• Andrén, A., 1990: Evaluation of a turbulence closure scheme suitable for air-pollution applications. J. Appl. Meteor., 29 , 224239.

  • Beardsley, R. C., , C. E. Dorman, , C. A. Friehe, , L. K. Rosenfeld, , and C. D. Winant, 1987: Local atmospheric forcing during the Coastal Ocean Dynamics Experiment. 1. A description of the marine boundary layer and atmospheric conditions over a northern California upwelling region. J. Geophys. Res., 92 , 14671488.

    • Search Google Scholar
    • Export Citation
  • Burk, S. D., , and W. T. Thompson, 1996: The summertime low-level jet and marine boundary layer structure along the California coast. Mon. Wea. Rev., 124 , 668686.

    • Search Google Scholar
    • Export Citation
  • Burk, S. D., , T. Haack, , and R. M. Samelson, 1999: Mesoscale simulation of supercritical, subcritical, and transcritical flow along coastal topography. J. Atmos. Sci., 56 , 27802795.

    • Search Google Scholar
    • Export Citation
  • Dorman, C. E., , and C. D. Winant, 1995: Buoy observations of the atmosphere along the west coast of the United States, 1981–1990. J. Geophys. Res., 100 , 1602916044.

    • Search Google Scholar
    • Export Citation
  • Dorman, C. E., , D. P. Rogers, , W. Nuss, , and W. T. Thompson, 1999: Adjustment of the summer marine boundary layer around Point Sur, California. Mon. Wea. Rev., 127 , 21432159.

    • Search Google Scholar
    • Export Citation
  • Edwards, K. A., , A. M. Rogerson, , C. D. Winant, , and D. P. Rogers, 2001: Adjustment of the marine atmospheric boundary layer to a coastal cape. J. Atmos. Sci., 58 , 15111528.

    • Search Google Scholar
    • Export Citation
  • Enger, L., 1990: Simulation of dispersion in moderately complex terrain—Part A. The fluid dynamic model. Atmos. Environ., 24A , 24312446.

    • Search Google Scholar
    • Export Citation
  • Haack, T., , S. D. Burk, , C. Dorman, , and D. Rogers, 2001: Supercritical flow interaction within the Cape Blanco–Cape Mendocino orographic complex. Mon. Wea. Rev., 129 , 688708.

    • Search Google Scholar
    • Export Citation
  • Hodur, R. M., 1997: The Naval Research Laboratory's Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125 , 14141430.

    • Search Google Scholar
    • Export Citation
  • Pielke, R. A., 1984: Mesoscale Meteorological Modeling. Academic Press, 599 pp.

  • Rogers, D. P., and Coauthors. 1998: Highlights of Coastal Waves 1996. Bull. Amer. Meteor. Soc., 79 , 13071326.

  • Rogerson, A. M., 1999: Transcritical flows in the coastal marine atmospheric boundary layer. J. Atmos. Sci., 56 , 27612779.

  • Samelson, R. M., 1992: Supercritical marine-layer flow along a smoothly varying coastline. J. Atmos. Sci., 49 , 15711584.

  • Söderberg, S., , and M. Tjernström, 2001: Supercritical channel flow in the coastal atmospheric boundary layer: Idealized numerical simulations. J. Geophys. Res., 106 , 1781117829.

    • Search Google Scholar
    • Export Citation
  • Ström, L., , M. Tjernström, , and D. P. Rogers, 2001: Observed dynamics of coastal flow at Cape Mendocino during Coastal Waves 1996. J. Atmos. Sci., 58 , 953977.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., 1987a: A study of flow over complex terrain using a three-dimensional model. A preliminary model evaluation focusing on stratus and fog. Ann. Geophys., 88 , 469486.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., . 1987b: A three-dimensional meso-γ-model for studies of stratiform boundary layer clouds: A model description. Rep. 85, Dept. of Meteorology, Uppsala University, Uppsala, Sweden, 44 pp.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., . 1999: The sensitivity of supercritical atmospheric boundary-layer flow along a coastal mountain barrier. Tellus, 51A , 880901.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., , and B. Grisogono, 2000: Simulations of supercritical flow around points and capes in a coastal atmosphere. J. Atmos. Sci., 57 , 108135.

    • Search Google Scholar
    • Export Citation
  • Winant, C. D., , C. E. Dorman, , C. A. Friehe, , and R. C. Beardsley, 1988: The marine layer off northern California: An example of supercritical channel flow. J. Atmos. Sci., 45 , 35883605.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    (a) A side view of the northern California terrain with some reference points. (b) The idealized representation of the northern California terrain used in this study. Maximum height of the idealized terrain is 1.2 km. The dashed line from south to north and the dashed–dotted line from west to east indicate the location of cross sections for later reference

  • View in gallery

    Contour plots from Ctrl at 1500 LST of (a) MBL depth (m), (b) maximum MBL wind speed (m s−1), and (c) the Froude number

  • View in gallery

    Vertical cross section from Ctrl at 1500 LST of potential temperature (K, dashed lines) and scalar wind speed (m s−1, solid lines) taken along the dashed line in Fig. 1b

  • View in gallery

    Contour plot of the initial Froude number from exp_restart at 1800 LST

  • View in gallery

    Maximum MBL wind speed (m s−1) at 1500 LST from exp_restart

  • View in gallery

    Contour plot of the Froude number from exp_restart at 0600 LST. Dashed lines indicate the area over which the averages in Figs. 8–10 are calculated

