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  • View in gallery

    A portion of the ERS-1 SAR image of Chesapeake Bay, containing the signatures of nocturnal-drainage-flow-forced exit jets. For geographical reference, north is indicated and the Patuxent River is labeled. The jet simulated in section 5 is marked.

  • View in gallery

    Hypothetical basin and corresponding exit jet. Basin and jet parameters are indicated. The shaded area represents that part of the jet visible on the SAR image.

  • View in gallery

    (a) Vertical and (b) horizontal view of the model domain.

  • View in gallery

    Plot of speed vs alongcenterline distance for a typical simulated exit jet.

  • View in gallery

    Plan view of speed for a typical simulated exit jet. Contours are m s−1.

  • View in gallery

    Model-simulated maximum velocity vs predicted maximum velocity for the entire range of jet simulations.

  • View in gallery

    Model-simulated jet lengths vs predicted jet lengths for the entire range of jet simulations.

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An Analysis of Exit-Flow Drainage Jets over the Chesapeake Bay

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  • a Applied Physics Laboratory, University of Washington, Seattle, Washington
  • | b Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

Synthetic aperture radar has shown great promise in detecting surface roughness patterns generated by atmospheric and oceanic features. Those roughness patterns that are the result of sea surface wind stress may be analyzed and related to characteristics of the atmospheric boundary layer. Previously reported examples of detectable atmospheric signatures include gravity waves and Rayleigh–Benard convection in cold-air outbreaks. In this paper, the results from an analysis of an image that contains the signatures of nocturnal-drainage-flow-forced exit jets along the western shore of Chesapeake Bay is presented. A regression analysis is performed that links the length of the surface stress patterns associated with these exit jets to the geometry of their source basins. This analysis differs from previous drainage-flow studies in that a population of drainage flows of varying sizes is studied under identical synoptic conditions. This large sample size provides a unique opportunity to examine the role that topography plays in forcing this kind of flow.

To complement the observational study, a two-dimensional, shallow-fluid model is developed to simulate the drainage-flow exit jets once they leave their source basins. This model allows simulation of the behavior of these flows over the entire range of forcing values observed in the image. This kind of analysis provides physical insight into the dynamics of these hybrid flows and a basis for the development of a similarity theory that relates the physically significant forcing parameters to the characteristic length and speed scales of this phenomenon. The lack of in situ observations unfortunately prevents a direct comparison between model results and observations; however, the model is shown to give characteristic jet length scales that are in reasonable agreement with values obtained from the image analysis.

Corresponding author address: Dr. Nathaniel S. Winstead, University of Washington, Applied Physics Lab., 1013 NE 40th St., Seattle, WA 98105.

winstead@apl.washington.edu

Abstract

Synthetic aperture radar has shown great promise in detecting surface roughness patterns generated by atmospheric and oceanic features. Those roughness patterns that are the result of sea surface wind stress may be analyzed and related to characteristics of the atmospheric boundary layer. Previously reported examples of detectable atmospheric signatures include gravity waves and Rayleigh–Benard convection in cold-air outbreaks. In this paper, the results from an analysis of an image that contains the signatures of nocturnal-drainage-flow-forced exit jets along the western shore of Chesapeake Bay is presented. A regression analysis is performed that links the length of the surface stress patterns associated with these exit jets to the geometry of their source basins. This analysis differs from previous drainage-flow studies in that a population of drainage flows of varying sizes is studied under identical synoptic conditions. This large sample size provides a unique opportunity to examine the role that topography plays in forcing this kind of flow.

To complement the observational study, a two-dimensional, shallow-fluid model is developed to simulate the drainage-flow exit jets once they leave their source basins. This model allows simulation of the behavior of these flows over the entire range of forcing values observed in the image. This kind of analysis provides physical insight into the dynamics of these hybrid flows and a basis for the development of a similarity theory that relates the physically significant forcing parameters to the characteristic length and speed scales of this phenomenon. The lack of in situ observations unfortunately prevents a direct comparison between model results and observations; however, the model is shown to give characteristic jet length scales that are in reasonable agreement with values obtained from the image analysis.

Corresponding author address: Dr. Nathaniel S. Winstead, University of Washington, Applied Physics Lab., 1013 NE 40th St., Seattle, WA 98105.

winstead@apl.washington.edu

Introduction

Synthetic aperture radar (SAR) has been proven to be a useful tool for studying the vast array of phenomena that govern the behavior of the ocean mixed layer and the convective marine atmospheric boundary layer (CMABL) (Alpers et al. 1981; Beal et al. 1981; Vesecky and Stewart 1982; Gerling 1986; Sikora et al. 1997). This utility exists because the spatial variation of the backscattered intensity field depicted in SAR imagery is related directly to the variation in the roughness of the ocean surface on scales comparable to the wavelength of radiation transmitted by SAR (Allan 1983). The local amplitude of the surface waves that produce this roughness depends on a broad range of oceanic and atmospheric processes and their interactions (e.g., Elachi 1987). Because these waves are driven by the surface stress and are locally modulated by wave–current interactions and surfactant slicks, SAR images frequently reveal features related to oceanographic processes such as current boundaries, internal waves, or tidal flows over bathymetry and to variations in the surface stress caused by atmospheric processes.

