Source
Parish and Duraisamy (2017), Physical Review Fluids 2: 014604, LIT-1259, read in full in NOTE-973 from arXiv:1611.03311v2. The closure is the paper’s §II.B, Eqs. 20–26, which it credits to Stinis (2006, 2012). The derivations below are the record’s.
What was actually shown
Setting. The paper’s Mori–Zwanzig identity (its Eq. 10) splits the evolution of a resolved variable φ_j into a Markovian term, a noise term and the memory integral
M_j(t) = ∫₀ᵗ e^{(t−s)L} P L e^{sQL} Q L φ₀ⱼ ds.
Its projection is P f(φ̂₀, φ̃₀) = f(φ̂₀, 0): the unresolved modes are set to zero (a Gaussian initial density in the zero-variance limit). Write g_j^{(m)}(t) = e^{tL} P L (QL)^{m+1} φ₀ⱼ, which needs only the resolved state at time t; g^{(0)} is the “first-order term” (its Eq. 31 for Burgers, Eq. 36 for Navier–Stokes).
Step 1: the closure. Define w_j^{(0)}(t) as M_j restricted to lags in [0, τ₀] (its Eq. 21, m = 0). Differentiating (its Eq. 22, with τ₀ constant) gives
dw^{(0)}/dt = g^{(0)}(t) − B(t) + w
where B(t) = e^{(t−τ₀)L} P L e^{τ₀QL} Q L φ₀ⱼ is the integrand at the far end of the window, which needs the orthogonal dynamics, and w^{(1)} is the same construction one order up. The trapezoidal rule on [t − τ₀, t] gives w^{(0)} ≈ (τ₀/2)(g^{(0)} + B), so B ≈ 2w^{(0)}/τ₀ − g^{(0)}. Substituting,
dw^{(0)}/dt = −(2/τ₀) w^{(0)} + 2 g^{(0)}(t) + w
the paper’s Eq. 26 at m = 0. The same steps one order up give Eq. 26 for each m, with τ_m and g^{(m)}. The hierarchy is closed by setting w^{(m+1)} = 0.
Step 2: it is a finite Markovian augmentation. After truncation, the reduced model is dφ̂/dt = R̂(φ̂) + w^{(0)} together with m + 1 linear relaxation equations, one per order, for each resolved mode. Every right-hand side depends only on the current resolved state and the current auxiliary variables. So the state (φ̂, w^{(0)}, …, w^{(m)}) of dimension (m + 2)N evolves Markovianly. That is the “coarse variable plus k memory coordinates” of QUESTION-222, with k = (m + 1)N.
Step 3: the kernel it implies. At m = 0 with w^{(1)} = 0 and w^{(0)}(0) = 0 (the paper’s initially resolved runs), the relaxation equation solves to
w^{(0)}(t) = 2 ∫₀ᵗ e^{−2(t−s)/τ₀} g^{(0)}(s) ds.
So the closure replaces the true memory by an exponential kernel of decay rate 2/τ₀ and amplitude 2 applied to the history of the first-order term. The kernel’s total weight is ∫₀^∞ 2e^{−2u/τ₀} du = τ₀.
Step 4: the quasi-steady limit is a rescaled t-model. If g^{(0)} changes little over a time τ₀, then w^{(0)}(t) → τ₀ g^{(0)}(t) for t ≫ τ₀. The paper’s t-model (its Eq. 20) is w = t g^{(0)}(t). The first-order finite-memory model therefore relaxes to the t-model with the elapsed time t replaced by the memory length τ₀. That is consistent with the paper’s finding that the t-model removes too much energy at late times: its coefficient grows without limit, while this one saturates. The paper does not draw this connection, though its Table 1 lists the t-model’s constant as “τ₀ = t”.
Step 5: the zero-lag amplitude is wrong for every τ₀. The exact memory integrand at lag zero is e^{tL} P L Q L φ₀ⱼ = g^{(0)}(t), with coefficient 1. The closure’s kernel at lag zero is 2. No choice of τ₀ changes this, because the factor 2 comes from the trapezoidal weights, not from τ₀.
