Effects of Horizontal Geometrical Spreading on the Parameterization of Orographic Gravity Wave Drag. Part II: Analytical Solutions

Stephen D. Eckermann Space Science Division, Naval Research Laboratory, Washington, D.C.

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Dave Broutman Computational Physics, Inc., Springfield, Virginia

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Harold Knight Computational Physics, Inc., Springfield, Virginia

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Abstract

Effects of horizontal geometrical spreading on the amplitude variation with height of linear three-dimensional hydrostatic orographic gravity waves (OGWs) are quantified via relevant simplifications to the governing transform relations, leading to analytical solutions. The analysis is restricted to elliptical Gaussian obstacles with principal axes aligned parallel and perpendicular to unidirectional shear flow and to vertical displacement and steepness amplitudes, given their relevance to OGW drag parameterizations in global models. Two solutions are derived: a “small l” solution in which horizontal wavenumbers l orthogonal to the flow are taken to be much smaller than those parallel to the flow, and a “single k” solution in which horizontal wavenumbers k parallel to the flow have a single mean value. The resulting analytical relations, valid for arbitrary vertical profiles of upstream winds and stability, depend only on the obstacle’s elliptical aspect ratio β and a normalized height coordinate incorporating wind and stability variations. These analytical approximations accurately reproduce the salient features of the exact numerical transform solutions. These include monotonic decreases with height that asymptotically approach z−1/2 forms at large z and strong β dependence in amplitude diminution with height. Steepness singularities close to the surface are shown to be a mathematical consequence of the Hilbert transform approach to deriving complex wavefield solutions. These approximate analytical solutions for horizontal geometrical spreading effects on wave amplitude highlight the importance of this missing physics for current parameterizations of OGW drag and offer an accurate and efficient means of incorporating some of these omitted effects.

Corresponding author address: Stephen Eckermann, Code 7631, Geospace Science and Technology Branch, Space Science Division, Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375. E-mail: stephen.eckermann@nrl.navy.mil

Abstract

Effects of horizontal geometrical spreading on the amplitude variation with height of linear three-dimensional hydrostatic orographic gravity waves (OGWs) are quantified via relevant simplifications to the governing transform relations, leading to analytical solutions. The analysis is restricted to elliptical Gaussian obstacles with principal axes aligned parallel and perpendicular to unidirectional shear flow and to vertical displacement and steepness amplitudes, given their relevance to OGW drag parameterizations in global models. Two solutions are derived: a “small l” solution in which horizontal wavenumbers l orthogonal to the flow are taken to be much smaller than those parallel to the flow, and a “single k” solution in which horizontal wavenumbers k parallel to the flow have a single mean value. The resulting analytical relations, valid for arbitrary vertical profiles of upstream winds and stability, depend only on the obstacle’s elliptical aspect ratio β and a normalized height coordinate incorporating wind and stability variations. These analytical approximations accurately reproduce the salient features of the exact numerical transform solutions. These include monotonic decreases with height that asymptotically approach z−1/2 forms at large z and strong β dependence in amplitude diminution with height. Steepness singularities close to the surface are shown to be a mathematical consequence of the Hilbert transform approach to deriving complex wavefield solutions. These approximate analytical solutions for horizontal geometrical spreading effects on wave amplitude highlight the importance of this missing physics for current parameterizations of OGW drag and offer an accurate and efficient means of incorporating some of these omitted effects.

Corresponding author address: Stephen Eckermann, Code 7631, Geospace Science and Technology Branch, Space Science Division, Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375. E-mail: stephen.eckermann@nrl.navy.mil

1. Introduction

In the first part of this work, Eckermann et al. (2015, hereafter Part I) quantified the effect of horizontal geometrical spreading on the vertical variation of peak amplitudes of three-dimensional hydrostatic gravity waves generated by flow over elliptical obstacles. On deriving numerical solutions to the exact Fourier-ray (FR) formulation of the problem, they used a modified inverse Fourier transform of those FR solutions to obtain a complex spatial wavefield containing local amplitude information. By selecting the maximum wavefield amplitude at each height, vertical profiles were constructed and then normalized to remove density and refraction effects, thereby isolating and quantifying the contribution due to horizontal geometrical spreading.

Their results showed large effects of horizontal geometrical spreading on wave-amplitude evolution with height for all but the most quasi-two-dimensional obstacles with long axes aligned orthogonal to the incident flow. Since horizontal geometrical spreading influences on wave amplitudes are absent from the current generation of orographic gravity wave drag (OGWD) parameterizations used in general circulation models (GCMs) of weather and climate, the findings of Part I call for efforts to include these effects in parameterizations. To do so requires a parameterization that is computationally cheap, yet accurate and general enough to apply over the entire parameter space of surface subgrid-scale terrain properties and arbitrary vertical profiles of atmospheric winds and temperatures, as occurs within a GCM. While the diagnostic FR-based numerical method of Part I is sufficiently general, it is both too expensive and impractical to incorporate into existing OGWD parameterizations.

Part I obtained numerical solutions for idealized upstream wind and stability profiles only. Nonetheless, those results showed some interesting properties that suggested a compact bulk parameterization of these effects for general wind and temperature profiles might be possible. For example, they found that variation of the effect with background wind and temperature could be entirely captured, at least for the simple problems that they considered, by remapping profiles to a transformed and dimensionless height coordinate. Another property was a remarkable insensitivity to the precise functional form of the elliptical obstacle but strong sensitivity to its elliptical aspect ratio.

Thus, motivated by these initial results of Part I, this companion paper also seeks to quantify the effects of horizontal geometrical spreading using the governing transform relations, as outlined in section 2. However, rather than pursuing numerical solutions to the exact FR formulation as in Part I, in section 3 we investigate potentially relevant simplifications to these equations that allow us to derive analytical solutions to this problem. Separate analytical solutions under a “small l” and “single k” approximation are derived, initially for upstream wind and stability profiles that do not vary with height, which reveal close agreement with corresponding values derived from numerical FR solutions. In section 4 we discuss how these analytical solutions generalize to arbitrary vertical profiles of upstream wind and stability and in section 5 compare their predictions to those from exact numerical FR solutions for problems of progressively greater complexity in the upstream wind and stability profiles. Section 6 discusses some additional effects requiring further research as possible pathways from our initial results, summarized in section 7, to a complete description of horizontal geometrical spreading for arbitrary subgrid-scale terrain and vertical profiles of vector winds and temperature, as is ultimately needed for OGWD parameterizations.

2. Theoretical framework

We consider upstream flow of horizontal velocity and stability impinging upon a three-dimensional obstacle of elevation and peak height . To focus solely on typical parameterization approaches for orographic gravity waves, we adopt the following simplifications, as described and justified in greater depth in, and using the same mathematical nomenclature as, Part I:
  • the linear limit ,

  • a hydrostatic nonrotating dispersion relation , and

  • winds aligned along the x direction .

