| 1. |
- Arneborg, Lars, 1969, et al.
(författare)
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MIXING IN FJORDS AND THE RELATION BETWEEN LOCAL ENERGY DISSIPATION AT A FJORD SILL AND RADIATED INTERNAL TIDES
- 2012
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Ingår i: TOS/ASLO/AGU 2012 Ocean Science Meeting abstract book, Salt Lake City, USA.
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Konferensbidrag (refereegranskat)abstract
- Diapycnal mixing in a sill fjord is vital for renewal of the deep water inside the sill. Tides can cause such mixing, both by local turbulence at the sill associated with supercritical baroclinic flow and internal hydraulic jumps, and by radiation of internal tides away from the sill that dissipate elsewhere and cause turbulence and mixing there. Previous studies tend to look at these two processes as independent of each others, whereas they in reality are closely linked: The internal tide generation depends on the hydraulic conditions at the fjord sill, and the internal hydraulic jump strength depends on the upstream and downstream radiated columnar disturbances which over time constitute the internal tides. An effort is done to link the hydraulic theory and the internal tide generation theory, and the result is compared to recent intensive observations over the Oslo fjord sill, including high-resolution microstructure profiler transects and mooring data on and inside the sill.
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| 2. |
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| 3. |
- Bengtsson, Tommy, et al.
(författare)
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Famines and mortality crises in 18th to 19th century southern Sweden
- 2011
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Ingår i: Genus: Journal of Population Sciences. - Rome : Università degli Studi di Roma La Sapienza. - 2035-5556. ; 67:2, s. 119-139
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Tidskriftsartikel (refereegranskat)abstract
- Causality is an important but complicated issue, not only within social sciences in general but also within economic and historical demography. Here we are dealing with two different, but related, problems of causality. The first is to what extent the impact of food prices on mortality is biased when selecting on years with mortality crises. The second concerns the problem of mixing factors that directly and indirectly have an impact on mortality. Dealing with the first problem, we compare the effects of food prices on child and adult mortality when selecting on mortality crises with a standard approach without selection. When dealing with the second problem we use the additive hazards model, in combination with dynamic path analysis, which allows for investigating the mediating effect of intermediate covariates in a causal framework. We use individual level data from the Scanian Economic Demographic Database for five rural parishes for the period 1766 to 1865. Data on food prices refers to the local area of these parishes. The statistical analyses are performed in the R statistical computing environment, especially with the aid of the package eha. The main findings are that selecting on mortality crises created a large bias in the direction of overestimating the impact of food prices and that that the direct effects of food prices are dominating.
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| 4. |
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| 5. |
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| 6. |
- Broström, Göran, 1942-
(författare)
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Event history analysis with R
- 2012. - 1
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Bok (refereegranskat)abstract
- With an emphasis on social science applications, Event History Analysis with R presents an introduction to survival and event history analysis using real-life examples. Keeping mathematical details to a minimum, the book covers key topics, including both discrete and continuous time data, parametric proportional hazards, and accelerated failure times.Features:Introduces parametric proportional hazards models with baseline distributions like the Weibull, Gompertz, Lognormal, and Piecewise constant hazard distributions, in addition to traditional Cox regressionPresents mathematical details as well as technical material in an appendixIncludes real examples with applications in demography, econometrics, and epidemiologyProvides a dedicated R package, eha, containing special treatments, including making cuts in the Lexis diagram, creating communal covariates, and creating period statisticsA much-needed primer, Event History Analysis with R is a didactically excellent resource for students and practitioners of applied event history and survival analysis.
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| 7. |
- Broström, Göran, 1942-, et al.
(författare)
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Generalized linear models with clustered data : fixed and random effects models
- 2011
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Ingår i: Computational Statistics & Data Analysis. - : Elsevier BV. - 0167-9473 .- 1872-7352. ; 55:12, s. 3123-3134
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Tidskriftsartikel (refereegranskat)abstract
- The statistical analysis of mixed effects models for binary and count data is investigated. In the statistical computing environment R, there are a few packages that estimate models of this kind. The packagelme4 is a de facto standard for mixed effects models. The packageglmmML allows non-normal distributions in the specification of random intercepts. It also allows for the estimation of a fixed effects model, assuming that all cluster intercepts are distinct fixed parameters; moreover, a bootstrapping technique is implemented to replace asymptotic analysis. The random intercepts model is fitted using a maximum likelihood estimator with adaptive Gauss–Hermite and Laplace quadrature approximations of the likelihood function. The fixed effects model is fitted through a profiling approach, which is necessary when the number of clusters is large. In a simulation study, the two approaches are compared. The fixed effects model has severe bias when the mixed effects variance is positive and the number of clusters is large.
