Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (2024)

Table of Contents
I Introduction II Direct cascade in two-dimensional Navier-Stokes equation III Numerical simulations of the direct cascade with friction IV Conclusions Acknowledgement References Appendix A Numerical integration of the 2D NS equation A.1 Pseudospectral GPU code A.2 Code implementation and Performance tests FAQs

V.J. Valadão1111Corresponding author: victor.dejesusvaladao@unito.it, G. Boffetta1, M. Crialesi-Esposito2, F. De Lillo1, and S. Musacchio11Dipartimento di Fisica and INFN - Università degli Studi di Torino, Via Pietro Giuria, 1, 10125 Torino TO, Italy.2DIEF, University of Modena and Reggio Emilia, 41125 Modena, Italy.

Abstract

It has been long known that the addition of linear friction on two-dimensional Navier-Stokes (NS) turbulence, often referred to as Ekman-Navier-Stokes (ENS) turbulence, induces strong intermittent fluctuations on small-scale vorticity. Such fluctuations are strong enough to be measurable at low-order statistics such as the energy or enstrophy spectrum. Simple heuristics lead to corrections in the spectrum which are proportional to the linear friction coefficient. In this work, we study the spectral correction by the implementation of a GPU-accelerated high-resolution numerical simulation of ENS covering a large range of Reynolds numbers. Among our findings, we highlight the importance of non-locality when comparing the expected results to the measured ones.

I Introduction

Turbulence is a complex and fascinating phenomenon that happens in nature atdifferent physics scales ranging from upper ocean dynamics[1], passing through planetary atmosphere[2], to interstellar media[3] and so on. The beauty of this phenomenon isnot restricted to its observation in nature, but also to the fact that a singleset of dynamic equations is responsible for all those complexities, theNavier-Stokes (NS) equations [4]. The non-linear andnon-local nature of the NS equations requires numerical techniques to solve itwithin a numerical precision, to advance the understanding of turbulencephenomena.

The need for larger and larger simulations has been growing over time at thesame rate as the hardware capabilities since numerical simulations are requiredto solve large and small scales at once [celani2007frontiers].Efficient numerical techniques such asthe pseudospectral method [5, 6] and theuse of massive CPU (Computing Processor Unit) parallelization[7, 8] became the usual paradigm oncomputational turbulence in the idealized case of hom*ogeneous-isotropicconditions. This combination relies on the fact thatmultidimensional Fast Fourier Transform (FFT), the basic core of thepseudospectral method, can be decomposed as a sequence of lower dimensionalFFTs [9]. The latter method slices the spatial domaininto many small packs of data and, through all-to-all CPU communicationsaccelerates the computation of the most intensive parts of the NS solvers byperforming concurrently low dimensional FFTs.

GPU (Graphical Processor Unit) started to be used as auxiliary devices (usuallycalled accelerators) to enhance computational gain on each spatially decomposedpiece of data as the GPUs have many more cores than single CPUs[10, 11]. GPU accelerators pushed thecomputation runtime to a point where almost all the computation time is spenton data transferring and communication among different devices[12, 13]. Fortunately, GPU manufacturerssuch as NVIDIA and AMD have been developing new devices with larger memoriesand/or faster inter-GPU connections, leading to huge accelerations incommunications. New GPU technologies and their applications on the mainsupercomputing centers enabled extensive turbulence simulations up to 327683superscript32768332768^{3}32768 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPTspatial gridpoints on feasible computational timescales [14].

Notwithstanding, several physical settings in the α𝛼\alphaitalic_α-turbulence model[15], thin layer turbulence[16, 17, 18, 19, 20],rotating and/or stratified turbulence[21, 22, 23], and manyothers, have almost identical dynamical equations and, in some cases, do notrequire extremely high spatial resolution to be studied. However, since thosesystems can display poor statistical convergence, they need to be solved forvery long times or need to cover a parameter space of large dimensions.

The most popular way of writing numerical code on GPU is by using CUDA, aparallel computing platform created by NVIDIA that provides drop-in acceleratedlibraries such as FFT and linear algebra libraries. One of its strong points isits availability of the most familiar computational languages such asC/C#/C++, Fortran, Java, Python, etc. However, the CUDA solution has aclosed code and works only for NVIDIA-based hardware, losing itsinteroperability among different hardware platforms and clusters. On the otherhand, the Hybrid Input-Output (HIT) is an open-source low-level language with asyntax that is similar to CUDA. Employing HIT or similar open-source languagessuch as OpenCL, developers can ensure the portability and maintainability oftheir code among heterogeneous platforms. In summary, even though GPU codingoptions and interfaces have evolved, it is still needed for inexperienceddevelopers to completely dive into one or more languages to be able to developheterogeneous applications on GPU.

From the point of view of applications, the use of GPU toaccelerate partial differential equations computations does not usuallyrequire to write completely new code, but simply to port anexisting one [24, 25]. If one isrestricted to NVIDIA platforms, a possible solution, is less efficient butmuch easier, is OpenACC, a directive-based programming model that simplifiesparallel GPU programming. Using compiler directives for C and Fortran, OpenACCallows developers to port their existing CPU code, specifying which partsshould be offloaded to the GPU for acceleration, and also supporting MessagePassing Interface (MPI) for multi-node, multi-GPU applications.

The focus of this work is to present a single GPU solution for a generalizedNavier-Stokes equation in two dimensions.Sec.II revisits the phenomenologyof 2D turbulence in the presence of linear large-scale friction, usuallyreferred to as Ekman friction. In Sec.III we introduce the standardpseudospectral method for solving the generalized equation showing someperformance tests. Sec.IV applies our numerical solver to revisit the problemof spectrum correction, focusing on the specific case of the correction to thescaling exponent of the energy spectrum due to the linear friction, concludingwith Sec.V where we discuss the results pointing directions of futureresearch.

