Vector cylindrical harmonics for low-dimensional convection models

Douglas H. Kelley1 and Eric G. Blackman2

1Department of Mechanical Engineering, University of Rochester, Rochester, NY, 14627, USA∗

2Department of Physics and Astronomy, University of Rochester, Rochester, NY, 14627, USA(Dated: October 10, 2018)

Approximate empirical models of thermal convection can allow us to identify the essential proper-ties of the flow in simplified form, and to produce empirical estimates using only a few parameters.Such “low-dimensional” empirical models can be constructed systematically by writing numerical orexperimental measurements as superpositions of a set of appropriate basis modes, a process knownas Galerkin projection. For three-dimensional convection in a cylinder, those basis modes should bevector-valued, mutually orthogonal, and defined in cylindrical coordinates. Here we construct sucha basis set and demonstrate that it has these desired properties and boundary conditions when theexact constraint of incompressibility is relaxed. We show its use for representing sample simulationdata and point out its potential for low-dimensional convection models.

INTRODUCTION

Given the inherent complexity of fluid flow, the scarcityof closed-form solutions to its equations of motion, andthe computational expense of large-scale numerical simu-lation, it is often practical to seek empirical models witha limited number of degrees of freedom, that is “low-dimensional” models. Ideally, such models would requireless input information than the full velocity field, butwould characterize the essential flow dynamics. The in-put could come from either simulation or experiment, andthe model could be used to forecast future flow states orto estimate the flow at the present time in regions whichhave not been measured directly. Here and throughout,we consider flows in the usual three spatial dimensions;all subsequent discussion of dimensionality refers to thephase space required to represent the system, or equiv-alently, the number of parameters used to represent theflow. Many low-dimensional models for fluid dynami-cal systems have been developed in previous work. Herewe will focus on a particular problem: building low-dimensional models of thermal convection in a cylindricalvessel from experimental measurements.

Historically, we might consider the first modern low-dimensional convection model to be that of Lorenz [1],whose three-parameter model for atmospheric convectiondisplayed such a strikingly sensitive dependence to initialconditions that it spurred a new understanding of fluiddynamics in terms of chaos and dynamical systems the-ory. Lorenz built his model by writing the equations ofmotion for convection in spectral form (that is, as a su-perposition of flow modes with different wave numbers)and truncating the equations to just three modes. Later,Howard and Krishnamurti [2] produced a six-mode trun-cation that captured more intricate dynamics, and Thif-feault and Horton [3] showed that retaining a seventhmode produced the lowest-dimensional truncated modelthat conserves total energy in the dissipationless limit.Many convection models have focused on the large-scalecirculation, whose shape is a single, large roll that nearly

fills the container. (The large-scale circulation is alsocalled the mean wind or the wind of turbulence.) Sreeni-vasan et al. [4] used laboratory measurements to builda model for the large-scale circulation and attributedreversals of the large-scale circulation to imbalance be-tween buoyancy and friction. Benzi [5] found that rever-sals were well-modeled by a set of stochastic differentialequations. Brown and Ahlers [6] also built a model fromstochastic ordinary differential equations, using one forflow strength and another for flow orientation, and foundpredictions consistent with their experimental measure-ments. Models of many fluid systems, including ther-mal convection systems, are frequently constructed us-ing proper orthogonal decomposition [7], balanced properorthogonal decomposition [8], and related techniques.Navarro et al. [9] constructed a low-dimensional modelfor convection in a cylinder whose upper and lower sur-faces counter-rotate and found that a model with 41 de-grees of freedom successfully reproduced the representa-tive states considered. Bailon-Cuba and Schumacher [10]constructed a low-dimensional model for convection in asquare domain by projecting simulation data onto a setof basis modes found via proper orthogonal decomposi-tion. Bailon-Cuba et al. [11] used a similar procedure toproduce a model applicable in a more complex domainhaving inlets, outlets, and localized heat sources.

If we have only experimental measurements as inputs,then model construction methods which require high-resolution a priori simulations spanning space and time(like proper orthogonal decomposition) are unavailableto us. We have been initially motivated by characteriz-ing the flow in liquid metal batteries [12–14], which arebuilt with cylindrical shapes having different aspect ra-tios than the most commonly studied convection systems,and which are subject additional physical processes; theirdominant flow features have not yet been characterizedwith low dimensional models. The general method thatwe describe herein has direct application to these sys-tems.

A rational and systematic procedure for producing low-

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dimensional models, without prior simulation results orknowledge of dominant flow features, is Galerkin projec-tion [15–17], in which numerical or experimental mea-surements are represented as a superposition of weightedbasis modes. The weights of the modes then expressthe information contained in the measurements, just asa Fourier transform expresses the information containedin the function from which it is produced. Truncatingthe projection to a small number of modes produces alow-dimensional model, in which only the degrees of free-dom associated with the remaining modes are considered.Galerkin projection also allows making estimates at lo-cations other than those where measurements were col-lected, by summing the weighted modes. Performing aGalerkin projection, however, requires an appropriate setof modes. Though similar considerations apply across adiverse array of other applications beyond fluid mechan-ics, there has been very little work on these methods incylindrical coordinates for three spatial dimensions. Oneexception is Wang et al. [18], in the context of imageprocessing.