The signatures of oceanographic processes in SAR imagery have been extensively studied since the launch of Seasat in 1978 (e.g., Beal et al. 1981; Gasparovic et al. 1988; Thompson and Jensen 1993; Nilsson and Tildesley 1995). The equally ubiquitous signatures of atmospheric processes have received much less attention until recent years, when it was recognized that atmospheric phenomena are important contributors to the observed variations in SAR backscatter intensity (Atlas 1994; Gerling 1986;Askari et al. 1993; Alpers and Brümmer 1994; Vachon et al. 1994; Sikora et al. 1995, 1997; Wackermann et al. 1996; Kalmykov et al. 1984; Velichko et al. 1989; Efremov et al. 1992; Trokhimovsky et al. 1994). These studies have shown that kilometer-scale eddies in the CMABL can be an important mechanism for feature generation in many SAR images. Additional features documented by SAR include mountain lee waves (Vachon et al. 1994) and nocturnal drainage flows (Alpers et al. 1998).

In this paper, a European remote sensor (ERS-1) SAR image that contains the signatures of nocturnal-drainage-flow-induced exit jets is analyzed to study the link between nocturnal-drainage-flow characteristics and basin geometry. Specifically, the length of the exit-jet signature on the ERS-1 image is linked to the geometry of the basin that is forcing the nocturnal drainage flow. To complement this observational study, a two-dimensional shallow-fluid model is developed, and its results then are used to find similarity relations for the exit-jet length and maximum velocity within the flow. These similarity relations provide valuable insight into the dynamics of nocturnal drainage flows as they spread onto flat terrain after leaving their source basins. They also provide a quantitative theory for predicting drainage-flow-exit-jet behavior from external parameter values.

Nocturnal drainage flows

Drainage flows occur most frequently during the evening and overnight hours when there is net radiational cooling (Whiteman 1990). This net radiation deficit creates cold, dense air near the surface. Over sloped terrain, gravity accelerates this air down the terrain gradient whence it either pools in deep valleys or flows out over flat terrain (Neff and King 1987; Whiteman 1990). Published studies of these flows are associated most often with mountainous terrain; however, because slopes as small as Δzx = 0.001, where z and x are vertical and horizontal distance, respectively, can generate drainage winds of 1–2 m s−1 (Brost and Wyngaard 1978; Mahrt 1981), nocturnal drainage flows are a common occurrence over virtually all land areas. The quantitative behavior of a particular nocturnal drainage flow depends on a variety of factors, including synoptic and mesoscale weather conditions and the geometry of the surrounding terrain. When nocturnal drainage flows that are generated within valleys exit their source basins and flow over flat terrain, they behave as exit jets with density current characteristics that include a leading front that propagates outward, typically at 2.5–3.5 m s−1 (Blumen 1984).

The relationship between ambient weather conditions and nocturnal-drainage-flow behavior has been extensively studied during the Atmospheric Studies in Complex Terrain (ASCOT) field experiments. The original ASCOT field experiments were performed during the summers and early autumns of 1979–81 (Orgill and Schreck 1985) to study the effect that mountain circulations have on the transport of power-plant emissions in the Geysers Geothermal Resource Area of northern California. The primary focus of these ASCOT studies was on determining the interactions of nocturnal drainage flow with synoptic and mesoscale processes. All of these studies found that, although nocturnal drainage flows occur almost every night, synoptic weather conditions, including cloud cover and ambient flow, can have a profound effect on their evolution (Fitzjarrald 1984; Horst and Doran 1986; Clements et al. 1989b; Whiteman 1990; Coulter and Gudiksen 1995; Fast et al. 1996). In particular, they found that the synoptic conditions most favorable for the formation of nocturnal drainage flows are clear skies and light winds. These synoptic conditions maximize radiational cooling, which is the dominant energy source for drainage flows.

Additional ASCOT studies have continued in the Colorado Rocky Mountains during the years following those initial studies (Clements et al. 1989a). Measurements from towers, tethersondes, ground-based sodars, and lidars located within major canyons in the ASCOT region have provided important insight into the evolution and vertical structure of these flows (e.g., Coulter and Martin 1986; Horst and Doran 1986; Dobosy et al. 1989; Clements et al. 1989b; Coulter and Gudiksen 1995; Doran 1996). These studies have provided an observational base from which information concerning the vertical structure, energetics, and dynamics of these flows inside the source basins can be extracted (e.g., McKee and O’Neal 1989; Whiteman et al. 1989). These studies indicate that nocturnal drainage flows are relatively shallow, with depths ranging from 10 to 400 m (Horst and Doran 1986; Neff and King 1987). They are stably stratified, and there is a residual mixed layer present above the drainage flow (Horst and Doran 1986; Clements et al. 1989a; Whiteman 1990). The maximum wind speeds obtained within the drainage flow vary with terrain characteristics and ambient weather conditions; however, alongvalley wind speeds of 1–8 m s−1 have been observed (Horst and Doran 1986; Clements et al. 1989a; Whiteman 1990; Doran 1996).

