an ensemble-based conventional turbulence model …

INTERNATIONAL JOURNAL OF c© 2018 Institute for ScientificNUMERICAL ANALYSIS AND MODELING Computing and InformationVolume 15, Number 4-5, Pages 492–519

AN ENSEMBLE-BASED CONVENTIONAL TURBULENCE

MODEL FOR FLUID-FLUID INTERACTION

JEFFREY M. CONNORS

(Communicated by L. Rebholz)

Dedicated to Professor William J. Layton on the occasion of his 60th birthday

Abstract. A statistical turbulence model is proposed for ensemble calculations with two fluidscoupled across a flat interface, motivated by atmosphere-ocean interaction. For applications, likeclimate research, the response of an equilibrium climate state to variations in forcings is importantto interrogate predictive capabilities of simulations. The method proposed here focuses on thecomputation of the ensemble mean-flow fluid velocities. In particular, a closure model is used forthe Reynolds stresses that accounts for the fluid behavior at the interface. The model is shown toconverge at long times to statistical equilibrium and an analogous, discrete result is shown for twonumerical methods. Some matrix assembly costs are reduced with this approach. Computationsare performed with monolithic (implicit) and partitioned coupling of the fluid velocities; the formerbeing too expensive for practical computing, but providing a point of comparison to see the effectof partitioning on the ensemble statistics. It is observed that the partitioned methods reproducethe mean-flow behavior well, but may introduce some long-time statistical bias.

Key words. Atmosphere-ocean, variance, statistical equilibrium

1. Introduction

Ensemble calculations with global circulation models (GCMs) are an impor-tant component of climate variability studies. Many long-time integrations areperformed to assess the average near-surface temperature change in response tochanges in forcings for the climate system at radiative equilibrium. There are mul-tiple sources of uncertainty in calculated responses. For a given computationalmodel, some studies focus on parametric uncertainty (see [8], 9.2.2.2). One wayto mitigate cost is to apply statistical models for simulation responses that requireonly a modest number of uncertain parameters and ensemble members, for examplein perturbed physics ensemble methods (e.g. [22]). Given a moderate number of en-sembles, this paper investigates another possible cost-reduction measure. Based onthe work of Jiang, Kaya and Layton [10], a method is proposed herein to computeensemble-mean flow states efficiently, by using a conventional (statistical) turbu-lence model (CTM). CTM models (like RANS [1] or k − ǫ [20]) seek to reduce thenumber of degrees of freedom required to resolve mean fluid behavior. The methodhas the additional benefit of reducing some matrix-assembly costs for the ensemblecomputations.

The focus is on atmosphere-ocean interaction (AOI). A key aspect of many AOImodels is to avoid resolving the boundary layers, instead relying on boundary con-ditions that conserve fluxes across the layers. In order to reduce the problem downbut retain this key mathematical detail, Connors, Howell and Layton investigateda model of two incompressible fluids coupled across a flat interface [5]. A similarmodel is adopted here, since it provides a convenient setting in which to incorporate

Received by the editors December 30, 2016, and in revised form, April 28, 2017.1991 Mathematics Subject Classification. 76F55,65Z05,65M60.

492

ENSEMBLE-BASED CONVENTIONAL TURBULENCE MODEL 493

the work of Jiang, Kaya and Layton. However, it is necessary to account for thespecial dynamics in AOI introduced by the differences in vertical versus horizontalscaling, and also by the interface boundary conditions.

1.1. A conventional turbulence model for coupled fluids. Consider an en-semble of velocities and pressures for flows in the atmosphere and ocean (domainsΩA,ΩO ⊂ R

d, d = 2, 3, respectively), satisfying

∂tuj −DA(uj) + uj · ∇uj +∇pj = fAj on ΩA × (0, T ],(1)

∇ · uj = 0 on ΩA × (0, T ],(2)

uj(x, t = 0) = u0j(x) on ΩA,(3)

∂tvj −DO(vj) + vj · ∇vj +∇qj = fOj on ΩO × (0, T ],(4)

∇ · vj = 0 on ΩO × (0, T ],(5)

vj(x, t = 0) = v0j (x) on ΩO.(6)

Here, DA(uj) and DO(vj) are viscosity terms. Let D(u) represent any d×d tensoror matrix, with entries D(u)ij , 1 ≤ i, j ≤ d. Define a decomposition by

(7)

D(u) = D(u)H +D(u)⊥,

(D(u)H

)ij=

(D(u))ij for i = 1, . . . , d− 1, j = 1, . . . , d,

0, otherwise

(D(u)⊥

)ij=

(D(u))ij for i = d, j = 1, . . . , d,

0, otherwise

Now let D(u) = (∇u+∇uT )/2 be specifically the viscous part of the Cauchy stresstensor. The diffusion terms are decomposed into horizontal and vertical terms:

DA(uj) = 2∇ ·(νA

HD(uj)H + νA

⊥D(uj)⊥),(8)

DO(vj) = 2∇ ·(νO

HD(vj)H + νO

⊥D(vj)⊥).(9)

The constants νAH > 0 and νO

H > 0 are horizontal diffusion parameters, whereasνA

⊥ > 0 and νO⊥ > 0 are (constant) vertical diffusion parameters. The horizontal

scale is much larger than the vertical scale for atmosphere-ocean simulations. Dueto the nature of flow features that result from this scale discrepancy, it is typi-cal in practice to treat horizontal and vertical diffusion processes differently (see,e.g. [21, 23]). While many other aspects of typical atmosphere-ocean models arenot included here, the above model retains the necessary mathematical features forthe investigation in this paper. Boundary conditions are discussed later.

We assume that j = 1, . . . , J and define the ensemble average, say a, of anycollection of J objects aj by

(10) a ≡< aj >≡1

J

J∑

j=1

aj .

A model for the ensemble-averaged mean flow is

∂tu−DA(u) + u · ∇u+∇p−∇ · R(u,u) = fA on ΩA × (0, T ],(11)

∇ · u = 0 on ΩA × (0, T ],(12)

u(x, t = 0) = u0(x) on ΩA,(13)

∂tv −DO(v) + v · ∇v +∇q −∇ ·R(v,v) = fO on ΩO × (0, T ],(14)

∇ · v = 0 on ΩO × (0, T ],(15)

v(x, t = 0) = v0(x) on ΩO,(16)

494 J. CONNORS

where the (yet to be closed) Reynolds stresses are denoted by

R(u,u) ≡< uj >< uj > − < ujuj > .

The key is what happens at the interface between the fluids, which are coupledacross a flat interface, ΓI . The coupling is based on the so-called “rigid-lid” hy-pothesis that has often been used for atmosphere-ocean coupling; that small-scalefluctuations can be eliminated by using a flat ocean surface model and still capturethe correct long-time statistical behavior. Also, the boundary layers of the atmo-sphere and ocean are not resolved but are instead replaced by using formulae for thebulk transfer of physical fluxes between the fluids. A full description of the prim-itive equations of the coupled atmosphere-ocean system and their mathematicalanalysis is found in the papers of Lions, Temam and Wang [16, 17, 18].

Let the outward-pointing normal vectors of unit length on the boundaries of ΩA

and ΩO be denoted by nA and nO, respectively. A generic unit vector τ is alsoused, defined by τ = (±1, 0) (d = 2) or τ = (τ1, τ2, 0) (d = 3). On the interfacethe fluids cannot penetrate and the transfer of horizontal momentum across theboundary layers is represented using a slip-with-friction condition:

(17)uj · nA = vj · nO = 0,

−ρAνA⊥nA · ∇uj · τ = ρOνO⊥nO · ∇vj · τ = ρAκj |uj − vj | (uj − vj) · τ

on ΓI × (0, T ]. The parameters ρA > 0 and ρO > 0 are densities. The friction pa-rameters κj > 0 are calculated in practice from bulk flux formulae that involve othervariables not considered in the above model, but which may be viewed as introduc-ing additional sources of uncertainty. These parameters will be time dependent,but for simplicity their dependence on x is neglected in this paper. Furthermore,these values should remain bounded away from zero and infinity. It is assumedherein that κj(t) ∈ C∞(0,∞) and

(18) 0 < κ0 ≤ κj(t) < κ∞ <∞, j = 1, . . . , J,

for all t ∈ [0,∞), where κ0 and κ∞ are independent of time and j.Upon ensemble averaging of (17), we see

(19)u · nA = v · nO = 0,

−ρAνA⊥nA · ∇u · τ = ρOνO⊥nO · ∇v · τ = ρA 〈κj |uj − vj | (uj − vj)〉 · τ ,

which adds another consideration for the closure problem, since (for example)

〈κj |uj − vj | (uj − vj)〉 · τ 6= κ |u− v| (u− v) · τ .

The focus of this paper is the computation of ensembles for the coupled fluid-fluid model, for which purpose it shall not be necessary to derive a theoreticalboundary condtion for the mean-flow transfer of momentum across the interface.It shall be shown that only individual realizations (uj ,vj) will need to be computed,for which purpose it suffices to apply the correct boundary conditions (17). Thenthe identities (19) will hold automatically at each discrete time level. The closureproblem associated with the mean-flow transfer of momentum across the interfaceis left as an interesting open problem for purposes of theoretical analysis.


Table 1. Choices for turbulent mixing lengths and turbulent energies.