The wave’s intrinsic frequency then simplifies to , and the gravity wave dispersion relation simplifies to
e1
where is the wavenumber vector and s is a sign parameter, set opposite to the sign of at the ground to ensure geophysically consistent sign conventions for group and phase velocities (Part I).
The numerical FR results of Part I showed that the effects of horizontal geometrical spreading on wave amplitudes were insensitive to the precise functional form for the elliptical obstacle : see their Figs. 5–7. Here it proves convenient mathematically to adopt the elliptical Gaussian form
e2
where, as in Part I, we quantify the obstacle’s ellipticity by the aspect ratio
e3
The Fourier transform of this Gaussian hill function is
e4
Under the aforementioned approximations, the FR solution for vertical displacement in (22) of Part I simplifies to (e.g., Broutman et al. 2002)
e5
where
e6
and is atmospheric density, while the corresponding steepness solution
e7
where
e8
e9
FR solutions for various state parameters X allow us to derive corresponding spatial wavefields via the inverse Fourier transform
e10
We note in passing here that, although application of (10) to the FR steepness solutions in (7)(9) yields well-behaved wavefields at , a singularity arises as , which is discussed in section 3d and is addressed in greater detail by Knight et al. (2015).
As in Part I, we consider two simple forms for S. The first, , yields the geophysical wavefield , which is real since is Hermitian. The second,
e11
yields the complex wavefield solution
e12
the real part of which returns (Part I). This complex solution [(12)] is of primary interest here, since yields local estimates of the peak wave amplitude , which Part I studied via vertical profiles of maximum values at each height; namely,
e13
Their numerical results showed that these maxima were generally located close to the obstacle peak at (see their Fig. 3). Above the obstacle peak, combining (10) and (11) with the FR vertical displacement solution [(5)] and the Gaussian form [(4)] yields
e14
where
e15
e16
In (14) we have represented (11) by choosing then restricting the integration limits for k to from (11) and doubling the FR values. Note that the values of are immaterial here since these waves are all nonpropagating (). Since this integration range for k retains the FR components with and removes all the FR components with , we set in deriving (16).

3. Uniform flow

Following Part I, we consider first the case in which U and N do not vary with height, such that and . Since (14) is derived from (11) by assuming , then the normalized height coordinate [(15)] simplifies to
e17
while (6) and (8) simplify to
e18

To make analytical progress in evaluating the integral (14), we must now deal with the dependence of the exponential argument in (16).

a. Two-dimensional limit

A two-dimensional ridgelike obstacle with b and implies , yielding from (1) given and . For vertical displacement, (10) and (11) yield
e19
For consistency of notation, we have retained a functional y dependence above, even though there is no y dependence in these particular solutions. Using the result [(7.4.2) of Abramowitz and Stegun (1972)]
e20
then (19) yields the analytical solution
e21
where erfc is the complementary error function [see section 7.1 of Abramowitz and Stegun (1972)]. Taking from (21) and differentiating with respect to z gives the wave steepness
e22
Thus, directly above the mountain, (21) yields
e23
which can also be derived by integrating (14) using (16). Likewise, (22) yields
e24
Part I quantified the vertical variation of wave amplitude due to horizontal geometrical spreading using the magnitude of normalized profiles, defined under the current notation and in their complex form as
e25
e26
Substituting (18) and (23) into (25), and (18) and (24) into (26), the normalized profile solutions in the two-dimensional limit are
e27
e28
These normalized profile magnitudes of unity indicate that there is no vertical amplitude variation due to horizontal geometrical spreading in the two-dimensional “ridge” limit. As we show later in section 6, this is consistent with the generation of plane waves with horizontal wavenumber vectors parallel to the flow, which propagate purely vertically owing to vanishing horizontal group velocities relative to the ground and so do not spread horizontally.

b. Small-l approximation

We now return to the three-dimensional case. Close to the mountain peak, we expect the main contributions to the integral (14) to come from horizontal wavenumbers that are directed approximately upstream and parallel to . This implies , so that we can approximate using the truncated Maclaurin series expansion
e29
The small-l approximation of (16) is then
e30
Substituting (30) into (14) and reversing the order of integration, we obtain
e31
Evaluating the l integral in (31) analytically using (20) leads to
e32
After some algebra, the k integral in (32) can be evaluated analytically as
e33
which, when substituted into (32) and using and [(7.1.16) and (7.1.2), respectively, of Abramowitz and Stegun (1972)], yields
e34
This is our small-l complex vertical displacement solution. The corresponding normalized form [(25)] is
e35

Note that the solutions (34) and (35) have no explicit dependence on a or b, only on the aspect ratio . Since , (34) reproduces the lower boundary condition . For , (35) reduces to the two-dimensional limit [(27)].

For , (35) simplifies to
e36
e37
which was derived using the asymptotic limit for large t of
e38
[(7.1.23) of Abramowitz and Stegun (1972)]. Thus the far-field vertical variation of vertical displacement amplitude due to horizontal geometrical spreading is
e39
implying a far-field height dependence at , where
e40
Note the strong quadratic β dependence of the far-field height limit in (40).

For , the limiting form [(37)] agrees with the stationary-phase solution of Smith (1980), when his (34) is expressed in complex form (see the final equation of his appendix IV) at . Smith’s solution is valid for and near the vertical axis. It agrees with our solution even though Smith (1980) used bell-shaped rather than Gaussian topography, confirming again the finding of Part I that these profiles are insensitive to the adopted three-dimensional functional form for .

We derive a corresponding solution for steepness by taking from (34) and differentiating with respect to . After some algebra and use of the relation
e41
the final expression in terms of the normalized wave steepness [(26)] is
e42

Figure 1 plots profiles of both (black curves) and (blue curves) for 11 different β values equispaced logarithmically between 10−1 and 10+1. Note that, for U0 = 10 m s−1 and N0 = 0.01 s−1, the values on the y axis in Fig. 1 can be interpreted as heights in kilometers. We see that horizontal geometrical spreading due to obstacle three dimensionality always leads to amplitude reductions relative to the height-invariant two-dimensional limits of unity in (27) and (28). The reductions are a strong function of β—largest for small-β obstacles and smallest for large-β obstacles—consistent with increases in β driving wave responses closer to the two-dimensional limit of no horizontal geometrical spreading derived in section 2a.

Fig. 1.
Fig. 1.

Profiles of normalized peak vertical displacement amplitudes (black curves) and normalized peak steepness amplitudes (blue curves), for discrete , where and . The βj = 10−1, 100, and 101 curves are labeled.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

At large , the vertical displacement and steepness curves in Fig. 1 overlay, which is not obvious from inspecting (42). On substituting the form for given by (37) into (42), the separate β-dependent terms in (42) exactly cancel, yielding the simple result , and thus , for , in agreement with the numerical FR results of Part I: see their Figs. 4, 7, and 11.