II Direct cascade in two-dimensional Navier-Stokes equation

It has long been known that studying the 2D NS equation is more than simply anacademic case study on the road to understanding turbulence phenomena. TheEarth’s atmosphere[26], the dynamics of oceans’surface[27], and geometrically confined flows[17, 19] are examples ofphysical systems where experiments and numerical simulations showfeature of two-dimensional turbulence, at least on a range of scales.To revisit the 2D phenomenology, we start with the incompressibleNavier-Stokes equation in two dimensions

{∂tvi+vj⁢∇jvi+∇iP=ν⁢∇2vi−μ⁢vi+Fi,∇ivi=0,casessubscript𝑡subscript𝑣𝑖subscript𝑣𝑗subscript∇𝑗subscript𝑣𝑖subscript∇𝑖𝑃𝜈superscript∇2subscript𝑣𝑖𝜇subscript𝑣𝑖subscript𝐹𝑖otherwisesubscript∇𝑖subscript𝑣𝑖0otherwise\begin{cases}\partial_{t}v_{i}+v_{j}\nabla_{j}v_{i}+\nabla_{i}P=\nu\nabla^{2}v%_{i}-\mu v_{i}+F_{i}\ ,&\\\nabla_{i}v_{i}=0\ ,\ \end{cases}{ start_ROW start_CELL ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_P = italic_ν ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_μ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , end_CELL start_CELL end_CELL end_ROW(1)

where repeated indices are implicitly summed and ∇i≡∂/∂xisubscript∇𝑖subscript𝑥𝑖\nabla_{i}\equiv\partial/\partial x_{i}∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≡ ∂ / ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the components of the gradient vector. The field vi⁢(x→,t)subscript𝑣𝑖→𝑥𝑡v_{i}(\vec{x},t)italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , italic_t )represents the fluid velocity, P⁢(x→,t)𝑃→𝑥𝑡P(\vec{x},t)italic_P ( over→ start_ARG italic_x end_ARG , italic_t ) is the pressure fieldwhose role is to ensure incompressibility and Fi⁢(x→,t)subscript𝐹𝑖→𝑥𝑡F_{i}(\vec{x},t)italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , italic_t ) representsan external forcing density.Two dissipative terms are in (1), one is the standardviscosity ν⁢∇2vi𝜈superscript∇2subscript𝑣𝑖\nu\nabla^{2}v_{i}italic_ν ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT which is active at small scales,while μ⁢vi𝜇subscript𝑣𝑖\mu v_{i}italic_μ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, often referred to as Ekman friction,remove energy at large scales and provide a statistically stationary state.This friction term has a different physical origin depending on the specificphysical model, e.g. layer-layer friction in stratified fluids[28] or air friction in the case ofsoap films [31].

2D incompressible NS equations can be conveniently rewritten in terms ofa pseudoscalar vorticity and the stream function. The former is given byω=ϵi⁢j⁢∇ivj𝜔subscriptitalic-ϵ𝑖𝑗subscript∇𝑖subscript𝑣𝑗\omega=\epsilon_{ij}\nabla_{i}v_{j}italic_ω = italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT while the latter is related to the velocityfield through vi=ϵi⁢j⁢∇jψsubscript𝑣𝑖subscriptitalic-ϵ𝑖𝑗subscript∇𝑗𝜓v_{i}=\epsilon_{ij}\nabla_{j}\psiitalic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ψ where in both cases,ϵi⁢jsubscriptitalic-ϵ𝑖𝑗\epsilon_{ij}italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is the full antisymmetric tensor.Clearly, we have ω=−∇2ψ𝜔superscript∇2𝜓\omega=-\nabla^{2}\psiitalic_ω = - ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ.By contracting (1) with ϵj⁢i⁢∇jsubscriptitalic-ϵ𝑗𝑖subscript∇𝑗\epsilon_{ji}\nabla_{j}italic_ϵ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT we get

∂tω+J⁢(ω,ψ)=ν⁢∇2ω−μ⁢ω+f,subscript𝑡𝜔𝐽𝜔𝜓𝜈superscript∇2𝜔𝜇𝜔𝑓\partial_{t}\omega+J(\omega,\psi)=\nu\nabla^{2}\omega-\mu\omega+f\ ,\ ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_ω + italic_J ( italic_ω , italic_ψ ) = italic_ν ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ω - italic_μ italic_ω + italic_f ,(2)

where the Jacobian isJ⁢(ω,ψ)=ϵi⁢j⁢∇iω⁢∇jψ=∇i(vi⁢ω)𝐽𝜔𝜓subscriptitalic-ϵ𝑖𝑗subscript∇𝑖𝜔subscript∇𝑗𝜓subscript∇𝑖subscript𝑣𝑖𝜔J(\omega,\psi)=\epsilon_{ij}\nabla_{i}\omega\nabla_{j}\psi=\nabla_{i}\left(v_{%i}\omega\right)italic_J ( italic_ω , italic_ψ ) = italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ω ∇ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ψ = ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ω )and f=ϵi⁢j⁢∇iFj𝑓subscriptitalic-ϵ𝑖𝑗subscript∇𝑖subscript𝐹𝑗f=\epsilon_{ij}\nabla_{i}F_{j}italic_f = italic_ϵ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is the 2D curl of the forcing.

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A convenient assumption for the study of hom*ogeneous-isotropic turbulenceis to assume that the forcing f⁢(x,t)𝑓𝑥𝑡f(x,t)italic_f ( italic_x , italic_t ) is a random functionwith zero mean and white-in-time correlations:

⟨f⁢(x→,t)⁢f⁢(y→,s)⟩=δ⁢(t−s)⁢ξ⁢(|x→−y→|),delimited-⟨⟩𝑓→𝑥𝑡𝑓→𝑦𝑠𝛿𝑡𝑠𝜉→𝑥→𝑦\left\langle f(\vec{x},t)f(\vec{y},s)\right\rangle=\delta(t-s)\xi(|\vec{x}-%\vec{y}|)\ ,\ ⟨ italic_f ( over→ start_ARG italic_x end_ARG , italic_t ) italic_f ( over→ start_ARG italic_y end_ARG , italic_s ) ⟩ = italic_δ ( italic_t - italic_s ) italic_ξ ( | over→ start_ARG italic_x end_ARG - over→ start_ARG italic_y end_ARG | ) ,(3)

where the spatial correlation function has support on a given scale,i.e. ξ⁢(|x→|≈ℓf)≈ξ0𝜉→𝑥subscriptℓ𝑓subscript𝜉0\xi(|\vec{x}|\approx\ell_{f})\approx\xi_{0}italic_ξ ( | over→ start_ARG italic_x end_ARG | ≈ roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) ≈ italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,and ξ⁢(|x→|≫ℓf)→0→𝜉much-greater-than→𝑥subscriptℓ𝑓0\xi(|\vec{x}|\gg\ell_{f})\rightarrow 0italic_ξ ( | over→ start_ARG italic_x end_ARG | ≫ roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) → 0.

