Sparse-Stochastic Model Reduction for 2D Euler Equations

Abstract

The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving these equations numerically can be extremely demanding. Several techniques to obtain fast and accurate simulations have been developed during the last decades. In this paper, we present a novel approach that combines recent developments in stochastic model reduction and conservative semi-discretization of the Euler equations. In particular, starting from the Zeitlin model on the 2-sphere, we derive reduced dynamics for large scales and we close the equations either deterministically or with a suitable stochastic term. Numerical experiments show that, after an initial turbulent regime, the influence of small scales to large scales is negligible, even though a non-zero transfer of energy among different modes is present.

Supplementary Information The online version contains supplementary material available athttps://github.com/cifanip/GLIFS

1 Introduction

The 2D Euler equations are a fundamental model for ideal fluids [10]. During the last two centuries, these equations have stimulated an intense activity both in terms of mathematics and physics (see for example the seminal works of Helmholtz and Arnol’d [14, 2]). In computational science and numerical analysis, retaining at a discrete level the rich non trivial structure of these equations is still a challenging problem [1, 19]. One main computational issue is the “curse of dimensionality”. Indeed, turbulent phenomena vary in different spatial and time scales and the distribution of energy over a vast range of scales of motion makes it computationally infeasible to fully resolve the flow in a numerical simulation. A well-established technique to mitigate large computational costs is large-eddy simulation (LES), where a spatial filter is applied to the governing equations after which only the large scales of motion are resolved [13, 21]. The filter may be defined explicitly, as a smoothing function where the filter width determines the level of detail left in the filtered solution and is chosen by the user, or implicitly, by coarsening the discretization operator. Either approach requires a closure model to represent the effect of the filtered scales on the unfiltered resolved scales. In this paper, we apply an explicit spectral cut-off filter to the 2D Euler equations, where the cut-off frequency is based on observed intrinsic scale separation. This filter leads to a discrete problem formulation in which matrix sparsity can be exploited to reduce computational costs. At the same time, this formulation still allows for an explicit representation of the effects of small scales on large scales. The equations are closed by a deterministic or stochastic model term, based on high-resolution measurements. Here, we choose to model small-scale flow features using a stochastic term mimicking high-resolution numerical data. Subsequently, we analyze the model performance by means of energy fluxes between the scales of motion. The proposed stochastic closures and assessment of energy fluxes in the high-resolution data can serve as a point of departure for further development of stochastic closure models.

A peculiar aspect of 2D ideal fluids is the presence of infinitely many conservation laws. In particular, as firstly described by Kraichnain [18], the conservation of energy and enstrophy (the $L^2$ norm of the curl of the velocity field) implies a double cascade phenomenon: the energy tends to move from small scales to large scales, whereas the enstrophy tends to follow the opposite direction. Hence, in terms of the curl of velocity, or vorticity, it is possible to clearly separate two regimes: one slowly evolving at large scales and one fast at small scales. This was shown to hold numerically in [20] for the Euler–Zeitlin model on the sphere. Theoretically, the study of non-deterministic fluid models for different regimes have gained interest in the SPDE community [12]. The equations studied in [12] and the results proved therein, show a precise connection between different space-time regimes with a reduced model for large scales. Indeed, it is shown that a suitable model for large scales is given by the so-called Stochastic Advection by Lie Transport (SALT) equations [15], in which a transport noise term models the infinitesimal action of the small scales on the large ones. Several numerical tests have shown the usefulness of the SALT equations as a powerful tool for model reduction [6, 9].

However, defining precisely what large and small scales are is still an open problem. In this paper, we present a criterion for defining large scales in terms of truncation of the Fourier expansion. We point out that other choices and interpretations of large and small scales are possible (see for example [20]). Let us first introduce the governing equations for the vorticity field $\omega$, defined on the 2-sphere $\mathbb {S}^2$ embedded in $\mathbb {R}^3$:

$$\displaystyle \begin{aligned} {} \begin{array}{ll} &\dot{\omega} = \lbrace\psi,\omega\rbrace\\ &\Delta \psi = \omega. \end{array} \end{aligned} $$

(1)

The Poisson bracket is defined as

$$\displaystyle \begin{aligned} \lbrace\psi,\omega\rbrace:=\nabla \psi\cdot\nabla^\perp\omega \end{aligned}$$

and the Laplacian is the Laplace–Beltrami operator on $\mathbb {S}^2$. As mention above, equations (1) have infinitely many first integrals: energy $H(\omega )=\frac {1}{2}\int _{\mathbb {S}^2}\psi \omega$, Casimirs $C_n(\omega )=\int _{\mathbb {S}^2}\omega ^n$, for $n\geq 1$, and angular momentum. Understanding the role played by these invariants is still an open problem, especially for the long-time evolution of the fluid [7].

