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- Author or Editor: Miguel A. C. Teixeira x

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## Abstract

A mechanism for the enhancement of the viscous dissipation rate of turbulent kinetic energy (TKE) in the oceanic boundary layer (OBL) is proposed, based on insights gained from rapid-distortion theory (RDT). In this mechanism, which complements mechanisms purely based on wave breaking, preexisting TKE is amplified and subsequently dissipated by the joint action of a mean Eulerian wind-induced shear current and the Stokes drift of surface waves, the same elements thought to be responsible for the generation of Langmuir circulations. Assuming that the TKE dissipation rate *ε* saturates to its equilibrium value over a time of the order one eddy turnover time of the turbulence, a new scaling expression, dependent on the turbulent Langmuir number, is derived for *ε*. For reasonable values of the input parameters, the new expression predicts an increase of the dissipation rate near the surface by orders of magnitude compared with usual surface-layer scaling estimates, consistent with available OBL data. These results establish on firmer grounds a suspected connection between two central OBL phenomena: dissipation enhancement and Langmuir circulations.

## Abstract

A mechanism for the enhancement of the viscous dissipation rate of turbulent kinetic energy (TKE) in the oceanic boundary layer (OBL) is proposed, based on insights gained from rapid-distortion theory (RDT). In this mechanism, which complements mechanisms purely based on wave breaking, preexisting TKE is amplified and subsequently dissipated by the joint action of a mean Eulerian wind-induced shear current and the Stokes drift of surface waves, the same elements thought to be responsible for the generation of Langmuir circulations. Assuming that the TKE dissipation rate *ε* saturates to its equilibrium value over a time of the order one eddy turnover time of the turbulence, a new scaling expression, dependent on the turbulent Langmuir number, is derived for *ε*. For reasonable values of the input parameters, the new expression predicts an increase of the dissipation rate near the surface by orders of magnitude compared with usual surface-layer scaling estimates, consistent with available OBL data. These results establish on firmer grounds a suspected connection between two central OBL phenomena: dissipation enhancement and Langmuir circulations.

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## Abstract

A simple analytical model is developed for the current induced by the wind and modified by surface wind waves in the oceanic surface layer, based on a first-order turbulence closure and including the effect of a vortex force representing the Stokes drift of the waves. The shear stress is partitioned between a component due to shear in the current, which is reduced at low turbulent Langmuir number La_{
t
}, and a wave-induced component, which decays over a depth proportional to the dominant wavelength *λ*
_{
w
}. The model reproduces the apparent reduction of the friction velocity and enhancement of the roughness length estimated from current profiles, detected in a number of studies. These effects are predicted to intensify as La_{
t
} decreases and are entirely attributed to nonbreaking surface waves. The current profile becomes flatter for low La_{
t
} owing to a smaller fraction of the total shear stress being supported by the current shear. Comparisons with the comprehensive dataset provided by the laboratory experiments of Cheung and Street show encouraging agreement, with the current speed normalized by the friction velocity decreasing as La_{
t
} decreases and *λ*
_{
w
} increases if the model is adjusted to reflect the effects of a full wave spectrum on the intensity and depth of penetration of the wave-induced stress. A version of the model where the shear stress decreases to zero over a depth consistent with the measurements accurately predicts the surface current speed. These results contribute toward developing physically based momentum flux parameterizations for the wave-affected boundary layer in ocean circulation models.

## Abstract

A simple analytical model is developed for the current induced by the wind and modified by surface wind waves in the oceanic surface layer, based on a first-order turbulence closure and including the effect of a vortex force representing the Stokes drift of the waves. The shear stress is partitioned between a component due to shear in the current, which is reduced at low turbulent Langmuir number La_{
t
}, and a wave-induced component, which decays over a depth proportional to the dominant wavelength *λ*
_{
w
}. The model reproduces the apparent reduction of the friction velocity and enhancement of the roughness length estimated from current profiles, detected in a number of studies. These effects are predicted to intensify as La_{
t
} decreases and are entirely attributed to nonbreaking surface waves. The current profile becomes flatter for low La_{
t
} owing to a smaller fraction of the total shear stress being supported by the current shear. Comparisons with the comprehensive dataset provided by the laboratory experiments of Cheung and Street show encouraging agreement, with the current speed normalized by the friction velocity decreasing as La_{
t
} decreases and *λ*
_{
w
} increases if the model is adjusted to reflect the effects of a full wave spectrum on the intensity and depth of penetration of the wave-induced stress. A version of the model where the shear stress decreases to zero over a depth consistent with the measurements accurately predicts the surface current speed. These results contribute toward developing physically based momentum flux parameterizations for the wave-affected boundary layer in ocean circulation models.

