Three-Dimensional Structure of Forced Gravity Waves and Lee Waves

664

JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 61

Three-Dimensional Structure of Forced Gravity Waves and Lee Waves R. D. SHARMAN National Center for Atmospheric Research,* Boulder, Colorado

M. G. WURTELE Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, and Center for Meteorology, University of California, Berkeley, Berkeley, California (Manuscript received 14 November 2002, in final form 18 October 2003) ABSTRACT The three-dimensional structure of lee waves is investigated using a combination of linear analysis and numerical simulation. The forcings are represented by flow over a single wave (monochromatic) in the alongstream direction but of limited extent in the cross-stream direction, and by flow over isolated obstacles. The flow structures considered are of constant static stability, and zero, positive, and negative basic-flow shears. Both nonhydrostatic and hydrostatic regimes are studied. Particular emphasis is placed on 1) the cross-stream structure of the waves, 2) the transition from three-dimensional to two-dimensional flow as the breadth of the obstacle is increased, 3) the criteria for three-dimensional nonhydrostatic to hydrostatic transitions, and 4) the effect of obstacle breadth-to-length aspect ratio on the wave drag for this linear system. It is shown that these aspects can in part be understood by relating the gravity waves produced by narrow-breadth obstacles to the ‘‘St. Andrew’s Cross’’ for hydrostatic and nonhydrostatic uniform flow and for hydrostatic shear flow.

1. Introduction A mountain range is typically composed of many individual peaks, each of which represents an isolated obstacle to flow impinging on it. When viewed from this perspective, the flow over a mountain range is actually a superposition of three-dimensional lee-wave structures produced by each obstacle, the actual leewave structure depending on the particular individual obstacle geometry and the nature of the local upstream environmental flow. Therefore, to gain an understanding of the complexity of the flow over mountain ranges one must first appreciate the variety of three-dimensional lee-wave structures that may be induced by flows of various atmospheric vertical structures over isolated obstacles of various shapes. Studies of lee waves generated by flow over topography have a long history going back to the early works of Scorer (1949) and Queney (1948); a recent comprehensive review of the subject is provided in Baines (1995). Three-dimensional linear analyses for uniform

* The National Center for Atmospheric Research is sponsored by the National Science Foundation. Corresponding author address: Dr. Robert Sharman, NCAR/RAP, P.O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected]

q 2004 American Meteorological Society

flow over isolated obstacles were originally provided by Wurtele (1957), Crapper (1959, 1962), and Janowitz (1984) for nonhydrostatic flows and by Smith (1980, 1988, 1989a,b) and Phillips (1984) for hydrostatic flows. Crapper (1962), Berkshire (1975), Sharman and Wurtele (1983), Gjevik and Marthinsen (1978), and Grubisˇic´ and Smolarkiewicz (1997) examined the effect of vertical variations in wind speed and stability of the environment, while Broad (1995, 1999) and Shutts (1995, 1998) consider the case of wind turning with height. Examples of finite amplitude studies, mostly concentrating on flow regimes as a function of nondimensional obstacle height and aspect ratio by using finite-difference simulations, are Smolarkiewicz and Rotunno (1989, 1990), Bauer et al. (2000), and Epifanio and Durran (2001). Despite these numerous studies, the fully three-dimensional structure of the lee-wave pattern and its relation to obstacle geometry is still a relatively unexplored area, even in idealized environments and for idealized obstacle shapes. At a conceptual level, to better understand the threedimensional structure of the lee wave pattern, the phase and group velocity relationships of three-dimensional gravity wave propagation in stratified fluids must be considered. These relations are discussed in detail, for example, in Lighthill (1978) and Baines (1995). Of importance here is the fact that in a quiescent fluid, the phase and group velocities are perpendicular; that is,

15 MARCH 2004

665

SHARMAN AND WURTELE

FIG. 1. Configuration of a sinusoidal lower boundary of finite breadth.

then be used to better understand the lee-wave structure forced by isolated obstacles, which of course excites an entire spectrum of waves. In order to connect to previous linear lee-wave analyses, wave amplitudes are assumed to be small, and therefore the use of linearized equations is appropriate. For three-dimensional unidirectional upstream flow, a single equation for the vertical velocity w may be derived [e.g., Baines 1995, his Eq. (4.13.7)] D 2 (¹ 2 w) 2 U zz D(w x ) 1 N 2¹ H2 w 5 0.

(1.1)

Here the notation is standard: the group velocity is parallel to surfaces of constant phase. This fact was elegantly demonstrated in the seminal paper of Mowbray and Rarity (1967; see also Lighthill 1978) by a series of tank experiments with an oscillating narrow source in a density stratified fluid. As a consequence of the group velocity being parallel to wave crests, the wave crests were observed to stretch out radially away from the source as a cross pattern of large amplitudes [referred to in Lighthill (1978) as ‘‘St. Andrew’s Cross’’], the branches of the cross making an angle u 5 cos 21 (v 0 /N) with the vertical, where v 0 is the oscillating source frequency and N is the Brunt– Va¨isa¨la¨ frequency. In the classical two-dimensional leewave problem with mean flow U perpendicular to a ridge, the cross does not exist because the waves are advected by the mean flow. However, in three dimensions, gravity waves also propagate laterally away from the obstacle and are not affected by the U advection. In this cross-stream direction the waves generated are therefore analogous to those observed in the Mowbray and Rarity experiments but with v 0 replaced by the intrinsic frequency 2kU. Provided the obstacle is of narrow breadth, the pattern in planes transverse to the mean flow will resemble the upper half of the cross. In this paper we link the traditional three-dimensional lee-wave analytic investigations with the dynamics manifested in St. Andrew’s Cross. In so doing we are able to better understand the cross-stream structure of threedimensional lee waves. The analyses also lead to the establishment of criteria for transition of the wave structures from their two-dimensional to their three-dimensional form and of criteria for three-dimensional nonhydrostatic to hydrostatic transitions. The wave dynamics involved are investigated by performing a systematic study of the structure of threedimensional gravity waves forced by flow over idealized topography. We begin by returning to first principles, and consider the simplest model containing the essential physics, namely, a single linear gravity wave propagating in a stratified fluid (constant stability and constant wind or constant wind shear, positive or negative). The single wave component is produced by flow over an obstacle in the shape of a single sinusoid in the alongstream direction of various widths—narrow to broad— in the cross-stream direction (Fig. 1). The results of these linear analyses for a single wave component can

D5

] ] 1 U(z) , ]t ]x

¹ 2 5 ¹ H2 1

]2 ]2 ]2 ]2 5 1 1 . ]z 2 ]x 2 ]y 2 ]z 2

The solution of (1.1) is subject to the usual boundary conditions, w 5 U]h(x, y)/]x evaluated at z 5 0,

(1.2)

where h(x, y) is the obstacle shape, and for large z, wave energy propagation must be upward. For idealized obstacle representations, such as a narrow-breadth sinusoid, the problem represented by (1.1) and (1.2) is to a considerable extent amenable to analysis. But for more complex obstacle shapes or upstream profiles, closed form solutions are not in general possible; for those situations numerical simulation of the governing equations may be used to provide a more complete description. In this paper we use a combination of linear analyses for a single gravity wave component augmented by numerical simulations to gain a better understanding of the three-dimensional lee-wave structure and its relation to obstacle geometry and environmental conditions. In section 2 three-dimensional analyses and simulations are presented for the constant wind and stability case, starting with a single wave component in x, and using these results to better understand the three-dimensional structure of waves generated by flow over isolated obstacles. A similar line of analyses and simulations is presented in section 3 for both positive (forward) and negative (backward) constant wind shear. Some implications for gravity wave drag parameterizations in numerical weather prediction models are discussed in section 4, and finally section 5 provides a summary and conclusions. 2. Constant wind and stability case a. Monochromatic forcings Consider first a single gravity wave component in the x direction of wavenumber k, propagating in a constant U, N environment, under steady-state conditions. This may be generated by flow over a sinusoid of wavenumber k in the along-stream direction x, and of arbitrary

666


breadth in the cross-stream direction y (cf. Fig. 1). This choice simplifies the analysis and helps elucidate the physics involved. Further, as pointed out by Welch et al. (2001), the use of periodic topography may be more physically meaningful in situations where individual peaks in a mountain chain may be influenced by wave motions created from peaks upwind. For this configuration, Eq. (1.1) for a single Fourier component wˆ(k, y, z) simplifies to k2w ˆ zz 2 (k s2 2 k 2 )w ˆ yy 1 k 2 (k s2 2 k 2 )w ˆ 5 0, (2.1)

resented by its Fourier transform in y as W(k, l, z), where l is the y wavenumber w ˆ (k, y, z) 5

h 5 ky,

E

`

1 2p

E

W(k, l, z) exp(ily) dl,

w ˆ (k, y, z) exp(2ily) dy,

(2.4b)

2`

then from (2.2) and the lower and upper boundary conditions, W 5 W0 (l) exp(ikx 1 imz ),

(2.5a)

m 5 1Ï1 1 s 2 ,

(2.5b)

s 5 l/k.