νAH

νA⊥

νOH

νO⊥

l ∆t < |(uHj )′| > ∆t < |(u⊥

j )′| > ∆t < |(vH

j )′| > ∆t < |(v⊥j )

′| >k′ ρA

2 < |(uHj )′|2 > ρA

2 < |(u⊥j )

′|2 > ρO

2 < |(vHj )′|2 > ρO

2 < |(v⊥j )

′|2 >

The critical issue is to model the behavior of the Reynolds stresses. Apply theeddy viscosity (EV) hypothesis away from boundaries and approximate

∇ · R(u,u) ≈ DAT (u) + model pressure terms, in ΩA(20)

DAT (u) ≡ 2∇ ·

(νA

HD(u)H + νA

⊥D(u)⊥

)

and ∇ · R(v,v) ≈ DOT (v) + model pressure terms, in ΩO,(21)

DOT (v) ≡ 2∇ ·

(νO

HD(v)H + νO

⊥D(v)⊥

).

Here, νAH, νO

H, νA

⊥, νO

⊥are turbulent viscosities in the bulk atmosphere and

ocean. Their dependence on turbulent fluctuations shall be prescribed via theKolmogorov-Prandtl relationship

νT =√2µl

√k′,(22)

where νT denotes a turbulent viscosity parameter, µ is a tuning parameter, l isa subscale mixing length and k′ is the kinetic energy associated with turbulentfluctuations.

The key is to choose l and k′ such that these values will naturally vanish nearboundaries, but away from boundaries they yield reasonable results to model sub-scale diffusion. First, define fluctuations a′j for data a1, . . . , aJ by

a′j ≡ aj− < aj >, j = 1, . . . , J.

Velocities will be decomposed into horizontal and vertical components:

uj = uHj + u⊥

j , vj = vHj + v⊥

j ,

where uHj ,v

Hj are the horizontal components and u⊥

j ,v⊥j are the vertical compo-

nents. Then the turbulent viscosities are specified by choosing l and k′ as shownin Table 1. The choices for l represent the (average) distance a turbulent, subscalefeature moves (either horizontally or vertically) in a time ∆t, shown in [9] to yieldgood results. The values for k′ represent the (average) kinetic energy density of asubscale eddy.

Note that on boundaries where a no-slip condition is imposed, these choicesof turbulent viscosities will automatically vanish. If a no-penetration conditionis imposed, such as at the atmosphere-ocean interface with rigid lid, the vertical

component of velocity still vanishes and thus drives the values of νA⊥

and νO⊥

tozero. As a result, from (20)-(21) it follows that

nA · R(u,u) ≈ 2νA⊥nA ·D(u)⊥ = 0

and nO ·R(v,v) ≈ 2νO⊥nO ·D(v)⊥ = 0

on ΓI . This is consistent with the behavior of the Reynolds stresses at the interface:

nA · R(u,u) = < nA · uj >< uj > − < (nA · uj)uj >= 0

and nO · R(v,v) = < nO · vj >< vj > − < (nO · vj)vj >= 0 on ΓI .

496 J. CONNORS

2. Mathematical preliminaries

Coordinates in space are denoted by x = (x1, . . . , xd). The domains are rectan-gles (d = 2) or boxes (d = 3) of the form

(23)

ΩA = (0, L1)× (0, HA) and ΩO = (0, L1)× (−HO, 0) (d = 2)

ΩA = (0, L1)× (0, L2)× (0, HA)

and ΩO = (0, L1)× (0, L2)× (−HO, 0) (d = 3).

The values HA > 0 and HO > 0 represent the height of ΩA and the depth ofΩO. The domain width in the lateral directions are 0 < L1, L2. The domains arecoupled across their shared interface ΓI = ΩA ∩ΩO, which lies in the plane xd = 0.Other boundaries are

(24)

Γt = ∂ΩA ∩ xd = HA (top of atmosphere)

Γb = ∂ΩO ∩ xd = −HO (bottom of ocean)

ΓAL = ∂ΩA \ (Γt ∪ ΓI) (lateral atmosphere boundaries)

ΓOL = ∂ΩO \ (Γb ∪ ΓI) (lateral ocean boundaries).

The velocities and pressures are denoted by

(25)uj = (u1,j, . . . , ud,j) : ΩA → R

d, pj : ΩA → R,

vj = (v1,j , . . . , vd,j) : ΩO → Rd, qj : ΩO → R.

Periodic lateral boundary conditions are chosen. At the atmosphere top and oceanbottom, a no-slip condition is imposed for the velocities:

(26) uj = 0 on Γt × (0, T ] and vj = 0 on Γb × (0, T ].

The ensemble-mean flows then have the same lateral, top and bottom boundaryconditions.

Remark 1. At the atmosphere top, the usual boundary conditions would be no-penetration and no-tangential-stress (also called “no-flux”). However, the no-slipcondition is used herein to simplify the analysis and presentation.

Some preliminary notation and results will be used for purposes of analysis.

Definition 1. Standard L2-inner-products are defined on each domain by

(u, u)A ≡∫

ΩA

u · u dx and (v, v)O ≡∫

ΩO

v · v dx.

Definition 2.

‖u‖p,2 ≡(∫ T

0

‖u‖p)1/p

, 1 ≤ p <∞,

‖u‖∞,2 ≡ ess sup0≤t≤T ‖u(t)‖.


Definition 3. Weak spaces for the velocities and pressures are defined with respectto the above boundary condtions.

XA ≡ cl(H1)d

u ∈

(C∞(ΩA)

)d |u(xi) = u(xi + Li), 1 ≤ i < d, u = 0 on Γt ∪ ΓI

.

XO ≡ cl(H1)d

v ∈

(C∞(ΩO)

)d |v(xi) = v(xi + Li), 1 ≤ i < d,

v = 0 on Γb, vd = 0 on ΓI

.

PA ≡ clH1

p ∈ C∞(ΩA) |

∫

ΩA

p dx = 0, p(xi) = p(xi + Li), 1 ≤ i < d

.

PO ≡ clH1

q ∈ C∞(ΩO) |

∫

ΩO

q dx = 0, q(xi) = q(xi + Li), 1 ≤ i < d

.

Divergence-free subspaces are needed for the velocities.

Definition 4.

VA ≡ u ∈ XA | (∇ · u, p)A = 0, ∀p ∈ PA(27)

VO ≡ v ∈ XO | (∇ · v, q)O = 0, ∀q ∈ PO(28)

Due to the boundary conditions on ΓI , it is necessary to work with the spaceL3(ΓI). It is well-known that traces of functions in XA and XO are well-defined inthis sense (see e.g. [6]). The next definition provides a compact notation.

Definition 5.

‖u‖I ≡(∫

ΓI

|u|3 dΓI

)1/3

.

Definition 6. Given functions u, u ∈ (H1(ΩA))d and v, v ∈ (H1(ΩO))

d, somebilinear forms are defined as follows.

aA(u, u) ≡ 2νAH

∫

ΩA

D(u)H : ∇u dx+ 2νA⊥

∫

ΩA

D(u)⊥ : ∇u dx(29)

aA(u, u) ≡ 2

∫

ΩA

νAHD(u)H : ∇u dx+ 2

∫

ΩA

νA⊥D(u)⊥ : ∇u dx(30)

aO(v, v) ≡ 2νOH

∫

ΩO

D(v)H : ∇v dx+ 2νO⊥

∫

ΩO

D(v)⊥ : ∇v dx(31)

aO(v, v) ≡ 2

∫

ΩO

νOHD(v)H : ∇v dx+ 2

∫

ΩO

νO⊥D(v)⊥ : ∇v dx.(32)

Lemma 1. Given functions (u, v) ∈ ((H1(ΩA))d, (H1(ΩO))

d) and (u,v) ∈ (VA, VO),it holds that

aA(u, u) = νAH

∫

ΩA

(∇u)H : (∇u)H dx+ νA⊥

∫

ΩA

(∇u)⊥ : (∇u)⊥ dx

aO(v, v) = νOH

∫

ΩO

(∇v)H : (∇v)H dx+ νO⊥

∫

ΩO

(∇v)⊥ : (∇v)⊥ dx.

Proof. It suffices to take u ∈ C∞(ΩA) ∩ VA and v ∈ C∞(ΩO) ∩ VO, since thesesubspaces are dense. Use the decomposition (7) and

(∇u)H : (∇u)⊥ = 0 = (∇u)H : (∇u)⊥

498 J. CONNORS

to write

aA(u, u) = 2νAH

∫

ΩA

D(u)H : ∇u dx+ 2νA⊥

∫

ΩA

D(u)⊥ : ∇u dx

= νAH

∫

ΩA

(∇u)H : (∇u)H +(∇uT

)H: (∇u)H dx

+ νA⊥

∫

ΩA

(∇u)⊥ : (∇u)⊥ +(∇uT

)⊥: (∇u)⊥ dx.

Integrate by parts to show that for each i = 1, . . . , d,

∑

j=1,...,d

∫

ΩA

∂xiuj∂xj

ui dx = −∫

ΩA

∂xi

∑

j=1,...,d

∂xjuj

ui dx = 0,

since u ∈ VA ⇒ ∇ · u = 0. It follows that

νAH

∫

ΩA

(∇uT

)H: (∇u)H dx = 0 = νA

⊥

∫

ΩA

(∇uT

)⊥: (∇u)⊥ dx,

and therefore

aA(u, u) = νAH

∫

ΩA

(∇u)H

: (∇u)Hdx+ νA

⊥

∫

ΩA

(∇u)⊥: (∇u)

⊥dx.

A similar analysis holds with v and v.

