LANS-α: Turbulence Closure Model

• LANS-α is an α-filtered turbulence model that preserves H¹ circulation while modifying subfilter-scale energy transfers.
• It uses an inverse Helmholtz filter to generate dual spectral behaviors, with k⁻¹ scaling in active regions and k⁺¹ in rigid bodies.
• Compared to Clark-α and Leray-α, LANS-α better captures intermittency and high-order statistics, though at the expense of spectral contamination in rigid regions.

The Lagrangian-Averaged Navier-Stokes-α (LANS-α) model is a turbulence closure and regularization approach for the incompressible Navier-Stokes equations, derived via Lagrangian averaging that preserves key physical invariants while introducing a smoothing operator of scale α. Within the large eddy simulation (LES) paradigm, LANS-α delivers a filtered velocity field that governs momentum while suppressing subfilter-scale (SFS) interactions, thereby altering the energy transfer and intermittency properties at small scales. The model stands out for its strict H¹-α circulation conservation, a mechanism that produces localized regions of rigid body-like flow and affects the spectral scaling and statistical features of turbulence. In magnetohydrodynamics (MHD), the extension to LAMHD-α protects superfilter-scale spectral fidelity and intermittent structure, benefiting from Lorentz-force-induced disruption of rigid body artifacts (Graham et al., 2010).

1. Subfilter-Scale Dynamics and the Helmholtz Filter

The LANS-α model begins with a spatial filtering of the velocity field via the inverse Helmholtz operator on a length scale α, resulting in a smoothed (filtered) velocity: $\overline{v} = (1 - \alpha^2 \nabla^2)^{-1} v$ This operator performs a convolution-like smoothing, penalizing spatial gradients at scales less than α. The Lagrangian-averaged perspective assumes that fluctuations below α are advected passively ("frozen" with the large-scale flow), which suppresses the transfer of energy through local SFS interactions:

• In the inertial range just below the filter, local SFS triad interactions are strongly diminished.
• Energy transfer remains possible via nonlocal interactions, but the direct cascade to smallest scales is impeded.
• This leads, for resolved wavenumbers $k \sim 1/\alpha$, to an energy spectrum in active regions scaling as $E_\alpha(k) \sim k^{-1}$.
• However, the lack of subfilter coupling enables the development of "frozen" regions (see §2), where spectral contamination occurs as $E_\alpha(k) \sim k^{+1}$.

This modification selectively changes the superfilter-scale energy distribution and the role of small-scale dynamics without introducing explicit eddy viscosity.

2. Circulation Conservation and Rigid Body Formation

A hallmark of LANS-α is its modified Kelvin circulation theorem:

• Circulation is conserved in the $H^1_\alpha$ (filtered) norm rather than the usual $L^2$ norm.
• The mathematical form ensures that, absent viscosity, small-scale circulation is "frozen in" and cannot be destroyed or reconfigured by turbulent interactions.
• This absolute circulation conservation enforces rigid body patches at subfilter scales. In these regions, the longitudinal velocity increments vanish:

$$\delta\overline{v}_\parallel(l) = 0 \implies E_\alpha(k) \sim k^{+1}$$

• Rigid bodies act as inert dynamical units: they have no internal scale-dependent transfer of energy and thus deviate from Kolmogorov-type cascade behavior.
• The global spectrum becomes a mixture: $k^{-1}$ scaling in "active" domains, $k^{+1}$ in rigid bodies, with contamination of the practical superfilter-scale energy distribution.

This phenomenon is a direct outcome of the stringency of the $H^1_\alpha$ circulation law, distinguishing LANS-α from other closures.

3. Comparison to Clark-α and Leray-α: SFS Physics and Stability

The Clark-α and Leray-α closures are obtained as truncations of LANS-α:

• Their subfilter stress tensors omit parts of the nonlinearity, leading to a relaxation of the circulation constraint.
• Consequently, these models do not manifest rigid body formation or $k^{+1}$ spectral artifacts.
• Instead, the superfilter spectrum is governed by the predicted $k^{-1}$ scaling in regions where cascade is allowed and no artificial contamination arises.
• While all models are stable at high Reynolds number, LANS-α best reproduces the statistical intermittency (see §4).
• The trade-off is that Clark-α and Leray-α achieve "cleaner" spectra at the expense of accurate representation of high-order statistics and some conservation properties.

