Vertical Heat Flux Research Articles

Dynamics and scaling of internally cooled convection

Our goal is to investigate fundamental properties of the system of internally cooled convection. The system consists of an upward thermal flux at the lower boundary, a mean temperature lapse-rate and a constant cooling term in the bulk with the bulk cooling in thermal equilibrium with the input heat flux. This simple model represents idealised dry convection in the atmospheric boundary layer, where the cooling mimics the radiative cooling to space notably through longwave radiation. We perform linear stability analysis of the model for different values of the mean stratification to derive the critical forcing above which the fluid is convectively unstable to small perturbations. The dynamic behavior of the fluid system is described and the scaling of various important measured quantities such as the total vertical convective heat flux and the upward mass flux is measured. We introduce a lapse-rate dependent dimensionless Rayleigh-number Raγ that determines the behavior of the system, finding that the convective heat-flux and mass-flux scale approximately as Raγ0.5 and Raγ0.7 respectively. The area-fraction of the domain that is occupied by upward and downward moving fluid and the skewness of the vertical velocity are studied to understand the asymmetry inherent in the system. We conclude with a short discussion on the relevance to atmospheric convection and the scope for further investigations of atmospheric convection using similar simplified approaches.

Diffusive-convection staircases in the polar oceans: the interplay between double diffusion and turbulence

Numerical simulations have been conducted to examine the structure of diffusive-convection staircases in the presence of vortical-mode-induced turbulent forcing. By modulating the input power $P$ and the background density ratio $R_\rho$ , we have identified three distinct types of staircase structures in these simulations: namely staircases maintained in the system driven by double-diffusion, by turbulence or by a combination of both double-diffusion and turbulence. While we showed that staircases maintained in the double-diffusion-dominated system are accurately characterised by the existing model originally proposed by Linden & Shirtcliffe (J. Fluid Mech., vol. 87, no. 3, 1978, pp. 417–432), we introduced new physical models to describe the staircase structures maintained in the turbulence-dominated system and the system driven by both turbulence and double-diffusion. Our integrated model reveals that turbulence fundamentally governs the entire life cycle of the diffusive-convection staircases, encompassing their formation, maintenance and eventual disruption in the Arctic Ocean's thermohaline staircases. While our previous work of Ma & Peltier (J. Fluid Mech., vol. 931, 2022b) demonstrated that turbulence could initiate the formation of Arctic staircases, these staircases are sustained by both turbulence and double-diffusion acting together after formation has occurred. Strong turbulence may disrupt staircase structures; however, the presence of weak turbulence could lead to unstable stratification within mixed layers of the staircases, as well as enhancing vertical heat and salt fluxes. Turbulence can even sustain a stable staircase structure factor when $R_\rho$ is relatively large, following a similar mechanism to the density staircases observed in laboratory experiments. Consequently, previous parameterisations (e.g. Kelley, J. Geophys. Res.: Oceans, vol. 95, no. C3, 1990, pp. 3365–3371) on the vertical heat flux across the diffusive-convection staircases may provide a significant underestimation of the heat transport by ignoring the influences of turbulence.

Observational Analysis of Vertical Heat Flux Caused by Typhoon‐Induced Near‐Inertial Waves Under the Modulation of Mesoscale Eddies

AbstractTropical cyclones (TCs) induce heat pump and cold suction in the upper ocean layer. However, limited research investigated whether this heat can be effectively transported into the deep ocean. The present study shows that seawater at Station S2 is anomalously warmer in the deep ocean layer than at S1 and S3 during Typhoon Kalmaegi (2014) in the South China Sea. The turbulence‐induced vertical heat flux is estimated based on fine‐scale mixing parameterization but it cannot explain the warming rate below the thermocline. Therefore, a new method is proposed to estimate the vertical velocity, allowing us to calculate the heat flux more accurately. To elucidate the underlying causes of the observed differences, we analyze horizontal velocity to examine the role of mesoscale eddies in modulating near‐inertial waves (NIWs)‐induced vertical heat flux. Station S1 is located inside a cyclonic eddy while S2 and S3 are within two distinct anticyclonic eddies. According to the “Chimney Effect” theory, cyclonic (anticyclonic) eddies tend to limit (enhance) the vertical propagation of the NIWs. While this theory explains the confined heat flux at S1, it fails to explain the differences observed at S2 and S3. Further examination of the vertical structures and intensities of the two anticyclonic eddies reveals that the eddy at S2 extends much deeper and is stronger than that at S3, allowing the NIWs to propagate and transport heat deeper at S2 than at S3. The study demonstrates the role of TC‐induced NIWs in deep ocean heat transport under the influence of mesoscale eddies.

