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- Author or Editor: Alastair D. Jenkins x

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## Abstract

A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of the Navier-Stokes equations in Lagrangian coordinates. The effects of rotation and of a constant eddy viscosity are included. Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface.

Some simple analytical solutions are presented. For small viscosities, a near-zero mean mass transport is obtained, in agreement with Ursell. Inertial oscillation are superimposed on the above mean solution, in agreement with Hasselmann and Pollard. In the case of a constant wind stress and a constant, horizontally homogeneous wave field, the steady-state results of Weber are reproduced (a surface drift current of about 3% of the wind speed, 23–30 deg to the right of the wind direction).

## Abstract

A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of the Navier-Stokes equations in Lagrangian coordinates. The effects of rotation and of a constant eddy viscosity are included. Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface.

Some simple analytical solutions are presented. For small viscosities, a near-zero mean mass transport is obtained, in agreement with Ursell. Inertial oscillation are superimposed on the above mean solution, in agreement with Hasselmann and Pollard. In the case of a constant wind stress and a constant, horizontally homogeneous wave field, the steady-state results of Weber are reproduced (a surface drift current of about 3% of the wind speed, 23–30 deg to the right of the wind direction).

## Abstract

The airflow above ocean waves is calculated using a quasi-linear model—one in which the effect of the waves on the mean flow is taken into account. The model uses curvilinear coordinates, in which one coordinate surface coincides with the instantaneous sea surface, and is consequently able to attain fine vertical resolution in the boundary layer just above the sea surface; the model equations are formulated in conservation-law form. The rates of energy and momentum input to the wave field are calculated from the oscillatory pressure and shear-stress components at the water surface. The equations are solved iteratively using a logarithmically spaced finite-difference mesh.

The effect of air turbulence is modeled using a vertically varying shear-stress–dependent eddy viscosity, which acts on the wave-correlated oscillatory motions as well as on the mean flow field. For infinitesimal waves the model agrees with the results of Conte and Miles as the Newtonian viscosity and eddy viscosity that act on the oscillatory motions are reduced toward zero, and it converges slowly toward the results of Jacobs' analytical eddy viscosity model as the drag coefficient is reduced.

In agreement with results from Janssen's simpler quasi-linear model, there is increased wave-induced drag for young wind seas with unidirectional JONSWAP spectra and Phillips constant proportional to the (−3/2) power of wave age. The present model gives similar values for wave drag and wave energy input to Janssen's, for the same values of roughness length and Phillips constant, and the spectral distribution of the rate of energy input to the waves is also in reasonable agreement. The variation of drag coefficient with wave age is quite close to the results obtained by Maat, Kraan, and Oost from analysis of HEXMAX field data.

## Abstract

The airflow above ocean waves is calculated using a quasi-linear model—one in which the effect of the waves on the mean flow is taken into account. The model uses curvilinear coordinates, in which one coordinate surface coincides with the instantaneous sea surface, and is consequently able to attain fine vertical resolution in the boundary layer just above the sea surface; the model equations are formulated in conservation-law form. The rates of energy and momentum input to the wave field are calculated from the oscillatory pressure and shear-stress components at the water surface. The equations are solved iteratively using a logarithmically spaced finite-difference mesh.

The effect of air turbulence is modeled using a vertically varying shear-stress–dependent eddy viscosity, which acts on the wave-correlated oscillatory motions as well as on the mean flow field. For infinitesimal waves the model agrees with the results of Conte and Miles as the Newtonian viscosity and eddy viscosity that act on the oscillatory motions are reduced toward zero, and it converges slowly toward the results of Jacobs' analytical eddy viscosity model as the drag coefficient is reduced.