Here W 0 (l) is the Fourier transform of wˆ(k, y, 0) derived from (1.2), (2.3), and (2.4b),

z 5 Ïk s2 2 k 2 z,

Eq. (2.1) becomes

W 0 (l) 5 2Ukh 0 A(l),

w ˆ zz 2 w ˆ hh 1 w ˆ 5 0.

(2.2)

Note that the vertical coordinate z in the hydrostatic limit (k 2 K k s2) becomes simply k s z. Therefore, for uniform flow the hydrostatic assumption, as in two dimensions, only increases the vertical scale of the disturbance and does not alter the dynamics. Equation (2.2) has the form of the much-studied Klein–Gordon equation. The properties of the solutions are well known (e.g., Whitham 1974; Gill 1982; Lighthill 1996). In particular, wave fronts are propagated along the coordinate lines z 6 h 5 constant, with negligible disturbance for z 2 , h 2 . The usual boundary conditions for the Klein–Gordon equation would be the assignment of the dependent variable and its vertical derivative at, say, z 5 0. In the lee-wave problem, however, as is well known, the w can be assigned at z 5 0, but the second condition has to be one that guarantees upward energy propagation of the disturbance from the forcing source. A simple forcing for w can be represented by flow over a single sinusoidal obstacle of a specified wavenumber k , k s and with an arbitrary profile in the y (cross-stream) direction: h(x, y) 5 h 0 a(y) coskx,

(2.3)

where h 0 is the amplitude of the sinusoidal and a(y) represents the cross-stream profile. If wˆ(k, y, z) is rep-

a(y) 5

5

for |y| , b, otherwise.

1 0

The solutions (2.8) confirm the conclusions derived from the known characteristics of the Klein–Gordon

(2.7)

This has the Fourier transform, A(l) 5 2 sinbl/l. For a narrow fin, A(l) → 2b, and by (2.6), W 0 is therefore a constant. Then inverting (2.5),

5 E

`

w(j, h, z ) 5 w0 Re sinj

exp(ish) cosmz ds

2`

E

6

`

1 cosj

exp(ish) sinmz ds ,

2`

where Re denotes the real part and w 0 5 2Ubh 0 k 2 /p, a constant. The two integrals can be evaluated in terms of Bessel functions of order 1 [e.g., Gradshteyn and Ryzhik 1994, their Eq. (3.876)]:

z cosj K1 (Ïh 2 2 z 2 ), Ïh 2 z 2 2

(2.6)

and A(l) is the Fourier transform of the obstacle forcing a(y). The solution (2.5) satisfies a sinusoidal boundary condition at z 5 0, and the positive radical ensures that the wave energy will be upward propagating. The case k . k s represents a nonpropagating wave and will not be considered further. First consider a sinusoid of narrow breadth, a fin resembling the serrated edge of a bread knife, represented mathematically as

p z w(j, h, z ) 5 2 w0 [sinj J1 (Ïz 2 2 h 2 ) 1 cosj Y1 (Ïz 2 2 h 2 )], 2 2 Ïz 2 h 2 w(j, h, z ) 5 w0

(2.4a)

2`

`

W(k, l, z) 5

where k s 5 N/U, otherwise subscript notation is used to indicate differentiation. If now the following scalings are adopted:

j 5 kx,

VOLUME 61

z 2 . h2

(2.8a)

z 2 , h2.

(2.8b)

equation (2.2): the wave disturbance propagates in the domain z 2 . h 2 , since the Bessel functions J1 and Y1

15 MARCH 2004

SHARMAN AND WURTELE

are oscillatory there. In the domain z 2 , h 2 the modified Bessel function K1 falls off exponentially with its argument. Thus the coordinate lines z 5 h, or z 5 2h, contain sharp wave fronts, mathematically singular lines, with the structure Y1 (Ïz 2 2 h 2 ) Ïz 2 h 2

2

→2

2 , p (z 2 h)(z 1 h)

z → 6h .

The singularity, which has the form of a first-order pole, has its source at z 5 h 5 0, and propagates along the characteristic lines z → 6h, that is, along the lines z → 6(k s2/k 2 2 1) 21/2 y. Equation (2.8) is the solution for w; the other variables are of the order ]p/]z 5 ]u/]y 5 y 5 w, where p is the perturbation pressure, and u and y are the x and y components of the perturbation velocity, respectively. So p and u contain less intense, that is, logarithmic, singularities. These solutions are a consequence of the group velocity being parallel to phase lines in (y, z) planes (e.g., Baines 1995, his section 4.13), and are, in fact, the upper half of the St. Andrew’s Cross mentioned in the introduction. Note that the solutions (2.8) were derived by Crapper (1959) in an equivalent form, but he did not discuss the nature or implications of the solution, and for his fully three-dimensional problem, he immediately attempts a k wavenumber integration of (2.8). Although the gravity wave forcing here was caused by flow over a narrow sinusoid, this phenomenon may also be produced by narrow oscillatory sources, as in the stratified tank experiments of Mowbray and Rarity (1967) or in the atmosphere, for example, in the cloud simulations of Lane et al. (2001) when cloud pulses impinged on the stratosphere. The solutions (2.8) are displayed in Fig. 2 for the case k s 5 10 23 m 21 with a dimensional forcing wavelength of L 5 3p km(k 5 2/3 3 10 23 m 21 ), so that k s /k 5 1.5. In dimensional terms, the singularity makes an angle with the vertical,

tanu 5 y/z 5 (k s2 /k 2 2 1)1/2 ,

667 (2.9)

which for our parameters is about 508. This is evident in Fig. 2a, which shows the solution in a (y, z) plane at an x position of kx 5 p/4 (x/L 5 0.25), so that both terms in (2.8a) contribute. Note the sharp demarcation at the singular line between the quiescent flow below and the propagating solutions above. Figure 2b shows the solution in an (x, y) plane at the level kz 5 10 in Fig. 2a. The intersections of the singular lines with any horizontal plane are, of course, parallel lines to the x axis, as shown in Fig. 2b, and from (2.9), the width of the disturbed strip in the (x, y) plane will increase with the elevation of that plane. Finally, Fig. 2c shows the solution in the y 5 0 plane and demonstrates the propagating solution in the wedge above the singular line. For more complex obstacle shapes or upstream profiles, closed form solutions are not in general possible, although approximations may be derived. For those situations, numerical simulation of the governing equations may be used to provide a more complete description. In this paper the simulation code described in Sharman and Wurtele (1983) is used to demonstrate the solutions. In brief, this model integrates the nonlinear, nonhydrostatic, Boussinesq three-dimensional equations of motion subject to specified initial conditions but with a lower boundary forcing provided by the simplified boundary condition (1.2). Although the exact boundary condition could be applied at the obstacle surface, its inclusion would unnecessarily complicate comparison to the linear analyses. However, the nonlinear equations of motion are retained in order to study nonlinear effects such as interaction of singularities with mean flow critical levels. The parameters of the simulations to be discussed are listed in Table 1. For reference, Fig. 3a shows the results of a simulation with this model using the same parameters as was used in the construction of Fig. 2, but using an obstacle with a Gaussian shape cross-stream profile,

FIG. 2. Vertical velocity fields derived from (2.8) in (a) the (y, z) plane at streamwise position kx 5 p/4, (b) the (x, y) plane for a dimensionless height kz 5 10, and (c) the y 5 0 (x, z) plane; L is the wavelength. In this and all subsequent figures, negative values are contoured using dashed lines; positive values using solid lines.

