Turbulence and mixing in a nearbottom convectively driven flow are examined by numerical simulations of a model problem: a statically unstable disturbance at a slope with inclination $\unicode[STIX]{x1D6FD}$ in a stable background with buoyancy frequency $N$ . The influence of slope angle and initial disturbance amplitude are quantified in a parametric study. The flow evolution involves energy exchange between four energy reservoirs, namely the mean and turbulent components of kinetic energy (KE) and available potential energy (APE). In contrast to the zeroslope case where the mean flow is negligible, the presence of a slope leads to a current that oscillates with $\unicode[STIX]{x1D714}=N\sin \unicode[STIX]{x1D6FD}$ and qualitatively changes the subsequent evolution of the initial density disturbance. The frequency, $N\sin \unicode[STIX]{x1D6FD}$ , and the initial speed of the current are predicted using linear theory. The energy transfer in the sloping cases is dominated by an oscillatory exchange between mean APE and mean KE with a transfer to turbulence at specific phases. In all simulated cases, the positive buoyancy flux during episodes of convective instability at the zerovelocity phase is the dominant contributor to turbulent kinetic energy (TKE) although the shear production becomes increasingly important with increasing $\unicode[STIX]{x1D6FD}$ . Energy that initially resides wholly in mean available potential energy is lost through conversion to turbulence and the subsequent dissipation of TKE and turbulent available potential energy. A key result is that, in contrast to the explosive loss of energy during the initial convective instability in the nonsloping case, the sloping cases exhibit a more gradual energy loss that is sustained over a long time interval. The slopeparallel oscillation introduces a new flow time scale $T=2\unicode[STIX]{x03C0}/(N\sin \unicode[STIX]{x1D6FD})$ and, consequently, the fraction of initial APE that is converted to turbulence during convective instability progressively decreases with increasing $\unicode[STIX]{x1D6FD}$ . For moderate slopes with $\unicode[STIX]{x1D6FD}<10^{\circ }$ , most of the net energy loss takes place during an initial, short ( $Nt\approx 20$ ) interval with periodic convective overturns. For steeper slopes, most of the energy loss takes place during a later, long ( $Nt>100$ ) interval when both shear and convective instability occur, and the energy loss rate is approximately constant. The mixing efficiency during the initial period dominated by convectively driven turbulence is found to be substantially higher (exceeds 0.5) than the widely used value of 0.2. The mixing efficiency at long time in the present problem of a convective overturn at a boundary varies between 0.24 and 0.3.
more »
« less
The Nusselt numbers of horizontal convection
In the problem of horizontal convection a nonuniform buoyancy, $b_{s}(x,y)$ , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\boldsymbol{J}$ , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ ; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $\unicode[STIX]{x1D705}$ is the molecular diffusivity of buoyancy $b$ . This connection between $\boldsymbol{J}$ and $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ justifies the definition of the horizontalconvective Nusselt number, $Nu$ , as the ratio of $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontalconvective Nusselt number. We investigate transient effects and show that $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ is the volumeaveraged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{s}(x,y)$ . In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of $\unicode[STIX]{x1D735}b^{2}$ demanded by $\unicode[STIX]{x1D705}\langle \unicode[STIX]{x1D735}b^{2}\rangle$ .
more »
« less
 Award ID(s):
 1657041
 NSFPAR ID:
 10169194
 Date Published:
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 894
 ISSN:
 00221120
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


null (Ed.)Let $f:X\rightarrow X$ be a continuous dynamical system on a compact metric space $X$ and let $\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$ be an $m$ dimensional continuous potential. The (generalized) rotation set $\text{Rot}(\unicode[STIX]{x1D6F7})$ is defined as the set of all $\unicode[STIX]{x1D707}$ integrals of $\unicode[STIX]{x1D6F7}$ , where $\unicode[STIX]{x1D707}$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $\unicode[STIX]{x210B}(w)$ to each $w\in \text{Rot}(\unicode[STIX]{x1D6F7})$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $f$ is a subshift of finite type. We prove that $\text{Rot}(\unicode[STIX]{x1D6F7})$ is computable and that $\unicode[STIX]{x210B}(w)$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $\unicode[STIX]{x210B}$ is not continuous on the boundary of the rotation set when considered as a function of $\unicode[STIX]{x1D6F7}$ and $w$ . In particular, $\unicode[STIX]{x210B}$ is, in general, not computable at the boundary of $\text{Rot}(\unicode[STIX]{x1D6F7})$ .more » « less

The central open question about Rayleigh–Bénard convection – buoyancydriven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling.more » « less

We investigate the behavior of higherform symmetries at variousquantum phase transitions. We consider discrete 1form symmetries, whichcan be either part of the generalized concept “categorical symmetry”(labelled as \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) )introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1form symmetry. We demonstrate that for many quantum phase transitionsinvolving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O 𝒞 ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim  \frac{A}{\epsilon} P + b \log P ⟨ ( log O 𝒞 ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 𝒞 is an operator defined associated with loop \mathcal{C} 𝒞 (or its interior \mathcal{A} 𝒜 ),which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)} Z N ( 1 ) 1form symmetry. P P is the perimeter of \mathcal{C} 𝒞 ,and the b \log P b log P term arises from the sharp corners of the loop \mathcal{C} 𝒞 ,which is consistent with recent numerics on a particular example. b b is a universal microscopicindependent number, which in (2+1)d ( 2 + 1 ) d is related to the universal conductivity at the quantum phasetransition. b b can be computed exactly for certain transitions using the dualitiesbetween (2+1)d ( 2 + 1 ) d conformal field theories developed in recent years. We also compute the"strange correlator" of O_\mathcal{C} O 𝒞 : S_{\mathcal{C}} = \langle 0  O_\mathcal{C}  1 \rangle / \langle 0  1 \rangle S 𝒞 = ⟨ 0  O 𝒞  1 ⟩ / ⟨ 0  1 ⟩ where 0\rangle  0 ⟩ and 1\rangle  1 ⟩ are manybody states with different topological nature.more » « less

We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $X=\mathbb{B}/\unicode[STIX]{x1D6E4}$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $n$ dimensional toroidal compactification $\overline{X}$ with boundary $D$ , $K_{\overline{X}}+(1\unicode[STIX]{x1D706})D$ is ample for $\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$ , and in particular that $K_{\overline{X}}$ is ample for $n\geqslant 6$ . By an independent algebraic argument, we prove that every ball quotient of dimension $n\geqslant 4$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture.more » « less