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## Abstract

Accurate predictions of the sea stateâ€“dependent airâ€“sea momentum flux require a thorough understanding of the wave boundary layer turbulence over surface waves. A set of momentum and energy equations is derived to formulate and analyze wave boundary layer turbulence. The equations are written in wave-following coordinates, and all variables are decomposed into horizontal mean, wave fluctuation, and turbulent fluctuation. The formulation defines the wave-induced stress as a sum of the wave fluctuation stress (because of the fluctuating velocity components) and a pressure stress (pressure acting on a tilted surface). The formulations can be constructed with different choices of mapping. Next, a large-eddy simulation result for wind over a sinusoidal wave train under a strongly forced condition is analyzed using the proposed formulation. The result clarifies how surface waves increase the effective roughness length and the drag coefficient. Specifically, the enhanced wave-induced stress close to the water surface reduces the turbulent stress (satisfying the momentum budget). The reduced turbulent stress is correlated with the reduced viscous dissipation rate of the turbulent kinetic energy. The latter is balanced by the reduced mean wind shear (satisfying the energy budget), which causes the equivalent surface roughness to increase. Interestingly, there is a small region farther above where the turbulent stress, dissipation rate, and mean wind shear are all enhanced. The observed strong correlation between the turbulent stress and the dissipation rate suggests that existing turbulence closure models that parameterize the latter based on the former are reasonably accurate.

## Abstract

Accurate predictions of the sea stateâ€“dependent airâ€“sea momentum flux require a thorough understanding of the wave boundary layer turbulence over surface waves. A set of momentum and energy equations is derived to formulate and analyze wave boundary layer turbulence. The equations are written in wave-following coordinates, and all variables are decomposed into horizontal mean, wave fluctuation, and turbulent fluctuation. The formulation defines the wave-induced stress as a sum of the wave fluctuation stress (because of the fluctuating velocity components) and a pressure stress (pressure acting on a tilted surface). The formulations can be constructed with different choices of mapping. Next, a large-eddy simulation result for wind over a sinusoidal wave train under a strongly forced condition is analyzed using the proposed formulation. The result clarifies how surface waves increase the effective roughness length and the drag coefficient. Specifically, the enhanced wave-induced stress close to the water surface reduces the turbulent stress (satisfying the momentum budget). The reduced turbulent stress is correlated with the reduced viscous dissipation rate of the turbulent kinetic energy. The latter is balanced by the reduced mean wind shear (satisfying the energy budget), which causes the equivalent surface roughness to increase. Interestingly, there is a small region farther above where the turbulent stress, dissipation rate, and mean wind shear are all enhanced. The observed strong correlation between the turbulent stress and the dissipation rate suggests that existing turbulence closure models that parameterize the latter based on the former are reasonably accurate.

## Abstract

A neutrally stratified turbulent airflow over a very young sea surface at a high-wind condition was investigated using large-eddy simulations. In such a state, the dominant drag at the sea surface occurs over breaking waves, and the relationship between the dominant drag and local instantaneous surface wind is highly stochastic and anisotropic. To model such a relationship, a bottom boundary stress parameterization was proposed for the very young sea surface resolving individual breakers. This parameterization was compared to the commonly used parameterization for isotropic surfaces. Over both the young sea and isotropic surfaces, the main near-surface turbulence structure was wall-attached, large-scale, quasi-streamwise vortices. Over the young sea surface, these vortices were more intense, and the near-surface mean velocity gradient was smaller. This is because the isotropic surface weakens the swirling motions of the vortices by spanwise drag. In contrast, the young sea surface exerts little spanwise drag and develops more intense vortices, resulting in greater turbulence and mixing. The vigorous turbulence decreases the mean velocity gradient in the roughness sublayer below the logarithmic layer. Thus, the enhancement of the airâ€“sea momentum flux (drag coefficient) due to breaking waves is caused not only by the streamwise form drag over individual breakers but also by the enhanced vortices. Furthermore, contrary to an assumption used in existing wave boundary layer models, the wave effect may extend as high as 10â€“20 times the breaking wave height.