668


VOLUME 61

TABLE 1. Parameters used in the numerical simulations. Obstacle U0 Run (m s21 ) 1a 1b 1c 1d 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 5a 5b 5c 5d 5e 6a 7a 7b 7c 7d 7e 7f 8

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

]U 0 /]z (s21 )

N (s21 )

h0 (m)

l or a (km)

b (km)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10.003 54 10.003 54 10.003 54 10.003 54 10.003 54 10.003 54 10.003 54 10.003 54 20.000 667 20.001 20.001 20.001 20.001 20.001 20.001 20.001

.01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01

250 250 250 250 500 500 500 500 500 500 500 500 500 500 500 500 100 100 100 250 250 250 250 250 100 250 250 250 250 250 250 250

3p 3p 3p 3p 1 1 1 1 1 1 5 5 5 5 5 5 3p 3p 9p 1 1 1 1 1 3p 1 1 1 1 1 1 5

0.1 1.0 2.0 5.0 0.1 1.0 3.0 5.0 8.0 10.0 1.0 5.0 15.0 25.0 40.0 50.0 0.1 3.0 0.1 0.3 1.0 5.0 8.0 10.0 0.1 0.1 1.0 3.0 5.0 8.0 10.0 5.0

h(x, y) 5 h 0 coskx exp(2y 2 /4b 2 )

(2.10)

[cf. (2.3)] with a rather narrow breadth b of 140 m, or kb ø 0.1. Figure 3a displays the vertical velocity in the (y, z) plane at kx 5 0, and it can be seen that the simulation reproduces the analytic solution. For any obstacle of finite breadth, as it necessarily is in the simulations, the singular line is actually a transition region from propagating wave solutions over the obstacle to evanescent solutions closer to the surface and to the sides of the obstacle. Within the transition region the wave amplitudes are a maximum, and the amplitude falls off rapidly outside this region. Naturally, the horizontal and vertical shears become large in these transition regions, and this is illustrated in Figs. 3b and 3c, showing the x component of vorticity (v x 5 ]w/]y 2 ]y/]z) and the Richardson number, Ri, respectively. Thus for flows over narrow obstacles, instabilities would be most likely to occur in the vicinity of the singular lines, that is, to the sides of the obstacle as opposed to directly over the obstacle. To study the effects of finite obstacle breadth, consider an obstacle with a cross-stream profile (2.10), which has the Fourier transform A(l) 5 2b(p)1/2 exp(2b 2 l 2 ).

(2.11)

Grid nx 3 ny 3 nz 64 64 64 64 513 513 513 513 513 513 513 513 513 513 513 513 64 64 64 257 257 257 257 257 128 513 513 513 513 513 513 513

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

128 128 128 128 256 256 256 256 256 256 256 256 256 256 256 256 128 128 128 256 256 256 256 256 128 256 256 256 256 256 512 256

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

101 101 101 101 181 181 181 181 181 181 181 181 181 181 181 181 76 76 76 76 76 76 76 76 101 81 81 81 81 81 81 81

Dx 3 Dy 3 Dz (m) 147 147 147 147 391 391 391 391 391 391 391 391 391 391 391 391 147 147 442 781 781 781 781 781 74 195 195 195 195 195 195 391

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

197 197 197 197 196 392 392 392 392 392 392 588 392 392 784 784 315 315 315 186 392 392 392 392 394 196 392 392 392 392 392 392

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

250 250 250 250 333 333 333 333 333 333 333 333 333 333 333 333 200 200 200 200 200 200 200 200 250 250 250 250 250 250 250 250

Dt (s)

Upper sponge (pts)

Fig. no.

10 10 10 10 20 20 20 20 20 20 20 20 20 20 20 20 1 1 2.5 9 8 8 8 8 5 10 10 10 10 10 10 10

40 40 40 40 81 81 81 81 81 81 81 81 81 81 81 81 30 30 32 27 27 27 27 27 30 36 36 36 36 36 36 33

3 4 4 4 5 — — 6 — — 7 — — — — — 8 8 8 9 — 9 9 — 10 11 — — — — 11 —

Inverting (2.5) and using this with (2.6), the formal solution is

E

`

w 5 (p)1/ 2 w0

exp(2k 2 b 2s 2 1 ish)

2`

3 sin(j 1 mz ) ds,

(2.12)

where s 5 l/k, as before. Numerical evaluation of the integral (2.12) for various values of kb shows that for kb * 2 (i.e., obstacle breadth greater than the sinusoid wavelength/p), the pattern is essentially two-dimensional with the approximate solution wø

1 2

p w0 y2 exp 2 2 sin(j 1 z ). kb 4b

(2.13)

Thus the solution is quasi two-dimensional containing horizontal phase lines in the (y, z) plane with amplitudes falling off exponentially away from the obstacle for y * 2b. Figure 4 illustrates the gradual transition from the narrow-breadth three-dimensional solutions with their strong cross-stream character to quasi-two-dimensional solutions produced by wider-breadth obstacles. Figures 4a and 4b (kb 5 1) show considerable resemblance to

15 MARCH 2004

SHARMAN AND WURTELE

669

FIG. 3. The (a) vertical velocity, (b) v x , and (c) Ri fields in the x 5 0 (y, z) plane derived from a numerical simulation for uniform flow (U 5 10 m s 21 , N 5 10 22 s 21 ) over a single sinusoid of dimensional wavelength L 5 3p km and of Gaussian-shaped cross section with a dimensional half-width of 0.14 km (kb ø 0.1).

the very narrow obstacle solutions already shown in Fig. 3, with maximum amplitudes in a very narrow transition region in the neighborhood of the singular line and with wave solutions above this region. For a rather large kb of 5, Figs. 4e and 4f show the solutions have the expected two-dimensional character. In Figs. 4c and 4d (kb 5 2) the singular lines in the (y, z) plane are not present, nor is there any suggestion of a concentrated intensity near the outer edge of the disturbance. However, the phase lines are still sloped, so that there is a transverse variation, but less so than in Figs. 4a and 4b. This is clearly a transition case. If the mean wind has a cross-stream component, with or without directional shear, the presence of singular lines is still apparent for narrow-breadth obstacles (not shown), but the slope of the transition region is no longer symmetric about y 5 0, and the slope of the upstream phase lines is more vertical than is the downstream phase lines. b. Extension to isolated obstacles If instead of a monochromatic forcing, the forcing is composed of several waves, or in the limit an entire spectrum of waves as might be forced by an isolated obstacle, each k in the forcing spectrum will generate singular lines at a unique angle given by (2.9). Qualitatively, the singularity introduced at the boundary, although present for each individual wave component k, is propagated along singular lines of different slope and is smoothed out by the continuous k spectrum introduced by the isolated obstacle. For very narrow length and breadth obstacles, existing theory does support this notion. A review of the existing theory is provided in the appendix, together with some new results concerning the nature of the phase lines in the (x, z) and (y, z) planes and their relation to the singular lines. But, as previous theoretical works on this subject do not present three-dimensional views of the solutions, the three-dimensional character of the wave pattern for various ob-

stacle geometries will be presented here through results from a series of numerical simulations. In these simulations the obstacle used to force the flow has the twodimensional Gaussian form h(x, y) 5 h 0 exp(2x 2 /4a 2 2 y 2 /4b 2 ),