  • View in gallery

    Vertical cross section from exp_restart of along-coast wind component (m s−1, solid) and MBL depth (dashed) taken along the upstream part of the dashed line in Fig. 1b at (a) 0600 LST and (b) 0900 LST. The solid gray line shows Fr along the cross section with values given on the right-hand side axis

  • View in gallery

    Diurnal cycle of shallow water gravity-wave phase speed and mean MBL wind speed for exp_restart. The numbers along the solid line show LST while the dashed line indicates where Fr = 1; subcritical area is upper left (corresponding to nighttime flow)

  • View in gallery

    Same quantities as in Fig. 8, but for (a) Ctrl and (b) exp_const_T. The numbers along the solid line in (b) show number of hours into the simulation

  • View in gallery

    Time evolution of the mean MBL wind speed and the phase speed from Ctrl along the upstream coastline

  • View in gallery

    Vertical cross sections of potential temperature (K, dashed lines) and scalar wind speed (m s−1, solid lines) from exp_restart, taken along the dashed–dotted line in Fig. 1b: (a) 0600 LST and (b) 1500 LST

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 9 9 3
PDF Downloads 2 2 0

Diurnal Cycle of Supercritical Along-Coast Flows

View More View Less
  • 1 Department of Meteorology, Stockholm University, Stockholm, Sweden
© Get Permissions
Full access

Abstract

In this study, a three-dimensional hydrostatic mesoscale model is used to address the transient behavior of supercritical along-coast flow. A control experiment and several sensitivity tests are performed in order to investigate the diurnal cycle of flow characteristics. An idealized representation of the northern California terrain is used, and the model results are interpreted within the reduced-gravity shallow water concept. In two preceding studies by the authors, this theory appeared to be violated since the flow accelerated along the coastline upstream of the change in coastline orientation, even though the flow was supercritical. Here, it is shown that the criticality of the flow for typical summertime conditions along the California coast actually varies diurnally. The gradual acceleration of the flow along the upstream coastline is established during a subcritical phase of the simulation; thus, the shallow water concept is not violated. Because the along-coast jet is primarily driven by the cross-coast baroclinicity, there will be a continuous variation in the strength of the jet. This in turn will affect the flow criticality and thus the flow is not only spatially, but also temporally, transcritical. The results here suggest that observations of quasi-steady-state supercritical flow in reality are not likely; transcritical flow along mountainous coastlines should be more prevalent.

Corresponding author address: Stefan Söderberg, Dept. of Meteorology, Stockholm University, Arrhenius Laboratory, SE-106 91 Stockholm, Sweden. Email: stefan@misu.su.se

Abstract

In this study, a three-dimensional hydrostatic mesoscale model is used to address the transient behavior of supercritical along-coast flow. A control experiment and several sensitivity tests are performed in order to investigate the diurnal cycle of flow characteristics. An idealized representation of the northern California terrain is used, and the model results are interpreted within the reduced-gravity shallow water concept. In two preceding studies by the authors, this theory appeared to be violated since the flow accelerated along the coastline upstream of the change in coastline orientation, even though the flow was supercritical. Here, it is shown that the criticality of the flow for typical summertime conditions along the California coast actually varies diurnally. The gradual acceleration of the flow along the upstream coastline is established during a subcritical phase of the simulation; thus, the shallow water concept is not violated. Because the along-coast jet is primarily driven by the cross-coast baroclinicity, there will be a continuous variation in the strength of the jet. This in turn will affect the flow criticality and thus the flow is not only spatially, but also temporally, transcritical. The results here suggest that observations of quasi-steady-state supercritical flow in reality are not likely; transcritical flow along mountainous coastlines should be more prevalent.

Corresponding author address: Stefan Söderberg, Dept. of Meteorology, Stockholm University, Arrhenius Laboratory, SE-106 91 Stockholm, Sweden. Email: stefan@misu.su.se

1. Introduction

In shallow water theory, the Froude number (Fr) is defined as the ratio of the flow speed U to the linear gravity wave phase speed c. When Fr is less than 1, the flow is subcritical, while, if Fr exceeds unity, the flow is referred to as supercritical. Physically supercritical flow means that the mass and wind field upstream of a local perturbation cannot adjust to this, since the phase speed of the waves responsible for the adjustment process is lower than the flow speed; the information is swept downstream by the flow.

One objective in the Coastal Ocean Dynamics Experiment (CODE) was to characterize the coastal marine boundary layer (MBL) off northern California. From the observations, Beardsley et al. (1987) suggest a conceptual model for the MBL. A strong capping inversion, sloping from west to east, separates the well-mixed MBL from a relatively warmer subsiding air mass aloft. Just below the inversion a nearshore jet is found. In particular they describe the diurnal variation of the coastal jet with stronger (weaker) winds during daytime (nighttime); this is due to the diurnal variation of the temperature contrast between land and sea, which alters the slope of the inversion toward the coast. Based on aircraft observations from one case during CODE, Winant et al. (1988) describe the dynamics of this coastal flow as a single-layer reduced-gravity shallow water flow past a blocking sidewall. Since the inversion along the U.S. west coast is strong, typically ΔΘ = 10–20 K, where Θ is potential temperature (Dorman and Winant 1995), and the MBL is typically well mixed, the use of this “simplified model” is motivated. They conclude that the MBL off northern California can be characterized as a supercritical channel flow with expansion fans and hydraulic jumps in the vicinity of important headlands. Samelson (1992) improved this theory using a numerical shallow water model, including rotation and simplified surface friction.

