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  • View in gallery

    Surface plots of the model terrain for (a) Cape Mendocino and (b) Point Sur, also showing some geographical references. Note that the whole model domain is not displayed.

  • View in gallery

    Plots of the model domain for (a) Cape Mendocino and (b) Point Sur, showing the coastal terrain (gray shading), gridpoint distribution (dots), and the flight patterns (solid) on the days in question. Also, a few locations are indicated.

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    Model data and measurements for an aircraft transect across the coast in the vicinity of Shelter Cove (see Fig. 1), at 1600 LT 7 June 1996. The measured data are averaged according to height above the surface and to horizontal position in relation to the coastal jet and is represented with symbols, while all the model profiles along the transect are shown, within each horizontal region, as dotted lines. The regions are far offshore (*), in the outer portions of the jet (+), in the jet core (○), and in the lee of the cape (×). The profiles are displaced horizontally around the profile in the jet core for clarity (temperature by 5°C each, humidity by 5 g kg−1 each, and the wind speed components by 10 m s−1 each); the x-axis ticks are valid for the jet core profile.

  • View in gallery

    Vertical cross sections from west to east through Shelter Cove (see Figs. 1, 2) at 1600 LT 7 June 1996 of simulated (a) scalar wind speed (m s−1) and (b) potential temperature (°C). Also shown is (c) a close-up of the simulated wind (dotted line and grayscale) and temperature fields (dashed line) and (d) the corresponding measurements.

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    Same as Fig. 4, but for a vertical cross section upstream of Cape Mendocino. (See Fig. 2.)

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    Horizontal cross section of the wind field at 1600 LT 7 June 1996 at 100 m. The plots show scalar wind speed (grayscale) and wind vectors from the simulations for two different area sizes. Also shown in (b) as bold arrows are the measured winds averaged for this height from the entire flight.

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    Horizontal plots of the simulated (a) maximum and (b) mean MABL scalar wind speed (m s−1), (c) the height to the inversion base (m), and (d) the Froude number for 1500 LT 7 June 1996.

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    Vertical cross section from south to north [(a), (b)] through Cape Mendocino and [(c), (d)] just off the tip of the cape, aligned with the upstream jet center line, showing [(a), (c)] scalar wind speed (m s−1) and [(b), (d)] potential temperature (°C).

  • View in gallery

    Details of the simulated MABL from south to north through Cape Mendocino showing (a) scalar wind speed (m s−1) at several different heights and (b) a vertical cross section of the MABL thermal structure (grayscale), with a dotted line also indicating the position of the heights in (a).

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    Horizontal plot of the simulated curl of the turbulent stress vector (106 m s−2) at the surface for the area around CapeMendocino.

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    Comparison of the depression of the measured SST (dots) and a normalized modeled stress vector curl (solid lines) across the coast (a) into Shelter Cove, (b) at Cape Mendocino, and (c) upstream of the cape (see Fig. 10). The model data are from three transects around the aircraft track. Also, in (d) the observed SST is plotted against a simulated stress vector curl that was interpolated to the observed positions using GPS positions from the aircraft.

  • View in gallery

    Horizontal cross sections showing the difference in wind speed (grayscale) and temperature (dashed lines) around Cape Mendocino between two simulations: one with a realistic SST distribution and one where SST was assumed to be constant at the background value (the control simulation). The figure displays two levels: (a) 4 m and (b) 200 m. The terrain is indicated by a thin solid line.

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    Simulated momentum budgets for [(a)–(d)] the u component (across coast) and [(e)–(h)] the υ component (along coast) of the wind around Cape Mendocino at 275 m at 1500 LT 7 June 1996. The subplots show [(a), (e)] the pressure gradient forcing, [(b), (f)] the geostrophic imbalance, [(c), (g)] the geostrophic plus turbulent friction imbalance, and [(d), (h)] the geostrophic plus acceleration imbalance (10−3 m s−2).

  • View in gallery

    Same as Fig. 13, but for the height 7 m.

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    Comparison of aircraft measured (solid) and simulated (dashed) profiles of (a) wind speed (m s−1), (b) wind direction (°), (c) potential temperature (K), and (d) specific humidity (g kg−1) around Point Sur on 17 June 1996. The measured profiles are taken at the beginning and end of a flight, a few hours apart, and are located a few tens of kilometers apart. The model profiles are taken at (x, y) = (71.5, 68.9), (71.5, 37.3), and (53, 81.1) km, respectively (x and y count from the southwest corner of the model domain—see Fig. 2).

  • View in gallery

    Vertical (x, z) across-shore cross sections at y = 68.9 km, close to Point Sur. Shown are (a) the along-shore wind component (m s−1; toward the reader), (b) potential temperature (°C), (c) TKE (m2 s−2; note that this figure frame is only up to 1000 m) and (d) vertical velocity (cm s−1; more intensive inland vertical velocities are not shown). The white areas in the lower-right corners outline the terrain.

  • View in gallery

    Vertical (y, z) along-shore cross sections of [(a), (c)] the along-shore wind component (m s−1; blowing from right to left) and [(b), (d)] potential temperature (°C) at [(a), (b)] x = 55 km and [(c), (d)] x = 77.7 km.

  • View in gallery

    Horizontal (x, y) cross section of some MABL characteristics: (a) the mean wind speed (m s−1; blowing parallel downcoast), (b) the mean TKE (m2 s−2), (c) the MABL depth (m; with isolines every 50 m), and (d) the mean vertical velocity (cm s−1; with isolines every 2 cm s−1).

  • View in gallery

    The MABL Fr summarizing the flow properties around Point Sur. A hydraulic jump (from Fr >1 to Fr <1) occurs upcoast of Point Sur, while downcoast the flow is supercritical, with max(Fr) >2.5. The expansion fans are launched offshore extending about 60 km out through the MABL.

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Simulations of Supercritical Flow around Points and Capes in a Coastal Atmosphere

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  • 1 Department of Meteorology, Stockholm University, Stockholm, Sweden
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Abstract

Fully 3D nonlinear model simulations for supercritical flow along locations at the California coast, at Cape Mendocino, and Point Sur, are presented. The model results are objectively and subjectively verified against measurements from the Coastal Waves 1996 experiment with good results. They are then analyzed in terms of the flow structure, the impact of the local terrain, the atmospheric forcing on the ocean surface, and the momentum budgets. It is verified that the flow is supercritical (Fr > 1) within a Rossby radius of deformation from the coast and that it can be treated as a reduced-gravity, shallow water flow bounded by a sidewall—the coastal mountain barrier. As the supercritical flow impinges on irregularities in the coastline orientation, expansion fans and hydraulic jumps appear. The modeled Froude number summarizes well the current understanding of the dynamics of these events. In contrast to inviscid, irrotational hydraulic flow, the expansion fans appear as curved lines of equal PBL depth and “lens-shaped” maxima in wind speed residing at the PBL slope. This is a consequence of the realistic treatment of turbulent friction. Modeled mean PBL vertical winds in the hydraulic features range ±∼1–2 cm s−1, while larger vertical winds (±∼5–10 cm s−1) are due to the flow impinging directly on the mountain barrier. Local terrain features at points or capes perturb the local flow significantly from the idealized case by emitting buoyancy waves. The momentum budget along straight portions of the coast reveals a semigeostrophic balance modified by surface friction. While being geostrophic in the across-coast direction, the along-coast momentum shows a balance between the pressure gradient force and the turbulent friction. In the expansion fans, the flow is ageostrophic, and the imbalance is distributed between turbulent friction and ageostrophic acceleration according to the magnitude of the former. There is also a good correspondence between the magnitude of the local curl of the surface stress vector and the measured depression in sea surface temperature (SST) in areas where the latter is large and the along-coast flow is relatively weak, implying that a substantial portion of the upwelling is driven locally. Supplying the measured SST in the numerical simulations, with a considerable depression along the coast, had only marginal feedback effects on the character of the flow.

Corresponding author address: Michael Tjernström, Dept. of Meteorology, Stockholm University, Arrhenius Laboratory, S-106 91 Stockholm, Sweden.