In agreement with results from Janssen's simpler quasi-linear model, there is increased wave-induced drag for young wind seas with unidirectional JONSWAP spectra and Phillips constant proportional to the (−3/2) power of wave age. The present model gives similar values for wave drag and wave energy input to Janssen's, for the same values of roughness length and Phillips constant, and the spectral distribution of the rate of energy input to the waves is also in reasonable agreement. The variation of drag coefficient with wave age is quite close to the results obtained by Maat, Kraan, and Oost from analysis of HEXMAX field data.

## Abstract

A simplified model is described for wave generation and air–sea momentum flux. The model is based upon the quasilinear theory employed by Fabrikant and Janssen, in which the mean flow is approximated to second order in the wave amplitude and fluctuating quantities are approximated to first order. The wave generation rate is computed using Miles' wave generation theory and the numerical method of Conte and Miles.

The computationally expensive iterative procedure used in Janssen's quasi-linear model is avoided by specifying in advance the vertical wind velocity profile as a combination of a logarithmic profile above height *z* = *z _{p}* and a square root profile below that height,

*z*being given by

_{p}*U*(

*z*) =

_{p}*c*(

*k*

_{p}), where

*U*is the wind velocity,

*k*is the wavenumber at the peak of the wave spectrum, and

_{p}*c*(

*k*) is the associated phase speed. In contrast to Janssen's model and his subsequent simple surface-layer model, no explicit surface roughness parameter is specified: the whole of the applied wind stress is assumed to be taken up by waves with the simple one-dimensional energy spectrum

_{p}*E*(

*k*) = (1/2)α

_{P}

*k*

^{−3}for wavenumbers

*k*between

*k*and ∞,

_{p}*E*(

*k*) = 0 for

*k*<

*k*. The square root part of the velocity profile is derived by assuming that the form of the sea surface is stochastically self-similar, and that the velocity profile is also self-similar. A self-similar velocity profile requires

_{p}*kz*(

_{c}*k*) to be independent of

*k*, where

*U*[

*z*(

_{c}*k*)] =

*c*(

*k*).

The model gives a rather crude approximation to the wave generation rate. The quantity σ^{−1}
*E*˙_{in}(*k*)/*E*(*k*), where σ is the wave angular frequency and *E*˙_{in}(*k*) is the rate of wave energy input, is independent of *k* for a spectrum of the form *E*(*k*) ∝ *k*
^{−3}. It is proportional to (*U*
_{*}/*c _{p}*)

^{2}if the Phillips parameter α

_{P}is constant and to (

*U*

_{*}/

*c*)

_{p}^{1/2}if α

_{P}∝ (

*c*/

_{p}*U*

_{*})

^{−3/2}where

*U*

_{*}is the friction velocity and

*c*=

_{p}*c*(

*k*).

_{p}Air–sea momentum flux, as described by the drag coefficient *C*
_{D}(10 m) = [*U*
_{*}/*U*(10 m)]^{2}, is represented rather better. For wave age *c _{p}*/

*U*

_{*}above 5–10 it decreases with increasing wave age if

*U*

_{*}is kept constant. The predicted drag coefficient, however, tends to decrease more rapidly with wave age for the older wind seas than field measurements indicate [if we assume α

_{P}∝ (

*c*/

_{p}*U*

_{*})

^{−3/2}]. For very young seas with

*c*/

_{p}*U*

_{*}< 5, the drag coefficient increases with wave age. The model predicts a significant variation of drag coefficient with wave age even when α

_{P}is constant. For sea states so “old” that

*z*> 10 m, the wind speed depends just on

_{p}*c*/

_{p}*U*

_{*}, and

*C*(10 m) increases again with wave age. This last, counterintuitive situation will be modified if we allow more of the air–sea momentum flux to be supported directly by turbulence.

_{D}## Abstract

A simplified model is described for wave generation and air–sea momentum flux. The model is based upon the quasilinear theory employed by Fabrikant and Janssen, in which the mean flow is approximated to second order in the wave amplitude and fluctuating quantities are approximated to first order. The wave generation rate is computed using Miles' wave generation theory and the numerical method of Conte and Miles.

