Unsteady Ekman--Stokes dynamics: implications for surface-wave induced drift of floating marine litter

We examine Stokes drift and wave‐induced transport of floating marine litter on the surface of a rotating ocean with a turbulent mixed layer. Due to Coriolis‐Stokes forcing and surface wave stress, a second‐order Eulerian‐mean flow forms, which must be added to the Stokes drift to obtain the correct wave‐induced Lagrangian velocity. We show that this wave‐driven Eulerian‐mean flow can be expressed as a convolution between the unsteady Stokes drift and an “Ekman‐Stokes kernel.” Using this convolution, we calculate the unsteady wave‐driven contribution to particle transport. We report significant differences in both direction and magnitude of transport when the Eulerian‐mean Ekman‐Stokes velocity is included

Prudence and Precaution for Natural Resource and Climate Uncertainty

Ploeg, F. van der [Promotor

The induced mean flow of surface, internal and interfacial gravity wave groups

Although the leading-order motion of waves is periodic - in other words backwards and forwards - many types of waves including those driven by gravity induce a mean flow as a higher-order effect. It is the induced mean flow of three types of gravity waves that this thesis examines: surface (part I), internal (part II) and interfacial gravity waves (part III). In particular, this thesis examines wave groups. Because they transport energy, momentum and other tracers, wave-induced mean flows have important consequences for climate, environment, air traffic, fisheries, offshore oil and other industries. In this thesis perturbation methods are used to develop a simplified understanding of the physics of the induced mean flow for each of these three types of gravity wave groups. Leading-order estimates of different transport quantities are developed. For surface gravity wave groups (part I), the induced mean flow consists of two compo- nents: the Stokes drift dominant near the surface and the Eulerian return flow acting in the opposite direction and dominant at depth. By considering subsequent orders in a separation of scales expansion and by comparing to the Fourier-space solutions of Longuet-Higgins and Stewart (1962), this thesis shows that the effects of frequency dis- persion can be ignored for deep-water waves with realistic bandwidths. An approximate depth scale is developed and validated above which the Stokes drift is dominant and below which the return flow wins: the transition depth. Results are extended to include the effects of finite depth and directional spreading. Internal gravity wave groups (part II) do not display Stokes drift, but a quantity analogous to Stokes transport for surface gravity waves can still be developed, termed the “divergent- flux induced flow” herein. The divergent-flux induced flow it itself a divergent flow and induces a response. In a three-dimensional geometry, the divergent-flux induced flow and the return flow form a balanced circulation in the horizontal plane with the former transporting fluid through the centre of the group and the latter acting in the opposite direction around the group. In a two-dimensional geometry, stratification inhibits a balanced circulation and a second type of waves are generated that travel far ahead and in the lee of the wave group. The results in the seminal work of Bretherton (1969b) are thus validated, explicit expressions for the response and return flow are developed and compared to numerical simulations in the two-dimensional case. Finally, for interfacial wave groups (part III) the induced mean flow is shown to behave analogously to the surface wave problem of part I. Exploring both pure interfacial waves in a channel with a closed lid and interacting surface and interfacial waves, expressions for the Stokes drift and return flow are found for different configurations with the mean set-up or set-down of the interface playing an important role.This thesis is not currently available in OR

Managing and Harnessing Volatile Oil Windfalls

Three funds are necessary to manage an oil windfall: intergenerational, liquidity, and investment funds. The optimal liquidity fund is bigger if the windfall lasts longer and oil price volatility, prudence, and the GDP share of oil rents are high and productivity growth is low. The paper applies the authors' theory to the windfalls of Norway, Iraq, and Ghana. The optimal size of Ghana's liquidity fund is tiny even with high prudence. Norway's liquidity fund is bigger than Ghana's. Iraq's liquidity fund is colossal relative to its intergenerational fund. Only with capital scarcity, part of the windfall should be used for investing to invest. The paper illustrates how this can speed up the process of development in Ghana despite domestic absorption constraints. © 2013 International Monetary Fund

The laminar seabed thermal boundary layer forced by propagating and standing free-surface waves

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves

The risk-adjusted carbon price

The social cost of carbon is the expected present value of damages from emitting one ton of carbon today. We use perturbation theory to derive an approximate tractable expression for this cost adjusted for climatic and economic risk. We allow for different aversion to risk and intertemporal fluctuations, skewness and dynamics in the risk distributions of climate sensitivity and the damage ratio, and correlated shocks. We identify prudence, insurance, and exposure effects, reproduce earlier analytical results, and offer analytical insights into numerical results on the effects of economic and damage ratio uncertainty and convex damages on the optimal carbon price.Environmental Fluid Mechanic

The influence of spectral bandwidth and shape on deep-water wave breaking onset

Deep-water surface wave breaking affects the transfer of mass, momentum, energy and heat between the air and sea. Understanding when and how the onset of wave breaking will occur remains a challenge. The mechanisms that form unforced steep waves, i.e. nonlinearity or dispersion, are thought to have a strong influence on the onset of wave breaking. In two dimensions and in deep water, spectral bandwidth is the main factor that affects the roles these mechanism play. Existing studies, in which the relationship between spectral bandwidth and wave breaking onset is investigated, present varied and sometimes conflicting results. We perform potential-flow simulations of two-dimensional focused wave groups on deep water to better understand this relationship, with the aim of reconciling existing studies. We show that the way in which steepness is defined may be the main source of confusion in the literature. Locally defined steepness at breaking onset reduces as a function of bandwidth, and globally defined (spectral) steepness increases. The relationship between global breaking onset steepness and spectral shape (using the parameters bandwidth and spectral skewness) is too complex to parameterise in a general way. However, we find that the local surface slope of maximally steep non-breaking waves, of all spectral bandwidths and shapes that we simulate, approaches a limit of. This slope-based threshold is simple to measure and may be used as an alternative to existing kinematic breaking onset thresholds. There is a potential link between slope-based and kinematic breaking onset thresholds, which future work should seek to better understand

The laminar seabed thermal boundary layer forced by propagating and standing free-surface waves

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.Environmental Fluid Mechanic

The Impact of the Spectral Tail on the Evolution of the Kurtosis of Random Seas

We perform simulations of random seas based on narrow-banded spectra with directional spreading. Our wavefields are spatially homogeneous and nonstationary in time. We truncate the spectral tail for the initial conditions at different cutoff wavenumbers to assess the impact of the spectral tail on the kurtosis and spectral evolution. We consider two cases based on truncation of the wavenumber tail at |k|/kp = 2.4 and |k|/kp = 6. Our simulations indicate that the peak kurtosis value increases if the tail is truncated at |k|/kp = 2.4 rather than |k|/kp = 6. For the case with a wavenumber cutoff at |k|/kp = 2.4, augmented kurtosis is accompanied by comparatively more aggressive spectral changes including redevelopment of the spectral tail. Similar trends are observed for the case with a wavenumber cutoff at |k|/kp = 6, but the spectral changes are less substantial. Thus, the spectral tail appears to play an important role in a form of spectral equilibrium that reduces spectral changes and decreases the peak kurtosis value. Our findings suggest that care should be taken when truncating the spectral tail for the purpose of simulations/experiments. We also find that the equation of Fedele (2015, “On the Kurtosis of Deep-Water Gravity Waves,” J. Fluid Mech., 782, pp. 25–36) provides an excellent estimate of the peak kurtosis value. However, the bandwidth parameter must account for the spectral tail to provide accurate estimates of the peak kurtosis.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Environmental Fluid Mechanic