Flow Patterns and Drag in Near-Critical Flow over Isolated Orography

Transcription

1 1DECEMBER 004 JOHNSON AND VILENSKI 909 Flow Patterns and Drag in Near-Critical Flow over Isolated Orography E. R. JOHNSON AND G. G. VILENSKI Department of Mathematics, University College London, London, United Kingdom (Manuscript received 17 June 003, in final form 4 June 004) ABSTRACT Since early manned space flight orographically forced cloud patterns have been described in terms of the single isolated shock structure of shallow-water flow or, equivalently, compressible fluid flow. Some of these observations show, behind an initial bow wave, a series of almost parallel wave crests. This paper considers the simplest extension of shallow-water theory that retains not only nonlinear steepening of waves but includes departures from hydrostatic balance, and thus wave dispersion, showing that the single shocks of shallow-water theory are transformed into multiple parallel finite-amplitude wave crests. The context of the discussion is the forced Kadomtsev Petviashvili equation from classical ship wave dynamics, which plays the same role in twodimensional near-critical fluid flow as the more familiar Korteweg de Vries equation in one-dimensional flow. The drag and flow regimes in near-critical flow over isolated orography are described in terms of the three governing parameters of the flow: the deviation of the flow speed from critical, the strength of nonhydrostatic effects, and the strength of orographic forcing. 1. Introduction Among the many observations of cloud patterns near orography some stand out as showing stationary disturbances upwind of the orography. Figure 1 shows a National Aeronautics and Space Administration (NASA) photograph of Guadalupe Island, Baja California, taken during the Gemini-V flight and described in Badgley et al. (1969). They note that a low layer of stratocumulus cloud is moving at 6 10 kt (3 5 m s 1 ) past the island whose peaks reach 4500 ft (137 m) and thereby project through, and interfere with, the cloud layer. They note that a bow wave spreads from the north end of the island, similar to waves formed by a ship moving through water, and further point out that, as these cloud features were photographed during four Gemini missions, they must be considered climatic features of the Guadalupe marine atmosphere, whose fluid dynamic details must be investigated for proper analyses of atmospheric and ocean flows. Figure shows a closer view of the island, from a NASA photograph, in August The parabolic bow wave is again clearly visible, bearing out the Badgley et al. observation that it is a climatic feature. Burk and Haack (000) describe similar wave clouds extending away from the Monterey Big Sur coastline of central California, noting that it is the orographic Corresponding author address: Dr. E. R. Johnson, Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom. e.johnson@ucl.ac.uk forcing and stationary appearance in Geostationary Operational Environment Satellite (GOES) image loops that set these waves apart from propagating wave phenomena. They present a schematic description based on isolated single, attached and detached, shocks forced by obstacles in supercritical flows. A more detailed model of these phenomena is given by Jiang and Smith (000) in the context of shallow-water flow over isolated orography, following discussion of atmospheric flows by Schär and Smith (1993a,b), Grubisic et al. (1995), Smith and Smith (1995), Smith et al. (1997), and Pan and Smith (1999). Jiang and Smith note that these models can capture such phenomena as long wakes in the lee of obstacles, quasi-steady vortices, and vortex shedding, observing that with realistic parameters a single-layer model as a representation of stratified airflow has even made some reasonable quantitative predictions such as vortex shedding period (Schär and Smith 1993b) and wake length. These studies assume that the environmental flow is subcritical, F 1 (where F is the Froude number, the ratio of the flow speed to the free-surface gravity wave speed), and one of the aims of Jiang and Smith (000) is to extend these studies to supercritical flow, F 1, noting that Samelson (199) and Samelson and Lentz (1994) have treated the shallow marine atmospheric boundary layer under a low-level inversion along the Californian coast as supercritical flow with a sidewall. Provided that the Froude number is sufficiently far from unity, flow patterns and drag for sufficiently low obstacles can be found by linearizing the flow field about an undisturbed upstream flow. However, the linear sys- 004 American Meteorological Society