An application of the Poincare-Friedrich inequality is used for the velocity spaces.The factor of 2 that appears in (34) below is for convenience in later application.

Lemma 2. There exists a constant α1 > 0 such that

(33) α1

‖∇u‖2 + ‖∇v‖2

≤ aA(u,u) + aO(v,v),

for any u ∈ VA and v ∈ VO. Furthermore, there exists a constant α2 > 0 such that

(34) 2α2

‖u‖2 + ‖v‖2

≤ α1

‖∇u‖2 + ‖∇v‖2

,

for any u ∈ VA and v ∈ VO.

Proof. It is clear from Lemma 1 that the constant α1 can be defined by

α1 = minνA

H , νA⊥, νO

H , νO⊥.

The rest follows from the observation that the Poincare-Friedrich inequality holdson each domain ΩA and ΩO.

The polarization identity is useful to decompose vector products of the form(u− v) · u into positive and negative parts.

Lemma 3 (Polarization identity). Given equal-size vectors u and v,

2(u− v) · u = |u|2 + |u− v|2 − |v|2.Ensembles of products can be manipulated using the next result to represent

them in terms of fluctuations and ensemble averages.

Lemma 4 (Product ensemble identities). Given two sets of scalars aj and bj,j = 1, . . . , J , it follows

(35) < ajbj >=< a′jb′j > +ab.

For two sets of vectors aj and bj, j = 1, . . . , J , it holds that

(36) < aj · bj >=< a′j · b′j > +a · b.


Proof. Since the averaging operator< · > distributes across addition and j-independentscalars can be pulled out,

< ajbj > =< a′jb′j > + < a′jb > + < ab′j > + < ab >

=< a′jb′j > + < a′j > b+ a < b′j > +ab.

Then (35) follows from noting that fluctuations average to zero;

< a′j >=< b′j >= 0.

The proof of (36) is similar.

The following result will help later to study the effects of the nonlinear frictionterms on the mean flow.

Lemma 5. Given a set of vectors aj, j = 1, . . . , J , it holds that⟨(|aj | aj)′ · a′j

⟩≥ 0.

Proof. Application of (36) gives

(37)

⟨(|aj | aj)′ · a′j

⟩= 〈|aj |aj · aj〉 − 〈|aj | aj〉 · 〈aj〉

=⟨|aj |3

⟩− 〈|aj | aj〉 · 〈aj〉 .

A lower bound may be found by first applying the Triangle and Minkowski inequal-ities to show that

|〈|aj | aj〉| =1

J

∣∣∣∣∣∣

J∑

j=1

|aj | aj

∣∣∣∣∣∣≤ 1

J

J∑

j=1

|aj |2 ≤ 1

J

J∑

j=1

1

1/3

J∑

j=1

|aj |3

2/3

=

1

J

J∑

j=1

|aj |3

2/3

=⟨|aj |3

⟩2/3.

Similarly, it holds that

|〈aj〉| =1

J

∣∣∣∣∣∣

J∑

j=1

aj

∣∣∣∣∣∣≤ 1

J

J∑

j=1

|aj | ≤1

J

J∑

j=1

1

2/3

J∑

j=1

|aj |3

1/3

=

1

J

J∑

j=1

|aj |3

1/3

=⟨|aj |3

⟩1/3.

The desired lower bound in (37) follows from the above results, via Cauchy-Schwarz:

|〈|aj | aj〉 · 〈aj〉| ≤⟨|aj |3

⟩1/3 ⟨|aj |3

⟩2/3=⟨|aj |3

⟩

⇒⟨(|aj | aj)′ · a′j

⟩≥⟨|aj |3

⟩−⟨|aj |3

⟩= 0.

The following monotonicity result is used later to analyze convergence of ensem-bles to statistical equilibrium.

Lemma 6 (Friction monotonicity). Define T : Rd → Rd by T (x) = |x|x, for all

x ∈ Rd, d ∈ N. Then T ∈ C1(Rd,Rd) and

(38) (T (x)− T (y)) · (x− y) ≥ 1

4|x− y|3, ∀x, y ∈ R

d.

500 J. CONNORS

Proof. At any point x, the derivative DT (x) : Rd → Rd is a linear map, defined for

all arguments y ∈ Rd by

(39) DT (x)(y) ≡

0, if x = 0x·y|x| x+ |x|y, otherwise.

It is a standard exercise in real analysis to verify this formula and that T ∈C1(Rd,Rd).

It remains to show (38) . Given two vectors x,y ∈ Rd, define a = y−x. Without

loss of generality, a 6= 0. One may write

(T (y)− T (x)) · (y − x) =

∫ 1

0

d

dsT (x+ sa) · a ds =

∫ 1

0

DT (x+ sa)(a) · a ds.

Note that x + sa = 0 is possible for at most one distinct value of s, so we mayignore this case in the above integral. It follows from (39) that

(T (y)− T (x)) · (y − x) =

∫ 1

0

(x+ sa) · a|x+ sa| (x+ sa) · a+ |x+ sa|a · a ds

=

∫ 1

0

|(x+ sa) · a|2|x+ sa| + |x+ sa||a|2 ds

≥ |a|2∫ 1

0

|x+ sa| ds ≥ 1

4|a|3.

The last inequality is equivalent to arguments in [11], page 131.

Remark 2. The value 1/4 that appears in (38) is probably not sharp.

3. Evolution of the model variance

Three sources of model uncertainties are considered: initial conditions, forcingterms and the friction parameters κj . Let ‖ · ‖ denote the standard L2-norm; thedomain is inferred by context. An arbitrary set of vector functions wj , j = 1, . . . , J ,are discussed on a generic domain.

Definition 7 (Variance). The variances are given by

V (wj) ≡⟨‖wj‖2

⟩− ‖w‖2 ,

V ((∇wj)H) ≡

⟨∥∥(∇wj)H∥∥2⟩−∥∥(∇w)H

∥∥2 ,

and V (∇w⊥j ) ≡

⟨∥∥(∇wj)⊥∥∥2⟩−∥∥(∇w)⊥

∥∥2 .

Note that the variance measures fluctuations:

Lemma 7.

(40)V (wj) =

⟨∥∥w′j

∥∥2⟩≥ 0, V ((∇wj)

H) =⟨∥∥((∇wj)

H)′∥∥2⟩≥ 0,

V ((∇wj)⊥) =

⟨∥∥((∇wj)⊥)′∥∥2⟩≥ 0.

Proof. Let Ω denote the domain for wj . Apply (36) as follows:⟨‖wj‖2

⟩=

⟨∫

Ω

wj ·wj dx

⟩=

∫

Ω

〈wj ·wj〉 dx =

∫

Ω

⟨w′

j ·w′j

⟩dx +

∫

Ω

w ·w dx

=

⟨∫

Ω

∣∣w′j

∣∣2 dx⟩+

∫

Ω

|w|2 dx =⟨‖w′

j‖2⟩+ ‖w‖2,


which implies that

V (wj) =⟨∥∥w′

j

∥∥2⟩.

The remainder of the proof is analogous.

Given that strong model solutions exist for each realization, then an energyequality will be satisfied for each realization and for the mean flows. These equationscan then be used to describe the evolution of the model variance.

Lemma 8. Given strong solutions (uj , pj) and (vj , qj) to the model equations (1)-(6) with boundary conditions described above, the following energy equations aresatisfied for j = 1, . . . , J :

d

dt

‖uj‖2 +

ρOρA

‖vj‖2+ aA(uj ,uj) +

ρOρA

aO(vj ,vj)(41)

+

∫

ΓI

κj |uj − vj |3 dΓI =

∫

ΩA

fAj · uj dx+ρOρA

∫

ΩO

fOj · vj dx,

d

dt

‖u‖2 + ρO

ρA‖v‖2

+

∫

ΓI

〈κj |uj − vj |(uj − vj)〉 · (u− v) dΓI(42)

+aA(u,u) +ρOρA

aO(v,v) =

∫

ΩA

fA · u dx+ρOρA

∫

ΩO

fO · v dx

−∫

ΩA

R(u,u) : ∇u dx− ρOρA

∫

ΩO

R(v,v) : ∇v dx.

Proof. Multiply through (1) by uj and integrate over ΩA. Also multiply through (4)by ρOvj/ρA and integrate over ΩO, adding the two equations together. Given theboundary condtions, it is easily shown that

∫

ΩA

uj ·∇uj ·uj dx =

∫

ΩO

vj ·∇vj ·vj dx = 0 =

∫

ΩA

∇pj ·uj dx =

∫

ΩO

∇qj ·vj dx.

For the diffusion terms, note that

−∫

ΩA

DA(uj) · uj dx− ρOρA

∫

ΩO

DO(vj) · vj dx = aA(uj ,uj) +ρOρA

aO(vj ,vj)

+

∫

ΓI

κj |uj − vj |(uj − vj) · uj dΓI −∫

ΓI

κj |uj − vj |(uj − vj) · vj dΓI

= aA(uj ,uj) +ρOρA

aO(vj ,vj) +

∫

ΓI

κj |uj − vj |3 dΓI .

The rest is standard to derive (41). For (42) one multiplies through (11) by u andthrough (14) by ρOv/ρA. Then follow the above procedure, applying the ensemble-averaged boundary conditions.