The Relationship between Vertical Kinematic Eddy Heat Flux, Air Temperature and Turbulent Kinetic Energy in Atmospheric Boundary Layer: Baghdad City

Studying the heat flux in the boundary layer and understanding its relationship to the various atmospheric variables reflects positively on understanding the nature of turbulence in this layer and thus understanding the nature of the spread and movement of pollutants and the transmission and distribution of energy in this layer, also, the vertical heat flux is considered a significant influence on the movement of buoyancy and stability in the boundary layer and its effect on the horizontal wind movement, and therefore it is considered one of the important studies indirectly involved in the estimates of wind energy production, pollutant diffusion, and turbulence. This study involved the calculation of the Eddy Heat Flux and turbulent kinetic energy across Baghdad city. Our investigation revealed a correlation between the Eddy Heat Flux, temperature, and Turbulent Kinetic Energy (TKE). The observations have been made for wind speed with three components (u, v, w) and temperature by using a fast-response anemometer. As for the atmospheric pressure data, it was obtained from the automatic weather station located in the department of Atmospheric Sciences at Mustansiriyah University. The range of observations extended through the duration of thirty days for 24 hours from 1st July 2016 to 30th July 2016 every second. The maximum Eddy Heat Flux value was 0.092 J/(m2.s) at 10:00 on 23th July 2016, while the minimum Eddy Heat Flux value was -0.013 J/(m2.s) at 21:00 hour on 25th July 2016, the negative sign refer to a change in the direction of heat transfer,. It was also found that there is a positive relationship between the eddy heat flux and temperatures with a correlation coefficient of 0.93, as well as between the eddy heat flux and the turbulent kinetic energy with a correlation coefficient of 0.8.

Re-estimation of the vertical sensible heat flux by determining the environmental temperature on a single-point tower measurement

Surface energy balance has always been a goal of those studying the Earth’s climate system. However, many studies have demonstrated that turbulent heat fluxes are usually underestimated by eddy covariance (EC) measurements, such that the energy balance is not closed. This study proposes a new perspective on calculating sensible heat flux based on the environmental temperature using EC. Using this approach, additional sensible heat fluxes were detected as outcomes of the vertical transportation of thermal structures in the atmospheric surface layer (ASL). For data obtained over a 40-day period over a grassland in Southern China, additional sensible heat flux observations exceeding 50 W m−2 were measured for 8 of the 40 days; smaller but still significant contributions were captured for another 11 days. In the proposed model, the difference between the mean and environmental temperature (∆T) and the local mean vertical velocity (w¯) serve as determinants for the additional flux, where the former can be deemed as the activity level of the thermal structures. A modeled underestimation of α[w¯] Ht of the total vertical sensible heat flux was revealed using our method, where α equals 3.55 for this study, Ht is the traditional EC results, and [w¯] is the non-dimensional w¯. Moreover, the additional flux usually showed large values in the daytime that were not detectable using the traditional EC method; this may help explain the energy imbalance problem in the ASL.

