3.1.3.2.1.1. The Problem¶

• The theoretical problem is difficult because we are modelling a turbulent airflow over a surface that varies in space and time.

• The experimental problem is difficult because most instrumentation cannot directly measure the growth of waves by wind.

3.1.3.2.1.2. The History¶

• Jeffreys (1924) started by assuming that airflow over a wave was sheltered on it’s lee side. This would give a pressure difference, so work was done by the wind. However, experiments on solid waves showed that this pressure difference was too small to account for the growth rates.

• Moved on to assume that waves were generated by resonance. Two approaches:

  • Miles (1957) “resonant interaction between pressure fluctuations and free surface waves”

  • Phillips (1957) “resonant forcing of surface waves by turbulent pressure fluctuations”

• Miles (1957) resonant interaction between pressure fluctuations and free surface waves was more promising because it implies exponential growth and it is of the order of the density ratio of air and water.

• Problems with Miles:

  • used quasi-laminar approach - airflow is inviscid and turbulence only maintains the shear flow.

  • neglected non-linear effects, such as wave-mean flow interaction (which are important at the critical height where the phase speed and wind speed match)

  • Experiments by Dobson (1971) show rate of energy transfer is an order of magnitude higher than Miles predicts

  • Recent experiments show Miles underpredicts energy transfer for low frequency waves with a wind speed close to phase speed

• Solutions to Miles’ problems, use mixing length turbulence model:

  • model turbulent boundary layer over a moving surface, using mixing length turbulence model for Reynolds stresses, for the effect of small scale turbulence on wave growth. This allows:

    • interaction of wave induced flow with mean flow

    • boundary layer turbulence

• What Miles didn’t show is then shown:

  • diffusion of momentum is so large with turbulence included that Miles critical mechanism is ineffective

  • in adverse winds or when waves are propagating faster than wind speed, damping is present (Miles showed no damping). However no field observations show damping - presumably because damping time scales are quite long.

• Problem with mixing length turbulence model:

  • assumes momentum transport caused by turbulence is the fastest process in the fluid - not justified for low frequency waves, which interact with large eddies, whose eddy turnover time is larger than the wave period

  • inadequate for slowly propagating waves. Far away from the surface, turbulence is slow moving and doesn’t have time to transfer momentum - causing truncation of the mixing length in the outer-layer. Energy transfer from wind to slow waves only occurs in a thin layer above the surface. The mechanism is non-separated sheltering - Reynolds stresses cause a thickening of the boundary layer on the leeside - This is similar to Jeffreys (1924)

• Use instead a rapid distortion approach:

  • critical layer mechanism is only relevant for very fast moving ocean waves

• Rapid distortion approach may overestimate the effect of eddies:

  • Use eddy resolving model, which shows a critical layer at a wide range of dimensionless phase speeds (but only done for low Re). A rapid fall off of the wave-induced stress was seen at the critical height.

  • In-situ observation also shows a critical layer (where the wave-induced stress shows a jump), but only for dimensionless phase speeds > 16.

  • reason for the over-estimation of eddies by rapid distortion approach: large eddies too slow to transport momentum - outer layer is approximately inviscid and only a thin layer above the surface applies

• Instead of rapid distortion approach, use truncated mixing length model with turbulent time scale:

  • Turbulent timescale redefined to give a longer timescale than rapid distortion model

  • Mixing length truncated

  • Resulting growth rate is a function of the Miles critical layer effect and a small damping term caused by the inner layer viscosity. This is ok for short wavelengths, but not for longer wavelengths.

3.1.3.2.2.1. The Problem¶

Quasi-laminar Miles theory is fairly good at predicting growth rate. However, it doesn’t include non-linearity.

Sources of non-linearity:

• In wind profile: Possible change in wind profile as ocean waves are evolving - deviations from the turbulent profile due to a flat plate are expected - profile controlled by both turbulent and wave-induced momentum flux. Momentum transfer from wind to wave can be so large that the wave-induced stress is a substantial fraction of the turbulent stress.

• In surface waves: Energy transfer from air to waves is probably affected by the sea state. Probably have strong coupling between the surface wave and the atmospheric turbulent boundary layer.