The core difference is summarized in the presence or absence of strict small-scale circulation conservation: | Model | Small-Scale Circulation | Rigid Bodies | Intermittency | Superfilter Spectrum | |------------|------------------------|--------------|--------------|---------------------| | LANS-α | Strict (H¹-α norm) | Yes | High | $k^{-1}$ & $k^{+1}$ | | Clark-α | Not strict | No | Weaker | $k^{-1}$ only | | Leray-α | Not strict | No | Weaker | $k^{-1}$ only |

4. Statistical Properties: Intermittency and Comparison to DNS

Intermittency, describing the "bursty" small-scale velocity fluctuations characteristic of turbulence, is captured in detail by LANS-α:

• DNS of Navier-Stokes turbulence reveals strong intermittency at small scales, often quantified by high-order structure functions.
• LANS-α, despite rigid body artefacts, most accurately recovers the scaling exponents and extreme-event statistics of the full system.
• In contrast, Clark-α and Leray-α, while spectrally unpolluted, underpredict these statistics, evidencing a limitation in modeling fidelity.
• Thus, LANS-α is, in the sense of high-order moments and rare events, closest to direct Navier-Stokes dynamics, though it pays a price in the spectral domain.

5. MHD Extension: LAMHD-α Mechanisms

The LAMHD-α model generalizes LANS-α to MHD via a parallel filtering of the magnetic field:

• The Lorentz force $j \times b$ introduces sources/sinks of circulation and facilitates both nonlocal (large$\to$small) and local SFS interactions.
• Circulation is no longer strictly conserved (even in the filtered norm), which prevents frozen-in rigid bodies.
• The result is a high-fidelity prediction of the superfilter energy spectra and intermittent structures (notably "current sheets" in MHD) even at high Re.
• LAMHD-α offers improved large-eddy simulation capabilities for MHD compared to under-resolved DNS, combining accurate spectral shape and intermittency capture.

This fundamentally distinguishes MHD regularizations (where additional forces break strict conservation) from their hydrodynamic counterparts.

6. Mathematical Framework

The essential model structure can be summarized by the following system for the incompressible case: $$\begin{align*} &\partial_t v + (\omega \times \overline{v}) = -\nabla \pi + \nu \nabla^2 v &\quad (1)\ &\nabla \cdot v = \nabla \cdot \overline{v} = 0\ &\overline{v} = (1 - \alpha^2 \nabla^2)^{-1} v &\quad (2)\ &\omega = \nabla \times v\ &\delta \overline{v}_\parallel(l) = 0 \text{ in rigid bodies} \implies E_\alpha(k) \sim k^1 &\quad (4)\ &\text{Elsewhere in active regions: } E_\alpha(k) \sim k^{-1} \end{align*}$$ For LES interpretations, the SFS stress tensor for LANS-α is: $$\overline{\tau}^\alpha_{ij} = (1 - \alpha^2 \nabla^2)^{-1} \alpha^2 [\partial_m \overline{v}_i \partial_m \overline{v}_j + \partial_m \overline{v}_i \partial_j \overline{v}_m - \partial_i \overline{v}_m \partial_j \overline{v}_m]$$ This set of equations specifies the flow of momentum coupled with a filtered velocity operand and encodes the selection of SFS physics that define the model’s unique properties.

7. Practical Implications and Model Selection

The LANS-α model is particularly instrumental for scenarios requiring high-order statistical accuracy and modeling of intermittent, bursty events at moderate computational cost relative to full DNS. However, the spurious emergence of rigid body regions and the resulting $k^1$ spectral contamination can limit its physical applicability, especially in sensitive spectral regions. Related truncation models (Clark-α, Leray-α) trade intermittency fidelity for spectral purity.

For MHD flows, the LAMHD-α extension avoids the main limitation—rigid body formation—by virtue of the Lorentz force. This leads to both improved spectral prediction and accurate intermittent structure at high Reynolds numbers, suggesting utility for MHD-LES applications where capturing both global statistics and small-scale intermittency is critical.

In summary, the LANS-α framework is a circulation-preserving, α-filtered regularization that induces a duality in spectral and statistical behavior, making it a powerful, though nuanced, tool for hydrodynamic and magnetohydrodynamic turbulence modeling (Graham et al., 2010).