Experimental Study on Effects of Moisture Content and Tree Height on Crown Fire Behaviors of Live Cypress Trees

ABSTRACT Once a fire occurs in urban green belts, the live foliage of some tree and shrub species with larger moisture content than dead fuel can support fire spread, which poses a great threat to the wildland-urban interface (WUI). Cypress tree is widely used as an ornamental vegetation worldwide. This work aims to study the fire and spread behaviors through cypress trees. Experiments were conducted using three tree heights between 0.8 m and 1.6 m and five values of fuel moisture content (FMC) between 30% and 160%. The fuel mass loss, flame shape, vertical temperature distribution and outward vertical heat fluxes were measured. The results showed that for a given tree height, a small enough FMC promoted the burning process and decreased both the maximum heat release rate (HRR) and the time to reach peak HRR compared to a larger FMC. The tree height correlated strongly with the flame height, while the influence of FMC was much smaller. Then, a correlation between the normalized maximum flame height and the normalized maximum HRR was established by taking the crown height as the characteristic length. Reducing the FMC did not necessarily promote the fire spread and trees with large FMCs can support relatively large flame front spread rates. The effect of the tree height on the radiant heat flux was greater than the effect of FMC.

Water Properties and Diffusive Convection in the Canada Basin

The aim of this study is to better understand diffusive convection (DC) and its role in the upper ocean dynamic environment and sea ice melting in the Canada Basin. Based on a moored dataset with 6737 profiles collected from August 2003 to August 2011 in the upper layer of the Canada Basin, DC between the warm and salty Atlantic Water (AW) and the colder and less salty Lower Halocline Water (LHW) were investigated. The moorings were designated at four stations: A, B, C, and D, located at the southwestern, southeastern, northeastern, and northwestern parts of the basin, respectively. During the observation period, the temperature, salinity, and depth of the AW and LHW exhibited unique temporal variations. The temperature and salinity of the AW varied among stations, with a decreasing trend from northwest to southeast, consistent with the propagation path of the AW in the Canada Basin. The temperature and salinity of the LHW were similar at all stations. The AW and LHW cores were located between depths of 320–500 m and 160–300 m, respectively, and both gradually deepened over time. Distinct DC staircase structures were observed between the AW and LHW, more pronounced at stations C and D than at stations A and B, which is speculated to be related to eddies at stations A and B during the observation period. The vertical heat fluxes through the DC staircase layer at stations C and D (FHc_C and FHc_D) were estimated using an empirical formula. FHc_C ranged from 0.05 to 0.94 W/m2, and FHc_D ranged from 0.05 to 0.6 W/m2, with the maximum probability value for both at approximately 0.2 W/m2. The effective diffusivities at these two stations (KT_C and KT_D) are similar, ranging from 2 × 10−6 to 3 × 10−5 m2/s, with the highest probability occurring at 6 × 10−6 m2/s. Both the probability density function of the heat flux and the effective diffusivity skewed towards larger values and obey a lognormal distribution, indicating turbulence intermittency of the DC staircase in the Canada Basin. These finding offers new insights into the heat transport and turbulence in the DC staircase, and then bring a deeper understanding of sea ice melting in the Canada Basin.

The Transition Layer and Remnant Transition Layer of the Western Arctic Ocean: Stratification, Vertical Diffusivity, and Pacific Summer Water Heat Fluxes

AbstractThe exchange of heat and other tracers between the ocean interior and sea ice cover depends on vertical diffusion through the thin and strongly stratified Arctic Ocean transition layer and remnant transition layer. Ice‐Tethered Profiler observations during 2004–2021 were used to characterize and investigate the active and remnant transition layers in the Beaufort Gyre region. Transition layers evolved seasonally, with the deepest, densest, and warmest transition layers observed in spring. In a composite view, the summer mixed layer shoaled over a timescale of approximately 2 weeks, leaving behind a remnant transition layer whose stratification had already begun to weaken. The stratification maximum of the remnant transition layer continued to weaken and broaden throughout the summer, including changes associated with storm events. The weakening stratification was used to estimate an effective vertical turbulent diffusivity which ranged from 10−8 to 10−6 m2/s for individual ITP records; a value of 6 × 10−7 m2/s is representative of the Beaufort Gyre region. Vertical heat fluxes through the active and remnant transition layers were estimated over 90‐day timescales. Springtime vertical heat fluxes increased from 0.3 ± 0.3 W/m2 during 2006–2011 to 1.1 ± 0.7 W/m2 during 2017–2021. Summertime vertical heat fluxes through the remnant transition layer were somewhat less, but also increased interannually. Vertical gradients of density during spring and summer, and so diffusion of density through the active and remnant transition layers, did not increase interannually. Pacific Summer Water warming has increased ocean to ice heat fluxes, which may continue to become more significant in the future.