The computationally expensive iterative procedure used in Janssen's quasi-linear model is avoided by specifying in advance the vertical wind velocity profile as a combination of a logarithmic profile above height *z* = *z _{p}* and a square root profile below that height,

*z*being given by

_{p}*U*(

*z*) =

_{p}*c*(

*k*

_{p}), where

*U*is the wind velocity,

*k*is the wavenumber at the peak of the wave spectrum, and

_{p}*c*(

*k*) is the associated phase speed. In contrast to Janssen's model and his subsequent simple surface-layer model, no explicit surface roughness parameter is specified: the whole of the applied wind stress is assumed to be taken up by waves with the simple one-dimensional energy spectrum

_{p}*E*(

*k*) = (1/2)α

_{P}

*k*

^{−3}for wavenumbers

*k*between

*k*and ∞,

_{p}*E*(

*k*) = 0 for

*k*<

*k*. The square root part of the velocity profile is derived by assuming that the form of the sea surface is stochastically self-similar, and that the velocity profile is also self-similar. A self-similar velocity profile requires

_{p}*kz*(

_{c}*k*) to be independent of

*k*, where

*U*[

*z*(

_{c}*k*)] =

*c*(

*k*).

The model gives a rather crude approximation to the wave generation rate. The quantity σ^{−1}
*E*˙_{in}(*k*)/*E*(*k*), where σ is the wave angular frequency and *E*˙_{in}(*k*) is the rate of wave energy input, is independent of *k* for a spectrum of the form *E*(*k*) ∝ *k*
^{−3}. It is proportional to (*U*
_{*}/*c _{p}*)

^{2}if the Phillips parameter α

_{P}is constant and to (

*U*

_{*}/

*c*)

_{p}^{1/2}if α

_{P}∝ (

*c*/

_{p}*U*

_{*})

^{−3/2}where

*U*

_{*}is the friction velocity and

*c*=

_{p}*c*(

*k*).

_{p}Air–sea momentum flux, as described by the drag coefficient *C*
_{D}(10 m) = [*U*
_{*}/*U*(10 m)]^{2}, is represented rather better. For wave age *c _{p}*/

*U*

_{*}above 5–10 it decreases with increasing wave age if

*U*

_{*}is kept constant. The predicted drag coefficient, however, tends to decrease more rapidly with wave age for the older wind seas than field measurements indicate [if we assume α

_{P}∝ (

*c*/

_{p}*U*

_{*})

^{−3/2}]. For very young seas with

*c*/

_{p}*U*

_{*}< 5, the drag coefficient increases with wave age. The model predicts a significant variation of drag coefficient with wave age even when α

_{P}is constant. For sea states so “old” that

*z*> 10 m, the wind speed depends just on

_{p}*c*/

_{p}*U*

_{*}, and

*C*(10 m) increases again with wave age. This last, counterintuitive situation will be modified if we allow more of the air–sea momentum flux to be supported directly by turbulence.

_{D}## Abstract

A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of a version of the Navier-Stokes equations in Lagrangian coordinates. The Coriolis effect and the effect of a depth-dependent eddy viscosity are included. (The eddy viscosity is taken to depend on the Lagrangian vertical coordinate *ĉ*.) Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. The vertical variation of the eddy viscosity causes an extra source term to appear in the equation for the evolution of the current profile. This additional source of momentum within the water column is exactly balanced by an extra term in the surface boundary condition, which in turn represents the contribution to wave dissipation caused by the eddy viscosity within the water column being different from its surface value.

