A Numerical Modeling Study of Warm Offshore Flow over Cool Water

VOLUME 133

MONTHLY WEATHER REVIEW

FEBRUARY 2005

A Numerical Modeling Study of Warm Offshore Flow over Cool Water ERIC D. SKYLLINGSTAD, ROGER M. SAMELSON, LARRY MAHRT,

AND

PHIL BARBOUR

College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon (Manuscript received 11 October 2003, in final form 8 July 2004) ABSTRACT Numerical simulations of boundary layer evolution in offshore flow of warm air over cool water are conducted and compared with aircraft observations of mean and turbulent fields made at Duck, North Carolina. Two models are used: a two-dimensional, high-resolution mesoscale model with a turbulent kinetic energy closure scheme, and a three-dimensional large-eddy simulation (LES) model that explicitly resolves the largest turbulent scales. Both models simulate general aspects of the decoupling of the weakly convective boundary layer from the surface, as it is advected offshore, and the formation of an internal boundary layer over the cool water. Two sets of experiments are performed, which indicate that complexities in upstream surface conditions play an important role in controlling the observed structure. The first (land–sea) experiments examine the transition from a rough surface having the same temperature as the ambient lower atmosphere, to a smooth ocean surface that is 5°C cooler. In the second (barrier island) experiment, a 4-km strip along the coastline having surface temperature 5°C warmer than the ambient atmosphere is introduced, to represent a narrow, heated barrier island present at the Duck site. In the land–sea case, it is found that the mesoscale model overpredicts turbulent intensity in the upper half of the boundary layer, forcing a deeper boundary layer. Both the mesoscale and LES models produce only a small change in the boundary layer shear and tend to decrease the momentum flux near the surface much more rapidly than the observations. Results from the barrier-island case are more in line with the observed momentum and turbulence structure, but still have a reduced momentum flux in the lower boundary layer in comparison with the observations. The authors find that turbulence in the LES model generated by convection over the heated land surface is stronger than in the mesoscale model, and tends to persist offshore for greater distances because of greater shear in the upper boundary layer winds. Analysis of the mesoscale model results suggests that better estimation of the mixing length could improve the turbulence closure in regions where the surface fluxes are changing rapidly.

1. Introduction Coastal weather is often dominated by heating and roughness differences introduced by the contrast between the land and sea surfaces. Sea and land breezes, for example, are forced by pressure differences between warm and cold air masses over the water and land. Other unique mesoscale circulations are produced by the interaction of the coastal boundary layer with coastal topography, such as the well-documented hydraulic behavior in regions of capes (Dorman 1985; Winant et al. 1988; Samelson and Lentz 1994; Burk et al. 1999). A more basic and less understood aspect of the meteorology at the land–sea interface concerns the evolution of the boundary layer as it crosses from the land to ocean surface. This involves the interaction of smallscale boundary layer turbulence with variations in sur-

Corresponding author address: Eric Skyllingstad, COAS, Oregon State University, 104 COAS Admin Bldg., Corvallis, OR 97331. E-mail: [email protected]

© 2005 American Meteorological Society

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face conditions and the associated mesoscale circulation. Over land, similarity theory and relatively abundant measurements have lead to parameterizations that offer a reasonable representation of vertical fluxes and average boundary layer structure under stationary, homogeneous conditions. Over the ocean, observation platforms are much more limited than land-based towers, with most measurements taken from aircraft, at least 15 m or more above the ocean surface. Consequently, even in homogeneous conditions, models of the turbulent boundary layer over the ocean have relied mostly on land-based measurements. Measurements and models of inhomogenous turbulent boundary layers, over either land or ocean, are comparatively less advanced. Our focus in this paper is on understanding the processes that control the evolution of the boundary layer as it transitions from land to a cool ocean surface. Our approach is to utilize both a mesoscale atmospheric model and a large-eddy simulation (LES) model to simulate a coastal boundary layer in an idealized offshore-flow configuration, and to compare the results with the observations described by Vickers et al. (2001).

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We will be especially interested in the decoupled upper boundary layer and the processes that control the local intensity of the turbulence. Offshore flow from heated or rough land conditions to a cold ocean surface presents a scenario that is especially difficult to simulate accurately. Under these conditions, a stable, internal boundary layer (IBL) tends to form in response to cooling by the surface. Formation of the IBL is controlled by a number of factors including the wind speed, upstream surface conditions such as roughness length and heat flux, and the water temperature. Observations of IBL formation over the ocean (Rogers et al. 1995; Friehe et al. 1991) show the formation of a weakly stable boundary layer with a balance between shear production of turbulence, buoyancy destruction, and frictional dissipation. Bulk stability, as measured by the local gradient Richardson number, indicates flow characteristics supporting a critical Richardson number of about 0.25 in these cases. These IBLs exhibit features of classical internal boundary layers, as surveyed in Garratt (1990). With larger changes of surface temperature, the turbulence may partially collapse within the IBL, referred to as quasi-frictional decoupling (Smedman et al. 1997a,b). Turbulence in the remnant land-based boundary layer above the IBL may become semidecoupled from the surface, as observed in offshore flow of warm air from land over colder water (Smedman et al. 1997a; Vickers et al. 2001) and flow from the Gulf Stream over cooler inland water (Mahrt and Vickers 2004). The latter study indicated that turbulence could intermittently regenerate in the residual mixed layer above the new surface inversion. Such flows violate the concept of a boundary layer. Reduced turbulence near the surface decreases the sea surface roughness, which accelerates the collapse of the turbulence (Plant et al. 1999; Mahrt et al. 2001a,b) in contrast to flow over land where the flow remains constant. Aloft, turbulence can persist because of advection from the upstream land surface and local shear generation. Fluxes in this elevated layer are not necessarily related to the surface forcing and may have a vertical structure far different from standard boundary layer profiles. Vickers et al. (2001, hereafter referred to as VMSC) presented a collection of aircraft case studies from the Shoaling Waves Experiment (SHOWEX) describing flow from a heated land surface over a cool ocean. These observations demonstrated a consistent structure with acceleration in the low-level winds and decaying turbulence at all levels (Fig. 1). Boundary layer fluxes generated over the land surface decayed gradually as the boundary layer was advected over the water. At the same time, near surface winds accelerated so that shear over the water was greatly reduced above a thin surface layer in comparison with the land surface. However, VMSC were not able to close the momentum budget for these cases, suggesting either errors in the measurements or a relatively large pressure gradient force. Based on the VMSC analysis, a conceptual model of

