High-wavenumber Modes Research Articles

Extraction of reflected waves from acoustic logging data using variation mode decomposition and curvelet transform

Extraction of reflected waves from acoustic logging data using variation mode decomposition and curvelet transform

Self-propulsion of a droplet induced by combined diffusiophoresis and Marangoni effects.

Chemically active droplets display complex self-propulsion behavior in homogeneous surfactant solutions, often influenced by the interplay between diffusiophoresis and Marangoni effects. Previous studies have primarily considered these effects separately or assumed axisymmetric motion. To understand the full hydrodynamics, we investigate the motion of a two-dimensional active droplet under their combined influences using weakly nonlinear analysis and numerical simulations. The impact of two key factors, the Péclet number ( ) and the mobility ratio between diffusiophoretic and Marangoni effects ( ), on droplet motion is explored. We establish a phase diagram in the space, categorizing the boundaries between four types of droplet states: stationary, steady motion, periodic/quasi-periodic motion, and chaotic motion. We find that the mobility ratio does not affect the critical for the onset of self-propulsion, but it significantly influences the stability of high-wavenumber modes as well as the droplet's velocity and trajectory. Scaling analysis reveals that in the high regime, the Marangoni and diffusiophoresis effects lead to distinct velocity scaling laws: and , respectively. When these effects are combined, the velocity scaling depends on the sign of the mobility ratio. In cases with a positive mobility ratio, the Marangoni effect dominates the scaling, whereas the negative diffusiophoretic effect leads to an increased thickness of the concentration boundary layer and a flattened scaling of the dropletvelocity.

Instabilities of rotating inclined buoyancy layers.

The boundary layer near a cooled inclined plate, which is immersed in a stably stratified fluid rotating about an axis parallel to the direction of gravity, is a model for katabatic flows at high latitudes. In this paper the base flow of such an inclined buoyancy layer is solved analytically for arbitrary Prandtl numbers. By applying linear stability analyses, five unstable modes are identified for both the fixed temperature and the isoflux boundary conditions, i.e., the stationary longitudinal roll (LR) mode, the oblique roll with low streamwise wave-number (OR-1) and high streamwise wave-number (OR-2) modes, and the Tolmien-Schlichting (TS) wave with low streamwise wave-number (TS-1) and high streamwise wave-number (TS-2) modes. It is indicated that the Coriolis effect induced by the rotation leads the critical modes to be three dimensional, and a larger tilt angle of the plate and stronger Coriolis effect cause both TS wave modes to be more unstable for both thermal boundary conditions. When the Coriolis effect is considered, the OR-1 and OR-2 modes are the most unstable mode at low and high tilt angles, respectively, but the TS-1 wave mode may be the most unstable one when the plate is nearly vertical. In addition, the spanwise phase velocities of the TS wave modes change directions as the tilt angle passes some threshold values for both thermal boundary conditions except for the TS-1 wave mode with a fixed temperature boundary condition, which propagates in the same spanwise direction for all explored tilt angles.

Ultra-light axions and the S 8 tension: joint constraints from the cosmic microwave background and galaxy clustering

We search for ultra-light axions as dark matter (DM) and dark energy particle candidates, for axion masses 10-32 eV ≤ m a ≤ 10-24 eV, by a joint analysis of cosmic microwave background (CMB) and galaxy clustering data — and consider if axions can resolve the tension in inferred values of the matter clustering parameter S 8. We give legacy constraints from Planck 2018 CMB data, improving 2015 limits on the axion density Ωa h 2 by up to a factor of three; CMB data from the Atacama Cosmology Telescope and the South Pole Telescope marginally weaken Planck bounds at m a = 10-25 eV, owing to lower (and theoretically-consistent) gravitational lensing signals. We jointly infer, from Planck CMB and full-shape galaxy power spectrum and bispectrum data from the Baryon Oscillation Spectroscopic Survey (BOSS), that axions are, today, < 10% of the DM for m a ≤ 10-26 eV and < 1% for 10-30 eV ≤ m a ≤ 10-28 eV. BOSS data strengthen limits, in particular at higher m a by probing high-wavenumber modes (k < 0.4h Mpc-1). BOSS alone finds a preference for axions at 2.7σ, for m a = 10-26 eV, but Planck disfavours this result. Nonetheless, axions in a window 10-28 eV ≤ m a ≤ 10-25 eV can improve consistency between CMB and galaxy clustering data, e.g., reducing the S 8 discrepancy from 2.7σ to 1.6σ, since these axions suppress structure growth at the 8h -1 Mpc scales to which S 8 is sensitive. We expect improved constraints with upcoming high-resolution CMB and galaxy lensing and future galaxy clustering data, where we will further assess if axions can restore cosmic concordance.