Email: michaelt@misu.su.se

Abstract

Fully 3D nonlinear model simulations for supercritical flow along locations at the California coast, at Cape Mendocino, and Point Sur, are presented. The model results are objectively and subjectively verified against measurements from the Coastal Waves 1996 experiment with good results. They are then analyzed in terms of the flow structure, the impact of the local terrain, the atmospheric forcing on the ocean surface, and the momentum budgets. It is verified that the flow is supercritical (Fr > 1) within a Rossby radius of deformation from the coast and that it can be treated as a reduced-gravity, shallow water flow bounded by a sidewall—the coastal mountain barrier. As the supercritical flow impinges on irregularities in the coastline orientation, expansion fans and hydraulic jumps appear. The modeled Froude number summarizes well the current understanding of the dynamics of these events. In contrast to inviscid, irrotational hydraulic flow, the expansion fans appear as curved lines of equal PBL depth and “lens-shaped” maxima in wind speed residing at the PBL slope. This is a consequence of the realistic treatment of turbulent friction. Modeled mean PBL vertical winds in the hydraulic features range ±∼1–2 cm s−1, while larger vertical winds (±∼5–10 cm s−1) are due to the flow impinging directly on the mountain barrier. Local terrain features at points or capes perturb the local flow significantly from the idealized case by emitting buoyancy waves. The momentum budget along straight portions of the coast reveals a semigeostrophic balance modified by surface friction. While being geostrophic in the across-coast direction, the along-coast momentum shows a balance between the pressure gradient force and the turbulent friction. In the expansion fans, the flow is ageostrophic, and the imbalance is distributed between turbulent friction and ageostrophic acceleration according to the magnitude of the former. There is also a good correspondence between the magnitude of the local curl of the surface stress vector and the measured depression in sea surface temperature (SST) in areas where the latter is large and the along-coast flow is relatively weak, implying that a substantial portion of the upwelling is driven locally. Supplying the measured SST in the numerical simulations, with a considerable depression along the coast, had only marginal feedback effects on the character of the flow.

Corresponding author address: Michael Tjernström, Dept. of Meteorology, Stockholm University, Arrhenius Laboratory, S-106 91 Stockholm, Sweden.

Email: michaelt@misu.su.se

1. Introduction

The meteorology of the coastal zone is important for many practical reasons, including environmental protection, the fishing industry, and both air and sea transport and recreation. Upwelling of cool nutrient-rich water at the coast is one important factor that is attributed to the structure and persistence of coastal wind systems (Kelly 1985). A large fraction of the population on the earth—approximately 50% of the U.S. population (National Research Council 1992)—lives in coastal areas and is directly influenced by coastal mesoscale phenomena. Conversely, activity by the coastal populations affects the coastal ocean, often via atmospheric transport of anthropogenic pollutants. To address with these issues, a good understanding of the atmospheric physics in the coastal zone is necessary.

A coastline forms a step change in the surface characteristics and, as a consequence, a variety of mesoscale phenomena occurs. A particular class of atmospheric conditions applies along coasts where the coastline is a barrier to the the marine atmospheric boundary layer (MABL) flow. This requires that the coastal orography be sufficiently high to be considered “steep,” for example, as expressed by the Burger number, Bu = (hml−1m)(Nf−1), where N is the Brunt–Väisälä frequency;lm is the distance over which the terrain increases to its maximum, hm; and f is the Coriolis parameter. This may be the case if the terrain is high (Beardsley et al. 1987), but may also be true for lower terrain if the MABL is shallow or sufficiently stably stratified (Tjernström and Grisogono 1996; Grisogono and Tjernström 1996). The former conditions are observed along the west coast of the United States. For much of this coast, the height of the coastal mountains exceeds 400 m, forming an almost continuous barrier from Oregon to southern California. The MABL depth during the summer is typically 300–400 m (Neiburger et. al. 1961) but varies between 30 and 800 m (Beardsley et al. 1987). It is often well mixed and is capped by a sharp temperature inversion. This inversion slopes upward to around ∼2000 m near Hawaii, representing the effect of the subsidence from the North Pacific subtropical high (Neiburger et al. 1961), but the strongest slope is downward toward the coast within a Rossby radius of deformation lR = Nhmf−1, O(10–100 km) due to the interplay with a thermally driven cross-coast flow (Beardsley et al. 1987). The sloping inversion introduces a mesoscale thermal wind with increasing strength approaching the coast. Although the background wind, between the North Pacific high and the thermal low over the southwestern United States, is only 5–10 m s−1 from the northwest, the wind speed in the coastal low-level jet often exceeds 20 m s−1 (Zemba and Friehe 1987; Burk and Thompson 1996;Cui et al. 1998). Since the terrain barrier impedes geostrophic adjustment, Overland (1984) argued that the flow under such conditions is semigeostrophic; geostrophic in the cross-shore component, while in the along-shore component the pressure gradient is balanced by either the ageostrophic acceleration, as for “gap winds,” or by surface friction. Samelson and Lentz (1994) confirmed these ideas in an analysis of buoy observations in a limited region around Point Arena (see Fig. 1). They found that the along-shore pressure gradient is balanced by surface friction but that cross-coast advection is sometimes important. Using a mesoscale model Cui et al. (1998) extended this analysis to the entire central coast of California. They found that within the near-coast zone (10–20 km), much of the variability along the coast in the along-coast wind component is attributable to a balance between the pressure gradient and the ageostrophic advection due to the variability in the coastline geometry, while the balance with the frictional drag prevails in a broader zone, O(100 km), farther offshore.

The strength of the inversion capping the MABL is typically Δϒ = 10–20 K in summer (Dorman and Winant 1995), and the inversion can be persistent for long periods, maintained from above by subsidence and from below by turbulent mixing. The density of the cooler MABL air is thus typically 5% higher than the air above the inversion, and the flow can often be treated as a single-layer, reduced-gravity flow with a free upper surface. An overview of such flows past a varying sidewall can be found in Baines (1995), for example. Buoyancy waves, with a phase speed c = [g(Δϒ/ϒ0)h]1/2, can form on the inversion, where g is the gravity, ϒ0 is the potential temperature of the lower layer, h is the layer depth, and g(Δϒ/ϒ0) is the “reduced gravity.” One role of these waves is to redistribute mass toward geostrophic balance. For a change in the coastline orientation at a moderate flow speed, U < c (or Fr = U/c < 1, where U is the wind speed and Fr is the Froude number), the flow passes the corner in a smooth fashion as buoyancy waves adjust the flow upstream. For a flow speed higher than the phase speed, Fr > 1, the flow is supercritical. Changes in the coastline orientation then have abrupt effects on the flow; all the adjustment must take place downstream since the gravity waves cannot propagate against the flow. If the sidewall turns away from the flow, an expansion fan forms: the layer depth decreases and the flow speed increases. If the sidewall turns into the flow, the partial blocking reduces the flow speed and a hydraulic jump may occur along a line where Fr ∼ 1. This was first observed in the coastal MABL by Winant et al. (1988) between Point Arena and Bodega Bay along the northern California coast, in aircraft measurements during the Coastal Ocean Dynamics Experiment (CODE). This case was analyzed by Winant et al. (1988) using single-layer, reduced-gravity theory. Samelson (1992) found that a model including rotation and surface friction improved the results. It explained the wind speed maximum at the coast with the layer-depth minimum a little downwind. As a consequence the expansion-fan critical lines, wind speeds, pressure, and layer depths form lens-shaped patterns in the lee rather than straight lines radiating from the bend in a fan-shaped pattern, as predicted by the irrotational, frictionless theory.

Even with the theoretical improvements, only the limited area around Point Arena had been adequately sampled to provide experimental evidence of supercritical coastal flows. It was thus uncertain how representative these conditions are for the west coast of the United States, or any other similar coast, although an analysis by Dorman and Winant (1995) indicates that conditions for supercritical flows could be frequent during summer. To provide data for more detailed studies, flights with an instrumented long-range research aircraft were performed out of Monterey, California, along the U.S. west coast in the summer of 1996 as part of the Coastal Waves 1996 program (Rogers et al. 1998). The flights spanned the U.S. coast from Point Conception in the south to Cape Blanco in the north. This paper presents a detailed analysis of model simulations based on two cases when supercritical flow was observed during this experiment;in the vicinity of Cape Mendocino (40.4°, −124.4°) on 7 June and at Point Sur (36.3°, −122°) on 17 June 1996 (see Fig. 1). The simulations were set up to reproduce the observations to a reasonable degree using as simple as possible initial conditions and background (synoptic scale) forcing. The resulting high-resolution, 3D model fields then lend themselves to a detailed analysis. It is assumed that modeled parameters or variables that cannot be directly verified by observations are also realistic, as a consequence of using a dynamically consistent numerical model.