2 910 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 FIG. 1. Low-layer stratocumulus cloud moving at 6 10 kt past Guadalupe Island, Baja California, Mexico. A bow wave spreads from the north end of the island. To the east of the island, behind the initial bow wave, can be seen three distinct crests parallel to the leading bow wave. tem is resonant at F 1 and the linear response is infinite there. Further, close to this critical flow the response to orographic forcing is sufficiently large to invalidate the linearization of the flow. Jiang and Smith (000) note that there are then two possible approaches to describing the flow in this transcritical regime. They use a hydraulic theory, solving the fully nonlinear shallow-water equations numerically and incorporating wave-breaking regions as discontinuities governed by the conservation of mass and momentum. This leads to flow patterns with isolated attached and detached shocks as in the schematic description of Burk and Haack (000). However, the GOES observations and nonhydrostatic numerical computations in Burk and Haack (000) show that the wave pattern does not consist of an isolated line, but of an initial line followed by a set of parallel lines of cloud. Similarly, Fig. 1 shows, to the east of Guadalupe Island three further lines of cloud parallel to the initial bow wave, and Fig. shows behind the initial bow wave a second wave crest extending across the flow parallel to the bow wave but separated from it by clear air. It is the purpose of the present paper to show that, by extending the shallow-water equations to include weak nonhydrostatic effects the single isolated shocks familiar from ship wave analogies, can split into multiple parallel crests similar to those observed. Jiang and Smith (000) point out, but do not pursue, this complementary approach of expanding the full nonhydrostatic Eulerian equations in the two small parameters A/H and H/L (for typical wave amplitudes A, depths H, and horizontal length scales L) and keeping only leading order terms. For one-dimensional flow over ridges this leads to the Korteweg devries equation and in two-dimensional flow over isolated orography to the two-dimensional KdV or Kadomtsev Petviashvili equation (Kadomtsev and Petviashvili 1970; Akylas 1994; Baines 1995; Johnson 1997). As originally derived, the FIG.. A similar observation of flow past Guadalupe Island. Here the parabolic shape of the initial bow wave is clear. Behind the initial wave is a separate wave parallel to the bow wave and elongated across the flow. forced Kadomtsev Petviashvili (fkp) equation gives the vertical the displacement of the free surface of a shallow flow, or the interface of a two-layer flow, but equally, following Grimshaw and Smyth (1986), it gives the horizontal and temporal variation of the amplitude of a vertical mode of a continuously stratified flow of finite depth. It can be written as tx xx ( ) xx xxxx yy M xx. (1) Nondimensionalization has been chosen here, without loss of generality, so that the coefficients of the nonlinear term and the cross-stream term are both unity. The parameter measures the difference between the speed of linear long gravity waves c and the speed of the oncoming flow U, taken to be from right to left, that is, in the negative x direction. When c U, is positive and the oncoming flow is subcritical; when c U, is negative and the oncoming flow is supercritical. The (always positive) parameter measures the departure from hydrostatic balance and thus the importance of dispersion in the flow. The function M(x, y) gives the shape of the orography, and M measures its height. Here the orography is taken to be isolated, so 0asx, y. The flow is taken to be initially undisturbed and to remain undisturbed at sufficiently large distance, as y ; (x, y, t) 0 as x ; (x, y, 0) 0. For a general stratified flow the coefficients that appear in the equation can be related to various integrals of the oncoming flow (Grimshaw and Smyth 1986).

3 1DECEMBER 004 JOHNSON AND VILENSKI 911 However, as noted by Baines (1995) and emphasized by Johnson (1997) and others for the KdV, the solutions of this equation should be, in some sense, generic for weakly forced flows near criticality, so the emphasis in the present work is to delineate the qualitative behavior of solutions to this equation for near-critical flows and to show that observed flows resemble these. The most important parameter in determining flow patterns remains, as in hydrostatic flow, the Froude number. Burk and Haack (000) discuss the difficulties involved in estimating Froude numbers for realistic flows and present a method that proves effective for their numerical computations. Placing a single numerical value on nonhydrostatic effects in an observed flow is similarly not straightforward; but in this case the simulations below show that even quite large changes in the strength of dispersion do not result in large qualitative changes in flow patterns. Sufficiently near to criticality, low orography can cause large flow perturbations upstream and to the side of the obstacle. Thus, it seems possible that for some flows the complications of separation, strong wakes, and wave breaking observed near high abrupt topography may be absent. In Figs. 1 and the dynamics near the island appear wavelike and it is only many island diameters downstream that a vortex wake becomes apparent. Previous analysis of (1) has been in the context of ship waves, and much of that has been restricted to examining flows in channels. However, there are significant