The water structure around chloride ion investigated from D2O ↔ H2O substitution effect

• Chlorine-correlated OD/OH stretch bands are investigated under H↔D substitution from 293 to 573 K. • A flexible water structure exists in the Cl − hydration shell. • H↔D substitution raises the non-tetrahedral hydrating-water configurations. The Raman spectra for (D 2 O + H 2 O)−NaCl solutions ( n H2O / n D2O = 0/1, 1/4, 1/1, 4/1 and 1/0) were measured at temperatures from 293 to 573 K. We further discuss the water structure in the solutions based on the solute-correlated (SC) OD/OH stretch bands obtained from the Multivariate Curve Resolution method. The most prominent effect of H↔D substitution is raising the relative intensity of the high-wavenumber modes. The ∼2660 cm −1 /∼3635 cm −1 mode can become the main peak when isotopic substitution ratio is 80% at temperatures over 493 K. The spectral features suggest a flexible water structure in the Cl − hydration shell where most hydrating water molecules are engaged in hydrogen bonding configurations of single donor (SD) and single hydrogen-bond water (SHW). The increase of isotopic substitution ratio leads to the structural transition of SD→SHW→FW (free water) so that FW is greatly increased at isotopic substitution ratio of 80% and temperatures over 493 K. Moreover, the estimated hydrogen bond number for hydrating water indicates a stronger D 2 O−Cl − interaction than H 2 O−Cl − interaction in the hydration shell.

Instability of a dusty vortex

We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M&gt;0$ . Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $m$ becomes unstable. This critical mass loading drops to zero as $m$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.

Generalised quasilinear approximations of turbulent channel flow. Part 2. Spanwise triadic scale interactions

Continuing from Part 1 (Hernándezet al.,J. Fluid Mech., vol. 936, 2022, A33), a generalised quasilinear (GQL) approximation is studied in turbulent channel flow using a flow decomposition defined with spanwise Fourier modes: the flow is decomposed into a set of low-wavenumber spanwise Fourier modes and the rest high-wavenumber modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wavenumber group is not able to do so, unlike the GQL models in Part 1, which place a minimal mathematical description for the self-sustaining process across all integral scales. Despite the important physical difference, it is shown that the GQL models in this study share some similarities with those in Part 1, i.e. the reduced multi-scale behaviour and anisotropic turbulent fluctuations. Furthermore, despite not being able to support the self-sustaining process in the high-wavenumber group, the GQL models in the present study are found to reproduce some key statistical features in the high-wavenumber group solely through the ‘scattering’ mechanism proposed by previous studies. Finally, using the nature of the GQL approximation, a further set of numerical experiments suppressing certain triadic nonlinear interactions are carried out. This unveils some key roles played by certain types of triadic interactions, including energy cascade and inverse energy transfer in the near-wall region. In particular, the inhibition of inverse energy transfer in the spanwise direction leads to suppression of the near-wall positive turbulent transport at large scales.

Dynamics of a High-Order Generalized Lorenz Model for Magnetoconvection

In this paper, we present a 6D generalized Lorenz model for magnetoconvection of an electrically conducting viscous fluid layer subjected to horizontally imposed uniform magnetic field. It generalizes the 4D generalized Lorenz model of Macek and Strumik [Phys. Rev. E 82, 027301 (2010)] taking into account high-wavenumber vertical Fourier modes of the temperature profile. These additional modes not only increase the feedback loop of the system but also subsequently affect the transitional processes. The boundedness, stability of solutions, bifurcation patterns enroute to chaos for the new 6D model are explored. Studies reveal that the stability of the quiescent state does not alter. But the stability of the steady convective state differs in comparison to the 4D model. The regions of aperiodic oscillation are suppressed which results in stabilization of the convective motion. Some new organized periodic structures embedded in chaotic domain appear in parameter space of the 6D model, and the transitional route to hyperchaos is altered owing to the inclusion of the high-order modes.