2. The field experiment

The Coastal Waves 1996 program is described in detail in Rogers et al. (1998). Extensive details from aircraft measurements around Point Sur are also presented in Dorman et al. (1999a). Conditions at Cape Mendocino are also described in Dorman and Rogers (1998). Also see Burk and Haack (1999, manuscript submitted to Mon. Wea. Rev.) for a study of special conditions around Point Sur. The program consisted of 1) a longer effort from May to September 1996 to obtain reliable average conditions and to capture the rare events of so-called southerly surges (Bond et al. 1996; Dorman 1997) and 2) a concentrated effort during June. Automated surface weather stations, drifting and fixed buoys (including the National Data Buoy Center buoy system), wind profilers, acoustic sounders, and radiosoundings were available along the coast from southern California to Oregon during the longer effort. During this period the University of North Carolina Piper Seneca III research aircraft (Bane et al. 1995) was also available.

During the intensive operations period, a primary measurement platform was the National Center for Atmospheric Research (NCAR)1 C-130 Hercules research aircraft. From 2 June to 1 July, the C-130 flew a total of 11 missions; 4 around Point Sur, 3 around Cape Mendocino, 3 around Point Conception, and 1 around Cape Blanco. The standard suite of data from this platform include winds, temperature, and humidity. Cloud liquid water, droplet, and aerosol spectra were also collected, as were atmospheric radiation and radiometric surface temperature. Most data are available at a low rate (1 Hz), while wind speed, temperature, and humidity are also sampled at a higher rate (25 Hz) for calculation of turbulence moments by eddy correlation technique. In addition, on this experiment, remote sensing of the MABL structure was provided by the Scanning Aerosol Backscatter Lidar, developed by the Atmospheric Technology Division of NCAR. Flight-level data from this platform forms the backbone for the simulations presented in this paper.

3. The numerical experiment

The numerical model is used as a numerical laboratory rather than as a forecast tool, and the aim is to study the flow in detail. The assumption is that when the model reproduces the observed conditions with sufficient accuracy, properties that were not, or could not, be measured will also be realistic due the the dynamic consistency in the model. Some of the simplified modeling techniques applied here are motivated by this aim. In this context, experimental and model data are treated in symbiosis. That the surface forcing of temperature and humidity and the background flow are prescribed and that the initialization technique is simplified must be viewed in this context. The focus of this paper is thus on the flow and not on the model.

The Department of Meteorology, Uppsula University meso-γ-scale model used in this study is a 3D hydrostatic model with a higher-order turbulence closure. The turbulence closure is an improved, consistent version of the “Level 2.5” closure (Mellor and Yamada 1982). It carries an improved description for the pressure redistribution terms (the “near-wall” correction) and an algorithm to keep second-order moments realizable (Andrén 1990). The model also includes routines for subgrid-scale condensation and radiation, as well as surface energy balance, but the latter is not used here for the sake of simplicity. Detailed descriptions are found in Tjernström (1987a,b), while shorter descriptions are found in Tjernström (1988a) and Enger (1990a).

The model has previously been applied in a variety of applications including terrain-induced flows (Tjernström 1987a, 1988a, 1989; Enger and Tjernström 1991;Enger 1990a; Enger et al. 1993; Koracin and Enger 1994; Grisogono 1995; Enger and Grisogono 1998), coastal flows (Tjernström and Grisogono 1996; Grisogono and Tjernström 1996; Grisogono et al. 1998; Cui et al. 1998), dispersion calculations (Enger 1983, 1986, 1990b), marine stratocumulus (Tjernström 1988b; Tjernström and Koracin 1995), and air chemistry (Svensson 1996a,b; Svensson 1998; Svensson and Klemm 1998). It has thus been thoroughly examined for a variety of flows and is well documented.

The vertical coordinate is transformed into a terrain-influenced coordinate system (Pielke 1984). The terrain was extracted from an ∼500-m resolution terrain database and was averaged to the model grid. For the Point Sur simulation, the grid was rotated to align the y axis of the model with the coast. The horizontal grid expands toward the lateral boundaries to achieve maximum resolution in the central parts of the domain while removing the boundaries from the area of interest. A simple radiative boundary condition is applied at the lateral boundaries, while a sponge layer is introduced at the model top (Grisogono 1995). This is a simpler and more cost-effective alternative to nesting several simulations with progressively decreasing domain sizes but will only work for steady synoptic conditions and for a flow that is dominated by local forcing. The vertical grid also expands, log-linearly, toward the model top. The maximum resolution is located close to the surface at the domain center, for Cape Mendocino chosen to be ∼20 km south of the tip of the cape and for Point Sur located ∼20 km southwest of the point. The domain sizes and grids, which are different for the two simulations, are given in Table 1.

Figure 1 shows the model terrain for the two locations, along with some geographical references, while Fig. 2 also shows the distribution of the model grid points and the flight track for the two experiments. With this horizontal resolution, the smallest resolvable horizontal scale is ∼4 km; the hydrostatic approximation will thus be invalid only for circulations with a depth similar to the model domain (cf., e.g., Pielke 1984). An additional restriction in the terrain-influenced coordinate system requires that the terrain slopes ≪45° (Pielke 1984). While the real terrain is certainly steeper locally, the resolved-scale terrain comes close to this limit only at a few locations. Neither model domain incorporates the Sierra Nevadas. This was found to be critical for some conditions in Cui et al. (1998); however, here the synoptic-scale flow is coast parallel or onshore at all levels, and this omission should not be critical.

The sea surface temperature (SST) was set constant in time: the offshore SST was 12°C. For the Cape Mendocino simulation, it was first set constant horizontally, but one simulation was performed using the observed SST, dropping by at most 6°C into Shelter Cove. For the Point Sur simulation, SST is lowest at the coast (10°C) and gradually increases offshore. The diurnal temperature variation of the inland soil surface was prescribed using a sinusoidal-type function with an amplitude of ±8°C and an average value of 18°C at sea level, decreasing with terrain height by 6 × 10−3 °C m−1. Similarly, specific humidity was specified at the surface using potential evaporation over the sea and a fraction (15%) of this value over land. These conditions were transferred onto the lower model boundary using matching surface-layer and roughness sublayer similarity theories. This allows coastal land surfaces to be influenced by local advection of moist and cold air from the sea, while also allowing for a strong experimental control over the surface forcing. To simplify, the roughness length over land was taken to be a function of terrain height. In the San Joaquin Valley, a small value, representative of agricultural areas, was used while the coastal mountains had higher values (reaching 1 m), representing the generally rougher terrain. The roughness of the sea surface was set constant corresponding to a neutral drag coefficient of CDN ∼ 1.5 × 10−3. Tests with different sets of values for the inland surface temperature forcing verify that the offshore coastal flow is relatively insensitive to exact details.

The pressure gradient terms in the equation of motion are decomposed into a resolved mesoscale part and a part representing the background (synoptic scale) flow (Pielke 1984). To simplify, the synoptic-scale flow was specified as a geostrophic wind. This was estimated from synoptic charts, aided by flight-level wind data aloft for the days in question, and was kept constant in time. Initial potential temperature and specific humidity profiles were taken as well mixed (constant values) in the MABL, capped with a strong inversion and stable stratification and constant (low) humidity aloft; values were estimated from flight-level data from the actual events.

The simulations were initialized using a dynamic initialization; the model is given horizontally homogeneous temperature, humidity, and wind fields and runs through a preintegration period when the model fields adjust gradually to a realistic quasi-balance. In particular, the turbulent kinetic energy (TKE) spins up during this time. Cui et al. (1998) found that buoyancy and buoyancy–inertia waves, generated by the initially unbalanced fields, propagated out of an even larger model domain than here within less than ∼12 h. The simulations presented here were initialized at 1800 LT the day preceding the events, and no data for the first 15 h are used.

4. Results

The lower-level synoptic-scale flow on both 7 and 17 June was similar, with a strong prevailing low-level geostrophic wind from almost due north being only slightly stronger on 17 June. This pattern was consistent through the lower troposphere and is evident both at the surface and in the 850-hPa analysis. The pressure patterns aloft were somewhat different on these days. On 7 June, the upper-level flow was quite weak, with an approaching trough far offshore. The wind speed decreased with height, while the direction veered to the southwest. On 17 June, there was also an eastward-moving trough aloft but closer to the coast, associated with a low pressure system over the northern U.S. west coast. As a consequence, the background wind speed does not decrease significantly with height and the wind direction was fairly constant with height throughout the lower half of the troposphere.

a. Model evaluation

Before analyzing model data, as a substitute for experimental data, the validity of the simulation must be established. This can be done in many different ways. First, it must be realized that there can be many types of model errors. Serious errors may arise if the model physics is unable to handle a particular flow type, for example, a hydrostatic simulation of a nonhydrostatic phenomenon. Less serious deviations may be due to inappropriate initial or boundary conditions. Then the model results can still be physically valid, even if the conformity to a particular set of measurements is not as good. The variability of the real atmosphere also has to be considered. The structure of the simulated flow will be compared to observations in later sections. Here a direct evaluation will be presented. Only the simulation for 7 June 1996 will be used since most of this flight appears within the model domain (see Fig. 2).