502 J. CONNORS

Theorem 1 (Variance evolution). Given an ensemble of strong model solutions,the model variance must satisfy the integral equation

V (uj(T )) +ρOρA

V (vj(T )) +

∫ T

0

νAHV ((∇uj(T ))

H) dt+

∫ T

0

νA⊥V ((∇uj(T ))

⊥) dt

+ρOρA

∫ T

0

νOHV ((∇vj(T ))

H) dt+ρOρA

∫ T

0

νO⊥V ((∇vj(T ))

⊥) dt

+

∫ T

0

∫

ΓI

⟨κj|uj − vj |(uj − vj)′ · (uj − vj)

′⟩dΓI dt

= V (u0j ) +

ρOρA

V (v0j ) +

∫ T

0

∫

ΩA

⟨fA

′j · u′

j

⟩+ρOρA

∫ T

0

∫

ΩO

⟨fO

′j · v′

j

⟩

+

∫ T

0

∫

ΩA

R(u,u) : ∇u dx dt+ρOρA

∫ T

0

∫

ΩO

R(v,v) : ∇v dx dt.

Proof. Integrate (41)-(42) in time and take the ensemble-average of the realizationenergy equations. Subtracting this from the time-integral of the mean-flow energyequation and applying Definition 7 yields

V (uj(T )) +ρOρA

V (vj(T )) +

∫ T

0

νAHV ((∇uj(T ))

H) dt+

∫ T

0


⊥) dt

+ρOρA

∫ T

0

νOHV ((∇vj(T ))

H) dt+ρOρA

∫ T

0


⊥) dt

+

∫ T

0

∫

ΓI

⟨κj |uj − vj |3

⟩− 〈κj |uj − vj |(uj − vj)〉 · (u− v) dΓI dt

= V (u0j ) +

ρOρA

V (v0j ) +

∫ T

0

∫

ΩA

(⟨fAj · uj

⟩− fA · u

)dx dt

+ρOρA

∫ T

0

∫

ΩO

(⟨fOj · vj

⟩− fO · v

)dx dt

+

∫ T

0

∫

ΩA


∫ T

0

∫

ΩO


Apply (36) to the forcing terms:⟨fAj · uj

⟩− fA · u =

⟨fA

′j · u′

j

⟩and

⟨fOj · vj

⟩− fO · v =

⟨fO

′j · v′

j

⟩.

In order to handle the interface terms, set γj = κj |uj−vj |(uj−vj) andwj = uj−vj .It holds that∫ T

0

∫

ΓI

⟨κj |uj − vj |3

⟩− 〈κj |uj − vj |(uj − vj)〉 · (u− v) dΓI dt

=

∫ T

0

∫

ΓI

⟨γj ·w′

j

⟩dΓI dt =

∫ T

0

∫

ΓI

⟨γ′j ·w′

j

⟩dΓI dt.

The last step holds since⟨γ ·w′

j

⟩= γ ·

⟨w′

j

⟩= 0. The desired result follows by

combining the above equations.

As a corollary, it may be shown that when perturbations are only introducedthrough the initial conditions, the Reynolds stresses must have a dissipative effecton the mean flow. This yields a proof (under certain conditions) of the so-calledBoussinesq assumption that partially motivates the closure models (20)-(21). This


analysis was performed in [10], but their analysis did not account for the extracoupling terms present with the atmosphere-ocean problem.

Corollary 1. Assume strong realizations (uj , pj) and (vj , qj) exist and the forc-ing terms satisfy fAj ∈ L∞(0, T ;L2(ΩA)) and fOj ∈ L∞(0, T ;L2(ΩO)), for j =

1, . . . , J . If κ′j = 0 and fA′j = fO

′j = 0 for j = 1, . . . , J , then

lim infT→∞

1

T

∫ T

0

∫

ΩA

R(u,u) : ∇u dx dt

+ρOρA

1

T

∫ T

0

∫

ΩO

R(v,v) : ∇v dx dt

≥ 0.

Proof. The assumptions on the forcing terms imply further that the variance sat-isfies V (uj) ∈ L∞(0, T ;L2(ΩA)) and that V (vj) ∈ L∞(0, T ;L2(ΩO)), for all j, bystandard arguments. Since also κ′j = 0 and fA

′j = fO

′j = 0, Theorem 1 implies that

V (uj(T )) +ρOρA

V (vj(T )) +

∫ T

0

νAHV ((∇uj(T ))

H) dt+

∫ T

0


⊥) dt

+ρOρA

∫ T

0

νOHV ((∇vj(T ))

H) dt+ρOρA

∫ T

0


⊥) dt

+

∫ T

0

∫

ΓI

κ⟨|uj − vj |(uj − vj)′ · (uj − vj)

′⟩dΓI dt

= V (u0j ) +

ρOρA

V (v0j ) +

∫ T

0

∫

ΩA


∫ T

0

∫

ΩO


Multiply through by 1/T and note that

1

TV (uj(T )) = O

(1

T

),

1

TV (vj(T )) = O

(1

T

),

1

TV (u0

j) = O(1

T

),

1

TV (v0

j ) = O(1

T

).

Also, taking aj = uj − vj in Lemma 5 yields

1

T

∫ T

0

∫

ΓI

κ⟨|uj − vj |(uj − vj)′ · (uj − vj)

′⟩dΓI dt ≥ 0.

The remaining variance terms are non-negative by Lemma 7.

Remark 3. It is expected that Corollary 1 would still hold if the data fluctuationsκ′j, fA

′j and fO

′j are small enough (in an appropriate sense), or vanish quickly

enough as T → ∞.

4. Leray-regularized realizations and properties

The model realizations shall employ a Leray-type regularization. That is, theensemble averaging operator < · > is assumed to have a smoothing effect on themean flow, so that replacement of the convecting velocity in the nonlinear term ofa realization will result in a regularized model, in the sense of Leray, [14, 15]. Therealizations satisfy

∂tuj −DA(uj)−DAT (uj) + u · ∇uj +∇pj = fAj on ΩA × (0, T ],(43)

∂tvj −DO(vj)−DOT (vj) + v · ∇vj +∇qj = fOj on ΩO × (0, T ].(44)

504 J. CONNORS

The notation for the realizations is reused here. The incompressibility, initial andboundary conditions are unchanged from before.

Note that upon taking the ensemble-average of (43)-(44) the mean flow satisfies

∂tu−DA(u)−DAT (u) + u · ∇u+∇p = fA on ΩA × (0, T ],

∂tv −DO(v) −DOT (v) + v · ∇v +∇q = fO on ΩO × (0, T ].

Comparison with (11)-(16) reveals that the mean flow for the Leray-regularizedensemble differs from that of the standard NSE model ensemble only due to theclosures (20)-(21) for the Reynolds stress terms.

In ensemble calculations for this regularized model, matrices for the convectionterms will not depend on the particular ensemble member. In practice, ensemblemembers would be advanced one time step in serial fashion, requiring one ensembleaveraging (a negligible cost) plus the recomputation of convection terms once pertime step. The matrix assembly cost is thus reduced significantly in proportion tothe (possibly large) ensemble size, as compared with standard ensemble methods.The resulting savings in run time will depend on the particular implementation andis left to future study.

Remark 4. The mean flow still satisfies (19) on ΓI . This is not closed for themean flow, so the realizations are a coupled system of J equations. Existence ofunique, strong solutions is assumed herein, in order to discuss algorithmic ideas. Atime-stepping method is applied later that decouples these equations numerically.

Unique, strong solutions for the realization equations are assumed to exist here-after and to satisfy the following variational problem. For 1 ≤ j ≤ J , (uj , pj) : t→(XA, PA) and (vj , qj) : t→ (XO, PO) satisfy (for a.e. t ∈ (0, T ])

(∂tuj , u)A + (u · ∇uj , u)A + aA(uj , u) + aA(uj , u)− (pj ,∇ · u)A

+κj

∫

ΓI

|uj − vj |(uj − vj) · u dΓI = (fAj , u)A, ∀u ∈ XA(45)

(∇ · uj , p)A = 0, ∀p ∈ PA,(46)

(∂tvj , v)O + (v · ∇vj , v)O + aO(vj , v) + aO(vj , v)− (qj ,∇ · v)O

−ρAρO

κj

∫

ΓI

|uj − vj |(uj − vj) · v dΓI = (fOj , v)O, ∀v ∈ XO(47)

(∇ · vj , q)O = 0, ∀q ∈ PO,(48)

with uj(t = 0) = u0j and vj(t = 0) = v0

j . Here, it is assumed also that ∇ · u0j =

∇·v0j = 0 and that uj ∈ L4(0, T ;XA) and vj ∈ L4(0, T ;XO). The time derivatives

satisfy ∂tuj ∈ L2(0, T ;XA∗) and ∂tvj ∈ L2(0, T ;XO

∗). This way to define strongsolutions is consistent with [13].

Lemma 9 (Long-time stability). Assume (uj , pj), (vj , qj) are strong solutions ofthe realization equations for 1 ≤ j ≤ J and let the turbulent viscosity coefficientsbe constants. If fAj ∈ L∞(0,∞;L2(ΩA)) and fOj ∈ L∞(0,∞;L2(ΩO)), then for


0 < t <∞,

‖uj(t)‖2 +ρOρA

‖vj(t)‖2 + α1

∫ t

0

e−α2(t−τ)

‖∇uj(τ)‖2 +

ρOρA

‖∇vj(τ)‖2dτ

+ 2

∫ t

0

e−α2(t−τ)κj‖(uj − vj)(τ)‖3I dτ

≤ e−α2t

‖u0

j‖2 +ρOρA

‖v0j‖2

+1− e−α2t

(α2)2

lim

T→∞‖fAj‖2∞,2 +

ρOρA

limT→∞

‖fOj‖2∞,2

.