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## Abstract

The analytical model proposed by Teixeira, Miranda, and Valente is modified to calculate the gravity wave drag exerted by a stratified flow over a 2D mountain ridge. The drag is found to be more strongly affected by the vertical variation of the background velocity than for an axisymmetric mountain. In the hydrostatic approximation, the corrections to the drag due to this effect do not depend on the detailed shape of the ridge as long as this is exactly 2D. Besides the drag, all the perturbed quantities of the flow at the surface, including the pressure, may be calculated analytically.

## Abstract

The analytical model proposed by Teixeira, Miranda, and Valente is modified to calculate the gravity wave drag exerted by a stratified flow over a 2D mountain ridge. The drag is found to be more strongly affected by the vertical variation of the background velocity than for an axisymmetric mountain. In the hydrostatic approximation, the corrections to the drag due to this effect do not depend on the detailed shape of the ridge as long as this is exactly 2D. Besides the drag, all the perturbed quantities of the flow at the surface, including the pressure, may be calculated analytically.

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## Abstract

The direct impact of mountain waves on the atmospheric circulation is due to the deposition of wave momentum at critical levels, or levels where the waves break. The first process is treated analytically in this study within the framework of linear theory. The variation of the momentum flux with height is investigated for relatively large shears, extending the authors’ previous calculations of the surface gravity wave drag to the whole atmosphere. A Wentzel–Kramers–Brillouin (WKB) approximation is used to treat inviscid, steady, nonrotating, hydrostatic flow with directional shear over a circular mesoscale mountain, for generic wind profiles. This approximation must be extended to third order to obtain momentum flux expressions that are accurate to second order. Since the momentum flux only varies because of wave filtering by critical levels, the application of contour integration techniques enables it to be expressed in terms of simple 1D integrals. On the other hand, the momentum flux divergence (which corresponds to the force on the atmosphere that must be represented in gravity wave drag parameterizations) is given in closed analytical form. The momentum flux expressions are tested for idealized wind profiles, where they become a function of the Richardson number (Ri). These expressions tend, for high Ri, to results by previous authors, where wind profile effects on the surface drag were neglected and critical levels acted as perfect absorbers. The linear results are compared with linear and nonlinear numerical simulations, showing a considerable improvement upon corresponding results derived for higher Ri.

## Abstract

The direct impact of mountain waves on the atmospheric circulation is due to the deposition of wave momentum at critical levels, or levels where the waves break. The first process is treated analytically in this study within the framework of linear theory. The variation of the momentum flux with height is investigated for relatively large shears, extending the authors’ previous calculations of the surface gravity wave drag to the whole atmosphere. A Wentzel–Kramers–Brillouin (WKB) approximation is used to treat inviscid, steady, nonrotating, hydrostatic flow with directional shear over a circular mesoscale mountain, for generic wind profiles. This approximation must be extended to third order to obtain momentum flux expressions that are accurate to second order. Since the momentum flux only varies because of wave filtering by critical levels, the application of contour integration techniques enables it to be expressed in terms of simple 1D integrals. On the other hand, the momentum flux divergence (which corresponds to the force on the atmosphere that must be represented in gravity wave drag parameterizations) is given in closed analytical form. The momentum flux expressions are tested for idealized wind profiles, where they become a function of the Richardson number (Ri). These expressions tend, for high Ri, to results by previous authors, where wind profile effects on the surface drag were neglected and critical levels acted as perfect absorbers. The linear results are compared with linear and nonlinear numerical simulations, showing a considerable improvement upon corresponding results derived for higher Ri.

