## Abstract

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

## 1. Introduction

Baines (1995) points out that a lack of understanding of the effects on the atmosphere of orographic features is a significant impediment to improvements in weather forecasting, noting that form drag (as distinct from surface frictional drag) contributes about 50% of the total drag of the atmosphere (e.g. Palmer *et al*. 1986) and that this drag is manifested in stratified effects such as internal gravity waves. Similar effects can be expected in oceanic flows. One approach to describing some aspects of these motions is through the consideration of forced weakly nonlinear long waves. When the leading order balance is between quadratic nonlinearity and dispersion, the dynamics of flow independent of the cross-stream coordinate (*y*, say), as in flow over a ridge, is typically governed by the well-known Korteweg–de Vries (KdV) equation. However Grimshaw *et al*. (2002*b*) point out that for larger waves, or for certain special configurations in stratified fluids, it has been found useful to include cubic nonlinearity, leading to the extended KdV (eKdV) equation. They note derivations in the review of Grimshaw (1997) and in the specialized applications of Holloway *et al*. (1997), Michallet & Barthelemy (1998) and Grimshaw *et al*. (1999). These studies and those of Melville & Helfrich (1987), Marchant & Smyth (1990) and Hanazaki (1992) demonstrate that solutions of the forced eKdV can differ sharply from those of the forced KdV, with, for example, stationary monotonic bores appearing in transcritical flows governed by the forced eKdV when only periodically generated solitary waves appear in the same regime for flows governed by the forced KdV.

When orographic features vary slowly in height across the flow direction a slow *y*-dependence appears in the flow and when this cross-stream variation is of the same order as the nonlinearity and dispersion the governing equation with quadratic nonlinearity becomes the two-dimensional KdV or Kadomtsev–Petviashvili (KP) equation. Johnson & Vilenski (2004; JV herein) describe orographically forced atmospheric waves in terms of solutions to the forced KP equation. However, applications to the atmosphere, oceans and experiments (Johnson *et al*. in press) tend to be for larger amplitudes and layered flows where, as noted by Grimshaw *et al*. (2002*b*) for one-dimensional motion, it is useful to include cubic nonlinearity, giving here the extended KP (eKP) equation. This paper thus considers the wavefield forced by near-critical flow over isolated orography when the layer depths and densities are such that cubic nonlinearity is important. Section 2 notes the non-dimensionalizations and scalings leading to the governing forced extended Kadomtsev–Petviashvili equation (feKP). Section 3 notes two properties of steady solutions of the feKP equation that prove useful in interpreting subsequent numerical integrations. Firstly, the flow supports upward and downward leaps, i.e. rapid localized changes in flow depth matching two different almost constant depths on either side of the leaps, denoted leaps here following the observations in Lawrence (1993) and discussion in Baines (1995) for two-layer flows and the analysis in the context of Rossby waves in Johnson & Clarke (1999). Secondly, as for the quadratic KP equation, steady two-dimensional supercritical solutions of the eKP can be related to unsteady one-dimensional solutions of the eKdV. This latter property is particularly useful in light of perhaps the most remarkable feature of the eKdV equation—the appearance in the temporal evolution of the eKdV, from sufficiently large initial conditions, of large-amplitude, wide, flat-topped solitons. Grimshaw *et al*. (2002*a*) discuss this evolution in detail and denote the waves ‘table-top’ solitons. The related manifestations in the steady feKP equation are table-top waves that have downstream profile like those of Grimshaw *et al*. (2002*a*) but extend significant distances across the flow with little change in profile and it is these features that are discussed in greatest detail below. Grimshaw *et al*. (2002*a*) note that observations of large amplitude internal waves in the ocean in Stanton & Ostrovsky (1998), Holloway *et al*. (2001) and Jeans & Sherwin (2001) can be interpreted in terms of table-top waves and although most of these appear to be propagating others may be attached to orographic features. Section 4 describes flow over finite length ridges, concentrating on subcritical flows and showing that it is the departure from criticality that determines most strongly the form of the flow. A table-top wave appears in the lee-wave wake behind the ridge for small subcriticality, altering the height of the transition from subcritical to supercritical flow over the ridge and leading to a well-defined maximum in wave drag in significantly subcritical flow. Section 5 shows that, remarkably, the table-top wave remains behind axisymmetric (in the present scaling) obstacles, extending equally as far across the flow as behind ridges and leading to a much smaller decrease in drag with decreasing ridge length than might otherwise been expected. Section 6 briefly discusses these results.

