An official website of the United States government
We extend the Matsuno–Gill model, originally developed on the equatorial$$\beta$$-plane, to the surface of the sphere. While on the$$\beta$$-plane the non-dimensional model contains a single parameter, the damping rate$$\gamma$$, on a sphere the model contains a second parameter, the rotation rate$$\epsilon ^{1/2}$$(Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the$$(\gamma, \epsilon ^{1/2})$$plane. We find that the$$\beta$$-plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of$$\gamma$$and$$\epsilon ^{1/2}$$. The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general$$\beta$$-plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply.
more »
« less
This content will become publicly available on May 10, 2026
Bridging the Rossby number gap in rapidly rotating thermal convection
Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers$${ {Ek}} \gtrsim 10^{-8}$$), while realistic$${Ek}$$for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($$ {Ek}\to 0$$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite$${Ek}$$has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for$${Ek}\to 0$$, recently introduced by Julienet al.(2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to$$ {Ek}=10^{-15}$$and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers ($${Ek}\approx 10^{-9}$$). We also identify an overshoot in the heat transport as$${Ek}$$is varied at fixed$$\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$$, where$$Ra$$is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small$${Ek}$$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.
more »
« less
- PAR ID:
- 10598464
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1010
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This study explores heat and turbulent modulation in three-dimensional multiphase Rayleigh–Bénard convection using direct numerical simulations. Two immiscible fluids with identical reference density undergo systematic variations in dispersed-phase volume fractions,$$0.0 \leq \varPhi \leq 0.5$$, and ratios of dynamic viscosity,$$\lambda _{\mu }$$, and thermal diffusivity,$$\lambda _{\alpha }$$, within the range$$[0.1\unicode{x2013}10]$$. The Rayleigh, Prandtl, Weber and Froude numbers are held constant at$$10^8$$,$$4$$,$$6000$$and$$1$$, respectively. Initially, when both fluids share the same properties, a 10 % Nusselt number increase is observed at the highest volume fractions. In this case, despite a reduction in turbulent kinetic energy, droplets enhance energy transfer to smaller scales, smaller than those of single-phase flow, promoting local mixing. By varying viscosity ratios, while maintaining a constant Rayleigh number based on the average mixture properties, the global heat transfer rises by approximately 25 % at$$\varPhi =0.2$$and$$\lambda _{\mu }=10$$. This is attributed to increased small-scale mixing and turbulence in the less viscous carrier phase. In addition, a dispersed phase with higher thermal diffusivity results in a 50 % reduction in the Nusselt number compared with the single-phase counterpart, owing to faster heat conduction and reduced droplet presence near walls. The study also addresses droplet-size distributions, confirming two distinct ranges dominated by coalescence and breakup with different scaling laws.more » « less
-
Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$$h$$, and systematically arranged to quantify the response to variable streamwise spacing,$$\delta _1$$, and spanwise spacing,$$\delta _2$$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$$k$$- and$$d$$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$$\delta _1$$and$$\delta _2$$. For$$\delta _2/\delta \lesssim 1$$and$$\gtrsim 2$$, where$$\delta$$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$$\delta _1$$regulates this transition, with secondary cell reversal occurring first for the largest$$k$$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$$z_{0,{Eff.}} = \alpha \sigma _h$$, where$$\alpha$$is a proportionality constant and$$\sigma _h$$is height root mean square. We report$$\alpha \approx 10^{-1}$$, the value reported over many decades for a broad range of rough surfaces, for$$k$$-type cases at small$$\delta _2$$, whereas the transition to$$d$$-type arrangements necessitates larger$$\delta _2$$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing.more » « less
-
Direct numerical simulations of spanwise-rotating turbulent channel flow with a parabolic bump on the bottom wall are employed to investigate the effects of rotation on flow separation. Four rotation rates,$$Ro_b := 2\varOmega H/U_b = \pm 0.42$$,$$\pm$$1.0, are compared with the non-rotating scenario. The mild adverse pressure gradient induced by the lee side of the bump allows for a variable pressure-induced separation. The separation region is reduced (increased) when the bump is on the anti-cyclonic (cyclonic) side of the channel, compared with the non-rotating separation. The total drag is reduced in all rotating cases. Through several mechanisms, rotation alters the onset of separation, reattachment and wake recovery. The mean momentum deficit is found to be the key. A physical interpretation of the ratio between the system rotation and mean shear vorticity,$$S:=\varOmega /\varOmega _s$$, provides the mechanisms regarding stability thresholds$$S=-0.5$$and$$-$$1. The rotation effects are explained accordingly, with reference to the dynamics of several flow structures. For anti-cyclonic separation, particularly, the interaction between the Taylor–Görtler vortices and hairpin vortices of wall-bounded turbulence is proven to be responsible for the breakdown of the separating shear layer. A generalized argument is made regarding the essential role of near-wall deceleration and resultant ejection of enhanced hairpin vortices in destabilizing an anti-cyclonic flow. This mechanism is anticipated to have broad impacts on other applications in analogy to rotating shear flows, such as thermal convection and boundary layers over concave walls.more » « less
-
Well-resolved direct numerical simulations (DNS) have been performed of the flow in a smooth circular pipe of radius$$R$$and axial length$$10{\rm \pi} R$$at friction Reynolds numbers up to$$Re_\tau =5200$$using the pseudo-spectral code OPENPIPEFLOW. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels. Small but distinct differences between various datasets are identified. The friction factor$$\lambda$$overshoots by$$2\,\%$$and undershoots by$$0.6\,\%$$the Prandtl friction law at low and high$$Re$$ranges, respectively. In addition,$$\lambda$$in our results is slightly higher than in Pirozzoliet al.(J. Fluid Mech., vol. 926, 2021, A28), but matches well the experiments in Furuichiet al.(Phys. Fluids, vol. 27, issue 9, 2015, 095108). The log-law indicator function, which is nearly indistinguishable between pipe and channel up to$$y^+=250$$, has not yet developed a plateau farther away from the wall in the pipes even for the$$Re_\tau =5200$$cases. The wall shear stress fluctuations and the inner peak of the axial turbulence intensity – which grow monotonically with$$Re_\tau$$– are lower in the pipe than in the channel, but the difference decreases with increasing$$Re_\tau$$. While the wall value is slightly lower in the channel than in the pipe at the same$$Re_\tau$$, the inner peak of the pressure fluctuation shows negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect-power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a$$k^{-1}$$dependence at an intermediate distance from the wall – also seen in the channel. In summary, these high-fidelity data enable us to provide better insights into the flow physics in the pipes as well as the similarity/difference among different types of wall turbulence.more » « less
