Inferring flow parameters and turbulent configuration with physics-informed data assimilation and spectral nudging

Inferring physical parameters of turbulent flows by assimilation of data measurements is an open challenge with key applications in meteorology, climate modeling, and astrophysics. Up to now, spectral nudging was applied for empirical data assimilation as a means to improve deterministic and statistical predictability in the presence of a restricted set of field measurements only. Here we explore under which conditions a nudging protocol can be used for two objectives: to unravel the value of the physical flow parameters and to reconstruct large-scale turbulent properties starting from a sparse set of information in space and in time. First, we apply nudging to quantitatively infer the unknown rotation rate and the shear mechanism for turbulent flows. Second, we show that a suitable spectral nudging is able to reconstruct the energy containing scales in rotating turbulence by using a blind setup, i.e., without any input about the external forcing mechanisms acting on the flow. Finally, we discuss the broad potentialities of nudging to other key applications for physics-informed data assimilation in environmental or applied flow configurations.

Figure 1

Diagram showing the setup of our numerical experiments. First, a reference simulation is performed (left). Second, a subset of data is filtered out of the reference field by keeping only data on a given subset of points in space and instant in times (middle). Third, we interpolate in time the input partial information and use it to nudge the evolution of a new field to reconstruct the missing data and to infer the correct physics parameters (right).

Figure 2

Nudging with different rotation rates. Simulations from setup INFER1 (see Table 1). Two-dimensional slices of the vorticity field $\mathbf{\omega }=\mathbf{\nabla }\times \mathbf{u}$ in the direction parallel to the rotation axis are shown for (a) the reference simulation with a rotation frequency ${\mathrm{\Omega }}_{\mathrm{ref}}$ and three nudged simulations performed with (b) $\mathrm{\Omega }=0$, (c) $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{ref}}$, and (d) $\mathrm{\Omega }=2{\mathrm{\Omega }}_{\mathrm{ref}}$. (e) Energy spectra of the reference simulation compared with error spectra $E_{\mathrm{\Delta }}\left(k\right)$ [see Eq. (4)] for different values of the rotation frequency $\mathrm{\Omega }$. All spectra were computed at the same instants of time. The shaded gray area indicates the modes where the nudging is acting. (f) Time evolution of $E(k,t)$ for $k=5$ for $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{ref}}$ and $\mathrm{\Omega }=0$, compared to the reference data. (g) Evolution of total energy for the reference field and for the three nudged simulations at changing $\mathrm{\Omega }$.

Figure 3

(a) Value of the mean error committed to reconstruct the reference field in the nudged window $C$ for two different scans of the phase-space parameters. Blue circles show the case with a fixed stirring mechanism and changing rotation rate $\mathrm{\Omega }$ (setup INFER1 in Table 1). Green triangles show the case with fixed rotation rate and changing intensity of the stirring parameter $\mathcal{S}$ (setup INFER2 in Table 1). Magenta squares show a further scan for $\mathrm{\Omega }$ following INFER1 but nudging all wave numbers up to $k=10$. In all cases a clear depth is measured only when the scanning values correspond to the ones used for the reference data ${\mathrm{\Omega }}_{\mathrm{ref}}$ and ${\mathcal{S}}_{\mathrm{ref}}$, respectively. Error bars for each data point were calculated by measuring the standard deviation of $C$. (b) Scan of $\mathrm{\Omega }$ performed without nudging but adding the forcing term (i.e., same as INFER1 but with $\alpha =0$ and $\mathcal{S}={\mathcal{S}}_{\mathrm{ref}}$).

Figure 4

Nudging for the case of rotating turbulence in the inverse energy cascade regime. Simulations are from setup PHYS1 (see Table 1). The nudged window is given by the gray area between $k_0=8$ and $k_1=20$. The reference spectrum $E_{\mathrm{ref}}$ and the nudged spectrum $E_u$ almost coincide for $k>8$, making it hard to discern between the two. Both the intensity of the forcing $\mathcal{S}$ and the rotation rate $\mathrm{\Omega }$ are zero in the nudged simulation, so all energy injection and anisotropic effects are coming from the nudging term. Notice the strongly reduced error spectrum $E_{\mathrm{\Delta }}\left(k\right)$ for a large set of wave numbers, indicating an optimal reconstruction quality.

Figure 5

(a) Probability density functions of the pointwise kinetic energy for the reference simulation $|{\mathbf{u}}_{\mathrm{ref}}{\left(\mathbf{x}\right)|}^2$ (solid black line), the nudged simulation ${\left|\mathbf{u}\left(\mathbf{x}\right)\right|}^2$ (circles), and the nudging input field $|{\mathcal{I}}_V{\mathbf{u}}_{\mathrm{ref}}{\left(\mathbf{x}\right)|}^2$ (triangles) for the inverse cascade experiment (setup PHYS1 in Table 1). (b)–(d) Two-dimensional slices of planes perpendicular to the rotation axis of the absolute velocity fields.