3.1.3.2.2.2. Observation¶

• Measurement confirms that the drag coefficient is dependent on the sea state.

• Quasi-linear theory of wind-wave generation keeps track of the evolution of the sea state and it’s effects on the wind profile and follows Miles theory.

• Parametrisation of roughness length in terms of wave-induced stress shows fair agreement with observed roughness. Can determine at every timestep what the momentum transfer is from air to waves.

3.1.3.2.2.3. Wave age¶

• Short waves are the fastest growing, so wave induced stress is largely determined by the high frequency spectrum. Also young windsea shows steeper short waves than old windsea. Nobody really know why this is. Four wave interactions and Stokes drift may be possible mechanisms.

3.1.3.2.2.4. Linearity¶

• Quasi-linear theory assumes short waves are linear, but they are probably steep. Therefore the non-linear process of air separation similar to Jeffreys (1924) may play a role in air-sea momentum transfer. This may explain the dependence of drag coefficient in sea state.

• Miles quasi-laminar theory assumed that each spectral component is linear. Tsimring (1983) studied two wave components and mean airflow finding that the effect is proportional to the air-sea density ratio times the square of the wave spectrum.

3.1.3.2.2.5. Evidence in the field of sea state dependence on drag coefficient¶

• For windsea, it is possible to find a convincing parametrisation of sea state dependence of the surface stress. The drag coefficient and dynamic roughness under mixed-sea conditions remain difficult to parametrise.

3.1.3.2.3.1. Problem¶

In previous comments we have assumed airflow is constant, but it isn’t. Wind varies with a timescale longer than the wind generated waves, but still below the resolution of meteorological models and may affect wave growth.

3.1.3.2.3.2. Solutions¶

• It is common to assume energy transfer is a function of the nominal wind speed, U and the phase speed, c of the component. The wave growth is non-linear wrt U-c.

  • For c > Um (mean value) there is no interaction between wind and waves.

  • For c close to Um, a positive fluctuation in wind speed will result in enhanced wave growth, but a negative fluctuation will not give rise to reduced growth. Growing waves act as a rectifier - the diode effect

  • How much they gain in wave height depends on the variability of the wind field (% rms deviation from Um).

• Gain in wave height also depends on correlation time scale of the fluctuating wind:

  • if the timescale of the gustiness <= integration timestep (or spatial step), growth curve will be smooth

  • if timescale of gustiness > integration timestep, growth curve will be oscillatory around mean growth curve

While there are good practical and theoretical reasons for considering gustiness, operational models don’t consider it, because there isn’t much information on gustiness of input wind fields and it complicates validation

3.1.3.2.4.1. Damping of Low Frequency Swells¶

• There is uncertainty regarding the damping rates of low frequency swells due to lack of field observations, because the timescales of these swells are quite long.

3.1.3.2.4.2. Momentum Transfer for High Wind Speeds¶

• There is some debate over whether the drag coefficient within a hurricane or typhoon increases or decreases with wind speed. The reason for this is the lack of observations and there are many complex mechanisms occurring simultaneously, and they affect each other e.g.:

  • momentum

  • heat fluxes

  • spray

Three reasons why drag might be reduced in a hurricane:

• Aerodynamic explanations of reduced drag in a hurricane follow the logic of the wind skipping over the crests, so the aerodynamic roughness is reduced.

• Also as spray returns to the sea, it removes small waves (which hold the greatest wave induced stress) hence drag can be reduced in this way.

• Also it is claimed that extremely young seas like in a hurricane would have reduced drag.

Argument using only spray:

• Spray may stay close to the surface, suppressing turbulence, inhibiting momentum transfer to the sea - reducing drag

• Spray may be accelerated by the mean airflow - increasing drag

3.1.3.2.4.3. Quality of Modelled Wind Fields¶

• At present no model of the atmospheric boundary layer can justify the level of wind variability measured in the field in certain conditions

3.1.3.3.2.1. Common formulations for spectral wave models: waves and currents¶

• The effect of the interaction between waves and currents on bottom stress is not completely solved - i.e. whether the interaction is weak or strong.