discrepancies between flow evolutions in confined geometries and the unbounded flows more relevant to atmospheric flows. Ertekin et al. (1986) shows that, if solitons develop in channel flow, the drag coefficient oscillates about a constant value and no steady solution appears; yet solutions are presented below that show unbounded steady flow with solitons. Katis and Akylas (1987) do consider unbounded domains but treat only exactly critical flow, showing that, unlike the straightcrested solitons in channels, the emerging solitary wave was curved, although their integration was for too short duration to determine whether the flow became steady. Longer time integrations below show that precisely critical flow lies in a region of parameter space where the flow does not become steady. Some aspects of unbounded flow have been considered in the numerical work by Li and Sclavounos (00), who use generalized Boussinesq equations to study nonhydrostatic shallow-water flow about three-dimensional obstacles at Froude numbers close to unity, concentrating mainly on subcritical and critical flows. They confirm earlier findings of Lee and Grimshaw (1990) and Choi et al. (1990) that the crests of the upstream advancing solitary waves have parabolic shapes in subcritical flow. For supercritical flow they observe trapped solitary waves some distance upstream of the obstacle and argued that these, too, have parabolic shape. Analytical results on flow about a slender body in Karpman (1975) also strongly suggest the possibility of steady solutions with oblique trapped solitary waves for supercritical flow. However, Karpman s work suggests that the flow pattern depends on the behavior of the characteristic lines of the related nondispersive steady operator, and the analysis of section b shows that sufficiently far from orography the solitary wave crests become straight (but obliquely aligned to the oncoming flow). Section presents asymptotic analysis showing the possibility of oblique steady-state solitons in unbounded flow in the limit when the amplitude of the topographic forcing is small. Section 3 extends these results through direct numerical integration of the governing equation, identifies typical flow regimes, and discusses the variation of drag with changes in the relevant parameters. Section 4 gives brief conclusions.. Steady supercritical flow over low orography Much analytical progress can be made for supercritical flow over low, but not infinitesimal, orography. In this case an asymptotic theory combining an inner hyperbolic region and an outer weakly nonlinear region described by a KdV equation along characteristics of the linear wave field can be developed. This approach has been extensively used in dispersive and nondispersive ship wave dynamics (i.e., see Karpman 1967; Whitham 1974, chapter 9; Karpman 1975; Mei 1976; and more recently Gurevich et al. 1995, 1996). The elements of this theory are noted here because of their value in describing the behavior seen later for larger orography. Let the orographic height M and the dispersion coefficient be proportional to a small parameter, say,, with finite and negative. Write M M,, O(1). Then, for steady flow (1) becomes yy xx ( ) xx xxxx M xx, () with the condition that there are no waves far upstream of the obstacle: x 0 at x. (3) a. Linear inner region Now consider a solution of () and (3) of the form, (x, y) u(x, y) o(). (4) Then u(x, y) satisfies the linearized system uyy uxx M xx, (5) u u 0 at x, (6) x with the solution (Jiang and Smith 000) M u(x, y) y x x y [x (y ), ] d [x (y ), ] d. (7)

4 91 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 Writing x (y ) and x (y ) in (7) gives M u(x, y) x (x ) x, y [ ] [ ] (x ) x, y d. (8) This completes the inner solution. However, the behavior of this solution at large distances determines the form of the outer flow. Thus, let x y so that (8) becomes, provided that the orography vanishes sufficiently rapidly at large distances, u(x, y) M ( ) [ x, x ] x[ ] ( ), d. Now x y when y 0 and x y when y 0, so ] ( ) x, d, y 0 M [ u(x, y) ( ) x, d, u 0. [ ] (9) This is the general far-field flow. It is useful to introduce a particular orography, both for definiteness and for the numerical computations. Thus, note that for exp(x /a y /b ) (10) expression (9) gives, with for y 0 and for y 0, abm u exp, 3/ (a b ) a b as x. (11) b. Nonlinear outer region In the outer region x and y are large. In this region introduce the scaling (x *, y * ) (x, y) so that x *, y * are O(1). Guided by the expansions (4), (9) look for a solution of the form (x, y) u(, x *, y * ) o(), where for y 0 and for y 0. Substituting this in () gives 3 u u u u 0, (1) x* y* 3 provided (x, y) decreases sufficiently fast at infinity. Here and elsewhere below in this section the upper and lower signs correspond to y * 0 and y * 0, respectively. Equation (11) matches the outer form of the solution in the inner domain, so [ x ] M ( ) u, d as x* 0, y* 0. (13) Let (y * x * ). Then (11) and (1) take the form 3 u u u 4 0, (14) 3 [ x ] M ( ) u( 0), d. (15) As noted by Gurevich et al. (1996) in the related