The Coastal Waves 1996 field experiment (CW96) was carried out in the summer of 1996 along the California coast (Rogers et al. 1998). One motivation for the CW96 was to study the supercriticality of the MBL along the California coast. Several cases of supercritical flow, with expansion fans and hydraulic jumps, were encountered during research missions with the National Center for Atmospheric Research (NCAR) C-130 research aircraft. Three cases near Cape Mendocino in northern California, on 7, 12, and 26 June 1996, have been extensively studied. Ström et al. (2001) investigate the observed dynamics and find that the supercriticality of the flow is instrumental for an understanding of these three cases. Edwards et al. (2001) study the 7 June case and compare observations with modeling results from a shallow water model (Rogerson 1999). The model captured the main features of the observed flow; within an expansion fan in the lee of a simplified cape, a depression of the MBL depth and an acceleration of the flow were reproduced. In a transcritical model run (with subcritical upstream conditions based on observations) it was observed that as the model flow spins up, the upstream flow becomes partially blocked by the cape. An upstream propagating jump forms, leaving a relatively deep and well-mixed MBL in its wake while propagating northward and exiting the model domain; thus, the cape influences the upstream region (Edwards et al. 2001). With simplified terrain and initial conditions, Burk et al. (1999) study the characteristics of along-coast stratified flow bounded by a sidewall with a nonhydrostatic compressible mesoscale model, the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997). The initial Fr is varied, which allows examination of different flow regimes. Consistent with shallow water model studies, the MBL depth decreases and the wind speed increases when the flow rounds convex bends (e.g., Samelson 1992; Rogerson 1999). They also find a flow response in the stratified layer above the MBL, which, however, cannot be explained by shallow water theory. From shallow water similarity solutions and linear theory Burk et al. (1999) conclude that the presence of a low-level jet is more important than a sharp boundary layer inversion to generate a supercritical flow response around coastal bends. Dorman et al. (1999) utilized COAMPS for comparisons of model results and measurements around Point Sur on 17 June 1996. Observations and model results show that the flow on this day is supercritical, which explains the observed wind distribution in the vicinity of Point Sur.

Tjernström and Grisogono (2000) performed numerical simulations based on CW96 observations on 7 and 17 June 1996. A hydrostatic meso-γ-scale model, the Department of Meteorology, Uppsala University (MIUU) model was used. A good agreement between observed and simulated fields were found, in particular the observed skewed-shaped jet was accurately simulated. Among the conclusions was that the collapse of the MBL into Shelter Cove downstream of Cape Mendocino is only partly due to expansion fan dynamics. The cape partially blocks the upstream flow and initiates a buoyancy wave that contributes to the collapse of the MBL and also causes the skewed shape of the jet. Burk and Thompson (1996) also recognized the importance of the cape. A pressure minimum and a near-surface wind speed maximum on the lee side of Cape Mendocino were modeled and found to be consistent with mountain wave theory.

In a sensitivity study, based on the experience of modeling the 7 June 1996 case, Tjernström (1999) found the expansion fan dynamics to be surprisingly persistent. Within realistic alterations to the 7 June case, the background conditions generated super- or transcritical conditions with an expansion fan at Cape Mendocino. In two simulations with altered terrain where the terrain at the cape or the entire cape was removed, the jet attained a more symmetric shape rather than the observed skewed shape. It was also concluded that the smooth curvature of the main coastal terrain (in the absence of the cape) is sufficient to excite an expansion fan. However, the MBL did not collapse, as in the cases when the terrain of the cape was included.

Tjernström (1999) observed an unexpected feature in the simulations with altered terrain. The wind speed started to increase far upstream of the bend in the coastline, even for supercritical flow. This is not expected from the linear shallow water theory and indeed does not appear in any of the previous shallow water investigations (e.g., Samelson 1992; Rogerson 1999). Neither does it appear in the study by Burk and Thompson (1996) or in Burk et al. (1999). The former simulation included the terrain at Cape Mendocino while the latter, although without an explicit cape, did not include any diurnal cycle. It was speculated in Tjernström (1999) that small continuous changes in the direction of the upstream coastline or along-coast changes in the height of the terrain might be the cause of this feature. However, Söderberg and Tjernström (2001, hereafter ST01) showed that this was not the case.

In ST01, a set of numerical simulations with smooth idealized terrain, fitted to the real terrain north of Cape Mendocino, was performed with the MIUU model in order to address hypotheses put forward in Tjernström (1999). Background initial conditions were taken the same as in Tjernström (1999), while the terrain was varied in a simple manner. In fact, the only simulations in ST01 that did not feature a gradual acceleration of the flow along the upstream coastline were the simulations where a simplified cape was inserted perpendicular to the main coastal range. Moreover, gravity wave breaking was found on the lee side of the cape; hence, the collapse of the MBL in the lee of Cape Mendocino is partly due to waves triggered by the cape. Thus, the cape is not only responsible for the collapse of the MBL in lee of the cape and a substantial upstream blocking, but also removes the gradual upstream acceleration of the flow found in simulations without the cape. Differences between shallow water models and a three-dimensional model were also illustrated by ST01. When the angle of the coastal bend was increased, this did not lead to only an increased wind speed. A warming of the upper layer and a strengthened stability, both upstream and downstream of the bend, were found when the angle was increased. Nevertheless, ST01 concluded that neither deviations from linearity or from the shallow water concept, although present and probably substantial downstream of the bend in the coastline, were the causes for the gradual acceleration of the flow found along the upstream coastline in simulations without a cape. It was hypothesized that the absence of a gradual upstream acceleration in the observations from Cape Mendocino is due to the presence of the cape and not due to the supercriticality of the flow.