In order to gain numerical insight on this question, V. Zeitlin proposed a spatial discretization of (1), which retains many of the first integrals above [22, 23]. The Euler–Zeitlin equations are defined as follows:

$$\displaystyle \begin{aligned} {} \begin{array}{ll} &\dot{W} = {}[{}P,W{}]{}\\ &\Delta_N P = W. \end{array} \end{aligned} $$

(2)

Here W is a $N\times N$ skew-Hermitian matrix with zero trace, that is, an element of the Lie algebra $\mathfrak {su}(N)$. The bracket $[P,W]$ is the usual matrix commutator and the discrete Laplacian $\Delta _N$ is defined such that its spectrum is a truncation of the spectrum of $\Delta$ [17]. As mentioned above, the Euler–Zeitlin equations possess the following integral of motions: energy $H(W)=\frac {1}{2}\mbox{Tr}(PW)$, Casimirs $C_n(W)=\mbox{Tr}(W^n)$, for $n = 2,\ldots , N$, and angular momentum. The core of the Zeitlin model is how the original vorticity $\omega$ and the discrete one W are linked. Indeed, the representation theory of $SU(2)$ provides a deep connection between the discrete Laplacian $\Delta _N$ and a particular basis $\lbrace T_{lm}\rbrace$ of $\mathfrak {su}(N)$, for $l=1,\ldots ,N-1$ and $m=-l,\ldots ,m$ [17, 4]:

• each $T_{lm}$ is an eigenvector of $\Delta _N$, with eigenvalue $-l(l+1)$,

• for each $N\geq 1$, there exists a linear map $p_N:C^\infty (\mathbb {S}^2)\rightarrow \mathfrak {su}(N)$, defined via the (real) spherical harmonics basis $\lbrace Y_{lm}\rbrace$ as $p_N(Y_{lm})=T_{lm}$, if and only if $l\leq N-1$,

• $\|p_N\lbrace \psi ,\omega \rbrace - N^{3/2}[p_N\psi ,p_N\omega ]\|\rightarrow 0$, for $N\rightarrow \infty$, where the norm is the operator norm.

The classical way to determine large and small scales is to choose a wave number $\overline {l}$ as a threshold for the large scales (see for example [3, 6]). In this work, we propose the following criterion to set the threshold $\overline {l}$. Consider a time scale in which the fluid’s energy spectrum profile has reached a stationary state. Then, typically (that is, out of equilibrium) the spectrum exhibits a double slope, which determines a kink at a certain wave number $\overline {l}$. Then, we define the large-scales $\overline {W}$ as the filtered vorticity with modes up to $\overline {l}$, obtaining a banded matrix. We propose three possible ways, both deterministic and stochastic, of closing the equations for $\overline {W}$, by choosing different interactions with the small scales. Finally, we provide numerical tests to assess the different models introduced.

2 Sparse-Stochastic Model Reduction

The Euler–Zeitlin equations (2) allow to study some typical features of the 2D fluids in the matrix language. In this section, we propose a way to reduce the complexity of the Eq. (2), by defining from W a sparse matrix $\overline {W}$ which retains the relevant large-scale information. Then, we show different ways of closing the equations for $\overline {W}$, by adding a suitable stochastic term.

In the Zeitlin model, the basis element $T_{lm}$ of $\mathfrak {su}(N)$ has non-zero entries only in the lower and upper $\pm m$ diagonal. If we look at the anti-diagonals, instead, we are looking at the components determining the value of the vorticity field at certain latitude bandwidth on $\mathbb {S}^2$, as shown in Fig. 1.

Fig. 1

Structure of the discrete vorticity W in the Zeitlin model

The large scales are typically chosen to be the modes such that l is smaller than a threshold level $\overline {l}$. In the Euler–Zeitlin model, this corresponds to considering the banded matrices limited in the diagonals $\pm l\leq \overline {l}$ and then removing the components corresponding to $l>\overline {l}$. The Poisson equation which defines the stream matrix P preserves this sparsity structure, since the basis elements $T_{lm}$, the eigenvectors of the Laplacian, are themselves sparse. However, the Lie bracket does not restrict to this space. Indeed at each time-step we have to project the vector field into the right space.