Variations in intensity of Bay of Bengal tropical cyclones with surface latent heat flux and other parameters

This study has been undertaken to find out the variation of central pressure (intensity) of intense Tropical Cyclones (TCs) with Sea Surface Temperature (SST), Mid-tropospheric Relative Humidity (MRH), Mid-tropospheric Instability (MI), Vertical Wind Shear (VWS), 200-hPa divergence, and Surface Latent Heat Flux (SLHF) during the lifetime intense TCs. This study also aims to determine the most crucial parameter which shows the highest correlation with central pressure (intensity) of intense TCs during their lifetime. Out of all these parameters, SLHF is highly correlated (R = 0.74) with the central pressure (intensity) of intense TCs. Increase and decrease of SLHF correspond to decrease and increase of TCs central pressure (increase and decrease in TCs intensity). The highest SLHF corresponds to the lowest central pressure (highest intensity).

A tri-point correlation method for ocean current estimation and its applications in near-bottom turbulent mixing on the shelf of the Northern South China Sea

We propose a tri-point correlation method to estimate ocean currents using moored temperature observation data. The transit time for two time series of temperature is determined with the cross correlation technique, enabling the formulations of an equation system to determine the current speed and angle. To validate our method, we utilize a high-resolution temperature dataset collected at the shelf bottom of the northern South China Sea and compare the inferred currents with direct measurement results with a current meter. The results demonstrate a high level of agreement between the inferred currents and the directly measured ones. Using this temperature dataset and the synchronously inferred currents, we further estimate near-bottom turbulent mixing, including diapycnal mixing and vertical heat flux. During the observation period, the mean values of the turbulent kinetic energy dissipation rate, turbulent diffusivity, and vertical heat flux are 3.8×10−8 m2/s3, 2.6×10−4 m2/s, and 10.0 W/m2, respectively. These inferred turbulent mixing quantities exhibit a positive correlation with the dynamics of internal waves.

The Inhomogeneity Effect. III. Weather Impacts on the Heat Flow of Hot Jupiters

The interior flux of a giant planet impacts atmospheric motion, and the atmosphere dictates the interior’s cooling. Here we use a non-hydrostatic general circulation model (Simulating Non-hydrostatic Atmospheres on Planets) coupled with a multi-stream multi-scattering radiative module (High-performance Atmospheric Radiation Package) to simulate the weather impacts on the heat flow of hot Jupiters. We found that the vertical heat flux is primarily transported by convection in the lower atmosphere and regulated by dynamics and radiation in the overlying radiation-circulation zone. The temperature inversion occurs on the dayside and reduces the upward radiative flux. The atmospheric dynamics relay the vertical heat transport until the radiation becomes efficient in the upper atmosphere. The cooling flux increases with atmospheric drag due to increased day–night contrast and spatial inhomogeneity. The temperature dependence of the infrared opacity greatly amplifies the opacity inhomogeneity. Although atmospheric circulation could transport heat downward in a narrow region above the radiative-convective boundary, the opacity inhomogeneity effect overcomes the dynamical effect and leads to a larger overall interior cooling than the local simulations with the same interior entropy and stellar flux. The enhancement depends critically on the equilibrium temperature, drag, and atmospheric opacity. In a strong-drag atmosphere hotter than 1600 K, a significant inhomogeneity effect in three-dimensional (3D) models can boost interior cooling several-fold compared to the 1D radiative-convective equilibrium models. This study confirms the analytical argument of the inhomogeneity effect in the companion papers by Zhang. It highlights the importance of using 3D atmospheric models in understanding the inflation mechanisms of hot Jupiters and giant planet evolution in general.