problem of water waves flowing past thin ships, when orographic forcing and dispersion are small, the far-field solution for the steady fkp equation is described by the KdV equation (13) whose independent variables are the characteristic variables of the corresponding linearized steady KP equation. Jiang and Smith (000) have numerically studied supercritical flow over isolated orography. When the height of their hill was sufficiently large, they observed two shock waves a bow shock in front of the obstacle and an oblique shock at its trailing edge, which they called a V jump. The main features of this pattern are paralleled here. Consider the nondispersive version of Eq. (13); that is, let 0 and, for definiteness, take y * 0 and exp[(x/a) (y/b) ]. Then the initial condition (14) is given by (10). The asymptotic solution to this problem for large is given by Whitham (1974, 48 50) and is known as an N wave. It describes the development of two shock waves one ahead and the other one downstream of the obstacle. For the chosen initial condition the function u has the positive jump of 4 A/ across the front shock A/ and the same but negative jump across the rear shock A/ [A ab M/ ( a b )]. Between these two u(, ) is linear in. Qualitatively this nondispersive solution of (13) complies with the shallow-water computations reported by Jiang and Smith (000) predicting flow deceleration in the bow shock, its further smooth acceleration, and subsequent deceleration through the V jump. When 0, dispersion effects this solution. Although the solution of (13) with 0 can be constructed analytically, it is more straightforward to solve (13) numerically. The quasi-spectral method of Fornberg and Driscoll (1999) proved highly efficient and accurate. A computation was carried out with the step in equal to , the number for Fourier modes equal to 8196, and the extent of the computational domain in from 300 to 300, roughly 00 times larger than the

5 1DECEMBER 004 JOHNSON AND VILENSKI 913 FIG. 3. Solution to the KdV equation (11) in the characteristic plane (, ). The flow is from right to left: exp[(x y )/(1 )], 0.1, 1.5, M 1. (a) Lines of u(, ) const. (b) Crosssectional plots of the solution u(, ) for 0, 5, 50. To avoid overlapping the curves 5, 50 were shifted upward by 3 and 6 units, respectively. extent of the obstacle. The numerical scheme was checked against known exact solutions and for 100 proved to be accurate to well within Figure 3a gives a typical contour plot of the solution (i.e., lines of constant u) and Fig. 3b gives the corre- sponding cross-sectional plots of the function u(, ). Here 0.1, 1.5, M 1, and exp[(x y )/(1 ]. The N wave structure gives way to the solitary wave (which travels to the right as increases) and a dispersive wave train some distance behind it. Note that for an isolated obstacle (as is the case here) the initial condition (14) has zero net mass in the sense that # u ( 0) d vanishes. Since Eq. (13) conserves mass for all, this requires a negative mass dispersive wave train to be generated when solitons (which have only positive mass) are generated. This corresponds closely to the example in Gurevich et al. (1996) of the flow pattern generate by flow past a vertical wall with a protrusion followed immediately by an equal and opposite indentation. Although the rear shock disperses, the linear region right ahead of it, predicted by the nondispersive theory, can still be identified in the solution to the KdV equation. 3. Fully nonlinear supercritical flow The numerical method of Fornberg and Driscoll (1999) extends directly to two dimensions and so allows (1) to be integrated numerically for order unity values of the normalized height M. The computational domain was chosen to be the rectangle x 00, y 50, with a characteristic size about 100 times larger than that of orography so that even at large time the periodicity of the numerical solution in x and y does not affect the solution. The typical number of Fourier modes in x and y varied between and 18 56, FIG. 4. Normalized drag C x /M versus orographic height M. The points marked as 1 and 3 Solitons correspond to the plots displayed in the Figs. 5a and 6, respectively. The point marked with the asterisk corresponds to the solution in Fig. 5b. respectively, and was chosen depending on the physical behavior of the solution. Comparison with exact solutions showed that this was usually enough to keep numerical errors well below 10 6 for times t 100. The orography was chosen to be of shape (10) with height M varying. a. Effects of orographic height and dispersion To compare results obtained in the previous section and the full unsteady integrations, first consider fixed 1.5, and let M and vary. For small bottom elevations (M 0.3) the flow soon becomes steady. The final flow pattern is close to that of the corresponding linear dispersive problem. The drag force is close to that of the linear flow, as can be seen in Fig. 4, which shows the plot of the drag force C M x x dx dy, normalized by M versus M. For the larger values of M nonlinear flow diverges from the linear one and the nonlinear drag rapidly increases. When the height of the hill M reaches a certain critical value (dependent on ), a solitary wave and a dispersive wave train some distance downstream it form. As the steady state is approached, the solitary wave takes the shape shown in the contour plots in Fig. 5a. This solution is the form described in Figs. 3a and 3b for the KdV equation (noting that the solution in Fig. 3 was obtained in the plane of characteristic variables and, whereas the one in Fig. 5a is in physical variables x and y). Emergence of the solitary wave is marked by the sharp rise of the normalized drag, which reaches its maximum at this point (Fig. 4). A similar study based on the Green Naghdi equations with further comparison with towing-tank ship model