The equations were solved numerically, using a constant wind stress and monochromatic wave field simultaneously applied in the same direction at time *t* = 0. The eddy viscosity ν was assumed to be proportional to depth, using Madsen's relation (ν = −k_{K}u_{*}ĉ, where k_{K} is von Kármán's constant, *u*
_{*} = (^&tau/ ρ)^{½} where ^τ is the wind stress and ρ is the water density), except for near the surface where it was modified to account empirically for the direct effects of breaking waves. Results for the Lagrangian mean current were in general agreement with observations; long-term average values ranged from 2.2% to 2.8% of the wind speed at the 10 m level, and were directed between 12° and 17° to the right of the wind and wave direction (in the Northern Hemisphere). The deviation of the current from the wind direction is closer to observed drift current observations than the corresponding results for a constant eddy viscosity. The Lagrangian mean current is surprisingly close to the current obtained from Madsen's theory, even though Madsen does not account explicitly for the effect of surface waves.

The theory can easily take account of random sea states. There are good prospects for coupling it with the output of a numerical model for surface gravity waves, using the wave model's input and dissipation source terms.

## Abstract

A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of a version of the Navier-Stokes equations in Lagrangian coordinates. The Coriolis effect and the effect of a depth-dependent eddy viscosity are included. (The eddy viscosity is taken to depend on the Lagrangian vertical coordinate *ĉ*.) Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. The vertical variation of the eddy viscosity causes an extra source term to appear in the equation for the evolution of the current profile. This additional source of momentum within the water column is exactly balanced by an extra term in the surface boundary condition, which in turn represents the contribution to wave dissipation caused by the eddy viscosity within the water column being different from its surface value.

The equations were solved numerically, using a constant wind stress and monochromatic wave field simultaneously applied in the same direction at time *t* = 0. The eddy viscosity ν was assumed to be proportional to depth, using Madsen's relation (ν = −k_{K}u_{*}ĉ, where k_{K} is von Kármán's constant, *u*
_{*} = (^&tau/ ρ)^{½} where ^τ is the wind stress and ρ is the water density), except for near the surface where it was modified to account empirically for the direct effects of breaking waves. Results for the Lagrangian mean current were in general agreement with observations; long-term average values ranged from 2.2% to 2.8% of the wind speed at the 10 m level, and were directed between 12° and 17° to the right of the wind and wave direction (in the Northern Hemisphere). The deviation of the current from the wind direction is closer to observed drift current observations than the corresponding results for a constant eddy viscosity. The Lagrangian mean current is surprisingly close to the current obtained from Madsen's theory, even though Madsen does not account explicitly for the effect of surface waves.

The theory can easily take account of random sea states. There are good prospects for coupling it with the output of a numerical model for surface gravity waves, using the wave model's input and dissipation source terms.

## Abstract

The phase-averaged energy evolution for random surface waves interacting with oceanic turbulence is investigated. The change in wave energy balances the change in the production of turbulent kinetic energy (TKE). Outside the surface viscous layer and the bottom boundary layer the turbulent flux is not related to the wave-induced shear so that eddy viscosity parameterizations cannot be applied. Instead, it is assumed that the wave motion and the turbulent fluxes are not correlated on the scale of the wave period. Using a generalized Lagrangian average it is found that the mean wave-induced shears, despite zero vorticity, yield a production of TKE as if the Stokes drift shear were a mean flow shear. This result provides a new interpretation of a previous derivation from phase-averaged equations by McWilliams et al. It is found that the present source or sink of wave energy is smaller but is still on the order of the empirically adjusted functions used for the dissipation of swell energy in operational wave models, as well as observations of that phenomenon by Snodgrass et al.

## Abstract

The phase-averaged energy evolution for random surface waves interacting with oceanic turbulence is investigated. The change in wave energy balances the change in the production of turbulent kinetic energy (TKE). Outside the surface viscous layer and the bottom boundary layer the turbulent flux is not related to the wave-induced shear so that eddy viscosity parameterizations cannot be applied. Instead, it is assumed that the wave motion and the turbulent fluxes are not correlated on the scale of the wave period. Using a generalized Lagrangian average it is found that the mean wave-induced shears, despite zero vorticity, yield a production of TKE as if the Stokes drift shear were a mean flow shear. This result provides a new interpretation of a previous derivation from phase-averaged equations by McWilliams et al. It is found that the present source or sink of wave energy is smaller but is still on the order of the empirically adjusted functions used for the dissipation of swell energy in operational wave models, as well as observations of that phenomenon by Snodgrass et al.