A research flight of several hours provides a wealth of data to compare with the model. However, each sample from the aircraft is collected at a different time and place; this paradox has to be resolved. In Fig. 3, an ensemble of data collected in a cross section west of Shelter Cove (Fig. 1) is shown; the C-130 flew a series of patterns here (Fig. 2). All the data from the aircraft collected in a narrow north–south interval was binned into four longitude intervals, and the data in each bin were averaged as a function of height. These data are compared to all the vertical profiles from the model appearing within each longitude interval, from one model output centered in time with respect to the measurements. The results for each interval are displaced around the data from within the coastal jet so that profiles from left to right in the figure are from west to east. Figure 3a shows the well-mixed offshore MABL and the slope of the inversion toward the coast in the two leftmost profiles, the steepening of this slope within the jet (third profile), and the collapse of the MABL in the near-shore region. It is obvious that the model is capable of describing this feature realistically. It is worth noting that the inversion layer appears to be as deep as the MABL and not the infinite step change required by shallow-water theory. The agreement in specific humidity is not as good (Fig. 3b). Only the near-shore profile is quite satisfactory. Surprisingly, the differences from east to west in the structure of the humidity profile (e.g., the vertical gradient, depth of moist layer, etc.) is less evident than in temperature, both in the simulation and in the measurements. The simulated humidity inversion is not nearly as sharp as in the measurements, and in the two offshore profiles the simulated MABL is too dry. The agreement in the wind speed components (Fig. 3c,d) is quite satisfactory. Only the simulated jet well offshore seems to be somewhat too shallow. The magnitude and the direction of the wind, however, appear quite realistic.

Using the time, Globel Postioning System (GPS) horizontal position and radar altitude from the aircraft, a “flight in the model” was performed (Svensson and Klemm 1998). One value of each model variable was interpolated, in space and time, for aircraft data samples at 0.1 Hz (note that the aircraft data was reduced to 0.1 Hz without filtering and includes the “true variability,” also part of the turbulent fluctuations that are not resolved explicitly in the ensemble averaged model variables). This is quite a difficult test—the model has to be correct both in space and time. Any uncertainty in the location of the aircraft will generate an error, as will small errors in simulated MABL depth. The latter will be particularly sensitive at the inversion. A number of statistical objective measures that objectively determine model performance (cf., e.g., Svensson 1998) are presented in Table 2 for this simulation. The correlation coefficients are generally quite high and the mean bias is acceptable. There seems to be quite a good agreement in general, although the problem with the too moist inversion layer is evident. The temperature is surprisingly good, although some “inversion-height effects” can be seen. The main wind component is somewhat low in the model, while the cross-coast wind speed shows no particular bias. The standard deviation of the observed and simulated variables should be of the same order of magnitude (σoσp), while the standard deviation of the error should be less than that of the observations (σerr < σo) for acceptable performance (Pielke 1984), which is the case here. The index of agreement (IOA) measures the agreement in the structure of the data in spite of biases. It should be close to unity, which is certainly fulfilled here. Undoubtedly, the model performance could be improved; however, it appears quite adequate for the purposes of this paper.

b. The Cape Mendocino simulation

Figures 4–8 illustrate the main flow features modeled for around midafternoon on 7 June 1996. Figures 4–6 also contain validating data from the C-130 aircraft measurements. Figure 4 shows west-to-east cross sections of scalar wind speed and potential temperature through Shelter Cove, downstream from Cape Mendocino, from the model and from the aircraft data at 1600 LT. It reveals clearly the sloping inversion in model, first from ∼700 km to ∼400 m over the first 150–180 km and then more steeply all the way to the surface as the MABL collapses over the 40–60 km closest to the coast (Figs. 4b,c). It is worth noting that all of the final collapse occurs in the lee of the cape, as the tip of the cape is located around x ∼ 0 km farther upstream. Associated to the sloping inversion is a wind speed maximum (Figs. 4a,c) with an absolute maximum aligned with the final, steeper collapse of the MABL; the simulated wind speed reaches ∼27 m s−1. Figure 5c shows an enlargement of the fields in Figs. 4a,b. This can be compared directly to the observed ensemble average (Fig. 4d), which is a composite of all the data collected in the flight pattern flown along this transect (I. Brooks 1997, personal communication) collected in ∼1 h, from around 1520 to 1610 LT. The agreement with the measurements is quite remarkable. Several features are seen: the steeply slanting inversion and the jet core aligned with the maximum slope; wind maximum greater than 25 m s−1 at ∼150–400 m in the simulation and at ∼100–450 m in the measurements; and the collapse of the MABL in the lee of the cape, with drastically reduced wind speeds, below 5 m s−1 in both simulation and measurements. This wind speed minimum is a significant feature in the climatology for this area (cf., e.g., Dorman et al. 1999b, manuscript submitted to Mon. Wea. Rev.). Figure 5 similarly shows the flow at a cross section immediately upstream of Cape Mendocino (see Fig. 2). Here the flow is slower, the MABL is deeper and more horizontally homogeneous, and the transition in the near-coast region is more square-shaped. The agreement is worse here than downstream of the cape; the simulated upstream MABL is somewhat too shallow and too well mixed. However, the simulated and observed structures are similar, with a flat coastal jet with wind speeds in the simulation reaching ∼18 m s−1 at ∼400 m and 16 m s−1 at ∼600 m in the observations, within a more or less horizontal inversion.

Figure 6 shows horizontal plane views of the wind speed at z ∼ 100 m from the model from 1500 LT. The flow decelerates along the coast directly upstream of the cape (Fig. 6a), as the terrain at the cape partially blocks the flow. At the tip of the cape, the flow accelerates and downstream, within the expansion fan, there is a lens-shaped wind speed maximum. In Fig. 6b the corresponding aircraft data is included for a smaller area as bold arrows. All the aircraft data from the flight (∼4 h) at the height interval of the model grid is averaged horizontally in 5 km × 5 km boxes. In particular, the agreement is very good south of Cape Mendocino. Upstream of the cape, the observed flow appears to be more controlled by the local terrain than in the model, and the simulated wind speeds are also higher than those measured.

1) Flow characteristics

Some flow properties of the simulated MABL are summarized in Fig. 7. The maximum wind speed (Fig. 7a) is the highest simulated wind speed in each model vertical. The mean MABL wind (Fig. 7b) is the mean wind below the inversion base. In both, the wind speed upstream of the cape increases by a few meters per second from offshore toward the coast, although there is a slight blocking and a retardation within 30–40 km directly upstream of the local terrain at Cape Mendocino. At the tip of the cape, an area with higher winds emanates and spreads downstream in a lens-shaped pattern with maximum winds reaching ∼28 m s−1. The prescribed background flow here is ∼15 m s−1, so the mesoscale acceleration is about a factor of 2. The height to the base of the inversion h is analyzed in Fig. 7c, where h is the height where the potential temperature is 1.5°C higher than the value at the lowest grid point. Upstream of the cape, it slopes by from ∼500 m to less than 400 m over ∼200 km toward the coast, somewhat more steeply over the last ∼30 km. At the cape and southward, a fan forms with a steepening MABL slope into the Shelter Cove area, where the MABL essentially collapses. Using the mean wind speed, the depth of the layer and the strength of the inversion (roughly 18°–20°C throughout the area), a Froude number (Fr) is calculated in Fig. 7d. Upstream of the cape, the flow is supercritical within ∼150 km, which corresponds well to the Rossby radius lR ∼ 150–200 km, using a coastal terrain height of 600–800 m and the stability in the inversion.