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## Abstract

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

## Abstract

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

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## Abstract

Linear nonhydrostatic theory is used to evaluate the drag produced by 3D trapped lee waves forced by an axisymmetric hill at a density interface. These waves occur at atmospheric temperature inversions, for example, at the top of the boundary layer, and contribute to low-level drag possibly misrepresented as turbulent form drag in large-scale numerical models. Unlike in 2D waves, the drag has contributions from a continuous range of wavenumbers forced by the topography, because the waves can vary their angle of incidence to match the resonance condition. This leads to nonzero drag for Froude numbers (Fr) both <1 and >1 and a drag maximum typically for Fr slightly below 1, with lower magnitude than in hydrostatic conditions owing to wave dispersion. These features are in good agreement with laboratory experiments using two axisymmetric obstacles, particularly for the lower obstacle, if the effects of a rigid lid above the upper layer and friction are taken into account. Quantitative agreement is less satisfactory for the higher obstacle, as flow nonlinearity increases. However, even in that case the model still largely outperforms both 3D hydrostatic and 2D nonhydrostatic theories, emphasizing the importance of both 3D and nonhydrostatic effects. The associated wave signatures are dominated by transverse waves for Fr lower than at the drag maximum, a dispersive “Kelvin ship-wave” pattern near the maximum, and divergent waves for Fr beyond the maximum. The minimum elevation at the density-interface depression existing immediately downstream of the obstacle is significantly correlated with the drag magnitude.

## Abstract

Linear nonhydrostatic theory is used to evaluate the drag produced by 3D trapped lee waves forced by an axisymmetric hill at a density interface. These waves occur at atmospheric temperature inversions, for example, at the top of the boundary layer, and contribute to low-level drag possibly misrepresented as turbulent form drag in large-scale numerical models. Unlike in 2D waves, the drag has contributions from a continuous range of wavenumbers forced by the topography, because the waves can vary their angle of incidence to match the resonance condition. This leads to nonzero drag for Froude numbers (Fr) both <1 and >1 and a drag maximum typically for Fr slightly below 1, with lower magnitude than in hydrostatic conditions owing to wave dispersion. These features are in good agreement with laboratory experiments using two axisymmetric obstacles, particularly for the lower obstacle, if the effects of a rigid lid above the upper layer and friction are taken into account. Quantitative agreement is less satisfactory for the higher obstacle, as flow nonlinearity increases. However, even in that case the model still largely outperforms both 3D hydrostatic and 2D nonhydrostatic theories, emphasizing the importance of both 3D and nonhydrostatic effects. The associated wave signatures are dominated by transverse waves for Fr lower than at the drag maximum, a dispersive “Kelvin ship-wave” pattern near the maximum, and divergent waves for Fr beyond the maximum. The minimum elevation at the density-interface depression existing immediately downstream of the obstacle is significantly correlated with the drag magnitude.

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## Abstract

This work studies nonhydrostatic effects (NHE) on the momentum flux of orographic gravity waves (OGWs) forced by isolated three-dimensional orography. Based on linear wave theory, an asymptotic expression for low horizonal Froude number [*U*, *V*) is the mean horizontal wind, *γ* and *a* are the orography anisotropy and half width, and *N* is the buoyancy frequency] is derived for the gravity wave momentum flux (GWMF) of vertically propagating waves. According to this asymptotic solution, which is quite accurate for any value of Fr, NHE can be divided into two terms (NHE1 and NHE2). The first term contains the high-frequency parts of the wave spectrum that are often mistaken as hydrostatic waves, and only depends on Fr*.* The second term arises from the difference between the dispersion relationships of hydrostatic and nonhydrostatic OGWs. Having an additional dependency on the horizontal wind direction and orography anisotropy, this term can change the GWMF direction. Examination of NHE for OGWs forced by both circular and elliptical orography reveals that the GWMF is reduced as Fr increases, at a faster rate than for two-dimensional OGWs forced by a ridge. At low Fr, the GWMF reduction is mostly attributed to the NHE2 term, whereas the NHE1 term starts to dominate above about Fr = 0.4. The behavior of NHE is mainly determined by Fr, while horizontal wind direction and orography anisotropy play a minor role. Implications of the asymptotic GWMF expression for the parameterization of nonhydrostatic OGWs in high-resolution and/or variable-resolution models are discussed.