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offshore flow from warm, rough land to cold, smooth water can be formulated. Boundary layer structure in the conceptual model begins over land with a shear and convectively driven boundary layer with winds increasing gradually from the surface to the boundary layer inversion. When the boundary layer is advected across the coast and offshore, turbulence decreases rapidly at the surface because of both decreased roughness and a reversal in the buoyancy flux from convective heating to cooling and the formation of a surface stratified layer. Moving offshore, the system divides into two regions of mixing. Near the surface, very weak turbulence is maintained by flow over the almost smooth ocean. Aloft, turbulence generated over land is advected out over the water and gradually decays. Two factors control the decay of the mixing region aloft. First, the direct forcing of turbulence by convection is cut off by cooling at the cool ocean surface. Second, shear produced by downward momentum flux over the rough land surface decreases as the flow accelerates above the surface stable layer. With these two reductions in sources, the turbulence decays from frictional dissipation with an e-folding distance of a few kilometers downwind from the coast for winds of ⬃12 m s⫺1. Numerically simulating the offshore-flow boundary layer observed by VMSC involves modeling the evolution of turbulent fields within an inhomogeneous mesoscale flow. To represent the unresolved turbulent fields, the mesoscale model used here relies on a version of the widely used Mellor and Yamada (1982) turbulence closure scheme (referred to hereafter as MY82), which calculates a flow-dependent budget of turbulent kinetic energy. The advantage of this closure scheme is that it is based on a turbulence budget equation derived from first principles. However, calculating this budget requires estimating turbulent eddy properties, including especially the turbulent length scale. Most schemes for estimating this length scale have been developed primarily from comparisons with land-based measurements under horizontally homogeneous conditions, and may not be as successful in the context of the offshore flow problem considered here. For example, schemes that link turbulence parameters directly with surface fluxes (e.g., Thompson and Burk 1991) are inconsistent with the presence of decoupled, elevated turbulence. Alternative turbulence closures based on vertical mixing coefficient profiles (e.g., Troen and Mahrt 1986) also assume a surface-based boundary layer; modified versions of the profile method may provide a means of dealing with elevated shear layers (see Mahrt and Vickers 2003). One outstanding feature of the offshore flow problem investigated by VMSC is the rapid adjustment that the boundary layer undergoes in crossing the coastline. The horizontal scale of the problem (⬃10 km) makes this a good candidate for the application of a LES model, which simulates both the boundary layer behavior and the turbulent scale processes. For example,

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FIG. 1. Vertical cross section of the offshore velocity (contours) and friction velocity, u* ⫽ (w⬘u⬘2 ⫹ w⬘v⬘2)1/4, (shaded) observed on (a) 11 Nov 1999 and (b) 18 Mar 1999 (from Vickers et al. 2001).

Glendening and Lin (2002) applied LES to examine the effects of surface roughness changes on the formation of the IBL. Our objectives in using the LES here were twofold: to compare turbulence parameters predicted by the LES to the parameterized fields generated by the mesoscale model simulations, and to determine why the mesoscale model simulations failed to reproduce some specific aspects of the observations. To make these comparisons, idealized simulations were performed with the LES model using a quasi-Lagrangian technique.

2. Models a. Mesoscale model We conduct numerical simulations using the Naval Research Laboratory (NRL) Coupled Ocean–Atmo-

sphere Prediction System (COAMPS) mesoscale atmospheric model (Hodur 1997). This model solves the nonhydrostatic, compressible, moist equations for atmospheric motion. It is implemented here in a twodimensional configuration, with variations cross-shore (x) and in the vertical (z), and no variations alongshore ( y). The basic x–z grid is 241 ⫻ 46 points, with uniform horizontal resolution of 200 m and vertical resolution increasing with height from 1 m at the surface to 40 m at the top of the model domain, at 3-km height. The simulations discussed below were generally limited to 4 h in duration. Surface boundary conditions were obtained from bulk-formula surface fluxes, using the standard COAMPS schemes with fixed land and ocean surface temperatures and either fixed or variable roughness lengths (Fairall et al. 1996). Details of the surface

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boundary conditions for individual simulations are given in the discussion below. Of particular interest in these simulations is the turbulence closure scheme, which is based on the MY82 formulation. In this scheme, the turbulent mixing length depends on the turbulent kinetic energy (TKE). The version implemented in COAMPS is the 1.5 order, level-2.5 scheme, which uses a prognostic equation for the TKE e,

冉

冊

⭸ ⭸U ⭸V ⭸e De ⫺ ⫺ w⬘v⬘ leSe ⫽ ⫺w⬘u⬘ Dt ⭸z ⭸z ⭸z ⭸z ⫹ ␤gw⬘␪⬘v ⫺

e 3Ⲑ2 , ⌳1

共1兲

where the second term on the left is the vertical diffusion of TKE, the first and second terms on the right are the shear production terms, the third term is the buoyancy production (or, frequently, destruction) term, the last term is the dissipation, and the notation is standard. The quantity ⌳1 ⫽ 6l, where l is the master length scale defined below. Shear production is approximated by an eddy viscosity, ⫺w⬘u⬘

冋冉 冊 冉 冊 册

⭸U ⭸V ⫺ w⬘v⬘ ⫽ Km ⭸z ⭸z

⭸U ⭸z

2

⫹

⭸V ⭸z

2

,

共2兲

and buoyant production is approximated using an eddy diffusion, ⭸␪v , ⭸z

␤gw⬘␪⬘v ⫽ Kh where

Km, h ⫽ Sm, hl 公2e.