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| 8. |
- Broström, Göran, et al.
(författare)
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Note on Coriolis-Stokes force and energy
- 2014
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Ingår i: Ocean Dynamics. - : Springer Science and Business Media LLC. - 1616-7341 .- 1616-7228. ; 64:7, s. 1039-1045
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Tidskriftsartikel (refereegranskat)abstract
- In this study, we consider the origin of the Coriolis-Stokes (CS) force in the wave-averaged momentum and energy equations and make a short analysis of possible energy input to the ocean circulation (i.e., Eulerian mean velocity) from the CS force. Essentially, we find that the CS force appears naturally when considering vertically integrated quantities and that the CS force will not provide any energy input into the system for this case. However, by including the "Hasselmann force", we show some inconsistencies regarding the vertical structure of the CS force in the Eulerian framework and find that there is a distinct vertical structure of the energy input and that the net input strongly depends on whether the wave zone is included in the analysis or not. We therefore question the introduction of the "Hasselmann force" into the system of equations, as the CS force appears naturally in the vertically integrated equations or when Lagrangian vertical coordinates are used.
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| 9. |
- Broström, Linus, et al.
(författare)
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Broad Consent
- 2011
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Ingår i: Translational Stem Cell Research: Issues Beyond the Debate on the Moral Status of the Human Embryo. - 9781607619581 ; , s. 237-250
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Bokkapitel (övrigt vetenskapligt/konstnärligt)
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| 10. |
- Christensen, Kai H., et al.
(författare)
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Surface wave measurements using a ship-mounted ultrasonic altimeter
- 2013
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Ingår i: Methods in Oceanography. - : Elsevier BV. - 2211-1220. ; 6, s. 1-15
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Tidskriftsartikel (refereegranskat)abstract
- We present a method for measuring one-dimensional surface wave spectra using a ship-mounted ultrasonic altimeter in combination with a motion correction device. The instruments are mounted at the bow of the ship and provide high-resolution, local, wave information. We present results from three recent field studies. The results are compared with data from a conventional waverider buoy and, when in-situ observations are not available, with wave model analyses and satellite altimetry. We find good agreement with regard to integrated parameters such as significant wave height and mean period. Comparison with a waverider demonstrates fair agreement with regard to spectral shape, but the representation of the low frequency part depends on the quality of the motion correction data.
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| 11. |
- Drivdal, Magnus, et al.
(författare)
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Wave mixing of buoyant particles: Statfjord A accident as a case study
- 2014
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Ingår i: Ocean Science. - : Copernicus GmbH. - 1812-0784 .- 1812-0792. ; 10, s. 977-991
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Tidskriftsartikel (refereegranskat)abstract
- This study focuses on how wave–current and wave–turbulence interactions modify the transport of buoy- ant particles in the ocean. Here the particles can repre- sent oil droplets, plastic particles, or plankton such as fish eggs and larvae. Using the General Ocean Turbulence Model (GOTM), modified to take surface wave effects into account, we investigate how the increased mixing by wave breaking and Stokes shear production, as well as the stronger veering by the Coriolis–Stokes force, affects the drift of the particles. The energy and momentum fluxes, as well as the Stokes drift, depend on the directional wave spectrum ob- tained from a wave model. As a first test, the depth and ve- locity scales from the model are compared with analytical solutions based on a constant eddy viscosity (i.e., classical Ekman theory). Secondly, the model is applied to a case in which we investigate the oil drift after an oil spill off the west coast of Norway in 2007. During this accident the aver- age net drift of oil was observed to be both slower and more deflected away from the wind direction than predicted by oil- drift models. In this case, using wind and wave forcing from the ERA Interim archive it is shown that the wave effects are important for the resultant drift and have the potential to im- prove drift forecasting.
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| 12. |
- Edvinsson, Sören, 1953-, et al.