6 914 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 FIG. 5. Contours of surface elevation in steady supercritical flow ( 1.5) over a hill of height M.0 centered on the origin. The insets show the surface elevation near the hill (a) with dispersion 0.1, and (b) with smaller dispersion FIG. 6. Hidden line plots of the surface elevation for a steady threesoliton flow with 1.5, M 4.0, 0.1. The flow is from right to left and the hill centered on the origin. tests was carried out by Ertekin et al. (1986) for a gravity flow in a channel induced by a moving localized threedimensional pressure forcing exerted at the the free surface. They noted the formation of two-dimensional solitary waves upstream of the forcing region and a complicated doubly corrugated dispersive wave train downstream. The process of straightening of these solitons was further studied by Hanazaki (1994). There is, however, a substantial difference between the structure of the solitary waves developing in a channel and those in an unbounded region considered here. As shown numerically by Li and Sclavounos (00), in contrast to a channel flow and one-dimensional theory, in three dimensions the supercritical soliton remains permanently trapped somewhat upstream of the forcing region and its crest appears to be bent parabolically. The analysis in section 3 shows, in fact, that the angle between of the solitary waves and the oncoming flow direction may be expected to be governed by the behavior of the characteristics of the nondispersive system. In the flows shown in Fig. 5 (and also in Fig. 6) in the far field this angle was very close to the angle of the characteristic direction of the steady nondispersive equation. In the weekly nonlinear case studied here it is largely determined by the detuning parameter. Further, in the absence of the channel walls no reflections or cross-stream standing waves are possible, and there is no physical mechanism to sustain a doubly corrugated downstream dispersive wave train. Instead, a KdV-like decaying dispersive wave train develops some distance downstream

7 1DECEMBER 004 JOHNSON AND VILENSKI 915 TABLE 1. A comparison between the number of solitons predicted by the KdV equation for various amplitudes of the forcing term and the number appearing in the full fkp simulations: 0.1. Amplitude KdV 1 3 Number of solitons fkp of the last solitary wave. The cross-sectional plots again reveal an almost linear decrease of the solution ahead of the dispersive wave train similar to that upstream of the rear shock in the nondispersive case. For higher orography nonlinear effects become more pronounced and the number of solitons increases. Figure 6 shows the structure of the three-soliton solution in the vicinity of the hill for M 4 and 0.1. The patterns now differ significantly from those predicted by nondispersive hydrostatic theories. Instead of single isolated shocks, a group of almost parallel wave crests appear to the side of the orography. In the neighborhood of the orography these have many similarities to the wave patterns of Fig. 1, offering the interpretation that behind an initial parabolic wave front to the east of Guadalupe Island, air descends and clouds disappear before reappearing in reascending air in the second wave crest, and so on. It similarly offers the interpretation that the dualwave-crest structure seen upwind of Guadalupe Island in Fig. could be a manifestation of a two-soliton flow, a structure absent from shallow-water theories. After the first soliton has appeared a subsequent increase of the height of the hill is accompanied by the gradual decrease of the normalized drag C x /M. Another maximum of drag appears at the amplitude corresponding to the three-soliton solution, but no maximum was found for two solitons. The emergence of the drag maximum depends on the relative position of the peak of the last soliton and the top of the hill. If the soliton is too far from the top, x is close to zero, and the value of the product x in the integrand for C x is also small. For sufficiently large M flow started impulsively from rest does not settle to a steady final state even though the forcing is steady. The flow oscillates about a mean state with periodic formation of solitary waves. The normalized drag shown in Fig. 4 for M 7 corresponds to the period-averaged value (as