An official website of the United States government
Convection is a ubiquitous process driving geophysical/astrophysical fluid flows, which are typically strongly constrained by planetary rotation on large scales. A celebrated model of such flows, rapidly rotating Rayleigh-Bénard convection, has been extensively studied in direct numerical simulations (DNS) and laboratory experiments, but the parameter values attainable by state-of-the-art methods are limited to moderately rapid rotation (Ekman numbers Ek≳10−8), while realistic geophysical/astrophysical Ek are significantly smaller. Asymptotically reduced equations of motion, the nonhydrostatic quasi-geostrophic equations (NHQGE), describing the flow evolution in the limit Ek→0, do not apply at finite rotation rates. The geophysical/astrophysical regime of small but finite Ek therefore remains currently inaccessible. Here, we introduce a new, numerically advantageous formulation of the Navier-Stokes-Boussinesq equations informed by the scalings valid for Ek→0, the \textit{Rescaled Rapidly Rotating incompressible Navier-Stokes Equations} (RRRiNSE). We solve the RRRiNSE using a spectral quasi-inverse method resulting in a sparse, fast algorithm to perform efficient DNS in this previously unattainable parameter regime. We validate our results against the literature across a range of Ek and demonstrate that the algorithmic approaches taken remain accurate and numerically stable at Ek as low as 10−15. Like the NHQGE, the RRRiNSE derive their efficiency from adequate conditioning, eliminating spurious growing modes that otherwise induce numerical instabilities at small Ek. We show that the time derivative of the mean temperature is inconsequential for accurately determining the Nusselt number in the stationary state, significantly reducing the required simulation time, and demonstrate that full DNS using RRRiNSE agree with the NHQGE at very small Ek.
more »
« less
Spontaneous generation of helical flows by salt fingers
We study the dynamics of salt fingers in the regime of slow salinity diffusion (small inverse Lewis number) and strong stratification (large density ratio), focusing on regimes relevant to Earth’s oceans. Using three-dimensional direct numerical simulations in periodic domains, we show that salt fingers exhibit rich, multiscale dynamics in this regime, with vertically elongated fingers that are twisted into helical shapes at large scales by mean flows and disrupted at small scales by isotropic eddies. We use a multiscale asymptotic analysis to motivate a reduced set of partial differential equations that filters internal gravity waves and removes inertia from all parts of the momentum equation except for the Reynolds stress that drives the helical mean flow. When simulated numerically, the reduced equations capture the same dynamics and fluxes as the full equations in the appropriate regime. The reduced equations enforce zero helicity in all fluctuations about the mean flow, implying that the symmetry-breaking helical flow is generated spontaneously by strictly non-helical fluctuations.
more »
« less
- Award ID(s):
- 2023499
- PAR ID:
- 10667373
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1020
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Turbulence in three dimensions (3D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here that the shear-flow-instability-supplied 3D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, and such inverse transfer is more efficient for turbulent fluctuations in 3D than in two dimensions (2D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in 3D than in 2D (≈70% vs ≈50%). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of 2D shear flow. The more widely occurring 3D turbulence is thus inherently different from the commonly studied 2D turbulence, despite both saturating via stable modes.more » « less
-
null (Ed.)Abstract We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.more » « less
-
Numerical models have been extensively used to understand and predict flow and reactive transport processes in the hyporheic zone. However, most models focus on fully saturated riverbeds without accounting for surface water stage fluctuations related to precipitation and flooding. To capture the complete picture of hyporheic processes in riverbeds and riverbanks, we developed a fully-coupled multiphase reactive transport solver using the Open Source Field Operation And Manipulation (OpenFOAM) platform. This solver captures surface water stage fluctuations and partially-saturated flow in fluvial sediment using VoF two-phase flow and extendedDarcy’s Law two-phase flow models for surface and subsurface domains, respectively. The transport models designed for partially saturated conditions in both domains are implemented. A geochemical reaction module, PhreeqcRM, is integrated into the solver to facilitate complex geochemical reaction networks. A two-way conservative flux boundary condition is implemented at the surface-subsurface interface to realistically map fluxes. The solver’s capability is illustrated through a variety of hyporheic-related problems across spatial scales. These include laboratory experiments and reactive transport in two and three dimensions, from the bedform scale to multiscale riverbeds and riverbanks with fluctuating surface water flow. This novel solver allows for quantifying dynamics in the hyporheic zone with fewer simplifications. Based on the code structure and parallel design of OpenFOAM, the solver can simulate large, three-dimensional (3D) multiscale cases. The code, examples, and pre- and post-processing scripts are all open source, providing community access to use and modify them as desired.more » « less
-
We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1017/jfm.2025.10641