(författare)
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Old age, health and social inequality : exploring the social patterns of mortality in 19th Century Northern Sweden
- 2012
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Ingår i: Demographic Research. - 1435-9871. ; 26, s. 23-
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Tidskriftsartikel (refereegranskat)abstract
- BACKGROUNDSocial position is one of the major determinants of health. Less is known about its effect in historical contexts. Previous studies have shown surprisingly small effects of social class in working age populations. Not much is known about social differences in health among the elderly in history.OBJECTIVEThe present paper analyses social differences in health among the elderly (60+) in the Sundsvall region in northern Sweden during the 19th century. We investigate whether social mortality differences are particularly apparent in old age when unpropertied groups lost their most important asset for survival: their capacity to work.METHODSThe data, representing 9,535 fatal events, are analysed using a Cox regression model, assuming proportional hazards.RESULTSSocial class had no significant effect for women during the pre-industrial period, while only those with unknown social position had higher mortality among men. During the industrial period female mortality was lowest in the skilled working class and highest in the upper class. Social position was not significant for men in the full model. Urban mortality was 30% higher for women and 59% higher for men during the pre-industrial period compared to the peripheral parishes.CONCLUSIONSThe results lead us to question the accepted 'fact' of social health differences as a historical constant. Higher social position did not lead to better survival, and social differences in mortality did not increase in old age, despite the fact that the elderly were a highly vulnerable group. Instead, the spatial aspects of mortality were important, particularly during the pre-industrial period.
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| 13. |
- Holmberg, Henrik, 1976-
(författare)
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Generalised linear models with clustered data
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In situations where a large data set is partitioned into many relativelysmall clusters, and where the members within a cluster have some common unmeasured characteristics, the number of parameters requiring estimation tends to increase with sample size if a fixed effects model is applied. This fact causes the assumptions underlying asymptotic results to be violated. The first paper in this thesis considers two possible solutions to this problem, a random intercepts model and a fixed effects model, where asymptoticsare replaced by a simple form of bootstrapping. A profiling approach is introduced in the fixed effects case, which makes it computationally efficient even with a huge number of clusters. The grouping effect is mainly seen as a nuisance in this paper. In the second paper the effect of misspecifying the distribution of the random effects in a generalised linear mixed model for binary data is studied. One problem with mixed effects models is that the distributional assumptions about the random effects are not easily checked from real data. Models with Gaussian, logistic and Cauchy distributional assumptions are used for parameter estimation on data simulated using the same three distributions. The effect of these assumptions on parameter estimation is presented. Two criteria for model selection are investigated, the Akaike information criterion and a criterion based on a chi-square statistic. The estimators for fixed effects parameters are quite robust against misspecification of the random effects distribution, at least with the distributions used in this paper. Even when the true random effects distribution is Cauchy, models assuming a Gaussian or a logistic distribution regularly produce estimates with less bias.
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| 14. |
- Holmberg, Henrik, 1976-
(författare)
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Generalized linear models with clustered data
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In situations where a large data set is partitioned into many relatively small groups, and where the members within a group have some common unmeasured characteristics, the number of parameters requiring estimation tends to increase with sample size if a fixed effects model is applied. This fact causes the assumptions underlying asymptotic results to be violated. The first paper in this thesis considers two possible solutions to this problem, a random intercepts model and a fixed effects model, where asymptotics are replaced by a simple form of bootstrapping. A profiling approach is introduced in the fixed effects case, which makes it computationally efficient even with a huge number of groups. The grouping effect is mainly seen as a nuisance in this paper. In the second paper the effect of misspecifying the distribution of the random effects in a generalized linear mixed model for binary data is studied. One problem with mixed effects models is that the distributional assumptions about the random effects are not easily checked from real data. Models with Gaussian, logistic and Cauchy distributional assumptions are used for parameter estimation on data simulated using the same three distributions. The eect of these assumptions on parameter estimation is presented. Two criteria for model selection are investigated, the Akaike information criterion and a criterion based on a X2 statistic. The estimators for fixed effects parameters are quite robust against misspecification of the random effects distribution, at least with the distributions used in this paper. Even when the true random effects distribution is Cauchy, models assuming a Gaussian or a logistic distribution regularly produce estimates with less bias. In the third paper the results from the first two papers are applied to infant mortality data. We found that there was significant clustering of infant mortality in the Skellefteå region in the years 1831-1890. An "ad hoc" method for comparing the magnitude of unexplained clustering after a model is applied is also presented. The last paper of this thesis is concerned with the problem of testing for spatial clustering caused by autocorrelation. A test that is robust against heteroscedasticity is proposed. In a simulation study the properties of the proposed statistic, K, are investigated. The power of the test based on K is compared to that of Moran's I in the simulation study. Both tests are then applied to mortality data from Swedish municipalities.
