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Received: 30 September 2021 Revised: 26 November 2021 Accepted: 1 December 2021

DOI: 10.1002/gamm.202200003

ORIGINAL PAPER

Data-driven identification of the spatiotemporal structure of

turbulent flows by streaming dynamic mode decomposition

Rui Yang1,2 Xuan Zhang1Philipp Reiter1Detlef Lohse1,2

Olga Shishkina1Moritz Linkmann3

1Max Planck Institute for Dynamics and

Self-Organisation, Göttingen, Germany

2Physics of Fluids Group, Max Planck

Center for Complex Fluid Dynamics,

MESA+Institute and J.M. Burgers Center

for Fluid Dynamics, University of Twente,

AE Enschede, The Netherlands

3School of Mathematics and Maxwell

Institute for Mathematical Sciences,

University of Edinburgh, Edinburgh, UK

Correspondence

Moritz Linkmann, School of Mathematics

and Maxwell Institute for Mathematical

Sciences, University of Edinburgh,

Edinburgh EH9 3FD, UK. Email:

Moritz.Linkmann@ed.ac.uk

Abstract

Streaming Dynamic Mode Decomposition (sDMD) is a low-storage version

of dynamic mode decomposition (DMD), a data-driven method to extract

spatiotemporal flow patterns. Streaming DMD avoids storing the entire data

sequence in memory by approximating the dynamic modes through incremen-

tal updates with new available data. In this paper, we use sDMD to identify and

extract dominant spatiotemporal structures of different turbulent flows, requir-

ing the analysis of large datasets. First, the efficiency and accuracy of sDMD

are compared to the classical DMD, using a publicly available test dataset that

consists of velocity field snapshots obtained by direct numerical simulation of a

wake flow behind a cylinder. Streaming DMD not only reliably reproduces the

most important dynamical features of the flow; our calculations also highlight

its advantage in terms of the required computational resources. We subsequently

use sDMD to analyse three different turbulent flows that all show some degree

of large-scale coherence: rapidly rotating Rayleigh–Bénard convection, horizon-

tal convection and the asymptotic suction boundary layer (ASBL). Structures of

different frequencies and spatial extent can be clearly separated, and the promi-

nent features of the dynamics are captured with just a few dynamic modes. In

summary, we demonstrate that sDMD is a powerful tool for the identification of

spatiotemporal structures in a wide range of turbulent flows.

KEYWORDS

data-driven method, dynamic mode decomposition, turbulent flows

1INTRODUCTION

Coherent structures at different spatial and temporal scales are a prominent feature of many turbulent fluid flows occur-

ring in nature and in engineering applications [11,20,62]. Examples include large-scale vortices, wakes, convection rolls

and thermal plumes in Rayleigh–Bénard convection (RBC) [1,32], Taylor rolls in Taylor–Couette flow [18], jets, traveling

waves, very-large scale motions [23,24] and low-momentum zones [35] that develop in wall-bounded turbulent boundary

layers (BLs). These structures are known to have manifold significant effects in turbulent flows, for instance influence

on heat and mass transport, the occurrence of extreme fluctuations or enhanced drag due to their interaction with

near-wall dynamics in turbulent BLs [27,34,38]. Improving our knowledge of multiscale spatiotemporal coherence and

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the

original work is properly cited.

© 2022 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.

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https://doi.org/10.1002/gamm.202200003

2of22 YANG .

the underlying physics is of paramount importance as it would lead to a better fundamental understanding of turbulence,

specifically in terms of model-building and turbulence control. However, the co-existence of several coherent structures

makes the identification and the extraction of particular spatiotemporal features difficult, which led to a growing need

for data-driven methods designed to identify and extract patterns.

Modal decomposition, as an umbrella term for a variety of structurally similar methods, identifies structures by

decomposing a given dataset in a suitable set of basis functions, or modes. Fourier analysis constitutes perhaps the most

well-known and widely used example of a modal decomposition technique. A more sophisticated example is Proper

Orthogonal Decomposition (POD) [4,5,41,44,58], where each mode describes a flow structure according to its energy con-

tent. However, as the POD modes do not produce a separated and compact signal in frequency space, they usually contain

more than one characteristic frequency and thus cannot yield information on frequential coherence.

Dynamic Mode Decomposition (DMD), by contrast, decomposes a dataset into spatiotemporal coherent structures

[50] with dynamic modes obtained as eigenmodes of a high-dimensional linear approximation of the dynamics. More

precisely, DMD has solid mathematical foundations in the context of nonlinear dynamical systems theory. Under

certain conditions it represents a finite-dimensional approximation of the Koopman operator [47,61], a linear but

infinite-dimensional representation of a nonlinear dynamical system [29,36,37]. DMD results have an intuitive physi-

cal interpretation as each dynamic mode corresponds to a single frequency and growth or decay rate. Therefore, it is a

well-suited data-driven method for the analysis of complex datasets and model reduction. Since its introduction by Peter

Schmid [50], DMD has had a history of successful applications in fluid dynamics such as obtaining low-dimensional

dynamic model of the cylinder wake flow [3,60], generating good initial guesses for unstable periodic orbits in turbulent

channel flow [40], flow control [7,42,45], aerodynamics [13], and more general in pattern recognition [6,26]. An overview

of the development of DMD extensions and applications thereof in the context of fluid dynamics is given in an upcoming

review article by Peter Schmid [49].

Most DMD applications consist of postprocessing a time series of experimental or computational data, where most

implementations process the entire data sequence at once. However, the size of highly resolved turbulent flow data usually

precludes saving or loading the entire dataset into memory. Therefore, only a few studies so far have applied DMD to

highly turbulent flows. These constraints can be circumvented by a DMD implementation that allows for incremental

data updates [2,19,63], such that the DMD calculation proceeds alongside the main data acquisition process such as Direct

Numerical Simulations (DNS) or real-time Particle Image Velocimetry. Streaming DMD (sDMD) [19] is such a method,

which requires only two data samples at a given instant in time and converges to the same results as classical DMD. In

what follows we focus on sDMD as a promising method for the analysis of turbulent flows.

The present article is intended to serve three purposes: (a) to demonstrate the applicability of streaming DMD across

different large datasets of highly turbulent flows relevant to fundamental science and engineering applications, (b) to

analyze the large-scale spatiotemporal dynamics of the flow in sub-domains of particular interest, (c) to demonstrate the

robustness of the algorithm with respect to different degrees of downsampling, which allows for analyses of very large

datasets to be carried out efficiently on local desktop machines.

The streaming version of the DMD algorithm [19] is applied to three datasets consisting of time series obtained in

DNS of three different turbulent flows: rapidly rotating RBC, horizontal convection (HC) and the ASBL. Despite their

physical differences, these three system share a few features that render them interesting and suitable as test cases.

We demonstrate the advantages of sDMD for the analysis of turbulent flows, with a particular focus on large-scale

spatiotemporal data features.

First, all three cases are paradigmatic examples of fluid-dynamic systems of interest in geophysical fluid dynamics

and engineering applications. Rapidly rotating RBC is of relevance whenever rotation and thermal convection are the

key physical processes [1,57], such as in the dynamics of planetary cores. HC [22,51,54,59] occurs in the ocean which is

mostly heated and cooled by its upper surface being in contact with the atmosphere. The ASBL [25,48] is a flat-plate BL

with a constant BL thickness in the streamwise direction. The latter is achieved by removing fluid through the pores in

the bottom plate, a well-known technique for BL stabilization. Furthermore, due to the constant BL thickness the ASBL

allows the application of techniques developed for parallel wall-bounded shear flows to an open flow.

Second, all three systems host spatiotemporally coherent structures. In rapidly rotating RBC, this is the boundary

zonal flow (BZF), a large-scale traveling wave structure confined to the lateral near-wall region [10,53,65]. HC features two

characteristic processes that operate on very different time scales, that is, plume emission and slow oscillatory dynamics

in the bulk [43], with the former one being an order of magnitude faster than the latter one. The ASBL shows coherent

low momentum zones in the free stream, as do many wall bounded shear flows and freely evolving BLs [35], in the present

dataset with a slow spanwise drift.