Impact of Atmospheric Cooling on the High‐Frequency Submesoscale Vertical Heat Flux

AbstractRecent simulations suggest that submesoscale motions with scales smaller than 30 km and frequencies greater than 1 day−1 drive upward vertical heat transport. These simulations have prompted us to revisit the mechanisms that explain high‐frequency (HF) vertical heat fluxes (VHFs) within the surface mixed layer (ML). Here, an idealized numerical simulation of a re‐entrant channel flow with an unbalanced submesoscale thermal front is used to analyze the impact of surface cooling on HF VHFs. Two types of simulations are analyzed: forced and unforced. The VHFs cospectrum analysis shows that surface diurnal cooling increases VHFs, reaching frequencies larger than 1 day−1. However, the fastest‐growing length scale of ML instabilities limits the extension of positive VHFs toward fine scales. Symmetric and gravitational instabilities are the main conduits producing ageostrophic HF and small‐scale structures, which in turn enhance upward VHFs across the diurnal frequency. A comparison between forced‐idealized simulations with the K‐profile parameterization scheme and a realistic regional simulation in the frequency‐wavenumber space, reveals that the two simulation types reproduce similar VHFs near the diurnal frequency. However, the realistic simulation displays higher VHFs than the forced‐idealized simulation. This study emphasizes that surface diurnal cooling significantly impacts HF VHFs. However, this impact is not sufficient to reach the HF VHFs estimated in realistic submesoscale‐permitting and tidal‐resolving simulations.

Large-scale coherent structure and canopy-level turbulence in the convective boundary layer over urban areas

Thermally unstable stratification alters the winds over urban areas and excites organized, large-scale motions (LSMs) in the atmospheric boundary layer (ABL). The current understanding of the interaction between urban turbulence and buoyancy-induced coherent structures is rather limited due to their multiscale, multiphysics nature. Nine sets of large-eddy simulation (LES) are performed to critically examine the influence of atmospheric instability on turbulent boundary layers (TBLs) interacting with idealized urban roughness elements. Varying the stratification from neutral to almost free convective conditions enables the coherent structures to transit from horizontal rolls to open cellular patterns. The aerodynamic roughness parameters tend to decrease in more unstable stratification, potentially a result of buoyancy effects that might enhance organized ejections and suppress energetic sweeps within the roughness sublayers (RSLs). Unexpectedly, roughness-induced sweeps dominate the vertical momentum fluxes within urban canopies even under the most convective condition. However, decoupling the momentum and heat transports in highly convective states results in a subtle influence of urban roughness on the vertical heat fluxes therein. The spatial distribution of canopy-level turbulent momentum fluxes is found to be strongly modulated by the overlying large-scale coherent structures. The modulation is enhanced with increasing instability. In addition, urban-like surfaces elevate the large-scale rolls and assist them in forming under a weaker stratification than in canonical settings.

A random 2D walk of a submerged free-floating disc in a convective layer

Free motions of a thermally insulating disc submerged in a liquid layer heated from below are studied experimentally. The disc is fixed at a certain height from the bottom of the convective cell, but can move freely in horizontal directions. Blocking of vertical motion and heat flux leads to the appearance of large-scale vortices, which in turn, drags the disc. The interplay of the large-scale circulation and the disc produces complex two-dimensional disc motions. The disc dynamics is mainly determined by large-scale modes: toroidal vortex with descending fluid in the centre and large-scale vortices along each coordinate, existing on the background of intense, but relatively short-lived small-scale convective cells. The wandering disc forms a cold spot above itself, which affects large-scale circulation but has little effect on cell convection at scales on the order of the layer thickness, which provide the main heat transport and dominate the upper part of the layer. Despite the essentially three-dimensional flow structure, which is fundamentally different from that of the convective flow in an elongated tank with one-dimensional disc motions, the travels along each coordinate look similar to one-dimensional motions.