In the present paper we construct a basis set appro-priate for low-dimensional models of fluid flow, includingconvection, in cylindrical geometry. We begin in § withthe mathematical background necessary for projectingconvection measurements onto a basis set, which identi-fies the required characteristics for the basis. In §, weshow how to construct a vector-valued basis set in whichevery mode satisfies no-slip boundary conditions and isorthogonal to every other mode, as desired for flow mod-eling. We provide a few examples of vector cylindricalharmonics and discuss the properties of the basis. In §,we demonstrate the use of the basis via Galerkin pro-jection of one velocity field, taken from a simulation ofconvection in a cylindrical container. Characteristics ofthe resulting modal weights are considered. By truncat-ing the projections to varying mode counts, we considerhow the fidelity of a low-dimensional model depends onits dimensionality (that is, the number of parameters),at least for this set of measurements. Finally, we closewith §, which summarizes our conclusions and points outopportunities for future work.

GALERKIN PROJECTION AND DESIREDCHARACTERISTICS OF BASIS

Given a set of scalar basis modes ψj , any scalar quan-tity f(xn) measured at locations xn (where each n labelsa different point in space) can be written as a superposi-tion of weighted modes

f(xn) =

Nj∑j=1

αjψj(xn), (1)

where Nj is the mode count and αj is the scalar weightof the jth mode. A Fourier series is one example of sucha superposition, in which the modes ψj are sinusoids. Afinite mode count Nj may not allow an exact match be-tween the measurements and their reproduction in termsof the basis modes. Then, the representation in Eq. 1 isonly approximate, and the squared error is

ε =

Nn∑n=1

f(xn)−Nj∑j=1

αjψj(xn)

2

,

where Nn is the number of measurements. It can beshown [19] that the error ε is minimized by the particularvalues of the weights αj that satisfy the matrix equations

Nn∑n=1

Nj∑j=1

ψj(xn)ψj(xn)αj =

Nn∑n=1

ψj(xn)f(xn), (2)

known as the “normal equations” of the linear, least-squares fit. Minimizing the error is mathematically iden-tical to maximizing the probability that the weights αjcorrectly model the measured data, assuming Gaussianerrors. It is often convenient to solve Eq. 2 via singu-lar value decomposition [20], and we require Nj ≤ Nn.Choosing Nj � Nn provides a systematic and naturaltechnique for constructing a low-dimensional model ofthe original measurements f(xn). Galerkin projection(also known as least-squares projection) requires, how-ever, that the basis modes ψj(xn) be orthogonal andspan the domain xn. Reproducing arbitrary measure-ments f(xn) to maximal accuracy also requires that thebasis set be complete and that Nj → ∞. However, aswe shall see, explicit demonstration of completeness isnot required for the method to yield practical results ina given application.

Our goal is to model velocity fields of a convectingsystem in a cylindrical vessel. Because velocity is a vec-tor, we require a vector-valued basis set. Considering acylindrical vessel, we seek a basis set defined in terms ofcylindrical coordinates (ρ, ϕ, z), made dimensionless sothat 0 ≤ ρ ≤ 1/2 and 0 ≤ z ≤ 1. (Cylindrical containersof other sizes and other aspect ratios can be accommo-dated by scaling.) Two vector basis modes ψj(ρ, ϕ, z)and ψk(ρ, ϕ, z) (here labeled generically with a singlesubscript) are orthogonal if and only if the volume in-tegral of their product vanishes for non-identical modes(j 6= k):∫

ψj ·ψk dV =

∫ 1

0

∫ 2π

0

∫ 1/2

0

ψj ·ψk ρ dρ dϕ dz = 0. (3)

Working with experimental measurements, we considerno-slip boundary conditions at the outer radius of thecylindrical volume. Thus we would prefer to ensuresuch a boundary condition by finding modes that sat-isfy ψk(ρ = 1/2) = ψk(z = 0) = ψk(z = 1) = 0 for any

3

k. In general, we also prefer a basis set that is completeso that it can represent any incompressible vector fieldin the spatial domain.

Though Galerkin projection is a well-known technique,we found it necessary to devise a new basis set after find-ing no existing basis sets with all of the desired character-istics for three-dimensional cylinders. Our approach wasinitially inspired by vector spherical harmonics [21, 22],widely used for spectral simulations in atmospheres [23],oceans [24], planetary cores [25], solar convection anddynamos [26, 27] and helioseismology [28] in sphericalcoordinates. Vector spherical harmonics are also usefulfor producing low-dimensional models from laboratorymeasurements of velocity fields and magnetic fields inspherical systems [29]. Vector spherical harmonics arevector-valued, mutually orthogonal, and have the properperiodic boundary conditions for representing functionson the surface of any sphere. Additionally, because vec-tor spherical harmonics are constructed via curls, eachharmonic individually has zero divergence, making themuseful for representing incompressible flows and magneticfields.