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Third, the size of the datasets presents challenges in all three cases that can be mitigated by the incremental nature of

sDMD. For rapidly rotating RBC and HC, the fine grids required to properly resolve the small-scale turbulent dynamics

result in large datasets, as usual for DNS of turbulent flows. In case of ASBL, a further difficulty lies in the slow dynamics

of the low-momentum zone, as an analysis thereof requires very long time series.

This article is organized as follows. Section 2 provides a summary of both, classical DMD [50] and streaming DMD

[19], where we highlight few subtle differences concerning technical steps and compare our implementations of DMD

and sDMD using a standard publicly available dataset—DNS of a developing von-Kármán vortex street. The main results

of our analysis concerning turbulent flows are contained in Section 3, beginning with rapidly rotating RBC in Section 3.1,

followed by HC in Section 3.2 and the asymptotic suction BL in Section 3.3. The paper ends with conclusions and an

outlook in Section 4.

2DYNAMIC MODE DECOMPOSITION

Before describing the specific features and advantages of sDMD [19], we briefly summarize the basic ideas and the classi-

cal singular value decomposition- (SVD) based DMD algorithm [50]. For simplicity we restrict ourselves here to the case

of equidistant data sequences, for a more general discussion see [31]. Consider a time series of spatially resolved measure-

ment results recorded at a fixed sampling rate 1∕Δtresulting in, say, Nequidistant snapshots. Let us further assume that

the possibly multidimensional data in each snapshot is flattened into a corresponding M-dimensional real vector, such

that the time series can be represented by an ordered sequence (xk){k=1,…,N}of column vectors xk∈RMfor k∈{1,…,N}.

In the present context xkwould represent the kth velocity field in a series of Nmeasurements; hence in particular for

highly resolved three-dimensional flow fields M=3N3

p,whereNpis the number of grid points, can quickly become very

large. We will come back to this point in due course.

The assumption DMD relies upon is the existence of a linear operator A∈RM×Mwhich approximates the nonlinear

dynamics across the interval Δt,thatis

xk+1=Axk+𝜺kfor all k∈{1,…,N−1}.(1)

Here, crucially, Adoes not depend on k. Finally, 𝜺kdenotes an error term that is assumed to be small. The validity of

this assumption depends to some extent on the ratio of the characteristic time scale of the observed nonlinear dynam-

ics and the sampling interval Δt, but most importantly on the potential to describe the dynamics by a linear surrogate

model (i.e., the degree of nonlinearity). In practice, Ais chosen by regression over the available data by least-squares

minimization of the 𝜺k[31]. Since the operator Adescribes the spatiotemporal dynamics of the system, its eigenvectors,

known as dynamic modes or somewhat tautologically DMD modes, may be used to disentangle complex spatiotempo-

ral dynamics and to construct low-dimensional models. In what follows we summarize how the dynamic modes may be

determined from the data sequence (xk){k=1,…,N}, following an SVD-based approach as this is what is mostly used in prac-

tice owing to numerical stability concerns with the more fundamental Krylov-subspace-type approach and for reasons of

computational cost reduction. Further details can be found in the original work by Reference [50] and the textbook by

Reference [31].

2.1 SVD-based DMD

For what follows it is convenient to combine data sequences that consist of N−1 samples and are shifted forwards in

time by Δt,thatis(xk){k=1,…,N−1}and (xk){k=2,…,N}, into M×(N−1)-dimensional matrices

X=XN−1

1=(xjk )∶=(x1x2···xN−1),(2)

Y=XN

2=(yjk )∶=(x2x3···xN),(3)

where j∈1,…,Mis the spatial index and kthe temporal index. Then Equation (1) implies

Y=AX +R,(4)

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where R=(𝜀jk )is the matrix of residuals. The best-fit solution for Awith respect to least-squares minimization of Ris

given by A=YX+,whereX+is the pseudo-inverse of X. In practice, and in particular in fluid dynamics, M≫Nas the

dimension Mof the spatial samples usually exceeds the number of temporal samples Nby far. Hence, A∈RM×Mis at

most of rank N−1, which calls for a lower-dimensional approximation of A, for instance, by restricting Ato act on a

subspace spanned by, say, rPOD modes obtained by calculating the compact SVD of X,

X=UX𝚺XWT

X,(5)

where the superscript Tdenotes the transpose. The truncation number ris bounded from above by the rank of the data

matrix X, which is at most N−1. The columns of UX∈RM×rand the rows of WX∈R(N−1)×rare orthogonal, and 𝚺X∈Rr×r

is a diagonal matrix containing the nonzero singular values of X. The matrix UXcontains the spatial structures of the data

sequence, that is, the POD modes are given by the columns of UX. Restricting Ato act on the subspace spanned by rPOD

modes gives rise to the definition of an auxiliary matrix

S∶= UT

XAUX=UT

XYWX𝚺−1

X∈Rr×r.(6)

This equation is to be interpreted in a least-squares optimal sense (hence the absence of the residual), it is obtained by

calculating Athrough the pseudo-inverse of X, which is calculated via SVD, and subsequently projecting Aonto the

r-dimensional subspace spanned by the POD modes, i.e., using the orthogonality of UX. The eigenvalues of Scorrespond

to a subset of the nonzero eigenvalues of A. For practical purposes we summarize the SVD-based DMD algorithm [50] as

follows:

•Collect Ntemporally equidistant samples {x1,x2,x3,…,xN},xj∈RM,j∈{1,…,N}.

•Build a matrix X∈RM×(N−1)out of the first (N−1)snapshots, according to Equation (2).

•Calculate the compact SVD of Xaccording to Equation (5).

•Build a matrix Y∈RM×(N−1)out of the last (N−1)snapshots, according to Equation (3) and combine it with the matri-

ces UXand WXto calculate the optimal representation Sof the linear mapping Ain the orthogonal basis given by the

POD modes according to Equation (6).

•Calculate the eigenvectors vkand eigenvalues 𝜆kof Sfor k∈{1,…,r}.

•Calculate the (projected) dynamic modes 𝜓k

𝜓k=Uvk.(7)

The data vector x, or, in the present context the velocity field, at time ts=sΔt,wheresis an integer, can then be

approximated using N′⩽rdynamic modes and their corresponding DMD eigenvalues

x(ts)=x(sΔt)≈

N′

k=1

bk𝜆s

k𝜓k,(8)

where bkare the components of the least-squares solution of this equation at s=0. The real and imaginary parts of the

logarithm of the eigenvalues 𝜆k

𝜔k=Im (ln (𝜆k))

Δt,𝜎

k=Re (ln (𝜆k))

Δt(9)

are the frequency and temporal growth or decay rate of the kth dynamic mode for k∈{1,…,r}, respectively. The accu-

racy of the approximation does not only depend on the number of dynamic modes used to reconstruct the data, it

also depends on the truncation number r, which determines the accuracy with which the projected dynamic modes

have been calculated. Several truncation criteria have been developed to determine a suitable value for r,suchas

Optimal Singular Value Hard Threshold [12], or to choose a number of nonzero mode coefficients from a larger

number of modes. The sparsity-promoting algorithm [26], does the latter. Kou and Zhang [30] also developed an

improved rank selection criterion for the most dominant DMD modes, which are determined based on the temporal

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history of each DMD mode and has better modal convergence compared to the classical DMD. So far, the dynamic

modes are ordered by amplitude, which may or may not result in a good reconstruction of the data, as the most

energetic modes may not be the most dynamically relevant ones, and a mode selection criterion is required. Concern-

ing low-dimensional data representation, sparsity-promotion [26] solves this impasse by minimizing the least-squares

error between the original and the reconstructed data over the available set of dynamic modes, and it includes an

L1-penalization term to restrict the number of active modes used for reconstruction. As the focus here is on flow

features on large spatiotemporal scales that represent statistically stationary dynamics, we only retain modes with eigen-

values that lie on or very close to the unit circle. The remaining modes are then ranked by frequency in ascending

order, and we restrict our attention on the first few low-frequency modes. We point out that this criterion may not

be adequate for flows that feature multiple important dynamical features with vastly different timescales, in which

case either temporal filtering or a different mode selection criterion may be required. We will come back to this point

inSec.3.2onHC.