(Invited) Molecular Models for Electrodeposition Inhibitors, Suppressors, and Anti-Suppressors

We evaluate the effect of chain length for a series of alkyl sulfonic acid additives on Cu electrodeposition by using a combination of electrochemical and Raman spectroscopic methods. The combination of these methods suggests that effective Cu electrodeposition acceleration processes require: 1) direct tethering of mercaptoalkylsulfonate species to the electrode, 2) partial desolvation of Cu2+ by the sulfonate group to minimize its solvent reorganization energy, and 3) stabilization of Cu+ adjacent to the electrode surface by addition of halide. Finally, we use the cis-trans ratio idea developed for SPS to examine the effect of inhibitors on SPS activity.We also investigate the behavior of Cu plating bath suppressor additives poly(ethylene glycol) (PEG) and poly(propylene glycol) (PPG) using normal Raman spectroscopy, surface-enhanced Raman spectroscopy (SERS), and electrochemical quartz crystal microbalance (QCM) measurements. Raman and SERS show a clear spectroscopic trend of increased intensity in higher wavenumber modes in the CH stretching region as the environment is changed from pure material to solution to surface for both PEG and PPG. The spectral changes associated with PEG are larger than those associated with PPG, suggesting that the relatively more hydrophilic PEG undergoes more conformational changes upon surface association relative to the more hydrophobic PPG. In the presence of Cl-, PPG forms a denser surface layer compared to PEG on a Cu underpotential deposition (UPD) layer on Au. These differences are consistent with the increased hydrophobicity of PPG relative to PEG.

Development of linear unstable modes in supersonic streamwise vortices using a weighted compact nonlinear scheme

Abstract Growth processes in supersonic streamwise vortices with linear unstable modes at Mach numbers 2.5 and 5.0 were numerically investigated using a weighted compact nonlinear scheme (WCNS) with three different accuracies. In the evolution of the inviscid linear unstable mode m = − 6 as a high-wavenumber mode, the growth rate and eigenfunction profiles of the mode numerically resolved were generally consistent with those obtained with the linear stability theory (LST) during the early transition stage, regardless of the computational accuracy. The numerical scheme used here can capture the growth of three-dimensional linear disturbance in structure. However, the numerical results obtained by nonlinear developments differed according to the accuracy. Among the different accuracies under the same grid resolution, the ninth-order accuracy scheme for the interpolation of primitive variables was able to capture small vortical structures at the downstream in the supersonic flow. In addition, the negative circulation generated and the total disturbance energy indicated that such high accuracy is effective in resolving the developed flow in supersonic vortices, even at moderate grid resolutions.

Gyrofluid modeling of magnetosphere in feedback instability

Feedback instability occurs in a coupling system of the magnetosphere and the ionosphere, and is a theoretical model explaining spontaneous development of the quiet aurora. In this study, we extend a model of the magnetosphere in the feedback instability to the gyrofluid model. This extension makes it possible to properly discuss kinetic effects, such as the finite Larmor radius effect, the Landau damping, and the mirror force, on the feedback instability in a framework of a fluid model. The derived model is applied to linear stability analysis of the low-frequency feedback instability in the absence of the mirror force. An analytical expression of the parallel electron momentum diffusion coefficient associated with the electron Landau damping is obtained by considering a Landau-fluid closure model. It is found that high-wavenumber modes are strongly affected by the finite ion Larmor radius effect and the electron Landau damping. In particular, a stabilizing effect of the electron Landau damping is dominant and stronger for high-parallel-wavenumber modes. This result indicates that the lowest-parallel-wavenumber mode primarily grows during the spontaneous development of the quiet aurora.

Transitional shock wave boundary layer interaction over a flexible panel

Abstract A three-dimensional transitional shock boundary layer interaction (SWBLI) over a finite-span flexible panel is investigated by performing direct numerical simulations (DNS). The laminar inflow is at Mach 2, on which an oblique shock of turn angle 5 . 6 2 ∘ is imposed. The shock impinges at the mid-chord length of the panel, giving rise to flow separation due to the induced adverse pressure gradient. The flow separation leads to streamline curvature in the laminar boundary layer, initiating a centrifugal instability similar to Gortler instability; where the nominally two-dimensional and steady SWBLI becomes three-dimensional and unsteady at a critical Reynolds number. The flow transition is characterized by the presence of counter-rotating streamwise-oriented Gortler-like vortices, engendering spanwise undulations for increasing Gortler number. The transitional SWBLI is analyzed over both the rigid and flexible panels by performing proper orthogonal decomposition (POD) in order to examine the onset of transition as well as the modal response of the flexible panel. The presence of flexible panel promotes flow transition, resulting in lower values of the critical Reynolds and Gortler numbers. Furthermore, at a post-critical Reynolds number, the interaction results in increased turbulence levels and exchange of energy at the low and high wavenumber modes.