(2.14)

which is consistent with that used in the analysis of Crapper (1962). Figure 5 shows the results of a simulation over a fairly narrow obstacle, with obstacle parameters h 0 5 500 m, a 5 1000 m, and b 5 100 m. Figures 5a and 5b present the simulation results in (y, z) planes 25 and 50 km downstream, respectively. Consistent with the doublet solution (A2), the phase lines are straight near the origin and have a parabolic shape at higher elevations. The singular lines present for a monochromatic forcing are replaced by a V-shaped zone of maximum amplitudes. In the appendix it is shown that for narrow length and breadth obstacles, the superposition of the many singular lines produced by the topographic spectrum of an isolated obstacle results in a band of maximum amplitudes in a transition region in the (y, z) plane centered at tanu 5 1Ï2 or u ø 358, regardless of the upstream values of U and N. Consistent with that prediction, Figs. 5a and 5b show maximum amplitudes near u ø 358. This pattern of large amplitudes in the transition region is very similar to Fig. 3a in Shutts (1998) for a constant wind and stability case computed by performing the necessary Fourier transforms numerically. Figure 5c presents the simulation results for the vertical velocity w in the y 5 0 (x, z) plane. The phase lines are circles centered at the origin as expected, but the amplitude has a series of secondary maxima in a tail-like structure near but above z 5 0, which is not captured by the asymptotic analysis. Figure 5d presents the solution in an (x, z) plane at y 5 110 km. Again, the results are consistent with the doublet approximate solution: phase lines parallel to z 5 0 near the surface, which steepen aloft, and with maximum amplitudes at elevations defined by the intersection of the maximum amplitudes in

670


VOLUME 61

FIG. 4. Comparisons of the vertical velocity fields derived from numerical simulations of uniform flow (U 5 10 m s 21 , N 5 10 22 s 21 ) over a single sinusoid of dimensional wavelength L 5 3p km and of various Gaussian-shaped cross-section half-widths b, in the (y, z) plane at streamwise position x/L 5 21/4, and in the (x, y) plane at a dimensional height z 5 2.5 km. (a),(b) b 5 1.4 km (kb ø 1); (c),(d) b 5 2.8 km (kb ø 2); (e),(f ) b 5 7 km (kb ø 5).

the (y, z) planes near u 5 1 358. Finally, Figs. 5e and 5f display the results for w and Ri, respectively, in the (x, y) plane at z 5 10 km. The downstream tail is visible, and consistent with the doublet solution, the phase lines are hyperbolic. Of significance for gravity wave parameterizations, note that Ri is decreased in a pattern consistent with the three-dimensional wave structure, and therefore wave breaking, if it occurs, may in fact be in regions downstream laterally away from the sides of the obstacle. The changes in the three-dimensional lee-wave structure as the obstacle becomes wider or longer are shown in Figs. 6 and 7, respectively. When b is increased to

5 km, so that b/a 5 5 but Na/U is still unity (Fig. 6), differences from the narrow-breadth obstacle become noticeable. Although the wave pattern in the y 5 0 plane (Fig. 6a) is similar in appearance to the narrower-breadth (Fig. 5a) obstacle, in both the (y, z) and (x, y) planes (Figs. 6b and 6c, respectively), the phase lines of the disturbance are essentially straight with maximum amplitudes downstream confined to a strip of width approximately equal to the obstacle breadth and with only a slight suggestion of the hyperbolic phase structure at the sides of the obstacle. The confinement of maximum amplitudes to the downstream strip is consistent with the analysis presented in section 2a for a single sinusoid

15 MARCH 2004

SHARMAN AND WURTELE

671

FIG. 5. Results from a numerical simulation of uniform flow (U 5 10 m s 21 , N 5 10 22 s 21 ) over a very narrow breadth Gaussian-shaped isolated elliptical obstacle [Eq. (2.14)] with a 5 1 km, b 5 0.1 km. (a) The vertical velocity field w in the x 5 25 km (y, z) plane; (b) w in the x 5 50 km (y, z) plane; (c) w in the y 5 0 center plane; (d) w in the y 5 10 km (x, z) plane; (e) w in the z 5 10 km (x, y) plane; and (f ) Ri in the z 5 10 km (x, y) plane.

wide-breadth obstacle, and is distinctly different from the narrow-breadth obstacle which has maximum wave amplitudes to the sides of the obstacle. In this regard the wave structure directly downstream of the obstacle is essentially the same as that that would be produced by flow over a two-dimensional ridge. This transition

value of the obstacle aspect ratio (b/a 5 5) is somewhat smaller than the value b/a 5 10 estimated by Epifanio and Durran (2001), but they considered finite height obstacles and a different obstacle shape. Figure 7 shows the pattern derived when a is increased to 5 km for a narrow-breadth obstacle of b 5 1 km, so

FIG. 6. Results from a numerical simulation of uniform flow (U 5 10 m s 21 , N 5 10 22 s 21 ) over a Gaussian-shaped isolated elliptical obstacle [Eq. (2.14)] with a 5 1 km, b 5 5 km. (a) The vertical velocity field w in the y 5 0 (x, z) plane; (b) w in the x 5 25 km (y, z) plane; (c) w in the z 5 10 km (x, y) plane.

672


VOLUME 61

FIG. 7. The same as in Fig. 6 except a 5 5 km, b 5 1 km, and (c) z 5 5 km (x, y) plane.

that b/a 5 0.2, but Na/U 5 5. In agreement with expectations for long obstacles, the solutions show a hydrostatic character. For example, in Fig. 7a maximum amplitudes in the (x, z) plane are confined to a region directly over the obstacle. In Fig. 7b the solutions in the (y, z) planes downstream display the transition region of maximum wave amplitudes with larger values of u than for the narrow-breadth case. In Fig. 7c the solutions in the (x, y) planes show hyperbolic phase lines with negligible disturbance downstream near the centerline (y 5 0) and are very similar to the hydrostatic solutions presented in Smith (1980). From the structure of the wave patterns for various obstacle lengths, we conclude that the obstacle length k s a solely determines the presence or absence of nonhydrostatic effects. In a series of other simulations (see Table 1) in which b was held fixed and a was varied, for the forcing (2.14), the transition to hydrostatic solutions occurred for k s a * 3, for both two- and threedimensional cases regardless of the value of b. Thus, in this linear system, the two nondimensional parameters that completely describe the character of the solutions are the aspect ratio b/a, and the nondimensional obstacle length k s a. Of course for obstacles of finite height, a third parameter, k s h 0 or Nh 0 /U, is also important (e.g., Baines 1995). 3. Shear flow In the previous section, solutions were derived for a constant wind and stability environment. In this section we relax the constraint of a constant wind and consider a background wind with constant vertical shear (positive or negative). Specifically, we consider a mean wind profile of the form

1

U(z) 5 U0 1 1

2

z , H

the upstream Ri. With this profile, for a monochromatic forcing with wavenumber k, Eq. (1.1) becomes w ˆ ZZ 2

1Z

Ri 2

2

21 w ˆ hh 1

1Z

Ri 2

2

21 w ˆ 5 0,

(3.2)

where

h 5 ky,

Z 5 kz,

and

Ri 5 N 2 H 2 /U 02 .

a. Positively sheared flow Under this restriction, if (3.2) is Fourier transformed in h according to (2.4b), the resulting equation for the transformed vertical velocity W(k, l, z) becomes Wzz 1

[

]

Ri(1 1 s 2 ) 2 (1 1 s 2 ) W 5 0, z2

(3.3)

where

s 5 l/k and z 5 kH [1 1 (z/H )]. This equation has the form of the two-dimensional (x, z) lee-wave equation analyzed by Wurtele et al. (1987), except that now the solution depends on the wavenumber l in the y direction, while the wavenumber k in the x direction is fixed. In that paper it was shown that the solution satisfying the condition of upward energy propagation is a modified Bessel function K of imaginary argument and imaginary order. By analogy, the solution to (3.3) can immediately be written as w 5 w0 (x)

1 2 z kH

1/ 2

E

`

2`

A(s)

K im (z Ï1 1 s 2 ) K im (kHÏ1 1 s 2 )