2.2 Streaming DMD

Classical DMD requires access to the entire data sequence at once, which precludes the analysis of large datasets due

to memory constraints. This applies to data of either a high degree of spatial or temporal complexity, where the former

results in high spatial dimensionality (large M) and the latter requires long time series (large N) to capture the temporal

features of the dynamics. Streaming DMD is a method for the calculation of the POD-projected linear operator Sbased

on incremental data updates that addresses this challenge by only requiring two data samples to be held in memory at

a given time [19]. In what follows we summarize this procedure; further details including processing steps that reduce

the effects of data contamination by noise can be found in the original work by [19]. Streaming DMD consists of two

conceptual parts, a low-storage calculation of S, and a scheme to update Susing new data samples based on the iterative

Gram–Schmidt orthogonalization.

Let us reconsider the data matrices Xand Ydefined in Equations (2) and (3) and write Equation (6) as

S=UT

XY(UT

XX)+=UT

XUỸ

Ỹ

X+=UT

XUỸ

Ỹ

XT(̃

X̃

XT)+=UT

XUYHG+

X,(10)

where ̃

Y∶= UT

YY∈RrY×N−1and ̃

X∶= UT

XX∈RrX×N−1, are the projected data matrices, and H∶= ̃

Ỹ

XT∈RrY×rXand GX=

̃

X̃

XT∈RrX×rX. The identity ̃

X+=̃

XT(̃

X̃

XT)+, which can be readily verified via SVD, was used in the penultimate step.

That is, now both data matrices Xand Yare projected onto orthogonal bases consisting of their respective left singular

vectors, the POD-modes, with truncation numbers rX⩽rank Xand rY⩽rank Y. The rearrangement carried out in the

penultimate step has the advantage that H∈RrY×rXand GX∈RrX×rX, which in itself is an improvement of classical DMD

in terms of memory usage as long as rX<Mand rY<M. Especially in fluid dynamics, this is often the case unless the

data is very noisy. We will come back to this issue in due course. However, the main advantage of the formulation in

Equation (10) lies in the fact that all matrices on the right-hand side of Equation (10) can be obtained incrementally

from a data stream using only two samples at a time. The matrices UXand UYcan be calculated incrementally from

the data stream by iterative Gram–Schmidt orthogonalization. After each orthogonalization step the updated orthogonal

matrices are then used to project the sample vectors onto the respective bases, and the matrices Hand GXare subsequently

constructed from the projected sample vectors. More precisely, consider for instance the kth pair of sample vectors xk

and yk=xk+1. The matrices UXand UY, which have been constructed incrementally from the previous data samples, are

now updated using xkand the newly available yk.Then, ̃

xk=UT

Xxkand ̃

yk=UT

Yykare calculated and we can update the

remaining matrices according to

H=

k

l=1

̃

yl̃

xT

land GX=

k

l=1

̃

xl̃

xT

l.(11)

Before proceeding to the calculations, a few comments are in order. First, the incremental nature of the method pre-

cludes the application of numerically more stable orthogonalization methods such as Householder reflections, and this

may affect the convergence properties of the method. Second, experimental noise may result in a drastic decrease in com-

putational efficiency as noise usually results in the data matrices being of high rank. In practice, this can be mitigated

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through an intermediate processing step, as explained in detail in Reference [19]. Third, we note that GX=𝚺X𝚺Xcontains

the squares of the nonzero singular values of X, as can be verified via SVD.

2.3 Validation

Before applying sDMD to the three aforementioned datasets, we first compare classical and streaming DMD implementa-

tions in terms of their respective memory consumption for a publicly available dataset [31] that has been extensively used

for testing and validation purposes in the literature [2,9]. Subsequently, we use this dataset to test DMD in conjunction

with a coarsening interpolation scheme designed to reduce the computational effort when analyzing data of high spatial

dimension M, as will be the case for turbulent flows.

Since the focus of the present work lies in the identification of the large-scale features which happen to be also the

energetically dominant structures of the system, usually represented by one or two of the dynamic modes with the highest

amplitudes, we do not apply any specific algorithm to order the dynamic modes or to determine the truncation number

r. Instead, different values of rwere tested to ensure convergence with respect to changes in the mode order and values

of the lowest frequencies, resulting in r=30 as a sufficient truncation number. For data where dynamical relevance and

energy content of the determined dynamic modes result in different mode ordering, more sophisticated methods such as

sparsity promotion [26] are required to obtain good low-dimensional data representations.

2.3.1 Comparison between sDMD and DMD

The dataset provided in Reference [31] consists of a time series of two-dimensional vorticity fields obtained by computer

simulation of the wake flow behind a cylinder for Reynolds number Re =UD∕𝜈=100, where U,D,and𝜈denote the

free-stream velocity, the diameter of the cylinder and the kinematic viscosity of the fluid. The dominant dynamics is

governed by periodic vortex shedding, therefore it is very well suited for DMD validation. The vortex-shedding frequency

can be expressed in nondimensional form through the Strouhal number St =fD∕U. Here, the Strouhal number is around

St =0.16. For details on the numerical method used to generate the data we refer to the original reference [31]. In total,

150 vorticity-field samples, separated by a time interval Δt=0.2, were analyzed.

The memory consumption of both methods, that is, the RAM usage of the code at a single iteration in case of sDMD

and for the full dataset for DMD, has been assessed by two comparisons. First we increased the number of samples for

a fixed state dimension, and secondly increased the state dimension for a fixed number of samples. In order to ensure

consistency, all tests were carried out on the same computer with an Intel i5-8250U CPU at 1.60GHz and 8GB RAM. We

expect the memory usage to increase with the number of data samples for DMD, as each additional data sample requires

the same additional amount of memory. However, the scaling should be nonlinear, as the efficiency of the SVD is not

linearly related to the data matrix size when the state dimension is much larger than the number of samples and as the

memory consumption due to matrix multiplications depends on the number of data samples. For sDMD, the memory

consumption should remain constant, as only two data samples are held in memory at a given time. Concerning the

memory consumption as a function of the state dimension, we expect a linear relation for both methods. The predictions

are confirmed by the data to a good approximation, as can be seen in Figure 1A for memory consumption as a function

of the number of data samples, and in Figure 1B as a function of the state dimension. In the former case, we indeed

observe nonlinear scaling of memory consumption as a function of the number of data samples for DMD, while the

memory consumption remains constant for sDMD. In the latter case, the memory consumption scales linearly with state

dimension. Compared to DMD, it is lower by a nearly constant offset for sDMD, which results from the larger number

of snapshots required by the DMD algorithm. For DMD all 150 snapshots are stored in memory, while sDMD requires

only two. Information on computational time and memory usage for DMD and sDMD calculations for 150 snapshots on

90 000 grid points is provided in Table 1.

2.3.2 Coarse interpolation for the analysis of high-dimensional data

Numerical simulations of highly turbulent flows require fine computational grids to accurately resolve the dynamics at

the small scales. This is not necessarily always due to a need to precisely measure small-scale quantities such as dissipation

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(A) (B)

FIGURE 1 Memory consumption of classical singular value decomposition-based dynamic mode decomposition (DMD) (red) and

streaming DMD (blue) as a function of (A) the number of samples with fixed state dimension and (B) the state dimension with fixed number

of samples for the vortex shedding data time series provided in Reference [31]. The solid black lines in (A) and (B) correspond to scaling

exponents 0.5 and 1, respectively

TABLE 1 Memory requirements for all datasets. For the cylinder flow dataset discussed in Section 2.3, 150 data snapshots on 90000

grid points have been used, and the wall time refers to the sDMD calculation

Dataset

Wall

time (s)

Memory

DMD (MB)

Memory

sDMD (MB) CPU

RAM

(GB)

RRB-Ra =108940 — 6201 Intel i5-4440 3.10 GHz 16

RRB-Ra =1092167 —4230 Intel i5-9400 2.90 GHz 16

HC-slow 453 — 2060 Intel i5-9400 2.90 GHz 16

HC-fast 2 — 37 Intel i5-9400 2.90 GHz 16

ASBL 44 — 204 Intel i7-8550U 1.80 GHz 16

Cylinder wake 8154 88 Intel i5-8250U 1.60 GHz 8

Abbreviations: ASBL, asymptotic suction boundary layer; DMD, dynamic mode decomposition; HC, horizontal convection; sDMD, streaming dynamic mode

decomposition.