Usually, we do not have any chance to guess the contribution of the small scales to the evolution of the large ones. However, we expect that after an initial turbulent transition, the fluid exhibits two clearly separated spatial scales. The hint for such a scenario is due to several numerical simulations of the Euler–Zeitlin equations [3, 20]. Eventually, the energy profile reaches a fixed configuration with two slopes. The first part of the spectrum represents the distribution of energy at large scales, whereas the second part the distribution of energy at small scales. Typically, the separation between large and small scales occurs at a wave number $\overline {l}\approx \sqrt {N}$. For wave numbers larger than $\overline {l}$ the energy spectrum has the characteristic slope of $l^{-1}$, which is the one of white noise, see Fig. 3. The universal nature of the small scales suggests a model reduction in terms of large-scale evolution combined with a stochastic term contribution. In Fig. 2, we show the procedure to get the two new fields $\overline {W}$ and $\widetilde {W}$. To define $\overline {W}$, we introduce the orthogonal projection $\pi$ onto the modes $l\leq \overline {l}$. The small-scale field $\widetilde {W}$ is defined as the linear combination of the basis elements $T_{lm}$, for $l>\overline {l}$ with coefficients $\beta ^{lm}$ as independent Brownian motions, with mean and variance obtained from the high-resolution DNS. This choice of coefficients ensures that the random field yields the same mean energy spectrum at the small scales as measured from the DNS. Additionally, applying the Kolmogorov–Smirnov and Anderson–Darling tests for normality to the high-resolution data suggests that the distribution of the basis coefficients for $T_{lm}$, for $l > \bar {l}$, is Gaussian.

Fig. 2

Filtering of large-scale and definition of random small scale vorticity $\widetilde {W}$, via the independent Brownian motions $\beta ^{lm}$

Fig. 3

Initial vorticity obtained via high-resolution DNS. Top left, the field W, top right, the filtered field $\overline {W}$, bottom left, $W-\overline {W}$, bottom right, the energy spectrum of W. Note the change of slope in the energy profile at $l\approx \sqrt {N}$

Hence, we define $\overline {W}:=\pi W$ and $\widetilde {W}:=\sum _{l=\overline {l}+1}^{N-1} \sum _{m=-l}^l \beta ^{lm}T_{lm}$. With these new fields, we essentially have three possible choices. The first one consists of a deterministic closure simply via the projection of the vector field onto the large scales:

$$\displaystyle \begin{aligned} {} \begin{array}{ll} &\dot{\overline{W}} =\pi [\overline{P},\overline{W}]\\ &\Delta_N\overline{P} = \overline{W}, \end{array} \end{aligned} $$

(3)

which we refer to as the no-model closure. The second model is the large scale enstrophy-preserving stochastic closure, which is up to the projection $\pi$ a type of SALT equation (see [15]):

$$\displaystyle \begin{aligned} {} \begin{array}{ll} &d\overline{W} = \pi[\overline{P},\overline{W}]dt + \sum_{l=\overline{l}+1}^{N-1} \sum_{m=-l}^l \frac{1}{-l(l+1)}\pi[T_{lm},\overline{W}]\circ d\beta^{lm}\\ &\Delta_N\overline{P} = \overline{W}. \end{array} \end{aligned} $$

(4)

We recall that the symbol $\circ$ denotes the Stratonovich integral and indeed we find that the large scale enstrophy is conserved, via

$$\displaystyle \begin{aligned} d\frac{1}{2}\mbox{Tr}(\overline{W}^2) & = \mbox{Tr}(\overline{W}d\overline{W})=\mbox{Tr}(\overline{W}\pi[\overline{P},\overline{W}])dt\\ & \quad + \sum_{l=\overline{l}+1}^{N-1} \sum_{m=-l}^l \frac{1}{-l(l+1)}\mbox{Tr}(\overline{W}\pi[T_{lm},\overline{W}])\circ d\beta^{lm}=0, \end{aligned} $$

being $\pi \overline {W}=\overline {W}$ and $[\overline {W},\overline {W}]=0$. Notice that the other large-scale Casimirs are not preserved, since in general $\pi \overline {W}^k\neq (\pi \overline {W})^k$, for $k>1$.

Finally, the third model is a large-scale energy-preserving stochastic closure (see [11] for its analysis and [16, 8] for more recent applications). We note that [16] introduces this stochastic closure as Stochastic Forcing by Lie Transport (SFLT) and provides its general definition. Here, we refer to this closure as energy-preserving noise (EPN) and adopt the following definition:

$$\displaystyle \begin{aligned} {} \begin{array}{ll} &d\overline{W} = \pi[\overline{P},\overline{W}]dt + \sum_{l=\overline{l}+1}^{N-1} \sum_{m=-l}^l \pi[\overline{P},T_{lm}]\circ d\beta^{lm}\\ &\Delta_N\overline{P} = \overline{W}. \end{array} \end{aligned} $$

(5)

The proof of conservation of the large scale energy for (5) is identical to the one of large scale enstrophy conservation by noticing that $\pi$ commutes with $\Delta _N$. The benefit of the stochastic models (4) and (5) compared to the no-model closure is that the stochastic models provide an explicit representation of the small-scale flow features and, doing so, aim to truthfully affect the evolution of the large scales of motion. No such representation is included in the no-model closure. In the next section, we perform a numerical test for the three different models (3), (4), (5), comparing them with the high-resolution DNS.