It follows that uj ∈ L∞(0,∞;L2(ΩA)) and vj ∈ L∞(0,∞;L2(ΩO)).

Proof. Define wj =√ρOvj/

√ρA. It follows from (45)-(48) that

1

2

d

dt

‖uj‖2 + ‖wj‖2

+ aA(uj ,uj) + aA(uj ,uj) + aO(wj ,wj) + aO(wj ,wj)

+κj‖uj − vj‖3I = (fAj ,uj)A +ρOρA

(fOj ,vj)O ≤ ‖fAj‖‖uj‖+ρOρA

‖fOj‖‖vj‖.

Since the turbulent viscosity parameters are constants, the analysis of Lemma 1and Lemma 2 may be applied to show that

aA(uj ,uj) + aO(wj ,wj) ≥ 0.

Bound the remaining viscous terms below by applying Lemma 2. The result is

1

2

d

dt

‖uj‖2 + ‖wj‖2

+ α2

‖uj‖2 + ‖wj‖2

+α1

2

‖∇uj‖2 + ‖∇wj‖2

+ κj‖uj − vj‖3I ≤ ‖fAj‖‖uj‖+ρOρA

‖fOj‖‖vj‖.

Bound the right-hand side above using Young’s inequality:

1

2

d

dt

‖uj‖2 + ‖wj‖2

+ α2

‖uj‖2 + ‖wj‖2

+α1

2

‖∇uj‖2 + ‖∇wj‖2

+κj‖uj − vj‖3I ≤ 1

2α2

‖fAj‖2 +

ρOρA

‖fOj‖2+α2

2

‖uj‖2 + ‖wj‖2

⇒ 1

2

d

dt

‖uj‖2 + ‖wj‖2

+α2

2

‖uj‖2 + ‖wj‖2

+α1

2

‖∇uj‖2 + ‖∇wj‖2

+κj‖uj − vj‖3I ≤ 1

2α2

‖fAj‖2 +

ρOρA

‖fOj‖2.

The remainder of the proof follows by using an integration factor.

The next result is needed to discuss convergence to statistical equilibrium.

Lemma 10. Let the mapping τ → e−α2(t−τ)f(τ) ∈ L1(0, t) satisfy∫ t

0

e−α2(t−τ)|f(τ)| dτ ≤ C <∞

for all 0 < t <∞, where α2 > 0 and C > 0 are independent of τ and t. Given anyg ∈ L∞(0,∞) such that lim supt→∞ |g(t)| = 0, it holds that

lim supt→∞

∫ t

0

e−α2(t−τ)|g(τ)||f(τ)| dτ = 0.

506 J. CONNORS

Proof. Without loss of generality, assume ‖g‖L∞ > 0. Let δ > 0 be arbitrary.Given 0 < s < t,

∫ t

0

e−α2(t−τ)|g(τ)||f(τ)| dτ ≤ ‖g‖L∞e−α2(t−s)

∫ s

0

e−α2(s−τ)|f(τ)| dτ

+ sups≤τ≤t

|g(τ)|∫ t

s

e−α2(t−τ)|f(τ)| dτ ≤ C‖g‖L∞e−α2(t−s) + C sups≤τ≤t

|g(τ)|.

Choose s so large that sups≤τ≤∞ |g(τ)| < δ/(2C) and then choose any t∗(δ) > s solarge that

e−α2(t−s) ≤ e−α2(t∗−s) <

δ

2C‖g‖L∞

,

for all t ≥ t∗. It follows that

0 ≤ supt≥t∗

∫ t

0

e−α2(t−τ)|g(τ)||f(τ)| dτ < δ,

which is the desired result.

Theorem 2. Let the turbulent viscosity coefficients be constants. Assume that forall 1 ≤ j ≤ J , fAj ∈ L∞(0,∞;L2(ΩA)), fOj ∈ L∞(0,∞;L2(ΩO)) and also that

(49) lim supt→∞

‖fA′j(t)‖ = lim sup

t→∞‖fO′

j(t)‖ = lim supt→∞

κ′j(t) = 0.

Given (ui, pi), (vi, qi) and (uj , pj), (vj , qj) are any strong solutions of the realiza-tion equations with 1 ≤ i, j ≤ J , it holds that

(50) lim supt→∞

‖(ui − uj)(t)‖2 +

ρOρA

‖(vi − vj)(t)‖2

= 0

and

(51) lim supt→∞

∫ t

0

e−α2(t−τ)

‖∇(ui − uj)(τ)‖2 +

ρOρA

‖∇(vi − vj)(τ)‖2dτ = 0.

Proof. Define a = ui − uj and b = vi − vj . It follows from (45)-(48) that

1

2

d

dt‖a‖2 +

∫

ΓI

κi|ui − vi|(ui − vi) · a dΓI −∫

ΓI

κj |uj − vj |(uj − vj) · a dΓI

+ aA(a, a) + aA(a, a) =(fAi − fAj , a

)A.

Next, decompose the data κi, κj , fAi and fAj in terms of fluctuations and means,then insert above and rearrange terms to see that

1

2

d

dt‖a‖2 +

∫

ΓI

κ|ui − vi|(ui − vi) · a dΓI −∫

ΓI

κ|uj − vj |(uj − vj) · a dΓI

+ aA(a, a) + aA(a, a) =(fA

′i − fA

′j , a)A

−∫

ΓI

κ′i|ui − vi|(ui − vi) · a dΓI +

∫

ΓI

κ′j|uj − vj |(uj − vj) · a dΓI .


On the domain ΩO an analogous equation is derived and added to the above result,which leads to

1

2

d

dt

‖a‖2 + ρO

ρA‖b‖2

+ aA(a, a) + aA(a, a) +

ρOρA

aO(b,b) +ρOρA

aO(b,b)

+

∫

ΓI

κ|ui − vi|(ui − vi) · (a− b) dΓI −∫

ΓI

κ|uj − vj |(uj − vj) · (a − b) dΓI

=−∫

ΓI

κ′i|ui − vi|(ui − vi) · (a− b) dΓI +

∫

ΓI

κ′j |uj − vj |(uj − vj) · (a− b) dΓI

+(fA

′i − fA

′j , a)A+ρOρA

(fO

′i − fO

′j ,b)O.

The interface terms on the left are treated by applying the monotonicity resultLemma 6, with x = ui − vi and y = uj − vj . Then x− y = a− b and

∫

ΓI

κ|ui − vi|(ui − vi) · (a− b) dΓI −∫

ΓI

κ|uj − vj |(uj − vj) · (a− b) dΓI

=

∫

ΓI

κ (|x|x− |y|y) · (a− b) dΓI =

∫

ΓI

κ (|x|x− |y|y) · (x− y) dΓI

≥ 1

4

∫

ΓI

κ |x− y|3 dΓI =1

4

∫

ΓI

κ |a− b|3 dΓI ≥ 0.

Next, bound the viscous terms below by applying Lemma 2. Upon combining theabove results, it holds that

1

2

d

dt

‖a‖2 + ρO

ρA‖b‖2

+ α2

‖a‖2 + ρO

ρA‖b‖2

(52)

+α1

2

‖∇a‖2 + ρO

ρA‖∇b‖2

≤(‖fA′

i‖+ ‖fA′j‖)‖a‖+ ρO

ρA

(‖fO′

i‖+ ‖fO′j‖)‖b‖

+|κ′i|∫

ΓI

|ui − vi|2|a− b| dΓI + |κ′j |∫

ΓI

|uj − vj |2|a− b| dΓI .

Young’s inequality is used to bound

(‖fA′

i‖+ ‖fA′j‖)‖a‖+ ρO

ρA

(‖fO′

i‖+ ‖fO′j‖)‖b‖(53)

≤ 2

α2

‖fA′

i‖2 + ‖fA′j‖2 +

ρOρA

‖fO′i‖2 +

ρOρA

‖fO′j‖2+α2

2

‖a‖2 + ρO

ρA‖b‖2

.

The interface terms on the right of (52) require more work. First, note that

|a− b| = |x− y| ≤ |ui − vi|+ |uj − vj |.

It follows from Young’s inequality that

|κ′i|∫

ΓI

|ui − vi|2|a− b| dΓI + |κ′j |∫

ΓI

|uj − vj |2|a− b| dΓI

≤ 5

3|κ′i|

∫

ΓI

|ui − vi|3 dΓI +5

3|κ′j |

∫

ΓI

|uj − vj |3 dΓI

+1

3|κ′i|

∫

ΓI

|uj − vj |3 dΓI +1

3|κ′j |

∫

ΓI

|ui − vi|3 dΓI .

508 J. CONNORS

Next, use the positivity requirement (18) to bound

|κ′i|∫

ΓI

|ui − vi|2|a− b| dΓI + |κ′j |∫

ΓI

|uj − vj |2|a− b| dΓI

≤5|κ′i|+ |κ′j |

3κ0κi

∫

ΓI

|ui − vi|3 dΓI +5|κ′j |+ |κ′i|

3κ0κj

∫

ΓI

|uj − vj |3 dΓI .

