The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

During the 1990s, a large number of new tidal atlases were developed, primarily to provide accurate tidal corrections for satellite altimetry applications. During this decade, the French tidal group (FTG), led by C. Le Provost, produced a series of finite element solutions (FES) tidal atlases, among which FES2004 is the latest release, computed from the tidal hydrodynamic equations and data assimilation. The aim of this paper is to review the state of the art of tidal modelling and the progress achieved during this past decade. The first sections summarise the general FTG approach to modelling the global tides. In the following sections, we introduce the FES2004 tidal atlas and validate the model against in situ and satellite data. We demonstrate the higher accuracy of the FES2004 release compared to earlier FES tidal atlases, and we recommend its use in tidal applications. The final section focuses on the new dissipation term added to the equations, which aims to account for the conversion of barotropic energy into internal tidal energy. There is a huge improvement in the hydrodynamic tidal solution and energy budget obtained when this term is taken into account.

https://hpiers.obspm.fr/combinaison/documentation/articles/Modelling_Ocean_Tides_Lyard_FES2004.pdf

Modelling the global ocean tides: modern insights from FES2004

▪ AbstractThe coexistence in the deep ocean of a finite, stable stratification, a strong meridional overturning circulation, and mesoscale eddies raises complex questions concerning the circulation energetics. In particular, small-scale mixing processes are necessary to resupply the potential energy removed in the interior by the overturning and eddy-generating process. A number of lines of evidence, none complete, suggest that the oceanic general circulation, far from being a heat engine, is almost wholly governed by the forcing of the wind field and secondarily by deep water tides. In detail however, the budget of mechanical energy input into the ocean is poorly constrained. The now inescapable conclusion that over most of the ocean significant “vertical” mixing is confined to topographically complex boundary areas implies a potentially radically different interior circulation than is possible with uniform mixing. Whether ocean circulation models, either simple box or full numerical ones, neither explic...

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Vertical mixing, energy, and the general circulation of the oceans

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

The influence of the bending rigidity of a flexible heaving wing on its propulsive performance in a two-dimensional imposed parallel flow is investigated in the inviscid limit. Potential flow theory is used to describe the flow over the flapping wing. The vortical wake of the wing is accounted for by the shedding of point vortices with unsteady intensity from the wing’s trailing edge. The trailing-edge flapping amplitude is shown to be maximal for a discrete set of values of the rigidity, at which a resonance occurs between the forcing frequency and a natural frequency of the system. A quantitative comparison of the position of these resonances with linear stability analysis results is presented. Such resonances induce maximum values of the mean developed thrust and power input. The flapping efficiency is also shown to be greatly enhanced by flexibility.

Resonance and propulsion performance of a heaving flexible wing

Using linear wave theory, the rate at which energy is converted into internal gravity waves by the interaction of the barotropic tide with topography in an ocean is calculated. Bell’s formula for the conversion rate is extended to the case of an ocean of finite depth H with weak two-dimensional topography h(x, y) and arbitrary buoyancy frequency N(z). Approximate solutions are computed using the WKB method, which reduce to the previous result for an ocean of infinite depth with constant stratification. The conversion rate for a finite-depth ocean can be substantially smaller than the infinite-ocean prediction when the length scale of the topography is of the same order as the horizontal wavelength of the internal tide. The conversion rate for two-dimensional Gaussian seamounts is calculated. Using observed statistics for the distribution of seamounts, the authors estimate 1/4 GW of conversion for a square of ocean floor of side 1000 km.

/pdf/conversion-of-the-barotropic-tide-4xp7dub3j4.pdf

Conversion of the Barotropic Tide

A two-dimensional model for the flapping of an elastic flag under axial flow is described. The vortical wake is accounted for by the shedding of discrete point vortices with unsteady intensity, enforcing the regularity condition at the flag's trailing edge. The stability of the flat state of rest as well as the characteristics of the flapping modes in the periodic regime are compared successfully to existing linear stability and experimental results. An analysis of the flapping regime shows the co-existence of direct kinematic waves, travelling along the flag in the same direction as the imposed flow, and reverse dynamic waves, travelling along the flag upstream from the trailing edge.

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Vortex shedding model of a flapping flag

We obtain an analytic solution for the generation of internal gravity waves by tidal flow past a vertical barrier of height $b$ in a uniformly stratified ocean of depth $h\,{>}\,b$ and buoyancy frequency $N$ . The radiated power (watts per metre of barrier) is $$\textstyle\frac{1}{4} \pi \rho_0 b^2 U^2 N \sqrt{1-(f/\omega)^2}M(b/h),$$ where $\rho_0$ is the mean density of seawater, $U \cos (\omega t)$ the tidal velocity, and $f$ the Coriolis frequency. The function $M(b/h)$ increases monotonically with $M(0)\,{=}\,1$ , $M(0.92)\,{=}\,2$ and $M(1)\,{=}\,\infty$ . As $b/h \,{\to}\, 1$ , $M$ diverges logarithmically and consequently the radiated power grows as $\ln[(h-b)/b]$ . We also calculate the conversion in a realistically stratified ocean with strongly non-uniform buoyancy frequency, $N(z)$ . A rough approximation to the radiated power in this case is $$\textstyle\frac{1}{4} \pi \rho_0 b^2 U^2 N(b) \sqrt{1- (f/\omega)^2} M(B/\pi),$$ where $N(b)$ is the buoyancy frequency at the tip of the ridge and $B$ is the height of the ridge in WKB coordinates. (The WKB coordinate is normalized so that the total depth of the ocean is $\pi$ .) The approximation above is an over-estimate of the actual radiation by as much as 20% when $B/\pi \,{\approx}\, 0.8$ . But the formula correctly indicates the strong dependence of conversion on stratification through the factor $N(b)$ .

Tidal conversion at a very steep ridge

This paper formulates a model of mixing in a stratified and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic ‘mixing-length’ arguments lead to a coupled set of parabolic differential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratification. The instability results in the formation of steps and interfaces in the buoyancy profile. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diffusion.The turbulent mixing process can take one of three forms depending on the strength of the initial stratification. When the stratification is weak, instability is not present and mixing smoothly homogenizes the buoyancy. At intermediate strengths of stratification, layers and interfaces form rapidly over a substantial interior region bounded by edge layers associated with the fluxless condition of the boundaries. The interior pattern subsequently develops more slowly as interfaces drift together and merge; simultaneously, the edge layers advance inexorably into the interior. Eventually the edge layers meet in the middle and the interior pattern of layers is erased. Any remaining structure subsequently decays smoothly to the homogeneous state. Both the weak and intermediate stratified cases are in agreement with experimental phenomenology. The model predicts a third case, with strong stratification, not yet found experimentally, where the central region is linearly stable and no steps form there. However, the edge layers are unstable; mixing fronts form and then erode into the interior.

Stefan G. Llewellyn Smith

Papers

Resonance and propulsion performance of a heaving flexible wing

Conversion of the Barotropic Tide

Vortex shedding model of a flapping flag

Tidal conversion at a very steep ridge

Dynamics of interfaces and layers in a stratified turbulent fluid