## Abstract

A neutrally stratified turbulent airflow over a very young sea surface at a high-wind condition was investigated using large-eddy simulations. In such a state, the dominant drag at the sea surface occurs over breaking waves, and the relationship between the dominant drag and local instantaneous surface wind is highly stochastic and anisotropic. To model such a relationship, a bottom boundary stress parameterization was proposed for the very young sea surface resolving individual breakers. This parameterization was compared to the commonly used parameterization for isotropic surfaces. Over both the young sea and isotropic surfaces, the main near-surface turbulence structure was wall-attached, large-scale, quasi-streamwise vortices. Over the young sea surface, these vortices were more intense, and the near-surface mean velocity gradient was smaller. This is because the isotropic surface weakens the swirling motions of the vortices by spanwise drag. In contrast, the young sea surface exerts little spanwise drag and develops more intense vortices, resulting in greater turbulence and mixing. The vigorous turbulence decreases the mean velocity gradient in the roughness sublayer below the logarithmic layer. Thus, the enhancement of the airâ€“sea momentum flux (drag coefficient) due to breaking waves is caused not only by the streamwise form drag over individual breakers but also by the enhanced vortices. Furthermore, contrary to an assumption used in existing wave boundary layer models, the wave effect may extend as high as 10â€“20 times the breaking wave height.

## Abstract

Planetary boundary layer (PBL) flows are known to exhibit fundamental differences depending on the relative combination of wind shear and buoyancy forces. These differences are not unexpected in that shear instabilities occur locally, while buoyancy force sets up vigorous thermals, which result in nonlocal transport of heat and momentum. At the same time, these two forces can act together to modify the flow field. In this study, four large-eddy simulations (LESs) spanning the shear and buoyancy flow regimes were generated; two correspond to the extreme cases of shear and buoyancy-driven PBLs, while the other two represent intermediate PBLs where both forces are important. The extreme cases are used to highlight and quantify the basic differences between shear and convective PBLs in 1) flow structures, 2) overall statistics, and 3) turbulent kinetic energy (TKE) budget distributions. Results from the two intermediate LES cases are used to develop and verify a velocity scaling and a TKE budget model, which are proposed for the intermediate PBL. The velocity variances and the variance fluxes (i.e., third moments) normalized by this velocity scaling are shown to become quantities on the order of one, and to lie mostly between those of the two extreme PBL cases. The proposed TKE budget model is shown to adequately reproduce the profiles of the TKE budget terms and the TKE.

## Abstract

Planetary boundary layer (PBL) flows are known to exhibit fundamental differences depending on the relative combination of wind shear and buoyancy forces. These differences are not unexpected in that shear instabilities occur locally, while buoyancy force sets up vigorous thermals, which result in nonlocal transport of heat and momentum. At the same time, these two forces can act together to modify the flow field. In this study, four large-eddy simulations (LESs) spanning the shear and buoyancy flow regimes were generated; two correspond to the extreme cases of shear and buoyancy-driven PBLs, while the other two represent intermediate PBLs where both forces are important. The extreme cases are used to highlight and quantify the basic differences between shear and convective PBLs in 1) flow structures, 2) overall statistics, and 3) turbulent kinetic energy (TKE) budget distributions. Results from the two intermediate LES cases are used to develop and verify a velocity scaling and a TKE budget model, which are proposed for the intermediate PBL. The velocity variances and the variance fluxes (i.e., third moments) normalized by this velocity scaling are shown to become quantities on the order of one, and to lie mostly between those of the two extreme PBL cases. The proposed TKE budget model is shown to adequately reproduce the profiles of the TKE budget terms and the TKE.

## Abstract

The effect of horizontal temperature heterogeneity of the underlying surface on the turbulence structure and mixing intensity in the stably stratified boundary layer (SBL) is analyzed using large-eddy simulation (LES). Idealized LESs of flows driven by fixed winds and homogeneous and heterogeneous surface temperatures are compared. The LES data are used to compute statistical moments, to estimate budgets of the turbulence kinetic energy (TKE), of the temperature variance and of the temperature flux, and to assess the relative importance of various terms in maintaining the budgets. Unlike most previous studies, the LES-based second-moment budgets are estimated with due regard for the subgrid-scale contributions.