One hypothesis offered in ST01 as the cause of the upstream acceleration of the flow was based on the fact that the initial profile used to start all simulations in the early evening on the day before the event is actually subcritical. The supercriticality of the flow then develops during the dynamic initialization of the model. The gradual acceleration along the upstream coastline might thus be established during the subcritical phase of the initialization and then disappear only very slowly, after the flow has turned supercritical. If this is the true mechanism, the real flow at Cape Mendocino may in reality be predominantly transcritical rather than supercritical, since it is expected that the flow in the coastal jet is weaker during nighttime. Daytime conditions, with a stronger jet and consequently supercritical flow, may then not prevail sufficiently long for a true quasi-steady supercritical flow to be established in the absence of blocking terrain at the cape.

While the first part of this study (ST01) focused on the sensitivity of along-coast supercritical flow to terrain forcing, the present study is set up to test the hypothesis described above. Here we address the transient behavior of this type of flow with the aim to find an explanation for the gradual acceleration of the flow along the upstream coastline. The outline for this paper is as follows. In section 2 the numerical experiments are described. The model results are presented in section 3 followed by a discussion with conclusions in section 4.

2. The experiment

a. The model

The model used in this study is the Uppsala University meso-γ-scale model, that is, the MIUU model. This is a three-dimensional hydrostatic, nonlinear primitive equations model with a higher-order turbulence closure. A terrain-influenced vertical coordinate system according to Pielke (1984, 118–125) is employed. Horizontally and vertically expanding grids are used to achieve high resolution in the area of interest. For a detailed description of the model we refer to Andrén (1990), Tjernström (1987a,b), and Enger (1990).

b. Analysis of model results

In the evaluation of model results, the reduced-gravity shallow water definition of Fr,
UgH−0.5
is used for simplicity. The flow speed in the layer is U, while H is the layer depth, and g′ is the reduced gravity, here expressed using the potential temperature; thus,
gg0
where g is the acceleration of gravity and Θ0 is the temperature of the lower layer. This analog has previously been used in a number of studies of supercritical flow along the U.S. west coast and has been shown to be useful when interpreting the physics of the flow. Here, H was taken as the height where Θ exceeds the surface value by 1.4 K (the MBL is typically well mixed, hence Θ is constant). The strength of the inversion was determined as the temperature difference over a layer above H, where the vertical temperature gradient exceeded a prescribed value. Together these parameters determine the linear shallow water gravity wave phase speed
cgH0.5
The ratio between the mean MBL wind speed, used as a proxy for U, and c was used to calculate Fr. The maximum MBL wind speed was taken as the maximum wind speed in each model vertical. In all simulations, the maximum MBL wind speed was found within the calculated MBL depth.

c. The numerical experiment

We use the same terrain as in the control run in ST01, an idealized representation of the northern California terrain. By fitting a simple parabolic function to an ensemble of east–west cross sections of the real terrain north of Cape Mendocino, a smooth upstream terrain, homogeneous in the north–south direction was generated. The real terrain of northern California and some reference points are shown in Fig. 1a, while the idealized terrain, with a maximum height of 1.2 km, is illustrated in Fig. 1b. At y ≈ 25 km there is a bend in the coastline. The model domain is 495 × 430 × 5 km3, resolved by 41 × 41 × 30 grid points. The maximum horizontal resolution is 2 × 2 km2 in the domain center at (x, y) = (0, 0) km. The vertical resolution is 1 m close to the surface, decreasing to ≈270 m toward the model top. All model results in this paper are presented in model coordinates.

Surface boundary conditions were applied as in ST01, Tjernström and Grisogono (2000), and Tjernström (1999). SST was held constant at 285.65 K (Burk and Thompson 1996; Tjernström and Grisogono 2000; Tjernström 1999) while a sinusoidal-type function prescribed the diurnal variation of the soil surface temperature. The surface humidity was prescribed to produce potential evaporation at the sea surface (reduced by salinity) and a fraction thereof (15%) over land.

All simulations, except one, were initialized using a dynamic initialization. At initial time, the model was given horizontally homogeneous fields of potential temperature and humidity. An initially 800-m-deep well-mixed MBL was capped by a 17 K and 4 g kg−1 inversion with stable stratification and constant (1 g kg−1) humidity aloft. A geostrophic wind, constant in time and in the horizontal but varying with height, prescribes the background flow. The geostrophic wind components (ug, υg) vary linearly from (−2, −15) m s−1 at the surface, to (4, 3.5) m s−1 at 2500 m, and (9, 7.5) m s−1 at the model top. The wind speed thus turns from northerly to southwesterly with height. The initial wind was set equal to the geostrophic wind, except in the PBL where a logarithmic wind profile was applied. With these profiles, using the method described in section 2b, the initial Froude number is Fr ≈ 0.36.

The control run (Ctrl) was initialized at 1800 LST with the conditions described above. Four sensitivity tests were performed; they varied as follows.