3 exp(ish) ds, where the order of the Bessel function is determined by

(3.1)

so that the mean shear is a constant, equal to U 0 /H, and U 5 U 0 at z 5 0. Note that H can be negative or positive. In this model if the stability N is constant, then so is

m 2 5 Ri(1 1 s 2 ) 2 1/4 ø Ri(1 1 s 2 ) and w 0 (x)A(s) represents the forcing at z 5 0. Evaluating this by Cauchy’s residue theorem gives resonance

15 MARCH 2004

SHARMAN AND WURTELE

lee-wave-type solutions determined by the zeros of the denominator (poles) at K im (kHÏ1 1 s 2j ) 5 0,

(3.4)

where the subscript j represents the jth zero or discrete wave. Wave trapping now occurs in the y direction, symmetric in 6y, and the eigenfunction solution for each discrete wave is of the form w j 5 w0 (x)

1 2 z kH

1/ 2

A(s j )

K i,m (z Ï1 1 s 2j ) d K (kHÏ1 1 s 2 )s5s j ds im

3 sin(s j h), and the total solution is the sum over all eigenfunctions w j . The sine term in this expression represents crossstream waves that are trapped by the shear of the basic flow, just as effectively as are downstream waves in two dimensions (e.g., Wurtele et al. 1987). They are the diverging wave components discussed in Sharman and Wurtele (1983), with the waves nearly aligned with the mean flow. It should be emphasized that these waves are nonhydrostatic in their dynamics. For larger wavelength forcings, such that k 2 K Ri/z 2 , the hydrostatic approximation applies and (3.2) reduces to Z 2w ˆ zz 2 Riw ˆ hh 1 Riw ˆ 5 0.

(3.5)

With the transformations on dependent and independent variables,

z 5 ÏRi ln(Z ),

w ˆ 5 ÏZw ˆ *,

(3.6)

(3.5) with the usual approximation Ri k 1 takes the form w ˆ *zz 2 w ˆ *hh 1 w ˆ * 5 0.

(3.7)

Thus, in the hydrostatic limit, the governing equation (3.7) has exactly the same form as (2.2) for uniform basic flow, and so the analysis of section 2 applies in the deformed coordinates (3.6). Wave trapping is therefore not allowed, which was to be expected under the hydrostatic assumption. However, consistent with the analysis of section 2, for a narrow-breadth sinusoid the hydrostatic solutions will contain singular lines along h → 6z, that is, along the curves ky 5 6ÏRi ln(kz)

(3.8)

with negligible disturbance outside the curves. The results of three simulations using this model are shown in Fig. 8 and confirm these characteristics. Here the upstream constant shear and stability are chosen so that the upstream Ri 5 8, and the forcing is provided by flow over the obstacle (2.10). Figures 8a and 8b show the patterns derived for flow over an obstacle of wavelength of 3p km and of narrow breadth (kb 5 0.1). In Figs. 8c and 8d the wavelength is the same but the obstacle breadth is increased to kb 5 2. In both cases, a ship wave–like pattern is obvious in the (x, y) plane,

673

with the waves spreading outward and downstream. From (3.4), using the parameters for these simulations, the wavelength of the cross-stream waves corresponding to the first zero is about 2.5 km. This is in good agreement with the simulation results in Figs. 8a and 8b. The diverging waves are damped by the wider obstacle and are therefore not evident in Figs. 8c and 8d. Figures 8e and 8f show the hydrostatic character when the obstacle is of narrow breadth (kb 5 0.1), but the forcing wavelength is increased to 9p km. The singular lines (3.8) are evident. For broader sinusoids, of course, the singular line disappears, as we saw in section 2. The extension to isolated obstacles for the linear profile (3.1) is difficult mathematically because of the complex nature of the Bessel eignfunctions K im . However, the dynamics involved are similar to that in which the wind increases exponentially with height, and for that profile, as shown by Sharman and Wurtele (1983), the eigenfunctions are the simpler Bessel functions of the first kind J of real order and argument. In that paper they derived (far field) solutions that are analogous to ship waves in water of finite depth, the upstream surface Richardson number, Ri 0 providing the effect of finite depth. For an upstream Ri 0 greater than a critical value (14.7 for their chosen parameters), both transverse and diverging wave systems composing the classical ship wave pattern are present with the pattern half-width depending on Ri 0 , whereas for values of Ri 0 less than the critical value, the transverse waves disappear, leaving only a diverging system of waves. It should be pointed out that these two wave systems, transverse and diverging, correspond to two distinct saddle points in the asymptotic analysis, unlike the single saddle point making up the three-dimensional wave structure in the uniform flow case. It therefore seems inappropriate to use the term ship wave in the latter (i.e., uniform flow) case. The difference in the upstream wind profiles (exponential vs linear) produces no essential difference in the dynamics; however, for the sake of completeness and since patterns in the (y, z) planes were not displayed in the Sharman and Wurtele paper, Fig. 9 displays the results of simulations for constant shear flow with an upstream Ri 5 8 (constant with height) over an isolated obstacle of the shape (2.14) for a narrow-breadth obstacle (a 5 1 km, b 5 0.3 km; Figs. 9a and 9b) and a wider-breadth obstacle (a 5 1 km, b 5 5 km; Figs. 9c and 9d). Both the transverse and diverging wave systems are apparent, with maximum amplitudes at the boundary of the wedge pattern. Consistent with the analysis of Sharman and Wurtele (1983), large-amplitude diverging waves are favored by narrow-breadth obstacles, while wider-breadth obstacles favor transverse waves. Note that the pattern in the (y, z) planes is very similar to that produced by the monochromatic forcings shown in Figs. 8a–d. b. Negatively sheared flow and critical levels Now consider flow with a linearly decreasing wind with height, so that the profile is still given by (3.1) but

674


VOLUME 61

FIG. 8. Vertical velocity fields derived from numerical simulation of constant shear flow with Ri 5 8 over a single sinusoid and of Gaussian-shaped cross section. (a)–(d) The forcing wavelength is 3p km; (e)–(f ) it is 9p km. In the top and bottom rows, the obstacle is narrow (kb 5 0.1); in the middle row, it is wider (kb 5 2). The left column displays w at streamwise position x/L 5 1/4 in the (y, z) plane; the right column displays w in the (x, y) plane at elevation z 5 1 km.

with a minus sign replacing the plus sign so that there is a mean flow critical level (U 5 0) at z 5 | H | . To simplify the analysis here, consistent with Grubisˇic´ and Smolarkiewicz (1997), a hydrostatic model of the disturbance is adopted. This seems justified, as the criticallevel effect is a hydrostatic one. The governing equation is again (3.7) where now

1

z 5 ÏRi ln 1 2

2

z , H

1

w ˆ 5 12

2

z H

1/ 2

w ˆ *.

(3.9)

The singular lines in the Cartesian system are then given by

ky 5 6ÏRi ln(1 2 z/H ),

(3.10)

that is, concave downward as the wave approaches the critical level from below, with phase-line compression in the vertical, just as in the two-dimensional case. From (2.8a) the solutions are of the form w ˆ 1* } J1 (Ïz 2 2 h 2 )

z , Ïz 2 h 2

w ˆ *2 } Y1 (Ïz 2 2 h 2 )

2

z . Ïz 2 h 2 2

(3.11)

15 MARCH 2004

SHARMAN AND WURTELE

675

FIG. 9. Vertical velocity fields derived from a numerical simulation of constant positive shear flow with Ri 5 8 over a Gaussian-shaped isolated obstacle (2.14) with a 5 1 km and of (a),(b) narrow breadth (b 5 300 m) and (c),(d) wider breadth (b 5 5 km). In (a), (c), w is displayed in the (x, y) plane at 5-km elevation; in (b), (d) w is displayed in the (y, z) plane 100 km downstream.