One might expect to produce a set of functions with thesame attractive features for incompressible flows in cylin-drical coordinates by using steps analogous to those forderiving vector spherical harmonics, but this proceduredoes not work. First, individual vector spherical har-monics do not satisfy no-slip boundary conditions at thewall of a spherical vessel. Rather, when vector sphericalharmonics are used in spectral simulations, radial bound-ary conditions are enforced by careful choice of the modalweights (analogous to αj in Eq. 1) Hollerbach [30]. In ourmathematical investigations we find that analogous basissets constructed in cylindrical coordinates suffer from thesame trouble satisfying boundary conditions. A secondproblem arises in cylindrical coordinates as well. Whenwe constructed basis sets in cylindrical coordinates usingcurls that enforce incompressibility, the requirements formodes to be mutually orthogonal were not easy to sat-isfy with any common functions. These two fundamentalproblems require that we construct a new basis in whichwe relax the exact constraint of incompressibility.

CONSTRUCTING VECTOR CYLINDRICALHARMONICS

We represent velocity fields in terms of separable scalarharmonics of the form

ψlmk = J1(2ζkρ) cos lϕsin lϕ sinmπz, (4)

where now we label each mode with three positive in-teger indices (k, l,m), J1 is the Bessel function of thefirst kind (Jn with n = 1) and ζk is the kth zero of J1.The azimuthal factor in the ψlmk can be constructed us-ing either a cosine or a sine, and we will use both modes

0 10 20

ρ

-0.5

0

0.5

1

J n(ρ

)

(a)

J0

J1

J2

J3

J4

J5

0 0.2 0.4

2ζkρ

-0.5

0

0.5

1

J 1 (

2ζ kρ)

(b)

k = 1

k = 2

k = 3

k = 4

k = 5

k = 6

FIG. 1. Bessel functions of the first kind.

FIG. 2. Scalar cylindrical harmonics as given by Eq. 4, plottedon the cylinder ρ = 0.45 and on the plane y = 0. (a–b), ψ2c3

1 .(c–d), ψ1c2

3 . (e–f), ψ3c12 .

when we reconstruct data. In Eq. 4, l, m and k are wavenumbers in the azimuthal, axial, and radial directionsrespectively. Figure 1 shows the situation: Bessel func-tions of the first kind oscillate as ρ varies. The radialfactor J1(2ζkρ) is scaled such that the no-slip boundarycondition is satisfied and there are k/2 oscillations overthe range 0 ≤ ρ ≤ 1. Here and below we will denotecosine and sine modes with the suffixes “c” and “s”, re-spectively. For example, ψ3c4

2 = J1(2ζ2ρ) cos 3ϕ sin 4πzand ψ2s3

1 = J1(2ζ1ρ) sin 2ϕ sin 3πz. A few of the scalarharmonics ψlmk are shown in Fig. 2.

We can use the scalar harmonics ψlmk to approximatevector-valued functions u in cylindrical coordinates by

4

summing over (k, l,m) in all three coordinate directions:

u(ρ, ϕ, z) =

∞∑k=1

∞∑l=−∞

∞∑m=1

(αlmk ρ+ βlmk ϕ+ γlmk z

)ψlmk ,

(5)where (ρ, ϕ, z) are unit vectors in the radial, azimuthal,and axial directions, respectively. We shall refer to eachρψlmk as a radial basis mode, each ϕψlmk as an azimuthalbasis mode, and each zψlmk as an axial basis mode. To-gether, the ρψlmk , ϕψlmk , and zψlmk form a set whichwe shall call the “vector cylindrical harmonics”. The(αlmk , βlmk , γlmk ) are dimensionless coefficients.

Such an approximation has many of the characteristicsdesired for constructing low-dimensional models of con-vection in a cylinder. The vector cylindrical harmonicsare defined in cylindrical coordinates. Each mode hasmagnitude zero at the boundaries z = 0, z = 1, andρ = 1/2, as is clear from Eq. 4. Each mode thereforesatisfies the required no-slip boundary condition, as doany superpositions of modes. Each mode satisfies theperiodicity boundary condition in the azimuthal direc-tion: ψlmk (ϕ = 2π) = ψlmk (ϕ = 0). Each mode is alsoorthogonal to every other. By scalar products,

ρψ · ϕψ = ϕψ · zψ = zψ · ρψ = 0,

where we have not written the indices (k, l,m) becausetheir values do not matter; Eq. 3 is satisfied for pairs ofmodes of different coordinate direction regardless of theirwave numbers. For pairs of modes of the same coordinatedirection, orthogonality is guaranteed by the facts that,if (k, l,m) 6= (p, q, r),∫ 1/2