In the inviscid, unforced limit, the model (2) conserved thekinetic energy E=⟨|𝒗|2⟩/2𝐸delimited-⟨⟩superscript𝒗22E=\left\langle|{\bm{v}}|^{2}\right\rangle/2italic_E = ⟨ | bold_italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ / 2 and the enstrophyZ=⟨ω2⟩/2𝑍delimited-⟨⟩superscript𝜔22Z=\left\langle\omega^{2}\right\rangle/2italic_Z = ⟨ italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ / 2, where the brackets indicate the average over the domain.In the presence of forcing and dissipations the energy/enstrophy balancesread

d⁢Ed⁢t=−2⁢ν⁢Z−2⁢μ⁢E+⟨𝐯⋅𝐅⟩=−εν−εμ+εI𝑑𝐸𝑑𝑡2𝜈𝑍2𝜇𝐸delimited-⟨⟩⋅𝐯𝐅subscript𝜀𝜈subscript𝜀𝜇subscript𝜀𝐼\frac{dE}{dt}=-2\nu Z-2\mu E+\left\langle{\bf v}\cdot{\bf F}\right\rangle=-%\varepsilon_{\nu}-\varepsilon_{\mu}+\varepsilon_{I}divide start_ARG italic_d italic_E end_ARG start_ARG italic_d italic_t end_ARG = - 2 italic_ν italic_Z - 2 italic_μ italic_E + ⟨ bold_v ⋅ bold_F ⟩ = - italic_ε start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT - italic_ε start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_ε start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT(4)

and

d⁢Zd⁢t=−2⁢ν⁢P−2⁢μ⁢Z+⟨ω⁢f⟩=−ην−ημ+ηI𝑑𝑍𝑑𝑡2𝜈𝑃2𝜇𝑍delimited-⟨⟩𝜔𝑓subscript𝜂𝜈subscript𝜂𝜇subscript𝜂𝐼\frac{dZ}{dt}=-2\nu P-2\mu Z+\left\langle\omega f\right\rangle=-\eta_{\nu}-%\eta_{\mu}+\eta_{I}divide start_ARG italic_d italic_Z end_ARG start_ARG italic_d italic_t end_ARG = - 2 italic_ν italic_P - 2 italic_μ italic_Z + ⟨ italic_ω italic_f ⟩ = - italic_η start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT(5)

The different terms in (4-5) define thecharacteristic scales of forcing ℓf=2⁢π⁢εI/ηIsubscriptℓ𝑓2𝜋subscript𝜀𝐼subscript𝜂𝐼\ell_{f}=2\pi\sqrt{\varepsilon_{I}/\eta_{I}}roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = 2 italic_π square-root start_ARG italic_ε start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT / italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT end_ARG,viscous dissipation ℓν=2⁢π⁢εν/ηνsubscriptℓ𝜈2𝜋subscript𝜀𝜈subscript𝜂𝜈\ell_{\nu}=2\pi\sqrt{\varepsilon_{\nu}/\eta_{\nu}}roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 2 italic_π square-root start_ARG italic_ε start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT / italic_η start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT end_ARGand friction ℓμ=2⁢π⁢εμ/ημsubscriptℓ𝜇2𝜋subscript𝜀𝜇subscript𝜂𝜇\ell_{\mu}=2\pi\sqrt{\varepsilon_{\mu}/\eta_{\mu}}roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = 2 italic_π square-root start_ARG italic_ε start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT / italic_η start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_ARG. Whenthese scales are well separated ℓν≪ℓf≪ℓμmuch-less-thansubscriptℓ𝜈subscriptℓ𝑓much-less-thansubscriptℓ𝜇\ell_{\nu}\ll\ell_{f}\ll\ell_{\mu}roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≪ roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ≪ roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPTand in stationary conditionsone expects the development of a direct enstrophy cascade in the inertialrange of scales ℓν≤ℓ≤ℓfsubscriptℓ𝜈ℓsubscriptℓ𝑓\ell_{\nu}\leq\ell\leq\ell_{f}roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≤ roman_ℓ ≤ roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT and an inverse energycascade in the scalesℓf≤ℓ≤ℓμsubscriptℓ𝑓ℓsubscriptℓ𝜇\ell_{f}\leq\ell\leq\ell_{\mu}roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ≤ roman_ℓ ≤ roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT [35] (see Appendix A).

The central statistical object in the classical theory of turbulenceis the energy spectrum E⁢(k)𝐸𝑘E(k)italic_E ( italic_k ) defined as ∫E⁢(k)⁢𝑑k=E𝐸𝑘differential-d𝑘𝐸\int E(k)dk=E∫ italic_E ( italic_k ) italic_d italic_k = italic_E.In the range of scales ℓν≤ℓ≤ℓfsubscriptℓ𝜈ℓsubscriptℓ𝑓\ell_{\nu}\leq\ell\leq\ell_{f}roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ≤ roman_ℓ ≤ roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPTdissipative effects are negligible and can assume a constant flux ofenstrophy ΠZ⁢(k)subscriptΠ𝑍𝑘\Pi_{Z}(k)roman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ). This can be expressed in terms of the energyspectrum as [35]

ΠZ⁢(k)=λk⁢E⁢(k)⁢k3subscriptΠ𝑍𝑘subscript𝜆𝑘𝐸𝑘superscript𝑘3\Pi_{Z}(k)=\lambda_{k}E(k)k^{3}roman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ) = italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_E ( italic_k ) italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT(6)

where λksubscript𝜆𝑘\lambda_{k}italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the characteristic frequency of deformation of eddiesat the scale 1/k1𝑘1/k1 / italic_k. Dimensionally one has

λk2=∫kfkE⁢(p)⁢p2⁢𝑑psuperscriptsubscript𝜆𝑘2superscriptsubscriptsubscript𝑘𝑓𝑘𝐸𝑝superscript𝑝2differential-d𝑝\lambda_{k}^{2}=\int_{k_{f}}^{k}E(p)p^{2}dpitalic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∫ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_E ( italic_p ) italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_p(7)

where kf=2⁢π/ℓfsubscript𝑘𝑓2𝜋subscriptℓ𝑓k_{f}=2\pi/\ell_{f}italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = 2 italic_π / roman_ℓ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT is the wavenumber associated to the forcingand the upper limit in the integral reflects that scales smaller than1/k1𝑘1/k1 / italic_k act incoherently.A scale-free solution that gives a constant enstrophy fluxΠZ⁢(k)=ηsubscriptΠ𝑍𝑘𝜂\Pi_{Z}(k)=\etaroman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ) = italic_η gives E⁢(k)≃η2/3⁢k−3similar-to-or-equals𝐸𝑘superscript𝜂23superscript𝑘3E(k)\simeq\eta^{2/3}k^{-3}italic_E ( italic_k ) ≃ italic_η start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT.However, this is not consistent sincethis gives, when inserted in (7) and (6),a log-dependent enstrophy flux.In other words, a scale-independent flux is not compatible with apure power-law energy spectrum.The correction proposed by Kraichnan is to include a log-correctionin the spectrum [36]