3 Numerical Simulations

In this section, we carry out a numerical experiment to study the performance of the models proposed in the previous section. The numerical experiment is conducted as follows. We set the high-resolution level at $N=128$. Then we generate a random initial condition and we run a high-resolution DNS. We stop the simulation once a stationary energy profile is reached (see Fig. 3). The solution at this point in time defines the initial condition of the reference solution and the ensuing model simulations.

From the DNS we select the large-scale threshold as wave number $\overline {l}\approx \sqrt {N}$, at which the kink in the energy spectrum appears. In our numerical simulation the kink is found to be at $\overline {l}=14$.

Remark 1

The kink in the energy spectrum must depend on the truncation level N. Indeed, it was shown that the tail of the energy distribution at small scales for conservative schemes, like the Euler–Zeitlin one, has a characteristic slope of $k^{-1}$. This would imply unbounded energy for $N\rightarrow \infty$, which contradicts the fact that we only consider vorticities with bounded energy. Therefore, the kink wave number between the two slopes in the energy profile must increase with N. Numerically, we have observed that it grows like $\sqrt {N}$.

Then, we define our large-scale field as $\overline {W}:=\pi W$, where $\pi$ denotes the orthogonal projection onto the modes for $l\leq \overline {l}$. The projection consists of two steps: first we extract the components up to $\overline {l}$ and then we generate the field $\overline {W}$. The cost of calculating each component is $\mathcal {O}(N)$ and since we need to repeat this operation $\overline {l}^2-1\approx N$ times, the total cost of extracting the components is $\mathcal {O}(N^2)$. Clearly, to construct the field $\overline {W}$ we have to perform $\mathcal {O}(N^2)$ operations. Hence, the total computational cost of the projection $\pi$ is $\mathcal {O}(N^2)$. Hence, given two matrices $A,B\in \mathfrak {su}(N)$, the cost of evaluating $\pi [\pi A,\pi B]$ is given by the evaluation of $\pi$ plus the cost of multiplying $\pi A$ and $\pi B$. Since we are interest only in the $\pm \overline {l}$ diagonals we need to perform $\mathcal {O}(N\overline {l})$ vector-vector multiplications of the cost $\mathcal {O}(\overline {l})$, which implies a total cost of $\mathcal {O}(N\overline {l}^2)\approx \mathcal {O}(N^2)$. The same cost $\mathcal {O}(N^2)$ holds for the stochastic term, since we can actually consider only $m=-\overline {l},\ldots ,\overline {l}$. We also define $\widetilde {W}$, as explained in the previous section. Finally, we define the reference solution as the high-resolution numerical simulation using the initial condition previously defined. The large-scale field obtained by projecting this initial condition serves as a starting point for the simulations where the small scales are modeled as described in Eqs. (3), (4), (5). All simulations are run for 250 time units from the initial condition. In our numerical simulations, the time integration is done via the Heun-type scheme adapted for the SDEs, with time-step $h=0.25$. In the following, we perform an ensemble of realizations for the stochastic closures. Each ensemble consists of 25 realizations, which is sufficient to show the qualitative difference between the proposed models.

We notice from Figs. 4 and 5 that the no-model solution and the mean SALT solution perform well compared to the reference solution, in terms of qualitative vorticity evolution as well as the kinetic energy spectrum. On the contrary, the energy-preserving scheme loses accuracy and a cascade of energy to lower wave numbers occurs. We attribute the observed difference between SALT and EPN to two causes. Firstly, the orders of magnitude of the stochastic terms entering the equations differ between the methods. The stochastic forcing types are parametrizations of components of the effect of small scales on large scales, respectively given by $\pi [\widetilde {P}, \overline {W}]$ and $\pi [\overline {P}, \widetilde {W}]$. Figure 6 shows these quantities for a high-resolution snapshot, which illustrates that the term parametrized by the EPN is significantly larger than the term that SALT parametrizes. Therefore, it is reasonable to expect that the proposed use of EPN leads to more substantial deviations from the no-model simulation than the use of SALT. In fact, the proposed use of SALT only leads to very small changes compared to the no-model setting. Secondly, in the energy-preserving scheme, no energy can leave the large scales. Hence, if the transfer of energy between different modes is non-zero, the conservation of the large-scale energy prevents the energy to flow from large scales to small scales, causing an extra accumulation of energy $l\approx \overline {l}$.