共3兲

In (1) and (3), Se and Sm,h are coefficients that depend on the gradient Richardson number (details are presented in MY82). The master length scale, l, is defined as l⫽

␬z , 1 ⫹ ␬z Ⲑ␭

共4兲

where the asymptotic length scale

冕公 冕公 z

␭⫽␣

e dz 共5兲

e dz

is proportional to the first moment of the vertical distribution of e, with ␣ ⫽ 0.1 in the original MY82 formulation. In our COAMPS implementation, ␣ is prescribed depending on the dimensionless quantity ␵1 ⫽ z1/L, where L is the Monin–Obukhov depth (in units of meters),

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⫺␪sfc u3 * L⫽ , g␬共w⬘T⬘兲sfc and z1 ⫽ 1 m. We set ␣ depending on the stability,

␣⫽

再

0.1 0.1 ⫺ 10␵1 0.3

␵1 ⬎ 0 ⫺0.02 ⬍ ␵1 ⬍ 0, ␵1 ⬍ ⫺0.02

共6兲

where ␵1 ⬎ 0 for stable conditions (surface cooling) and ␵1 ⬍ 0 for unstable conditions (surface heating). The standard COAMPS scheme follows a similar prescription, but sets z1 equal to the first gridpoint height and effectively assumes z1 ⫽ 10 m; for the vertical grid used here, this yielded an ␣ value that was too small, causing overly weak turbulence in convective conditions (see section 2c).

b. Large-eddy simulation model The LES model used here is described in Skyllingstad (2003) and based on Deardorff (1980) with a subgrid-scale parameterization by Ducros et al. (1996). The model x–y–z grid was 256 ⫻ 256 ⫻ 120, with grid spacing of 4 m in all directions. This domain size is just large enough to encompass the maximum eddy scales generated by convective heating given the initial boundary layer depths considered here. Experiments with larger domain sizes (not presented) demonstrated that allowing for multiple convective circulations does not substantially change the average resolved fluxes generated by the LES model. Simulations were initialized with a constant vertical potential temperature gradient and geostrophic offshore wind of 14 m s⫺1, and allowed to spin up over a 3-h period. Surface roughness length over land was set to 0.1 m, representing a typical surface. Surface fluxes were parameterized using the Louis (1979) scheme as in the COAMPS model. Changes in air temperature from radiative effects or larger-scale mesoscale circulations were not considered. Air temperature and surface temperature were initially set to the same value. Simulations of the coastal zone were made by assuming a Lagrangian model framework moving with the geostrophic wind. Changes in the surface forcing were imposed simultaneously across the model domain, as an approximation to the more gradual change that would occur as the boundary layer moved over the coast. Assuming a 8–10 m s⫺1 average boundary layer airspeed, it would only take about 2–3 min for the air contained in the model domain to pass over the coast, so this approximation is reasonable. Sea surface temperature was set to the same value as in the mesoscale experiments (described below). The roughness length over water in the LES model was based on output values from COAMPS (which uses a variable roughness following Fairall et al. 1996). Comparisons between the LES and mesoscale model

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fields are performed by horizontally averaging the LES output. This process removes the turbulent variations that are resolved in the LES model, but parameterized in the mesoscale model.

c. Preliminary numerical experiments The master length scale l depends on the scaling factor, ␣, through Eq. (5). For the standard COAMPS scheme, ␣ was limited to values very close to 0.1 for our vertical grid, and consequently convective conditions generated much less mixing than comparable LES simulations. To correct this discrepancy, we modified the specification of ␣ as described above, and performed a series of sensitivity experiments similar to the land–sea case described below, but with periodic lateral boundary conditions and a land surface only, and with a range of surface temperatures representing neutral to unstable (convective) conditions. Results from these experiments indicate that the ␣ formula proposed above yields a homogeneous boundary layer structure generally consistent with results from the LES model. These experiments also served to confirm that the open boundary conditions in the mesoscale model simulations described below did not have a large effect on the boundary layer depth or average properties.

3. Land–sea case a. Model configurations The VCMS observations and analysis of offshore flow included only limited consideration of the upstream conditions, and effectively assumed a quasiequilibrium structure for the boundary layer over land. The first set of simulations that we present here proceeds from this starting point, and considers offshore flow from a homogenous, rough land surface to a relatively cold, smooth ocean surface. The land surface temperature in this experiment is set equal to the initial boundary layer temperature so that the land surface heat flux is near zero. The resulting boundary layer is weakly stable with near linear shear. We refer to this case as the land–sea scenario. The mesoscale model was initialized with an idealized vertical structure similar to observations taken on 18 March 1999 (Fig. 1). The initial potential temperature, ␪, was set to 287 K at the surface, with a linear increase with height to 293 K at 350 m. Above 350 m, the increase was set to 1 K per 150 m. Surface conditions over land were imposed with ␪ ⫽ 288 K and surface roughness, zo, set to 0.1 m for a distance of 20 km upstream from the coastline. From x ⫽ 20 km to the outflow boundary at x ⫽ 48 km, surface ␪ was set to 283 K and zo to the value predicted by Charnock’s relationship (about 5 ⫻ 10⫺5 m). Winds were initially set to be in geostrophic balance over the entire domain, with a velocity of 14 m s⫺1. The mesoscale model simulations were conducted for 4–6 h, long enough for the bound-

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ary layer system to reach an approximate equilibrium state. A similar setup was prescribed for the LES simulations, but in the quasi-Lagrangian configuration described above, with the changes in the surface forcing imposed as a function of time instead of space. The simulation was initiated with 4 h of land forcing, with surface ␪ ⫽ 288 K and zo ⫽ 0.1 m, which allowed for the development of a near-equilibrium boundary layer structure. Ocean surface boundary conditions were then imposed in the land–sea case, with a surface ␪ ⫽ 283 K and a fixed roughness length of 5 ⫻ 10⫺5 m.