The critical level is z → 2`, for which value the solutions (3.11) approach

1

w ˆ *1 → |z |21/ 2 cos |z | 2

3p 4

2

(3.12)

and a similar form for w ˆ *2 with a sine term. Thus the vertical velocity vanishes with increasingly rapid oscillations, just as in the two-dimensional case (e.g., Bretherton 1966). However, in Cartesian coordinates it can easily be shown that the horizontal velocity perturbation approaches in magnitude

)

) )1

z |u9| → 1 2 H

21/ 2

2)

z ln 1 2 H

21/ 2

,

(3.13)

thus increasing without limit, but owing to the logarithmic factor, more slowly than in the two-dimensional case. This is due to the concentration of energy in the singular lines that diverts energy from the critical level and delays the onset of breakdown below the critical level. The logarithmic coordinate in the vertical is un-

changed from the two-dimensional model. It accommodates the approach to the critical level, with packing of the phase lines and the falloff of the vertical group velocity. The square root factor provides for the vanishing of the vertical velocity, and the singularity in the horizontal velocity, consistent with the phase-line pattern. The effect of the third dimension enters away from the critical level, and the three-dimensional dynamics are analogous to that associated with uniform flow, including the propagation of singularities, but in the coordinate system defined above. Figure 10 displays the results of a three-dimensional simulation of flow over a single sinusoid of narrow breadth (b 5 0.1 km) with a critical level at 15-km elevation. The wave generated at the lower boundary propagates vertically, but at 3000 time steps, the threedimensional wave in the center plane (y 5 0) is still intact (Fig. 10a). However, the singular lines are just starting to approach the critical level (Fig. 10b) and are beginning to cause compression of the phase lines below it (Fig. 10c). By 5000 time steps (Figs. 10d and 10e) substantial reflections occur below the singular lines, but the classical breakdown process arising from the

676


VOLUME 61

FIG. 10. Results from a numerical simulation of a constant negative shear flow with a critical level at 15-km elevation and forced by flow over a sinusoid of dimensional wavelength L 5 3p km and Gaussian-shaped cross section with kb 5 0.1. The horizontal perturbation velocity field, u9 after 3000 time steps (15 000 s) in (a) the y 5 0 plane, (b) the x 5 0 plane, and (c) the y 5 20 km (x, z) plane. The vertical velocity field w after 5000 time steps (25 000 s) in the (d) x 5 0 plane and (e) y 5 25 km (x, z) plane. (f ) The vertical velocity field after 6000 time steps (30 000 s) in the y 5 0 plane.

interaction of the propagating gravity wave with the critical level in the y 5 0 plane does not occur until after about 6000 time steps (Fig. 10f). This is a considerable delay from the two-dimensional result, where overturning occurs in only 3000 time steps (not shown). This delay is due to the concentration of energy in the singular lines, causing a lateral deflection of energy as the wave approaches the critical level from below. When the forcing sinusoid is broader, as discussed in section 2a, the singularity disappears, and the disturbance is free to approach the critical level, as in the two-dimensional case. The presence of the singular lines [Eq. (3.11)] obvious in the monochromatic forcing is present in the downstream flow over an isolated obstacle as well, and produces a similar result, namely, it causes a delay of breakdown under the critical level. This is demonstrated by the results of a numerical simulation of negatively sheared flow (3.1) with a critical level at 10-km elevation (Fig. 11). Figures 11a–c display the results of the simulation for a narrow-breadth (a 5 1 km, b 5 0.1 km) obstacle [Eq. (2.14)] at a time early in the breakdown process (2500 time steps, 25 000 s). The

disturbance pattern shown in the y 5 0 plane is similar to the pattern produced by flow over a single sinusoid with the breakdown into smaller wavelength cells as the disturbance approaches the critical level (see Wurtele et al. 1996). The now familiar V-shaped pattern in the (y, z) plane is evident in the v x field (Fig. 11b), but with cross-roll instabilities [noted in the simulations of Winters and D’Asaro (1994) and others] appearing below the critical level. The parabolic pattern predicted by the linear analyses of Grubisˇic´ and Smolarkiewicz (1997) is shown in the (x, y) plane (Fig. 11c), but the pattern is interrupted by the along-stream rolls. As the obstacle breadth is increased, the breakdown is more rapid, consistent with the single sinusoid results. This is demonstrated in Figs. 11d–f when the obstacle breadth is increased to 10 km with all other parameters (including resolutions) unchanged. Here the time displayed is only 1000 time steps (10 000 s) and as shown in Fig. 11d for the y 5 0 center plane, the breakdown process is already well under way. As expected for the widerbreadth obstacle, the V-shaped pattern is replaced by a two-dimensional pattern in the (y, z) plane, with compression and some cross-stream splitting under the crit-

15 MARCH 2004

677

SHARMAN AND WURTELE

FIG. 11. Results from numerical simulations of a constant negative shear flow with a critical level at 10-km elevation forced by flow over an isolated elliptical obstacle of Gaussian shape with (a)–(c) a 5 1 km, b 5 0.1 km after 2500 time steps (25 000 s) and (d)–(f ) a 5 1 km, b 5 10 km after 1000 time steps (10 000 s). In (a), (d) w is displayed in the y 5 0 center plane; v x is displayed in (y, z) plane at (b) x 5 15 km and (e) x 5 5 km; w is displayed in the z 5 5 km (x, y) plane in (e), (f ).

ical level (Fig. 11e). At this early time the downstream wave pattern in the (x, y) plane has not had time to establish (Fig. 11f). 4. Implications for gravity wave drag parameterizations Both the analyses and simulations presented here were obtained using an obstacle-induced vertical velocity specified at z 5 0, as opposed to along the obstacle surface z 5 h(x, y). In other words, the inverse Froude number Nh 0 /U is very small, and therefore we have neglected nonlinear leeside effects such as vortex shedding and flow splitting. In situations where the inverse Froude number is large, substantial changes to the dynamics can take place which may overwhelm the effects described here (e.g., Baines 1995). Nevertheless, the linear regime may still be applicable in some situations, and it is therefore of interest to examine the effect of obstacle breadth on gravity wave drag, especially for different upstream flow structures. For the purposes of presentation, in order to remove the explicit effects of the obstacle height and breadth, we define a nondimensional drag as the longitudinal

wave drag D x normalized by D 3DH , the 3D hydrostatic wave drag value for uniform flow. The 2D hydrostatic wave drag ` dh D2DH 5 p(x, z) dx| z50 (4.1) dx 2` can be evaluated using the relation pˆ(k, 0) 5 ir 0 NUhˆ (k), where r 0 is the mean density at z 5 0. Inverting this, substituting into (4.1) and changing the order of integration gives r NU 2 1` D2DH 5 0 |hˆ (k)| 2 k dk. (4.2) p 2` For the 2D Gaussian obstacle (2.14) with b → `, hˆ (k) 5 2h 0 a(p)1/2 exp(2a 2 k 2 ), and (4.2) evaluates to r 0 U 0Nh 20 per unit breadth, 4/p times larger than the corresponding value for the bell-shaped obstacle. In three dimensions the corresponding expression for drag is (e.g., Phillips 1984) 1` 1` ]h(x, y) D3DH 5 p(x, y, z) dx dy| z50 ]x 2` 2`

E

E

E E 5

r 0 NU 2 4p 2

E E 1`

2`

1`

2`

k 2|hˆ (k, l)| 2 dk dl. (k 2 1 l 2 )1/ 2

(4.3)

678


FIG. 12. Values of the normalized drag D*x defined by Eq. (4.4) for various N, U, and obstacle geometries as a function of the obstacle aspect ratio, b/a. Here, ● 5 U, N constant, a 5 1 km (runs 2 in Table 1); C 5 U, N constant, a 5 5 km (runs 3 in Table 1); m 5 negative shear flow with a critical level at z 5 10 km (runs 7 in Table 1); and D 5 positive shear flow (runs 5 in Table 1).