Fig. 4

Evolution of the large scales over 250 time units, via the different models proposed and the high-resolution simulation. No model corresponds to (3), SALT to (4) and EPN to (5). The shown results using SALT and EPN are the mean of an ensemble each consisting of 25 independent realizations

Fig. 5

Energy spectra at $t=250$ using SALT (left) and EPN (right), both computed from an ensemble of 25 independent realizations and compared to the reference and no-model energy spectra. Note that the standard deviation of the ensemble obtained using SALT is too small to discern in this figure

Fig. 6

Components of the evolution of the reference vorticity projected onto the large scales, generated from the reference vorticity field at $t=250$. Note that the ranges of the color bars vary per field, which highlights the difference in magnitudes between the fields

In order to check this thesis, we compute the energy transfer among different modes in the high-resolution DNS. Let us consider the energy at a level l:

$$\displaystyle \begin{aligned} E(l)=\frac{1}{2}\sum_{m=-l}^l \frac{\omega_{lm}^2}{l(l+1)}.\end{aligned}$$

Then, the energy variation in time is given by

$$\displaystyle \begin{aligned} \frac{dE(l)}{dt}=\sum_{m=-l}^l \frac{\omega_{lm}[P,W]_{lm}}{l(l+1)}.\end{aligned}$$

Let $F(l):=|\frac {dE(l)}{dt}|$ be the absolute value of the energy transfer due to the nonlinearity of the vector field $[P,W]$. In Fig. 7, we plot the energy transfer contributions of the four possible couplings of large and small scales. We notice that the transfer of energy between large and small scales is non-zero. In particular, the main driver of the energy for the components of $\overline {W}$ is the vector field $[\overline {P},\overline {W}]$, whereas for small scales it is $[\overline {P},\widetilde {W}]$.

Fig. 7

Energy transfer among different modes computed from a snapshot of the high-resolution simulation. Left, energy transfer among modes at large scales, right, at small scales

The plots in Fig. 7 show that the term $[\overline {P},\widetilde {W}]$ becomes more and more relevant in the energy flux at large scales, while approaching the threshold level $\bar l$. Therefore, one might expect the stochastic model (5), which takes into account this term too, to be more accurate than (3) or (4). However, form Figs. 4 and 5 this does not seem the case. Indeed, the way we model the term $[\overline {P},\widetilde {W}]$ in (5) prevents the energy from flowing from large scales to small scales. However, also in (3) there is no energy flow from large scales to small scales. Therefore, we suggest that at large scales the term $[\overline {P},\widetilde {W}]$ is responsible for redistributing the energy from lower to higher frequencies and in absence of a dissipation mechanism the rearranging of energy diverges from the correct spectrum.

The fact that (3) and (4) perform equally well is quite surprising and indicates that the small scales do not affect the large scales much when a stationary energy profile is reached. However, for much longer times than those we have run, it is possible that the effect of small scales on large scales become more relevant and (4) becomes more accurate than (3) in terms of large-scale dynamics. Further investigations on this aspect are ongoing research and will be presented in future work.

4 Conclusions and Outlook

In this paper, we have presented a possible strategy to reduce the complexity of the Euler–Zeitlin model, while performing long-time simulations. Numerical evidence shows that the Euler–Zeitlin equations exhibit a clear separation of scales such that the large-scale dynamics are quite robust to different couplings with small scales, either deterministic or stochastic. Interestingly, the energy-preserving scheme we have defined shows that the energy at large scales cannot be exactly conserved. This means that large and small scales are never completely decoupled, even when the energy spectrum profile reaches a stationary regime. This indicates that for very long times a non-zero transfer of energy among different scales is present.

The Zeitlin model has been criticized for unrealistic conservation of enstrophy and other Casimirs at a finite level of truncation N. Our result shows that this issue can be understood such that the Euler–Zeitlin equations are quite robust and precise in describing large scales, which means for wave numbers $l\approx \sqrt {N}$. On the other hand, the remaining modes are themselves a model for the small scales, which correctly mimic the energy flux among different modes.

In conclusion, we have shown that the Zeitlin model can be a useful tool for simulating long-time large-scale dynamics. In future work, we aim to perform more systematic simulations using the parallelized code developed in [5] and available on https://github.com/cifanip/GLIFS. Additionally, further analysis of energy and enstrophy transfers between the scales of motion may serve to derive tailored data-driven stochastic closure models for the Euler-Zeitlin equations.