b. Model results In the land–sea mesoscale simulation (Fig. 2), the upstream conditions after 6 h consist of a weakly convective, 300-m-deep boundary layer that is well-mixed in potential temperature and has nearly constant vertical shear in cross-shore velocity (u component) above 50 m. Turbulence is confined primarily to the boundary layer, where TKE decreases uniformly from 0.45 m2 s⫺2 near the surface to negligible values above the boundary layer. After leaving the coastline, turbulence intensity decreases rapidly near the water surface in response to the development of stable stratification and smaller surface roughness, with maximum turbulence values occurring at progressively higher altitudes as the boundary layer moves over the water. The transitions at the land–sea boundary in the turbulent fields, for example, TKE and friction velocity u (defined as the * square root of turbulent momentum flux: u ⫽ (w⬘u⬘2 ⫹ * 2 1/4 w⬘v⬘ ) ), are more dramatic than those in the mean fields (Fig. 3), as should be anticipated from general considerations. At the surface, where the stable IBL develops over the cool water, TKE and u decrease * abruptly to less than one-third of their upstream values. This transition takes place in less than 1-km distance offshore. The maximum values of the turbulent fields, which occur at heights of 100–200 m, also decline rapidly offshore, with those for TKE and u decreasing by * half in roughly 10 km from the coastline, while the u component of velocity accelerates by only 0.5 m s⫺1 when passing over the coastline. Results from the land–sea LES case (Fig. 4) are very similar to the mesoscale model results (Fig. 3). In both simulations, u and TKE drop rapidly offshore at the * surface, while decreasing more slowly aloft. However, the LES model does not reduce u as quickly in the * lowest 50 m. For example, at 20-m height, the LES u * ⫺1 decreases to 0.15 m s at a point roughly 5 km offshore, whereas the mesoscale u falls to below 0.15 m * s⫺1 about 2 km from the coastline. Some of this difference may be related to the use of a vertically uniform advection velocity in the quasi-Lagrangian LES model configuration. In the upper boundary layer, turbulence penetrates to greater depths in the mesoscale model than in the LES model; u decreases to 0.15 m s⫺1 at a * height of 225 m in the LES, versus 280 m in the meso-

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FIG. 2. Vertical cross sections of (a) potential temperature, (b) u wind component, and (c) TKE from the land–sea mesoscale model simulation at hour 4.

scale model. Offshore vertical gradients of u are also * somewhat smaller below 100 m in the LES model than in the mesoscale model, indicating less momentum flux convergence (momentum flux is negative), which would tend to decrease the acceleration of offshore flow. However, the effect is small, as in both cases the u wind component increases only slightly offshore. The cross-shore gradients in boundary layer depth in the two simulations are both weak, but differ in sign. Near surface acceleration in the mesoscale model is produced as the boundary layer moves over the smoother ocean surface, generating a net divergence and weak downward vertical motion. This mesoscale effect is missing in the LES model, which has periodic

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FIG. 3. Vertical cross sections of (a) TKE, (b) u , and (c) u wind * component near the simulated coast (x ⫽ 20 km) from the land– sea mesoscale model simulation at hour 4.

boundaries and zero average vertical velocity. Consequently, the boundary layer depth in the LES model instead increases slightly over the ocean, because of entrainment. Other differences between the mesoscale model and the LES results are apparent in offshore vertical profiles of potential temperature and TKE (Fig. 5). In particular, the LES TKE is largest near the bottom of the boundary layer whereas the mesoscale model TKE is largest toward the top of the boundary layer. The net effect of higher TKE aloft in the mesoscale model is a deepening of the boundary layer. In general, the two simulations appear to resemble each other more closely than either resembles the observations (Fig. 1), which have stronger mean flow accelerations and weaker vertical gradients in u . There * are a variety of possible explanations for these latter

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FIG. 4. As in Fig. 3, but for the land–sea LES case. Each vertical profile represents an instantaneous horizontal average over the LES domain, and the horizontal axis has been converted from time to distance using a uniform 14 m s⫺1 translation velocity. The simulated coast is at 20 km.

differences, including unknown mesoscale pressure gradients and aliased transient motions. We argue below (section 4) that some of the systematic differences between the modeled and observed fields are associated with surface heating over a barrier island that is not represented in the land–sea simulations.

c. TKE budgets Before proceeding with the analysis of the more complex simulations that include the barrier-island surface heating, it is useful to examine and compare the meso-

FIG. 5. Vertical profiles of (a) u component, (b) potential temperature, and (c) TKE at roughly 4 km offshore in the land–sea case. The vertical grid for the mesoscale model (x) is shown in (a).

scale and LES model representations of the turbulent dynamics in the simpler land–sea case. In the mesoscale model, TKE is computed using the scheme outlined above in Eqs. (1)–(5). The turbulent fluxes in the TKE

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budget Eq. (1) are not directly simulated, but are estimated based on the MY82 parameterization. In the case of the LES model, we can directly calculate the

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TKE budget terms because a significant portion of the turbulence flow field is resolved by the model. For the LES model, TKE is determined using

⭸E ⭸ui ␳⬘ ⭸ui ⭸ ˜ 兲, ⫽ ⫺ui u3 ⫺ u3 · g ⫹ 具u ⬙i u ⬙3典 ⫺ 共u E ⫹ ui 具u ⬙i u ⬙3典 ⫹ u3P ⭸t ⭸x3 ␳o ⭸xj ⭸x3 3 共IV兲 共I兲 共II兲 共III兲 where

具u ⬙i u ⬙j 典 ⫽ ⫺Km

冉

冊

p 2 ⭸ui ⭸uj ⫹ , P˜ ⫽ ⫹ q2, ⭸uj ⭸ui ␳o 3

overbars represent horizontal domain averages, angle brackets represent subgrid averaging, double primes denote subgrid-scale fields, E represents the LES TKE field (as opposed to e for the mesoscale model TKE), p is the pressure, ␳o is the background density, and q is the subgrid-scale TKE, which is solved as part of the pressure field. Subscripts 1, 2, and 3 represent directions x, y, and z respectively, and summation over repeated subscripts is implied in (7). Terms in (7) are (I) shear production, (II) buoyant production or destruction, (III) dissipation, and (IV) vertical transport. Overall, the budget profiles in the offshore region from the two models (Fig. 6) are quite similar. Turbulence in both models is forced by shear production and balanced for the most part by dissipation. Buoyancy destruction comes in to play near the top of the boundary layer where turbulent entrainment of warmer air expends TKE, and above the stable IBL, where upward transport of cooler air removes TKE. Significant differences between the two models are noted in the details of the TKE budget term profiles. For example, in comparison with the mesoscale model, the LES model displays higher shear production below 120 m and smaller