The SBL over a heterogeneous surface is more turbulent with larger variances (and TKE), is better vertically mixed, and is deeper compared to its homogeneous counterpart. The most striking difference between the cases is exhibited in the temperature variance and its budget. Because of surface heterogeneity, the turbulent transport term (divergence of the third-order moment) not only redistributes the temperature variance vertically but is a net gain. The increase in the temperature variance near the heterogeneous surface explains the reduced magnitude of the downward buoyancy flux and the ensuing increase in TKE that leads to more vigorous mixing. Analysis of the temperature flux budget shows that the transport term contributes to net production/destruction. Importantly, the role of the third-order transport cannot be elucidated if the budgets are computed based solely on resolved-scale fields. Implications for modeling (parameterizing) the SBL over thermally heterogeneous surfaces are discussed.

## Abstract

The effect of horizontal temperature heterogeneity of the underlying surface on the turbulence structure and mixing intensity in the stably stratified boundary layer (SBL) is analyzed using large-eddy simulation (LES). Idealized LESs of flows driven by fixed winds and homogeneous and heterogeneous surface temperatures are compared. The LES data are used to compute statistical moments, to estimate budgets of the turbulence kinetic energy (TKE), of the temperature variance and of the temperature flux, and to assess the relative importance of various terms in maintaining the budgets. Unlike most previous studies, the LES-based second-moment budgets are estimated with due regard for the subgrid-scale contributions.

The SBL over a heterogeneous surface is more turbulent with larger variances (and TKE), is better vertically mixed, and is deeper compared to its homogeneous counterpart. The most striking difference between the cases is exhibited in the temperature variance and its budget. Because of surface heterogeneity, the turbulent transport term (divergence of the third-order moment) not only redistributes the temperature variance vertically but is a net gain. The increase in the temperature variance near the heterogeneous surface explains the reduced magnitude of the downward buoyancy flux and the ensuing increase in TKE that leads to more vigorous mixing. Analysis of the temperature flux budget shows that the transport term contributes to net production/destruction. Importantly, the role of the third-order transport cannot be elucidated if the budgets are computed based solely on resolved-scale fields. Implications for modeling (parameterizing) the SBL over thermally heterogeneous surfaces are discussed.

## Abstract

Large-eddy simulation (LES) is used to investigate how dominant breaking waves in the ocean under hurricane-force winds affect the drag and near-surface airflow turbulence. The LES explicitly resolves the wake turbulence produced by dominant-scale breakers. Effects of unresolved roughness such as short breakers, nonbreaking waves, and sea foam are modeled as the subgrid-scale drag. Compared to the laboratory conditions previously studied using the same method, dominant-scale breakers in open-ocean conditions are less frequent, and the subgrid-scale drag is more significant. Nevertheless, dominant-scale breakers are more fully exposed to high winds and produce more intense wakes individually. As a result, they support a large portion of the total drag and significantly influence the turbulence for many ocean conditions that are likely to occur. The intense wake turbulence is characterized by flow separation, upward bursts of wind, and upward flux of the turbulent kinetic energy (TKE), all of which may influence sea spray dispersion. Similarly to the findings in the laboratory conditions, high production of wake turbulence shortcuts the inertial energy cascade, causes high TKE dissipation, and contributes to the reduction of the drag coefficient. The results also indicate that if the drag coefficient decreases with increasing wind at very high winds, as some recent observations suggest, then the unresolved roughness must also decrease.