This result and (53) provide an upper bound for the right-hand side of (52). Aftersome algebra, one obtains

d

dt

‖a‖2 + ρO

ρA‖b‖2

+ α2

‖a‖2 + ρO

ρA‖b‖2

+ α1

‖∇a‖2 + ρO

ρA‖∇b‖2

≤ 4

α2

‖fA′

i‖2 + ‖fA′j‖2 +

ρOρA

‖fO′i‖2 +

ρOρA

‖fO′j‖2

+ 25|κ′i|+ |κ′j |

3κ0κi‖ui − vi‖3I + 2

5|κ′j|+ |κ′i|3κ0

κj‖uj − vj‖3I

An integration factor may be used here to find that

(54)

‖a(t)‖2 + ρOρA

‖b(t)‖2 + α1

∫ t

0

e−α2(t−τ)

‖∇a(τ)‖2 + ρO

ρA‖∇b(τ)‖2

dτ

≤ e−α2t

‖a(0)‖2 + ρO

ρA‖b(0)‖2

+4

α2

∫ t

0

e−α2(t−τ)

‖fA′

i‖2 + ‖fA′j‖2 +

ρOρA

‖fO′i‖2 +

ρOρA

‖fO′j‖2dτ

+2

3κ0

∫ t

0

e−α2(t−τ)(5|κ′i|+ |κ′j |)κi‖ui − vi‖3I dτ

+2

3κ0

∫ t

0

e−α2(t−τ)(5|κ′j|+ |κ′i|)κj‖uj − vj‖3I dτ.

Due to (49) and the boundedness of the data fAj and fOj , it holds that

τ → g1(τ) ≡ ‖fA′i(τ)‖2 + ‖fA′

j(τ)‖2 +ρOρA

‖fO′i(τ)‖2 +

ρOρA

‖fO′j(τ)‖2 ∈ L∞(0,∞)

such that lim supτ→∞ |g1(τ)| = 0. Therefore, Lemma 10 may be applied with g = g1and f(τ) = 1 to show that

lim supt→∞

∫ t

0

e−α2(t−τ)

‖fA′

i‖2 + ‖fA′j‖2 +

ρOρA

‖fO′i‖2 +

ρOρA

‖fO′j‖2dτ = 0.

In order to show that the interface terms also vanish in the limit, first note thatdue to Lemma 9, for any 1 ≤ j ≤ J

τ → f(τ) ≡ e−α2(t−τ)κj(τ)‖(uj − vj)(τ)‖3Isatisfies the assumptions of Lemma 10. Due to (49) and the uniform boundednessof the data κi and κj , it then follows from Lemma 10 that

lim supt→∞

∫ t

0

e−α2(t−τ)(5|κ′i|+ |κ′j |)κi‖ui − vi‖3I dτ

= lim supt→∞

∫ t

0

e−α2(t−τ)(5|κ′j |+ |κ′i|)κj‖uj − vj‖3I dτ = 0.

The desired results are now evident from (54).


5. Numerical methods and variance analysis

CTM realizations are to be computed using three methods that differ only in thetreatment of the coupling across the interface. One method uses implicit coupling ofvelocities, called the monolithic method, which is analogous to solving atmosphereand ocean equations simultaneously as one large system. In practice, partitionedmethods are commonly used, which decouple the velocities for the atmosphere andocean systems. Monolithic coupling is used here only to provide a benchmark pointof comparison for the partitioned methods.

The partitioned methods are based on the work by Connors, Howell and Lay-ton [5], and they provide unconditional numerical stability. The proof of stabilityis omitted for brevity, but it is a simple consequence of combining [10], Theorem4.6 and [5], Lemma 3.1. However, with a partitioned method the friction does notautomatically satisfy the monotonicity property (see Lemma 6), which may inhibitconvergence to equilibrium. For comparison, one of the two partitioned variantscalculates the friction based on the approximation

κj |uj − vj |(uj − vj) ≈ κj |u− v|(uj − vj),

which changes the mean flow and realizations, but is proved to guarantee conver-gence to statistical equilibrium.

5.1. Numerical methods to approximate realizations. Let ΩA and ΩO haveassociated conforming, triangular or tetrahedral meshes with mesh sizes hA andhO, respectively, defined as the maximum diameter of a mesh element. Conformingfinite element spaces are defined by using Taylor-Hood pairs, resulting in spaces

(XA, PA) and (XO, PO). These elements are known to satisfy the so-called inf-supor LBB condition (see [7, 2]) and have enjoyed wide-spread use for fluid com-putations. A uniform time step is used with size ∆t > 0. Given a functiong(t) ∈ C[0,∞), let gn ≈ g(tn = n∆t) denote an approximation at the discretetime levels n = 0, 1, . . .. Define explicitly skew-symmetrized trilinear forms for theconvection terms by

cA(u,uj ; u) ≡1

2

∫

ΩA

u · ∇uj · u dx− 1

2

∫

ΩA

u · ∇u · uj dx

and cO(v,vj ; v) ≡1

2

∫

ΩO

v · ∇vj · v dx− 1

2

∫

ΩO

v · ∇v · vj dx.

Define functions µnj and µn

j on ΓI by

µnj ≡ κj(t

n)|unj − vn

j | and µnj ≡ κj(t

n)|un − vn|,for 1 ≤ j ≤ J and all n. The three coupling approaches are referred to as M(monolithic) and then P1 and P2 (partitioned). The friction terms are defined asfollows.

IA(un+1j ,vn+1

j ,unj ,v

nj

)=

µn+1j

(un+1j − vn+1

j

)(M)

√µnj

(√µnj u

n+1j −

√µn−1j vn

j

)(P1)

√µnj

(√µnj u

n+1j −

√µn−1j vn

j

)(P2)

IO(un+1j ,vn+1

j ,unj ,v

nj

)=

µn+1j

(vn+1j − un+1

j

)(M)

√µnj

(√µnj v

n+1j −

√µn−1j un

j

)(P1)

√µnj

(√µnj v

n+1j −

√µn−1j un

j

)(P2)

510 J. CONNORS

Table 2. Choices for turbulent mixing lengths and turbulent energies.

νAH

νA⊥

l ∆t < |(E(uHj )n+1)′| > ∆t < |(E(u⊥

j )n+1)′| >

k′ ρA

2 < |(E(uHj )n+1)′|2 > ρA

2 < |(E(u⊥j )

n+1)′|2 >νO

HνO

⊥

l ∆t < |(E(vHj )n+1)′| > ∆t < |(E(v⊥

j )n+1)′| >

k′ ρO

2 < |(E(vHj )n+1)′|2 > ρO

2 < |(E(v⊥j )

n+1)′|2 >

Note that the coupling is linearized with both methods. Also, extrapolation is usedto linearize the convecting velocity for efficiency. To this end, given numbers gn,n = 0, 1, . . ., define E(g)n+1 ≡ 2gn − gn−1. Then numerical approximations for

the realizations are sought by finding (unj , p

nj ) ∈ (XA, PA) and (vn

j , qnj ) ∈ (XO, PO)

satisfying

(55)

1

2∆t

(3un+1

j − 4unj + un−1

j , u)A+ cA(E(u)n+1,un+1

j , u)

+ aA(un+1j , u) + aA(u

n+1j , u)− (pn+1

j ,∇ · u)A

+

∫

ΓI

IA(un+1j ,vn+1

j ,unj ,v

nj

)· u dΓI

+ γA(∇ · un+1

j ,∇ · u)A= (fA

n+1j , u)A, ∀u ∈ XA

(∇ · un+1j , p)A = 0, ∀p ∈ PA,

(56)

1

2∆t(3vn+1

j − 4vnj + vn−1

j , v)O + cO(E(v)n+1,vn+1j , v)

+ρAρO

∫

ΓI

IO(un+1j ,vn+1

j ,unj ,v

nj

)· v dΓI

+ aO(vn+1j , v) + aO(v

n+1j , v)− (qn+1

j ,∇ · v)O+ γO

(∇ · vn+1

j ,∇ · v)O= (fO

n+1j , v)O, ∀v ∈ XO

(∇ · vn+1j , q)O = 0, ∀q ∈ PO,

with initial data u0j ∈ XA and v0

j ∈ XO. The constants γA > 0 and γO > 0 areknown as “grad-div stabilization parameters” and help to improve mass conserva-tion (see e.g. [3, 19]), since Taylor-Hood elements do not admit a divergence-freevelocity space. Values for these parameters are to be determined by experiment.The turbulent viscosity coefficients are chosen as shown in Table 2.

Note that the coefficients for these linear systems do not depend on j except forthe terms for the flux coupling over ΓI . But these latter terms only affect a relativelysmall number of the total matrix entries. Furthermore, the partitioned methods arelinearized; compared to nonlinear methods applied to the true realization equations,the approach in this paper is very fast to run.

5.2. Evolution of variance for the methods. In order to simplify the analysisin this section, variance is only introduced through the initial conditions. In general,it is expected that the variance for the discrete approximations M and P2 will vanishas time evolves so long as the fluctuations κ′j , fA

′j and fO

′j vanish fast enough in

time. However, for method P1 this behavior may be inhibited and it is not knownif the variance vanishes at long times.


Lemma 11. Let κj = κ > 0 for all j = 1, . . . , J . For methods M and P2, thereexist numbers Rn ≥ 0 and Sn ≥ 0 for n = 1, . . . , N , which depend on the method,such that

(57)

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)·(un+1j

)′⟩dΓI

+

∫

ΓI

⟨IO(un+1j ,vn+1

j ,unj ,v

nj

)·(vn+1j

)′⟩dΓI

= Sn+1 +Rn+1 −Rn, n = 1, . . . , N − 1.