Near the ground, Fig. 1 shows that the steepness amplitudes given by (42) diverge from the vertical displacement amplitudes because of the dependence of the second term in (42), which becomes singular as . The numerical FR steepness solutions of Part I also showed divergent behavior near the ground. We revisit this behavior of the small-l and numerical FR steepness amplitudes in section 3d.

c. Single-k approximation

We can further simplify the dependence in (1) and (16) by assuming that the wavefield response is characterized by a single dominant k that is of the order of the mountain width in the x direction, a, such that
e43
where γ is a scaling constant of order unity. Then the truncated Maclaurin series [(29)] simplifies to
e44
leading to a single-k approximation to the exponential argument [(16)] of
e45
e46
Substituting (46) into (14) yields a Gaussian integral in , which can be evaluated using (20) to yield the single-k vertical displacement solution. Expressed in the normalized form [(25)], the result, after some algebraic manipulation, is
e47
This solution reproduces the lower boundary condition . As , (47) reduces to the limit [(27)]. For , (47) becomes
e48
Comparing (48) with the corresponding limit of the small-l solution [(37)], we get equality [i.e., ] in the limit on setting . With this choice of γ, the single-k wavenumber in (43) becomes . Using (4), it can be shown that this value is the mean k value that arises as a weighted average of the terrain elevation spectrum along the x direction: that is,
e49
We derive a corresponding single-k steepness solution by differentiating with respect to . In the normalized form [(26)], the result is
e50
For , (50) and the limiting forms in (48) and (39) yield
e51
for . This approximate equality between the vertical displacement and steepness amplitudes at is consistent with the FR solutions of Part I and the small-l solutions plotted in Fig. 1, which all superimpose onto a universal form at large z for a given β.

Figure 2 plots the real parts of the complex small-l, single-k, and numerical FR solutions above the hill as well as the magnitude of the numerical FR solution , for a case with U0 = 15 m s−1 and N0 = 0.02 s−1. For , the curves in Fig. 2a are all in close agreement, except very near the ground. Note the order-of-magnitude reductions in amplitude due to horizontal geometrical spreading. For , the small-l and FR solutions in Fig. 2b compare very well, whereas the single-k solution slightly overestimates amplitudes. Note that the limiting forms where small-l and single-k solutions superimpose occur only at heights where . From (40), zff = 11 m for the obstacle, whereas zff = 48 km for the obstacle. Since at most heights in Fig. 2a, the curves superimpose, whereas, since in Fig. 2b, small near-field differences emerge among the solutions.

Fig. 2.
Fig. 2.

Vertical profiles of the real part of the numerical FR solution (blue), the real part of the small-l solution (open circles), the real part of the single-k solution (red), and the magnitude (amplitude) of the numerical FR solution (thick black line), for z = 0–20 km for unsheared upstream flow of U0 = 15 m s−1 and N0 = 0.02 s−1 impinging on Gaussian terrain [(2)] of (a) and (b) .

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

Figure 3 compares the magnitudes of the small-l and single-k normalized vertical displacement solutions with the numerical FR results of Part I. There is excellent agreement among the FR results and the approximate analytical solutions for all β and . The only noticeable differences occur very close to the ground for small-β obstacles. This finding is consistent with the results presented in Fig. 3 of Part I, which showed that maximum amplitudes for obstacles occurred some distance downstream of the obstacle peak, before returning closer to the obstacle peak at larger z. Recall that our analytical solutions are derived above the obstacle peak at .

Fig. 3.
Fig. 3.

Profiles of normalized peak vertical displacement amplitudes, showing from the numerical FR experiments of Part I (gray curves), (red curves) and (green curves), for β values ranging from (a) 1/8 to (i) 8.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

The analytical solutions accurately reproduce both the shape and the magnitude of the universal form of the numerical FR profiles at large z. This far-field dependence of the horizontal geometrical spreading effect is given in our theoretical relations (39), (48), and (51). The intrinsic dependence of these analytical solutions also explains why the numerical FR solutions in Part I varied with when plotted as a function of z but all collapsed onto a single common profile for a given β when replotted as a function of . Figure 3 also shows that both the small-l and single-k solutions accurately capture the changes in the profiles as the obstacle aspect ratio β varies.

d. Steepness amplitude as

Figure 4 repeats the same presentation as Fig. 3 for profiles of peak steepness amplitude. As for η, all the normalized amplitude profiles agree closely at upper levels. Nearer the ground, the curves diverge more, and all attain values greater than unity near the surface, with the effect more pronounced at smaller β.

Fig. 4.
Fig. 4.

Profiles of normalized peak steepness amplitudes, showing from the numerical FR experiments of Part I (gray curves), (red curves) and (green curves), for β values ranging from (a) 1/8 to (i) 8.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

As noted in section 3b, as , the small-l steepness solution becomes singular because of the dependence of the second term in (42). By contrast, the single-k steepness solution remains finite near the ground, asymptotically approaching a limiting value of as according to (50). For , this limiting value is near unity, as evident in Fig. 4i, for example, but becomes much larger for , as evident in Fig. 4a.

It can be shown analytically that the FR amplitude solution for steepness also diverges as . Differentiating the Boussinesq form in (14) with respect to z yields the corresponding steepness relation
e52
where m is defined as in (1) with . We derived (52) by taking the z derivative inside the integrand, which is valid for , even though the k integral is improper with a singular integrand at : for theoretical justification, see Knight et al. (2015).

In the limit , the term in (52) diverges as , while the and terms oscillate infinitely rapidly. As , these rapid oscillations cause cancellations that allow the real part of the integral to converge. However, the imaginary part containing the term becomes singular.

Knight et al. (2015) show that for , as and , the real and imaginary components of the solution to (52) are given by
e53
e54
Note that Knight et al. (2015) restricted integration to positive k in their integrals, whereas we restricted to negative k in (14), which leads to some sign differences between their relations and ours. While the real component [(53)] remains finite, the imaginary term [(54)] contains a logarithmic singularity as that dominates the amplitude estimate very close to the ground. The form of this singularity differs from the singularity of the small-l approximation and finite limit of the single-k approximation as .

Numerical verification of this predicted divergence as is complicated by rapid oscillations of the Fourier integrand as , requiring very fine resolution in the discrete Fourier transform. The black dotted–dashed curve in Fig. 5 shows complex steepness magnitudes derived from numerical FR solutions using a Fourier transform calculation discretized logarithmically in k with 4 × 104 grid points and a fundamental wavenumber of 5 × 104 km. As the imaginary part of the FR solution (green curve in Fig. 5) reproduces the theoretical divergence of (54).

Fig. 5.
Fig. 5.

Normalized steepness amplitude near the ground for unsheared flow of U0 = 10 m s−1 and N0 = 0.01 s−1 impinging on a Gaussian obstacle. Dotted–dashed black line is the amplitude from a high-resolution numerical FR solution. Red circles show the small-z theoretical behavior of the real (blue line) and imaginary (green line) components given by (53) and (54), respectively.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

To summarize, the anomalous behavior of the complex steepness solutions as is associated with a singularity at in the governing integral relation [(52)]. Because of this singularity, the steepness solution near is sensitive to the approximations used to evaluate the Fourier integral. For the full numerical FR solutions, grows as as , whereupon the imaginary component [(54)] diverges as —a tendency verified numerically at small z in Fig. 5. In the small-l approximation [(42)], grows as as , and both the real and imaginary parts diverge as . In the single-k approximation [(50)], m is independent of k, and so does not diverge as . In the single-k approximation, the real part of vanishes at —in contrast to the real part of the Fourier solution, which is given by (53).