Downstream of the cape, the coast turns away from the flow roughly at an angle of β = 34°. From inviscid nonrotational flow theory (cf., e.g., Baines 1995), an expansion fan should form at the cape. For an upstream Fr0 = 1.2, the Mach angle α0 = ∼−55°. This is in rough agreement with the present results, but only very close to the cape. There are also other deviations from this theory: if the flow is assumed to be parallel to the local coast well up- and downstream of the cape, it has to undergo a change in direction Δγ after passing the cape. The theoretical Froude number should then approach Fr ∼ 4 as Δγβ, while the wind should increase monotonically downstream and toward the coast. Froude values significantly larger than this are simulated close to the coast, downstream of Cape Mendocino and Point Arena (the white areas in Fig. 7d), while the wind speed decreases inside of the jet and toward the coast. This suggests that the simulated MABL is much more shallow than that predicted by the simple hydraulic theory. Furthermore, the simulated expansion fan is curved and the wind speed maximum occurs where the MABL depth has contracted to ∼200 m and then decreases again, while the layer thickness continues to decrease all the way toward the coast. The wind speed also decreases downstream, and the acceleration within the expansion fan is only a fraction of that predicted by momentum and mass conservation considerations, due only to the reduction of the layer thickness. Turbulent friction and rotational effects are partial explanations for these deviations in wind speed and PBL depth. Samelson (1992) predicted this shape of the wind maximum when including friction in his model, while rotation had a smaller effect. However, including friction also increased the MABL depth in the expansion fan in his simpler model; this is still another indication that the MABL close to the coast is more shallow here than predicted by the theory.

One reason for these deviations is the fact that a real coast is seldom a homogeneous sidewall required by this theory. For example, the terrain at Cape Mendocino appears to play an important role for the wind regime in the lee, the wind minimum at Shelter Cove. There are two ridges at the cape, oriented roughly perpendicular to the flow and ∼400–600 m high (see Fig. 1a). Still within the framework of the simple single-layer reduced-gravity flow, a flow with an upstream FrO ∼ 1.2 and a nondimensional depth H = h/hm ∼ 1 over an infinite ridge would result in a partial blocking with no hydraulic jump in the lee (e.g., Baines 1995). An upstream hydraulic jump may appear, which could be either stationary or propagating upstream. In either case, it would be difficult to resolve such a feature in the present model. South–north cross sections of wind speed and temperature, through the terrain at the cape and just off the tip of the cape, respectively, are shown in Fig. 8. The single-wave structure associated to the cross-terrain flow is quite evident. The flow upstream is about 500 m deep with a coastal jet at 22 m s−1, but while approaching the cape, the flow speed is reduced to less than half (Fig. 8a). In the immediate lee, the MABL depth is virtually eliminated but recovers to ∼200 m farther downstream (Fig. 8b). The 13°C isotherm is lowered about 200 m, comparing upstream and downstream conditions, and the temperature at 400 m increases ∼7°C downstream. This heating must be because of the subsidence in the wave, which is even clearer somewhat farther west in Figs. 8c,d.

The partial blocking of the upstream MABL is evident in Fig. 9, showing this area in more detail. The two lower model levels in Fig. 9a (15 and 173 m; note that model levels are roughly parallel to the terrain) remain within the MABL air. A significant retardation of the wind speed is evident within 50 km upstream of the first crest. The highest model level plotted here (744 m) remains above the MABL and is hardly affected. However, the width of these ridges is not sufficient to be treated as infinite and the retardation is thus most marked close to the coast. It also becomes more marked when the flow passes in over the rougher land area, upstream of the actual barrier, due to increased turbulence due to surface friction. However, the upstream distance being affected by the ridges remains independent of the height.

2) Atmospheric forcing on the ocean surface

It is well established that a northerly flow along a west coast on the northern hemisphere forces ocean surface water offshore due to Ekman transport. The well-known consequence, upwelling, is important in causing the relatively cold sea surface along the U.S. west coast (Kelly 1985). This background upwelling is a 2D (in the vertical) circulation in the ocean, but additional upwelling is associated with a positive curl of the surface wind stress vector ξτ = k ·  × τ (cf., e.g., Gill 1982). Here, τ is the wind stress vector, easily obtained from the model closure (Cui et al. 1998). Figure 10 shows the model-derived ξτ around and in the lee of Cape Mendocino at 1500 LT. There is an area of enhanced values upstream of the cape (∼0.5 × 10−5 m s−2) and in a band along the cape, but the main maximum covers the lee region, with maximum values exceeding ∼3 × 10−5 m s−2. It is easy to realize that a region of enhanced stress curl must exist on the “left” side of a wind speed (stress) maximum, here on the coastal side of the jetthat becomes detached from the local coast when the coast turns away from the flow. Comparing this to the pattern of SST is not straightforward. First, the background upwelling may mask any contribution by a locally positive stress curl. Second, the stress curl can only be considered a source function for additional upwelling. The subsequent transport of the colder water is much more complex and requires a more complex model.

Still, in Fig. 11, simulated ξτ are compared with patterns of the SST reduction (ΔSST) from an offshore value of SST ∼ 12°C, as measured by the downward-looking radiometers on the C-130 aircraft. These measurements were taken at three east–west transects (see Fig. 10). The values of ξτ were normalized as ξ*τ = ΔSSTmξτξ−1τm, where ξτm is the maximum stress curl and ΔSSTm is the maximum SST depression in each transect, respectively. The largest depressions (ΔSSTm) are −5°, −5.5°, and −3°C, while ξτm are 1.7, 4.2, and 0.15 × 10−5 from south to north, respectively. The agreement in the shape of these profiles in the transect through Shelter Cove (Fig. 11a) is surprising and indicates the relative importance of ξτ for the SST pattern here. This may be expected since as ξτ is large, while the along-shore wind at the coast is weaker here. At the cape (Fig. 11b), the observed SST depression appears to have two modes and the latter corresponds in location to the maximum stress curl. Upstream of the cape (Fig. 11c), the observed SST depression appears over a broad horizontal zone with little or no correspondence to ξτ. Here, ξτ is smaller, and the effect of the background upwelling, along the almost straight coastline upstream of the cape, is expected to be relatively larger. Admittedly, there is a great deal of ambiguity in this simple comparison; however, Fig. 12d shows the measured SST from almost the entire flight (flight legs above the MABL and over land are excluded) plotted against interpolated values of ξτ from the model. Indeed, the SST is low also at locations with small ξτ; however, at all locations where ξτ is large the SST is substantially reduced. This may be interpreted as follows. Along the entire coastline, upwelling occurs due to the forcing of the down-coast flow, and the SSTs are thus lower at the coast. In areas where the local curl of the stress vector is large, this imposes an additional upwelling and the largest SST depressions are found where ξτ is large. Indeed, the lowest SSTs tend to be found in the lee of capes or points.

In view of these results, two very valid questions may be raised: why were these simulations carried out with a constant SST, and does the actual SST depression feed back into the atmospheric flow in any significant way? To investigate this, one additional simulation was carried out using the observed SST along the coast. The differences in wind speed and temperature across the transect in Fig. 4 are small (not shown); the differences in wind speed are smaller than ±1.5 m s−1. The largest differences are in the core of the jet, thus the relative difference is small. Temperature differences are confined to altitudes below 100 m. Figure 12 shows differences in wind speed and temperature in plane view for two heights. At 4 m there is a significant difference, with a reduction in the wind speed by ∼4 m s−1 at the upstream edge of the cape and by ∼1–2 m s−1 along the SST minimum. The reduction upstream of the cape is probably caused by increased blocking due to the increased low-level stability. The temperature difference is greater than 3°C in the lee of the cape, where the SST depression is the largest (∼ΔSST −6°C). At 275-m height the wind speed increases by ∼2 m s−1 in a band aligned with the jet, while the temperature deficit is marginal, less than 0.5°C. Recalling Figs. 4–6, this actually improves somewhat details in the simulations. The feedback to the atmosphere, however, seems to be marginal in this case, and the reason is probably that this flow is so strongly determined by its hydraulics that even rather large changes in the lower boundary conditions do not alter the main character of the flow very much.

3) Momentum budgets

One property that may be useful to analyze from model output, rather than from experimental data, is the momentum budgets. Budgets are quite difficult to estimate from experimental data but can reveal significant features in the flow dynamics. The budget equations for the east–west and north–south components are written in a compact form as dUa + Px + Cu + Fu = 0 and dVa + Py + Cυ + Fυ = 0, respectively. The first term in each equation is the ageostrophic acceleration (the Lagrangian time derivative), the second term is the pressure gradient force (synoptic scale plus mesoscale), the third term is the Coriolis term, and the last term is the vertical momentum–flux divergence due to turbulence. All the terms were calculated using the same algorithms as in the actual model simulations, averaging over 10 time steps (2.5 min).

Evaluating these budgets rapidly becomes quite complex. Figures 13 and 14 show the budgets over the ocean for both velocity components at two model heights, 275 and 7 m, at 1500 LT. The subplots in Figs. 13a–d and 14a–d show the across-shore momentum budget (associated with the u component of the wind) and Figs. 13e–h and 14e–h show the along-shore momentum budget (associated with the υ component of the wind). Panels (a) and (e) in both figures show the total pressure-gradient forcing, while the subplots (b) and (f) similarly show the deviation from the steady-state geostrophic balance. The remaining subplots show other imbalances; (c) and (g) show the imbalance when only turbulence is added to the geostrophic balance, while (d) and (h) show the imbalance when only the ageostrophic acceleration is added to the geostrophic balance—the inviscid equation.