E⁢(k)≃ην2/3⁢k−3⁢[ln⁡(k/kf)]−1/3similar-to-or-equals𝐸𝑘superscriptsubscript𝜂𝜈23superscript𝑘3superscriptdelimited-[]𝑘subscript𝑘𝑓13E(k)\simeq\eta_{\nu}^{2/3}k^{-3}\left[\ln(k/k_{f})\right]^{-1/3}italic_E ( italic_k ) ≃ italic_η start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 / 3 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT [ roman_ln ( italic_k / italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) ] start_POSTSUPERSCRIPT - 1 / 3 end_POSTSUPERSCRIPT(8)

which now gives a scale-independent enstrophy flux.

One important remark for the following is that the spectrum(8) gives a log-dependency of the frequencies λksubscript𝜆𝑘\lambda_{k}italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPTon the wavenumber. Therefore the direct cascade of enstrophyis at the border of locality in the sense that all the scalescontribute with the same rate to the transfer of enstrophy.

While viscous dissipation simply acts as a high-wavenumber cut-offof the constant enstrophy flux and the Kraichnan spectrum (8),the role of friction is more subtle as it produces a correction to theexponent of the spectrum and even the breakdown of self-similar scaling[40, bernard2000influence].This effect is strictly related to the non-local property of thecascade. Indeed in the presence of friction one can write a simpleexpression for the rate of enstrophy transfer [35]

d⁢ΠZ⁢(k)d⁢k=−μ⁢k2⁢E⁢(k)𝑑subscriptΠ𝑍𝑘𝑑𝑘𝜇superscript𝑘2𝐸𝑘\frac{d\Pi_{Z}(k)}{dk}=-\mu k^{2}E(k)divide start_ARG italic_d roman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ) end_ARG start_ARG italic_d italic_k end_ARG = - italic_μ italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_E ( italic_k )(9)

which states that part of the flux is removed in the cascade at arate proportional to the friction coefficient μ𝜇\muitalic_μ.Using now (6) with a constant deformation rateλk=λkfsubscript𝜆𝑘subscript𝜆subscript𝑘𝑓\lambda_{k}=\lambda_{k_{f}}italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT one immediately obtain the solution

E⁢(k)∼k−(3+ξ)similar-to𝐸𝑘superscript𝑘3𝜉E(k)\sim k^{-(3+\xi)}italic_E ( italic_k ) ∼ italic_k start_POSTSUPERSCRIPT - ( 3 + italic_ξ ) end_POSTSUPERSCRIPT(10)

with the correction in the power-law scaling

ξ=μλkf.𝜉𝜇subscript𝜆subscript𝑘𝑓\xi=\frac{\mu}{\lambda_{k_{f}}}\,.italic_ξ = divide start_ARG italic_μ end_ARG start_ARG italic_λ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG .(11)

A posteriori, the use of a constant deformation rate λkfsubscript𝜆subscript𝑘𝑓\lambda_{k_{f}}italic_λ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPTis justified by the fact that inserting (10) (with any ξ>0𝜉0\xi>0italic_ξ > 0)into (7) produces a deformation frequency which decreases with k𝑘kitalic_kand therefore kfsubscript𝑘𝑓k_{f}italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT is the most efficient scale in the transfer of enstrophy.

We remark that the above argument can be made more rigorousin the physical space where λkfsubscript𝜆subscript𝑘𝑓\lambda_{k_{f}}italic_λ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT is replaced bythe Lyapunov exponent of the smooth flow. By taking into accountit* finite-time fluctuations one predicts the breakdown of self-similarscaling and the production of intermittency in the statistics ofthe vorticity field [40] which has been observed in numerical simulations [37].

III Numerical simulations of the direct cascade with friction

We tested the prediction of the previous Section, and in particular thecorrection (10) to the energy spectrum in the presence of friction,by means of extensive direct numerical simulations of the 2D NS equations(2) at very high resolutionsusing a pseudo-spectral code implemented on Nvidia GPU usingthe directive-based programming model OpenACC.Some details about the code and its performances can be foundin AppendixA.

Simulations are done in a square box of size Lx=Ly=2⁢πsubscript𝐿𝑥subscript𝐿𝑦2𝜋L_{x}=L_{y}=2\piitalic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 2 italic_π,with regular grid N=Nx=Ny𝑁subscript𝑁𝑥subscript𝑁𝑦N=N_{x}=N_{y}italic_N = italic_N start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT with forcing correlation givenby (3).Three sets of simulations have been done with different resolutionand viscosity ν𝜈\nuitalic_ν, with different values of the friction coefficientμ𝜇\muitalic_μ in each set.

RunN𝑁Nitalic_Nν𝜈\nuitalic_νkf±Δ⁢kplus-or-minussubscript𝑘𝑓Δ𝑘k_{f}\pm\Delta kitalic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ± roman_Δ italic_kηIsubscript𝜂𝐼\eta_{I}italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPTR⁢eν𝑅subscript𝑒𝜈Re_{\nu}italic_R italic_e start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPTkmax⁢ℓνsubscript𝑘subscriptℓ𝜈k_{\max}\ell_{\nu}italic_k start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPTμ×102𝜇superscript102\mu\times 10^{2}italic_μ × 10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
A40962×10−52superscript1052\times 10^{-5}2 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT8±1plus-or-minus818\pm 18 ± 19.615655844.191,4,7,10,20,30,40,50,60,80
B81925×10−65superscript1065\times 10^{-6}5 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT16±1plus-or-minus16116\pm 116 ± 134.5601004633.384,6,10,20,30,40,50,60,80,100
C163841.25×10−61.25superscript1061.25\times 10^{-6}1.25 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT32±1plus-or-minus32132\pm 132 ± 1114.7501498772.776,12,18,36,48,60,72,96,120

We changed the forcing scale to allow, for the simulations at the highestresolution, the development of a narrow inverse cascade to study its effect onthe phenomenology of the direct cascade.Table 1 shows the most relevant parameters of our simulations.