opposed to the other points where the flow becomes steady). The instantaneous value of C x /M for M 7 oscillates about its mean with amplitude equal to slightly less than 10% of the mean value. This flow unsteadiness is accompanied by the development of a pronounced region of subcritical flow upstream of the hill. Even when the orographic forcing amplitude is much greater than unity, qualitative agreement between the solution to the full fkp equation and its far-field KdV counterpart (13) (which holds for small M) remains good. This can be seen by comparing the number of solitons that arise in the above computations on the basis of the scattering problem for Eq. (13) with the initial data derived from the solution to fkp equation on the line of symmetry y 0. The results are summarized in Table 1. For the computations performed here, the number of solitons predicted on the basis of (13) proved accurate for M 3. For larger amplitudes M strong nonlinearity in the fkp simulations away from the centerline causes deviation from the KdV predictions. If the height of the hill M is held fixed and the dispersion coefficient is decreased, the role of nonlinearity becomes more pronounced and the number of solitons in the solution increases. This can be seen from the comparison of Figs. 5a and 5b, which show the contour plots of the solutions obtained for 0.1 and 0.03, respectively (the rest of the parameters being identical). Thus, the effect of the decrease of the dispersion coefficient on the structure of the wave pattern is qualitatively similar to the increase of the orographic height. However, its influence on the value of the normalized drag for the case of sufficiently large is relatively mild, as is clear from Fig. 4 (the value of the normalized drag for 0.03 is designated by the asterisk). b. Deviations from criticality As increases from negative to positive values in Eq. (1), the flow slows down, passing through a transition from super- to subcritical flow. Steady nondispersive flows satisfying (1) are supercritical where (x, y) /, critical on the curve (x, y) / and subcritical where (x, y) /. The steady nondispersive form of (1) is hyperbolic in the supercritical region and elliptic in the subcritical region. The role of the parameter is similar to that of the excess of the Froude number over unity in the conventional shallowwater theory. V-wave regime: If the height of the obstacle M is moderate and the detuning parameter is large and negative, the flow rapidly approaches a steady state and the near-field flow is close to that of linear supercritical theory (although, as in section b, far downstream of the obstacle a solitary wave always develops). This is shown in Fig. 7, which gives the dependence of the normalized drag on computed on the basis of linear and nonlinear theories for the fixed value of the obstacle height M. The flow structure characteristic of this regime is given in Fig. 7b. Although the wave pattern is substantially modified by dispersion, it is qualitatively similar to the V-wave regime of shallow-water theory and is labeled as such in Fig. 7a. Soliton regime: As increases (with M held fixed) toward zero, the solution becomes essentially nonlinear and the normalized drag begins to rapidly deviate from its linear value. This is accompanied by the emergence of solitons as in Fig. 7c. The pattern is closely related to the bow-shock and V-shock regime of the nondispersive theory (Jiang and Smith 000) with dispersion 1 4

8 916 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 FIG. 7. The normalized drag versus detuning parameter, where asterisks are nonlinear computations, and triangles are the linear dispersive value. (a) A schematic regime diagram for flow past D orography based on the fkp model. (b) Cross-sectional plots of surface elevation versus x for different values of y. To avoid overlapping each line is shifted upward from the previous one; M,.0, 0.1. (c) As in (b) but 1.0, two-soliton solution. (d) As in (b) but 0.5, unsteady upstream propagating modulated wave train at t 75. (e) Shape of the critical line for the same flow as in (d) but at the earlier time t 5; a pronounced subcritical region is visible. (f) As in (e) but at time t 75; the upstream boundary of the subcritical region has expanded farther upstream. (g) Temporal behavior of the normalized drag for the flows of (d), (e), and (f). (h) as in (b) but 0.5, subcritical regime. (i) Shape of the critical line for the flow in (h); the flow is subcritical in the bulk of the domain. spreading the bowshock into solitary waves and the V shock into a rear wave train. Modulated unsteady wave train regime: This regime is related to the subcritical flow of Li and Sclavounos (00). When is sufficiently close to zero (for M this is approximately the region ), the flow does not settle to a steady state. At early times a pronounced subcritical region develops upstream of the obstacle (Fig. 7e). This regime is characterized by the successive generation of upstream propagating waves (as in Fig. 7d) that push the boundary of the subcritical flow further upstream, as in