exploit (i) mode slaving and (ii) minimal flow units.

• BUT. . . “Only in a very limited sense can coherent structures ‘explain’ thebehavior of near-wall turbulent flows. [...] The goal of developing a quan-titative theory of near-wall turbulence based on dynamical interaction of asmall number of structures has not been attained, and is likely unattain-able.” (S. Pope, Turbulent Flows, p.323)

• What if bounds are poor or don’t exist. . . ? Possibility of introducingphysically (or asymptotically) motivated constraints? (R. Kerswell)

“Turbulence” at Ra = 7924, k = π

Convection in a porous layer 271

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Figure 4. Snapshots of the temperature field for (a) Ra= 315, (b) 500, (c) 1255, (d) 1581,(e) 5000, (f ) 7924.

and drifting upward (downward). We refer to the dynamics in this Rayleigh numberregime as ‘turbulent’, at least in the sense that it is spatially and temporally chaotic(presumably) displaying dynamics over a range of length scales. We also observe arobust ‘anomalous’ Nu–Ra scaling to emerge in this high-Ra range. The best fittingpower law to the data has an exponent very close 0.9, clearly distinct from the classicalNu ! Ra scaling law. The anomalous scaling regime we observe is relatively small –on the order of a single decade – and so the deviation from exponent 1 could be