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| 15. |
- Holmberg, Henrik, et al.
(författare)
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On statistical methods for clustering : A case study on infant mortality, northern Sweden, 1831-1890
- 2012
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Ingår i: Biodemography and Social Biology. - Routledge : Informa UK Limited. - 1948-5565 .- 1948-5573. ; 58, s. 173-184
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Tidskriftsartikel (refereegranskat)abstract
- This article considers the interfamily clustering of infant mortality (defined as mortal- ity during the first year of life). We developed and evaluated statistical tools to detect clustering and a measure to quantify excess clustering for nineteenth-century data from Skellefteå, Sweden. The detection was performed using the standard methods of gener- alized linear models and logistic regression. The index of clustering was constructed by comparing the observed numbers of families with specific numbers of deaths to the cor- responding observed numbers, after correcting for explanatory variables. To the best of our knowledge, no clustering index of this kind has ever been created.
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| 16. |
- Norberg, Margareta, et al.
(författare)
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Who is using snus? - Time trends, socioeconomic and geographic characteristics of snus users in the ageing Swedish population
- 2011
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Ingår i: BMC Public Health. - London : BioMed Central. - 1471-2458. ; 11, s. 929-
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Tidskriftsartikel (refereegranskat)abstract
- BACKGROUND: The prevalence of smoking in Sweden has decreased in recent decades, and is now among the lowest in the world. During the same period, the use of Swedish moist oral snuff, a smokeless tobacco called snus, has increased. Few studies have evaluated time trends of the socioeconomic and geographic characteristics of snus users in Sweden. This paper contributes to filling that gap.METHODS: This study utilized the Linnaeus Database, which links national registers with comprehensive individual data on socioeconomic status (SES) to health data from a large ongoing health survey, the Västerbotten Intervention Programme (VIP). The VIP targets the entire middle-aged population of Västerbotten county at ages 40, 50 and 60 years with yearly cross-sectional surveys including self-reported data on tobacco habits. Time trends of snus use among 92,563 VIP-participants across different areas of residence and smoking groups were investigated graphically. Logistic regression was performed to estimate the associations between SES and geographical variables and current use versus non-use of snus.RESULTS: Overall, in parallel to decreasing smoking, the increasing trend of snus use in this middle-aged population continues, particularly in 40-year-olds. In both genders, the highest prevalence of snus use was observed among previous smokers. The prevalence of snus use also increased over time among smokers, and was consistently higher compared to those who had never smoked. Among males - both those who had never smoked and previous smokers - low education (OR 1.21, 95%CI 1.06-1.40 and OR 1.28, 95%CI 1.14-1.43), living alone (OR 1.16, 95%CI 1.07-1.27 and OR 1.13, 95%ci 1.04-1.23), low income and living in rural areas was associated with using snus, while this was not seen among male current smokers. Among women, living alone was associated with using snus irrespective of smoking habits. Among female smokers, the OR for snus use increased with higher education.CONCLUSIONS: A disadvantaged social profile and also higher prevalence in rural areas is observed among male snus users who had never smoked or were previous smokers. Among male smokers there was no association between SES and use of snus. The prevalence of snus use among women is increasing, but is still considerably lower than that of men. The association between snus and SES characteristics is less pronounced among women, although snus is clearly linked to living alone. These patterns should be taken into consideration in tobacco control policies.
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| 17. |
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| 18. |
- Rohrs, J., et al.