Proof. Define S1 = 0 for both methods. For method M, the relevant coupling termssatisfy∫

ΓI

⟨µn+1j

(un+1j − vn+1

j

)·(un+1j

)′⟩dΓI −

∫

ΓI

⟨µn+1j

(un+1j − vn+1

j

)·(vn+1j

)′⟩dΓI

= κ

∫

ΓI

⟨∣∣un+1j − vn+1

j

∣∣ (un+1j − vn+1

j

)·(un+1j − vn+1

j

)′⟩dΓI

= κ

∫

ΓI

⟨∣∣un+1j − vn+1

j

∣∣ (un+1j − vn+1

j

)′ ·(un+1j − vn+1

j

)′⟩dΓI .

It follows from Lemma 5 that

κ

∫

ΓI

⟨∣∣un+1j − vn+1

j

∣∣ (un+1j − vn+1

j

)′ ·(un+1j − vn+1

j

)′⟩dΓI ≡ Sn+1 ≥ 0

for n = 1, . . . , N − 1. For method M, take Rn = 0 for all n.Note that for method P2, the numbers µn

j satisfy µnj = κ(tn)|un − vn|, which

does not depend on j. Then the interface terms satisfy∫

ΓI

⟨(√µnj u

n+1j −

√µn−1j vn

j

)·(√

µnj u

n+1j

)′⟩dΓI

+

∫

ΓI

⟨(√µnj v

n+1j −

√µn−1j un

j

)·(√

µnj v

n+1j

)′⟩dΓI

=

∫

ΓI

⟨(√µnj u

n+1j

)′−(√

µn−1j vn

j

)′

·(√

µnj u

n+1j

)′⟩dΓI

+

∫

ΓI

⟨(√µnj v

n+1j

)′−(√

µn−1j un

j

)′

·(√

µnj v

n+1j

)′⟩dΓI .

The polarization identity may be applied to write the above terms in the formSn+1 +Rn+1 −Rn, where

Rn+1 =1

2

∫

ΓI

µnj

⟨∣∣∣(un+1j

)′∣∣∣2⟩+ µn

j

⟨∣∣∣(vn+1j

)′∣∣∣2⟩dΓI ≥ 0,

Sn+1 =1

2

∫

ΓI

⟨∣∣∣∣√µnj

(un+1j

)′ −√µn−1j

(vnj

)′∣∣∣∣2⟩dΓI

+1

2

∫

ΓI

⟨∣∣∣∣√µnj

(vn+1j

)′ −√µn−1j

(unj

)′∣∣∣∣2⟩dΓI ≥ 0.

It can be shown that Lemma 11 need not hold in a purely algebraic sense formethod P1. As a result, spurious variance might be created numerically for methodP1 through the coupling terms and inhibit convergence to statistical equilibrium.

512 J. CONNORS

Theorem 3. Let the turbulent viscosities be constants. Assume κ′j = 0, fA′j = 0

and fO′j = 0 identically, for all j. Then the discrete solutions for methods M and

P2 satisfy

limn→∞

(V(unj

)+ρOρA

V(vnj

))= lim

n→∞

(V(∇un

j

)+ρOρA

V(∇vn

j

))= 0,

for all j.

Proof. Insert u = un+1j and q = pn+1

j in (55), apply the polarization identity,multiply through by ∆t and take the ensemble average to see that

1

4

⟨∥∥un+1j

∥∥2⟩+

1

4

⟨∥∥2un+1j − un

j

∥∥2⟩− 1

4

⟨∥∥unj

∥∥2⟩− 1

4

⟨∥∥2unj − un−1

j

∥∥2⟩

+1

4

⟨∥∥un+1j − 2un

j + un−1j

∥∥2⟩

+∆t⟨aA(u

n+1j ,un+1

j )⟩+∆t

⟨aA(u

n+1j ,un+1

j )⟩+∆t γA

⟨∥∥∇ · un+1j

∥∥2⟩

+∆t

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)· un+1

j

⟩dΓI = ∆t

⟨(fA

n+1j ,un+1

j )A⟩.

On the other hand, take the ensemble average of (55) before inserting u = u. Thensimilar arguments may be used to show that

1

4

∥∥un+1∥∥2 + 1

4

∥∥2un+1 − un∥∥2 − 1

4‖un‖2 − 1

4

∥∥2un − un−1∥∥2

+1

4

∥∥un+1 − 2un + un−1∥∥2

+∆t aA(un+1,un+1) + ∆t aA(u

n+1,un+1) + ∆t γA∥∥∇ · un+1

∥∥2

+∆t

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)⟩· un+1 dΓI = ∆t (fA

n+1,un+1)A.

Now subtract this from the previous equation and apply Lemma 7 and (36) to seethat

(58)

1

4V(un+1j

)+

1

4V(2un+1

j − unj

)− 1

4V(unj

)− 1

4V(2un

j − un−1j

)

+1

4V(un+1j − 2un

j + un−1j

)+∆t γA

⟨∥∥∇ · (un+1j )′

∥∥2⟩

+∆t (νAH + νA

H)V((∇un+1

j )H)+∆t (νA

⊥ + νA⊥)V

((∇un+1

j )⊥)

+∆t

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)·(un+1j

)′⟩dΓI = 0.

Note that the forcing terms cancel by assumption. The analogous steps may beapplied with (56) and the result added to (58). For the viscous terms, bound belowusing Lemma 7 and Lemma 2. For a more compact notation, define

ωn+1 ≡ V(un+1j

)+ρOρA

V(vn+1j

)

βn+1 ≡ V(∇un+1

j

)+ρOρA

V(∇vn+1

j

)

λn+1 ≡ V(2un+1

j − unj

)+ρOρA

V(2vn+1

j − vnj

)

Λn+1 ≡ V(un+1j − 2un

j + un−1j

)+ρOρA

V(vn+1j − 2vn

j + vn−1j

)

Ωn+1 ≡⟨γA∥∥∇ · (un+1

j )′∥∥2⟩+ρOρA

⟨γO∥∥∇ · (vn+1

j )′∥∥2⟩.


In summary, it holds that

1

4

(ωn+1 + λn+1 − ωn − λn

)+

1

4Λn+1 +

∆t

2

2Ωn+1 + α2ω

n+1 + α1βn+1

+∆t

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)· un+1

j

⟩dΓI

−∆t

∫

ΓI

⟨IA(un+1j ,vn+1

j ,unj ,v

nj

)⟩· un+1 dΓI

+∆t

∫

ΓI

⟨IO(un+1j ,vn+1

j ,unj ,v

nj

)· vn+1

j

⟩dΓI

−∆t

∫

ΓI

⟨IO(un+1j ,vn+1

j ,unj ,v

nj

)⟩· vn+1 dΓI ≤ 0.

Next, apply Lemma 11 for the interface terms, sum over the time index and multiplythrough by 4. The result is(59)

ωN + λN + 4∆t RN + 2∆t

N−1∑

n=1

2Sn+1 + 2Ωn+1 +

1

2Λn+1 + α2ω

n+1 + α1βn+1

≤ λ1 + ω1 + 4∆t R1.

In the limit as N → ∞, it is necessary that

limn→∞

ωn = limn→∞

βn = 0.

This is the desired result.

6. Computational testing

Here, heat transport is considered under conditions of (approximate) radiativeequilibrium, as an example of an important process in AOI research [8]. Basedon Theorem 2, it is likely that the conclusions of Theorem 3 still hold when ensem-bles are run with uncertainty in forcings, fAj and fOj , and friction coefficients, κj.The statistical convergence of the CTM methods is investigated in this section.

The methods M, P1 and P2 will be tested in 2D domains ΩA = [0, 10L]× [0, L]and ΩO = [0, 10L]× [−L, 0], where L = 500 is a length scale in meters. The compu-tational meshes in both domains are uniform Delaunay-Voronoi triangulations withmesh size h = 50

√2 (meters). Time steps are uniform, with size ∆t = 5 (seconds).

These parameters are fixed throughout the computations below.

6.1. Heat convection model background state. A background flow state iscomputed in order to provide a uniform point of comparison for the CTM models.Whereas ensemble indices take on values j = 1, 2, . . ., the background state will beassociated with index j = 0. Buoyancy-driven forcings for the momentum are used,which take the form

fA0 =⟨0,−g(1− βA(θ0 − θ0))

⟩and fO0 =

⟨0,−g(1− βO(ψ0 − ψ0))

⟩.

Here, θ0 and ψ0 are temperatures in Kelvin (K), g = 9.81m · s−2 is gravitationalacceleration and βA = 3.43 · 10−3 (K−1), βO = 2.07 · 10−4 (K−1) are coefficients ofthermal expansion for air and sea water. The temperature averages over domainsΩA and ΩO are denoted by θ0 and ψ0, respectively.