## Abstract

Large-eddy simulation (LES) is used to investigate how dominant breaking waves in the ocean under hurricane-force winds affect the drag and near-surface airflow turbulence. The LES explicitly resolves the wake turbulence produced by dominant-scale breakers. Effects of unresolved roughness such as short breakers, nonbreaking waves, and sea foam are modeled as the subgrid-scale drag. Compared to the laboratory conditions previously studied using the same method, dominant-scale breakers in open-ocean conditions are less frequent, and the subgrid-scale drag is more significant. Nevertheless, dominant-scale breakers are more fully exposed to high winds and produce more intense wakes individually. As a result, they support a large portion of the total drag and significantly influence the turbulence for many ocean conditions that are likely to occur. The intense wake turbulence is characterized by flow separation, upward bursts of wind, and upward flux of the turbulent kinetic energy (TKE), all of which may influence sea spray dispersion. Similarly to the findings in the laboratory conditions, high production of wake turbulence shortcuts the inertial energy cascade, causes high TKE dissipation, and contributes to the reduction of the drag coefficient. The results also indicate that if the drag coefficient decreases with increasing wind at very high winds, as some recent observations suggest, then the unresolved roughness must also decrease.

## Abstract

The effects of breaking waves on near-surface wind turbulence and drag coefficient are investigated using large-eddy simulation. The impact of intermittent and transient wave breaking events (over a range of scales) is modeled as localized form drag, which generates airflow separation bubbles downstream. The simulations are performed for very young sea conditions under high winds, comparable to previous laboratory experiments in hurricane-strength winds. The results for the drag coefficient in high winds range between about 0.002 and 0.003. In such conditions more than 90% of the total airâ€“sea momentum flux is due to the form drag of breakers; that is, the contributions of the nonbreaking wave form drag and the surface viscous stress are small. Detailed analysis shows that the breaker form drag impedes the shear production of the turbulent kinetic energy (TKE) near the surface and, instead, produces a large amount of small-scale wake turbulence by transferring energy from large-scale motions (such as mean wind and gusts). This process shortcuts the inertial energy cascade and results in large TKE dissipation (integrated over the surface layer) normalized by friction velocity cubed. Consequently, the large production of wake turbulence by breakers in high winds results in the small drag coefficient obtained in this study. The results also suggest that common parameterizations for the mean wind profile and the TKE dissipation inside the wave boundary layer, used in previous Reynolds-averaged Navierâ€“Stokes models, may not be valid.

## Abstract

The effects of breaking waves on near-surface wind turbulence and drag coefficient are investigated using large-eddy simulation. The impact of intermittent and transient wave breaking events (over a range of scales) is modeled as localized form drag, which generates airflow separation bubbles downstream. The simulations are performed for very young sea conditions under high winds, comparable to previous laboratory experiments in hurricane-strength winds. The results for the drag coefficient in high winds range between about 0.002 and 0.003. In such conditions more than 90% of the total airâ€“sea momentum flux is due to the form drag of breakers; that is, the contributions of the nonbreaking wave form drag and the surface viscous stress are small. Detailed analysis shows that the breaker form drag impedes the shear production of the turbulent kinetic energy (TKE) near the surface and, instead, produces a large amount of small-scale wake turbulence by transferring energy from large-scale motions (such as mean wind and gusts). This process shortcuts the inertial energy cascade and results in large TKE dissipation (integrated over the surface layer) normalized by friction velocity cubed. Consequently, the large production of wake turbulence by breakers in high winds results in the small drag coefficient obtained in this study. The results also suggest that common parameterizations for the mean wind profile and the TKE dissipation inside the wave boundary layer, used in previous Reynolds-averaged Navierâ€“Stokes models, may not be valid.

## Abstract

Direct numerical simulations (DNS) of turbulent Couette flow are combined with Lagrangian point-particle tracking to investigate the effects of a dispersed phase on bulk passive heat transport when the two phases can exchange both momentum and sensible heat. The idealized setup allows a fixed number of particles, without the influence of gravity, to be transported by carrier-phase motions across the mean velocity and temperature gradients that exist between the solid boundaries of turbulent Couette flow. In this way, the setup serves as a model of spray in a shear-dominated layer in the immediate vicinity of the water surface and provides insight into the ability of spray to enhance sensible heat fluxes. The authors find that the dispersed phase contributes a relatively large amount of vertical heat transport and increases the total heat flux across the domain by 25% or greater. Particles that accumulate in regions associated with wall-normal ejections efficiently carry heat across the channel. Furthermore, the authors find that the relative contribution of the dispersed-phase heat flux becomes larger with Reynolds number, suggesting an importance at atmospheric scales.