or correlation functions of high order, it is also a requirement for numerical stability. The required large number of grid

points results in a high memory load even for a single sample, which quickly becomes prohibitive even for sDMD. This

calls for a reliable downsampling strategy to interpolate the data on coarser grids, in particular when the focus is on

large-scale structures. In what follows we analyze the robustness of sDMD with respect to different degrees of spatial

downsampling, using the same vortex shedding dataset of the previous subsection. The downsampling was carried out

by successively decreasing the original number of grid points uniformly, that is by merging nearby grid points, beginning

with 90 000 grid points down to a minimum of five grid points. The effect of downsampling is assessed by considering

two observables, the DMD eigenvalues and the time-averaged reconstruction error, defined as

𝜀2∶= v(ts)−

N′

k=1

bk𝜆s

k𝜓k2,(12)

where vis the downsampled vorticity field here, and the angled brackets denote a time average. The results are sum-

marized in Figure 2, with Figure 2A,B showing the streaming DMD eigenvalues for the original data and after different

degrees of downsampling, Figure 2C the reconstruction error as a function of the state dimension for classical DMD and

sDMD, and Figure 2D presenting visualizations of a sample of the reconstructed vorticity fields after different degrees of

downsampling. A number of observations can be made from Figure 2. The DMD eigenvalues, which need to lie on the

unit circle as the dynamics are nonlinear and statistically stationary [21], are remarkably robust under the downsam-

pling procedure. This can be expected as the Koopman operator can be approximated for any observable function. Thus, a

coarsening of the grid represents a change in the observable, which should not have a large impact on the eigenfunctions.

8of22 YANG .

(A) (B)

(E)

(C) (D)

FIGURE 2 (A) Streaming dynamic mode decomposition (sDMD) eigenvalues for the original data (green circles) and after different

degrees of downsampling. (B) Magnification of the blue region in (A). (C) Error of the Strouhal number as a function of the state dimension

for the downsampled data for classical DMD and sDMD. (D) The time-averaged reconstruction error as defined in Equation (12) as a function

of the state dimension for the downsampled data for classical DMD and sDMD. (E) The instantaneous flow field on 90000, 900 and 100 grid

points, respectively, from top to bottom

As can be seen from the data shown in Figure 2A,B, a reduction by three orders of magnitude in the state dimension results

in almost the same values for the DMD eigenvalues. Significant qualitative differences in the eigenvalues occur only after

drastic downsampling from 90 000 to less than 10 data points. A more quantitative comparison is achieved by considering

the difference 𝜀1between the Strouhal number and the dimensionless frequency of the second dynamic mode as a func-

tion of the state dimension presented in Figure 2C. As can be seen from the figure, the Strouhal number is reproduced

very accurately using only 25 data points. This is particularly striking in view of the unsurprisingly large reconstruction

error 𝜀2of order 10−3to 10−2, for the corresponding downsampled data, as shown in Figure 2D. According to the data

presented in the figure, converged results for the reconstruction of the full vorticity field requires a state dimension of

least 9000 points. The finite residual for higher resolved data is then due to truncation in the DMD algorithm. The visu-

alizations of the reconstructed vorticity fields in Figure 2E give a visual impression of the effect the downsampling has

on the reconstructed data. As expected, the large-scale spatial coherence is still present in the downsampled data. Since

the focus is on the detection of large-scale coherent structures, like the vortex street in this case, and since the coarsen-

ing interpolation results in the removal of small-scale spatial structures, the downsampling has very little effect on the

results, as expected.

3RESULTS

Having validated our implementation of sDMD in conjunction with downsampling on publicly available data, we now

apply the method to three different flows, rapidly rotating RBC, HC, and the ASBL. We chose these three examples in order

to demonstrate sDMD to be a useful tool for the analysis of different turbulent flows in terms of their main spatiotemporal

structure.

YANG . 9of22

In rapidly rotating RBC, the anticyclonic circulation in the bulk is surrounded by a cyclonic layer close to the horizon-

tal cell walls, and the aim is to identify this large-scale flow pattern. HC lends itself well as a test case for the distinction of

different spatiotemporal structures, as the dynamics is largely governed by two instabilities that operate on different time

scales. The Rayleigh–Taylor instability leads to fast periodic plume generation close to the boundary while an oscillatory

instability in the bulk results in much slower periodic dynamics in the bulk. The respective frequencies associated with

these two processes differ by an order of magnitude. Similar to canonical wall-bounded parallel shear flows and spatially

developing BLs, the ASBL features long-lived large-scale coherent motion. Here the aim is to identify the correspond-

ing spatiotemporal structure. The slow dynamics requires very long time series, which makes this example particularly

suitable for the application of sDMD.

All datasets were obtained by DNS at parameter values corresponding to turbulent flow. Further details on the numer-

ical methods and parameter values will be given in the following subsections. A summary of computational details such

as wall time and memory consumption for the DMD or sDMD calculations is provided in Table 1 for all datasets.

3.1 Rapidly rotating RBC

3.1.1 Fluid structures

In rotating RBC, a fluid is confined between a heated bottom plate and a cooled top plate and is rotated around a vertical

axis. It is a paradigmatic problem to study many geophysical and astrophysical phenomena in the laboratory, for example,

convective motion occurring in the oceans, the atmosphere, in the outer layer of stars, or in the metallic core of planets.

In rotating RBC laboratory experiment, the fluid is laterally confined. The centrifugal force can be neglected, provided

the Froude number is small, and then only the Coriolis force is considered. The interplay of the occurring buoyancy and

Coriolis forces, however, may yield highly complex flows with very distinct flow structures whose nature strongly depends

on the control parameters. Without rotation or with slow rotation, a distinct feature of turbulent RBC is the emergence

of the Large-Scale Circulation (LSC) of fluid. For rapid rotation, however, a mean flow with cyclonic azimuthal velocity

near the boundary, the BZF, develops close to the side walls, surrounding a core region of anticyclonic mean flow. The

viscous Ekman BLs near the plates induce an anticyclonic circulation with radial outflow in horizontal planes, which is

balanced by the vertical velocity in a thin annular region near the sidewall, where cyclonic vorticity is concentrated. The

Taylor–Proudman effect induced by rapid rotation tends to homogenize the flow in the vertical direction, resulting in an

anticyclonic mean flow in the core region throughout the height. The temperature pattern near the vertical wall, however,

moves anticyclonically within the BZF and is likely connected to the thin anticyclonic Ekman layers at the top and bottom

plates. The interesting part of the BZF flow structure is, that it has an organized and predominant pattern as mean flow,

although the instantaneous flow is turbulent in the whole domain, and consists of active and complex vortex motion.

This grants sDMD big potential, as it can find the dominant modes quickly and thus reconstruct the global statistics at

very low cost. In addition, the BZF has special drift feature, which could test how well sDMD could capture the temporal

evolution of a flow. Thus the aim here is to recover the BZF via sDMD.

3.1.2 Dynamic equations & control parameters

We consider RBC in a vertical cylinder rotating with uniform angular velocity Ωabout the vertical axis. The governing

equations of the problem are the incompressible Navier–Stokes equations in the Oberbeck–Boussinesq approximation,

coupled with the temperature equation, given here in dimensionless form

𝜕tu+(u⋅∇)u+∇p=Pr ∕Ra∇2u−Ro−1̂

z×u+T̂

z,(13)

𝜕tT+(u⋅∇)T=1∕(Pr Ra)∇2T,(14)

∇⋅u=0.(15)

which are nondimensionalized by using the fluid layer height H, the temperature differences between heated bottom and

cooled top plates Δ=T+−T−, and the free-fall velocity uff =𝛼gΔH,with𝛼denoting the isobaric expansion coefficient,

10 of 22 YANG .

gthe acceleration due to gravity. The Rayleigh number, Ra, describes the strength of the thermal buoyancy force, the

Prandtl number, Pr, the ratio of viscosity and diffusivity, and the convective Rossby number, Ro, is a measure for the

rotation rate. They are defined as

Ra ≡𝛼gΔH3∕(𝜅𝜈),Pr ≡𝜈∕𝜅, Ro ≡𝛼gΔH∕(2ΩH),(16)

where 𝜅the thermal diffusivity, 𝜈the kinematic viscosity, and Ωthe angular rotation speed. Equations (13)–(15) were

stepped forward in time using the finite-volume code [28,55,65]. For the temperature we impose Dirichlet

boundary conditions (isothermal) on the top and bottom plates and Neumann conditions (adiabatic) on the lateral walls.