4. Extension to vertically varying N and U

Remarkable general properties of our small-l and single-k solutions emerge on repeating the derivations of section 3 for vertically varying and profiles of arbitrary form. We do not repeat those derivations here but simply state the result: that the same set of small-l and single-k solutions derived in sections 3b and 3c also emerge, but with , , and replaced by their general forms: (15), (6), and (8), respectively.

Thus, our theory predicts that the effects of horizontal geometrical spreading on vertical variation of the largest local wavefield amplitudes can be accurately captured using compact and convenient functional forms that depend only on and β. The effects of vertical variations in background winds and stability are entirely captured within z′ as defined in (15). There is also no dependence on the fine details of the obstacle’s shape, apart from the ellipticity as defined by β. All these properties were inferred qualitatively from the numerical FR solutions of Part I and are now reproduced by these analytical solutions to simplified forms of the governing FR integrals.

5. Detailed comparisons with FR solutions

We now compare our small-l and single-k solutions, as derived in section 3 and generalized in section 4, with exact results derived from numerical FR integrations as in Part I, for a series of problems with progressively greater and more complex vertical structure in the upstream and profiles.

a. Constant shear

We consider first the constant wind shear problem modeled in Part I, in which winds take the form
e55
where C is the constant vertical shear, , and . Equations (6) and (8) become
e56
e57
As in Part I, we refer to and as the refraction terms, since they quantify how refraction of m values due to vertical variations in U and N modifies amplitudes so as to conserve wave action. For in (55), the normalized height coordinate [(15)] can be evaluated analytically as
e58

Following Part I, we consider solutions for four different C values of −0.05, 0, +0.1, and 1.0 m s−1 km−1. Figure 6 compares the numerical FR results of Part I with the corresponding small-l and single-k analytical predictions. In Fig. 6 we have performed the comparisons in terms of the total geometrical spreading and refraction influence, given for vertical displacement (Figs. 6a–c) by and for steepness (Figs. 6d–f) by . Close agreement among the numerical FR solutions (black) and the small-l (red) and single-k (green) analytical solutions is evident at all heights, for each β, and for different magnitudes and sign of C.

Fig. 6.
Fig. 6.

Profiles of peak vertical displacement amplitude in the normalized form for β values of (a) 1/3, (b) 1, and (c) 3. Curves are for C = −0.05 (solid), 0 (dotted), +0.1 (dashed), and +1.0 m s−1 km−1 (dotted–dashed). Black curves are numerical FR solutions, and red and green curves are the small-l and single-k analytical solutions, respectively. (d)–(f) Corresponding curves for peak steepness amplitudes .

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

b. Constant shear with tropopause-like N

Next, we consider a slight extension in which linear reverse shear is combined with a tropopause-like stability jump, as shown in Figs. 7a and 7b. The resulting normalized height coordinate [(15)] is plotted in Fig. 7c, which shows an increased gradient above approximately 12 km due to the increase in N. Equations (6) and (8) become
e59
e60
Profiles of and , which quantify the refraction effects on wave amplitude, are plotted in Fig. 7d. Horizontal geometrical spreading profiles are specified in Fig. 7e using in (35) and in (42) for β values of 1/8, 1, and 8. Profiles of the total geometrical spreading and refraction effect, given by the products and , are plotted in Fig. 7f.
Fig. 7.
Fig. 7.

Vertical profiles of (a) and (b) that yield corresponding profiles of (c) , (d) (black) and (blue), (e) (black) and (blue) for β values of 1/8 (solid lines), 1 (dotted lines), and 8 (dashed lines), and (f) and .

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

Figure 8 compares the real parts of the complex single-k and small-l analytical steepness solutions to results from numerical FR integrations, with both the real (wavefield) component as well as peak amplitudes derived from the Hilbert-transformed FR solution shown. For , there is close agreement among the FR, small-l, and single-k solutions away from . Since , then the profiles are in the range where they assume a common range dependence given by (39) and (51). Note also the divergence of the solutions as . By contrast, zf ≈ 50 km for the obstacle and so these curves are in the range where solutions can diverge slightly, as evident in the bottom panel of Fig. 8, where the small-l solution is closer to the FR results than the single-k solution. Overall, however, there is excellent agreement and consistency among the curves.

Fig. 8.
Fig. 8.

Normalized wave steepness amplitudes as a function of height directly above an elliptical obstacle of (top) and (bottom) , showing the real part (blue dashed curve) and magnitude (thick black solid curve) of the numerical FR solution and the real part of the complex small-l analytical solution [(42)], plotted as thin black line with open circles, and of the complex single-k solution [(50)], plotted as red line with crosses.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

c. Geophysical and profiles

Finally, we consider a case with geophysical and profiles, plotted in Figs. 9a and 9b, respectively, for z = 0–90 km. These profiles were taken from the data assimilation products of Eckermann et al. (2009) at 0000 UTC 15 June 2009 at an oceanic location (45°S, 165°E) upstream of the Southern Alps of New Zealand. The resulting in (15) is plotted in Fig. 9c. Figure 9d plots profiles of the effect of refraction on vertical displacement amplitudes as given by in (59) and on steepness amplitudes as given by in (60).

Fig. 9.
Fig. 9.

Profiles of (a) and (b) from data assimilation fields of Eckermann et al. (2009) upstream of New Zealand (45°S, 165°E) on 15 Jun 2009. These yield corresponding profiles of (c) and (d) (black) and (blue). (e) Normalized peak amplitudes of η (black) and (blue) derived from numerical FR experiments, for β values of 1/3 (solid), 1 (dotted), and 3 (dashed), which define the total geometrical spreading (GS) and refraction effects. (f) Profiles in (e) divided by the corresponding refraction profiles in (d) to isolate the horizontal GS effect in the FR solutions. (g),(h) Corresponding small-l analytical solutions for (g) and and (h) and . Note the near equality between numerical FR results in (e) and (f) and the corresponding small-l predictions in (g) and (h).

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

We evaluated numerical FR solutions for the wavefields that result as these and profiles impinge on elliptical obstacles of β = 1/3, 1, and 3, and then computed maximum wavefield amplitudes at each height, as in Part I. The resulting normalized amplitude profiles for vertical displacement and steepness are plotted in Fig. 9e. As in Part I, dividing these profiles by the corresponding and curve in Fig. 9d quantifies the horizontal geometrical spreading contributions, which are plotted in Fig. 9f.

Figures 9g and 9h are corresponding predictions using the small-l analytical relations, in (35), and in (42), evaluated using the , , and profiles in Figs. 9a–c. These functions predict the horizontal geometrical spreading effect in Fig. 9h and show excellent agreement with the corresponding numerical FR solutions in Fig. 9f. The net influences of geometrical spreading and refraction, given by and , are plotted in Fig. 9g and again show close agreement with the profiles in Fig. 9e derived directly from numerical FR integrations.