共7兲

shear production above ⬃160 m. Also, values of the buoyant production near the boundary layer top are noticeable larger in the mesoscale model, suggesting that entrainment rates are too large in the MY82 scheme. The relative strength of the shear production term in the mesoscale model, in comparison with the LES, is consistent with stronger TKE values in the mesoscale model (Fig. 5) and may explain the deeper boundary layer in the mesoscale model. A more direct measure of the boundary layer growth rate can be established by examining the eddy viscosity coefficient, Km. Eddy viscosity is computed from (3) for the mesoscale model, and estimated using u⬘w⬘ KLES ⫽ ⫺ ⭸u Ⲑ⭸z for the LES model. Values of the mixing coefficient (Fig. 7) from the simulations are consistent with the TKE profiles shown above, indicating stronger vertical mixing in the upper boundary layer in the mesoscale model relative to the LES. Below ⬃60 m height, the LES model shows stronger mixing, again in agreement with the TKE profiles.

d. Master length scale and LES mixing length A number of factors could cause the TKE and Km profile differences in Figs. 5–7. For example, large ed-

FIG. 6. Domain horizontally averaged terms from the TKE budget from (a) the mesoscale model and (b) the LES model. Definitions of terms are provided in the text.

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FIG. 7. Vertical profile of eddy viscosity, Km, at 4 km offshore. The profile from the LES model is the horizontal average.

dies that directly transport flow properties over the depth of the boundary layer are not adequately represented in the standard MY82 scheme (1)–(4). An important parameter related to the eddy scale and the parameterized eddy diffusion is the master length scale, l, defined by (4)–(6). We can estimate the turbulent mixing length from the LES results by using the definition of dissipation in (1), and substituting the values of E and dissipation calculated from the LES model, lLES ⫽

E3Ⲑ2 , 6␧

FIG. 8. Length scale lLES (labeled LES) estimated from the LES model as described in the text, and master length scale l from the mesoscale model (labeled Mesoscale).

e. Buoyancy and shear mixing lengths In the present simulations, the master length scale l is largest near the top of the boundary layer because of the vertical dependence assumed in (4)–(6). Other approaches for calculating l include methods that are not dependent on ␭, but are tied to the local TKE and stratification. For example, Andre et al. (1978) derives an eddy scale dependent on the local Brunt–Vaisälä frequency, N: lb ⫽ 0.75

where ⭸ui ␧ ⫽ 具u ⬙i u ⬙3 典 . ⭸xj The values and vertical distribution of l in the mesoscale model are significantly different from the resulting LES estimates (Fig. 8). Specifically, lLES is larger in the lower boundary layer and decreases gradually when moving upward to the boundary layer top. In contrast, the mesoscale model l increases rapidly near the surface and then more gradually through the boundary layer until reaching the boundary layer top, where it drops off rapidly. The location of the maximum l in the mesoscale simulations is close to the peak values for TKE and u , indicating that the vertical structure of this mix* ing length may be affecting the distribution of the boundary layer turbulence.

where N⫽

公E N

冑

g ⭸␪ . ␪ ⭸z

,

共8兲

The idea behind this scaling is that the square root of the TKE is an approximation to the eddy vertical velocity, while the local stratification, represented by N, limits the vertical extent of the motion; the coefficient of 0.75 accounts for E being larger than w⬘2. Scaled versions of (8) have been used as a maximum length scale in higher-order MY82 schemes such as proposed in Galperin et al. (1988). A second alternative limiting length scale can be obtained by considering the local effect of mean vertical shear on turbulent eddies: shear in the background wind will tend to advect the top of the eddy downstream at a different rate than the bot-

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tom of the eddy, effectively tearing the eddy in half. A characteristic scale of this type is ls ⫽ 0.75

公E , ⭸UⲐ⭸z

共9兲

where we again multiply by 0.75 to equate E to w⬘2. Similar scales have been suggested by Smedman et al. (1995) and Hunt et al. (1988). Comparison of the dissipation-estimated length scale from the LES with lb and ls also calculated from the LES (Fig. 9) shows reasonable agreement for the shear length scale ls, but poor agreement for the buoyancy length scale lb, except near the top of the boundary layer. This result is consistent with the shear-dominated turbulence indicated by the TKE budget analysis and the relatively weak stratification that exists in the remnant decoupled boundary layer over the ocean surface. Near the boundary layer top, both shear and buoyancy begin to control turbulent eddy scales, hence the closer agreement between all of the length scales. Similarity between the LES dissipation length scale and l s throughout most of the boundary layer suggests that local shear controls the size of turbulent eddies. In contrast, the standard mesoscale parameterization assumes that eddy scales are set by the integrated boundary layer turbulence as represented in (4)–(6). Motivated by these considerations, we conducted ad-

FIG. 9. Length scale lLES (labeled LES) and buoyancy and shear length scales, lb and ls, estimated from the LES model as described in the text, along with the master length scale l from the mesoscale model (labeled Mesoscale).

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ditional simulations in which the master length scale was limited by ls as calculated from the mesoscale shear and TKE. Results from this test did not show a noticeable improvement, because the boundary layer shear decreased, yielding a greater ls. However, given that the MY82 scheme includes many adjustable parameters that are tuned to provide optimal results with the TKE length scale, it should not be surprising that a simple, physically motivated alteration of a single component of the scheme may not improve a given simulation. A comprehensive attempt to improve the parameterization scheme is beyond the scope of the present study. In addition, we have found that in the present case, specific details of the upstream conditions strongly influence the offshore flow evolution, and proceed here instead to a more detailed consideration of this dependence.