The mesoscale perturbation in the pressure is distributed differently with respect to the cape in the two momentum equations. In the u component (Figs. 13a and 14a), there is a dipole pattern around the cape, while in the υ component (Figs. 13e and 14e) the maximum is located almost due west of the cape, with minima upstream and downstream. The magnitude of the maximum pressure forcing in the vicinity of the cape is a factor ∼5 larger than upstream. The patterns of the deviation from geostrophic balance are similar in shape to the pressure-gradient forcing patterns. This is a direct manifestation of the flow supercriticality; hydraulic effects govern the scale of the flow, generating a large Rossby number, and the flow accelerates much faster than the geostrophic adjustment process can keep up—the flow becomes ageostrophic.

The u component at 275 m is an approximate geostrophic balance (Fig. 13b) north of the cape, improving with offshore distance, except in the blocking area within ∼50 km north of the cape, while at 7 m friction is required for balance north of the cape (Fig. 14c). South of the cape, the flow is ageostrophic at both levels with a maximum deviation directly south of the cape. The magnitudes of the of the imbalance reach ∼50% and ∼80% of the pressure gradient, respectively (Figs. 13b and 14b). Adding the turbulent friction at the higher height (Fig. 13c) changes little, while at the lower height it improves the balance (Fig. 14c); the imbalance is reduced to 10%–30% of the pressure forcing. Adding only the ageostrophic acceleration removes essentially all the imbalance at the higher height (Fig. 13d), but at 7 m the ageostrophic acceleration is less important (Figs. 15c,d). Friction is a factor of ∼2 larger than the acceleration here.

The υ component is ageostrophic practically everywhere (Figs. 13f and 14f), and the magnitude of the ageostrophic terms is of the same size as the pressure gradient term (cf., e.g., Figs. 13e,f and 14e,f). At 275 m upstream of the cape, the Coriolis forcing is nearly zero (Figs. 13e,f), and friction and ageostrophic acceleration are both of about the same magnitude (Figs. 13g,h). Closer to the surface, friction roughly balances the pressure forcing upstream and offshore downstream of the cape (Fig. 14g). In the expansion fan at the higher height, the ageostrophic acceleration dominates over friction closer to the cape, while friction becomes equally important farther south. Into the lee of the cape, the acceleration dominates. At the lower height (Figs. 14g,h), friction dominates over acceleration to the west, and vice versa in the Shelter Cove region farther east.

The conditions upstream of the cape are similar to those described by Samelson and Lentz (1994) and by Cui et al. (1998), while the conditions in the vicinity of the cape and in the expansion fan are more complex. Samelson and Lentz (1994) note that the u component is geostrophic, while in the υ component the pressure gradient is often balanced by friction but sometimes by acceleration. Their estimates were made closer to the coast and in a very limited area. Cui et al. (1998) found a similar balance as they attributed most of the variability close to the coast to the ageostrophic acceleration. It is unclear if expansion fans were a factor in these two studies. Here, neither component is in geostrophic balance in the expansion fan and the friction term seems to “govern” the acceleration; that is, friction always reduces the imbalance, and the relative importance of the ageostrophic acceleration terms is smaller when the friction is relatively larger.

c. The Point Sur simulation

The case presented here is based on data collected on 17 June 1996 and was also described in Rogers et al. (1998) and Dorman et al. (1999a), who present a detailed analysis of the MABL dynamics around Point Sur. They reveal a hydraulic jump downstream of Monterey and significant accelerations in the lee of Point Sur. The MABL thickness is between 400 m offshore and about 50 m in the lee of Point Sur. From the observations, most of the MABL is supercritical, with Fr ∼ 1–2 (extreme values 0.7–2.8). They finally conclude that the overall flow structure was compatible with a hydraulic supercritical expansion fan [also indicated in Rogers et al. (1998)] in the lee of Point Sur. Most of their findings are confirmed here, while some new details will be added. First, the model results are here compared to some observations, then the across- and along-shore MABL structure, and finally the horizontal MABL properties are analyzed.

Only the simulation with a realistic SST gradient, with the lowest SST at the coast, will be presented here. The MABL structure for this case varies very slowly over time, and all model results will be shown for a single time in the late morning. The model performance is illustrated in Fig. 15, showing observed and modeled profiles around Point Sur. The intensity and the position of the downcoast jet (Figs. 15a,b) and the inversion (Figs. 15c,d) are well captured by the model; the heights are underestimated by ∼50 m. The model produces somewhat less momentum mixing in the MABL, and there is a small bias in the direction of about 15°. The inversion strength is overestimated by 1.5°C. The flow aloft is simulated reasonably well, even though the model is somewhat moister. In general, the model performance, compared to the observation around Point Sur, concurs with that around Cape Mendocino; a further direct model validation for this case will not be presented here.

Figure 16 shows an across-shore section of the along-shore wind, potential temperature, TKE, and vertical velocity at Point Sur. The highest speed of the low-level downcoast jet, with a maximum of 23–24 m s−1 (Fig. 16a), is embedded into the middle of the capping inversion (Fig. 16b). This jet has a “tongue” of the highest winds, following the inversion down to the coast. As in the observations (Dorman et al. 1999a), the warmest and slowest air is above the coast, and the potential temperature gradient fans out while approaching the steep terrain. The TKE field (Fig. 16c) attains its MABL maximum in the offshore surface layer, still sustaining substantial shear from underneath the jet. It decreases monotonously onshore, while being suppressed from above by the strong capping inversion. An arbitrarily small value of TKE, 0.02 m2 s−2, is used here to map the slope of the stably stratified and sheared layer immediately overlaying the MABL. Close to the coast, the MABL turbulence is also suppressed by significant downward motion, down to −6 cm s−1 (Fig. 16d). Note that in most of the lower marine troposphere w ∼ ±1 cm s−1. The MABL collapse at the coast is further signified by the upward motion over the terrain, enhancing the local differential vertical velocity: ∂w/∂x ≫ 0. The Richardson number, Ri (not shown), corresponding to Fig. 16 is Ri ⩽ 0.1 for the bulk of the MABL (roughly z < 310 m, x < 55 km). While above this region there is a rather uniform and sharp increase of Ri (Ri ∼ 5 around 450–500 m), a step wise nonuniform increase of Ri takes place toward onshore (65 km < x < 76 km).

The lower tropospheric along-shore structure is revealed in Fig. 17, showing the along-shore wind speed component and potential temperature along x = 55 km and x = 77.7 km, respectively. The downcoast jet a small distance offshore, just below 500 m (Fig. 17a, V < −19 m s−1), relaxes while approaching Point Sur from the northwest. Further downwind (y < 40 km), the jet descends while still maintaining a relatively high velocity (V < −17 m s−1); however, the observations revealed a more complete jet recovery farther downstream (Rogers et al. 1998). Note the vertical perturbation in velocity downstream of the deceleration area (y < 50 km, z < 1500 m). Here, there is another deceleration area (V > −13 m s−1 for y < 10 km and z ∼ 630 m) above the main jet branch, topped by a weaker and thicker elevated jet branch (900 < z < 1500 m). A more complicated structure occurs onshore over Point Sur. The low-level jet, here V < −14 m s−1, decelerates and branches before impinging on Point Sur (Fig. 17c). The lower, more intensive branch recovers in the lee of the semibarrier, although at a lower height. The higher, weaker branch spreads out over a broader area, but it still apparently interacts with the underlying terrain, presumably via evanescent buoyancy waves. The elevated along-shore wind speed extreme is V = −12.8 m s−1 at 1250 m. Between these jet branches, there is a minimum down wind, with V ∼ −5 m s−1 around z ∼ 400 m, and another some kilometers upwind from Point Sur (a circular pattern), with V > −7 m s−1. The alternating pattern of maxima and minima fades in intensity and smears with height, which is typical for evanescent waves in an inhomogeneous medium. Figure 17d shows the vicinity of the hydraulic jump in front of and the MABL collapse at Point Sur (this cross section does not go right across the jump).