  • Two simulations using the same setup as in Ctrl, except that the initial LST was varied. One simulation started at 0000 LST (exp_ini00) and one at 0600 LST (exp_ini06), thus shortening the initialization period from 21 to 15 and 9 h, respectively. The purpose of the experiments was to ascertain that the observed model behavior in Tjernström (1999) and in ST01 was not an artifact of the model initialization.
  • One simulation (exp_restart), initialized with results from Ctrl after 24 h of simulation (at 1800 LST the day of the event). Vertical west–east cross sections of all prognostic variables at the coastal bend (y ≈ 25 km) were distributed to all model cross sections upstream of the bend. Downstream of the bend, the variables were left unchanged. The model was then restarted with this new 3D distribution of the prognostic variables. By initializing the model this way we were certain that the flow was supercritical from the very beginning of the simulation. It allowed us to test if the subcritical initial profile, used to initialize all other experiments, was the cause of the upstream acceleration of the flow.
  • One simulation (exp_const_T) with the same initial conditions as in Ctrl, but where the radiation routine in the model was switched off and the soil surface temperature was held constant in time at its diurnal maximum. Differences between a steady-state forcing simulation and simulations with a realistic diurnal cycle are then investigated, for example, the time needed for the initial flow to adjust to the forcing.
A summary of the experiments is given in Table 1.

3. Results

a. The control run

The results from the control run will only briefly be discussed here; a more thorough presentation is found in ST01. In Fig. 2, the MBL depth (m), maximum MBL wind speed (m s−1), and Fr at 1500 LST (21 h into the simulation) are presented. Upstream of the bend, the MBL depth decreases gently toward the coastline and is fairly homogeneous in the north–south directions. From the bend and southward, the decrease in MBL depth is more dramatic. Here the contour lines of the MBL depth form a fan-shaped pattern (Fig. 2a). In response to the decreased MBL depth downstream of the change in the coastline orientation, the flow accelerates to a maximum of ≈22 m s−1 (Fig. 2b). Note the gradual acceleration of the flow along the upstream coastline. This is clearly not in agreement with linear shallow water theory since the flow is supercritical in the greater part of the model domain (Fig. 2c).

Shown in Fig. 3, for the same time as in Fig. 2, are vertical cross sections of potential temperature (K) and scalar wind speed (m s−1) taken along the dashed line indicated in Fig. 1b. South of the bend (y < 25 km) there is a substantial warming of the atmosphere between the MBL top and ≈1.5 km; note also the wavelike structure in the isotherms. Thus, as in Burk et al. (1999), there is a response to the coastline bend in the stratified atmosphere above the MBL. Furthermore, this cross section reveals a significant increase in vertical wind shear below z ≈ 1 km south of the bend. The acceleration of the flow starting upstream of the bend is clearly illustrated also here. Several hypotheses for this feature were offered and rejected in ST01, the remaining simulations in this paper were conducted to explain this feature.

b. Varying the initial LST of the simulation

It was found that exp_ini00 and exp_ini06 at 1500 LST displayed similar flow features as Ctrl (not shown). The straight forward conclusion from these two experiments is that shortening the initialization from 21 to 9 h does not affect the results appreciably, and does not remove the gradual upstream acceleration of the flow, which therefore still remains unexplained.

c. Restart at 1800 LST

Figure 4 displays Fr at initial time (1800 LST) for exp_restart. With the modified fields, the flow in exp_restart is now supercritical along the upstream coastline at initial time. This allows us to test if subcritical initial conditions can be the cause of the upstream acceleration. After 21 h of simulation, at 1500 LST the following day, the MBL flow attains a maximum wind speed of nearly 22 m s−1 (Fig. 5). However, again, the gradual acceleration of the flow along the upstream coastline is present although the flow is supercritical (not shown). Since the initial flow is supercritical we here must conclude that this feature, present here and in all simulations without capes in ST01, is not due to the fact that the initial profile used in Ctrl is subcritical.

d. Constant surface temperature

In this steady-state forcing simulation exp_const_T the MBL wind speed accelerates in time and attains a maximum of nearly 23 m s−1 after ≈10 h of simulation time (not shown). During this time, the model spins up and the flow adjusts to the constant forcing. The upstream acceleration of the flow was present also in this experiment. Real steady-state flow is, however, never reached in this simulation. This is believed to be due to the lateral boundary formulation applied in the model. This allows for perturbations in the form of gravity waves generated inside the limited domain to propagate out of the model also at inflow points. Since the model is not nested within a larger model, no information on changing the background conditions on a larger scale is imported, and thus the information about the adjustment process is lost forever. It also means that the sloping inversion all the way from Hawaii to the California coast is not enforced.

e. The modeled diurnal cycle of the flow criticality

The previous sections show that the gradual upstream acceleration is present when the model is initialized with subcritical conditions using constant forcing at typical daytime conditions, with supercritical initial conditions and a realistic diurnal cycle, but also regardless of the length of the initialization period with subcritical initial conditions. This points to the presence of subcritical flow, at any time during the simulation, being an important factor. We thus study the diurnal cycle of the flow in exp_restart in more detail. In contrast to Ctrl, no gradual acceleration of the flow along the upstream coastline took place in exp_restart during the first hours of the simulation (not shown). We believe this is due to the fact that in exp_restart the initial flow is supercritical, while it is subcritical in Ctrl. The MBL wind speed in exp_restart actually decreases with time. This is, however, consistent with the decrease in the temperature contrast between land and sea since the surface temperature over land decreases with time during these first hours of the simulation. This decreases the local MBL slope from the west toward the coast, which is in agreement with observations (Beardsley et al. 1987). Thus, the along-coast wind speed decreases with time while the nearshore MBL depth increases slightly; the combined effect will be a decrease in Fr with time. At 0600 LST the flow is actually subcritical upstream of the bend, and the flow has become transcritical (Fig. 6).