4. Barrier-island case a. Model configurations In the preceding section, substantial differences were found between the land–sea simulations and the VCMS observations. In this section, we examine the possibility that some of these discrepancies may be in the representation of upstream conditions, rather than internal model dynamics. A relatively complex upstream history of the offshore-flowing boundary layer air can be inferred for the VMSC observations: a weakly convective continental boundary layer well upstream from the coastline initially passed over a large embayment, where it developed characteristics of a weakly stable boundary layer, and then passed over a narrow (4-km) barrier island, where relatively strong surface heating and a rough surface boundary forced a strong development of turbulence before the air moved across the coastline and over the colder Atlantic ocean (VMSC, their Figs. 1 and 2; see also Sun et al. 2001, their Fig. 4). Instead of a uniform upstream land surface, we now introduce a short section of surface heating to represent the effect of the barrier island. In this barrier-island case, we focus on the effects of the strong heating pulse generated by the narrow island surface, and the subsequent downstream adjustment. Far upstream, we imposed the same land surface conditions in the barrier-island simulations as in the land– sea case, with ␪ ⫽ 288 K and zo ⫽ 0.1 m in the mesoscale model. This upstream region extended from the inflow boundary to x ⫽ 16 km and represented the near-neutral boundary layer structure upstream from the barrier island. The barrier island was represented by raising the surface temperature to ␪ ⫽ 293 K in the region 16 km ⬍ x ⬍ 20 km. Ocean (x ⬎ 20 km) surface conditions were the same as in the land–sea case. For the LES barrier-island case, passage of the air volume over the barrier island was represented by increasing the surface ␪ to 293 K for a period of 8 min

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before imposing the ocean boundary conditions The 8-min period is roughly the time it would take an air volume to move 4 km at the 8 m s⫺1 airspeed near the middle of the boundary layer.

b. Mesoscale model results Upstream from the heated barrier-island region, the simulated boundary layer is very similar to the land–sea case, with a shear-forced boundary layer extending up to ⬃300 m (Fig. 10). When the boundary layer moves over the barrier island at x ⫽ 16 km, the flow undergoes an abrupt change as convective turbulence develops

FIG. 10. Vertical cross sections of (a) potential temperature, (b) u wind component, and (c) TKE from the mesoscale model barrier-island simulation at hour 6. The heated strip of land is located between x ⫽ 16 and 20 km.

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and TKE increases from ⬃0.4 to ⬃1.3 m2 s⫺2. At the same time, the offshore flow speed increases throughout the boundary layer as slower moving air is transported vertically by the parameterized turbulence. At the coastline, the surface temperature again changes rapidly, and the sign of the surface heat flux reverses, as the warm boundary layer air is advected offshore over the cool water. A shallow, stable, IBL develops near the surface with characteristics much like the land–sea case described earlier. Above the stable layer, potential temperature in the advected warm layer remains nearly uniform, while the height of the inversion capping the warm layer decreases slowly offshore. The cross-shore velocity increases offshore by ⬃1–2 m s⫺1 between 50 and 150 m, with the increase occurring progressively closer to the coast at successively lower levels. Near the top of the boundary layer, the cross-shore velocity and potential temperature displays only a slight change as the boundary layer passes the barrier-island heat source, indicating that the width of the island (or the duration of heating) limits the depth to which the enhanced turbulence penetrates. Offshore, weak subsidence forced by the cross-shore velocity acceleration accounts for the slight decrease in the warm layer depth. Comparison of the model friction velocity u and * mean offshore velocity u (Fig. 11) with the aircraft observations (Fig. 1) indicates that the barrier-island simulation reproduces several important qualitative features of the offshore flow more accurately than the land–sea case. These include the region of elevated u * around 150–250 m altitude over the ocean, above the decoupled boundary layer, and the acceleration of the mean offshore flow, and corresponding decrease in mean vertical shear, between 50 and 250 m. More detailed examination, however, reveals substantial remaining differences between the model and observations. The offshore decrease in u is much more rapid in * the model than in the observations, leading to a more significant vertical gradient of u above the IBL. The * model maintains significant mean vertical shear offshore, while the observations show almost no mean vertical shear in the lowest 100 m at 5 km offshore. Even some basic, qualitative features differ: in the model, vertical profiles of u over the ocean surface have a * well-defined minimum in the upper part of the IBL, near 50-m altitude, while no such minimum appears in any of the aircraft observations. Some of these discrepancies may result from insufficient knowledge of upstream conditions, alongshore gradients, or other uncertainties or deficiencies in the measurements and model initialization. However, similar contrasts in modeled and observed cross-shore and vertical gradients of turbulent and mean quantities in and immediately above the IBL were found in several related cases. These features appear to be robust differences between the observations and the model response.

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FIG. 11. Vertical cross-section plots of (a) TKE, (b) u*, and (c) u wind component made the simulated coastline at x ⫽ 20 km from the mesoscale model barrier-island simulation at hour 6.

c. LES model results In the barrier-island case, the LES model produces an offshore boundary layer structure with general qualitative features that are broadly similar to the observations and mesoscale model results, with an elevated region of turbulence over the cool ocean (Fig. 12). Some features of the observations appear more accurately reproduced in the LES results than in the mesoscale model results. In particular, the LES model produces a higher u value below 50-m height and a less * pronounced minimum near 50 m, relative to the mesoscale model. Both TKE and u are more uniform with * height in the LES results than in the mesoscale model. In the latter, the elevated turbulent region appears as a narrow band with peak values that are located near the

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FIG. 12. Vertical cross sections of (a) TKE, (b) u , and (c) u * velocity component from the LES barrier-island simulation. Each vertical profile represents an instantaneous horizontal average over the LES domain, and the horizontal axis has been converted from time to distance using a uniform 14 m s⫺1 translation velocity. The simulated coast is at 20 km.

top of the simulated boundary layer between 200 and 300 m, while in the observations and the LES model, the turbulence is more uniformly distributed with height. In the LES results, boundary layer heating over the island homogenizes the momentum field by combining slow-moving near-surface air with faster-moving air aloft. Near the surface, the flow increases in speed from about 4 to 7 m s⫺1 within a few kilometers of the coastline. At heights between 50 and 180 m, the flow initially decreases in speed as momentum is removed by down-