The small-l profile solutions for η and in Fig. 9g are replotted in Figs. 10a and 10b, respectively, as colored curves with the density term included to yield geophysical amplitude variations with height: and , respectively. As discussed in Part I, existing OGWD parameterizations adopt the two-dimensional limit of section 3a, leading to the solutions (27) and (28) of (i.e., no horizontal geometrical spreading). The solid gray curves show these two-dimensional solutions: specifically, and . The black curves in Fig. 10 show threshold amplitudes for wave breaking. As discussed in section 2a of Part I, wave breaking occurs at a threshold steepness amplitude of α, where , implying a corresponding breaking amplitude for vertical displacement. Expressing these thresholds in normalized form requires choices for α and and are plotted in Fig. 10 using and hm = 500 m.

Fig. 10.
Fig. 10.

Profiles of (a) peak vertical displacement amplitudes normalized by the surface value hm = 500 m, showing for (red), (green), (blue), and (gray), taken from Fig. 9g, where . Black curve shows the normalized breaking amplitude using and dotted curves show amplitude profiles after imposition of wave breaking via linear saturation. (b) Corresponding profile sets for peak steepness amplitudes normalized by the surface value, along with the normalized breaking amplitude (black). (c) Corresponding drag profiles.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

Wave breaking first occurs at the lowest height , where these black curves intercept the relevant amplitude curve. Despite large differences in profile shapes, inspection of Figs. 10a and 10b reveals that both vertical displacement and steepness curves give identical values. The gray curve that omits horizontal geometrical spreading predicts breaking initially at zb ≈ 20 km. By contrast, our small-l solutions that quantify the density, refraction, and geometrical spreading effects yield zb ≈ 67 km for , zb ≈ 71 km for , and no wave breaking (zb > 90 km) for .

The dotted curves in Figs. 10a and 10b show peak amplitude profiles after imposition of wave breaking, as reviewed in section 2a of Part I. On assuming in each case a vertical flux of horizontal wavefield momentum generated from the obstacle of , then we can use the saturated and unsaturated profiles in Fig. 10a to compute the wave-induced drag force [see (16) of Part I]. Drag profiles are plotted in Fig. 10c for each of the four solutions and show several interesting properties.

First, the gray curve in Fig. 10c, based on a two-dimensional solution without horizontal geometrical spreading, produces a large burst of drag at z ≈ 20 km that disappears entirely from the solutions with horizontal geometrical spreading included. As discussed in Part I, solutions without horizontal geometrical spreading are used in all current OGWD parameterizations. Part I also notes that excessive parameterized OGWD in the lower stratosphere leads to errors in many GCMs (e.g., Klinker and Sardeshmukh 1992; Milton and Wilson 1996), and so OGWD tendencies are often artificially reduced or eliminated in GCMs above the tropopause (e.g., Hogan and Brody 1993; Norton and Thuburn 1999; Scinocca et al. 2008). Figure 10c suggests that inclusion of the missing physics of horizontal geometrical spreading yields geophysical (rather than ad hoc) reductions of lower-stratospheric OGWD that can reduce this source of GCM error. Figure 10c also shows that at upper levels, where several of these wave solutions break within the same height intervals, almost identical drag profiles are produced.

6. Discussion

Dispersive three-dimensional group propagation into progressively broader horizontal areas leads to reductions in local OGW amplitudes—a process that we refer to as horizontal geometrical spreading. Given the hydrostatic dispersion relation [(1)], an OGW packet has a vertical group velocity , where , and a ground-based group velocity in the downstream direction x of
e61
For rays initialized at the origin (), the downstream displacement of an OGW group at altitude z, using (15), is
e62
The corresponding lateral group displacement
e63
which tightly couples lateral and downstream group displacement loci into a parabolic arc at any given (Smith 1980).
The earliest OGWD parameterizations assumed two-dimensional plane hydrostatic OGWs with horizontal wavenumber vectors aligned parallel to the upstream flow vector (Palmer et al. 1986; McFarlane 1987), yielding from (62) and (63) and thus no horizontal geometrical spreading, as verified in section 3a: see (27) and (28). This result in turn allows the OGWD to be parameterized within a vertical column directly above a GCM grid cell. Amplitude evolution with height was parameterized using WKB solutions to a two-dimensional OGW equation [see, e.g., section 2 of both Palmer et al. (1986) and McFarlane (1987)], which for vertical displacement takes the form [see (13) of Part I]
e64
The m and N dependences in (64) quantify the effects of refraction, as in (6).

Those initial two-dimensional OGWD parameterizations have since been replaced by a new generation that parameterizes both OGWD and near-surface drag due to flow over explicitly three-dimensional subgrid-scale orography (e.g., Lott and Miller 1997; Scinocca and McFarlane 2000; Webster et al. 2003). Since generally for the resulting three-dimensional wavefields, then from (62) and (63). Thus contemporary OGWD parameterizations can no longer automatically discount horizontal geometrical spreading effects on local wave amplitudes or horizontal propagation of OGWs out of a vertical GCM column. Nonetheless, parameterizations continue to use the WKB result [(64)] within vertical GCM columns and so, in effect, assume that these three-dimensional modifications have negligible net impacts on OGWD.

This work and that of Part I have highlighted significant effects of horizontal geometrical spreading on OGW amplitude evolution with height. Here we discuss how these missing geometrical spreading effects might be included within current OGWD parameterizations, and then go on to investigate briefly the issue of possible lateral propagation of OGW momentum flux out of vertical GCM columns.

a. Adding horizontal geometrical spreading terms to OGWD parameterizations

The numerical results of Part I and their analytical approximations derived in sections 3 and 4 revealed a clean separation between local OGW amplitude variations due to horizontal geometrical spreading and refraction. Thus the missing effects of horizontal geometrical spreading could be included within existing OGWD parameterizations by simply scaling their current WKB amplitudes in (64) to yield a new amplitude
e65
Indeed, our derivations in sections 3 and 4 revealed that the horizontal geometrical spreading term in (65) could be efficiently parameterized using either our small-l solution or single-k solution , given their accurate reproduction of exact transform results for arbitrary , , and β, as was shown in section 5. However, several comments and caveats pertain to such an application, which we now discuss.

Small β obstacles yield small values of and thus large reductions in parameterized OGW amplitudes via (65). Thus, inclusion of these terms via (64) within OGWD parameterizations already implemented within GCMs will change OGWD profiles (e.g., Fig. 10c) and thus may require some initial recalibration and tuning of various “free” parameters within the parameterization to reproduce the first-order distributions of OGWD required by the GCM. This is neither unusual nor problematic given that GCMs routinely retune their parameterized OGWD in response to changes in model resolution and physics (e.g., Scinocca et al. 2008).