In the early morning hours, the temperature contrast between land and sea increases again; this steepens the slope of the inversion toward the coast, which in turn decreases the nearshore MBL depth and enhances the alongshore wind speed. When studying north–south vertical cross sections of the along-coast wind component at different times, it turned out that the upstream flow below ≈500 m gradually accelerated from the bend and northward. This does not violate shallow water theory since the flow is now subcritical along the upstream coastline (see Fig. 6). As the wind speed increases and the MBL depth decreases with time, the flow slowly turns supercritical from the bend and upstream. Thus, the acceleration of the flow along the upstream coastline is established before the flow has turned supercritical again. The increase in the along-coast wind component and decrease in MBL depth with time are illustrated in Figs. 7a (at 0600 LST) and 7b (at 0900 LST). Also shown is the along-coast value of Fr (solid gray line). The north–south cross section is taken along the upstream part of the dashed line in Fig. 1b.

The diurnal variation of the flow criticality from exp_restart is illustrated in Fig. 8, where the phase speed (m s−1) is plotted versus the mean MBL wind speed (m s−1). To capture a mean state of the upstream conditions at a given time, the values are taken as a mean over a rectangular area close to the coast (see Fig. 6). The width of the area is ≈10 km and stretches from ≈25 km upstream of the bend (y ≈ 50 km) to the northern model boundary. The dashed line in Fig. 8 indicates where the two quantities are equal; that is, Fr = 1. Lower right from this line, the flow is supercritical. At initial time, the flow is supercritical, since it was initialized supercritical, but the mean MBL wind speed decreases with time while the phase speed increases. Between 2100 and 2200 LST the flow upstream of the bend turns subcritical and remains so until ≈0900 LST the day after. While the flow is subcritical, the gradual acceleration of the flow along the upstream coastline can take place without violating shallow water theory since shallow water gravity waves can now propagate upstream. When the flow becomes supercritical once again, it remains so until the end of the simulation (after 24 h, at 1800 LST the day after the simulation was initialized). Using single vertical profiles along the upstream coastline to calculate the phase speed and mean MBL wind speed did not alter the main characteristics of Fig. 8. Vertical profiles far upstream of the bend will, however, change the time of the transition from subcritical flow to supercritical flow and vice versa. Nevertheless, the diurnal cycle is also evident here, but the exact timing of the transitions in Fig. 8 should not be taken too literally.

The same quantities as in Fig. 8 are plotted for Ctrl and exp_const_T in Figs. 9a and 9b, respectively. The diurnal cycle of the criticality in Ctrl is clearly seen here (Fig. 9a). The initially subcritical flow turns supercritical around noon (1200 LST) the day after the simulation was started. In the evening (≈2200 LST) the flow becomes subcritical as in exp_restart. As the temperature contrast between land and sea increases in the morning, the flow once again turns supercritical until the end of the simulation at 1800 LST, 48 h after the initialization of the model. In contrast to Ctrl, approximately 6 h of integration is needed for exp_const_T to spin up into a supercritical flow, which then remains until the end of the simulation (Fig. 9b).

In exp_ini00 and exp_ini06, we found that the flow first became supercritical at the same local time, around noon the same day as they were initialized. Thus, we conclude that a sufficiently large temperature gradient between land and sea, present around noon, is required for the flow to turn supercritical.

From the analysis of simulations with a diurnal cycle it is clear that there is a substantial diurnal variation of the flow characteristics. This can be explained by the fact that the along-coast jet is driven by the cross-coast baroclinicity. The strength of the jet will therefore respond to any changes in this. Since the cross-coast baroclinicity varies with the surface temperature the nearshore flow will continuously undergo changes; thus the flow will never reach steady state.

4. Discussion

Based on observations, a number of studies of supercritical flow along the northern California coast have previously been carried out with different numerical models. A common thread for all these studies is that whether or not a shallow water model has been used, the model results have been interpreted within the reduced-gravity shallow water concept.

In our studies of supercritical coastal flows, the gradual acceleration of the flow along the upstream coastline has been a puzzle. Here we have shown that the criticality of the flow for typical summertime conditions along the California coast actually varies diurnally. The upstream acceleration is established during a subcritical phase of the simulation, and we therefore conclude that this feature does not violate the reduced-gravity shallow water theory. Previous studies have shown that subcritical flow along mountainous coastlines turns supercritical when rounding capes. Here we have found that the flow never reaches steady state since the cross-coast baroclinicity continuously varies. Thus, the flow is not only spatially but also temporally transcritical.

Figure 10 displays the time evolution of the mean MBL wind speed (m s−1) and the phase speed (m s−1) plotted in Fig. 9a during a 24-h period. It is clear that the variation of the mean MBL wind speed is greater than the variation of the phase speed. In particular, the mean MBL wind speed increases during daytime and decreases during nighttime, while no diurnal trend in the phase speed is apparent. To some extent this confirms the conclusion in Burk et al. (1999) that the presence of a low-level jet is more important than a sharp boundary layer inversion to generate a supercritical flow response around coastal bends.