0

J1(2ζkρ) J1(2ζpρ)ρ dρ = 0∫ 2π

0

cos lϕ cos qϕ dϕ = 0∫ 2π

0

sin lϕ sin qϕ dϕ = 0∫ 2π

0

cos lϕ sin qϕ dϕ = 0∫ 1

0

sinmπz sin rπz dz = 0;

again, Eq. 3 is satisfied.If the vector cylindrical harmonics formed a complete

basis set, then the approximation stated in Eq. 5 is exact:as long as an infinite number of modes are included in thesum, any vector-valued function defined in the cylindricaldomain considered here can be represented without er-ror. Still, even an incomplete basis can represent a largevariety of flows of practical interest with good fidelity.The set of all cosines and sines, used in the azimuthalfactor, is known to be complete. The set of all sines,used in the axial factor, is also known to be complete for

no-slip boundaries. For the radial factor, we expect theBessel functions to be able to match functions of arbi-trary spatial frequency because of the oscillatory natureof the J1(kρ). However, the envelope of their oscillationalways decreases with radius ρ, as shown in Fig. 1, whichis similar to Fig 1a of Wang et al. [18]. Thus functionswhose amplitude is minimum at small radius ρ ∼ 0 maynot be well-matched with Bessel functions. In general,construction and rigorous proof of a complete basis set isa challenge beyond the present scope.

EXAMPLE USING CONVECTION SIMULATIONRESULTS

To test the applicability of vector cylindrical harmonicsfor representing data, we consider an example. J. Schu-macher has graciously provided us with results of a nu-merical experiment in which he and colleagues solved theBoussinesq equations for thermal convection in a cylin-der of height H and diameter d = H. The side wallwas thermally insulated, such that the heat flux throughthe wall, or equivalently the radial temperature gradientat the wall, was zero. The temperatures of the top andbottom were held constant and uniform. All boundarieswere no-slip (zero velocity). The equations were solvedusing a spectral element method with the Nek5000 pack-age. The Rayleigh number (dimensionless buoyancy) wasRa = 107. The Prandtl number was Pr = ν/α = 0.7(the value for water), where ν is the kinematic viscos-ity and α is the thermal diffusivity. We will examine asingle snapshot from the simulation, with all three ve-locity components tabulated on a cylindrical grid of size97× 193× 128 in ρ, ϕ, and z, respectively, equivalent to7.2 × 106 individual measurements. An illustrative partof the three-dimensional snapshot is is plotted in Fig. 3.

We need not retain all information from this high-resolution, high-dimensional simulation to demonstratethe use of vector cylindrical harmonics for constructinglow-dimensional models. Rather, we retain only everyfourth grid point in each direction; part of the reduced-resolution snapshot is shown in Fig. 3b. We reiterate thatthe entire three-dimensional snapshot includes more datathan the two surfaces displayed in the figure. The methoddescribed below could be readily applied to the originalsnapshot at full resolution, given sufficient computationalresources.

Using the reduced data set as an example, we pro-ceed with Galerkin projection. We can write the velocitymeasurements u(xn), as a superposition of vector cylin-drical harmonics according to Eq. 5, where each of thewave numbers (k, l,m) is summed over some (possiblyinfinite) set of integers. The weights αlmk , βlmk , and γlmkcan be then calculated if we rewrite the normal equations(Eq. 2) more explicitly in terms of the vector cylindrical

5

FIG. 3. (a) Example simulation results. Speed on the sur-faces y = 0 and the ρ = 0.9H are indicated in color. Gravitypoints downward. (b) The same results, plotted at reducedresolution used for projection onto vector cylindrical harmon-ics.

harmonic modes, specifically,∑n,k,l,m

(αlmk ρ+ βlmk ϕ+ γlmk z

)ψlmk · (ρ+ ϕ+ z)ψqrp

=∑n

(ρ+ ϕ+ z)ψqrp u(xn), (6)

and we remind that ψqrp = ψqrp (xn).

As an example, we allow modes with 1 ≤ k ≤ kmax, 0 ≤l ≤ lmax, and 1 ≤ m ≤ mmax and (kmax, lmax,mmax) =(16, 10, 12). Solving Eq. 6 with u(xn) given by thereduced-resolution simulation results, we arrive at thevalues of αlmk , βlmk , and γlmk , the strongest 10% of whichare plotted in Fig. 4. These weights provide a spectralrepresentation of the snapshot because each weight cor-responds to one vector cylindrical harmonic mode, andeach mode has a well-defined wave number in each spatialdirection. Examining the weights directly can thereforeprovide insight into the spectral structure of the snap-shot, often revealing features that are more difficult tovisualize when working with the original measurements.For example, the αlmk have large magnitudes for widerange of all three wave numbers (k, l,m), indicating thatthe radial velocity component of the snapshot containsa broad range of spatial frequencies in the radial, az-imuthal, and axial directions. The βlmk rarely have largemagnitudes when k or l is large, indicating that the az-imuthal velocity component of the snapshot varies moregradually in the radial and azimuthal directions. Largeβlmk for large m, however, indicate that the azimuthalvelocity component varies over short length scales in theaxial direction. In contrast, the γlmk have large magni-

2

4

10

6

8

m

10

l

12

5

k

10

0 0

(a), αk

lcm

2

4

10

6

8

m

10

l

12

5

k

10

0 0

(b), αk

lsm

2

4

10

6

8

m

10

l

12

5

k

10

0 0

(c), βk

lcm

2

4

10

6

8

m

10

l

12

5

k

10

0 0

(d), βk

lsm

2

4

10

6

8m

10

l

12

5

k

10

0 0

(e), γk

lcm

2

4

10

6

8

m

10

l

12

5

k

10

0 0

(f), γk

lsm

-0.02 0 0.02

weight (arb)