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ward turbulent transport and enhanced surface drag in the convective region over the island. Reversal of the heat flux at the coastline rapidly changes the strength of turbulence in the lower boundary layer and decreases this downward momentum flux. However, the boundary layer aloft is not immediately affected by the changes at the coastline, and turbulence above roughly 80 m continues to transport momentum downward. The net effect of this momentum flux convergence is an increase in the wind speed at lower levels, between roughly 50- and 150-m altitude, for a distance of about 10 km offshore, and a slight decrease in wind speed at upper levels, between roughly 150 and 350 m. Increasing winds above the offshore stable boundary layer has often been attributed to cross-isobaric acceleration produced by the reduction in surface friction. Our results indicate that downward turbulent momentum flux may also play an important role in accelerating the low-level flow. Much of the abrupt change in the mesoscalesimulated turbulence at the coastline can be attributed to the surface flux control of the mixing length using (4)–(6). Unlike the mesoscale model, turbulent eddy scales in the LES gradually increase as the convective boundary layer develops over the island, and then gradually decrease as the decoupled boundary layer moves over the cold ocean. In contrast, the mesoscale length scale changes simultaneously throughout the boundary layer depth between small values over the cool surface and large values over the simulated heated island. Detailed differences between the LES and the mesoscale model results are shown by profiles of the average fields 4 km from the coastline (Fig. 13). Vertical shear in both cases is reduced by the island heating, however, the LES profile shows slightly stronger shear near the top of the profile. Potential temperature in the mesoscale model case is well mixed up to a height of ⬃280 m, whereas the LES profile shows significant restratification. Much of the restratification is likely caused by the higher TKE values in the LES model, which are forced by large convective eddies that propagate offshore from the heated strip, as shown by crosssection plots of the vertical velocity and potential temperature (Fig. 14). Although horizontal advection of TKE in the mesoscale model accounts for some of the eddy effect, the parameterized turbulent heat flux is much less than the resolved eddy field prediction of the LES model. Figure 14 also shows that turbulent eddies are occasionally transporting cooler (warmer) air upward (downward), which is consistent with the buoyancy destruction term in the TKE budget. Comparison of the LES results with the observations indicates a better agreement than for the mesoscale model simulation discussed above. One likely reason for this is the stronger transport of momentum and heat by convective eddies in the LES case in the decoupled turbulent layer over the ocean. For example, the near-

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FIG. 13. Vertical profiles of (a) u component, (b) potential temperature, and (c) TKE at 4 km offshore from the barrier-island case.

surface air temperature has cooled considerably (1°– 2°C) from the ambient boundary layer potential temperature by a distance of 2 km (Fig. 14), but turbulent updrafts and downdrafts advected from the land surface are still directly transporting air from near the sur-

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FIG. 14. Vertical cross sections of (a) potential temperature, and (b) vertical velocity from the LES barrier-island simulation after 5 min of surface cooling (about 2.5 km offshore).

face to heights of 300 m. Analysis of the TKE budget terms from the two models shows that in both simulations, turbulence production is almost entirely forced by the shear term and balanced by dissipation, which both have absolute maxima near the top of the boundary layer (Fig. 15). However, the intensity of these terms in the LES model is more than double the me-

soscale model values. In the mesoscale model, convective mixing of the momentum fields over the barrier island reduces the background shear too rapidly. Consequently, shear production over the water is much less in the mesoscale model in comparison with the LES. Stronger turbulence in the LES model (see Fig. 13) produces greater turbulent heat transport, as indicated

FIG. 15. Horizontally averaged terms from the TKE budget from (a) the mesoscale model and (b) the LES model for the barrier-island case. Definitions of terms are provided in the text.

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by the stronger buoyancy destruction in comparison with the mesoscale model. Finally, vertical transport of turbulence by the large eddies in the LES produces a larger turbulent energy export from upper levels downward to the lower half of the boundary layer. Comparison of the mixing lengths calculated from the LES and mesoscale model (Fig. 16) are consistent with the outcome of the TKE budget analysis. Overall, the LES model indicates a mixing length maximum at about 75-m height (the secondary maximum at ⬃330 m is an artifact of the dissipation decreasing to very small values), whereas the mesoscale model places the maximum near the boundary layer top at 300 m. The maximum LES mixing length is ⬃2 times greater than the mesoscale value at the same height. Again, the shearbased length scale is very similar to the LES result, indicating that turbulent scales are controlled by the background shear in this case. These results suggest that the mixing length parameterization used in the mesoscale model has a dominant effect on the vertical turbulence profile, forcing unrealistic boundary layer behavior when surface conditions change rapidly. Over land, convective heating in the mesoscale model generates an immediate increase in the mixing length, which leads to a rapid increase in turbulence throughout the boundary layer. Results from the LES, however, indicate that the effect of surface heating takes more time to evolve, such that the strongest turbulence is not generated until the bound-

FIG. 16. Mixing length as estimated from the LES model using the dissipation rate (LES), shear length scale (ls), and as calculated in the mesoscale model.

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ary layer has started moving over the cooler water surface. It might be hoped that some improvement would result if a constant ␣ were used in the mesoscale model instead of the surface-flux dependent (4)–(6). However, additional experiments (not shown) indicate that use of a constant, intermediate value of ␣ gives reduced turbulence during strong heating and increased turbulence under neutral conditions, and is inadequate when both stable and unstable conditions occur simultaneously.