The final body of research into nocturnal drainage flows has centered around the modeling of these flows and their dynamic structure (e.g., Manins and Sawford 1979; Mahrt 1982; Doran 1996; Bader and Horst 1990;Fast 1995; Pan and Smith 1999). These studies ranged from a combination of observations with model simulations (Doran 1996; Manins and Sawford 1979) to a detailed analysis of the equations that govern slope flow within the valley (Mahrt 1982). Most of the modeling studies were performed inside the basins so as to capture the complex terrain interactions that dominate these flows during their intrabasin stage. Doran (1996), however, reported on mesoscale simulations of the flow behavior outside both the Coal Creek and El Dorado canyons in Colorado. The purpose of his study was to determine the effect that the drainage-flow-forced exit jets had on the dispersion and transport of the Rocky Flats Power Plant emissions. Most modeling studies have focused on the flow associated with individual basins such as these and have included the basin within the model simulation domain, thus requiring a model with complex terrain capabilities. Although models of this kind provide valuable insight into drainage flows while they are inside their source regions, the requirement for complexity limits their ability to simulate exit jets from a large population of basins. Such an ability is necessary for a comprehensive study of the dynamics of these flows after they leave their source basins.

Because they are initiated with both density and momentum differences from their surroundings, such drainage-flow-forced exit jets behave as hybrids of gravity currents (Simpson 1969, 1987) and conventional two-dimensional exit jets similar to those described by Tennekes and Lumley (1994). The gravity-current aspects of these flows can cause acceleration after the exit jet leaves the basin. This acceleration occurs as potential energy is converted to kinetic energy during the gravity-current collapse. The collapse also gives the drainage-flow exit jets a fan-shaped appearance when viewed from above. Although observations of such flows are relatively scarce, this behavior has been observed, both in Germany (Pamperin and Stilke 1985) and in the United States (Whiteman 1990). Pamperin and Stilke (1985) (in Whiteman 1990) found that drainage flows accelerated from 7 m s−1 inside a valley to a maximum speed of 13 m s−1 some distance beyond the mouth.

Image properties

The image analyzed is an ERS-1 (wavelength 5.6 cm) SAR image of Chesapeake Bay taken at 0300 UTC 9 May 1992. Figure 1 shows this image. North is 12.5° counterclockwise from up on the image, and the pixel spacing is 12.5 m. The features of interest are the areas of brightness extending eastward from the western shore of the bay. The population of such features contains a wide variety of shapes and lengths, but most of the structures have a sharp brightness cutoff at their eastern end, reflecting the minimum sensitivity of SAR. The length of these structures as measured along the centerline from the shore to this sharp reduction in brightness ranges from 251 to 4941 m. United States Geological Survey (USGS) 7.5-min topographic maps indicate that each of these areas of brightness is clearly linked to the mouth of one of the ravines, valleys, or creek beds that dominate the terrain along the western shore of Chesapeake Bay in the region imaged by SAR.

The spatial correspondence between the mouths of the basins along the western shore of Chesapeake Bay with the SAR image signatures strongly suggests that the terrain along the western shore is instrumental in forcing the winds responsible for these surface stress signatures on the image. Similar structures have been seen on SAR images of the eastern coast of Italy (Alpers and Gross 1998) and have been attributed to katabatic winds. It is our hypothesis that these signatures are the outflow signatures of nocturnal-drainage-flow-forced exit jets.

Synoptic weather conditions on 9 May 1992 were characterized by a weak pressure gradient, clear skies, and calm winds, conditions ideal for strong radiational cooling. Many observation stations around Chesapeake Bay were reporting nocturnal radiation fog at 0300 UTC, demonstrating that significant radiational cooling already was occurring at that time. The ambient potential temperature at the Patuxent Naval Air Station at the time of the image was approximately 285 K, 3 K lower than the previous daily observed maximum temperatures for the day. This temperature decrease, coupled with the weak ambient synoptic flow, indicates not only that conditions were right for nocturnal drainage flows but also that such flows would be relatively undisturbed by larger-scale circulations. Thus, terrain characteristics rather than ambient flow conditions will be the dominant determiners of the characteristics of each of the observed drainage flows.

Image analysis

The primary purpose of the observational study is to determine this relationship between exit-jet length and the geometric dimensions of the originating basins. In the population of exit jets captured by the ERS-1 SAR image, each exit jet is conclusively linked to its basin of origin by matching the latitude and longitude of the exit jet with the latitude and longitude of the mouth of the originating basin. A total of 17 exit jets are discarded from the sample either because they are associated with two adjacent basin mouths or are clearly affected by towns and other humanmade structures. The former are eliminated because of the uncertainty in source basin characteristics, the latter because the radiational cooling and surface roughness characteristics are likely to be significantly different from those of the remaining population.

For each of the remaining 31 exit jets, the length LJ of the brightness pattern associated with each exit jet is used to quantify the size of that feature. The jet length is affected by the sensitivity of the SAR instrument in that the length represents the distance beyond which the wind is not fast enough to generate sufficient roughness to provide backscatter power above the SAR instrument’s detection limit. As a consequence, there is a limiting isotach Uj beyond which SAR is not sensitive enough to detect the backscattered power (see Fig. 2). Under neutral conditions, this value is between 1.5 and 2.0 m s−1 (Stoffelen and Anderson 1993). Thus, the jet length reported here is the distance from the basin mouth to the furthermost detectable isotach.