Fig.1 shows some snapshots of the vorticity field taken fromnumerical simulations at different resolutions.The size of the largest vortices observed in the flow corresponds to theforcing scale which is reduced increasing the resolution, as indicatedin Table1. By zooming the runs by the factor correspondingto the different forcing scales, we see indeed that the vorticesare rescaled approximatively to the same scale.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (1)
Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (2)

The enstrophy balance (5) is shown in Fig.2 for allthe simulations in stationary conditions.Remarkably, this quantity is independent on theresolution (i.e. on the viscosity) and depends on the dimensionless parameterμ⁢ηI1/3𝜇superscriptsubscript𝜂𝐼13\mu\eta_{I}^{1/3}italic_μ italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT only. We see that for μ⁢ηI1/3≳0.2greater-than-or-equivalent-to𝜇superscriptsubscript𝜂𝐼130.2\mu\eta_{I}^{1/3}\gtrsim 0.2italic_μ italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT ≳ 0.2all the enstrophy in the cascade is removed by friction term.

Figure3 shows the time-averaged energy spectra forthe different simulations of the set C at the highest resolution.In all cases, in the direct cascade range, the spectrum displays a power-lawscaling steeper than k−3superscript𝑘3k^{-3}italic_k start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT and the steepness increases for largervalues of the friction coefficient.The runs with the smallest values of friction display a short inverse cascadeat wavenumber k<kf𝑘subscript𝑘𝑓k<k_{f}italic_k < italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT with an exponent close to thedimensional prediction k−5/3superscript𝑘53k^{-5/3}italic_k start_POSTSUPERSCRIPT - 5 / 3 end_POSTSUPERSCRIPT.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (3)
Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (4)

In order to measure the correction ξ⁢(μ)𝜉𝜇\xi(\mu)italic_ξ ( italic_μ ) to the scaling exponent wefitted E⁢(k)⁢k3𝐸𝑘superscript𝑘3E(k)k^{3}italic_E ( italic_k ) italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT with a pure power-law behavior. We found that this procedureis not very robust since it depends on the range of wavenumbers chosenfor the fit. Indeed, a non-power-law behavior is observed for wavenumberk≳kfgreater-than-or-equivalent-to𝑘subscript𝑘𝑓k\gtrsim k_{f}italic_k ≳ italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT, as it is evident from Fig.3.Moreover, we find that even for finite μ𝜇\muitalic_μ the relation (6)between the spectrum and the enstrophy flux requires the introductionof a logarithmiccorrection. This is evident from Fig.4 where we plot the ratioE(k)k3ln(k/kf)1/3/ΠZ(k)E(k)k^{3}\ln(k/k_{f})^{1/3}/\Pi_{Z}(k)italic_E ( italic_k ) italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_ln ( italic_k / italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT / roman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ) together withE⁢(k)⁢k3/ΠZ⁢(k)𝐸𝑘superscript𝑘3subscriptΠ𝑍𝑘E(k)k^{3}/\Pi_{Z}(k)italic_E ( italic_k ) italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT / roman_Π start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_k ) for run C. It is evident that while thelatter quantity is never constant, the incorporation of the logarithmicterm produces a constant ratio on the inertial range of scales.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (5)

We therefore measure the exponent correction ξ⁢(μ)𝜉𝜇\xi(\mu)italic_ξ ( italic_μ ) by a power-lawfit of the compensated energy spectra E(k)k3ln(k/kf)1/3E(k)k^{3}\ln(k/k_{f})^{1/3}italic_E ( italic_k ) italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_ln ( italic_k / italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT.The results are shown in Fig.5 for all our simulations.We find that the correction is indeed linear with the friction coefficientm⁢u𝑚𝑢muitalic_m italic_u, as predicted by (11) with a different slope for the differentsets of simulations.The deformation rate λkfsubscript𝜆subscript𝑘𝑓\lambda_{k_{f}}italic_λ start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT in (11) is dimensionally aninverse time and this suggests to plot the exponents of the different runsas a function of the dimensionless friction μ/ηI1/3𝜇superscriptsubscript𝜂𝐼13\mu/\eta_{I}^{1/3}italic_μ / italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT.In Fig.5 we see that indeed this produces an almost perfectcollapse of the data from the different runs.We remark that the above rescaling is not a unique possibility: Indeedan inverse time of the flow is also given by ην1/3superscriptsubscript𝜂𝜈13\eta_{\nu}^{1/3}italic_η start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT orZ1/2superscript𝑍12Z^{1/2}italic_Z start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT but we find that data collapse is observed with the proposedcompensation only.

IV Conclusions

In this paper, we offer a GPU-accelerated solution for high-resolution numerical simulations of generalized convection-diffusion equations in two dimensions. The use of a single GPU permits huge accelerations due to the absence of inter-GPU communication. This paradigm shift permitted an acceleration of about 40404040 times respectively to the CPU code. Since GPU memories are the huge bottleneck of the application, the best use for our code are problems that do not require too big spatial resolution but still need to solve many large time scales. One among many examples is the study of non-equilibrium fluctuation on SQG turbulence [47]. Indeed, the latter work used a preliminary version of the present code.

For our physical application, we studied the 2D NS equations in the presence of Ekman friction, revisiting low-resolution results on its energy spectra that exhibit a steeper power-law decay than predicted by dimensional analysis, consistent with the presence of intermittency. This steepness increases linearly with friction, aligning with the theoretical prediction E⁢(k)∼k−(3+ξ)similar-to𝐸𝑘superscript𝑘3𝜉E(k)\sim k^{-(3+\xi)}italic_E ( italic_k ) ∼ italic_k start_POSTSUPERSCRIPT - ( 3 + italic_ξ ) end_POSTSUPERSCRIPT, where ξ=2⁢μ/λ¯𝜉2𝜇¯𝜆\xi=2\mu/\bar{\lambda}italic_ξ = 2 italic_μ / over¯ start_ARG italic_λ end_ARG put forward by [37]. Furthermore, in our simulations, we permitted the existence of an inverse energy cascade that, in principle, should invalidate the arguments leading to the linear dependency of the correction on the friction parameter. For the latter, we notice that as long the turbulent state is not of a condensate, the linear prediction holds.