5. Conclusions We have used a mesoscale model (COAMPS) and an LES model to simulate basic aspects of the offshoreflow boundary layer structure observed by VMSC. The flow problem consisted of a planetary boundary layer passing over a land surface before encountering a cold ocean surface. Two sets of experiments were conducted, the first (denoted the land–sea case) having a land surface with negligible surface heat flux, and the second (denoted the barrier-island case) having surface temperatures 5°C warmer than the upstream atmosphere over a 4-km strip of land adjacent to the coast. In the land–sea case, comparison of the mesoscale and LES results showed differences between the model fields over the cold ocean, primarily in the turbulent quantities. The mesoscale model tended to produce a deeper boundary layer with turbulent kinetic energy having a maximum near the boundary layer top. In contrast, the LES model had a maximum in turbulence in the bottom half of the boundary layer and a more rapid decay in turbulence near the boundary layer top. These differences between the mesoscale and LES turbulence fields could be traced in part to the mixing length prescribed in the turbulence closure scheme used in the mesoscale model. Mixing length in the standard MY82 scheme is based on the vertical integral of turbulent kinetic energy and yields a relatively constant value as a function of height. For comparison, we diagnosed an estimate of the mixing length from the LES model by using the MY82 formula for dissipation rate and computing an equivalent mixing length from the simulated LES dissipation rate. In general, the LES length scale exhibited a peak near the middle of the boundary layer with a value similar to the standard MY82 scheme prediction. However, near the boundary layer top, the LES value decreased to about half the mesoscale model value. Longer length scales in the mesoscale model prediction near the top of the boundary layer appear to force enhanced turbulence and more rapid boundary layer growth in comparison with the LES results. The barrier-island simulations, which included a convective boundary layer over a portion of the land surface, produced a dramatically different boundary layer structure that was more in line with the aircraft obser-

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vations. We found that both models were able to duplicate the gross, qualitative features observed by the aircraft. However, details of the flow over the cold water, including the turbulent momentum flux, were reproduced less accurately by the mesoscale model. Although the LES model produced results closer to the observations, it overpredicted the strength of the turbulence and the turbulent momentum flux. The mesoscale model produced an overly abrupt change in the turbulent kinetic energy and momentum flux as the boundary layer air passed over the coastline. Collapse of turbulence over the cold water in the mesoscale model was forced by decreased vertical shear brought on by upstream convective mixing, and the reduction in the mixing length, which is dependent on the local surface heat and momentum fluxes. As in the nonconvective case, the mixing length predicted in the mesoscale model exhibited a gradual increase from the surface up to the boundary layer top. The LES mixing length, however, showed a distinct maximum in the decoupled boundary layer that was ⬃2 times larger than the mesoscale value. One conclusion from this study is that turbulent closure schemes that assume a surface-based boundary layer with an increasing mixing length with height, such as the standard COAMPS implementation of MY82, are not appropriate for these offshore flow conditions, and in general will not be well suited for conditions having variable stratification and partially decoupled boundary layers. When stratification controls the turbulent eddy scaling, mixing length estimates based on the strength of the local turbulence and stratification may provide a more accurate prediction of the boundary layer structure. However, in the case presented here the buoyancy length was much larger than the dissipation based length scale, probably because of the weak vertical temperature gradient. We found that a more realistic length scale for the land–sea case could be computed using the vertical shear of the mean wind in place of the stratification. This result is consistent with the TKE budget analysis, which showed that the buoyancy term was very small in comparison with the shear production. Unfortunately, a simple application of the shear based mixing length theory in the Mellor and Yamada (1982) Y82 scheme did not provide noticeable improvement. It is possible that more experimentation with local length scales such as the vertical shear scale presented here could lead to improvements to MY82. However, this would require a recalibration of many prescribed constants and ultimately might not be feasible because of the strong connection between local shear and TKE production. A more appropriate solution might be to consider a two-equation closure that predicts the mixing length, such as one of the higher-order models discussed in Mellor and Yamada (1982). Using a predictive equation for the mixing length would allow for the transport of turbulence length scale properties from the

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convective land region to the decoupled boundary layer. This method should also eliminate the need to tie the mixing length parameterization directly to the surface fluxes, which in the present scheme generates unrealistic, simultaneous changes in turbulence through the depth of the boundary layer when surface conditions change abruptly. In addition, no assumptions would be needed concerning the vertical profile of mixing length, allowing representation of multiple boundary layer structures that are not necessarily tied to surface forcing. Acknowledgments. This research was supported by the Office of Naval Research, Grant N00014-01-1-0231 and N00014-01-1-0138, Code 322 MM and OM. We are grateful to J. Doyle for his valuable assistance during the early stages of this study and Dean Vickers for providing figures from SHOWEX. REFERENCES Andre, J. C., G. De Moor, P. Lacarrere, G. Therry, and R. du Vachat, 1978: Modeling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci., 35, 1861–1883. Burk, S., T. Haack, and R. Samelson, 1999: Mesoscale simulation of supercritical, subcritical, and transcritical flow along coastal topography. J. Atmos. Sci., 56, 2780–2795. Deardorff, J. W., 1980: Stratocumulus-capped mixed layers derived from a three-dimensional model. Bound.-Layer Meteor., 18, 495–527. Dorman, C. E., 1985: Evidence of Kelvin waves in California’s marine layer and related eddy generation. Mon. Wea. Rev., 113, 827–839. Ducros, F., P. Comte, and M. Lesieur, 1996: Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech., 326, 1–36. Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young, 1996: Bulk parameterization of air-sea fluxes for tropical ocean-global atmospheric coupled-ocean atmosphere response experiment. J. Geophys. Res., 101, 3747– 3764. Friehe, C. A., and Coauthors, 1991: Air-sea fluxes and surface layer temperatures around a sea-surface temperature front. J. Geophys. Res., 96, 8593–8609. Galperin, B., L. H. Kantha, S. Hassid, and A. Rosati, 1988: A quasi-equilibrium turbulent energy model for geophysical flows. J. Atmos. Sci., 45, 55–62. Garratt, J. R., 1990: The internal boundary layer—A review. Bound.-Layer Meteor., 50, 171–203. Glendening, J. W., and C.-L. Lin, 2002: Large eddy simulation of internal boundary layers created by a change in surface roughness. J. Atmos. Sci., 59, 1697–1711. Hodur, R., 1997: The Naval Research Laboratory’s Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 135, 1414–1430. Hunt, J. C. R., D. D. Strech, and R. E. Britter, 1988: Length scales in stable stratified turbulent flows and their use in turbulence models. Stable Stratified Flow and Dense Gas Dispersion, J. S. Puttock, Ed., Institute of Mathematics and Its Applications Conference Series, Vol. 15, University Press, 285–322. Louis, J.-F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17, 187–202. Mahrt, L., and D. Vickers, 2003: Formulation of turbulence fluxes in the stable boundary layer. J. Atmos. Sci., 60, 2538–2548.

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