FIG. 4. Spectral representation of the simulation snapshotshown in Fig. 3. (a) Weights αlcm

k corresponding to thestrongest 10% of cosine modes in the ρ direction, indicatedin color. (b) Weights αlsm

k corresponding to the strongest10% of sine modes in the ρ direction, indicated in color. (c)Weights βlcm

k corresponding to the strongest 10% of cosinemodes in the ϕ direction, indicated in color. (d) Weightsβlsmk corresponding to the strongest 10% of sine modes in theϕ direction, indicated in color. (e) Weights γlcm

k correspond-ing to the strongest 10% of cosine modes in the z direction,indicated in color. (f) Weights γlsm

k corresponding to thestrongest 10% of sine modes in the z direction, indicated incolor.

tude only for small m, indicating that the axial veloc-ity component varies gradually in the axial direction. Itvaries over short length scales, however, in both the ra-dial and azimuthal directions, as shown by large valuesof γlmk for large k and l. Finally, though the spectra arenot identical for sine and cosine modes, they follow thesame general trends.

Once calculated, the weights αlmk , βlmk , and γlmk can beused in Eq. 5 to reproduce u(xn). With a finite set of vec-tor cylindrical harmonics, the reproduction is imperfect,

6

FIG. 5. Example simulation results projected onto vec-tor cylindrical harmonics, plotted as in Fig. 3. With(kmax, lmax,mmax) = (16, 10, 12), the reproduction closelymatches the original measurements.

but may still provide a useful model; we will denote thereproduction ur(xn) to distinguish it from the originalfield u(xn). We used the weights plotted in Fig. 4 to pro-duce ur(xn), and the results are shown in Fig. 5. As inFig. 3, the plot illustrates a three-dimensional snapshotby displaying the speed on two surfaces in the volume.Comparing to the original velocity field (Fig. 3), we seethat though the match is not exact, the reproductionclosely resembles the original data, demonstrating thatvector cylindrical harmonics as defined in Eqs. 4 and 5can reproduce a typical convection simulation with goodfidelity. Though we have not proved completeness, andcompleteness is required for representing arbitrary func-tions, a basis set may be robust for specific practical ap-plications even without being strictly complete [18]. Atleast in this example, vector cylindrical harmonics arerobust in that sense.

We built the reproduction shown in Fig. 5 to demon-strate that the vector cylindrical harmonics are robustfor producing low-dimensional models. The reproductionshown does have lower dimensionality than the measure-ments from which it was built, which comprised threevelocity components at 3.8× 104 locations — 1.12× 105

measurements altogether — compared to 1.21× 104 har-monics in the reproduction. The dimensionality of thesystem has been reduced by an order of magnitude, withminimal loss in fidelity. Still, a reproduction using 104

modes is more complex than the low-dimensional modelswe intend to build. Our goal is to continue reducing thedimensionality as long as the dominant flow features canbe retained. To do so, we must determine the requireddimensionality.

Figure 6 shows a series of reproductions of decreas-

FIG. 6. Example simulation results projected onto vec-tor cylindrical harmonics, with decreasing dimensional-ity. Each reproduction is plotted as in Fig. 3, andeach uses fewer harmonics than the one before. (a)(kmax, lmax,mmax) = (16, 10, 12): 12,096 harmonics (Identi-cal to Fig. 5). (b) (kmax, lmax,mmax) = (13, 8, 10): 6630harmonics. (c) (kmax, lmax,mmax) = (9, 6, 7): 2457 har-monics. (d) (kmax, lmax,mmax) = (5, 4, 5): 675 harmon-ics. (e) (kmax, lmax,mmax) = (4, 3, 4): 336 harmonics. (f)(kmax, lmax,mmax) = (3, 2, 3): 135 harmonics. The fidelity ofthe reproduction decreases monotonically with dimensional-ity.

ing dimensionality, beginning the with same reproduc-tion shown in Fig. 5, and continuing to reproductionswith as few as 135 harmonics, as described in detail inthe figure caption. In all cases, harmonics with the lowestwave numbers are retained, whereas more and more har-monics with high wave numbers are dropped with eachsubsequent reproduction. The maximum wave numbersin the radial, azimuthal, and axial directions are all re-duced from each reproduction to the next. As expected,the features with high spatial frequency fade as the maxi-mum wave numbers are decreased. When few harmonicsare used, reproductions tend to be inaccurate near theρ = 0 axis, consistent with the fact that ψlmk (ρ = 0) = 0according to Eq. 4 and Fig 1. However, the high-speedregion near the top center of the snapshot is retained formost of the reproductions, and the high-speed region atleft is retained for all. Even a reproduction that reduces1.12 × 105 numerical measurements to 135 modes canprovide useful insight into the flow state in an industrialsetting.