All boundaries are assumed to be impenetrable and no-slip, that is, the velocity field vanishes at all boundaries.

3.1.3 Numerical details

We consider datasets for two different Rayleigh numbers, Ra =108and Ra =109, the remaining control parameters are

Pr =0.8,Ro =0.1. The resolution of the original datasets is Nr×N𝜙×Nz=100 ×256 ×380 for Ra =108and Nr×N𝜙×

Nz=192 ×512 ×820 for Ra =109, according to [56], where Nr,N𝜙and Nzdenote the number of grid points in radial,

azimuthal and vertical direction, respectively. Grid nodes are nonequidistant in both the radial and vertical directions,

being clustered near the boundaries to resolve thermal and velocity BLs [64]. The velocity fields from both datasets are

sampled for a time period of 200 free-fall time units with a sampling interval of of Δt=1.25, resulting in 160 samples in

total. Both datasets are spatially downsampled for the sDMD analysis, by a four-fold and a 16-fold reduction in the number

of data points, respectively, resulting in a spatial resolution of Nr×N𝜙×Nz=100 ×128 ×190 for Ra =108and Nr×

N𝜙×Nz=96 ×128 ×410 for Ra =109. The truncation number is r=40 in both cases. In what follows we first describe

eigenvalue spectra and the generic spatial features that can be extracted with the first few dynamic modes for the case Ra =

108. Subsequently, we consider the temporal features for both Ra =108and Ra =109, providing a quantitative comparison

of the time scales associated with the global structure obtained directly from the DNS data and calculated from the lowest

DMD frequencies.

3.1.4 Streaming DMD

The DMD eigenvalues obtained for the two cases Ra =108and Ra =109are presented in Figure 3A,B, respectively. As can

be seen from the data shown in the figure, the eigenvalues lie on or close to the unit circle for a truncation number r=40.

We only consider converged modes corresponding to eigenvalues on the unit circle, that is, the mean flow and the dynamic

mode corresponding to the next lowest frequency indicated by the blue dots. The first three dynamic modes, that is the

mean flow and the complex conjugate pair of dynamic modes corresponding to the lowest frequency, obtained from the

Ra =108-dataset are visualized in Figure 4 in terms of temperature and azimuthal velocity. The temperature field of the

first dynamic mode shown in Figure 4A resembles the mean temperature profile, the corresponding azimuthal velocity

field (Figure 4B) consists of anticyclonic motion in the bulk and cyclonic motion close to the sidewall. As expected, the

first dominant mode corresponds to a base or mean flow. However, this mode is evidently dynamically not important

without temporal change. The principal mode is the second one, presented in Figure 4C, and the BZF [65] is clearly

visible, both in the temperature (left) and azimuthal velocity (right).

Even though the flow is turbulent, its large-scale spatial structure can be reconstructed nicely with only a few modes

as demonstrated by the visualization of the azimuthal velocity in the lower half of the RBC-cell presented in Figure 5. We

point out that much of the small-scale dynamics and thereby accuracy in the representation is lost through the down-

sampling procedure, and applying sDMD without coarse interpolation but with larger memory consumption may be

advisable if the focus is on a more detailed reconstruction of the flow. Here, we focus only on the large-scale structure.

The comparison is carried out for two velocity fields which have been sampled about 30 free-fall times apart in order to

guarantee sufficiently decorrelated samples. The originals are shown in Figure 5A,D, respectively. Figure 5B,E contains

the reconstructions using the first three modes, and Figure 5C,F the reconstructions from the first nine modes for the two

samples, respectively. These examples demonstrate consistently that even though the main features of the flow can be

captured by the mean flow and the BZF, a fair amount of detail is missing and its inclusion requires a few more modes.

A much better reconstruction can be achieved with as little as nine modes.

YANG . 11 of 22

(A) (B)

FIGURE 3 Dynamic mode decomposition eigenvalues for rapidly rotating Rayleigh–Bénard convection. (A) Ra =108,(B)Ra=109.

The eigenvalues shown in blue (three) correspond to the modes used in the reconstruction shown in Figure 5 (B) and (E)

(A) (B) (C)

FIGURE 4 The first two dynamic modes for Pr =0.8, Ra =108and Ro =0.1. (A) The first mode of the anticyclonic drifting

temperature field in the full cell (left) and in the bottom half of the cell (right). (B) The first mode of the azimuthal velocity field in the full

cell (left) and in the bottom half of the cell (right). (C) The sum of the second mode and its complex conjugate, the third mode, of the

temperature (left) and the azimuthal velocity field (right) in the bottom half of the cell

(A) (B) (C)

(D) (E) (F)

FIGURE 5 Reconstructed azimuthal velocity field for two velocity-field samples for Pr =0.8, Ra =108and Ro =0.1. (A),(D) original

field, (B),(E) reconstruction with three dynamic modes, and (C),(F) reconstruction with nine dynamic modes

12 of 22 YANG .

(A) (B) (C) (D)

Modes 1–3 Modes 1–3 Modes 1–3

FIGURE 6 Time evolution of temperature and vertical velocity. (A) Schematic setup. The red circle indicates the location where

temperature and velocity were measured. (B) temperature field and (C) vertical velocity field for Ra =108, and (D) temperature field for

Ra =109. The original fields are shown in the left panels and the right panels correspond to the reconstructed field using three dynamic

modes. The color scale varies from minimum values indicated in blue to maximum values indicated in magenta for the respective fields,

given by temperatures at the top and bottom plates in (B) and (D), and [−uff ∕2,uff ∕2]with uff ≡𝛼gΔHbeing the free-fall velocity in (C)

Having discussed the identification of the dominant spatial feature of the flow, the BZF, we now focus on its tem-

poral structure. Figure 6 presents spatiotemporally resolved diagrams of the dynamics in a ring located at half-height

z=H∕2 and at radial location r=rumax

𝜙, where the maximum azimuthal velocity is observed, as indicated by the red

circle in the schematic drawing shown in Figure 6A. The time evolution of the temperature and vertical velocity fields

of the Ra =108-dataset are presented in Figure 6B,C, respectively, while Figure 6D corresponds to the time-evolution

of the temperature field at Ra =109. The original data is shown in the left panels of the respective visualizations and

the data reconstructed from the first three dynamic modes is shown in the right panels. Visual comparison of the

left and right panels confirms again the zonal flow pattern can be clearly captured with only the first three dynamic

modes. Furthermore, the visualizations clearly identify the BZF as a traveling wave with strongly correlated tempera-

ture and vertical velocity fields as can be seen by comparison of Figure 6B,C. The traveling wave structure of the BZF

is also present at higher Ra, as can be seen in Figure 6D. As such, it seems to be a robust feature of the BZF in the

Rayleigh-number range considered here. However, according to the visualization the dynamics appear to be slightly more

complex at Ra =109than at Ra =108, hence it remains to be seen to what extent the travelling wave dynamics persist

with increasing Ra.

We now provide a qualitative comparison of time scales between the full data and a reconstruction using only the

mean flow and the dynamic modes representing statistically stationary dynamics on the longest time scales presented

qualitatively in Figure 6. For the case Ra =108, the BZF period obtained from the data is 74 free-fall time units, which

compares well with the period of 72.96 free-fall time units corresponding to the and third DMD mode. The relative error

between the two amounts to 1.4 %. The comparison at higher Rayleigh number for the case Ra =109results in a larger

relative error, the period obtained from the data of 53 free-fall time units compares with a relative error of 8.5 %to the

DMD-period of 57.5 free-fall time units. The latter has been calculated using the expression for DMD frequencies in

Equation (9).

In summary, the most prominent spatiotemporal features of rapidly rotating RBC can be identified through sDMD,

with the BZF emerging as the dominant dynamic mode. The cyclonic motion of the fluid reflected in the azimuthal

velocity and the anticyclonic motion of the flow pattern reflected in the temperature and vertical velocity as well as their

frequencies are fully reproduced by only the first three dynamic modes. These results firmly establish sDMD as a powerful

tool for the extraction of dominant coherent structures in turbulent rapidly rotating RBC.