3.1.3.3.2.2. Bottom roughness model for movable beds¶

• Hydraulic roughness is a combination of:

  • skin friction on individual grains

  • form drag due to bed forms

• Different experiments can produce different roughness values, by orders of magnitude, for large roughness elements or steep ripples, momentum transfer is dominated by vortex-shedding rather than random turbulence.

3.1.3.4.1.1. Spectral dissipation due to wave breaking¶

Two main approaches for modelling spectral dissipation:

• prior to breaking

  • probability model

  • quasi-saturated model

• after wave breaking

  • weak in the mean model

  • effective turbulent viscosity note: eddy viscosity approach is not generally accepted

  • bubble model

Validation is needed:

• radar

• aerial images

What does breaking depend on? There are many theories:

• balance of source terms - balance of wind generation and breaking dissipation

• spectral dissipation is a function of frequency relative to peak

• wave breaking has a maximum in the wavenumber domain

• max spectral restoration occurs at intermediate wavelengths, caused by breaking of larger dominant waves

• spectral dissipation is a function of wavenumber and spectral energy input

• whitecapping dissipation depends on larger scales

• two phase bubble dissipation also present

• dissipation effect measured by comparing non-breaking with breaking wave spectrum to determine the dissipation due to breaking

• breaking dependence on wind direction - directional spectrum

Conclusions:

1. No theoretical consensus exists concerning spectral dissipation of wave energy due to wave breaking

2. Theory strongly disagrees with experiment

3. Experimental results disagree with each other

3.1.3.4.1.2. Wave-turbulence interactions¶

• For wave periods > 10s viscosity has negligible effect - swells once generated were supposed to dissipate slowly due to the action of the wind, via the “sheltering theory” Jeffreys (1925)

• The sheltering theory was then replaced with eddy viscosity analogies. However, no evidence supports the idea that wave induced velocity profiles are unstable and become turbulent - except for the surface viscous layer (a few mm thick) and the wave bottom boundary layer. Therefore except in these two boundary layers, no theory justifies the use of eddy viscosity.

• Instead, apply rapid distortion theory to model the stretching of eddies by wave motion or consider the scattering of waves by turbulence. Assume:

  • Eddy turnover time is less than wave period or

  • The strain rate of turbulence by wave motion is more than that of the turbulence by itself

  • … and the turbulent velocity is much less than the wave-induced velocity

• A large part of the momentum flux may be carried by long-lived and stable Langmuir rolls, so the weak modulation of the turbulent flux by the waves is possible.

• Turbulence is probably strongest at the peak of the wave groups where the Stokes drift is largest. This effect produces turbulence at larger depths than that produced by whitecaps.

3.1.3.4.1.3. Wave-wave modulations¶

• Exchange of energy and momentum takes place between the wind the long waves via the short waves and not between the waves and the ocean circulation and the turbulence.

• Modulations of wind stress can be significant:

  • If the wind stress modulations are directed against the long waves, it dissipates them

  • If the wind stress modulations are directed with the long waves, it amplifies them

• Modelling using “rapid distortion theory” of the air showed that the wind stress effect dissipates swells.

• As well as wind stress modulation, there is also surface roughness modulation, which may contribute to the growth of the waves in the wind direction.

3.1.3.7.2.1. Time Domain¶

• Usually time-domain Boussinesq models

• Limited to spatial scales of around 10 wavelengths and operational nearshore wave prediction, as computational demands are prohibitive for large applications. Because:

  • Phase-resolving boundary conditions are usually not available

  • The need for wave field statistics - random directionally spread waves (as opposed to a single realisation) requires computation of multiple realisations

3.1.3.7.2.2. Spectral Domain¶

• (Complex) amplitude evolution or spectral models

• Superposition of plane waves (Fourier modes)

• Fourier models are attractive because:

  • They provide a natural continuation of the deep water approach.

  • Suited to handling processes of a statistical nature, e.g. dissipation and wind input.