To establish the quantitative relationship between the exit-jet length estimated from the image and the terrain characteristics of the originating basin, the following basin geometry parameters are tested as candidate predictors for exit-jet length: basin area A, gap-opening width d, gap-opening flow depth H, and headwater height HH. Each of these predictors is indicated in Fig. 2. The basin area was estimated from the USGS 7.5-min topographic map series by digitizing the map and calculating the area enclosed by the highest basin-bounding contour for each valley. The square root of the basin area also was included as a predictor to serve as an unambiguous measure of the basin length scale. Table 1 summarizes the range of values of the jet lengths and each of the candidate predictors.

A stepwise, multiple, linear regression analysis was performed to relate these basin statistics to the observed exit-jet lengths. Predictors were accepted or rejected using the t statistic (Walpole and Myers 1993). For a sample size of 31, the threshold value for acceptance at the 95% confidence level is 1.68. The results indicate that the square root of the basin area was linked most closely to the length of the jets (t statistic = 16.2). The gap width was also accepted as a predictor (t statistic = 3.8). The remaining predictors, including the raw basin area, gap depth, and headwater height, were rejected at this confidence level. The following regression equation results:
LJAd.
Equation (1) yields an adjusted variance R2 of 0.90, with a standard error of 214 m. The mean value of LJ as measured by the SAR image analysis is 886.7 m. This regression analysis indicates a strong relationship between the square root of basin area and exit-jet length. Thus, larger basins lead to longer jets. Although (1) may capture the general form of the relationship between basin characteristics and exit-jet length, the specific coefficients in (1) are weather, time, and instrument dependent and should not be applied generally.
This statistical relationship between the exit-jet signatures in Chesapeake Bay and the basin terrain characteristics can be interpreted based on the underlying physics of drainage flows. Inside the source region, the predominant energy source for the generation of the nocturnal drainage flow is the rate of radiational cooling (W m−2). For a given cooling rate, the volume production rate of cooled air per unit of surface area is given by
i1520-0450-39-8-1269-e2
where P is the production rate of cooled air per unit area (m3 s−1 m−2), ρ is the density of the air in the valley, Cp is the specific heat of air at constant pressure, and Δθ is the potential temperature change. Multiplying (2) by A yields
i1520-0450-39-8-1269-e3
the volume production rate of cooled air. If it is Assumed that all of the cooled air produced inside the basin flows out via the basin mouth, the rate at which this cooled air leaves the mouth of the basin must be equal to (3). This volume flow rate of cooled air from the basin is given by
MU0Hd,
where U0 is the drainage flow speed at the gap. Thus the following relationship links the production rate of cold air inside the valley to the outflow from the basin:
i1520-0450-39-8-1269-e5
With the assumption that ρ, Δθ, and are the same over all of the source regions, the basin-to-basin variability in U0 in (5) depends on A, d, and H. The fact that the regression analysis picked the square root of the basin area over the basin area suggests that jet length must depend nonlinearly on U0 and, possibly, the other parameters in (5). This nonlinearity is shown to be the case in the similarity theory developed in section 6.

Modeling study

To gain insight into the dynamics of drainage flows once they leave their basins of origin, a two-dimensional, reduced-gravity, shallow-fluid model with bulk aerodynamic surface drag is used to simulate a large and diverse population of these features. This kind of model was selected because it captures the essential physics, as demonstrated below, and has been shown in previous modeling studies to capture the physics of these kinds of flows (Pan and Smith 1999; Mahrt 1982) without being computationally intensive. Thus, it can be used to generate a large population of exit jets for a broad range of external parameter values. A large population is needed to provide the quantitative database required for development of similarity relations for exit-jet length and speed. The model equations consist of a u (east–west) momentum equation, a υ (north–south) momentum equation, and a continuity equation [see, e.g., Pielke (1984) for a derivation of the reduced-gravity system of equations]:
i1520-0450-39-8-1269-e6
where K is the horizontal diffusion coefficient, t is time, w is vertical wind speed, h is the depth of the lower fluid, the primes on the wind components denote turbulent fluctuations, and g′ is reduced gravity:
i1520-0450-39-8-1269-e9
with θ1 being the temperature of the lower fluid, θ2 being the temperature of the upper fluid, and g being gravity. The turbulence flux convergence terms in (6) and (7) are parameterized using a bulk aerodynamic drag law (Stull 1988) as
i1520-0450-39-8-1269-e10
where CD is the drag coefficient, and |ũ| is the wind speed. Here, CD is set to a constant value of 0.0013 after Fairall et al. (1996). Note that (6)–(8) assume no ambient flow, and, hence, the velocities and heights in the above equation set are perturbation quantities. The model uses the Arakawa-C staggered grid (Arakawa and Lamb 1977) system, a centered-in-time, centered-in-space finite-difference integration scheme (Haltiner and Williams 1980; Pielke 1984). An Asselin time filter is used with γ = 0.25, where γ is the time filter parameter (Asselin 1972).