Aggregated Negative Feedback in a Generalized Lorenz Model

In this study, we first present a generalized Lorenz model (LM) with [Formula: see text] modes, where [Formula: see text] is an odd number that is greater than three. The generalized LM (GLM) is derived based on a successive extension of the nonlinear feedback loop (NFL) with additional high wavenumber modes. By performing a linear stability analysis with [Formula: see text] and [Formula: see text], we illustrate that: (1) within the 3D, 5D, and 7D LMs, the appearance of unstable nontrivial critical points requires a larger Rayleigh parameter in a higher-dimensional LM and (2) within the 9DLM, nontrivial critical points are stable. By comparing the GLM with various numbers of modes, we discuss the aggregated negative feedback enabled by the extended NFL and its role in stabilizing solutions in high-dimensional LMs. Our analysis indicates that the 9DLM is the lowest order generalized LM with stable nontrivial critical points for all Rayleigh parameters greater than one. As shown by calculations of the ensemble Lyapunov exponent, the 9DLM still produces chaotic solutions. Within the 9DLM, a larger critical value for the Rayleigh parameter, [Formula: see text], is required for the onset of chaos as compared to a [Formula: see text] for the 3DLM, a [Formula: see text] for the 5DLM, and a [Formula: see text] for the 7DLM. In association with stable nontrivial critical points that may lead to steady-state solutions, the appearance of chaotic orbits indicates the important role of a saddle point at the origin in producing the sensitive dependence of solutions on initial conditions. The 9DLM displays the coexistence of chaotic and steady-state solutions at moderate Rayleigh parameters and the coexistence of limit cycle and steady-state solutions at large Rayleigh parameters. The first kind of coexistence appears within a smaller range of Rayleigh parameters in lower-dimensional LMs (i.e. [Formula: see text] within the 3DLM) but in a wider range of Rayleigh parameters within the 9DLM (i.e. [Formula: see text]). The second kind of coexistence has never been reported in high-dimensional Lorenz systems.

Role of electron inertia and electron/ion finite Larmor radius effects in low-beta, magneto-Rayleigh-Taylor instability

The magneto-Rayleigh-Taylor (MRT) instability has been investigated in great detail in previous work using magnetohydrodynamic and kinetic models for low-beta plasmas. The work presented here extends previous studies of this instability to regimes where finite-Larmor-Radius (FLR) effects may be important. Comparisons of the MRT instability are made using a 5-moment and a 10-moment two-fluid model, the two fluids being ions and electrons. The 5-moment model includes Hall stabilization, whereas the 10-moment model includes Hall and FLR stabilization. Results are presented for these two models using different electron mass to understand the role of electron inertia in the late-time nonlinear evolution of the MRT instability. For the 5-moment model, the late-time nonlinear MRT evolution does not significantly depend on the electron inertia. However, when FLR stabilization is important, the 10-moment results show that a lower ion-to-electron mass ratio (i.e., larger electron inertia) under-predicts the energy in high-wavenumber modes due to larger FLR stabilization.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

Reduced modeling of porous media convection in a minimal flow unit at large Rayleigh number

Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale cellular motions generally are localized within the thermal and vorticity boundary layers near the upper and lower walls, while in the interior, the flow exhibits persistent large-scale structures and only a few low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first strategy, the domain is decomposed into two near-wall regions and one interior region. Our results confirm that suppressing the interior high-wavenumber modes has negligible impact on the essential structural features and transport properties of the flow. In the second strategy, a hybrid reduced model is constructed by using Galerkin projection onto a fully \emph{a priori} eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini \emph{et al.} (\emph{Physica~D}, vol. 240, 2011, pp. 241--248) to large $Ra$. The results indicate that the near-wall upper-bound eigenmodes can economically represent the small-scale rolls within the exquisitely-thin thermal boundary layers. Relative to DNS, the hybrid algorithm enables over an order-of-magnitude increase in computational efficiency with only a modest loss of accuracy.

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenumber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.