## Abstract

The lowest order sigma-transformed momentum equation given by Mellor takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent because of inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure *p* and coordinate transformation function *s* to first order in parameters that define the large-scale evolution of the wave field, such as the bottom slope. Unfortunately, there is no analytical expression for *p* and *s* at that order. A numerical correction method is thus proposed and verified. Alternative coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inconsistency nor do they require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directly relevant to the mean flow evolution.

## Abstract

The lowest order sigma-transformed momentum equation given by Mellor takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent because of inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure *p* and coordinate transformation function *s* to first order in parameters that define the large-scale evolution of the wave field, such as the bottom slope. Unfortunately, there is no analytical expression for *p* and *s* at that order. A numerical correction method is thus proposed and verified. Alternative coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inconsistency nor do they require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directly relevant to the mean flow evolution.

## Abstract

The Innovative Strategies for Observations in the Arctic Atmospheric Boundary Layer Program (ISOBAR) is a research project investigating stable atmospheric boundary layer (SBL) processes, whose representation still poses significant challenges in state-of-the-art numerical weather prediction (NWP) models. In ISOBAR ground-based flux and profile observations are combined with boundary layer remote sensing methods and the extensive usage of different unmanned aircraft systems (UAS). During February 2017 and 2018 we carried out two major field campaigns over the sea ice of the northern Baltic Sea, close to the Finnish island of Hailuoto at 65°N. In total 14 intensive observational periods (IOPs) resulted in extensive SBL datasets with unprecedented spatiotemporal resolution, which will form the basis for various numerical modeling experiments. First results from the campaigns indicate numerous very stable boundary layer (VSBL) cases, characterized by strong stratification, weak winds, and clear skies, and give detailed insight in the temporal evolution and vertical structure of the entire SBL. The SBL is subject to rapid changes in its vertical structure, responding to a variety of different processes. In particular, we study cases involving a shear instability associated with a low-level jet, a rapid strong cooling event observed a few meters above ground, and a strong wave-breaking event that triggers intensive near-surface turbulence. Furthermore, we use observations from one IOP to validate three different atmospheric models. The unique finescale observations resulting from the ISOBAR observational approach will aid future research activities, focusing on a better understanding of the SBL and its implementation in numerical models.

## Abstract

The Innovative Strategies for Observations in the Arctic Atmospheric Boundary Layer Program (ISOBAR) is a research project investigating stable atmospheric boundary layer (SBL) processes, whose representation still poses significant challenges in state-of-the-art numerical weather prediction (NWP) models. In ISOBAR ground-based flux and profile observations are combined with boundary layer remote sensing methods and the extensive usage of different unmanned aircraft systems (UAS). During February 2017 and 2018 we carried out two major field campaigns over the sea ice of the northern Baltic Sea, close to the Finnish island of Hailuoto at 65°N. In total 14 intensive observational periods (IOPs) resulted in extensive SBL datasets with unprecedented spatiotemporal resolution, which will form the basis for various numerical modeling experiments. First results from the campaigns indicate numerous very stable boundary layer (VSBL) cases, characterized by strong stratification, weak winds, and clear skies, and give detailed insight in the temporal evolution and vertical structure of the entire SBL. The SBL is subject to rapid changes in its vertical structure, responding to a variety of different processes. In particular, we study cases involving a shear instability associated with a low-level jet, a rapid strong cooling event observed a few meters above ground, and a strong wave-breaking event that triggers intensive near-surface turbulence. Furthermore, we use observations from one IOP to validate three different atmospheric models. The unique finescale observations resulting from the ISOBAR observational approach will aid future research activities, focusing on a better understanding of the SBL and its implementation in numerical models.