A second-order horizontal diffusion term is included to control numerical instability (Haltiner and Williams 1980; Pielke 1984). The diffusion coefficient is set to 20 m2 s−1, the minimum value found to yield computational stability. This value is of the same order as a typical value of HU0 for the simulations performed here. Thus, internal, horizontal diffusion is not negligible. Several experiments were performed for different values of K to determine the effect that the diffusion term has on the resulting length and maximum velocity of the exit jet. It was found that, for basins that were less than 8Δx wide (160 m for all simulations reported here), exit-jet lengths and maximum velocities varied by as much as 10% as K changed by 50%. For basins 8Δx and wider, variations in jet length and maximum velocity were less than 0.5% through the same range of K. As a result, all simulations reported here were performed on basins 8Δx and greater and thus were negligibly sensitive to K.

The domain of the model is shown in Fig. 3. Figure 3a depicts the vertical structure of the model, and Fig. 3b depicts the horizontal domain as a 3-km × 3-km square with open boundaries on the east, north, and south, and a rigid wall with a gap representing the basin entrance on the west. The horizontal grid spacing is 20 m for all of the exit-jet simulations. The model is configured with user-specified values for the inflow velocity in the gap, the gap width, the potential temperature difference, and the flow depth at the gap. The model then is integrated until it reaches steady state, and the resulting wind field is analyzed.

To verify that the model is capable of accurately reproducing gravity current propagation behavior, the height field was initialized with a positive height perturbation (domain wavenumber of 0.5). This perturbation had an amplitude of 30 m and had a sinusoidal shape in the x direction. The perturbation was constant in the y direction. The initial velocity fields were zero. The model then was integrated, and the propagation behavior of the height perturbation was examined as its leading front propagated toward the boundaries. Empirical studies in wave tanks and observations in the atmosphere (Keulegan 1957, 1958; Simpson 1969) have shown that the phase speeds c of the resulting density currents obey the following law:
i1520-0450-39-8-1269-e12
The parameter k has a value of 0.78 for oceanic salinity fronts (Keulegan 1957, 1958). In the atmosphere, the value of k is less certain; however, Simpson (1969) summarized observations that indicate that k ranges from 0.38 to 0.9 and presented laboratory results that indicated a value of 0.78. The phase speed given by the model for a g′ of 0.17 m s−2 is approximately 1.8 m s−1, giving a value of 0.79 for k. The value of g′ used in this calculation was determined from a Δθ of 5 K and a θ2 of 285 K. Thus, the model accurately captures gravity current propagation behavior observed in the laboratory, atmosphere, and ocean. This success is important, because drainage-flow exit jets have a significant gravity current component.

The current application of this model is twofold. First, it is used to demonstrate that basins of the size found along Chesapeake Bay are capable of producing SAR-detectable exit-jet signatures and that the length and maximum speed of the resulting exit jet are within the range of both the drainage flows observed in the image and those reported elsewhere. Second, and key to the remainder of this study, the model is used to generate a large sample set of exit jets for conditions that span the physically realistic controlling parameter space. This dataset is used to analyze the dynamics of these flows and leads to similarity theories for the maximum velocity that the drainage flow attains and for the length of the exit jet.

To demonstrate that the basins along the western shore of Chesapeake Bay are sufficiently deep to generate SAR-detectable nocturnal-drainage-flow signatures, basin parameter values that correspond to one of the more typical Chesapeake Bay drainage basins were used to initialize the model. The inflow depth of the gap was 18.3 m and the gap width was 200 m. This basin is indicated in Fig. 1. Because there were no in situ observations of the wind field inside these valleys, the actual inflow velocity is not known. Therefore, (12) was used to provide an estimate of U0, which is 1.07 m s−1. The value of g′ used in this calculation was 0.17 m s−2, the same value used above. An along–centerline plot of the resulting steady-state exit-jet speed is shown in Fig. 4. Initially, there is acceleration as the flow collapses and spreads laterally. In this case, the maximum velocity within the exit jet is 2.32 m s−1. Thereafter, the speed decreases rapidly with distance along the centerline. The ERS-1 satellite is capable of detecting the backscattered signal from the surface roughness generated by winds of roughly 1.5–2.0 m s−1 (Stoffelen and Anderson 1993). These isotachs yield lengths of 881 and 541 m, respectively, values well within the observed range of jet lengths from the ERS-1 image. The maximum velocity of 2.32 m s−1 is also within the range of speeds for nocturnal drainage flows as reported in previous studies. Although this value is on the slow end of the 1–14-m s−1 range reported in the literature, this result is not surprising given the relatively shallow depth of the Chesapeake Bay basins.

Figure 5 shows the plan view of wind speed over the model domain for this exit jet. The shape of the exit jet is similar to those of the exit jets seen on the ERS-1 image (Fig. 1). The contours are labeled every 0.3 m s−1. The lateral spread of the gravity current and the subsequent acceleration of the flow are indicated by the widening of the jet with downstream distance from the mouth. SAR-detected exit jets also show this characteristic, although there is considerable variability in the amount of downstream broadening for the Chesapeake Bay exit jets. A similar range of exit-jet shapes is found in the isotachs from the entire range of model runs, which suggests that the observed shape depends on the ratio of the maximum jet speed to the SAR sensitivity limit. Second-order effects of terrain details also may contribute.

Exit-jet dynamics

In this section, the problem of exit-jet dynamics is addressed by using results from a large set of simulations generated by the model described above. The goal of this analysis is to develop similarity theories for exit-jet length and intensity based on the controlling external parameters.

