Statistically relevant conserved quantities for truncated quasigeostrophic flow

Abstract

Systematic applications of ideas from equilibrium statistical mechanics lead to promising strategies for assessing the unresolved scales of motion in many problems in science and engineering. A scientific debate over more than the last 25 years involves which conserved quantities among the formally infinite list are statistically relevant for the large-scale equilibrium statistical behavior. Here this important issue is addressed by using suitable discrete numerical approximations for geophysical flows with many conserved quantities as a numerical laboratory. The results of numerical experiments are presented here for these truncated geophysical flows with topography in a suitable regime. These experiments establish that the integrated third power of potential vorticity besides the familiar constraints of energy, circulation, and enstrophy (the integrated second power) is statistically relevant in this regime for the coarse-grained equilibrium statistical behavior at large scales. Furthermore, the integrated higher powers of potential vorticity larger than three are statistically irrelevant for the large-scale equilibrium statistical behavior in the examples studied here.

Geophysical flows provide an important prototype context for developing methods for underresolved models in science and engineering. The wide range of unresolved scales of motion with highly inhomogeneous behavior for the atmosphere and ocean on Earth as well as for other planets such as Jupiter require statistical strategies to assess the nontrivial impact of these unresolved features on the larger scales. Systematic applications of ideas from equilibrium statistical mechanics to these inhomogeneous geophysical flows have lead to promising statistical strategies for the ocean (1–4), the atmosphere (5, 6), and the giant planets such as Jupiter (7) in agreement with contemporary observations. These different equilibrium statistical theories all attempt to predict the coarse-grained behavior at large scales through use of some of the formally infinite list of conserved quantities for idealized geophysical flow; this has been a topic of continuous debate over more than the last 25 years in the fluid dynamics community (8–14) and represents the main topic of the present work. Here the results of numerical experiments are presented for truncated geophysical flows with topography in a suitable regime, which establish that the integrated third power of potential vorticity besides the familiar constraints of energy, circulation, and enstrophy is statistically relevant for the coarse-grained equilibrium statistical behavior at large scales. Furthermore, in this regime of fluid motion the integrated higher powers of potential vorticity larger than three are statistically irrelevant for the large-scale equilibrium statistical behavior. The simplest geophysical model used here is barotropic two-dimensional flow with topography in periodic geometry, which is described by the equations

In Eqs. 1, q is the potential vorticity, ω = Δψ is the relative vorticity, v→ is the incompressible fluid velocity, ψ is the stream function, and h is the prescribed topography. When h is 0, Eqs. 1 become the equations for two-dimensional incompressible flow. Nonzero topography often has profound impact on the large-scale flow (11–14). Here and below we assume a 2π-periodic geometry in both the x and y variables, which are also denoted by x₁, x₂ whenever convenient. Eqs. 1 conserve kinetic energy,

as well as the infinite number of conserved quantities,

In Eqs. 2 and 3, ∫_T²⋅ denotes integration over the period domain in periodic geometry, where the total circulation, Q₁, satisfies Q₁ ≡ 0. The quadratic conserved quantity, the enstrophy

is singled out in some statistical theories for large-scale flow (8, 10–12, 14, 15) as having special significance, whereas the higher generalized enstrophies, Q_p(q), for p ≥ 3 are ignored in these theories. Other researchers (16) claim that the entire infinite list of conserved quantities in Eq. 3 is statistically significant for describing the coarse-grained features at large scales. A third group (17, 18) invokes the central role of point vortex dynamics with three conserved quantities in large-scale statistical behavior but with Q_p(q) formally infinite for p ≥ 2. Finally, a fourth statistical approach has evolved recently (2, 3, 7, 19) where the energy and circulation are imposed on the large-scale flow, whereas a suitable prior distribution is used to encode the potential vorticity fluctuations at small scales. These various statistical theories, their relative strength and weakness, new statistical approaches, and potential applications are discussed by Majda and Wang (13) in a recent monograph that contains many additional references and discussion. One way to address these issues is to study the statistical behavior of discrete approximations to Eqs. 1 conserving the energy in Eq. 2, and in addition other discrete approximations to the generalized enstrophies in Eq. 3. This is the main topic in subsequent sections of this article.