(författare)
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Observation-based evaluation of surface wave effects on currents and trajectory forecasts
- 2012
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Ingår i: Ocean Dynamics. - : Springer Science and Business Media LLC. - 1616-7341 .- 1616-7228. ; 62:10-12, s. 1519-1533
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Tidskriftsartikel (refereegranskat)abstract
- Knowledge of upper ocean currents is needed for trajectory forecasts and is essential for search and rescue operations and oil spill mitigation. This paper addresses effects of surface waves on ocean currents and drifter trajectories using in situ observations. The data set includes colocated measurements of directional wave spectra from a wave rider buoy, ocean currents measured by acoustic Doppler current profilers (ADCPs), as well as data from two types of tracking buoys that sample the currents at two different depths. The ADCP measures the Eulerian current at one point, as modelled by an ocean general circulation model, while the tracking buoys are advected by the Lagrangian current that includes the wave-induced Stokes drift. Based on our observations, we assess the importance of two different wave effects: (a) forcing of the ocean current by wave-induced surface fluxes and the Coriolis-Stokes force, and (b) advection of surface drifters by wave motion, that is the Stokes drift. Recent theoretical developments provide a framework for including these wave effects in ocean model systems. The order of magnitude of the Stokes drift is the same as the Eulerian current judging from the available data. The wave-induced momentum and turbulent kinetic energy fluxes are estimated and shown to be significant. Similarly, the wave-induced Coriolis-Stokes force is significant over time scales related to the inertial period. Surface drifter trajectories were analysed and could be reproduced using the observations of currents, waves and wind. Waves were found to have a significant contribution to the trajectories, and we conclude that adding wave effects in ocean model systems is likely to increase predictability of surface drifter trajectories. The relative importance of the Stokes drift was twice as large as the direct wind drag for the used surface drifter.
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| 19. |
- Rohrs, J., et al.
(författare)
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Wave-induced transport and vertical mixing of pelagic eggs and larvae
- 2014
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Ingår i: Limnology and Oceanography. - : Wiley. - 0024-3590. ; 59:4, s. 1213-1227
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Tidskriftsartikel (refereegranskat)abstract
- The transport of pelagic plankton by wind-driven ocean currents and surface gravity waves is investigated for the example of Northeast Arctic cod eggs and larvae on the coast of northern Norway. Previous studies indicate that the wave-induced drift (i.e., Stokes drift) is relevant for the transport of particles in the upper ocean. We use an ocean general circulation model together with a numerical wave prediction model and a Lagrangian particle tracking model to calculate trajectories of fish eggs and larvae. Waves are considered not only for particle drift but also for the air-sea momentum flux, its contribution to the Coriolis force, and vertical mixing. The sample species provides the advantage that many of its physical and behavioral properties are well known (e.g., egg buoyancy), allowing investigation of vertical particle displacement by turbulent mixing in response to wind forcing and wave breaking. The approach accounting for particle mixing by breaking waves enhances agreement between observed and modeled egg profiles. Results also show a general shoreward transport of particles by the Stokes drift. This wave drift exhibits a more constant direction than the Eulerian current and hence stabilizes particle diffusion to favor a dominant direction. For the case of Northeast Arctic cod, waves concentrated model eggs and larvae on average 1.5 km closer to shore, which is 20% of their total distance to the coast. This increases the residence time of first-feeding larvae close to the spawning areas compared to earlier models.
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| 20. |
- Ulander, Martin, 1980-
(författare)
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Psychometric aspects of obstructive sleep apnea syndrome
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Introduction: Obstructive sleep apnea (OSA) is a common chronic disorder consisting of episodes with impaired breathing due to obstruction of the upper airways. Treatment with Continuous Positive Airway Pressure (CPAP) is a potentially effective treatment, but adherence is low. Several potential factors affecting adherence, e.g., subjective sleepiness and personality, are only quantifiable through questionnaires. Better knowledge about psychometric properties of such questionnaires might improve future research on CPAP adherence and thus lead to better treatment options.Aim: Study I: To describe the devlopment and initial testing of the Side Effects of CPAP treatment Inventory (SECI) questionnaire. Study II: To describe the prevalence of Type D personality in OSAS patients with CPAP treatment longer than 6 months and the association with self-reported side effects and adherence. Study III: To study whether any of the items in the Epworth Sleepiness Scale (ESS) exhibit differential item functioning and, if so, to which degree. Study IV: To examine the evolution of CPAP side effects over time; and prospectively assess correlations between early CPAP side effects and treatment adherence.Patients and Methods: In study I, SECI items were based on a literature review, an expert panel and interviews with patients. It was then mailed to 329 CPAP-treated OSAS patients. Based on this, a principal component analysis was performed, and SECI results were compared between adherent and non-adherent patients. In study II, the population consisted of 247 OSAS patients with ongoing CPAP treatment. The DS14 was used to assess the prevalence of type D personality, and SECI and adherence data from medical records were used to correlate Type D personality to side effects and adherence. In study III, the population consisted of pooled data from 1,167 subjects who had completed the ESS in five other studies. Ordinal regression and Rasch analysis were used to assess the existence of differential item functioning for age and gender. The cutoff for age was 65 years in the Rasch analysis. In study IV, SECI was sent to 186 subjects with newly diagnosed OSAS three times during the first year on CPAP. SECI results were followed over time within subjects, and were correlated to treatment dropout during the first year and machine usage time after 6 months.Results: SECI provides a valid and reliable instrument to measure side effects, and non-adherent patients have higher scores (i.e., were more bothered by side effects) than adherent patients (study I). Type D personality was prevalent in approximately 30 % of CPAP treated OSAS patients, and was associated to poorer objective and subjective adherence as well as more side effects (study II). Differential item functioning was present in items 3, 4 and 8 for age in both DIF analyses, and to gender in item 8 the Rasch analysis (study III). Dry mouth and increased number of awakenings were consistently associated to poorer adherence in CPAP treated patients. Side effects both emerged and resolved over time (study IV).Conclusions: Differences in previous research regarding side effects and CPAP adherence might be explained by differences in how side effects and adherence are defined. While some side effects are related to adherence, others are not. Side effects are furthermore not stable over time, and might be related to personality. ESS scores are also related to CPAP adherence according to previous research, but might be affected by other factors than sleepiness, such as age and possibly gender.