To support the finding of predominantly evanescent buoyancy waves above the MABL, a linear estimation for the wave-amplitude decrease can be made. In the linear hydrostatic regime, the wave amplitude aligned with the unperturbed wind perpendicular to the barrier is proportional to the maximum barrier height h0 and the buoyancy frequency within the WKB validity υ(z) ∼ N(z)h0. Hence, for two levels, υ1/υ2 = [(∂ϒ/∂z)1/(∂ϒ/∂z)2]1/2 (Gill 1982). If the unperturbed velocity is V ∼ −11 m s−1 (Fig. 18c), then υ1 = |V + 12.8| m s−1 = 1.8 m s−1 and (∂ϒ/∂z)1 = 3 K/(500 m) for the upper perturbation (at z ∼ 1250 m). Similarly, υ2 = |V + 7| m s−1 = 4 m s−1 and (∂ϒ/∂z)2 = 10 K/(400 m) for the lower perturbation (at z ∼ 520 m). Then υ1/υ2 = 1.8/4 = 0.45, which is close to [(∂ϒ/∂z)1/(∂ϒ/∂z)2]1/2 = [(3/500)/(10/400)]1/2 = 0.49. An equivalent estimation for the phase could be done. In reality, the evanescent waves are 3D, probably nonlinear (e.g., due to internal reflections), mixed with internal modes, nonhydrostatic waves, and a further more detailed analysis is beyond the scope of this paper. It suffices here to identify their presence and effects on the lower troposphere. The observations (Dorman et al. 1999a) reveal a pattern of alternating areas with wind maxima and minima in the vertical along-coast cross section that qualitatively resembles these waves.

The vertically averaged MABL structure is illustrated in Fig. 19, showing the mean wind speed, the mean TKE, the inversion height, and the vertical wind speed. While the MABL mean wind and depth (Figs. 18a,c) resemble the observations (Dorman et al. 1999a), the TKE and vertical velocity (Figs. 18b,d) are new results, only obtainable with this resolution and coverage via numerical modeling. Figure 18 suggests the presence of two expansion fans: one weaker, located upcoast from Monterey Bay, and the stronger one past Point Sur (cf., e.g., Fig. 19d). The near-shore mean MABL wind decelerates while approaching Point Sur and accelerates in the lee (Fig. 18a), while the mean TKE (Fig. 18b) decreases, nonmonotonically, toward the coast as the MABL thins (Fig. 18c). Inshore, between the expansion fan and the lee of Point Sur, the mean TKE alternates a few times; this is a subtle dynamical feature of the MABL related to the wave generation at Point Sur. A more detailed analysis would probably require a nonhydrostatic treatment. Two more details in the MABL distribution are shown in Fig. 18c. First, the MABL thickens downwind inside Monterey Bay from about 100 to over 200 m. Second, as the thin and fast-moving MABL flow diverges horizontally a few tens of kilometers in the lee of Point Sur, it impinges on the steep coast north of Point Piedra Blancas. Due to the wall effect, the MABL rises locally (follow the 100-m isoline). In this area, upward motions up to 6 cm s−1 are generated (Fig. 18d). While most of the MABL has very low mean vertical speed, stronger downward motions follow immediately behind both expansion fans. The strongest mean downward motion (about −5.5 cm s−1) is behind Point Sur. Another upward motion, with a maximum reaching 3 cm s−1, is associated with the hydraulic jump before Point Sur.

The dynamic structure of the MABL around Point Sur on this day can be condensed using the Froude number (Fig. 19). This compares favorably to Froude estimates obtained from the observation (Dorman et al. 1999a); furthermore, it extends the Froude field in a dynamically consistent way over adjacent areas not covered by the low-level aircraft measurement. Consequently, using numerical model results is a useful way to extend our understanding of the coastal summertime MABL dynamics. Obviously, the MABL is supercritical almost everywhere. The local Froude maxima are found in the lee of the coastal points. Note the hydraulic jump upcoast of Point Sur with minimum Fr ∼ 0.8–0.9, while downcoast there is a supercritical flow with the maximum Fr > 2.5. The expansion fans are launched at the coast and extend offshore about 60 km. As in the Cape Mendocino simulation, the flow isolines (the sine of the angle) only approximately coincide with the inverse Froude isolines. As a curiosity, the modeled far off shore upcoast wind was fitted to, with a reasonable agreement (not shown), the analytical Ekman layer profile with a gradually varying eddy diffusivity (Berger and Grisogono 1998)

5. Discussion and conclusions

Fully 3D numerical simulations, performed for two situations with supercritical flow past sharp bends in a coastline are presented. The simulations were performed with the hydrostatic MIUU mesoscale model, including a higher-order turbulence parameterization. The simulations are compared with measurements during the Coastal Waves 1996 experiments from research flights collected by the NCAR C-130 research aircraft around Cape Mendocino and Point Sur (Rogers et al. 1998; Dorman et al. 1999a). The model performance is evaluated both by comparing measured and simulated fields for the two cases and through an objective evaluation, reconstructing a research flight around Cape Mendocino in the model domain. The simulations were set up to use the model as a numerical laboratory, and thus external synoptic-scale forcing and forcing from the land surface were highly idealized. Still, the model reproduced the measurements adequately, and the model fields were then analyzed to highlight some of the observed features.

Even though the depth of the inversion layer is sometimes as deep as the MABL, the flow can be characterized to a first-order approximation as a supercritical, single-layer, reduced-gravity flow past a varying sidewall, as in, for example, Samelson (1992) and Winant et al. (1988). Upstream of capes or points, the flow is typically supercritical, Fr > 1, within a Rossby radius of deformation. This is due to the coastal jet that forms as the inversion slopes steeply down toward the coast. At Cape Mendocino this jet is more localized close to the coast, resulting in a more narrow coast-parallel zone of supercritical flow with subcritical flow well offshore, while in the Point Sur case the wind maximum is broader and the flow is supercritical almost everywhere in the model domain. As this flow “hits” bends in the coastline, expansion fans appears. Although the initial angle of the expansion fans agrees well with the theoretical estimates based on hydraulic theory, MABL-depth isolines are curved and the associated wind speed maximum appears in lens-shaped patterns, rather than in straight lines as in the hydraulic theory. This is attributed to the effect of turbulence and also, to a lesser extent, to the Coriolis force. It is clear that the modeled Froude number summarizes well the current understanding of this type of flow.

At Cape Mendocino, the local terrain at the cape is oriented in two ridges across the flow, which appears to be of vital importance to the local flow. This terrain height is approximately equal to the MABL depth, causing a partial blocking of the flow. This forces a single lee-wave pattern, bringing down higher temperatures and lower wind speeds from aloft, directly in the lee of the cape. The collapse of the MABL in the lee of the cape and the associated extreme minimum in wind speed, into the Shelter Cove area, is thus a combined effect of the expansion fan dynamics and this lee-wave formation.

At Monterey Bay and Point Sur, the coast is oriented more northwest–southeast, and with the northerly background flow, the along-coast jet is broader and the flow is supercritical almost everywhere. The MABL is 400–500 m offshore and steeply collapses to ∼50 m at the coastline within the final 20 km. This is partly caused by the expansion fan dynamics but also forced by vertical circulations due to the blocking effect of the coastal terrain. Two expansion fans are formed: a weak fan originates at the terrain upstream of Monterey Bay, off Santa Cruz, but reaches only a short distance offshore, while a stronger structure originates at Point Sur that extends ∼60 km offshore. Between these, an area with a lower Froude number emanates in Monterey Bay and extends ∼60 km offshore. At Monterey Bay Fr < 1 locally and a weak hydraulic jump occurs. Inside the bay itself the flow is subcritical, and the MABL depth also increases gradually southward. These features are most clearly seen in the vertical velocity pattern. When the flow impinges on Point Sur from the northwest, an evanescent buoyancy wave forms that perturbs the jet structure downstream into alternating wind speed maxima and minima in the vertical.

The forcing on the ocean surface was investigated for the Cape Mendocino simulation. The local curl of the surface stress vector was found to correspond qualitatively to the observed depression in SST, in particular, in the Shelter Cove area. This is because the background upwelling here is less dominant, due to lower winds at the coast, and the local effect is stronger. However, in an additional simulation for the Cape Mendocino case, with a more realistic SST distribution estimated from the measurements, the original results were only slightly perturbed. This may be summarized as follows: the interplay between the synoptic-scale flow, the MABL properties, and the coastal terrain contribute to the flow structure, with an intense supercritical coastal jet that develops into an expansion fan at Cape Mendocino. The forcing on the coastal ocean favors upwelling along the entire coast, which is intensified by the expansion fan dynamics in the lee of the cape. This changes the lower boundary for the MABL (Enriquez and Friehe 1997). However, this influence, although significant locally, is marginal in the sense that it does not alter the main flow structure. It seems that for these conditions, which cause a large local forcing on the ocean, the forcing on the flow itself is so strong that it becomes insensitive to the details in the local surface forcing. It is left open to speculation if the opposite can also be true. Are there situations with less dominating atmospheric dynamics that still cause significant forcing on the ocean surface so that the feedback is relatively more important?