The diurnal variation of the alongshore jet that Beardsley et al. (1987) suggest in their conceptual model of the MBL off northern California is reproduced here. In Fig. 11, vertical cross sections of potential temperature (K) and scalar wind speed (m s−1) from exp_restart are shown. The cross section is taken from west to east at y ≈ 75 km along the dashed–dotted line in Fig. 1b. In the morning at 0600 LST, a relatively weak along-coast jet is found far offshore (Fig. 11a). The inversion slopes gently toward the coastline and the nearshore MBL is ≈400 m deep. In the afternoon at 1500 LST the jet is considerably stronger and attached to the main coastal mountains (Fig. 11b). The slope of the inversion toward the coastline is steeper than in the morning, and the depth of the nearshore MBL is also smaller, as suggested by Beardsley et al. (1987). Burk and Thompson (1996) also addressed the diurnal variation in position and intensity of the jet. Their simulated change in position displayed similar behavior as here although the timing of the jet maximum occurred later than here (2200 LST versus 1700 LST in Ctrl).

The observed diurnal cycle of the flow criticality and the finding that the presence of subcritical flow at any time during the simulation being an important factor gave inspiration to repeat exp_restart, but holding the soil surface temperature constant. If the previous conclusions are true, we then expect that the flow should remain supercritical since the flow is supercritical at initial time and no diurnal cycle is present. It, however, turned out that whether or not the flow remains supercritical is crucially dependent on the time at which the soil surface temperature is fixed (not shown). The maximum baroclinicity occurs right after noon while the maximum flow speed occurs in the late afternoon in these simulations; thus there is a time lag between the warmest land surface temperature (maximum baroclinicity) and the maximum wind speed along the coast. When the temperature was fixed at its value at 1800 LST (same time as the model was restarted with modified initial conditions), the flow turned subcritical after a few hours of integration. When the temperature was fixed at its maximum value (just after noon), this resulted in an imbalance between the mass and wind fields. But, the flow remained supercritical until the end of the simulation (24 h). This illuminates that a large temperature gradient between land and sea is not only needed for the flow to turn supercritical, but also for the flow to remain supercritical, which is consistent with our previous findings; the flow criticality varies diurnally.

From the first part of this study (ST01) it was evident that the terrain to a great extent determines the flow characteristics. The time needed for an initially subcritical flow to become supercritical varies substantially with different terrain, using the same background conditions. Moreover, in an experiment using a coastline without the bend, the flow never turned supercritical. This highlights the importance of using a high-resolution model to simulate the flow along coastlines with complex terrain. Not resolving the terrain properly will result in an incorrect flow structure. On the other hand, it also illustrates the difficulty of initializing high-resolution models. The results here suggest that proper quasi-steady-state supercritical flow may be difficult to find in reality; transcritical flow should be much more prevalent. The reason for observed flows, like that on 7 June during CW96, to bear resemblance to such a state is likely to be the effects of the shape of the coastline and the terrain. When the coastline protrude into the flow, as in the experiments with simplified capes in ST01, it will become partially blocked. This blocking, not the supercriticality of the flow, will prevent the upstream acceleration to take place during the nocturnal part of the day, when the flow is typically subcritical.

The effect of two closely spaced capes on supercritical flow is investigated by Haack et al. (2001) in a real-data COAMPS forecast around Cape Blanco and Cape Mendocino. It is found that the flow responses around each cape cannot be analyzed independently. In terms of Fr, the Cape Blanco expansion fan (upstream of Cape Mendocino) obtains its largest magnitude during the nighttime. This, however, appears to be an effect of the blocking terrain at Cape Mendocino rather than a direct effect of the cross-coast baroclinicity, which determines the flow velocity along the coast and thus the flow criticality in the absence of blocking terrain.

Finally, deviations from the reduced-gravity shallow water concept will be present when fully stratified flows along mountainous coastlines are considered. One important issue highlighted in this study is the diurnal cycle of the flow criticality. Nevertheless, we still believe that the reduced-gravity shallow water concept is a useful analog to interpret observations and model results of flows along mountainous coastlines.

Acknowledgments

The authors are grateful for discussions and comments from Stephen Burk, Tracy Haack, and Branko Grisogono.

REFERENCES

  • Andrén, A., 1990: Evaluation of a turbulence closure scheme suitable for air-pollution applications. J. Appl. Meteor., 29 , 224239.

  • Beardsley, R. C., , C. E. Dorman, , C. A. Friehe, , L. K. Rosenfeld, , and C. D. Winant, 1987: Local atmospheric forcing during the Coastal Ocean Dynamics Experiment. 1. A description of the marine boundary layer and atmospheric conditions over a northern California upwelling region. J. Geophys. Res., 92 , 14671488.

    • Search Google Scholar
    • Export Citation
  • Burk, S. D., , and W. T. Thompson, 1996: The summertime low-level jet and marine boundary layer structure along the California coast. Mon. Wea. Rev., 124 , 668686.

    • Search Google Scholar
    • Export Citation
  • Burk, S. D., , T. Haack, , and R. M. Samelson, 1999: Mesoscale simulation of supercritical, subcritical, and transcritical flow along coastal topography. J. Atmos. Sci., 56 , 27802795.

    • Search Google Scholar
    • Export Citation
  • Dorman, C. E., , and C. D. Winant, 1995: Buoy observations of the atmosphere along the west coast of the United States, 1981–1990. J. Geophys. Res., 100 , 1602916044.

    • Search Google Scholar
    • Export Citation
  • Dorman, C. E., , D. P. Rogers, , W. Nuss, , and W. T. Thompson, 1999: Adjustment of the summer marine boundary layer around Point Sur, California. Mon. Wea. Rev., 127 , 21432159.