Traditional Spectral Truncation and Equilibrium Statistical Theory

The potential vorticity, q, in Eqs. 1 is truncated by projection onto (2M + 1) × (2M + 1) Fourier modes by the projection operator

The traditional aliased truncation of the quasigeostrophic equations in Eqs. 1 for these Fourier coefficients q̂_k, |k₁| ≤ M, |k₂| ≤ M, is given by

In Eq. 6 the finite domain of the Fourier coefficients is extended to (4M + 1) × (4M + 1) by using the periodicity rule q̂_k = q̂_k+2M+1. It is well known (10, 12, 13, 20) that the traditional spectral truncation in Eq. 6 conserves energy, circulation, and enstrophy from Eqs. 2–4 but in general none of the higher-order invariants in Eq. 3 for p ≥ 3; this last fact has been demonstrated in recent numerical experiments (20, 21). The energy–enstrophy statistical theory that utilizes only these two conserved quantities for the truncation in Eq. 6 predicts a Gaussian probability measure for the equilibrium statistical behavior of the Fourier modes (10–13). Recall that Gaussian measures are uniquely characterized by their mean and variance. Given externally prescribed mean values for the energy and enstrophy, the mean of this probability distribution for q̂_k is determined by the linear equation

while there is equipartition of pseudoenergy of perturbations around the mean state in Eq. 7 with a constant variance α > 0. Here the pseudoenergy Fourier coefficients, p̂_k, are given by

with ω̂_k being the Fourier coefficients of relative vorticity. The constants, μ and α, completely characterizing the Gaussian measure, are determined by the values of E and ℰ. A wide variety of numerical experiments with the truncation in Eq. 6 (12, 13, 15) confirm the predictions of the equilibrium statistical theory from Eqs. 7 and 8. However, these results cannot address the fundamental issue for the statistical relevance of the higher-order invariants in Eq. 3 for p ≥ 3, because they fail to be conserved by the approximation in Eq. 6 over long time intervals of integration. Here, the equilibrium statistical predictions through Eqs. 7 and 8 are regarded as a null hypothesis in the tests reported below for the role of the higher-order invariants.

Spectral Truncation with Many Conserved Quantities

Here, following the important observation by Zeitlin (22), we consider the sine-bracket truncation as an approximation to the quasigeostrophic dynamics in Eqs. 1 through the spectral approximation in Eq. 6. This finite dimensional set of equations for the Fourier coefficients is given by

with the same convention of periodicity as in Eq. 6. The truncation in Eqs. 9 conserves the energy in Eq. 2 as the Hamiltonian. In addition, the sine-bracket truncation in Eqs. 9 conserves 2M invariants of the form

The Casimir invariant C₂ is a multiple of the enstrophy, because A(i, −i) = 0. However, the higher Casimir invariants C_N for 3 ≤ N ≤ 2M are suitable regularizations of those in Eq. 3 for 3 ≤ p ≤ 2M. A pedagogical detailed discussion of Eqs. 9 and 10 as well as related numerical issues used below can be found in refs. 13, 20, and 21. In particular, to guarantee conservation of the invariants in Eqs. 10 within round-off error under time discretization, McLachlan's (23) symplectic integrator is combined with a suitable version of second-order accurate Strang splitting in time (20, 21). This last modification of the basic algorithm in ref. 23 is crucial for accurate energy conservation and results in a 10⁻⁴ decrease in relative errors for the energy with the time steps used below for only twice the computational expense.