Regarding the infinite time Lyapunov exponent λ¯¯𝜆\bar{\lambda}over¯ start_ARG italic_λ end_ARG, our results show that it is the inverse of twice the large scale time τf=ηI−1/3subscript𝜏𝑓superscriptsubscript𝜂𝐼13\tau_{f}=\eta_{I}^{-1/3}italic_τ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 3 end_POSTSUPERSCRIPT, instead of the small scale quantities. Future investigations could delve deeper into the specific mechanisms through which large-scale dynamics influence λ¯¯𝜆\bar{\lambda}over¯ start_ARG italic_λ end_ARG by the proper estimation of the Cramer function, potentially shedding light on fundamental aspects of turbulence intermittency and the passiveness of vorticity at small scales. In this sense, adding the implementation of passive scalar, eulerian/lagrangian tracers, and possible multiple GPU implementations on the gTurbo code is likely to be developed in the near future.

Acknowledgement

This work has been supported by Italian Research Center on High Performance Computing Big Data andQuantum Computing (ICSC), project funded by European Union - NextGenerationEU -and National Recovery and Resilience Plan (NRRP) - Mission 4 Component 2 withinthe activities of Spoke 3 (Astrophysics and Cosmos Observations). We acknowledge HPC CINECA for computing resources within the INFN-CINECA GrantINFN24-FieldTurb.

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*

Appendix A Numerical integration of the 2D NS equation

To build numerical solvers for a broader class of turbulent models, we rewrite(2) in a more general formulation,

(∂t+ℒν,μn,m)⁢ω+J⁢(ω,ψ)=f,subscript𝑡superscriptsubscriptℒ𝜈𝜇𝑛𝑚𝜔𝐽𝜔𝜓𝑓(\partial_{t}+\mathcal{L}_{\nu,\mu}^{n,m})\omega+J(\omega,\psi)=f\ ,\ ( ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT + caligraphic_L start_POSTSUBSCRIPT italic_ν , italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , italic_m end_POSTSUPERSCRIPT ) italic_ω + italic_J ( italic_ω , italic_ψ ) = italic_f ,(12)

where we introduced a generalized linear dissipative operator,

ℒν,μn,m≡(−1)n⁢ν2⁢n⁢∇2⁢(n+1)+(−1)m⁢μ2⁢m⁢∇−2⁢m,superscriptsubscriptℒ𝜈𝜇𝑛𝑚superscript1𝑛subscript𝜈2𝑛superscript∇2𝑛1superscript1𝑚subscript𝜇2𝑚superscript∇2𝑚\mathcal{L}_{\nu,\mu}^{n,m}\equiv(-1)^{n}\nu_{2n}\nabla^{2(n+1)}+(-1)^{m}\mu_{%2m}\nabla^{-2m}\ ,\ caligraphic_L start_POSTSUBSCRIPT italic_ν , italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , italic_m end_POSTSUPERSCRIPT ≡ ( - 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_ν start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ∇ start_POSTSUPERSCRIPT 2 ( italic_n + 1 ) end_POSTSUPERSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT 2 italic_m end_POSTSUBSCRIPT ∇ start_POSTSUPERSCRIPT - 2 italic_m end_POSTSUPERSCRIPT ,(13)

representing a positive-diagonal operator in the Fourier spaceℒ^ν,μn,m⁢(k)=ν2⁢n⁢k2⁢(n+1)+μ2⁢m⁢k−2⁢msuperscriptsubscript^ℒ𝜈𝜇𝑛𝑚𝑘subscript𝜈2𝑛superscript𝑘2𝑛1subscript𝜇2𝑚superscript𝑘2𝑚\hat{\mathcal{L}}_{\nu,\mu}^{n,m}(k)=\nu_{2n}k^{2(n+1)}+\mu_{2m}k^{-2m}over^ start_ARG caligraphic_L end_ARG start_POSTSUBSCRIPT italic_ν , italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n , italic_m end_POSTSUPERSCRIPT ( italic_k ) = italic_ν start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT 2 ( italic_n + 1 ) end_POSTSUPERSCRIPT + italic_μ start_POSTSUBSCRIPT 2 italic_m end_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT - 2 italic_m end_POSTSUPERSCRIPT.Although this paper is devoted to the study of the direct cascade in 2DNS turbulence, the equation (12) contains a whole class ofturbulence models known as α𝛼\alphaitalic_α-turbulence [15].The definition of thisclass of model is better understood through the relation between thegeneralized vorticity ω⁢(𝒙,t)𝜔𝒙𝑡\omega({\bm{x}},t)italic_ω ( bold_italic_x , italic_t ) and the stream functionψ⁢(𝒙,t)𝜓𝒙𝑡\psi({\bm{x}},t)italic_ψ ( bold_italic_x , italic_t ), represented in the Fourier space through

ω^⁢(𝒌,t)=|𝒌|α⁢ψ^⁢(𝒌,t).^𝜔𝒌𝑡superscript𝒌𝛼^𝜓𝒌𝑡\hat{\omega}({\bm{k}},t)=|{\bm{k}}|^{\alpha}\hat{\psi}({\bm{k}},t)\ .\ over^ start_ARG italic_ω end_ARG ( bold_italic_k , italic_t ) = | bold_italic_k | start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT over^ start_ARG italic_ψ end_ARG ( bold_italic_k , italic_t ) .(14)

In the following, we will discuss the particular case α=2𝛼2\alpha=2italic_α = 2 butthe scheme can be adapted to any value of α𝛼\alphaitalic_α.

The generalized dissipative operator has the role discussed inSectionII, i.e. to provide stationary states and preventingcondensate formations. For m=n=0𝑚𝑛0m=n=0italic_m = italic_n = 0 one recovers the standardfriction/viscosity terms, while for m,n>0𝑚𝑛0m,n>0italic_m , italic_n > 0 depending on the orders n𝑛nitalic_n and m𝑚mitalic_mof the dissipative operator, the coefficients μ𝜇\muitalic_μ and ν𝜈\nuitalic_ν have differentdimensional roles and can dissipate over a more narrow range ofscales. For example, hyperviscosity (n>0𝑛0n>0italic_n > 0) is used to diminish the action ofdissipation on the dissipative subrange, leading to extended inertial ranges atthe cost of a bigger thermalization effect (bottleneck) of high wavenumber[38, 39].Moreover, one reason to introduce hypofriction (m>0𝑚0m>0italic_m > 0)instead of normal friction is to avoid the correction to theenstrophy cascade discussed in SectionII.