Large-scale structures in RBC at Ra =107and Pr =0.7 without rotation and in a cubic domain have recently been

identified through Koopman analysis [14]. In cubic geometry, eight LSC states occur in RBC, four long-lived diagonal

configurations with sojourn times of O(1000)free-fall time units and four wall-aligned configurations, which are vis-

ited shortly during transitions between the diagonal configurations. All LSC states have been reconstructed using three

Koopman modes, a pair of complex conjugate modes whose frequency should approximate the time scale of one cycle

YANG . 13 of 22

(A) (B)

FIGURE 7 Sketch of (A) horizontal convection adapted from [43] and (B) front view of the setup. Only the shaded area (B) is used for

streaming dynamic mode decomposition. The inset shows a snapshot of the temperature field for Ra =1011 and Pr =10. There, the grey

arrows indicate the motion of the periodically detaching plumes; the dark arrow indicates the oscillatory motion inside the bulk region

through all four diagonal configurations, and a real mode representing the mean flow. In this context, the present results

suggest that a representation of the BZF through Koopman eigenfunctions should also be possible.

3.2 Horizontal convection

3.2.1 Fluid structures

HC, similarly to RBC, is driven by thermal buoyancy. However, in HC heating and cooling are applied to different parts

of the same horizontal surface. In our case, the heated plate is located in the center and the cooled plates are placed at

both ends, as shown in Figure 7A. This setup is relevant for many geophysical and astrophysical flows [51,59] and engi-

neering applications [17], in particular concerning the large-scale overturning circulation of the ocean as heat is supplied

to and removed from the ocean predominantly through its upper surface, where the ocean contacts the atmosphere. The

dimensionless control parameters are similar to RBC, that is the Rayleigh number, the Prandtl number and the aspect

ratio Γ,

Ra ≡𝛼gΔL3∕(𝜅𝜈),Pr ≡𝜈∕𝜅, Γ≡L∕H=10,

where the characteristic length scale Lis the half-cell length. The governing equations are again the incompress-

ible Navier–Stokes equations in the Oberbeck–Boussinesq approximation, and a temperature equation stated in

Equations (13)-(15), but without the Coriolis term in the momentum equation.

For the parameters Ra =1011 and Pr =10 it was observed that sheared plumes, originated by a Rayleigh–Taylor insta-

bility, periodically arise above the heated plate and travel toward the center [43]. However, another time-dependent

feature that emerges is the oscillatory instability that breaks symmetry inside the bulk region, see Figure 7B. So there

is a fast periodic emission of thermal plumes close to the boundary and a slow periodic oscillation in the bulk region.

That is, the HC has coexisting dynamics on very different time scales. Streaming DMD is used in conjunction with tem-

poral filtering to provide separate low-dimensional reconstructions of these coexisting dynamics. Temporal filtering is

required here for reasons of computational efficiency. In principle, it is possible to extract both time scales from a sin-

gle analysis. However, the dataset would become very large and the calculations slow. This is because the snapshot

spacing must be small enough to detect the small period and we need to process a large number of snapshots along a

trajectory long enough to capture the large period. To obtain converged results in particular for the larger period, the

truncation number must be increased accordingly, resulting in much larger matrices to be processed at each iteration

step.

14 of 22 YANG .

(A) (B)

FIGURE 8 Streaming dynamic mode decomposition eigenvalues for horizontal convection with (A) fast plume emission and (B) slow

oscillatory flow. The eigenvalues shown in blue (three) corresponds to the modes used in the reconstruction shown in Figure 9

3.2.2 Dynamic equations & numerical details

The dataset consists of velocity fields obtained in the DNS for a rectangular geometry, as shown in the schematic draw-

ing in Figure 7A. The temperature boundary conditions at the bottom plate are 𝜃=0.5 for 0 ≤x≤0.1and𝜃=−0.5 for

0.9≤x≤1, all the other walls are adiabatic. No-slip boundary conditions are imposed at all walls for the velocity field.

The calculations were carried out using the code, as in the previous section. Further details can be found in

Reference [43]. The original grid is Nx×Ny×Nz=1026 ×66 ×98, where Nx,Ny,andNz, denote the number of grid points

in the mean-flow x-direction, the spanwise y-direction and the z-direction, which is normal to the heated and cooled bot-

tom plates, respectively. Though, since plumes and oscillations are concentrated above the heated plate, we extract only

the dynamically most important data inside the shaded domain, shown in Figure 7B, with Nx×Ny×Nz=200 ×66 ×98.

The truncation number ris set to 80 to ensure the dominant modes can be captured properly. Since the plume emission

motion is more than ten times faster than the oscillatory flow, a small time interval is needed to capture the fast plume

emission while a large number of velocity-field samples is required to simultaneously identify the slow oscillations. To

save computational resources, we decouple the two tasks and use two datasets comprised of 200 snapshots each, sam-

pled at different time intervals: 0.1 free-fall time units for the fast plume emission and 0.5 free-fall time units for the slow

oscillatory flow. As discussed in the previous section, it is in principle possible to extract both phenomena from a single

dataset.

3.2.3 Streaming DMD

Figure 8 presents the DMD spectra for fast plume emission in panel (A) and the slow oscillatory flow in panel (B). Similar

to the RRBC DMD spectra shown in Figure 3, the eigenvalues lie on or close to the unit circle. In both cases the focus is

on the slow dynamics in the respective datasets obtained by the two sampling procedure outlined in the previous section,

we focus on the lowest obtained frequencies. The eigenvalues corresponding to the first five lowest frequencies lie in fact

on the unit circle for both cases, with the data sampled at larger intervals shown in Figure 8B being converged at higher

frequencies as well.

The temporal structure of the original temperature field and the temperature field reconstructed from the first three

dynamic modes is shown in Figure 9 using horizontal slices located that the spanwise middle of the domain, y=H∕2, and

at different heights. Figure 9A,B contains visualizations of the original field at z=0.1H, to capture fast plume emission,

and at z=0.8H, to capture slow oscillations, respectively, and Figure 9C,D presents the corresponding reconstructions.

A visual comparison of the original and the reconstructed fields qualitatively shows that sDMD can clearly distinguish

the two dominant spatiotemporal structures, with the first three dynamic modes identifying the fast motion of the plume

emission for the dataset sampled at 0.1 free-fall time units (Figure 9A,C), and the first three dynamic modes capturing

the slow oscillatory mode for the dataset sampled at 0.5 free-fall time units (Figure 9B,D). The frequencies obtained from

the DNS data and the sDMD calculations are compared with the DMD frequencies calculated according to Equation (9).

YANG . 15 of 22

(A) (B)

(C) (D)

FIGURE 9 (A, B) Time evolution of the temperature of the original flow at a horizontal slice located at y=H∕2 and at the height (A)

z=0.1H, to capture plume emission, and (B) z=0.8H, to capture oscillations. (C, D) Time evolution of the temperature of the reconstructed

field with the first 3 dominant modes. It is noted that the reconstruction (C) and (D) are based on different snapshot intervals

For the first case, the second and third dynamic modes have a period of 1.58 free-fall time units according to Equation

(9), which fits the period of 1.6 free-fall time units of the fast plume emission determined from the original DNS data. For

the second case, the period corresponding to the second and third dynamic mode is 16.98 free-fall time units according

to Equation (9), which matches very well the period of the slow oscillation observed in the original dataset, measured

to be 16.8 free-fall time units. In both cases, the relative error between the time scales obtained from the full data and

via DMD is about 1 %. The agreement between the sDMD results and the DNS data, and the distinct identification and

separation of the two dominant spatiotemporal structures with frequencies that differ by an order of magnitude, gives

further confidence in the capability of DMD to capture the relevant processes, be it in the temporal or spatial framework.