Figs. 7e and 7f, which show the location of critical line (x, y) / at t 5 and t 75, respectively (M, 0.5). This is not seen in hydrostatic flow. The wave dynamics here are qualitatively similar to those described by modulation theory for the forced KdV equation in the near-resonant flow regime (Smyth 1987) where, again, steady forcing does not lead to steady flow. The normalized drag does not become steady but fluctuates periodically with time, as shown in Fig. 7g. It is these mean values that are shown by the asterisks in the drag plot of Fig. 7. Subcritical regime: As nears zero, the amplitude of the drag oscillations rapidly decreases (Fig. 7g shows the largest amplitudes found in the computations). For sufficiently large the flow near the obstacle becomes almost steady. Figure 7h shows a typical flow pattern and Fig. 7i the shape of the critical line: The flow is subcritical everywhere apart from a small region immediately downstream of the obstacle and in the dispersive wave train. 4. Conclusions It has been shown that the inclusion of dispersion causes the single isolated shock structure of shallow-

9 1DECEMBER 004 JOHNSON AND VILENSKI 917 FIG. 8. Hidden line plots of the surface elevation for steady flow over an elliptic ridge of aspect ratio 6 with 1.15, M.0, 0.1. The flow is from right to left. (a) Streamlined ridge, (b) ridge positioned at 45 to the incoming flow, and (c) ridge oriented normally to the incoming flow. water and compressible flow theory to becomes a series of parallel crests or solitons. These patterns then offer an interpretation of the observed wave lines noted here in the flow past Guadalupe Island where Fig. 1 shows, to the east of Guadalupe Island, three additional lines of cloud parallel to the initial bow wave and Fig. shows, behind the initial bow wave, a second wave crest extending across the flow parallel to the bow wave, but separated from it by clear air. The discussion of the types of flow patterns that can appear in near-critical flow over low orography has been presented based on accurate numerical integrations of the forced KP equation, the simplest model to retain at leading order the effects of cross-stream flow variations, nonlinear wave steepening, and wave dispersion due to deviations from hydrostatic balance. The three independent parameters of the problem can be taken as the strength of the forcing, the strength of dispersion, and the deviation of the flow from criticality. As in hydrostatic flow it is this last parameter that has the strongest qualitative effect on the types of patterns seen. Figure 7 summarizes the flow regimes that appear at large times when a flow that is steady and uniform at large distance passes over isolated orography as a function of the deviation from critically and qualitatively unchanged over significant variations in the strength of dispersion and forcing. The discussion to date has been confined to orography that has similar cross-stream and along-stream dimensions in normalized coordinates [although in practice this means that the obstacles are elongated across the oncoming flow (Johnson 1997)]. For nonaxisymmetric orography the flow patterns can be expected to vary with the direction of the oncoming flow. Figure 8 shows surface elevations for supercritical flow over elliptical orography with the aspect ratio 6, with its major axis at 0, 45, and 90 to the oncoming flow. Comparison with Fig. 5a at the same criticality shows an overall qualitative similarity with some novel aspects. The disturbance due to the slender streamlined orography of Fig. 8a is initially small near the orography but steepens to form a solitary wave in the far field. This behavior follows immediately from the asymptotic description of section : even an arbitrarily small-slope zero-mass perturbation to the KdV equation eventually evolves into a single solitary wave. At 45 incidence (Fig. 8b) the perturbation near the orography increases and becomes asymmetric. The larger disturbance to the right (looking downstream) breaks into three solitons, whereas the perturbation to the left is sufficient to force only one soliton. This pattern has many features of the cloud pattern in Fig. 1 where there appears to be a bow wave and three wave crests to the east of the island and simply the bow wave to the west (corresponding here to incidence of