• Similar to time-domain models, these are deterministic models and Monte Carlo simulations are needed for wave statistics

3.1.3.8.1.1. The Problem¶

Errors are caused by:

• CFL number, which quantifies the number of grid spaces traversed by a packet of energy in one timestep. If ocean scale wind-forcing is present, the resolution depends on the scale of the meteorological features and the wave sensitivity to those features.

• Relative resolution (i.e. the grid density w.r.t. what we are trying to resolve). Some geographic features may be completely left out in a 0.5 degree to 1 degree resolution global model.

3.1.3.8.1.2. Existing Solutions¶

• Just increase the grid density (!) from 1 degree to 0.5 degrees. This increases the computing time and the required temporal resolution (to maintain stability).

• High spectral resolution is mostly needed for swells and swell attenuation is really linear.

• Unstructured grids could provide efficient use of grid density, although may not be less diffusive.

• High resolution is needed when we are near the shore

3.1.3.8.1.3. Future Solutions¶

• Unstructured grids are expected to become more prevalent, since scales of variation are shorter near the shoreline. Offshore wave fields only vary on the scale larger than that of the wind field.

3.1.3.8.2.1. The Problem¶

Numerical diffusion is the smearing of wave energy during propagation due to discretisation. It is not to be confused with dissipation which implies energy loss. It is caused by the even-ordered truncation error in the finite difference scheme. It is different from numerical dispersion which is caused by the odd-ordered truncation error. It depends on:

• CFL number

• May also depend on propagation direction

3.1.3.8.2.2. Existing Solutions¶

There are two effects here:

1. Minimising numerical diffusion in the 1D case:

• Second order leapfrog scheme, which has zero numerical diffusion (WAM)

• Ultimate Quick-EST scheme and limiter, which is third order in 1D (WW3). Total variance diminishing limiters can be used to control wiggles.

2. The efficacy of the extension to 2D:

• Solve for both dimensions simultaneously

• Propagate each dimension in sequence (this decreases the order of accuracy)

• Implicit schemes in 2D, these are less efficient at oceanic scales (SWAN)

• Semi-Lagrangian schemes

3.1.3.8.2.3. Future Solutions¶

Semi-Lagrangian schemes are an attractive alternative to traditional Euler schemes. However, there are two difficulties:

1. Assuring mass conservation is generally less straightforward than with Euler schemes

2. Source/sink terms must be applied along the ray at the Lagrangian stage - doing this computationally efficiently is a challenge.

3.1.3.8.3.1. The Problem¶

The “Garden sprinkler effect”. This where we can’t resolve the spectral/frequency content and is due to numerical dispersion caused by truncation error. This is due to the odd ordered truncation error. It depends on:

• CFL number

• Relative resolution

Numerical dispersion can result in non-physical “wiggles” in the solution. We can counteract numerical dispersion by:

• Adding diffusion artificially

• Formulating diffusion and dispersion in roughly equal amounts.

3.1.3.8.3.2. Existing Solutions¶

• Nobody wants to increase spectral resolution for this - due to computational cost.

• Controllable diffusion can be added

• Grid point averaging can be used

• Or an angular diffusive operator

3.1.3.8.3.3. Future Solutions¶

• Let source/sink terms dictate frequency resolution and increase directional resolution as computing resources allow

• Numerics and physics needs to become coupled through some parametrisations of physical processes.

3.1.3.8.4.1. The Problem¶

1st order Euler method requires time steps of around a few minutes, which is too computationally expensive. Various solutions such as semi-implicit methods with limiters have been tried, but are sensitive to time step size, especially for initial wave growth.

3.1.3.8.4.2. Existing Solutions¶

Three solutions:

1. Dynamically adjustment of timestep using a limiter for the maximum timestep. For large scale applications this is good. However, for small scale applications (where wind and wave changes occur rapidly over the domain), this is bad because of the small timesteps involved.

2. Limiter made proportional to step size - but this could prevent convergence. Can also remove limiter proportional to step size.

3. Spreading numerical method - semi-analytical solution for integration source term, which includes wind-wave input, dissipation term, and exact non-linear energy transfer.

3.1.3.8.4.3. Future Solutions¶

• Alternative non-convergent limiters

• Prototype for convergent limiter with reduced time step dependencies