The first step in this analysis thus is to determine which external parameters govern the internal behavior of drainage-flow exit jets. The two internal parameters for which a similarity theory is sought are the exit-jet length LJ and the maximum centerline velocity Umax. The former is chosen for its direct link to the basin observations discussed in section 4; the latter is chosen to be a velocity scale that represents the speed (and hence intensity) of the flow. Because of the dependence of LJ on the isotach used to define LJ, that theory also can be used to predict the decay of centerline wind speed beyond the point at which Umax occurs.

The controlling external flow parameters must cover both aspects of drainage-flow exit jets: behavior at the gap and subsequent gravity current effects. These controlling external parameters can be grouped according to which aspect of the physics they describe. The external parameters that are related directly to gravity current dynamics include H and g′ (Holton 1992). The parameters that are related to traditional exit jet dynamics are d and U0 (Tennekes and Lumley 1994). Last, UJ is included because it is intrinsic to the definition of exit-jet length.

From these controlling external parameters, the following dimensionless products are derived via the Buckingham pi theorem (Kundu 1990):
i1520-0450-39-8-1269-e13
Both similarity relations for the internal parameters will be functions of combinations of these external dimensionless products as required by the Buckingham pi theorem (Kundu 1990).

Maximum-velocity similarity theory

The following dimensionless product contains the first internal parameter Umax:
i1520-0450-39-8-1269-e16
The similarity relation for Π4 must be some combination of Π1, Π2, and Π3. The resulting functional form is, moreover, subject to physical limits based upon the asymptotic behavior of drainage-flow exit jets. These limits provide adequate constraints on the functional form to yield a relationship that fits the data well.

The first limit is that as U0 approaches infinity Umax also must approach infinity. The necessity of this limit is obvious; however, the behavior of Umax as U0 approaches 0 is not as simple. If there is no gravity current contribution to the flow, then as U0 approaches 0 Umax also approaches 0. This limit applies only to pure momentum-forced exit jets. For the hybrid flows described here, as U0 approaches 0 Umax must remain a function of gH to capture the conversion from potential to kinetic energy as the flow layer collapses. The similarity relation for maximum velocity must capture this behavior.

The second important limit is that as gH approaches infinity Umax also approaches a constant value beyond which a hydraulic jump will occur. In this modeling study, hydraulic jumps were not simulated, and, hence, this behavior is not captured in the similarity theory reported here. Clearly, because hydraulic jumps do occur in the atmosphere, it will be necessary in future work to expand the similarity theory to incorporate this kind of behavior. The opposite limit for gH approaching 0 is less simple. To determine this limit, it is necessary to return to first principles. If one considers the forces that dictate Umax for a steady-state exit jet, it is found that the hydrostatic pressure gradient force is opposed by friction. If there is no gravity current component to the flow (i.e., no hydrostatic pressure gradient), then the friction always will act to slow the jet, and the maximum velocity will be U0. In contrast, if gH is nonzero yet still very small, the gravity current acceleration is much smaller than frictional deceleration. In that case, the initial velocity is still the maximum. This reasoning suggests that there must be sufficient gravity current acceleration to exceed frictional deceleration for Umax to exceed U0. Thus, the appropriate limit is that as gH approaches 0 Umax reaches U0 for some nonzero values gH.

The final important limiting behavior that must be captured is the behavior of Umax as the gap width increases relative to the depth of the flow. As the gap width approaches infinity the flow becomes a quasi-one-dimensional gravity current. Thus, as d approaches infinity the dependence of Umax on the width of the gap must disappear. The opposite limit is somewhat more complicated. Clearly, if the gap width is identically 0, there can be no jet and Umax must be 0. For narrow, deep gaps, however, the lateral spread of the flow as the gravity current exits the basin contributes to the value of Umax. Thus, although the limit of Umax as d approaches 0 is 0, this limit is reached via a discontinuity. The appropriate behavior is thus for the gap width–to–gap height ratio to remain significant as width approaches 0.

A functional form that satisfies the above limits is given by the following equation:
i1520-0450-39-8-1269-e17
Equation (17) depends only on the gravity current speed gH, the basin-mouth aspect ratio d/H, and the initial velocity at the mouth of the basin. Thus, (17) indicates that hybrid gravity current exit-jet flows attain a maximum velocity that is a function of both momentum jet and gravity current dynamics. The U0 term represents the contribution of the initial momentum, and the more complex second term represents the acceleration from this value caused by gravity current dynamics.

To determine the coefficients and exponents in (17), the shallow-fluid model described above was used to generate a large and diverse population (225 simulations) of gravity current exit jets. For each of the external nondimensional products described above, the parameter values were distributed over a large range of external parameter space. Histograms of the dimensionless external parameters were examined to ensure that the parameter space was spanned adequately. A realistic range of g′ was found by varying Δθ from 1 to 5 K. The velocity at the gap was found by applying the gravity current equation (12). Additional runs, with U0 being 0.5 and 1.0 m s−1 slower and faster than the gravity current speed, also were included. Because of the relatively low gravity current speeds, this range of values proved sufficient to capture the jet-induced departures from pure gravity current behavior. From these simulations, the maximum centerline velocity was recorded for each simulation.