Choice of function spaces for thermodynamic variables in mixed finite‐element methods

We study the dispersion properties of three choices for the buoyancy space in a mixed finite‐element discretization of geophysical fluid flow equations. The problem is analogous to that of the staggering of the buoyancy variable in finite‐difference discretizations. Discrete dispersion relations of the two‐dimensional linear gravity wave equations are computed. By comparison with the analytical result, the best choice for the buoyancy space basis functions is found to be the horizontally discontinuous, vertically continuous option. This is also the space used for the vertical component of the velocity. At lowest polynomial order, this arrangement mirrors the Charney–Phillips vertical staggering known to have good dispersion properties in finite‐difference models. A fully discontinuous space for the buoyancy corresponding to the Lorenz finite‐difference staggering at lowest order gives zero phase velocity for high vertical wavenumber modes. A fully continuous space, the natural choice for scalar variables in a mixed finite‐element framework, with degrees of freedom of buoyancy and vertical velocity horizontally staggered at lowest order, is found to entail zero phase velocity modes at the large horizontal wavenumber end of the spectrum. Corroborating the theoretical insights, numerical results obtained on gravity wave propagation with fully continuous buoyancy highlight the presence of a computational mode in the poorly resolved part of the spectrum that fails to propagate horizontally. The spurious signal is not removed in test runs with higher‐order polynomial basis functions. Runs at higher order also highlight additional oscillations, an issue that is shown to be mitigated by partial mass‐lumping. In light of the findings and with a view to coupling the dynamical core to physical parametrizations that often force near the horizontal grid scale, the use of the fully continuous space should be avoided in favour of the horizontally discontinuous, vertically continuous space.

Raman and QCM Studies of PPG and PEG Adsorption on Cu Electrode Surfaces

We investigate the behavior of Cu plating bath suppressor additives poly(ethylene glycol) (PEG) and poly(propylene glycol) (PPG) using normal Raman spectroscopy, surface-enhanced Raman spectroscopy (SERS), and electrochemical quartz crystal microbalance (QCM) measurements. Raman and SERS show a clear spectroscopic trend of increased intensity in higher wavenumber modes in the CH stretching region as the environment is changed from pure material to solution to surface for both PEG and PPG. The spectral changes associated with PEG are larger than those associated with PPG, suggesting that the relatively more hydrophilic PEG undergoes more conformational changes upon surface association relative to the more hydrophobic PPG. Calculations show that the observed spectroscopic trend is associated with increased gauche character in the polymer backbone. QCM measurements show PEG adsorbs to the surface only in the presence of Cl−, while PPG adsorbs to the surface both with and without Cl− present. In the presence of Cl−, PPG forms a denser surface layer (0.598 μg/cm2) compared to PEG (0.336 μg/cm2) on a Cu underpotential deposition (UPD) layer on Au. These differences are consistent with the increased hydrophobicity of PPG relative to PEG.

Damping Characteristics of Horizontal Laplacian Diffusion Filters

Horizontally diffusive computational damping terms are frequently employed in 3D atmospheric simulation models to enhance stability and to suppress small-scale noise. In configuring these filters, it is desirable that damping effects are concentrated on the smaller-scale disturbances close to the grid scale and that the dissipation is spatially isotropic. On Cartesian meshes, the isotropy of the damping can vary greatly depending on the numerical formulation of the horizontal filter. The most isotropic behavior appears to result from recursive application of a 2D Laplacian that combines both along-axis and diagonal contributions. Also, the recursive application of 1D Laplacians in each coordinate direction provides better isotropy than the recursive application of the 2D Laplacian represented with a five-point operator. Increased isotropy also permits a larger maximum diffusivity, which may be beneficial in certain filter applications. On hexagonal and triangular meshes, Laplacian operators exhibit excellent isotropy, owing to the more isotropic nature of the meshes. However, previous research has established that straightforward application of the Laplacian may yield a diffusion operator that damps both resolved physical modes and unresolved high-wavenumber (aliased) modes, but it does not converge to the proper analytic behavior. Special averaging is then required to recover an accurate representation for the Laplacian. A consequence of this averaging is that the resulting filters do not act on the aliased modes (the checkerboard mode in particular) and thus employing the unaveraged diffusion operators may be preferable. The damping characteristics and stability constraints are derived for both the unaveraged and averaged Laplacian filters for C-grid staggering on these meshes.