This can be easily evaluated for the Gaussian-shaped circular obstacle (2.14) with a 5 b as (p/2) 3/2 r 0UNh 20a (with units of force), but for arbitrary obstacle ellipticity (4.3) must be evaluated by numeric cubature as done by Phillips (1984). Figure 12 displays some values of the normalized drag D*x D*x 5

Dx D3DH

E E 1`

5

2`

1`

2`

p(x, y, z)

]h0 (x, y) dx dy| z50 ]x

D3DH

(4.4)

with the numerator derived from the previously presented simulations, and the denominator evaluated by numeric cubature of (4.3) as a function of obstacle aspect ratio, b/a, for three different environments. Note these results represent the linearized wave drag and do not include regimes where the waves may break aloft. Figure 12 extends the results presented in Phillips (1984, his Fig. 8) to aspect ratios less than unity. Note that the normalized drag for the long obstacle (a 5 5 km) in the uniform flow case is nearly unity for all values of the obstacle breadth, reinforcing earlier observations of the hydrostatic character of the flow fields for obstacle lengths *3 km. All other essentially nonhydrostatic configurations displayed have substantially lower values of D*x and all configurations show markedly different values for very narrow obstacle breadths, but leveling off to nearly constant values for b/a * 3. The strong train of trapped lee waves present in the constant shear

VOLUME 61

cases tends to enhance the values of D*x , but the effects of the critical level in the negatively sheared cases tends to lower the values. Note that these critical-level cases are for a relatively large upstream Ri (5100), so that the effects of Ri in reducing drag as discussed by Grubisˇic´ and Smolarkiewicz (1997) are not likely present. But the obstacle length is small (a 5 1 km), so we infer that the smallness of the D*x is due to purely nonhydrostatic effects. In fact another critical-level simulation with the same upstream Ri but with a circular Gaussian obstacle of a 5 5 km (run 8 in Table 1, not shown) gave a D*x of 0.98. All these factors add uncertainties to GWD parameterizations that are based on nonhydrostatic assumptions. Further, for very narrow-breadth obstacles, wave breaking, if it occurs, will likely be in the vicinity of the singular line(s), that is, maximum momentum deposition occurs to the sides of the obstacle rather than over it. 5. Summary and conclusions The flow over isolated peaks and ridges within a mountain chain offer a rich variety of lee-wave regimes depending on the individual obstacle geometry and structure of the upstream environment. To better understand the spectrum of possible lee-wave structures, the nature of flow of a stratified fluid of idealized vertical structure and over idealized obstacle shapes was investigated using a combination of linear analysis and numerical simulations. Emphasis was placed on gaining a better understanding of 1) the cross-stream structure of the three-dimensional lee wave, 2) the transition from three-dimensional to two-dimensional flow as the breadth of the obstacle increases, 3) the transition from nonhydrostatic to hydrostatic behavior, and 4) the effect of obstacle breadth-to-length aspect ratio on the wave drag for this linear system. The flow over isolated obstacles, which contains a spectrum of forcing wavelengths, can be understood by reference to analyses of the simpler configuration of a single wave (monochromatic) forcing. In particular, the structure of three-dimensional lee waves is related to the presence of outward-sloping singular lines which form a V-shaped region in the (y, z) plane that defines the location of maximum wave amplitudes. The singular lines were shown to exist for both hydrostatic and nonhydrostatic uniform flow, and hydrostatic shear flow; they are the analog of the upper half of the ‘‘St. Andrew’s Cross’’ first demonstrated in stratified tank experiments with oscillating sources. For uniform flow the lines are straight, for shear flow; they are curved. The lines become a zone of maximum amplitude that gradually disappears and becomes a quasi-two-dimensional pattern as the obstacle breadth is progressively increased. The lines induce large shears and potential for breaking in the vicinity of the lines (i.e., to the sides of the obstacle rather than above it), and for backward

15 MARCH 2004

679

SHARMAN AND WURTELE

shear flow with a critical level, they divert energy away from the critical level and delay the onset of instability under the critical level. For the linear systems considered here, the wave pattern is completely determined by the aspect ratio, b/a, and the nondimensional obstacle length, Na/U. For the Gaussian obstacle shape considered here, the transition to two-dimensional patterns occurs for b/a * 5, and the transition from nonhydrostatic to hydrostaticlike solutions occurs for obstacle lengths Na/U * 3, regardless of the obstacle breadth. Although the flow configurations considered are simple, namely, unidirectional flow with no directional shear, the results hold provided the directional shear is not too large, but no systematic study was attempted to define the transition to the directional shear solutions provided by Broad (1995) and Shutts (1998). Finally, the gravity wave drag as measured by the drag ratio D*x for narrow-breadth obstacles was found to be considerably altered from that associated with even moderate-breadth obstacles. This fact, together with three-dimensional wave spreading effects and the presence of singular lines, should be considered in future gravity wave drag parameterization formulations. Acknowledgments. We are grateful to Teddie Keller, Piotr Smolarkiewicz, and Todd Lane for their comments which helped improve the manuscript. We are also grateful to the three anonymous reviewers for their careful reading of the manuscript and their helpful comments. APPENDIX Review of Theory of Nonhydrostatic Flow over Isolated Obstacles Approximations for uniform flow over an infinitely narrow obstacle in x and y (expressed mathematically as a doublet or source–sink dipole—that is, a point source of upward velocity next to a point sink of downward velocity when the distance between them approaches zero) were derived by Wurtele (1957), Crapper (1959), and Janowitz (1984). All used the saddle point or stationary phase methods and are therefore generally valid far downstream from the doublet and all obtained consistent expressions for wave amplitudes and phase lines downstream (x . 0) of the obstacle, and have the form w(x, y, z) 5 A sinf, where A5

2UQ(r 4 1 x 2 y 2 )1/ 2 x 2 z 2 , pk s r 4 R 3

f5

k s zR , r

(A.1)

r2 5 y2 1 z2, R2 5 x2 1 y2 1 z2,

(A.2)

and Q is the doublet strength. For large x, lines of constant phase satisfy the relation,

f5

k s zR k xz ø 2 s 2 1/ 2 5 constant. r (y 1 z )

(A.3)

Although this result (A.3) was derived laboriously by asymptotic evaluations of the double Fourier integrals, the same result can be derived simply by consideration of the shape of phase surfaces as dictated by the singular lines present in the monochromatic forcing solutions presented in section 2. From (2.9), for positive y the pole at k 5 k p say, is located along the line (k s2 2 k p2 )1/2 z 2 ky 5 0.

(A.4)

And therefore the phase is

f 5 kP x 5

k s zx . (y 1 z 2 )1/ 2 2

(A.5)

From (A.3) or (A.5) lines of constant phase in (x, y) planes can immediately be identified as rectangular hyperbolas k s2 x 2 y2 2 2 5 1, 2 f0 z0 where the constant phase f 5 f 0 at z 5 z 0 , as derived by Wurtele (1957) and Crapper (1959). The shape of the (nonhydrostatic) phase lines in the (x, z) and (y, z) planes were not discussed in the earlier papers. In the y 5 0(x, z) plane the phase lines, as in two dimensions, are simply circles centered at the origin. For small z, the phase lines become increasingly vertical downstream since (x 2 1 z 2 )1/2 ø x 1 z 2 /2x. However, for any (x, z) plane with | y | 5 | y 0 | . 0, this configuration is altered. From (A.2) lines of constant phase, C, obey the relation k s2 z 2 (x 2 1 y 20 1 z 2 ) 5 C(y 20 1 z 2 ). For small z K y 0 , z } (x 2 1 y 20) 21/2 , that is, the phase lines become parallel to the surface z 5 0. The larger the value of y 0 the deeper the layer of parallel lines. The shape in the (y, z) plane may be derived from (A.5): lines of constant phase satisfy z2 ø

C 2y2 k s2 x 2 2 C 2

provided C 2 , k s2x 2 . These are straight lines emanating from the origin y 5 z 5 0. Thus the shape of the phase lines in the (x, z) and (y, z) planes downstream of an isolated obstacle are completely consistent with singular lines generated by a monochromatic forcing with a wavenumber given by (A.4). For larger values of the phase C, such that C 2 , k s2x 2 , this equivalency breaks down. In this case (A.2) gives the parabolic shape z 2 ø C 2 /k s2 2 x 2 2 y 2 . Therefore, surfaces of constant phase