The coefficients in (17) were determined by applying a nonlinear least squares fit to this model-generated dataset. This least squares optimization was implemented using a robust genetic algorithm based on our modification of Eschelman and Schaffer (1993), as described by Babb et al. (1999). The resulting equation, including the coefficients and exponent values, is
i1520-0450-39-8-1269-e18
Figure 6 shows a plot of observed maximum velocity versus the similarity-predicted maximum velocity. Equation (18) explains 99.7% of the variance, with a mean error of 0.05 m s−1 (1.7% of the average value of Umax in this dataset).

The resulting form indicates approximately equal contributions from the initial momentum of the exit jet and the gravity current aspect of the jet. The second term has very little impact on the maximum velocity for the range of gap widths examined in this study. This effect is strongest for narrow gaps and becomes negligible for wider gaps.

Exit-jet-length similarity theory

The second internal parameter that we wish to express as a function of the external parameters discussed above is the exit-jet length, represented by the following dimensionless product:
i1520-0450-39-8-1269-e19
Again with invocation of the Buckingham pi theorem, the similarity relation for Π5 must be a function of some combination of Π1, Π2, and Π3. Application of the methods described in the previous section yielded a similarity theory in which considerable functional complexity was required to handle the limiting case where U0gH. Because this complex behavior already is captured in the similarity relation for Umax, Π4 can be used in place of Π2 to yield a much simpler form of the LJ similarity relation without loss of generality. This form also obeys all of the required physical limits and fits the model results very well. This simpler form is used; however, if one desires to express LJ solely in terms of external parameters, one need only substitute from the Umax similarity relation [(18)].

Just as in section 6a, the functional form for jet length is subject to limits based upon asymptotic behavior. First, as UJ approaches infinity, exit-jet length must approach 0, because the maximum speed must be finite. Conversely, as UJ approaches 0, exit-jet length must become infinite (assuming that Umax > 0). An additional limit is that as either d or H approaches infinity the exit-jet length also must approach infinity. This limit is a consequence of the outflow arguments presented in section 3c, because an infinite width or an infinite depth implies an infinite outflow rate. Therefore, as the outflow rate increases, exit-jet length also must increase. Conversely, as either d or H approach 0, the exit-jet length also must approach 0. The final limit is that as Umax approaches infinity jet length also must approach infinity. Similarly, if Umax approaches 0, jet length also must approach 0.

The following functional form obeys these physical limits and so serves as a candidate similarity relation for exit-jet length:
i1520-0450-39-8-1269-e20

The same procedure is used to fit the coefficients in (20) as was used to fit those in (18). The jet length was measured along the centerline from the mouth of the gap to several downstream isotachs (5.0, 4.0, 3.0, 2.5, 2.0, and 1.5 m s−1). A total of 598 length measurements are obtained in this manner.

The resulting similarity relation, including the coefficient and exponent value, is
i1520-0450-39-8-1269-e21
Figure 7 shows the actual jet lengths versus the similarity-predicted jet lengths; 99.7% of the variance of exit-jet length is explained by (21), with a mean standard error of 27 m, approximately 1.6% of the average jet length.

Each of the terms in (21) provides valuable physical insight into the dynamics of gravity current exit jets. The first term indicates that the downstream length of exit jets depends on the aspect ratio of the basin mouth. The greater the width-to-depth ratio, the longer the jet, an indication of the role played by downstream gravity current collapse in determining jet length. All of the basin mouths in the modeling study were wider than they were deep so this term was always greater than 1. The second and fourth terms capture the sensitivity of the jet length to its own definition. Here UJ can be adjusted to reflect either instrument sensitivity or an isotach of particular interest.

Another interesting aspect of the dynamics of these hybrid flows is the relationship between lateral spread and downstream length. In particular, as Δθ increases, g′ increases, enhancing the hydrostatic, horizontal pressure gradient. Interestingly, the model results indicated that as the Δθ increases the jet length decreases, all else being equal. This finding suggests that as Δθ increases the primary effect is to increase the lateral spread at the expense of downstream length.

Sensitivity tests

Although the above similarity relations capture the behavior of the velocity and length scales associated with these hybrid density current, momentum-jet flows, the actual utility of these relations for the purposes of predicting exit-jet characteristics depends on the sensitivity of (18) and (21) to the external parameter values. If the relations are overly sensitive to errors in the external parameter measurements, operational use of (18) and (21) may be limited. For each similarity relation, sensitivity tests are performed to determine the accuracy to which each external parameter must be measured still to explain 90% of the variance of Umax and LJ, respectively. These tests were performed by allowing for the addition of random fluctuations of each of the external parameters and then examining the effect on the adjusted R2 value for the prediction of the internal variables from the similarity relations.

The results indicated that one needs to know the external parameters U0, d, H, and Δθ to within only approximately 30% still to explain 90% of the variance in maximum velocity. A total of 10 trials (of 225 cases) were performed, allowing for 30% fluctuations in each of the four parameters in order to determine the magnitude of the mean error introduced by these fluctuations. The mean error in maximum velocity prediction for these trials was 0.26 m s−1, indicating an increase in the mean error to 10%. Similar trials were performed to determine which external parameters introduce the most error into the predictions for maximum velocity. As expected from the exponent in (18), it was found that the gap width has little or no effect on the prediction for maximum velocity, but the remaining variables are significant.