We can go beyond qualitative examination of snap-

7

shots by quantifying the fidelity of reproduction. Onemeasure of the fidelity of a low-dimensional model is theratio of the root-mean-square velocity of the model tothe root-mean-square velocity of the original data,

〈u2r〉1/2

〈u2〉1/2, (7)

where the brackets 〈·〉 signify averaging over the spatialdomain. A perfect model captures all motions, such thatthe velocity ratio is unity. A real model does not ex-actly match the input data; rather, at different locationsit either underestimates or overestimates the speed. Anymode in the infinite sum in Eq. 5 that has a nonzeroweight contains some spectral power, and if that mode isneglected when the sum is truncated, its spectral poweris lost. In low-dimensional models for which many modesare necessarily neglected, the speed is underestimatedmore often than it is overestimated, such that the velocityratio defined in Eq. 7 falls below unity, dropping to zeroin the extreme case of all modes being eliminated. Speedcan occasionally be overestimated, however, because ofaliasing onto modes with lower spatial frequency.

A second measure of fidelity is the normalized error,

〈(ur − u)2〉1/2

〈u2〉1/2.

The normalized error would be zero for a perfect modelthat matches the original data exactly. To identify asuitable minimal model that captures basic features ofthe dynamics, both the velocity ratio and the normalizederror must be considered in light of the count of modesretained in the reproduction,

Nr = 3kmax(2lmax + 1)mmax,

which is necessarily less than the number of measure-ments: Nr ≤ Nn.

Figure 7 shows the velocity ratio and normalized errorfor the six reproductions plotted in Fig. 6. As the modecount Nr (or equivalently, the ratio Nr/Nn) decreases,the measured fidelity of the reproductions behaves as ex-pected. With fewer modes, the velocity ratio is lower,and the normalized error is higher.

Choosing the number of modes to retain in a low-dimensional model requires striking a balance betweenthe required fidelity and the complexity of informationto be retained (as well as the complexity of the measure-ments required). The balance will vary depending on theapplication at hand, so by quantifying fidelity, modelerscan make an informed decision. Figure 7 shows that bothvelocity ratio and normalized error are monotonic withmode count — but neither is linear. An astute choice ofmode count will consider that fact. For example, Fig. 7ashows that the velocity ratio stays above 80% with just675 modes, for which Nr/Nn = 0.6% and the complexity

00.050.1

Nr / N

n

0.6

0.8

1

<u

r2>

1/2

/ <

u2>

1/2

(a)

00.050.1

Nr / N

n

0.2

0.4

0.6

0.8

<(u

r-u)

2>

1/2

/ <

u2>

1/2

(b)

0500010000

Nr

0500010000

Nr

FIG. 7. Fidelity and information content of the six repro-ductions shown in Fig. 6. (a) Velocity ratio. (b) Normalizederror. Both are plotted against mode count Nr and normal-ized mode count Nr/Nn, which decrease from left to right. Asmode count decreases, velocity ratio decreases monotonicallyand normalized error increases monotonically.

of information retained has been reduced by more thantwo orders of magnitude.

Constructing a useful low-dimensional model requiresnot only considering the total mode count Nr, but themaximum wave number in each spatial direction, sincedifferent data sets by definition have different spatialstructure, and may have features that are either broador narrow in any of the three spatial directions, thereforepresenting different spectra. We can examine the spec-tral content of the velocity snapshot shown in Fig. 3 ineach spatial dimension by varying kmax, lmax, and mmax

independently. Figure 8 shows the results. We first heldlmax = 6 and mmax = 10 constant, varying kmax, thenheld kmax = 15 and mmax = 10 constant, varying lmax,and finally held kmax = 15 and lmax = 6 constant, vary-ing mmax. To speed the calculations, we used every tenthgrid point from the original snapshot. The velocity ratio,normalized error, and mode count is plotted in each case.Again we find that the velocity ratio decreases monoton-ically as the mode count decreases. With this snapshot,the effect is stronger for kmax and mmax than for lmax.Setting mmax < 2 has an especially strong effect, suggest-ing that there is substantial energy in flow shapes withhigher-order symmetry than the m = 1 mean wind.

Likewise, we find that the normalized error increasesmonotonically as the mode count decreases, but not al-ways at the same rate, as shown in Fig. 8(d–f). Normal-ized error increases fastest in the same ranges of wavenumber where velocity ratio decreases fastest, confirm-ing the observation that the effect of removing additionalmodes from the model is not always the same.

CONCLUSIONS

Summary

We have constructed a set of basis functions appropri-ate for representing velocity fields in cylindrical coordi-

8

0.5

0.6

0.7

0.8

0.9<

ur2>

1/2

/ <

u2>

1/2

(a) (b) (c)

0.2

0.4

0.6

0.8

<(u

r-u)

2>

1/2

/ <

u2>

1/2

(d) (e) (f)

010

kmax

0

0.1

0.2

0.3

Nr /

Nn

(g)

0510

lmax

(h)

0510

mmax

(i)