A.1 Pseudospectral GPU code

We developed and tested an original pseudospectral code to integratethe general model on Nvidia hardware.Pseudospectral schemes are widely used in numerical studies of turbulencebecause of their accuracy in derivatives and the simplicity to invertthe Laplace equation.Another practical advantage is that most of the resources in pseudospectralscheme is used to compute the Fast Fourier Transforms (FFT) necessary tomove back and forth from Fourier space (where derivatives are computed)to physical space (where products and other nonlinear terms are evaluated).Therefore, to make the code efficient for a given architecture,it is (almost) sufficient to have an efficient FFT.

The numerical code gTurbo2D uses a standard Runge-Kutta (RK) scheme totime advance the solution with exact integration of the linear terms.In the simple case of a second-order RK scheme, the evolution of thevorticity field in (12) is given by(…^^…\hat{...}over^ start_ARG … end_ARG represents the Fourier transform of the quantity)

ω^⁢(𝐤,t+d⁢t)=e−ℒ^⁢d⁢t⁢ω^⁢(𝐤,t)+e−ℒ^⁢d⁢t/2⁢N^⁢(e−ℒ^⁢d⁢t/2⁢ω^′)⁢d⁢t^𝜔𝐤𝑡𝑑𝑡superscript𝑒^ℒ𝑑𝑡^𝜔𝐤𝑡superscript𝑒^ℒ𝑑𝑡2^𝑁superscript𝑒^ℒ𝑑𝑡2superscript^𝜔′𝑑𝑡\hat{\omega}({\bf k},t+dt)=e^{-\hat{\mathcal{L}}dt}\hat{\omega}({\bf k},t)+e^{%-\hat{\mathcal{L}}dt/2}\hat{N}\left(e^{-\hat{\mathcal{L}}dt/2}\hat{\omega}^{%\prime}\right)dtover^ start_ARG italic_ω end_ARG ( bold_k , italic_t + italic_d italic_t ) = italic_e start_POSTSUPERSCRIPT - over^ start_ARG caligraphic_L end_ARG italic_d italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_ω end_ARG ( bold_k , italic_t ) + italic_e start_POSTSUPERSCRIPT - over^ start_ARG caligraphic_L end_ARG italic_d italic_t / 2 end_POSTSUPERSCRIPT over^ start_ARG italic_N end_ARG ( italic_e start_POSTSUPERSCRIPT - over^ start_ARG caligraphic_L end_ARG italic_d italic_t / 2 end_POSTSUPERSCRIPT over^ start_ARG italic_ω end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_d italic_t(15)

where

ω^′=ω^⁢(𝐤,t)+N^⁢(ω^⁢(𝐤,t))⁢d⁢t/2superscript^𝜔′^𝜔𝐤𝑡^𝑁^𝜔𝐤𝑡𝑑𝑡2\hat{\omega}^{\prime}=\hat{\omega}({\bf k},t)+\hat{N}\left(\hat{\omega}({\bf k%},t)\right)dt/2over^ start_ARG italic_ω end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = over^ start_ARG italic_ω end_ARG ( bold_k , italic_t ) + over^ start_ARG italic_N end_ARG ( over^ start_ARG italic_ω end_ARG ( bold_k , italic_t ) ) italic_d italic_t / 2(16)

The evaluation of the nonlinear term N^^𝑁\hat{N}over^ start_ARG italic_N end_ARG is partially done inthe physical space (in order to avoid the computation of convolutions).In the present implementation of the code the evaluation of the nonlinearterm is done as follows. From the vorticity field in Fourier space thecode computes the stream function by inverting (14).The two components of the velocity v^isubscript^𝑣𝑖\hat{v}_{i}over^ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are then obtained fromthe derivatives of ψ^^𝜓\hat{\psi}over^ start_ARG italic_ψ end_ARG and then transformed in the physicalspace together with the vorticity (this step requires 3 inverse FFTs).The products (vi⁢ω)subscript𝑣𝑖𝜔(v_{i}\omega)( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ω ) are computed (and stored in the samearrays of the velocity) and transformed back in Fourier space(this requires 2 direct FFTs). Finally the divergence of (vi⁢ω)^^subscript𝑣𝑖𝜔\hat{(v_{i}\omega)}over^ start_ARG ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ω ) end_ARGis computed and stored in the original array.Therefore the evaluation of the nonlinear term requires 5 FFTs and eachstep of the n-order RK scheme requires 5⁢n5𝑛5n5 italic_n FFTs.

A.2 Code implementation and Performance tests

The code gTurbo2D is written in Fortran 90 with OpenACC, which enablesthe use of Nvidia hardware through compiler directives.Simulations are done on Leonardo machine, a pre-exascale Tier-0supercomputer where,each of the 3456 computing nodes is composed of a single-socket processor of32-core at 2.60GHz, 512 GB of RAM and, 4 Nvidia A100 GPUs of 64GB eachconnected by NVLink 3.0.The version of gTurbo2D used for this work is a single GPU code while themulti-GPU version is under development.We remark that the study of 2D turbulence requires much less memory than3D (a single scalar field in two dimensions) and the remarkable resolutionof N2=327682superscript𝑁2superscript327682N^{2}=32768^{2}italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 32768 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT grid points can be reached on a single GPU.However, large resolutions require very small time steps and thereforethe resolution is limited not only by the memory but also by thespeed of the code.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (6)
Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (7)

The left panel of Figure 2 shows the total GPU memory usage in Gb(circles) as a function of the resolution N𝑁Nitalic_N. For moderate resolutionN≲2000less-than-or-similar-to𝑁2000N\lesssim 2000italic_N ≲ 2000 the memory usage is almost independent of the resolutionsince most of the memory is used to store the libraries, the kernel, andthe resolution-independent variables. For larger resolutions, the memoryused to store the 2D fields dominates and therefore it is proportional toN2superscript𝑁2N^{2}italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.Remarkably, we observe a similar behavior for the mean elapsed time.This can be explained by the relative smallness of the problem comparedto the GPU parallelization capacity. Indeed, not all the registers on theGPUs are required to fully parallelize the computation which thereforeis performed in a time independent of the resolution. For larger resolution,the computational time grows proportionally to the amount of computationrequired for the time step, i.e. to N2superscript𝑁2N^{2}italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (8)