## Abstract

This work studies nonhydrostatic effects (NHE) on the momentum flux of orographic gravity waves (OGWs) forced by isolated three-dimensional orography. Based on linear wave theory, an asymptotic expression for low horizonal Froude number [*U*, *V*) is the mean horizontal wind, *γ* and *a* are the orography anisotropy and half width, and *N* is the buoyancy frequency] is derived for the gravity wave momentum flux (GWMF) of vertically propagating waves. According to this asymptotic solution, which is quite accurate for any value of Fr, NHE can be divided into two terms (NHE1 and NHE2). The first term contains the high-frequency parts of the wave spectrum that are often mistaken as hydrostatic waves, and only depends on Fr*.* The second term arises from the difference between the dispersion relationships of hydrostatic and nonhydrostatic OGWs. Having an additional dependency on the horizontal wind direction and orography anisotropy, this term can change the GWMF direction. Examination of NHE for OGWs forced by both circular and elliptical orography reveals that the GWMF is reduced as Fr increases, at a faster rate than for two-dimensional OGWs forced by a ridge. At low Fr, the GWMF reduction is mostly attributed to the NHE2 term, whereas the NHE1 term starts to dominate above about Fr = 0.4. The behavior of NHE is mainly determined by Fr, while horizontal wind direction and orography anisotropy play a minor role. Implications of the asymptotic GWMF expression for the parameterization of nonhydrostatic OGWs in high-resolution and/or variable-resolution models are discussed.

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## Abstract

An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri^{−1}, decreasing as Ri decreases for a wind that varies linearly with height, and increasing as Ri decreases for a wind that rotates with height maintaining its magnitude. In these two cases the surface drag is predicted to be aligned with the surface wind. When one of the wind components varies linearly with height and the other is constant, the surface drag is misaligned with the surface wind, especially for relatively small Ri. All these results are shown to be in fairly good agreement with numerical simulations of mesoscale nonhydrostatic models, for high and even moderate values of Ri.

## Abstract

An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri^{−1}, decreasing as Ri decreases for a wind that varies linearly with height, and increasing as Ri decreases for a wind that rotates with height maintaining its magnitude. In these two cases the surface drag is predicted to be aligned with the surface wind. When one of the wind components varies linearly with height and the other is constant, the surface drag is misaligned with the surface wind, especially for relatively small Ri. All these results are shown to be in fairly good agreement with numerical simulations of mesoscale nonhydrostatic models, for high and even moderate values of Ri.

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## Abstract

The drag produced by 2D orographic gravity waves trapped at a temperature inversion and waves propagating in the stably stratified layer existing above are explicitly calculated using linear theory, for a two-layer atmosphere with neutral static stability near the surface, mimicking a well-mixed boundary layer. For realistic values of the flow parameters, trapped-lee-wave drag, which is given by a closed analytical expression, is comparable to propagating-wave drag, especially in moderately to strongly nonhydrostatic conditions. In resonant flow, both drag components substantially exceed the single-layer hydrostatic drag estimate used in most parameterization schemes. Both drag components are optimally amplified for a relatively low-level inversion and Froude numbers Fr ≈ 1. While propagating-wave drag is maximized for approximately hydrostatic flow, trapped-lee-wave drag is maximized for *l*
_{2}
*a* = *O*(1) (where *l*
_{2} is the Scorer parameter in the stable layer and *a* is the mountain width). This roughly happens when the horizontal scale of trapped lee waves matches that of the mountain slope. The drag behavior as a function of Fr for *l*
_{2}
*H* = 0.5 (where *H* is the inversion height) and different values of *l*
_{2}
*a* shows good agreement with numerical simulations. Regions of parameter space with high trapped-lee-wave drag correlate reasonably well with those where lee-wave rotors were found to occur in previous nonlinear numerical simulations including frictional effects. This suggests that trapped-lee-wave drag, besides giving a relevant contribution to low-level drag exerted on the atmosphere, may also be useful to diagnose lee-rotor formation.