## Abstract

Direct numerical simulations (DNS) of turbulent Couette flow are combined with Lagrangian point-particle tracking to investigate the effects of a dispersed phase on bulk passive heat transport when the two phases can exchange both momentum and sensible heat. The idealized setup allows a fixed number of particles, without the influence of gravity, to be transported by carrier-phase motions across the mean velocity and temperature gradients that exist between the solid boundaries of turbulent Couette flow. In this way, the setup serves as a model of spray in a shear-dominated layer in the immediate vicinity of the water surface and provides insight into the ability of spray to enhance sensible heat fluxes. The authors find that the dispersed phase contributes a relatively large amount of vertical heat transport and increases the total heat flux across the domain by 25% or greater. Particles that accumulate in regions associated with wall-normal ejections efficiently carry heat across the channel. Furthermore, the authors find that the relative contribution of the dispersed-phase heat flux becomes larger with Reynolds number, suggesting an importance at atmospheric scales.

## Abstract

A massively parallel large-eddy simulation (LES) code for planetary boundary layers (PBLs) that utilizes pseudospectral differencing in horizontal planes and solves an elliptic pressure equation is described. As an application, this code is used to examine the numerical convergence of the three-dimensional time-dependent simulations of a weakly sheared daytime convective PBL on meshes varying from 32^{3} to 1024^{3} grid points. Based on the variation of the second-order statistics, energy spectra, and entrainment statistics, LES solutions converge provided there is adequate separation between the energy-containing eddies and those near the filter cutoff scale. For the convective PBL studied, the majority of the low-order moment statistics (means, variances, and fluxes) become grid independent when the ratio *z _{i}*/(

*C*Î”

_{s}*) > 310, where*

_{f}*z*is the boundary layer height, Î”

_{i}*is the filter cutoff scale, and*

_{f}*C*is the Smagorinsky constant. In this regime, the spectra show clear Kolmogorov inertial subrange scaling. The bulk entrainment rate determined from the time variation of the boundary layer height

_{s}*w*=

_{e}*dz*/

_{i}*dt*is a sensitive measure of the LES solution convergence;

*w*becomes grid independent when the vertical grid resolution is able to capture both the mean structure of the overlying inversion and the turbulence. For all mesh resolutions used, the vertical temperature flux profile varies linearly over the interior of the boundary layer and the minimum temperature flux is approximately âˆ’0.2 of the surface heat flux. Thus, these metrics are inadequate measures of solution convergence. The variation of the vertical velocity skewness and third-order moments expose the LESâ€™s sensitivity to grid resolution.

_{e}## Abstract

A massively parallel large-eddy simulation (LES) code for planetary boundary layers (PBLs) that utilizes pseudospectral differencing in horizontal planes and solves an elliptic pressure equation is described. As an application, this code is used to examine the numerical convergence of the three-dimensional time-dependent simulations of a weakly sheared daytime convective PBL on meshes varying from 32^{3} to 1024^{3} grid points. Based on the variation of the second-order statistics, energy spectra, and entrainment statistics, LES solutions converge provided there is adequate separation between the energy-containing eddies and those near the filter cutoff scale. For the convective PBL studied, the majority of the low-order moment statistics (means, variances, and fluxes) become grid independent when the ratio *z _{i}*/(

*C*Î”

_{s}*) > 310, where*

_{f}*z*is the boundary layer height, Î”

_{i}*is the filter cutoff scale, and*

_{f}*C*is the Smagorinsky constant. In this regime, the spectra show clear Kolmogorov inertial subrange scaling. The bulk entrainment rate determined from the time variation of the boundary layer height

_{s}*w*=

_{e}*dz*/

_{i}*dt*is a sensitive measure of the LES solution convergence;

*w*becomes grid independent when the vertical grid resolution is able to capture both the mean structure of the overlying inversion and the turbulence. For all mesh resolutions used, the vertical temperature flux profile varies linearly over the interior of the boundary layer and the minimum temperature flux is approximately âˆ’0.2 of the surface heat flux. Thus, these metrics are inadequate measures of solution convergence. The variation of the vertical velocity skewness and third-order moments expose the LESâ€™s sensitivity to grid resolution.