FIG. 8. Fidelity and information content of low-dimensionalmodels, varying maximum wave numbers systematically. (a)Velocity ratio for models with lmax = 6, mmax = 10, and1 ≤ kmax ≤ 15. (b) Velocity ratio for models with kmax =15, mmax = 10, and 1 ≤ lmax ≤ 6. (c) Velocity ratio formodels with kmax = 15, lmax = 6, and 1 ≤ mmax ≤ 10. (d)Normalized error for models with lmax = 6, mmax = 10, and1 ≤ kmax ≤ 15. (e) Normalized error for models with kmax =15, mmax = 10, and 1 ≤ lmax ≤ 6. (f) Normalized error formodels with kmax = 15, lmax = 6, and 1 ≤ mmax ≤ 10. (g)Normalized mode count for models with lmax = 6, mmax = 10,and 1 ≤ kmax ≤ 15. (h) Normalized mode count for modelswith kmax = 15, mmax = 10, and 1 ≤ lmax ≤ 6. (i) Normalizedmode count for models with kmax = 15, lmax = 6, and 1 ≤mmax ≤ 10. In all plots, the mode count decreases fromleft to right. Decreasing the mode count always decreasesthe velocity ratio and increases the normalized error, but notalways by the same rate.

nates, and we call them “vector cylindrical harmonics”.The functions are Bessel functions in the radial direction,sines and cosines in the azimuthal direction, and sines inthe axial direction. Every function in the basis is orthog-onal to every other according to Eq. 3 and satisfies no-slipboundary conditions at the vessel walls. The versatilityof our approach is facilitated by the absence of imposingan exact constraint of incompressibility.

We have demonstrated the use of this basis set by rep-resenting a velocity snapshot of a simulation of thermalconvection in a cylinder. We used least-squares projec-tion to determine the set of modal weights αlmk , βlmk ,and γlmk that best model the simulation results. Thoseweights give a spectral representation of the snapshot,and we have used them to characterize its spectral con-tent. We have demonstrated the use of the basis set for

making low-dimensional models by varying the numberof basis modes used to represent the simulation results.As expected, including fewer modes produces a simplermodel with less fidelity.

However, the fidelity of the model varies nonlinearlywith mode count; some modes capture more informationthan others. Proving that the vector cylindrical harmon-ics comprise a complete basis set is beyond the scope ofthis paper. Completeness is not be necessary for the basisto be robust for practical application, however [18].

Applications and Outlook

Our original motivation for developing these generalvector cylindrical harmonics was for the future applica-tion to characterizing convection and magnetoconvectionin cylindrical liquid metal batteries, given measurementsof the flow using ultrasound velocimetry. Liquid metalbatteries [12–14] are a new technology intended for stor-ing large amounts of electrical energy, thereby allowingwidespread incorporation of intermittent wind and solarpower on the world’s electrical grids. Because the batter-ies are built with a liquid electrolyte and two liquid metalelectrodes, fluid flow can affect battery performance [31].Flow is driven by thermal gradients, since liquid metalsrequire high temperatures. Flow is also driven by theelectrical currents running through the batteries, whosemagnitudes are on the order of 100 mA/cm2 [32]. Wehope that low-dimensional models will give battery de-signers and operators useful information about batterycharge state, electrode uniformity, and battery health,all of which can be affected by flow. Magnetic fieldsmight also be expressed in terms of the vector cylindri-cal harmonics, though finding a set of weights to makethe reproduction have divergence as near zero as possiblewould be important.

Practical questions will require attention in futurework to construct low-dimensional modes using vectorcylindrical harmonics. Already we have raised the ques-tion of how many modes should be retained, and findinga definitive answer will depend on characteristics of theflow being measured as well as the uses intended for themodel being constructed. There is also the question ofwhich modes should be retained. Above we have simplyretained the modes with lowest wave number, but ourmeasurements show that the lowest modes are not al-ways the modes that capture the most information. Re-taining a non-consecutive set of modes would sometimesbe useful.

Having chosen a set of modes, one should ask whichmeasurements are most useful. Solving the normal equa-tions (Eq. 6), either directly or via singular value decom-position, involves inverting the characteristic matrix thatappears on the left-hand side of the equations. If the ma-trix is singular, it cannot be inverted, and Galerkin pro-

9

jection fails. If the matrix is nearly singular, inversionproduces substantial numerical error, and Galerkin pro-jection produces a poor representation of the measure-ments. The matrix can be characterized by its conditionnumber, that is, the ratio of its largest eigenvalue to itssmallest eigenvalue. Condition numbers near unity sig-nify a matrix which is far from singular, and thereforeinvertible with negligible error. The characteristic ma-trix in Eq. 6 has elements calculated by evaluating allthe basis modes at all the measurement locations. Thusits condition number — and therefore the quality of therepresentation it produces — depends on both the choiceof modes and the choice of measurement locations. Wehave seen in our own past work that optimizing the con-dition number is not trivial [33]. Perhaps future workcan find an optimization algorithm.

Finally, the vector cylindrical harmonics may be usefulnot only for producing models of simulation results, butfor constructing spectral and/or pseudospectral simula-tions. In simple geometries like cylinders, spectral simu-lations are more accurate than simulations based on finitedifferences or finite elements, given the same computa-tional resources. The vector spherical harmonics are usedwidely for simulating atmospheric flow [23], flow in astro-physical objects [28], and magnetic fields of stars [26, 27](though, as mentioned above, radial boundary conditionsare satisfied by appropriate choice of modal weights, notby the modes themselves). Similarly, spectral simulationsusing the vector cylindrical harmonics could be applied toa wide variety of cylindrical systems, including Rayleigh-Benard convection [34]) and flow in astrophysical disks[35]).