Numerical Experiments Demonstrating the Statistical Relevance of C₃(q) at Large Scales

In the numerical experiments reported here, the prescribed values of the initial data were chosen such that the energy and enstrophy were fixed at E = 7 and ℰ = 20, whereas the value of the third invariant, C₃(q) from Eqs. 10, normalized through Ĉ₃(q) = C₃(q)/ℰ^3/2, varied through the four values, Ĉ₃ = 0, 2, 4, and 6. A standard constrained optimization problem was set up to find initial data with many Fourier modes satisfying these constraints (21). The higher-order invariants, C_N, for 4 ≤ N ≤ 2M were not specified by predetermined values for the initial data, although they are automatically conserved within round-off error for the algorithm in Eqs. 9. The values of energy and enstrophy, E = 7 and ℰ = 20, guarantee a suitable “negative temperature regime” (11–13) for the energy–enstrophy equilibrium statistical theory summarized in Eqs. 7 and 8 with μ ≅ −0.9 and α ≅ 20. The numerical experiments using Eqs. 9 reported below use M = 11 such that there are 528 active Fourier modes and 22 nontrivial conserved quantities in the dynamics. In all numerical experiments reported below, the equations in Eqs. 9 were calculated for times of order 10⁴ with a time step, Δt = 0.01 with a maximum relative error in energy of 1.2·10⁻⁶. Statistics were gathered through long-time averaging of individual solutions with

denoting the averaging window. After several initial experiments, it was determined that initial times T₀ with T₀ ≥ 10³ were sufficiently long to begin the time-averaging procedure. Three different cases of barotropic geophysical flow from Eqs. 1 with the discretization from Eqs. 9 were studied with the above parameters for the initial data: No topography, large-scale random topography with nonzero Fourier modes at |k|² = 2, 4, and the deterministic layered topography, h = 0.2 cos(x) + 0.4 cos(2x). This last case has the advantage in display that the energy–enstrophy theory in Eqs. 7 and 8 predicts a large-scale mean flow that is also layered, so ψ̄(x) is a function of the x variable alone, for any value of μ; thus, significant departures from such a layered structure in x in stream-function plots provide strong visual evidence for departures in the large-scale statistical predictions beyond the energy–enstrophy theory. Most of the results are reported below for this case to take advantage of this fact. Because the focus here is on the statistical relevance of the higher-order invariants in Eqs. 3 or 10 for equilibrium statistical theories, a higher-order invariant is called statistically relevant at large scales if it affects single-point spatial statistics such as the large-scale mean, energy spectrum, and probability distribution function (PDF) centered about the mean. For example, the effect of the higher-order invariants for Eqs. 9 on two-point spatial correlations is ignored here. With all of this background, next results are reported below for the mixing properties of the system, the energy spectrum, the mean stream function, the mean potential vorticity, and the PDF for potential vorticity.

Mixing and Decay of Temporal Correlations

The normalized temporal correlation functions averaged over the two groups of Fourier modes |k|² = 1 and |k|² = 2 are depicted in Fig. 1 for the four cases with varying Ĉ₃ values for layered topography. These are the correlation functions with the slowest decay rate among all the groups of 528 Fourier modes with significantly faster decay for time correlations of the higher modes not depicted here. Obviously, the longest tails for the correlations occur for |k|² = 1. The two graphs in Fig. 1 each have four curves with the cases Ĉ₃ = 0 demonstrating the most rapid decay of correlations, Ĉ₃ = 2 and 4 exhibiting clear decay of correlations, and the case Ĉ₃ = 6 with |k|² = 1 showing marginal decay of correlations to <15% of the value with zero lag at time lags of 10³. The correlation function in this case eventually decays to 0 with time lags of order 2 × 10³ but the similar plot for |k|² = 1 with Ĉ₃ = 8, not depicted here, exhibits no decay of correlations. Thus the value Ĉ₃ = 6 is near the boundary of the parameter regime with decay of temporal correlations. The decay of correlations of Fourier modes is the most primitive test for mixing in phase space of an individual solution (13, 24–26) and justifies the use of the time averages in Eq. 11 in calculating statistics of the mean flow and energy spectrum. The other cases with no topography and random topography have similar behavior in the decay of temporal correlations (21).