_{e}## Abstract

The effect of longwave radiative cooling at the planetary boundary layer (PBL) top in determining the entrainment rate was examined in this study. The entrainment rate equation that accounts for longwave radiative cooling can be formally derived from the following steps: 1) Derive the mean buoyancy budget in the thin layer between the averaged entrainment flux (i.e., the minimum buoyancy flux) level and the top of the entrainment zone where turbulent fluxes vanish. 2) Use the Deardorff entrainment closure assumption that the entrainment buoyancy flux is proportional to the vertically averaged buoyancy flux over the whole PBL, which is a generalized form of a widely accepted entrainment closure for the surface-heated convective PBL. This leads to an entrainment velocity that depends linearly on both the inverse of the interfacial Richardson number and the radiative flux divergence above the entrainment buoyancy flux level.

The relative importance of these two terms was examined through large eddy simulations (LESs) of several smoke-cloud-topped PBLs with various radiative forcings, radiative properties, temperature inversion strengths, and numerical advective schemes. These PBLs are driven only by cloud-top radiative cooling. The LESs were performed with a fine-grid nesting layer in the entrainment zone where the grid size is about 16 m and 8 m in the horizontal and vertical, respectively, which should be sufficient to resolve entrainment processes, at least for cases with weak capping inversions. The LESs showed that the contribution to entrainment rate from the radiative flux divergence term was either larger than or about equal to that of the interfacial Richardson number term. The LESs also showed that this radiative flux divergence occurs *within* cloudy regions below the local cloud tops.

The analysis was extended to the stratocumulus-topped PBL by using a stratocumulus-like smoke layer, and the result showed that the radiative flux contribution to the entrainment rate was still significant. Based on the physical understanding that this portion of the radiative flux divergence occurs within the upper half of cloud (or smoke) hummocks, the authors were able to analytically derive a relationship that links the radiative flux divergence to the cloud-top fluctuations, which were then empirically related to the interfacial Richardson number.

## Abstract

The effect of longwave radiative cooling at the planetary boundary layer (PBL) top in determining the entrainment rate was examined in this study. The entrainment rate equation that accounts for longwave radiative cooling can be formally derived from the following steps: 1) Derive the mean buoyancy budget in the thin layer between the averaged entrainment flux (i.e., the minimum buoyancy flux) level and the top of the entrainment zone where turbulent fluxes vanish. 2) Use the Deardorff entrainment closure assumption that the entrainment buoyancy flux is proportional to the vertically averaged buoyancy flux over the whole PBL, which is a generalized form of a widely accepted entrainment closure for the surface-heated convective PBL. This leads to an entrainment velocity that depends linearly on both the inverse of the interfacial Richardson number and the radiative flux divergence above the entrainment buoyancy flux level.

The relative importance of these two terms was examined through large eddy simulations (LESs) of several smoke-cloud-topped PBLs with various radiative forcings, radiative properties, temperature inversion strengths, and numerical advective schemes. These PBLs are driven only by cloud-top radiative cooling. The LESs were performed with a fine-grid nesting layer in the entrainment zone where the grid size is about 16 m and 8 m in the horizontal and vertical, respectively, which should be sufficient to resolve entrainment processes, at least for cases with weak capping inversions. The LESs showed that the contribution to entrainment rate from the radiative flux divergence term was either larger than or about equal to that of the interfacial Richardson number term. The LESs also showed that this radiative flux divergence occurs *within* cloudy regions below the local cloud tops.

The analysis was extended to the stratocumulus-topped PBL by using a stratocumulus-like smoke layer, and the result showed that the radiative flux contribution to the entrainment rate was still significant. Based on the physical understanding that this portion of the radiative flux divergence occurs within the upper half of cloud (or smoke) hummocks, the authors were able to analytically derive a relationship that links the radiative flux divergence to the cloud-top fluctuations, which were then empirically related to the interfacial Richardson number.