The momentum budgets were also investigated for the Cape Mendocino simulation. In the undisturbed flow upstream of the kink in the coastline, the flow is quasigeostrophic in the cross-shore momentum equation with the expected modification due to turbulence in the MABL. In the along-shore momentum, there is an approximate balance between the pressure gradient and turbulent friction, as the along-coast Coriolis force is small in the roughly coast-parallel flow. In the expansion fan, the flow in both components is highly ageostrophic. The imbalance is typically 50%–100% in magnitude, as compared to the pressure gradient forcing. The latter is instead balanced by turbulent friction and acceleration, with the latter being larger when the former is small and vice versa. Thus, close to the surface, both friction and acceleration are important, while close to the MABL top, ageostrophic acceleration dominates.

Acknowledgments

This study was sponsored by the Office of Naval Research through Grant N00014-96-1-0002. The authors are grateful to Dr. David Rogers for providing the aircraft data and to Clive Dorman, Gunilla Svensson, Steve Burk, and Trace Haack for their help and many suggestions. We also thank Dr. Ian Brooks for supplying the gridded data for the composite cross section from the measurements. Thanks also to the NCAR flight crew. This work was performed while MT was visiting the California Institute of Technology on a sabbatical, and he is grateful to Professor John Seinfeld for providing the necessary facilities.

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Fig. 1.
Fig. 1.

Surface plots of the model terrain for (a) Cape Mendocino and (b) Point Sur, also showing some geographical references. Note that the whole model domain is not displayed.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 2.
Fig. 2.

Plots of the model domain for (a) Cape Mendocino and (b) Point Sur, showing the coastal terrain (gray shading), gridpoint distribution (dots), and the flight patterns (solid) on the days in question. Also, a few locations are indicated.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 3.
Fig. 3.

Model data and measurements for an aircraft transect across the coast in the vicinity of Shelter Cove (see Fig. 1), at 1600 LT 7 June 1996. The measured data are averaged according to height above the surface and to horizontal position in relation to the coastal jet and is represented with symbols, while all the model profiles along the transect are shown, within each horizontal region, as dotted lines. The regions are far offshore (*), in the outer portions of the jet (+), in the jet core (○), and in the lee of the cape (×). The profiles are displaced horizontally around the profile in the jet core for clarity (temperature by 5°C each, humidity by 5 g kg−1 each, and the wind speed components by 10 m s−1 each); the x-axis ticks are valid for the jet core profile.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 4.
Fig. 4.

Vertical cross sections from west to east through Shelter Cove (see Figs. 1, 2) at 1600 LT 7 June 1996 of simulated (a) scalar wind speed (m s−1) and (b) potential temperature (°C). Also shown is (c) a close-up of the simulated wind (dotted line and grayscale) and temperature fields (dashed line) and (d) the corresponding measurements.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 5.
Fig. 5.

Same as Fig. 4, but for a vertical cross section upstream of Cape Mendocino. (See Fig. 2.)

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 6.
Fig. 6.

Horizontal cross section of the wind field at 1600 LT 7 June 1996 at 100 m. The plots show scalar wind speed (grayscale) and wind vectors from the simulations for two different area sizes. Also shown in (b) as bold arrows are the measured winds averaged for this height from the entire flight.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 7.
Fig. 7.

Horizontal plots of the simulated (a) maximum and (b) mean MABL scalar wind speed (m s−1), (c) the height to the inversion base (m), and (d) the Froude number for 1500 LT 7 June 1996.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 8.
Fig. 8.

Vertical cross section from south to north [(a), (b)] through Cape Mendocino and [(c), (d)] just off the tip of the cape, aligned with the upstream jet center line, showing [(a), (c)] scalar wind speed (m s−1) and [(b), (d)] potential temperature (°C).

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 9.
Fig. 9.

Details of the simulated MABL from south to north through Cape Mendocino showing (a) scalar wind speed (m s−1) at several different heights and (b) a vertical cross section of the MABL thermal structure (grayscale), with a dotted line also indicating the position of the heights in (a).

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 10.
Fig. 10.

Horizontal plot of the simulated curl of the turbulent stress vector (106 m s−2) at the surface for the area around CapeMendocino.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 11.
Fig. 11.

Comparison of the depression of the measured SST (dots) and a normalized modeled stress vector curl (solid lines) across the coast (a) into Shelter Cove, (b) at Cape Mendocino, and (c) upstream of the cape (see Fig. 10). The model data are from three transects around the aircraft track. Also, in (d) the observed SST is plotted against a simulated stress vector curl that was interpolated to the observed positions using GPS positions from the aircraft.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 12.
Fig. 12.

Horizontal cross sections showing the difference in wind speed (grayscale) and temperature (dashed lines) around Cape Mendocino between two simulations: one with a realistic SST distribution and one where SST was assumed to be constant at the background value (the control simulation). The figure displays two levels: (a) 4 m and (b) 200 m. The terrain is indicated by a thin solid line.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 13.
Fig. 13.

Simulated momentum budgets for [(a)–(d)] the u component (across coast) and [(e)–(h)] the υ component (along coast) of the wind around Cape Mendocino at 275 m at 1500 LT 7 June 1996. The subplots show [(a), (e)] the pressure gradient forcing, [(b), (f)] the geostrophic imbalance, [(c), (g)] the geostrophic plus turbulent friction imbalance, and [(d), (h)] the geostrophic plus acceleration imbalance (10−3 m s−2).

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 14.
Fig. 14.

Same as Fig. 13, but for the height 7 m.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 15.
Fig. 15.

Comparison of aircraft measured (solid) and simulated (dashed) profiles of (a) wind speed (m s−1), (b) wind direction (°), (c) potential temperature (K), and (d) specific humidity (g kg−1) around Point Sur on 17 June 1996. The measured profiles are taken at the beginning and end of a flight, a few hours apart, and are located a few tens of kilometers apart. The model profiles are taken at (x, y) = (71.5, 68.9), (71.5, 37.3), and (53, 81.1) km, respectively (x and y count from the southwest corner of the model domain—see Fig. 2).

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 16.
Fig. 16.

Vertical (x, z) across-shore cross sections at y = 68.9 km, close to Point Sur. Shown are (a) the along-shore wind component (m s−1; toward the reader), (b) potential temperature (°C), (c) TKE (m2 s−2; note that this figure frame is only up to 1000 m) and (d) vertical velocity (cm s−1; more intensive inland vertical velocities are not shown). The white areas in the lower-right corners outline the terrain.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 17.
Fig. 17.

Vertical (y, z) along-shore cross sections of [(a), (c)] the along-shore wind component (m s−1; blowing from right to left) and [(b), (d)] potential temperature (°C) at [(a), (b)] x = 55 km and [(c), (d)] x = 77.7 km.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 18.
Fig. 18.

Horizontal (x, y) cross section of some MABL characteristics: (a) the mean wind speed (m s−1; blowing parallel downcoast), (b) the mean TKE (m2 s−2), (c) the MABL depth (m; with isolines every 50 m), and (d) the mean vertical velocity (cm s−1; with isolines every 2 cm s−1).

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Fig. 19.
Fig. 19.

The MABL Fr summarizing the flow properties around Point Sur. A hydraulic jump (from Fr >1 to Fr <1) occurs upcoast of Point Sur, while downcoast the flow is supercritical, with max(Fr) >2.5. The expansion fans are launched offshore extending about 60 km out through the MABL.

Citation: Journal of the Atmospheric Sciences 57, 1; 10.1175/1520-0469(2000)057<0108:SOSFAP>2.0.CO;2

Table 1.

Domain size, number of grid points, and maximum resolution for the two grids used in the model runs. Note that the grid in the Point Sur simulation was rotated ∼45° counterclockwise, i.e., the x coordinate is aligned with the southwest–northeast direction and the y coordinate with the northwest–southeast direction.

Table 1.
Table 2.

Summary of the model validation for the Cape Mendocino simulation showing the mean bias (B), the fractional bias (Bf), the correlation coefficient (r), the standard deviation of the error (σerr), the fractional standard deviation of the error (σferr), the ratio of the standard deviations for the observed and simulated values (σo/σp), the ratio of the standard deviation of the observation to the standard deviation of the error (σo/σerr), and the IOA.

Table 2.
1

NCAR is supported by the National Science Foundation.

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