10 918 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 45). For a finite ridge normal to the incoming flow (Fig. 8c), the flow is quasi one-dimensional near the ridge, but away from the ridge similar to the far-field wave pattern forced by isolated orography. The inclusion of nonhydrostatic effects breaks the single isolated strong shocks of hydrostatic flow into a series of almost parallel stationary solitons of elevation. These appear to fit well the observations of cloud patterns presented in Fig. 1. of Burk and Haack. Behind the initial bow wave disturbance, where the oncoming flow passes a point of land, lie almost parallel bands of cloud. Burk and Haack note that these clouds are not typical boundary layer roll vortices as they align more nearly perpendicular than parallel to the wind. They tend to appear repeatedly in the same coastal locations and take on a similar orientation relative to the coastline. Edinger (1966) notes that such clouds are almost stationary both temporally and spatially with considerable up- and downdrafts. The earth s rotation can become important in flows with moderate speeds and horizontal extent of the order of tens to hundreds of kilometers. Close to the obstacle disturbances remain similar to those described here but rotation causes the solitons generated near the obstacle to decay over cross-stream distances on the order of a Rossby radius (Vilenski and Johnson 004). Acknowledgments. This work was funded by the U.K. Natural Environment Research Council Grant NER/A/ S/000/0133 whose support is gratefully acknowledged. ERJ is indebted to the School of Mathematical Sciences at Monash University for their hospitality while some of this work was carried out. GGV acknowledges helpful conversations with Dr. Sergei Timoshin. We are grateful to the referees of a previous version of this work for their helpful comments. REFERENCES Akylas, T. R., 1994: Three-dimensional long water-wave phenomena. Annu. Rev. Fluid Mech., 6, Badgley, P. C., L. Miloy, and L. Childs, 1969: Oceans from Space. Gulf Publishing, 34 pp. Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 48 pp. Burk, S. D., and T. Haack, 000: The dynamics of wave clouds upwind of coastal orography. Mon. Wea. Rev., 18, Choi, H. S., K. J. Bai, J. W. Kim, and H. Cho, 1990: Nonlinear free surface waves due to a ship moving near the critical speed in a shallow water. Proc. 18th Symp. on Naval Hydrodynamics, Ann Arbor, MI, National Research Council, Edinger, J. G., 1966: Wave clouds in the marine layer upwind of Point Sal, California. J. Appl. Meteor., 5, Ertekin, R. C., W. C. Webster, and J. V. Wehausen, 1986: Waves caused by moving disturbance in a shallow channel of finite width. J. Fluid Mech., 169, Fornberg, B., and T. A. Driscoll, 1999: A fast spectral algorithm for nonlinear wave equations with linear dispersion. J. Comput. Phys., 155, Grimshaw, R. H. J., and N. Smyth, 1986: Resonant flow of a stratified fluid over topography. J. Fluid Mech., 9, Grubisic, V., R. B. Smith, and C. Schär, 1995: The effect of bottom friction on shallow-water flow past an isolated obstacle. J. Atmos. Sci., 5, Gurevich, A. V., A. L. Krylov, V. V. Khodorovsky, and G. A. El, 1995: Supersonic flow past bodies in dispersive hydrodynamics. J. Exp. Theor. Phys., 80, ,,, and, 1996: Supersonic flow past finite-length bodies in dispersive hydrodynamics. J. Exp. Theor. Phys., 8, Jiang, Q., and R. B. Smith, 000: V-waves, bow shocks, and wakes in supercritical hydrostatic flow. J. Fluid Mech., 406, Johnson, R. S., 1997: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, 445 pp. Hanazaki, H., 1994: On the three-dimensional waves excited by topography in the flow of a stratified fluid. J. Fluid Mech., 63, Kadomtsev, B. B., and V. I. Petviashvili, 1970: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl., 15, Karpman, V. I., 1967: The structure of two-dimensional flow around bodies in dispersive media. Sov. Phys. JETP, 3, , 1975: Nonlinear Waves in Dispersive Media. Pergamon Press, 31 pp. Katis, C., and T. R. Akylas, 1987: On the excitation of long nonlinear water waves by a moving pressure distribution. Part. Threedimensional effects. J. Fluid Mech., 177, Lee, S. J., and R. H. J. Grimshaw, 1990: Upstream-advancing waves generated by three-dimensional moving disturbances. Phys. Fluids, A, Li, Y., and P. D. Sclavounos, 00: Three-dimensional nonlinear solitary waves in shallow water generated by an advancing disturbance. J. Fluid Mech., 470, Mei, C. C., 1976: Flow around a thin body moving in shallow water. J. Fluid Mech., 77, Pan, F., and R. Smith, 1999: Gap winds and wakes: SAR observations and numerical simulations. J. Atmos. Sci., 56, Samelson, R., 199: Supercritical marine-layer flow along a smoothly varying coastline. J. Atmos. Sci., 49, , and S. Lentz, 1994: The horizontal momentum balance in the marine atmospheric boundary layer during CODE-. J. Atmos. Sci., 51, Schär, C., and R. B. Smith, 1993a: Shallow-water flow past an isolated topography. Part I: Vorticity production and wake formation. J. Atmos. Sci., 50, , and, 1993b: Shallow-water flow past an isolated topography. Part II: Transition to vortex shedding. J. Atmos. Sci., 50, , and D. R. Durran, 1997: Vortex formation and vortex shedding in continuously stratified flows past isolated topography. J. Atmos. Sci., 54, Smith, R. B., and D. F. Smith, 1995: Pseudoviscid wake formation by mountains in shallow water flow with a drifting vortex. J. Atmos. Sci., 5, , A. C. Gleason, P. A. Gluhosky, and V. Grubisic, 1997: The wake of St. Vincent. J. Atmos. Sci., 54, Smyth, N. F., 1987: Modulation theory solution for resonant flow over topography. Proc. Roy. Soc. London A, 409, Vilenski, G. G., and E. R. Johnson, 004: Near-critical free surface rotating flow over topography. Proc. Roy. Soc. London A, 460, Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley-Interscience, 636 pp.