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Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(Δt)^2$ leapfrog finite difference scheme run backward in time

Alfred S. Carasso

Abstract

For the 2D incompressible Navier-Stokes equations, with given hypothetical non smooth data at time $T > 0 $that may not correspond to an actual solution at time $T$, a previously developed stabilized backward marching explicit leapfrog finite difference scheme is applied to these data, to find initial values at time $t = 0$ that can evolve into useful approximations to the given data at time $T$. That may not always be possible. Similar data assimilation problems, involving other dissipative systems, are of considerable interest in the geophysical sciences, and are commonly solved using computationally intensive methods based on neural networks informed by machine learning. Successful solution of ill-posed time-reversed Navier-Stokes equations is limited by uncertainty estimates, based on logarithmic convexity, that place limits on the value of $T > 0$. In computational experiments involving satellite images of hurricanes and other meteorological phenomena, the present method is shown to produce successful solutions at values of $T > 0$, that are several orders of magnitude larger than would be expected, based on the best-known uncertainty estimates. However, unsuccessful examples are also given. The present self-contained paper outlines the stabilizing technique, based on applying a compensating smoothing operator at each time step, and stresses the important differences between data assimilation, and backward recovery, in ill-posed time reversed problems for dissipative equations. While theorems are stated without proof, the reader is referred to a previous paper, on Navier-Stokes backward recovery, where these proofs can be found.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(Δt)^2$ leapfrog finite difference scheme run backward in time

Abstract

For the 2D incompressible Navier-Stokes equations, with given hypothetical non smooth data at time that may not correspond to an actual solution at time , a previously developed stabilized backward marching explicit leapfrog finite difference scheme is applied to these data, to find initial values at time that can evolve into useful approximations to the given data at time . That may not always be possible. Similar data assimilation problems, involving other dissipative systems, are of considerable interest in the geophysical sciences, and are commonly solved using computationally intensive methods based on neural networks informed by machine learning. Successful solution of ill-posed time-reversed Navier-Stokes equations is limited by uncertainty estimates, based on logarithmic convexity, that place limits on the value of . In computational experiments involving satellite images of hurricanes and other meteorological phenomena, the present method is shown to produce successful solutions at values of , that are several orders of magnitude larger than would be expected, based on the best-known uncertainty estimates. However, unsuccessful examples are also given. The present self-contained paper outlines the stabilizing technique, based on applying a compensating smoothing operator at each time step, and stresses the important differences between data assimilation, and backward recovery, in ill-posed time reversed problems for dissipative equations. While theorems are stated without proof, the reader is referred to a previous paper, on Navier-Stokes backward recovery, where these proofs can be found.

Paper Structure

This paper contains 10 sections, 4 theorems, 59 equations, 6 figures.

Table of Contents

  1. Introduction
  2. 2D Navier-Stokes equations in the unit square
  3. An example: the 2004 Hurricane Ivan stream function
  4. Uncertainty in backward in time Navier-Stokes reconstruction
  5. Stabilized leapfrog scheme for the 2D Navier-Stokes Equations
  6. Fourier stability analysis in linearized problem
  7. Application to data assimilation
  8. Leapfrog nonlinear computational instability and the RAW filter
  9. Nonlinear computational experiments
  10. Concluding Remarks

Key Result

Lemma 1

Let $\lambda_{j,k},~\sigma_{j,k},$ be as in Eq. (eq:1.0023), and let $~g_{j,k}$ be as in Eq. (eq:3.003). Choose a positive integer $J$ such that if $\lambda_J=4 \pi^2 \nu J$, we have With $p > 1,$ choose $\gamma \geq 4(\lambda_J)^{1-p}$ in Eq. (eq:1.0023). Then, Hence, and, for $n=1,2,\cdots,N$, Therefore, with this choice of $(\gamma, p)$, the linear leapfrog scheme in Eq. (eq:3.0019), is unc

Figures (6)

  • Figure 1: Non-smooth Hurricane Ivan data leads to challenging Navier-Stokes data assimilation problem. Here, with $\nu=0.01,~U_{max}=111, ~RE=11,100,$ and $\sup_{\Omega}\{|\omega(x,y)|\approx 1.5\times10^5$, theoretical uncertainty estimates in Eq. (\ref{['eq:3c']}) indicate that useful identification of initial values, given the above hypothetical data at some $T > 0$, might not be feasible for $T > 1.0 \times 10^{-9}$.
  • Figure 2: As hypothetical data at various times $T > 0$, leftmost column features the August 2016 NASA Suomi NPP Satellite image of hurricanes Madeline and Lester approaching Hawaii. Middle column presents the corresponding initial values, obtained by solving the 2D Navier-Stokes equations backward in time. Rightmost column presents the forward evolution to time $T$ of these initial values, obtained by solving the same equations forward in time. Assimilation is successful if rightmost image is a useful approximation to leftmost image.
  • Figure 3: As hypothetical data at various times $T > 0$, leftmost column features the September 1999 NASA Landsat 7 image of a von Karman vortex street in clouds off the Juan Fernandez islands. Middle column presents corresponding initial values, obtained by solving the 2D Navier-Stokes equations backward in time. Rightmost column presents the forward evolution to time $T$ of these initial values, obtained by solving the same equations forward in time. Assimilation is successful if rightmost image is a useful approximation to leftmost image.
  • Figure 4: As hypothetical data at various times $T > 0$, leftmost column features the September 2004 NASA Aqua Satellite image of Hurricane Ivan off Florida coast. Middle column presents the corresponding initial values, obtained by solving the 2D Navier-Stokes equations backward in time. Rightmost column presents the forward evolution to time $T$ of these initial values, obtained by solving the same equations forward in time. Assimilation is successful if rightmost image is a useful approximation to leftmost image.
  • Figure 5: As hypothetical data at various times $T > 0,$ leftmost column features the 1951 USAF Resolution Chart. Middle column presents the corresponding initial values, obtained by solving the 2D Navier-Stokes equations backward in time. Rightmost column presents the forward evolution to time $T$ of these initial values, obtained by solving the same equations forward in time. Assimilation is successful if rightmost image is a useful approximation to leftmost image.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2