Fig 1.

Averaged temporal correlations for |k|² = 1 and |k|² = 2 for varying values of skewness, Ĉ₃ = 0, 2, 4, 6.

Large-Scale Mean Flow

Here the large-scale mean flow is calculated from the numerical solution by processing the time average of the stream function, ψ̄ = 〈ψ〉_τ and q̄ = 〈q〉_τ at individual spatial points and comparing the results. If the energy and enstrophy are the only important conserved quantities, then the energy–enstrophy statistical theory in Eq. 7 predicts that 〈ψ〉_τ and 〈q〉_τ should become increasingly collinear as τ increases. This is quantified by measuring the graphical correlation between 〈ψ〉_τ and 〈q〉_τ, defined by

with (f, g)₀ = ∫_T² fg, |f| = (f, f). Note that solutions of the energy–enstrophy statistical theory in Eq. 7 with μ < 0 are collinear and satisfy C(ψ̄_μ, q̄_μ) = −1. In Fig. 2, the behavior of the graphical correlation C(〈ψ〉_τ, 〈q〉_τ) is depicted as a function of the averaging window, τ, in Eq. 11 for 10³ ≤ τ ≤ 10⁴ for the four cases with Ĉ₃ = 0, 2, 4, and 6 for the layered topography. The graphical correlation for the case with Ĉ₃ = 0 clearly converges to the value −1, whereas the graphical correlations for the other cases with nonzero third invariant clearly level off for the largest values of τ and clearly do not asymptote to −1; increasing values of Ĉ₃ lead to increasing values of the graphical correlation. Fig. 2 provides quantitative evidence that the coarse-grained large-scale mean flows do not satisfy the predictions of the energy–enstrophy statistical theory unless the invariant Ĉ₃ vanishes identically. To confirm this, in Fig. 3 we present scatter plots of the large-scale mean stream function 〈ψ〉_τ and mean potential vorticity, 〈q〉_τ, for the largest averaging window, τ = 10⁴, from Fig. 2 for the four cases with varying Ĉ₃ values. For the case with Ĉ₃ = 0 in Fig. 3, the scatter plot shows collinear behavior, whereas the scatter plots for the cases with increasing skewness display a coherent nonlinear relationship between ψ̄ and q̄ with increasing curvature as Ĉ₃ increases through the values 2, 4, and 6. Fig. 3 provides powerful evidence for the nontrivial effect of the third invariant, Ĉ₃, on the large-scale coarse-grained statistical mean flow. This nontrivial effect of Ĉ₃ is clearly evident in the contour plots for the coarse-grained stream functions depicted in Fig. 4, plotted in each case over one period interval in x and y. For Ĉ₃ = 0 the stream function is clearly layered in y as predicted by the energy–enstrophy statistical theory. This information from Fig. 4 combined with that in Figs. 2 and 3 for the case with Ĉ₃ = 0 indicates that as regards the coarse-grained mean flow, the energy–enstrophy statistical theory predicts the behavior; thus, with Ĉ₃ = 0, the other 19 conserved quantities, Ĉ_N, 4 ≤ N ≤ 22 in Eqs. 10 are statistically irrelevant for predicting the large-scale mean flow. On the other hand, the streamline contours in Fig. 4 b–d indicate stronger, morelocalized regions of closed stream lines associated with negative values of the stream function for the cases with Ĉ₃ = 2, 4, and 6. Recall that closed stream lines with negative stream function (pressure) correspond to positive (cyclonic) relative vorticity. Thus, the stream line contours in Fig. 4 indicate the presence of coherent structures through positive cyclonic vortices for positive values of the regularized third moment Ĉ₃; for the case with Ĉ₃ = 2, there are both cyclonic and anticyclonic vortices, whereas for the cases with Ĉ₃ = 4 and 6, there are only cyclonic vortices in the mean stream function with increasingly weaker anticyclonic flow without coherent vortices. This is further strong evidence demonstrating the statistical relevance of Ĉ₃ in determining the coarse-grained large-scale mean flow.