The wavefields given via (10) from our FR solutions [(5)] are exact three-dimensional linear hydrostatic OGW solutions governed by the dispersion relation [(1)]. Conversely, (64) is derived by approximating locally using WKB methods (Lindzen 1981; Palmer et al. 1986; McFarlane 1987), but those WKB approximations prove to be formally invalid at all heights above the mountain since stationary phase requirements are violated (Shutts 1998; Broutman et al. 2001, 2004). By contrast, our and solutions are not WKB approximations, but instead model the largest local amplitudes of the exact solutions derived from the hydrostatic FR solution [(5)] using (10). In the far-field () our small-l solution [(37)] and single-k solution [(48)] yield associated vertical phase variations given by , as illustrated in Fig. 2, which, from the definition (15), implies an effective for use in (64). The point here is that, while the mathematical modifications to (64) from adding the geometrical spreading term via (65) are minor, this change in fact moves the parameterization equations away from simplified (and formally invalid) local WKB approximations toward newer and more accurate descriptions of the dominant amplitude and phase variations of three-dimensional wavefields provided by exact transform solutions. The latter is a more appropriate basis for parameterizing OGWD owing to flow over three-dimensional subgrid-scale terrain.

Our work here has focused on vertically sheared flow parallel to a principal axis of an elliptical obstacle ( or ), whereas GCMs must parameterize OGWD for flows aligned at any angle to the obstacle axes. Part I presented numerical results for uniform flows aligned at various angles φ to the obstacle’s principal short axis. At any altitude z, the effect of horizontal geometrical spreading on maximum wavefield amplitudes varied monotonically with φ between limiting values at β () and (). Since our work provides accurate analytical approximations for the vertical variation of these limiting values, vertical profiles for any φ could be approximated using φ-dependent weighted averages of these limiting solutions. We experimented with several φ-dependent interpolations and found reasonable fits to the numerical FR results in Fig. 8 of Part I. Future work could address this issue systematically using both exact numerical and approximate analytical solutions to the FR integrals, as here, but for arbitrary φ. A complication relative to the current approach is that obstacle rotation destroys the mathematical symmetry of the problem, leading to wavefield asymmetries either side of and maximum wavefield amplitudes that can migrate significant distances from the obstacle peak.

A related issue arises when the wind vector rotates with altitude. Again, the symmetry of the three-dimensional OGW ship-wave response is lost as directional critical levels are progressively swept out in one arm of the diverging wake (Shutts 1998), below which breaking and OGWD can occur (Broad 1999). Wave groups in these regions have horizontal wavenumber vectors aligned nearly orthogonal to the local wind vector and, thus, from (62) can propagate long distances downstream of the mountain (Shutts 1998). Thus, further work may be needed to parameterize geometrical spreading and OGWD in directionally sheared flows. For example, geometrical spreading in the downstream wake regions can yield amplitude reductions with height greater than the asymptotic reductions that we have derived near the mountain (Shutts 1998; Broutman et al. 2002). Note, however, that use of these -dependent forms in downstream portions of the wavefield would still constitute a far better approximation than the assumption of no horizontal geometrical spreading currently used in OGWD parameterizations.

b. Ramifications for vertical column parameterizations

Existing OGWD parameterizations all assume that flux deposition leading to OGWD occurs entirely within a vertical column extending directly above the GCM grid cell containing the subgrid-scale orography. The large influences of horizontal geometrical spreading on vertical amplitude evolution of three-dimensional OGWs found here and in Part I in turn imply generally significant horizontal group displacements , which might exceed typical GCM gridcell dimensions and call into question the efficacy of vertical column parameterizations.

However, Part I found that the largest wavefield amplitudes most prone to breaking were located close to the mountain for almost all β and z (see their Fig. 3)—a result that not only simplified our route to analytical solutions here but also validates to some extent the vertical column approach to parameterizing OGWD even for three-dimensional subgrid-scale terrain. As discussed in section 3b, this occurs because the wavefield near the mountain is dominated by wave groups with , which have small ground-based horizontal group velocities and displacements according to (61)(63), so that they spread less in the horizontal and thus attain the largest local amplitudes most prone to breaking and drag generation.

Conversely, wave groups with horizontal wavenumber vectors aligned at angles to the flow are less prone to breaking since they spread more in propagating significant distances downstream of the mountain. Could these waves invalidate vertical column approaches to parameterizing OGWD? To investigate this, we pose a simple question: for a GCM grid cell of horizontal dimensions , how much OGW momentum flux escapes the side boundaries of the vertical column before breaking or reaching the model’s upper boundary? To quantify this for the current problem, the numerical FR algorithm in Part I was modified to derive vector FR velocity solutions . Then their Hilbert transforms given by (10) and (11) were used to derive zonal and meridional momentum fluxes per unit mass, and , using (9) and (10) of Eckermann et al. (2010). Finally, we used these local flux solutions to compute the momentum flux at each z contained within an column area centered directly over the obstacle peak: that is,
e66
Note that for all L and z in the current problem. Since we are interested in the flux fractions inside and outside the column, we compute the fractional relation
e67
where is the total wavefield flux given by (66) as . These numerical FR solutions were computed without viscosity (: see the appendix of Part I) so that remains constant with height (Vosper and Mobbs 1998).

Figure 11 plots for obstacles ranging in β from 1/8 to 8. As in Part I, for every β the minimum value of a or b is 10 km, so that, from (3), yields a = 80 km and b = 10 km while yields a = 10 km and b = 80 km. Thus, L values in Fig. 11 range from much larger to somewhat smaller than the obstacle widths, both of which can occur in OGWD parameterizations (e.g., Lott and Miller 1997). Results are shown for two uniform flow experiments with U0 = 10 m s−1 (black contours) and 30 m s−1 (gray contours), with N0 = 0.01 s−1 in both cases. The results show that, for a given vertical column, there is progressively more leakage of momentum flux out of the side boundaries with increasing height and that the effect is more pronounced for smaller β.

Fig. 11.
Fig. 11.

Contours of (%) showing the fraction of zonal momentum flux within an column for β values ranging from (a) 1/8 to (i) 8. Results are plotted as a function of L and for U0 = 10 (black curves) and U0 = 30 m s−1 (gray curves); N0 = 0.01 s−1 in each case. See text for further details.

Citation: Journal of the Atmospheric Sciences 72, 6; 10.1175/JAS-D-14-0148.1

Note also that the U0 = 10 and 30 m s−1 results superimpose in Fig. 11 when plotted using owing to the dependence of horizontal group displacements in (62) and (63). If we further assume that an obstacle of launches rays with and , then (62) yields . Thus, just as for far-field spreading effects on wave amplitude given by (39) and (48), group propagation outside the side boundaries of an grid cell increases quadratically with decreasing β, qualitatively consistent with the panel-to-panel differences in Fig. 11.

It is also clear from (62) and (63) that the largest horizontal displacements occur for wave groups whose horizontal wavenumber vectors are aligned nearly orthogonal to the flow . From (61) these groups are primarily advected by , and since they also propagate slowly in the vertical (small ), the accumulated downstream advection can be large. Note that, in addition to potentially violating the vertical column approximation, such slow waves may also violate a second parameterization assumption that waves propagate from the surface to the GCM’s upper boundary within one model time step. However, such waves should also be highly susceptible to viscous dissipation. To test this, we repeated these FR simulations using ν = 1 m2 s−1—a value smaller than typical background levels of turbulent kinematic viscosity in the atmosphere (e.g., Fig. 7 of Marks and Eckermann 1995). The corresponding results (not shown) reveal, as expected, that the contours at large in Fig. 11 move to the left as slow waves are preferentially damped and thus produce more of their OGWD lower down prior to attaining larger higher up that may move them out of parameterization columns. Infrared radiative cooling will have similar preferential damping effects on these waves (Zhu 1993).