Two separate sensitivity tests for the length predictive equation were performed. First, the observed maximum velocity from the model was replaced with the predicted maximum velocity from (18) to determine the error induced in predicted lengths when the maximum velocity is not known ahead of time and must be predicted from the external variables described above. Results from this sensitivity test indicated that, although the mean error in jet length did double from 27 to 48 m, the percentage of variance explained decreased by less than 0.5%. This result suggests that one still can obtain reasonably accurate predictions of jet length without observations of the maximum velocity. Thus, knowledge of the flow properties at the gap is sufficient to achieve reasonable values of the jet length downstream.

The second sensitivity study performed was similar in nature to that performed on (18). Random fluctuations of 10%, 7%, and 5% were allowed for each of the external parameters, d, H, Δθ, UJ, and Umax. The results indicated that (21) is very sensitive to random fluctuations in each of the external parameters except H. In particular, fluctuations of 7% are sufficient to reduce the percentage of variance explained to 90%. Unfortunately, the variables responsible for the majority of this sensitivity are the two velocity scales, UJ and Umax. The limiting isotach is not likely to be a problem, because one simply may choose a downstream isotach and predict all length scales in terms of one limiting isotach; however, the sensitivity to the maximum velocity is much more difficult to overcome. Fortunately, the fact that (18) is not very sensitive to external parameters suggests that it actually is better to predict the maximum velocity of the flow rather than to try to measure it directly.

Conclusions

An analysis of nocturnal-drainage-flow-forced exit jets off the western shore of Chesapeake Bay was presented. A regression analysis of 31 such structures indicated that 90% of exit-jet length could be accounted for by the square root of the area of the basin generating the flow and the gap width. This relationship between basin geometry and exit-jet length indicates that larger basins generate longer exit-jet signatures but that the relation is nonlinear. Wider gap widths also contribute, but much less significantly.

In addition to the observational study, a two-dimensional, shallow-fluid, reduced gravity model with surface drag was used to simulate exit jets. It was shown that this model incorporates physical descriptions sufficient enough to reproduce accurately both previously published gravity current propagation behavior and these observed jet structures. The model then was used to generate a large population of sample exit jets spanning the entire range of realistic initial values for the controlling external parameters of such jets. Dimensional analysis and asymptotic behavior analysis produced similarity relations for the length and maximum centerline velocity of such jets. For exit-jet length, the similarity relations explained 99.7% of the variance of the model-simulated jet length. For maximum velocity, the similarity relation also explained 99.7% of the variance of the model results. Although these results are promising and indicate that the form of the similarity relations adequately captures the behavior of these flows, an extensive field project combining in situ measurements with coincident SAR overpasses is necessary to determine the ultimate, practical utility of these relations. As a first step toward this goal, a sensitivity analysis of the similarity relations was performed.

The sensitivity of the two similarity relations was examined to determine the practical utility of the relations for operational use. It was found that the accuracy of the maximum velocity similarity relation was not particularly sensitive to the accuracy of the external parameters. Conversely, the length similarity relation was found to be very sensitive to the value of the two velocity input parameters. It also was shown that this sensitivity could be alleviated by using the maximum velocities obtained from the predictive equation rather than the observed values. Thus, the similarity theories provide a practical means of predicting drainage-flow exit-jet length and intensity given gap width, gap depth, gap exit velocity, and reduced gravity (g′).

Acknowledgments

The authors would like to thank Donald Thompson and Robert Beal for their valuable technical discussions and for the processing of the ERS-1 SAR image presented in this paper. Presentations by numerous ASCOT researchers helped to motivate this work. This work was funded through ONR Grants N00014-92-J-1585, N00014-96-1-0375, and N00014-96-1-0978.

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Fig. 1.
Fig. 1.

A portion of the ERS-1 SAR image of Chesapeake Bay, containing the signatures of nocturnal-drainage-flow-forced exit jets. For geographical reference, north is indicated and the Patuxent River is labeled. The jet simulated in section 5 is marked.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 2.
Fig. 2.

Hypothetical basin and corresponding exit jet. Basin and jet parameters are indicated. The shaded area represents that part of the jet visible on the SAR image.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 3.
Fig. 3.

(a) Vertical and (b) horizontal view of the model domain.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 4.
Fig. 4.

Plot of speed vs alongcenterline distance for a typical simulated exit jet.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 5.
Fig. 5.

Plan view of speed for a typical simulated exit jet. Contours are m s−1.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 6.
Fig. 6.

Model-simulated maximum velocity vs predicted maximum velocity for the entire range of jet simulations.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Fig. 7.
Fig. 7.

Model-simulated jet lengths vs predicted jet lengths for the entire range of jet simulations.

Citation: Journal of Applied Meteorology 39, 8; 10.1175/1520-0450(2000)039<1269:AAOEFD>2.0.CO;2

Table 1.

Summary of basin characteristics and jet lengths. The jet lengths were measured from the ERS-1 SAR image (Fig. 1), and the basin characteristics were measured from the USGS 7.5-min topographic maps of the Chesapeake Bay region.

Table 1.
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