Step 6: a linear test. For ẋ = a₁₁x + a₁₂y, ẏ = a₂₁x + a₂₂y with a₂₂ < 0 and y(0) = 0, the projection gives g^{(0)} = a₁₂a₂₁ x(t), and the exact memory is a₁₂a₂₁ ∫₀ᵗ e^{a₂₂s} x(t − s) ds: one exponential, with amplitude 1 and rate |a₂₂|. The augmentation w′ = a₂₂w + a₁₂a₂₁x (w is a₁₂y) is exact with one coordinate. The closure’s kernel is 2a₁₂a₂₁e^{−2s/τ₀}. It matches the exact kernel’s total weight when τ₀ = 1/|a₂₂|, and then decays at twice the true rate. No τ₀ matches both amplitude and rate. Checked numerically with (a₁₁, a₁₂, a₂₁, a₂₂) = (−0.5, 1, −2, −3) and x(0) = 1 over t ∈ [0, 6]. The exact one-coordinate augmentation has zero error. The finite-memory closure’s maximum error in x is 4.7 × 10⁻² at τ₀ = 1/3, and 3.5 × 10⁻² at its best τ₀ ≈ 0.30. The t-model’s is 1.2 × 10⁻¹. A Markovian closure with the exact kernel’s total weight, x′ = (a₁₁ + a₁₂a₂₁/|a₂₂|)x, gives 9.8 × 10⁻².
What could have failed: the closure could have reproduced the exact kernel in the linear case where one memory coordinate suffices, and Step 6 would then have been a check that its form is right. It does not. Its form is fixed by the quadrature.
What this does not say
- Not that the closure’s error is bounded. The paper justifies dropping w^{(m+1)} only “if the support or magnitude of the integrand decreases with the repeated application of QL”, and gives no bound. The trapezoidal error is stated as O(τ₀²), with no constant. τ₀ is fitted to DNS for Burgers and set to 0.1 for Navier–Stokes. So this is a Markovian augmentation without the controlled prediction error QUESTION-222 asks for.
- Not that the true memory has finite support. The paper’s linear analysis (its Eqs. 15–19) shows that the memory integrand decays like e^{Λs} for the eigenvalues Λ of the unresolved block. Exponential decay is not finite support. “Finite memory” is an approximation the paper motivates, not a property it shows.
- Not that it is a statistical closure. Under the zero-variance projection, P is evaluation at φ̃ = 0. Then P commutes with every function of the state, so the paper’s “casual” commuting of the nonlinear Markovian term is exact, and the projected equation is just the full dynamics started from a fully resolved state. Nothing is averaged. The closures approximate one deterministic trajectory’s memory. Applied to flows whose unresolved modes are not zero at the start (the paper’s isotropic turbulence and channel runs), they are used outside the setting they were derived in.
- Nothing about which coarse variable to use. The cutoff is a Fourier wavenumber chosen by the modeller, and the paper calls the memory “numerical memory, as opposed to physical memory”, whose length depends “both on the physics of the problem at hand and the level of coarse-graining”.
Connections
- QUESTION-222: answers in part. It asks for a class in which the reduced process has a finite Markovian augmentation with bounded prediction error, with k and the error in terms of the eliminated dynamics’ spectrum. This shows the turbulence literature’s working answer has the shape asked for (k = (m + 1)N memory coordinates) and lacks the bound, and that its kernel form comes from a quadrature rule, not from the spectrum. A5’s linear example in the question is Step 6’s system. There the exact augmentation is trivial (keep y), and the finite-memory closure still misses it.
- The augmenting coordinate is the subgrid force. w^{(0)} is the closure’s estimate of i k τ^{SGS}, the subgrid force on each resolved mode (the paper’s Fig. 3 caption). In physical space (its Appendix C, Eq. 46, pressure neglected), the divergence of the subgrid stress becomes a field that relaxes at rate 2/τ₀, driven by the resolved field and the Leonard stress. A5’s “additional internal memory coordinate” is here a named physical quantity, the stress the unresolved scales exert.
- THEORY-257. Dynamical independence conditions on the coarse variable’s own history, so a coarse variable with memory need not fail it. This closure gives one way such a history could be summarised by a finite state. Whether the record’s criteria should be applied to φ̂ or to (φ̂, w) is what QUESTION-222 asks, and this account does not decide it.
- Chorin, Hald and Kupferman (LIT-1245, LIT-1253, registered unread) are the optimal-prediction source of the projection and of the t-model. Their readings may say whether a bound exists for any class.