In summary, the portions of the wavefield most prone to break within three-dimensional hydrostatic OGWs are located near the obstacle peak at all altitudes. Thus, with appropriate modifications to include geometrical spreading effects as in (65), existing vertical column parameterizations remain well suited to parameterizing the breaking of these regions of the wavefield and the OGWD that results. Wave groups with horizontal wavenumber vectors aligned at angles to the flow spread more and may present greater challenges to existing vertical column parameterizations. However, relevant physical damping mechanisms such as turbulent mixing and radiative cooling act preferentially on these waves and may significantly dissipate their momentum fluxes at lower altitudes before they have propagated too far vertically and laterally.

7. Summary

Motivated by reproducible results derived from numerical FR experiments in Part I, we have derived analytical solutions for the variation with height of peak wavefield amplitudes due to horizontal geometrical spreading of three-dimensional hydrostatic OGWs produced by linear flow over an elliptical obstacle of aspect ratio β. A small-l and a single-k approximation to key terms in the governing FR integrals each yielded solutions that were valid for arbitrary vertical profiles of wind and stability . These analytical solutions accurately reproduced characteristic features of the numerical FR results. These include 1) a strong β dependence to the profiles, 2) encapsulation of dependences on and within the transformed height variable given by (15), and 3) a universal far-field dependence given by (39) at , where has a quadratic β dependence given by (40). Close agreement among these analytical solutions and numerical FR results was demonstrated for a range of different and profiles and β values. These comparisons confirmed the findings of Part I that horizontal geometrical spreading is a large effect of clear relevance to the parameterization of OGWD (see Fig. 10). Since existing OGWD parameterizations omit these effects, the analytical solutions presented here appear to be both accurate and efficient enough to incorporate into existing OGWD parameterizations as a convenient initial description of this important missing physics. Future work should investigate possible extensions of our theory to arbitrary obstacle orientations and to directionally sheared flows.

Acknowledgments

This research was supported by the Office of Naval Research (ONR) via the Departmental Research Initiative “Unified Physical Parameterizations for Seasonal Prediction” and by the Chief of Naval Research via the Naval Research Laboratory (NRL) 6.1 Accelerated Research Initiative “The Boundary Paradox.”

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1998: Stationary gravity-wave structure in flows with directional wind shear. Quart. J. Roy. Meteor. Soc., 124, 14211442, doi:10.1002/qj.49712454905.

    • Search Google Scholar
    • Export Citation
  • Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32, 348364, doi:10.1111/j.2153-3490.1980.tb00962.x.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., and S. D. Mobbs, 1998: Momentum fluxes due to three-dimensional gravity-waves: Implications for measurements and numerical modelling. Quart. J. Roy. Meteor. Soc., 124, 27552769, doi:10.1002/qj.49712455211.

    • Search Google Scholar
    • Export Citation
  • Webster, S., A. R. Brown, D. R. Cameron, and C. P. Jones, 2003: Improvements to the representation of orography in the Met Office Unified Model. Quart. J. Roy. Meteor. Soc., 129, 19892010, doi:10.1256/qj.02.133.

    • Search Google Scholar
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  • Zhu, X., 1993: Radiative damping revisited: Parameterization of damping rate in the middle atmosphere. J. Atmos. Sci., 50, 30083021, doi:10.1175/1520-0469(1993)050<3008:RDRPOD>2.0.CO;2.

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Save
  • Abramowitz, M., and I. A. Stegun, 1972: Handbook of Mathematical Functions and with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, Vol. 55, U.S. Department of Commerce, National Bureau of Standards, 1046 pp.

  • Broad, A. S., 1999: Do orographic gravity waves break in flows with uniform wind direction turning with height? Quart. J. Roy. Meteor. Soc., 125, 16951714, doi:10.1002/qj.49712555711.

    • Search Google Scholar
    • Export Citation
  • Broutman, D., J. W. Rottman, and S. D. Eckermann, 2001: A hybrid method for wave propagation from a localized source, with application to mountain waves. Quart. J. Roy. Meteor. Soc., 127, 129146, doi:10.1002/qj.49712757108.

    • Search Google Scholar
    • Export Citation
  • Broutman, D., J. W. Rottman, and S. D. Eckermann, 2002: Maslov’s method for stationary hydrostatic mountain waves. Quart. J. Roy. Meteor. Soc., 128, 11591172, doi:10.1256/003590002320373247.

    • Search Google Scholar
    • Export Citation
  • Broutman, D., J. W. Rottman, and S. D. Eckermann, 2004: Ray methods for internal waves in the atmosphere and ocean. Annu. Rev. Fluid Mech., 36, 233253, doi:10.1146/annurev.fluid.36.050802.122022.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., and Coauthors, 2009: High-altitude data assimilation system experiments for the northern summer mesosphere season of 2007. J. Atmos. Sol.-Terr. Phys., 71, 531551, doi:10.1016/j.jastp.2008.09.036.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., J. Lindeman, D. Broutman, J. Ma, and Z. Boybeyi, 2010: Momentum fluxes of gravity waves generated by variable Froude number flow over three-dimensional obstacles. J. Atmos. Sci., 67, 22602278, doi:10.1175/2010JAS3375.1.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., J. Ma, and D. Broutman, 2015: Effects of horizontal geometrical spreading on the parameterization of orographic gravity wave drag. Part I: Numerical transform solutions. J. Atmos. Sci., 72, 23302347.

    • Search Google Scholar
    • Export Citation
  • Hogan, T. F., and L. R. Brody, 1993: Sensitivity studies of the navy’s global forecast model parameterizations and evaluation of improvements to NOGAPS. Mon. Wea. Rev., 121, 23732395, doi:10.1175/1520-0493(1993)121<2373:SSOTNG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Klinker, E., and P. D. Sardeshmukh, 1992: The diagnosis of mechanical dissipation in the atmosphere from large-scale balance requirements. J. Atmos. Sci., 49, 608627, doi:10.1175/1520-0469(1992)049<0608:TDOMDI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Knight, H., D. Broutman, and S. D. Eckermann, 2015: Integral expressions for mountain wave steepness. Wave Motion, doi:10.1016/j.wavemoti.2015.01.003, in press.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86, 97079714, doi:10.1029/JC086iC10p09707.

    • Search Google Scholar
    • Export Citation
  • Lott, F., and M. J. Miller, 1997: A new subgrid-scale orographic drag parametrization: Its formulation and testing. Quart. J. Roy. Meteor. Soc., 123, 101127, doi:10.1002/qj.49712353704.