3.3 Asymptotic suction boundary layer

3.3.1 Fluid flow

The ASBL is an open flow that develops over a flat bottom plate in the presence of suction through that plate. In con-

sequence, the BL thickness remains constant in the streamwise direction, and the ASBL shares certain properties with

parallel shear flows and spatially developing BLs. In the DNS, the ASBL is emulated by a plane Couette setup using a

high simulation domain. That is, we consider a fluid located in a wide gap between two parallel plates as shown schemat-

ically in Figure 10. The bottom plate is stationary and the fluid is set in motion through the top plate moving in the

x-direction with velocity U∞. The latter corresponds to the free-stream velocity of the open flow. The flow is assumed to

be incompressible and the conditions isothermal such that the density can be regarded as constant.

The occurrence of large-scale persistent coherent flow structures of long streamwise extent is one of the striking fea-

tures in turbulent BLs, and ASBL is no exception. We attempt to describe the dynamics of such a large-scale structure

16 of 22 YANG .

FIGURE 10 Schematic drawing of the asymptotic suction boundary layer in numerical simulations. The lower plate is stationary and

the fluid is set in motion by the upper plate that moves in x-direction with velocity U∞, representing the free-stream velocity of the emulated

open flow. Fluid is removed through a porous bottom plate with velocity VS, to guarantee conservation of mass, fluid enters the system at the

same speed through a porous top plate. In numerical simulations, this is realized uniformly through boundary conditions on the wall-normal

component of the velocity field

in a long time series using a small number of dynamic modes. In order to alleviate the computational effort, the simula-

tions were carried out at moderate Reynolds number using a short computational domain in the streamwise direction and

the sampled flow fields were averaged in streamwise direction. As such, the analysed two-dimensional fields obtained

by streamwise averaging adequately represent the three-dimensional fields at least concerning the large-scale dynamics

with streamwise coherence that is of interest here.

3.3.2 Governing equations & numerical details

Expressed in units of the free-stream velocity, the laminar flow is given by

U=

1−e−yVs∕𝜈

−Vs∕U∞

0

,(17)

where Vsis the suction velocity and 𝜈is the kinematic viscosity. The deviations uof the laminar flow are then described

by the dimensionless equations

𝜕tu+u⋅∇u+U⋅∇u+u⋅∇U+∇p−Re−1Δu=0,∇⋅u=0,(18)

where pis the pressure divided by the constant density 𝜌and Re =U∞𝛿∕𝜈the Reynolds number based on the free-stream

velocity, the laminar displacement thickness 𝛿=𝜈∕Vsand the kinematic viscosity 𝜈of the fluid.

The DNS data was generated with the open-source code channelflow2.0 [15,16]. Equations 18 are solved numer-

ically in a rectangular domain Ω=[−Lx∕2,Lx∕2]×[0,H]×[−Lz∕2,Lz∕2]as schematically shown in Figure 10, with

periodic boundary conditions in the streamwise x- and the spanwise z-directions and no-slip boundary conditions in the

wall-normal y-direction. Channelflow2.0 uses the standard pseudospectral technique with 2∕3rd dealiasing in stream-

and spanwise directions, where the spatial discretization is by Fourier expansions in the homogeneous directions and

a Chebyshev expansion in the y-direction. The temporal discretisation is given by a third-order semi-implicit backward

differentiation scheme (SBDF3). Details of the DNS dataset are summarized in Table 2.

YANG . 17 of 22

TABLE 2 Details of the ASBL simulations discussed in Section 3.3 The Reynolds number based on the free-stream velocity U∞and the

laminar displacement thickness 𝛿is denoted by Re, Re𝜏=u𝜏𝛿0.99∕𝜈is the friction Reynolds number, with u𝜏=𝜏w∕𝜌and 𝜏wbeing the

shear stress at the bottom wall, 𝜌the density, 𝜈the kinematic viscosity, 𝛿0.99 ≈18.5𝛿the boundary layer thickness, Lx,H,andLzare the

length, height, and width of the simulation domain, Nx,Ny,andNzthe number of grid points in x,y,andz-directions, respectively, Δx+and

Δz+the grid resolution in wall units taking into account two-thirds dealiasing in stream- and spanwise directions, Δtthe sampling interval

and Nthe number of samples.

Re Re𝝉𝝉w∕𝝆U2

∞Lx∕𝜹H∕𝜹Lz∕𝜹NxNyNz𝚫x+𝚫z+𝚫t∕(𝜹∕U∞)N

1000 320 0.0003 4𝜋20 4.6𝜋64 161 96 5.1 3.9 20 203

(A) (B) (C)

FIGURE 11 Spatiotemporal structure and reconstruction of large-scale dynamics in the asymptotic suction boundary layer. (A)

Representative original velocity-field sample taken at t=3920𝛿∕U∞. The color coding indicates the streamwise-averaged deviation uxfrom

laminar flow in streamwise direction and the arrows the streamwise-averaged cross-flow. A slow large-scale coherent structure is clearly

visible. (B) Time evolution of the original flow at the center of the coherent structure at y∗≈3𝛿. (C) Time evolution of the reconstructed flow

from the mean flow and the lowest frequency, that is, the first three dynamic modes at y∗≈3𝛿

3.3.3 Streaming DMD

Figure 11A shows the deviations from the laminar flow averaged in the streamwise direction of a typical data sample.

A large-scale coherent region that is localized in the spanwise direction and extends from about 2𝛿to 7𝛿in wall-normal

direction is clearly visible. This structure moves slower than the laminar flow and is accompanied by near-wall small-scale

regions where the flow is faster than the laminar flow. The slow large-scale structure drifts through the simulation

domain in spanwise direction. It takes the large-scale coherent structure T=(2070𝛿∕U∞)time units to traverse the sim-

ulation area once. The shift time scale Tand its relative error of about 6%have been determined by minimizing the

L2-distance between two velocity-field samples at time tand t+T′over T′. This was repeated for several pairs of data

snapshots separated by T′. The time scale of the spanwise shift is also clearly discernible through the periodic pattern

in the spatiotemporal evolution of the flow at a fixed distance y∕𝛿=3 from the bottom plate shown in Figure 11B. Dur-

ing that time it varies in intensity, as can be seen when considering the diagonal structure visible in the spatiotemporal

evolution, it does not disappear completely and it is difficult to discern other patterns in its dynamics.

The aim is to reconstruct the large spatiotemporal scales of the dynamics, that is, the spatial extent of the slow

large-scale structure, its slow spanwise drift and its dynamics, with a few dynamic modes. Capturing the latter two requires

a very long time series and as such the application of streaming DMD as opposed to classical DMD, as not all data can

be stored in memory at the same time. Following a convergence study for the first few lowest frequencies, the truncation

number was set to r=150, and the sampling interval to 20𝛿∕U∞. This sampling interval results in about 200 velocity-field

snapshots to be analyzed (see Table 2). The lowest nonzero DMD-frequency obtained from Equation (9) results in a time

scale TsDMD =2156𝛿∕U∞, which falls within the error margins of the time scale obtained by the aformentioned mini-

mization procedure, T=(2070𝛿∕U∞). We find that three dynamic modes, that is, the mean flow and the two complex

conjugate modes corresponding to the lowest frequency, reproduce the slow spanwise drift with time scale TsDMD ≈T,

18 of 22 YANG .

(A) (B) (C) (D)

FIGURE 12 Spectrum, spatiotemporal structure and reconstruction of large-scale dynamics in the asymptotic suction boundary layer

in a frame co-moving with the large-scale structure. (A) Reconstruction of the flow at t=3920𝛿∕U∞, using two dynamic modes. Time

evolution of the reconstructed flow from the first (B) three and (C) nine dynamic modes at the center of the large-scale structure at y∗≈3𝛿,

which is located in the co-moving frame at z∕𝛿≈1.5𝜋. (D) Spectrum. The eigenvalues corresponding to the modes used in the reconstruction

are shown in green (mean flow and two complex conjugates) and red (mean flow and four complex conjugates)

here demonstrated by comparison of the spatiotemporal evolution of the reconstructed flow shown in Figure 11C with

that of the original data shown in Figure 11B.