The authors acknowledge J. Schumacher for providingthe example simulation results and H. Aluie for help-ful conversations. This work was partially supported bythe National Science Foundation under award numberCBET-1552182.

∗ d.h.kelley@rochester.edu[1] E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).[2] L. N. Howard and R. Krishnamurti, J. Fluid Mech. 170,

385 (1986).[3] J.-L. Thiffeault and W. Horton, Phys. Fluids 8, 1715

(1996).[4] K. R. Sreenivasan, A. Bershadskii, and J. J. Niemela,

Phys. Rev. E 65, 056306 (2002).[5] R. Benzi, Phys. Rev. Lett. 95, 024502 (2005).[6] E. Brown and G. Ahlers, Phys. Rev. Lett. 98, 134501

(2007).[7] G. Berkooz and P. Holmes, Annu. Rev. Fluid Mech. 25,

539 (1993).

[8] C. W. Rowley, Int. J. Bifurc. Chaos 15, 997 (2005).[9] M. C. Navarro, L. M. Witkowski, L. S. Tuckerman, and

P. Le Quere, Phys. Rev. E 81, 036323 (2010).[10] J. Bailon-Cuba and J. Schumacher, Phys. Fluids 23,

(2011).[11] J. Bailon-Cuba, O. Shishkina, C. Wagner, and J. Schu-

macher, Phys. Fluids 24, 107101 (2012).[12] D. J. Bradwell, H. Kim, A. H. C. Sirk, and D. R. Sad-

oway, J. Am. Chem. Soc. 134, 1895 (2012).[13] H. Kim, D. A. Boysen, J. M. Newhouse, B. L. Spatocco,

B. Chung, P. J. Burke, D. J. Bradwell, K. Jiang, A. A.Tomaszowska, K. Wang, W. Wei, L. A. Ortiz, S. A. Bar-riga, S. M. Poizeau, and D. R. Sadoway, Chem. Rev.113, 2075 (2013).

[14] K. Wang, K. Jiang, B. Chung, T. Ouchi, P. J. Burke,D. A. Boysen, D. J. Bradwell, H. Kim, U. Muecke, andD. R. Sadoway, Nature 514, 348 (2014).

[15] G. Berkooz and E. S. Titi, Phys. Lett. A 174, 94 (1993).[16] D. Rempfer, Theoret. Comput. Fluid Dynamics 14, 75

(2000).[17] C. W. Rowley, T. Colonius, and R. M. Murray, Physica

D 189, 115 (2004).[18] Q. Wang, O. Ronneberger, and H. Burkhardt, IEEE T.

Pattern Anal. 31, 1715 (2009).[19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and

B. P. Flannery, Numerical Recipes: The Art of ScientificComputing, 3rd ed. (Cambridge University Press, Lon-don, 2007).

[20] G. H. Golub and C. Reinsch, in Linear Algebra (SpringerBerlin Heidelberg, Berlin, Heidelberg, 1971) pp. 134–151.

[21] E. L. Hill, Am. J. Phys. 22, 211 (1954).[22] E. C. Bullard and H. Gellman, Phil. Trans. R. Soc. Lon-

don A 247, 213 (1954).[23] H. Volland, Surv. Geophys. 17, 101 (1996).[24] T. N. Krishnamurti, H. S. Bedi, V. Hardiker, and

L. Watson-Ramaswamy, An Introduction to Global Spec-tral Modeling, Atmospheric and Oceanographic SciencesLibrary, Vol. 35 (Springer-Verlag, New York, 2006).

[25] G. A. Glatzmaier and P. H. Roberts, Science 274, 1887(1996).

[26] P. Charbonneau, Annu. Rev. Astron. Astrophys. 52, 251(2014).

[27] V. V. Pipin and A. A. Pevtsov, Astrophys. J. 789, 21(2014).

[28] S. Hanasoge, L. Gizon, and K. R. Sreenivasan, arXiv ,191 (2015), 1503.07961.

[29] D. H. Kelley, S. A. Triana, D. S. Zimmerman, and D. P.Lathrop, Phys. Rev. E 81, 026311 (2010).

[30] R. Hollerbach, Int. J. Num. Meth. Fluids 32, 773 (2000).[31] D. H. Kelley and D. R. Sadoway, Phys. Fluids 26, 057102

(2014).[32] A. Perez and D. H. Kelley, J. Vis. Exp. , e52622 (2015).[33] D. H. Kelley, Rotating, hydromagnetic laboratory experi-

ment modelling planetary cores, Ph.D. thesis, Universityof Maryland (2009).

[34] P. Urban, P. Hanzelka, T. Kralik, V. Musilova, A. Srnka,and L. Skrbek, Phys. Rev. Lett. 109, 154301 (2012).

[35] J. Frank, A. King, and D. J. Raine, “Accretion Powerin Astrophysics: Third Edition,” (2002).