## Abstract

The drag produced by 2D orographic gravity waves trapped at a temperature inversion and waves propagating in the stably stratified layer existing above are explicitly calculated using linear theory, for a two-layer atmosphere with neutral static stability near the surface, mimicking a well-mixed boundary layer. For realistic values of the flow parameters, trapped-lee-wave drag, which is given by a closed analytical expression, is comparable to propagating-wave drag, especially in moderately to strongly nonhydrostatic conditions. In resonant flow, both drag components substantially exceed the single-layer hydrostatic drag estimate used in most parameterization schemes. Both drag components are optimally amplified for a relatively low-level inversion and Froude numbers Fr ≈ 1. While propagating-wave drag is maximized for approximately hydrostatic flow, trapped-lee-wave drag is maximized for *l*
_{2}
*a* = *O*(1) (where *l*
_{2} is the Scorer parameter in the stable layer and *a* is the mountain width). This roughly happens when the horizontal scale of trapped lee waves matches that of the mountain slope. The drag behavior as a function of Fr for *l*
_{2}
*H* = 0.5 (where *H* is the inversion height) and different values of *l*
_{2}
*a* shows good agreement with numerical simulations. Regions of parameter space with high trapped-lee-wave drag correlate reasonably well with those where lee-wave rotors were found to occur in previous nonlinear numerical simulations including frictional effects. This suggests that trapped-lee-wave drag, besides giving a relevant contribution to low-level drag exerted on the atmosphere, may also be useful to diagnose lee-rotor formation.

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## Abstract

Mountain-wave turbulence in the presence of directional wind shear over the Rocky Mountains in Colorado is investigated. Pilot reports (PIREPs) are used to select cases in which moderate or severe turbulence encounters were reported in combination with significant directional wind shear in the upstream sounding from Grand Junction, Colorado (GJT). For a selected case, semi-idealized numerical simulations are carried out using the WRF-ARW atmospheric model, initialized with the GJT atmospheric sounding and a realistic but truncated orography profile. To isolate the role of directional wind shear in causing wave breaking, sensitivity tests are performed to exclude the variation of the atmospheric stability with height, the speed shear, and the mountain amplitude as dominant wave breaking mechanisms. Significant downwind transport of instabilities is detected in horizontal flow cross sections, resulting in mountain-wave-induced turbulence occurring at large horizontal distances from the first wave breaking point (and from the orography that generates the waves). The existence of an asymptotic wake, as predicted by Shutts for directional shear flows, is hypothesized to be responsible for this downwind transport. Critical levels induced by directional wind shear are further studied by taking 2D power spectra of the magnitude of the horizontal velocity perturbation field. In these spectra, a rotation of the most energetic wave modes with the background wind, as well as perpendicularity between the background wind vector and the wavenumber vector of those modes at critical levels, can be found, which is consistent with the mechanism expected to lead to wave breaking in directional shear flows.

## Abstract

Mountain-wave turbulence in the presence of directional wind shear over the Rocky Mountains in Colorado is investigated. Pilot reports (PIREPs) are used to select cases in which moderate or severe turbulence encounters were reported in combination with significant directional wind shear in the upstream sounding from Grand Junction, Colorado (GJT). For a selected case, semi-idealized numerical simulations are carried out using the WRF-ARW atmospheric model, initialized with the GJT atmospheric sounding and a realistic but truncated orography profile. To isolate the role of directional wind shear in causing wave breaking, sensitivity tests are performed to exclude the variation of the atmospheric stability with height, the speed shear, and the mountain amplitude as dominant wave breaking mechanisms. Significant downwind transport of instabilities is detected in horizontal flow cross sections, resulting in mountain-wave-induced turbulence occurring at large horizontal distances from the first wave breaking point (and from the orography that generates the waves). The existence of an asymptotic wake, as predicted by Shutts for directional shear flows, is hypothesized to be responsible for this downwind transport. Critical levels induced by directional wind shear are further studied by taking 2D power spectra of the magnitude of the horizontal velocity perturbation field. In these spectra, a rotation of the most energetic wave modes with the background wind, as well as perpendicularity between the background wind vector and the wavenumber vector of those modes at critical levels, can be found, which is consistent with the mechanism expected to lead to wave breaking in directional shear flows.