Figure 7 shows the percentage of time spent on each step of the timeintegration, step 7 was not included since it is a simple repetition of steps 1to 6. One should note that the most computationally intensive part is due tothe forward and backward FFTs (steps 2 and 4) that account for more than 75%percent7575\%75 %of the computational time. However, the importance of the forward and backwardtransforms is different since their subroutines are called with differentfrequencies. Besides, we decided to move the normalization to the forwardtransform since it has fewer calls per timestep. Although the integratorstability depends intrinsically on the physical properties of the system inquestion, we observed some practical advantages of using RK4EL in some testedcases. For α=1𝛼1\alpha=1italic_α = 1, the governing equations represent a model known as theSurface Quasi Geostrophic (SQG) model[30, 41], which has important applicationson atmospheric [27] and ocean flows[2]. On the one hand, the RK2EL scheme is roughly twiceas fast as RK4EL since it requires half the number of repetitions. On the otherhand, for N=8192𝑁8192N=8192italic_N = 8192, we were able to increase the timestep by a factor of almost5 using the fourth-order scheme. For a simulation with fixed physical timeT=Nt⁢d⁢t𝑇subscript𝑁𝑡𝑑𝑡T=N_{t}dtitalic_T = italic_N start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_d italic_t, one can have a speedup of approximately 2.52.52.52.5 on the totalsimulation time.

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions (2024)

FAQs

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions? ›

Spectrum correction on Ekman-Navier-Stokes equation in two-dimensions. It has been long known that the addition of linear friction on two-dimensional Navier-Stokes (NS) turbulence, often referred to as Ekman-Navier-Stokes (ENS) turbulence, induces strong intermittent fluctuations on small-scale vorticity.

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What is 2D Navier-Stokes equation? ›

The continuity equation of 2D incompressible steady-state flow in a differential form can be written as: The 2D Navier–Stokes equations explain the momentum conservation of incompressible fluid. There are two momentum equations corresponding to the velocity components in the x and y directions of 2D flow.

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Why is Navier-Stokes equations unsolvable? ›

The Navier–Stokes equations are nonlinear because the terms in the equations do not have a simple linear relationship with each other. This means that the equations cannot be solved using traditional linear techniques, and more advanced methods must be used instead.

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What is the significance of the Navier-Stokes equation? ›

The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing.

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What is the Navier-Stokes equation for viscous flows? ›

The Navier-Stokes equation is the dynamical equation of fluid in the classical fluid mechanics. M ρ τ M ∂ 2 ∂ t 2 v = - ∇ p + η ∇ 2 v + 1 3 η + ζ ∇ ∇ · v .

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What is the continuity equation for two dimensions? ›

Short Answer. Answer: The continuity equation for steady two-dimensional flow with constant properties is ∂ u ∂ x + ∂ v ∂ y = 0 , where: 1.

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What are the limitations of the Navier-Stokes equation? ›

The fundamental limitation of the Navier-Stokes equation follows from the fact that this is a dynamic equation. Therefore, this equation is structured so that the projections of its left and right sides onto the binormal direction of the trajectory of the fluid particle are always identically zero.

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What are the 7 unsolvable equations? ›

The seven problems are the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equations, P versus NP, the Poincaré Conjecture, the Riemann Hypothesis, and the Yang-Mills Theory.

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What is the hardest unsolvable math equation? ›

World's Most Puzzling Unsolved Math Problems
  1. Riemann Hypothesis. ...
  2. Birch and Swinnerton-Dyer Conjecture. ...
  3. Hodge Conjecture. ...
  4. Navier-Stokes Equations. ...
  5. Yang-Mills Existence and Mass Gap. ...
  6. P vs NP Problem. ...
  7. Collatz Conjecture.

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Is Navier Stokes a millennium problem? ›

The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré ...

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What are the real life applications of Navier-Stokes equation? ›

Applications of Navier Stokes Equations

They could be applied to model ocean currents, weather, air flow around wings, and the flow of water in pipes.

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What is the conclusion of the Navier-Stokes equation? ›

11.9 Conclusion

In this section we have derived the famous Navier-Stokes equation which is the equation for the conservation of momentum. As this equation is a vector equation, it will give rise to a total of three equations that we require for solving the field variables.

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How accurate are Navier-Stokes equations? ›

tl:dr The Navier-Stokes equations are only as good as the constitutive equation used to describe the fluid. By picking a constitutive equation, (like that the stress is directly proportional to the rate of strain) you are limiting yourself to the fluids that are well-described by that constitutive eq…

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What are the assumptions for the Navier-Stokes equation? ›

In order to apply this to the Navier–Stokes equations, three assumptions were made by Stokes: The stress tensor is a linear function of the strain rate tensor or equivalently the velocity gradient. The fluid is isotropic. For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure results).

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What is the simplest form of the Navier-Stokes equation? ›

For an incompressible fluid, the density is constant. Setting the derivative of density equal to zero and dividing through by a constant ρ, we obtain the simplest form of the equation ∇⋅→v=0.

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How do you reduce Navier-Stokes theorem? ›

The reduction of the Navier-Stokes equation to the form ∂2u/∂x2 + ∂(uv)/∂y = -1/ρ * ∂p/∂x + ν * (∂2u/∂y2) can be obtained by combining the x-component and y-component equations and using the continuity equation.

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Navier–Stokes equations - WikipediaWikipediahttps://en.wikipedia.org ›

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and ...
The motion of an incompressible Newtonian fluid with viscosity μ, density ρ, and subject to hydrostatic pressure p and uniform gravity with acceleration g can b...
The Navier-Stokes equations are a set of partial differential equations describing the motion of viscous fluid substances, deriving from Newton's second law...

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What is the equation for 2d incompressible flow? ›

X-component of velocity in a 2-D incompressible flow is given by u=y2+4xy.

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What is the Navier-Stokes equation simply explained? ›

What Are the Navier-Stokes Equations? Navier-Stokes equations are partial differential equations that govern the motion of incompressible fluids. These equations constitute the basic equations of fluid mechanics. The movement of fluid in the physical domain is driven by various properties.

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What is the equation for Navier Stokes in 1d? ›

So, (ρ, u)(t, x) = (ρ0(x + t), (1 + g(t))((ρ0(t))/(ρ0(x + t))) − 1) is the solution to compressible Navier-Stokes equations (1) and (2) with the initial data (3) and boundary condition (4).

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