A Cross-Plane Design for Wearable Thermoelectric Generators with High Stretchability and Output Performance.

Stretchable wearable thermoelectric (TE) generators (WTEGs) without compromising output performance for real wearables have attracted much attention recently. Herein, a 3D thermoelectric generator with biaxial stretchability is constructed on the device level. Ultraflexible inorganic Ag/Ag2 Se strips are sewn into the soft purl-knit fabric, in which the thermoelectric legs are aligned in the direction of vertical heat flux. A stable and sufficient temperature gradient of 5.2 °C across the WTEG is therefore achieved when contacted with the wrist at a room temperature of 26.3 °C. The prepared TEG generates a high power density of 10.02W m-2 at a vertical temperature gradient of 40 K. Meanwhile, the reliable energy harvesting promises a variation of less than 10% under the biaxial stretching up to 70% strain via leveraging the combined effects of the stretchability of knit fabric and geometry of TE strips. The knit fabric-supported TEG enables a seamless conformation to the skin as well as efficient body heat harvesting, which can provide sustainable energy to low power consumption wearable electronics.

New insights on wind turbine wakes from large‐eddy simulation: Wake contraction, dual nature, and temperature effects

AbstractLarge‐eddy simulation (LES) has been adopted to study wind turbine wakes because it can capture fine‐scale details of turbulent wind flows and interactions with wind turbines. Here, we use the LES version of the Weather Research and Forecasting (WRF) model with an actuator disk model to gain insights on several wake effects that have been traditionally difficult to measure. The first finding is that the wake has a “dual nature,” meaning that the wind speed deficit behaves differently from the added turbulent kinetic energy (TKE) and the two are not co‐located in space. For example, the wind speed deficit peaks at hub height and reaches the ground within 8D (D is the rotor diameter), but added TKE peaks near the rotor tip and generally remains aloft. Second, temperature changes near the ground are driven by the added TKE in the rotor area and by atmospheric stability. The combination of these two factors determines the sign and intensity of the vertical heat flux divergence below the rotor, with convergence and warming associated with stable conditions and weak divergence and modest cooling with unstable conditions. Third, wakes do not expand indefinitely, as suggested by similarity theory applied to the wind speed deficit, but eventually stop expanding and actually contract, at different rates depending on atmospheric stability. The implication of these findings is that, in order to study wakes, it is not sufficient to focus on wind speed deficit alone, because TKE is also important and yet behaves differently from the wind speed deficit.

Transferring heat accumulated by asphalt pavement from inside to outside through carbon fibers

On the Qinghai-Tibet Plateau, intense solar irradiation will cause serious heat accumulation in asphalt pavement. Downward heat transfer will disturb the permafrost under subgrade, thus leading to a series of subgrade and pavement diseases. This makes construction of asphalt pavement in the permafrost regions of the Qinghai-Tibet Plateau a world-class challenge. This paper proposes for the first time a transversal heat transfer channel in asphalt pavement, which can quickly transfer heat accumulated in pavement to road shoulders. Finite element software ABAQUS was used to simulate the rutting plate with high axial thermal conductivity carbon fibers inserted transversely. Under the constant heat load of 500 W/m2, the heat flux near the carbon fibers showed the characteristic of convergence. The maximum transverse heat flux in the carbon fibers reached 4.97×107W/m2, which was 14 times that in the asphalt mixture. Meanwhile, the maximum vertical heat flux in the carbon fibers reduced to 2.62×106W/m2, which was 12 % less than the asphalt mixture. The effective thermal conductivity of composite specimens with different carbon fiber parameters was studied through control experiments. When the axial thermal conductivity, volume fraction and number of carbon fiber bundles increased to 100 W/m·°C, 10.72 % and 4 bundles respectively, the transverse ETC of composite specimens increased to 4.05 W/m·°C, 7.09 W/m·°C and 16.18 W/m·°C respectively. They were 3.12 times, 5.45 times and 12.45 times of the control group. The radial thermal conductivity of carbon fibers has a significant effect on the vertical ETC. When it decreased to 0.2 W/m·°C, the vertical ETC decreased by 6.15 % compared with the control group. The temperature variation trend of specimens were measured by a laboratory irradiation experiment. The maximum temperature difference between the control group and the experimental group at 0 mm was 5.81 °C, and the average temperature difference within 2 h was 5.14 °C. The maximum temperature difference at 50 mm was 3.84 °C and the average temperature difference within 2 h was 3.24 °C. The results showed that the transverse heat transfer channel can reduce downward heat transfer. The application of the channels will enhance transverse heat release and relieve heat accumulation of asphalt pavement. This can create a low temperature state for pavement structure and has a positive effect on maintaining the stability of the permafrost regions on the Qinghai-Tibet Plateau.