Fig 2.

The graphical correlation, C(〈ψ〉_τ, 〈q〉_τ), as a function of the averaging window, τ; Ĉ₃ = 0, 2, 4, 6.

Fig 3.

The scatter plots q̄ vs. ψ̄ for the 23 × 23 sine-bracket truncation, layered topography, Ĉ₃ = 0, 2, 4, 6.

Fig 4.

The contour plots of the mean stream function, 23 × 23 sine-bracket truncation, layered topography, Ĉ₃ = 0, 2, 4, 6.

The Energy Spectrum

Recall that the energy–enstrophy theory predicts the equipartition of the pseudoenergy variables p̂_k defined in Eq. 8 for perturbations of the mean flow in Eq. 7. In Fig. 5 the energy spectrum of the pseudoenergy variables p̂_k from Eq. 8 is plotted for perturbations of the mean flow calculated from the numerical output as in Figs. 2–4 above for the three cases with no topography, random topography, and layered topography with Ĉ₃ varying for 0, 4, and 6. It is remarkable that this energy spectrum is virtually identical for the three different large-scale topographies and fixed values of Ĉ₃ with E = 7 and ℰ = 20; this indicates a potentially universal feature of the spectrum for fluctuations that depends on Ĉ₃ but is independent of the large-scale mean flow. In particular, for the case in Fig. 5 with Ĉ₃ = 0, equipartition of pseudoenergy is confirmed as predicted by the energy–enstrophy statistical theory; this is a further numerical confirmation of the statistical irrelevance of the 19 higher-order invariants Ĉ_N, 4 ≤ N ≤ 22, for the energy spectrum. The cases with Ĉ₃ = 4 and 6 in Fig. 5 clearly demonstrate the statistical relevance of this invariant in determining the energy spectrum with more pseudoenergy concentrated at small wave numbers and lower pseudoenergy at large wave numbers in a statistically identical fashion for the three cases with varying topography.

Fig 5.

The pseudoenergy spectrum, 23 × 23 sine-bracket truncation, different topographies, Ĉ₃ = 0, 4, 6. The pseudoenergy spectrum does not depend on topography.

PDF of Potential Vorticity

Although the mean stream function emphasizes the role of Ĉ₃ in determining the coarse-grained large-scale flow, the PDF for potential vorticity highlights the effect of Ĉ₃ on the small-scale fluctuations. Here this PDF is determined by a standard bin-counting algorithm (27) as time evolves with evaluation at a grid of spatial points. The PDF of potential vorticity is depicted in Fig. 6 for the case with layered topography and Ĉ₃ = 0, 2, 4, and 6. The variance and skewness of this PDF for varying Ĉ₃ values are recorded in Table 1. The main trends apparent from Fig. 6 and Table 1 is the nontrivial increase in the positive skewness of the PDF for potential vorticity as Ĉ₃ increases coupled with the nearly Gaussian PDF for potential vorticity for Ĉ₃ = 0, which is predicted from the energy–enstrophy statistical theory. Once again this last result suggests the statistical irrelevance of the invariants Ĉ_N, 4 ≤ N ≤ 22 in the regime with Ĉ₃ = 0 for E = 7 and ℰ = 20.

Fig 6.

The PDFs of the potential vorticity for Ĉ₃ = 0, 2, 4, and 6. Note the increasing skewness in PDFs as Ĉ₃ increases.

Table 1.