    • Search Google Scholar
    • Export Citation
  • Marks, C. J., and S. D. Eckermann, 1995: A three-dimensional nonhydrostatic ray-tracing model for gravity waves: Formulation and preliminary results for the middle atmosphere. J. Atmos. Sci., 52, 19591984, doi:10.1175/1520-0469(1995)052<1959:ATDNRT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • McFarlane, N. A., 1987: The effect of orographically excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci., 44, 17751800, doi:10.1175/1520-0469(1987)044<1775:TEOOEG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Milton, S. F., and C. A. Wilson, 1996: The impact of parameterized subgrid-scale orographic forcing on systematic errors in a global NWP model. Mon. Wea. Rev., 124, 20232045, doi:10.1175/1520-0493(1996)124<2023:TIOPSS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Norton, W. A., and J. Thuburn, 1999: Sensitivity of mesospheric mean flow, planetary waves, and tides to strength of gravity wave drag. J. Geophys. Res., 104, 30 89730 912, doi:10.1029/1999JD900961.

    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., G. J. Shutts, and R. Swinbank, 1986: Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parametrization. Quart. J. Roy. Meteor. Soc., 112, 10011039, doi:10.1002/qj.49711247406.

    • Search Google Scholar
    • Export Citation
  • Scinocca, J. F., and N. A. McFarlane, 2000: The parameterization of drag induced by stratified flow over anisotropic orography. Quart. J. Roy. Meteor. Soc., 126, 23532393, doi:10.1002/qj.49712656802.

    • Search Google Scholar
    • Export Citation
  • Scinocca, J. F., N. A. McFarlane, M. Lazare, J. Li, and D. Plummer, 2008: The CCCma third generation AGCM and its extension into the middle atmosphere. Atmos. Chem. Phys., 8, 70557074, doi:10.5194/acp-8-7055-2008.

    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1998: Stationary gravity-wave structure in flows with directional wind shear. Quart. J. Roy. Meteor. Soc., 124, 14211442, doi:10.1002/qj.49712454905.

    • Search Google Scholar
    • Export Citation
  • Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32, 348364, doi:10.1111/j.2153-3490.1980.tb00962.x.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., and S. D. Mobbs, 1998: Momentum fluxes due to three-dimensional gravity-waves: Implications for measurements and numerical modelling. Quart. J. Roy. Meteor. Soc., 124, 27552769, doi:10.1002/qj.49712455211.

    • Search Google Scholar
    • Export Citation
  • Webster, S., A. R. Brown, D. R. Cameron, and C. P. Jones, 2003: Improvements to the representation of orography in the Met Office Unified Model. Quart. J. Roy. Meteor. Soc., 129, 19892010, doi:10.1256/qj.02.133.

    • Search Google Scholar
    • Export Citation
  • Zhu, X., 1993: Radiative damping revisited: Parameterization of damping rate in the middle atmosphere. J. Atmos. Sci., 50, 30083021, doi:10.1175/1520-0469(1993)050<3008:RDRPOD>2.0.CO;2.

    • Search Google Scholar
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  • Fig. 1.

    Profiles of normalized peak vertical displacement amplitudes (black curves) and normalized peak steepness amplitudes (blue curves), for discrete , where and . The βj = 10−1, 100, and 101 curves are labeled.

  • Fig. 2.

    Vertical profiles of the real part of the numerical FR solution (blue), the real part of the small-l solution (open circles), the real part of the single-k solution (red), and the magnitude (amplitude) of the numerical FR solution (thick black line), for z = 0–20 km for unsheared upstream flow of U0 = 15 m s−1 and N0 = 0.02 s−1 impinging on Gaussian terrain [(2)] of (a) and (b) .

  • Fig. 3.

    Profiles of normalized peak vertical displacement amplitudes, showing from the numerical FR experiments of Part I (gray curves), (red curves) and (green curves), for β values ranging from (a) 1/8 to (i) 8.

  • Fig. 4.

    Profiles of normalized peak steepness amplitudes, showing from the numerical FR experiments of Part I (gray curves), (red curves) and (green curves), for β values ranging from (a) 1/8 to (i) 8.

  • Fig. 5.

    Normalized steepness amplitude near the ground for unsheared flow of U0 = 10 m s−1 and N0 = 0.01 s−1 impinging on a Gaussian obstacle. Dotted–dashed black line is the amplitude from a high-resolution numerical FR solution. Red circles show the small-z theoretical behavior of the real (blue line) and imaginary (green line) components given by (53) and (54), respectively.

  • Fig. 6.

    Profiles of peak vertical displacement amplitude in the normalized form for β values of (a) 1/3, (b) 1, and (c) 3. Curves are for C = −0.05 (solid), 0 (dotted), +0.1 (dashed), and +1.0 m s−1 km−1 (dotted–dashed). Black curves are numerical FR solutions, and red and green curves are the small-l and single-k analytical solutions, respectively. (d)–(f) Corresponding curves for peak steepness amplitudes .

  • Fig. 7.

    Vertical profiles of (a) and (b) that yield corresponding profiles of (c) , (d) (black) and (blue), (e) (black) and (blue) for β values of 1/8 (solid lines), 1 (dotted lines), and 8 (dashed lines), and (f) and .

  • Fig. 8.

    Normalized wave steepness amplitudes as a function of height directly above an elliptical obstacle of (top) and (bottom) , showing the real part (blue dashed curve) and magnitude (thick black solid curve) of the numerical FR solution and the real part of the complex small-l analytical solution [(42)], plotted as thin black line with open circles, and of the complex single-k solution [(50)], plotted as red line with crosses.

  • Fig. 9.

    Profiles of (a) and (b) from data assimilation fields of Eckermann et al. (2009) upstream of New Zealand (45°S, 165°E) on 15 Jun 2009. These yield corresponding profiles of (c) and (d) (black) and (blue). (e) Normalized peak amplitudes of η (black) and (blue) derived from numerical FR experiments, for β values of 1/3 (solid), 1 (dotted), and 3 (dashed), which define the total geometrical spreading (GS) and refraction effects. (f) Profiles in (e) divided by the corresponding refraction profiles in (d) to isolate the horizontal GS effect in the FR solutions. (g),(h) Corresponding small-l analytical solutions for (g) and and (h) and . Note the near equality between numerical FR results in (e) and (f) and the corresponding small-l predictions in (g) and (h).

  • Fig. 10.

    Profiles of (a) peak vertical displacement amplitudes normalized by the surface value hm = 500 m, showing for (red), (green), (blue), and (gray), taken from Fig. 9g, where . Black curve shows the normalized breaking amplitude using and dotted curves show amplitude profiles after imposition of wave breaking via linear saturation. (b) Corresponding profile sets for peak steepness amplitudes normalized by the surface value, along with the normalized breaking amplitude (black). (c) Corresponding drag profiles.

  • Fig. 11.

    Contours of (%) showing the fraction of zonal momentum flux within an column for β values ranging from (a) 1/8 to (i) 8. Results are plotted as a function of L and for U0 = 10 (black curves) and U0 = 30 m s−1 (gray curves); N0 = 0.01 s−1 in each case. See text for further details.

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