680


VOLUME 61

FIG. A1. Shape of the 12p phase surface for lee waves as given by Janowitz’s (1984) asymptotic analysis.

slope downward and the phase surfaces become more compressed near the surface with increasing x. For x small and z large, z ø C, that is, the phase lines are horizontal and straight. As an example, Fig. A1 displays the three-dimensional configuration of a constant phase surface. The amplitude in the y 5 0 plane is, from (A.1), A5

2UQx 2 2UQ sin 2g 5 , 3 pk s R pk s R

where g is the angle from the zenith z 5 0 in the (x, z) plane. For a given zenith angle this falls off as R 21 and has its maximum near y 5 p/2 (z 5 0). The asymptotic analysis is not valid at z 5 0, where the boundary condition is w 5 0. It is not clear from the analysis how close to the z 5 0 plane (A.1) can be applied, but the implication is that maximum amplitudes downstream must exist for some small z. In any case, the three-dimensional solution in the y 5 0 plane has a behavior distinctly different from the two-dimensional solution for a ridge, which has its maximum amplitude along the line g 5 p/4 [e.g., Baines 1995, his Eq. (5.2.19)]. Consequently, for small y and z, the wave amplitudes may actually be larger downstream of an isolated obstacle of narrow width and breadth than over a two-dimensional ridge of the same width. The amplitude in (y, z) planes downstream for larger x is approximately 2UQz 2 y 2UQ cos 2u sinu A; 5 , pk s r 4 pk s r where u is defined by (2.9). The disturbance thus dies out as r 21 from the origin and has a maximum at tanu 5 1Ï2 or u ø 358, regardless of the value of k s . All this agrees qualitatively with what we would expect from the narrow sinusoid solutions (2.8) but with a limited obstacle in x. The singularity introduced at the boundary, although present for each individual component k, is propagated along singular lines of different slope and is smoothed out by the continuous k spectrum

FIG. A2. Location of the lines of maximum amplitude of lee waves as given by Crapper’s (1962) asymptotic analysis [his Eq. (46)] for various nondimensional obstacle breadths k s b.

introduced by the isolated obstacle. From (2.9) this implies a band of wavenumbers centered at k ø (2/3)1/2 k s gives the maximum response for the doublet forcing. The extension to obstacles of finite length and breadth is mathematically difficult. Crapper (1959, 1962) did obtain approximate solutions for circular and elliptical obstacles; however, the solutions he obtained are too complicated to offer much insight into the character of the solutions for various obstacle aspect ratios. Further, there are some severe restrictions on the range of validity of his asymptotic expressions. For example, Crapper’s (1962) expressions for the wave displacement [his Eqs. (46) and (47)] are limited to large x, small z (z 2 / b 2 x K 1), and small y (b 2 x 2 y/2z 2 K 1). However, his expressions do offer some guidance concerning the effect of the obstacle breadth. Maximum wave amplitudes derived from his Eq. (46) for various obstacle breadths b are plotted in Fig. A2, which shows that, consistent with the monochromatic solutions, the maximum wave amplitudes are to the sides of the obstacle for small b, but move to a position over the obstacle as b increases. REFERENCES Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 482 pp. Bauer, M. H., G. J. Mayr, I. Vergeiner, and H. Pichler, 2000: Strongly nonlinear flow over and around a three-dimensional mountain as a function of the horizontal aspect ratio. J. Atmos. Sci., 57, 3971–3991. Berkshire, F. H., 1975: Critical levels in a three-dimensional stratified shear flow. Pure Appl. Geophys., 113, 561–568. Bretherton, F. P., 1966: The propagation of groups of internal waves in a shear flow. Quart. J. Roy. Meteor. Soc., 92, 466–480. Broad, A. S., 1995: Linear theory of momentum fluxes in 3-D flows with the turning of the mean wind with height. Quart. J. Roy. Meteor. Soc., 121, 1891–1902.

15 MARCH 2004

SHARMAN AND WURTELE

——, 1999: Do orographic gravity waves break in flows with uniform wind direction with height? Quart. J. Roy. Meteor. Soc., 125, 1695–1714. Crapper, G. D., 1959: A three-dimensional solution for waves in the lee of mountains. J. Fluid Mech., 6, 51–76. ——, 1962: Waves in the lee of a mountain with elliptical contours. Philos. Trans. Roy. Soc. London, 254A, 601–623. Epifanio, C. C., and D. R. Durran, 2001: Three-dimensional effects in high-drag-state flows over long ridges. J. Atmos. Sci., 58, 1051–1065. Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp. Gjevik, B., and T. Marthinsen, 1978: Three-dimensional lee-wave pattern. Quart. J. Roy. Meteor. Soc., 104, 947–957. Gradshteyn, I. S., and I. M. Ryzhik, 1994: Tables of Integral, Series and Products. 5th ed. Academic Press, 1086 pp. Grubisˇic´, V., and P. K. Smolarkiewicz, 1997: The effect of critical levels on 3D orographic flows: Linear regime. J. Atmos. Sci., 54, 1943–1960. Janowitz, G. S., 1984: Lee waves in three-dimensional stratified flow. J. Fluid Mech., 148, 97–108. Lane, T. P., M. J. Reeder, and T. L. Clark, 2001: Numerical modeling of gravity wave generation by deep tropical convection. J. Atmos. Sci., 58, 1249–1274. Lighthill, J. M., 1978: Waves in Fluids. Cambridge University Press, 504 pp. ——, 1996: Internal waves and related initial-value problems. Dyn. Atmos. Oceans, 23, 3–17. Mowbray, D. E., and B. S. H. Rarity, 1967: A theoretical and experimental investigation of the phase configuration of internal gravity waves of small amplitude in a density stratified liquid. J. Fluid Mech., 28, 1–16. Phillips, D. S., 1984: Analytic surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci., 41, 1073–1084. Queney, P., 1948: The problem of airflow over mountains: A summary of theoretical studies. Bull. Amer. Meteor. Soc., 29, 16–26.

681

Scorer, R. S., 1949: The theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc., 75, 41–56. Sharman, R. D., and M. G. Wurtele, 1983: Ship waves and lee waves. J. Atmos. Sci., 40, 396–427. Shutts, G. J., 1995: Gravity-wave drag parameterization over complex terrain: The effect of critical level absorption in directional wind shear. Quart. J. Roy. Meteor. Soc., 121, 1005–1021. ——, 1998: Stationary gravity wave structure in flows with directional wind shear. Quart. J. Roy. Meteor. Soc., 124, 1421–1442. Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32, 348–364. ——, 1988: Linear theory of stratified flow past an isolated mountain in isosteric coordinates. J. Atmos. Sci., 45, 3889–3896. ——, 1989a: Mountain induced stagnation points in hydrostatic flow. Tellus, 41A, 270–274. ——, 1989b: Hydrostatic airflow over mountains. Advances in Geophysics, Vol. 31, Academic Press, 1–41. Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci., 46, 1154–1164. ——, and ——, 1990: Low Froude number flow past three-dimensional obstacles. Part II: Upwind flow reversal zone. J. Atmos. Sci., 47, 1498–1511. Welch, W. T., P. K. Smolarkiewicz, R. Rotunno, and B. A. Boville, 2001: The large-scale effects of flow over periodic mesoscale topography. J. Atmos. Sci., 58, 1477–1492. Whitham, G. B., 1974: Linear and Nonlinear Waves. Wiley-Interscience, 636 pp. Winters, K. B., and E. A. D’Asaro, 1994: Three-dimensional wave instability near a critical level. J. Fluid Mech., 272, 255–284. Wurtele, M. G., 1957: The three-dimensional lee wave. Beitr. Phys. Atmos., 29, 242–252. ——, R. D. Sharman, and T. L. Keller, 1987: Analysis and simulations of a troposphere–stratosphere gravity wave model. Part I. J. Atmos. Sci., 44, 3269–3281. ——, A. Datta, and R. D. Sharman, 1996: The propagation of gravity– inertia waves and lee waves under a critical level. J. Atmos. Sci., 53, 1505–1523.