Variance and skewness of PDF of the potential vorticity

Ĉ3	Variance	Skewness
0	65.67	−1.37·10−2
2	56.14	0.3114
4	50.82	0.7752
6	48.08	1.441

Equilibrium Statistical Predictions of the Nonlinear Mean State

The graphs in Figs. 3 and 6 indicate that increasing the third invariant Ĉ₃ systematically results in simultaneously an increasing skewness in the PDF for potential vorticity, which represents small-scale fluctuations and increasing nonlinearity in the scatter plot of mean stream function versus mean potential vorticity reflecting large-scale behavior. A recently developed equilibrium statistical theory (2, 3, 7, 13, 19) begins with the PDF for the potential vorticity reflecting small-scale fluctuations as a prior distribution and predicts the equilibrium coarse-grained large-scale mean flow in the continuum limit through a specific large-scale functional relation, q̄ = G(ψ̄). Here the nonlinear function G depends on the large-scale energy and circulation constraints in a specific fashion (2, 3, 7, 13, 19). In particular, in ref. 7 it is shown that a skewed prior distribution for small-scale potential vorticity fluctuations given by the centered gamma distribution with mean zero, variance σ, and skewness ɛ yields the equilibrium statistical prediction for the large-scale mean flow,

In Eq. 13, θ and γ are Lagrange multipliers to satisfy the energy and zero circulation constraints. This modeling strategy successfully predicts the large-scale coherent spots on Jupiter from the small-scale gamma prior distribution motivated by recent observations from the Galileo mission (7). Here we address the much more limited issue: If one approximates the skewed PDFs for potential vorticity in Fig. 6 by the centered gamma distributions with the same skewness and flatness in Table 1 as Ĉ₃ values vary, are there values of the Lagrange multipliers θ, γ, such that the nonlinear large-scale q̄ ↔ ψ̄ relation in Eq. 13 fits the data in Fig. 3? The answer is yes, and the results are the curved lines overlayed on Fig. 3. Although these results are encouraging, the reader is warned that they are not a confirmation of the equilibrium statistical theory directly: If θ and γ are used as Lagrange multipliers to match the energy, E = 7, and zero circulation, the vorticity–stream relations show significantly less curvature, and the stream functions are significantly nonlinear but without closed stream lines as depicted in Fig. 4. This is not surprising, because simulations with 23 × 23 Fourier modes are extremely far from the continuum limit that requires strict spatial scale separation between the large-scale flow and the small-scale potential vorticity fluctuations (19, 28). Nevertheless, such trends are already evident in Figs. 3 and 6, Table 1, and Eq. 13.

Summary Discussion

Through unambiguous numerical experiments with the discrete approximation in Eqs. 9 with many conserved quantities, it has been demonstrated that the third nonlinear invariant, Ĉ₃, is statistically relevant at large scales for truncated quasigeostrophic flow with topography. The statistical relevance of Ĉ₃ has been demonstrated for the mean stream function and mean potential vorticity, the pseudoenergy spectrum, and the PDF for potential vorticity in a self-consistent fashion. For example, systematic increases in Ĉ₃ lead to a systematic increase in nonlinearity for the mean flow with large-scale coherent positive vortices as well as a systematic increase in the positive skewness of the PDF for potential vorticity. In contrast, numerical simulations not depicted here (see ref. 21) with the traditional spectral truncation in Eq. 6 with only the energy and enstrophy as conserved quantities reproduce the equilibrium statistical predictions in Eqs. 7 and 8 for any of the initial values of Ĉ₃. Also, the simulations with Eqs. 9 reported above strongly suggest that the other 19 conserved quantities, Ĉ_N, 4 ≤ N ≤ 22, are statistically irrelevant when Ĉ₃ = 0 in the regime studied here. The issues of statistically relevant conserved quantities have been studied recently by us and others in a much more complete fashion for a simpler model involving suitable Galerkin truncations of the Burgers–Hopf equations (20, 24, 25, 27) including comparison and prediction with equilibrium microcanonical Monte Carlo simulations. We hope that the present contribution as well as this recent work inspires other theoreticians to address the fundamental issues of statistically relevant conserved quantities for geophysical flows.

Abbreviations

• PDF, probability distribution function