3.3.4 Streaming DMD in a co-moving frame

The detected spanwise drift, however, is not dynamically relevant as it is merely a continuous shift symmetry allowed

by the periodic boundary conditions in spanwise direction. In fact, DMD is known to perform poorly in presence of

continuous symmetries [31], the drift can lead to spurious modes [52], for instance. Therefore, and in order to obtain

further information on the large-scale dynamics of the ASBL, the spanwise drift was removed from the data sequence by

spatially shifting each data sample in the sequence an appropriate distance in spanwise direction. That is, a re-analysis of

the data was carried out in a co-moving reference frame, similar to the approach taken by Rowley and Marsden [46]. Here,

the constant shift velocity has been determined by the aforementioned L2-minimization, while Rowley and Marsden used

a reconstruction equation to calculate a time-dependent shift velocity. Very recently, by combination of the method of

slices for symmetry reduction [8] with DMD, a new approach to remove time-dependent continuous symmetries has been

devised [33].

In order to analyze the large-scale dynamics of the flow, we focus on flow reconstruction using low-frequency modes.

Figure 12A presents a reconstruction of the flow at t=3920𝛿∕U∞, the time at which the full data sample shown in

Figure 11A was taken, using the mean flow and the complex conjugate pair of dynamic modes with the lowest frequency.

The DMD spectrum resulting from the calculation in the co-moving frame is provided in Figure 12D, where the eigenval-

ues corresponding to the modes used in the aforementioned reconstruction are highlighted in green. As can be seen from

the data reconstruction in Figure 12A three dynamic modes are sufficient to reproduce the large-scale coherent struc-

ture. However, this reconstruction does not reproduce the flow in the neighboring regions of the coherent structure very

well. For instance, a secondary low-momentum zone that extends further into the free stream and the vortex pattern of

the original cross-flow (see Figure 11A) are not captured. We found that adding more modes, up to the highest frequency

with corresponding eigenvalues still on the unit circle as indicated in red and blue in Figure 12D, has very little effect (not

shown). A more adequate reconstruction could only be achieved at r=200, where five dynamic modes were sufficient

to describe the aforementioned features (not shown). However, as the dataset only comprises of 203 snapshots, r=200

is likely to result in overfitting.

The distinctive maxima and minima that are present in the spatiotemporal pattern of the flow evolution shown in

Figure 11B suggests the presence of slow periodicity in the background and faster dynamics within the structure’s core

located at z∕𝛿≈1.5𝜋in the co-moving frame. We find that the former can already be captured with the first two dynamic

modes as can be seen form the corresponding spatiotemporal representation of the flow shown in Figure 11C. Adding

three pairs of complex conjugate modes—those with eigenvalues shown in red in Figure 12D—results in a representation

YANG . 19 of 22

of the background flow that changes little when reconstructed with higher frequency modes, see Figure 12B for a spa-

tiotemporal representation of the flow reconstructed with nine modes. The faster dynamics of the structure’s core requires

a higher-dimensional description. In this context we recapitulate that the structure never completely disappears, hence

the fast dynamics encoded in the higher-frequency modes represent fluctuations in intensity on top of a persistent flow

feature.

4CONCLUSION & OUTLOOK

In this paper we demonstrated the applicability of streaming DMD [19], an efficient low-storage version of the classi-

cal SVD-based DMD [50], for the analysis of turbulent flows that show a certain degree of spatiotemporal coherence. As

turbulent flow dataset can be substantial in size, we propose a to couple the DMD calculation with prior downsampling,

which is appropriate only if the focus is on the large-scale spatiotemporal dynamics, which we focussed on here. We first

validate the proposed combination of downsampling with streaming DMD by comparing it to the classical SVD-based

DMD [50], based on the example of the flow past a cylinder for Re =100. The comparison shows that the obtained

streaming dynamic modes and eigenvalues match well with those computed from a postprocessing implementation of

the SVD-based DMD given enough truncation modes. However, streaming DMD can handle considerably larger datasets

with less computational costs compared to the SVD-based DMD, thanks to the feature of incremental data updating,

which only requires two data samples to be held in memory at a given time.

The objective of this study was to extract the main dynamic features with an efficient data-driven method and use the

resulting information for a low-dimensional reconstruction of the flow. We considered three examples, namely rapidly

rotating turbulent RBC, horizontal convection, and asymptotic suction BL. For rapidly rotating turbulent RBC, a dom-

inant zonal flow pattern, the boundary zonal flow, was identified through the first two dynamic modes. Similarly, for

HC two processes that operate on different time scales could be clearly classified in terms of dynamic modes: The sec-

ond dynamic mode captures the slow oscillatory dynamics in the bulk while the third dynamic mode describes the much

faster process of thermal plume emission. Finally, for ASBL a distinctive coherent low-momentum zone that travels

through the simulation domain in spanwise direction can be well described by the only first two dynamic modes once a

streamwise drift is removed from the data by calculating dynamic modes in a co-moving frame of reference. To describe

dynamical features of the coherent low-momentum zone, a few more frequencies must be included. These examples show

that the incrementally updated DMD algorithm can successfully decompose the dominant structures with correspond-

ing frequencies and modes. This establishes sDMD as an accurate and efficient method to identify and capture dominant

spatiotemporal features from large datasets of highly turbulent flows.

As DMD decomposes datasets into coherent structures based on characteristic frequencies, it is especially useful

for the analysis of flows featuring large-scale coherent structures and periodic motion. The advantages of the sDMD

algorithm, both in terms of low-storage and potential real-time implementation, will make DMD available in numerous

contexts where it would have been infeasible previously. This includes in particular the analysis of massive datasets that

cannot completely reside in memory. One such application, for instance, concerns the search for unstable periodic orbits

in turbulent flows, where a classical DMD-based approach has been successfully applied at moderate Reynolds number

[40]. sDMD may constitute a step forward in extending the applicability of this method to higher Reynolds numbers.

In further steps, sDMD can be applied to different turbulent flow datasets, to investigate in detail the ability to decom-

pose coherent flow structures. One disadvantage of streaming DMD is that the truncation number of snapshots to achieve

a similar reconstruction accuracy as the SVD-based DMD is typically larger. Here, as we focussed on statistically station-

ary large spatiotemporal scales only, apart from convergence tests we have not considered the effect of the truncation

number but have restricted our attention on the analysis of the first few modes corresponding to statistically stationary

dynamics ranked by frequency in ascending order. The truncation number is, however, important and should be quanti-

tatively considered when applying sDMD to flow field reconstruction or decomposition of complex turbulent flows with

multifrequency temporal structures. Similarly, the mode ordering by frequency used here may not be the optimal choice

for all datasets. An efficient and robust criterion needs to be introduced for rank selection.

Finally we wish to mention that not only DMD but also some other approaches, based on or related to DMD, might be

very efficient in extraction and analysis of the dynamics of the turbulent superstructures. While in the DMD we apply lin-

ear transformations to obtain modes out of snapshots and vice versa, a natural extension of the DMD would be to employ

nonlinear transformations instead. This can be realized either via application of hand-picked nonlinear functions (e.g. to

use so-called extended DMD—eDMD [61]), by calculation of the full Koopman modes (see Reference [14] in the context

20 of 22 YANG .

of convection), or by training a deep neural network (e.g. to use deep Koopman models [39]). A multilayer convolutional

neural network appears to be a good candidate for such a task. In general, (un)supervised deep learning seems to be very

promising technique for extraction and analysis of the global dynamics of the turbulent flow superstructures. Further to

this, kernel methods or tensor-based reformulations of DMD exist that are also well suited for high-dimensional datasets.

A more detailed consideration of these alternate approaches is beyond the scope of this article and application of these

advanced methods for the turbulent superstructure analysis remains a challenge for future studies.

ACKNOWLEDGMENTS

This work is supported by the Max Planck Center for Complex Fluid Dynamics, the Priority Programme SPP 1881 “Turbu-

lent Superstructures" of the Deutsche Forschungsgemeinschaft (DFG) under grants Sh405/7 and Li3694/1 and DFG grants

Sh405/8 and Sh405/10. The authors acknowledge the Leibniz Supercomputing Centre (LRZ) and the Lichtenberg high

performance computer of the TU Darmstadt for providing computing time. This work used the ARCHER UK National

Supercomputing Service (http://www.archer.ac.uk).

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How to cite this article: R. Yang, X. Zhang, P. Reiter, D. Lohse, O. Shishkina, and M. Linkmann, Data-driven

identification of the spatiotemporal structure of turbulent flows by streaming dynamic mode decomposition,

GAMM-Mitteilungen. (2022), e202200003. https://doi.org/10.1002/gamm.202200003