The Effect of Submeso Motions on the Budgets of the Mean Turbulent Kinetic Energy and Temperature Variance in the Stable Atmospheric Surface Layer

By considering turbulence observations in the atmospheric stable surface layer over complex terrain, we study the effect of submeso motions on the budgets of the mean turbulent kinetic energy (TKE) and (half) the temperature variance. Different averaging times are considered (i.e., 100 s and 30 min), to filter out or retain the submeso contributions to second-order moments. Furthermore, results are interpreted by introducing four parameters that express the relative submeso contribution to the TKE, the temperature variance, and the vertical fluxes of heat and momentum. Four regimes are identified according to these four submeso parameters and the budgets are evaluated for these regimes. A balance among production, buoyancy (for the TKE) and dissipation occurs for the two regimes characterized by small submeso contribution to the fluxes, while an unbalance occurs for the other two regimes, where the submeso contribution to the fluxes is large. Instead, the budgets are independent of the magnitude of the submeso contribution to the TKE and the temperature variance.

Rigorous scaling laws for internally heated convection at infinite Prandtl number

We prove rigorous scaling laws for measures of the vertical heat transport enhancement in two models of convection driven by uniform internal heating at infinite Prandtl number. In the first model, a layer of incompressible fluid is bounded by horizontal plates held at the same constant temperature and convection reduces the fraction of the total dimensionless heat input per unit volume and time escaping the layer through the bottom boundary. We prove that this fraction decreases no faster than O(R−2), where R is a “flux” Rayleigh number quantifying the strength of the internal heating relative to diffusion. The second model, instead, has a perfectly insulating bottom boundary, so all heat must escape through the top one. In this case, we prove that the Nusselt number, defined as the ratio of the total-to-conductive vertical heat flux, grows no faster than O(R4). These power-law bounds improve on exponential results available for fluids with finite Prandtl number. The proof combines the background method with a minimum principle for the fluid’s temperature and with Hardy–Rellich inequalities to exploit the link between the vertical velocity and temperature available at infinite Prandtl number.

Winter Vertical Diffusion Rates in the Arctic Ocean, Estimated From 7Be Measurements and Dissipation Rate Profiles

AbstractOcean turbulent mixing is a key process affecting the uptake and redistribution of heat, carbon, nutrients, oxygen and other dissolved gasses. Vertical turbulent diffusivity sets the rates of water mass transformations and ocean mixing, and is intrinsically an average quantity over process time scales. Estimates based on microstructure profiling, however, are typically obtained as averages over individual profiles. How representative such averaged diffusivities are, remains unexplored in the quiescent Arctic Ocean. Here, we compare upper ocean vertical diffusivities in winter, derived from the 7Be tracer‐based approach to those estimated from direct turbulence measurements during the year‐long Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) expedition, 2019–2020. We found that diffusivity estimates from both methods agree within their respective measurement uncertainties. Diffusivity estimates obtained from dissipation rate profiles are sensitive to the averaging method applied, and the processing and analysis of similar data sets must take this sensitivity into account. Our findings indicate low characteristic diffusivities around 10−6 m2 s−1 and correspondingly low vertical heat fluxes.