514 J. CONNORS

The temperatures will be numerically approximated as solutions of the heatconvection model

∂tθ0 − αA∆θ0 + u0 · ∇θ0 = 0 on ΩA × (0, T ],

∂tψ0 − αO∆ψ0 + v0 · ∇ψ0 = 0 on ΩO × (0, T ],

with initial conditions discussed later. The (positive, constant) parameters αA

and αO are thermal diffusivity parameters. These values are defined in termsof the Prandtl numbers (see [12]) for air (PrA) and seawater (PrO) by settingαA = νA

H/PrA and αO = νOH/PrO. The solutions are horizontally periodic, as

described for the velocity and pressure variables. No heat flux is allowed throughthe bottom of ΩO. Heat may be transferred across the fluid-fluid interface andthrough the model top (top of ΩA), according to the boundary conditions

−cAρAαA∇θ0 · nA = Cir(θ0 − 285), on Γt,

−cAρAαA∇θ0 · nA = Q, on ΓI ,

and cOρOαO∇ψj · nO = Q, on ΓI ,

with heat flux Q on the interface consisting of three parts:

Q = 30(1− Calb)(1 + cos((x − 150)π/150)) (solar radiative flux)

+ Cir(θ0 − ψ0) (longwave radiative flux)

+ Csen|u0 − v0|(θ0 − ψ0) (sensible heat flux).

The constants cA and cO are the specific heats of air and seawater, respectively.The condition on the model top drives the temperature toward the value 285K byallowing heating (or cooling) if θ0 is below (or above) this value. The net heat fluxQ can be either upward or downward, depending on the net solar radiative flux andthe relative temperatures across the interface. The parameter Calb is the albedo ofthe ocean. The solar heating is allowed to vary in space, otherwise even heating inthese tests would inhibit circulation.

The numerical approximation for temperature uses globally-continuous, piece-wise quadratic finite elements in space and BDF-2 in time, analogous to method Mfor the coupling terms, except that the velocities are also extrapolated there;

|u0 − v0|(θ0 − ψ0)|t=tn+1 ≈ |E(un+10 )− E(vn+1

0 )|(θn+10 − ψn+1

0 ).

This decouples the temperature and velocity at each time step. A similar methodwas studied in [4] and shown to be numerically stable; the main difference is theBDF-2 time stepping, which can be analyzed as in [10]. The temperature fields areonly computed as described here for the background state. Perturbed temperaturefields are discussed later.

6.2. Model spin-up. A spin-up step was used to generate initial conditions forsubsequent testing, with the goal of first bringing the background state near ra-diative equilibrium. This was achieved by starting with zero initial velocity andpressure, and uniform temperatures of 285K in ΩA and 300K in ΩO. Method Mwas used to spin up the background state. Parameter values are shown in Table 3.

During spin-up, solar heating induces buoyancy-driven currents. The lower do-main loses heat into the upper domain, due to the initial temperature differential.Since heat in the upper domain can radiate through the model top, the averagetemperatures in both domains stabilize. The problem was run for 20,000 timesteps. Upon completion, the time-rate of change of average temperature in eachsubdomain was found to be below 10−7 in size. Although the system achieves an


Table 3. Physical parameter values in SI units for the background state.

νAH νA

⊥ νOH νO

⊥ PrA PrO γA γO1 20 1/15 20/15 0.713 7.2 1000 1000

ρA ρO cA cO κ0 Cir Csen Calb

1.2041 1025.0 1004.9 3993.0 0.5 6.0322 0.0011 0.1

approximate radiative equilibrium, the dynamics are not steady, as may be seen bylooking at the total kinetic energies over time. These are given by

ρA2

∫

ΩA

|u|2 dx andρO2

∫

ΩO

|v|2 dx,

shown in Figure 1. The time step size is constrained by the faster dynamics in ΩA.

Figure 1. Kinetic energy during spin-up for the last 500 timesteps in domains ΩA (left) and ΩO (right). The system is approx-imately at radiative equilibrium, but not stationary.

Temperature and velocity streamlines are shown in Figure 2. Although the sys-tem is quite simple compared to an atmosphere-ocean simulation, some similaritiesexist. Faster convective currents persist in ΩA, compared to those in ΩO. The ki-netic energy in ΩO is still considerable, as a result of the larger density, ρO ≈ ρA·103.A test could be done with a wider range of flow scales by decreasing the kinematicviscosities or by increasing the length scale, L. The parameter choices in this reportare limited by the computational cost.

Figure 2. Streamlines and temperatures in ΩA (top) and ΩO

(bottom) are shown for the initial conditions generated via spin-up.The flow exhibits a moderate range of flow scales.

516 J. CONNORS

6.3. Convergence to statistical equilibrium. The calculation of mean-flowkinetic energy (an example quantity of interest) and the convergence to statisticalequilibrium are investigated with the CTM methods M, P1 and P2. Uncertainty isintroduced through the Boussinesq forcing and the friction parameter. The initialconditions, shown in Figure 2, are not perturbed and do not contribute to themodel variance. The constant parameters in Table 3 are used here. Ten-memberensembles are used, with uncertain data expressed in terms of the numbers

δj =

−j, j = 1, . . . , 5j − 5, j = 6, . . . , 10

.

The friction parameters are κj = κ0 + 0.01 δj h(t)κ0, j = 1, . . . , 10. Note that∑10j=1 δj = 0. The mean friction parameter is thus κ = κ0.

The simulation is run for 2000 time steps, which yields a final time of 105 s. Thefunction h(t) is given by

0 ≤ h(t) =t2

2.5 · 107 exp(2− t/2500) ≤ 1,

which satisfies 0 ≤ h(t) ≤ h(5000) = 1 and decays to h(105) ≈ 10−14, so that (49)will hold.

Perturbed forcings are constructed to induce small-scale velocity fluctuations.The background temperature fields are perturbed to drive the flows, according to

fAj =⟨0,−g(1− βA(θ0 − θ0 + (5 · 10−5)δj h(t)

2 cos(π x/250) sin(π y/250)))⟩

fOj =⟨0,−g(1− βO(ψ0 − ψ0 + (5 · 10−6)δj h(t)

2 cos(π x/250) sin(π y/250)))⟩.

The full heat convection model is only used to calculate the background state(j = 0) at each time step. The ensemble only includes j = 1, . . . , 10, which areperturbations of the background state. The forcings for the ensemble membersdo not depend on the velocities uj or vj , since this would present a fundamentaldeviation from the theory in this paper. The relative sizes of perturbations for κj,fAj and fOj reflect an attempt to balance out the contributions to model variancefrom these sources.

In addition to the CTM methods, a straight ensemble calculation is includedhere for comparison. That is, the numerical method equivalent to method M, butwithout using the mean-flow velocity for the convection terms and without anyturbulent viscosity. Variance calculations in the L2-norm are shown in Figure 3,and kinetic energy is shown in Figure 4. For the CTM methods, the turbulentviscosities were calculated using (22) with µ = 1.0 in ΩA, µ = 0.5 in ΩO, and l,k′ defined in Table 2. These µ values were found to be roughly optimal for themean-flow kinetic energy using method M to match the mean-flow kinetic energywithout CTM as best as possible.

The model variance is consistent amongst CTM methods, but reduced comparedto the variance without CTM, and vanishing in all cases. This is consistent withthe implication of Theorem 2 and with Theorem 3. The kinetic energy is onlyshown on time intervals where variance is significant; details are hard to see onthe global time scale. The CTM methods all track the reference mean-flow kineticenergy (without CTM) very well, in particular when the mean flow deviates signif-icantly from the background state. When the variance is small, the mean-flow andbackground energies nearly coincide, but the partitioned methods predict a slightlydifferent result in these regions, shown in Figure 5. Methods P1 and P2 predict thesame energy when the variance in small, which is expected since these methods areequivalent when the variance is zero.


Figure 3. The variance (L2-norm) in domains ΩA (left) and ΩO

(right). As expected, the CTMmethods have reduced the variance.

Figure 4. The background state and mean-flow kinetic energiesin domains ΩA (left) and ΩO (right). The CTMs closely track themean-flow behavior as it deviates from the background state.

Figure 5. The background state and mean-flow kinetic energiesin domains ΩA (left) and ΩO (right). The model variances arevery small for the range of times shown here. The kinetic energyof the background state coincides with that of the mean-flow stateswithout CTM and with method M, but methods P1 and P2 displaya slightly different behavior.

Since the difference between the monolithic and partitioned methods is only thenumerical coupling, the results suggest that the coupling method can introducea statistical bias. In scientific AOI codes, this sort of bias would be challengingto quantify since (1) monolithic coupling is not available and (2) the couplingfrequency is limited, due to the computational cost. Method P1 is more interestingthan methods M or P2, for practical purposes. Aside from the issue of a statisticalbias, the results shown here indicate that method P1 may (with parameter tuning)do a good job of reproducing mean-flow statistics.

518 J. CONNORS

7. Conclusions

A statistical turbulence model was proposed for ensemble calculations with twocoupled fluids that enables some reduction in matrix assembly costs. The closuremodel accounts for the behavior of the Reynolds stress terms at the interface.The analyses of [10] were extended to account for the fluid coupling conditions,showing a proof of the Boussinesq hypothesis and squeezing of fluid trajectorieswith the turbulence model. Some numerical methods were proposed, includingimplicit and partitioned coupling, and it was proved for two methods that thediscrete variance must vanish at long times. The analogous proof for the thirdpartitioned method, which is of the most practical interest, is not known. But incomputations this method was observed to predict almost exactly the same behavioras the other methods. Furthermore, the computations indicate an excellent abilityfor the turbulence methods to reproduce the correct ensemble mean-flow behavior.The partitioned methods may introduce a statistical bias at long times, which wassmall in the computational example.

Acknowledgments

The author thanks Professor William J. Layton for providing feedback regardingthe development of this work.

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Department of Mathematics, University of Connecticut at Avery Point, Groton, CT, 06340,USA

E-mail : [email protected]: http://www.math.uconn.edu/∼connors/

an ensemble-based conventional turbulence model …

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