    • Search Google Scholar
    • Export Citation
  • Edwards, K. A., , A. M. Rogerson, , C. D. Winant, , and D. P. Rogers, 2001: Adjustment of the marine atmospheric boundary layer to a coastal cape. J. Atmos. Sci., 58 , 15111528.

    • Search Google Scholar
    • Export Citation
  • Enger, L., 1990: Simulation of dispersion in moderately complex terrain—Part A. The fluid dynamic model. Atmos. Environ., 24A , 24312446.

    • Search Google Scholar
    • Export Citation
  • Haack, T., , S. D. Burk, , C. Dorman, , and D. Rogers, 2001: Supercritical flow interaction within the Cape Blanco–Cape Mendocino orographic complex. Mon. Wea. Rev., 129 , 688708.

    • Search Google Scholar
    • Export Citation
  • Hodur, R. M., 1997: The Naval Research Laboratory's Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125 , 14141430.

    • Search Google Scholar
    • Export Citation
  • Pielke, R. A., 1984: Mesoscale Meteorological Modeling. Academic Press, 599 pp.

  • Rogers, D. P., and Coauthors. 1998: Highlights of Coastal Waves 1996. Bull. Amer. Meteor. Soc., 79 , 13071326.

  • Rogerson, A. M., 1999: Transcritical flows in the coastal marine atmospheric boundary layer. J. Atmos. Sci., 56 , 27612779.

  • Samelson, R. M., 1992: Supercritical marine-layer flow along a smoothly varying coastline. J. Atmos. Sci., 49 , 15711584.

  • Söderberg, S., , and M. Tjernström, 2001: Supercritical channel flow in the coastal atmospheric boundary layer: Idealized numerical simulations. J. Geophys. Res., 106 , 1781117829.

    • Search Google Scholar
    • Export Citation
  • Ström, L., , M. Tjernström, , and D. P. Rogers, 2001: Observed dynamics of coastal flow at Cape Mendocino during Coastal Waves 1996. J. Atmos. Sci., 58 , 953977.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., 1987a: A study of flow over complex terrain using a three-dimensional model. A preliminary model evaluation focusing on stratus and fog. Ann. Geophys., 88 , 469486.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., . 1987b: A three-dimensional meso-γ-model for studies of stratiform boundary layer clouds: A model description. Rep. 85, Dept. of Meteorology, Uppsala University, Uppsala, Sweden, 44 pp.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., . 1999: The sensitivity of supercritical atmospheric boundary-layer flow along a coastal mountain barrier. Tellus, 51A , 880901.

    • Search Google Scholar
    • Export Citation
  • Tjernström, M., , and B. Grisogono, 2000: Simulations of supercritical flow around points and capes in a coastal atmosphere. J. Atmos. Sci., 57 , 108135.

    • Search Google Scholar
    • Export Citation
  • Winant, C. D., , C. E. Dorman, , C. A. Friehe, , and R. C. Beardsley, 1988: The marine layer off northern California: An example of supercritical channel flow. J. Atmos. Sci., 45 , 35883605.

    • Search Google Scholar
    • Export Citation
Fig. 1.
Fig. 1.

(a) A side view of the northern California terrain with some reference points. (b) The idealized representation of the northern California terrain used in this study. Maximum height of the idealized terrain is 1.2 km. The dashed line from south to north and the dashed–dotted line from west to east indicate the location of cross sections for later reference

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 2.
Fig. 2.

Contour plots from Ctrl at 1500 LST of (a) MBL depth (m), (b) maximum MBL wind speed (m s−1), and (c) the Froude number

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 3.
Fig. 3.

Vertical cross section from Ctrl at 1500 LST of potential temperature (K, dashed lines) and scalar wind speed (m s−1, solid lines) taken along the dashed line in Fig. 1b

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 4.
Fig. 4.

Contour plot of the initial Froude number from exp_restart at 1800 LST

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 5.
Fig. 5.

Maximum MBL wind speed (m s−1) at 1500 LST from exp_restart

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 6.
Fig. 6.

Contour plot of the Froude number from exp_restart at 0600 LST. Dashed lines indicate the area over which the averages in Figs. 8–10 are calculated

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 7.
Fig. 7.

Vertical cross section from exp_restart of along-coast wind component (m s−1, solid) and MBL depth (dashed) taken along the upstream part of the dashed line in Fig. 1b at (a) 0600 LST and (b) 0900 LST. The solid gray line shows Fr along the cross section with values given on the right-hand side axis

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 8.
Fig. 8.

Diurnal cycle of shallow water gravity-wave phase speed and mean MBL wind speed for exp_restart. The numbers along the solid line show LST while the dashed line indicates where Fr = 1; subcritical area is upper left (corresponding to nighttime flow)

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 9.
Fig. 9.

Same quantities as in Fig. 8, but for (a) Ctrl and (b) exp_const_T. The numbers along the solid line in (b) show number of hours into the simulation

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 10.
Fig. 10.

Time evolution of the mean MBL wind speed and the phase speed from Ctrl along the upstream coastline

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Fig. 11.
Fig. 11.

Vertical cross sections of potential temperature (K, dashed lines) and scalar wind speed (m s−1, solid lines) from exp_restart, taken along the dashed–dotted line in Fig. 1b: (a) 0600 LST and (b) 1500 LST

Citation: Journal of the Atmospheric Sciences 59, 17; 10.1175/1520-0469(2002)059<2615:DCOSAC>2.0.CO;2

Table 1. 

Summary of the experiments conducted here

Table 1. 
Save