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| 21. |
- Weber, J. E. H., et al.
(författare)
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Stokes drift in internal equatorial Kelvin waves: Continuous stratification versus two-layer models
- 2014
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Ingår i: Journal of Physical Oceanography. - 0022-3670. ; 44:2, s. 591-599
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Tidskriftsartikel (refereegranskat)abstract
- The Stokes drift in long internal equatorial Kelvin waves is investigated theoretically for an inviscid fluid of constant depth. While the Stokes drift in irrotational waves is positive everywhere in the fluid, that is, directed along the phase velocity, this is not always the case for internal Kelvin waves, which possess vorticity. For constant Brunt-Väisälä frequency, the Stokes drift in such waves is sinusoidal in the vertical with a negative value in the middle of the layer for the first baroclinic mode. For a pycnocline that is typical of the equatorial Pacific, this study finds for the first mode that the largest negative Stokes drift velocity occurs near the depth where the Brunt-Väisälä frequency has its maximum. Here, estimated drift values are found to be on the same order of magnitude as those observed in the Pacific Equatorial Undercurrent at the same level. In contrast, a two-layer model with constant density in each layer yields a positive Stokes drift in both layers. This contradicts the fact that, as shown in this paper, the vertically integrated Stokes drift (the Stokes flux) must be zero for arbitrary Brunt-Väisälä frequency. © 2014 American Meteorological Society.
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| 22. |
- Wåhlin, Anna, 1970, et al.
(författare)
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Horizontal convection in water heated by infrared radiation and cooled by evaporation : scaling analysis and experimental results
- 2010
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Ingår i: Tellus. Series A, Dynamic meteorology and oceanography. - : Stockholm University Press. - 0280-6495 .- 1600-0870. ; 62:2, s. 154-169
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Tidskriftsartikel (refereegranskat)abstract
- An experimental study of horizontal convection with a free surface has been conducted. Fresh water was heated from above by an infrared lamp placed at one end of a tank, and cooled by evaporation as the water moved away from the heat source. The heat radiated from the lamp was absorbed in a thin (less than 1 mm) layer next to the surface, and then advected and diffused away from the lamp region. Latent heat loss dominated the surface cooling processes and accounted for at least 80% of the energy loss. The velocity and temperature fields were recorded with PIV technology, thermometers and an infrared camera. In similarity with previous horizontal convection experiments the measurements showed a closed circulation with a gradually cooling surface current moving away from the lamp. Below the surface current the water was stably stratified with a comparatively thick and slow return current. The thickness and speed, and hence the mass transport, of the surface- and the return current increased with distance from the lamp. The latent cooling at the free surface gives a heat flux which increases with the temperature difference between the surface water and the air above it. Hence the surface temperature relaxes towards an equilibrium value, for which the heat flux is zero. The main new result is a scaling law, taking into account this relaxation boundary condition for the surface temperature. The new scaling includes a (relaxation) length scale for the surface temperature, equivalent to the distance the surface current travels before it has lost the heat that was gained underneath the lamp. The length scale increases with the forcing strength and the (molecular) thermal diffusivity but decreases with the strength of the relaxation. Numerical simulations of this problem for a shallow tank have also been performed. The velocity and temperature in the laboratory and numerical experiments agree with the scaling laws in the upper part of the tank, but not in the lower.
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