## 2. The non-dimensionalization and governing equation

Consider two-layer, inviscid, incompressible rigid-lid flow past an isolated three-dimensional obstacle, where the layers have constant densities *ρ*_{±} and undisturbed depths *H*_{±}. Let the far-field flow be uniform with speed *U*_{∞} and take Cartesian axes *x*^{*}*y*^{*}*z*^{*} with *x*^{*} in the direction of the flow at large distance, *z*^{*} vertical and at the interface height. Let the obstacle shape be given by , so *L* is a typical horizontal scale for the motion and *ϵ*≪1 gives the obstacle height as a fraction of the lower layer depth (figure 1). Then, following Karpman (1975), Grimshaw & Melville (1989), Baines (1995) and Grimshaw *et al*. (1998), introduce the non-dimensional variables, stretched in *y*^{*} and time *t*^{*},(2.1)with velocity components and pressure(2.2)with *g* gravitational acceleration, and let the interface be given by . Four non-dimensional groups appear in the governing equations. These can be taken as the aspect ratio *H*_{−}/*L*, depth ratio *H*=*H*_{+}/*H*_{−}, density ratio *s*=*ρ*_{+}/*ρ*_{−}<1 and upstream lower-layer Froude number . The non-dimensional speed of long infinitesimal interfacial waves is given by . Now consider shallow, transcritical flow when cubic nonlinearity is important by writing(2.3)where *κ*, *H*, *b*, *a*=*O*(1) as *ϵ*→0. The parameter *κ* gives the strength of non-hydrostatic dispersion and in the limit *κ*→0 the problem reduces to one of two-layer hydrostatic shallow-water flow. The parameter *b* measures the small deviation (of order *ϵ*^{2/3}) of the oncoming flow speed from the interfacial longwave speed. Linear waves are thus almost stationary relative to the orography and so almost resonant with the orographic forcing. The oncoming flow is subcritical for *b*<0 (*F*<*U*_{0}), supercritical for *b*>0 (*F*>*U*_{0}) and critical if *b*=0 (*F*=*U*_{0}). Now look for solutions with leading order(2.4)

As expected from discussions of the extended Korteweg–de Vries equation, interfacial displacements are significantly larger here than the *O*(*ϵ*^{1/2}) displacement in the one-layer non-hydrostatic problem (JV; Vilenski & Johnson 2004). Substituting into the Euler equations gives(2.5)where the interface displacement satisfies the forced eKP (feKP) equation:(2.6)with , , and . For the two-layer flow here *q*, *d* and *c* are positive but *a* can take either sign. In the numerical integrations of §§4 and 5 the flow is taken to be set impulsively in motion from rest and so, given the finite speed of propagation of disturbances, the initial and boundary conditions for (2.6) are(2.7)

## 3. Leaps and the far-field in steady flow

The most remarkable feature of the asymptotically steady flow fields obtained in §§4 and 5 is the appearance in many parameter regimes of stationary finite-amplitude flat-topped waves of large cross-stream extent with strong leading edge (upward) and trailing edge (downward) leaps. Flow-wise cross-sections of these two-dimensional features show many features similar to the one-dimensional table-top waves of Grimshaw *et al*. (2002*a*). Arguments similar to those for a related one-dimensional problem in Johnson & Clarke (1999) show that in the limit of small dispersion (*c*≪1) leap widths are of order *c*^{1/2} and a model of a leap can be obtained straightforwardly. Suppose a leap is centred on a smooth curved line *ℓ* across which the solution of the steady feKP equation (2.6) jumps abruptly. Near *ℓ* introduce tangential and normal coordinates *s* and *η* through(3.1)where is the normal to *ℓ* at the point (*x*′, *y*′) on *ℓ*. Then at leading order in *c* for steady flow (2.6) becomes(3.2)where variations along *ℓ* appear only parametrically in the changes in direction of the normal. Let *h*_{±}(*s*) be the limiting interface heights on either side of the leap in the outer solution so(3.3)

Then, as for the steadily propagating bores of Kakutani & Yamasaki (1978) and Miles (1979), the internal structure is given by(3.4)where *η*=0 has been taken as the otherwise indeterminate centre of the leap. For *a*≠0(3.5)where(3.6)so, for *γ* real, *ζ*≤−2/3 or *ζ*>0. If quadratic term nonlinearity is absent (*a*=0) then *γ*=−1 in (3.6) and (3.4) becomes(3.7)

For real solutions *μ* is positive so the slope of *ℓ* must satisfy (d*y*′/d*x*′)^{2}>*b*/*d*, which holds automatically in subcritical flow (where *b*<0) but imposes a minimum slope on leaps in supercritical flow. If a leap in supercritical flow curves towards this critical slope its height decreases to zero. For all *a* (3.4) allows both upward and downward leaps.

Following Karpman (1975) and Gurevich *et al*. (1995, 1996), JV and Vilenski & Johnson (2004) note that the outer steady flow field of the two-dimensional KP equation can be described for weak forcing in supercritical flow in terms of solutions of the associated unsteady one-dimensional KdV equation with the cross-stream coordinate *y* in the two-dimensional problem corresponding to a slow time *τ* in unsteady problem. A similar reduction is possible in supercritical flow here. Equation (2.6) has steady solutions satisfying in the far-field, to leading order in *a*≪1, the unsteady one-dimensional eKdV equation(3.8)where *τ*=*a*^{2}*y* is the cross-stream time-like variable and a coordinate constant along characteristics of the linear problem (and thus present only in supercritical flow). Grimshaw *et al*. (2002*a*) discuss the solutions of (3.8) pointing out that the precise form of the solitary waves emerging from bounded initial data follows from the relevant scattering problem and that in particular sufficiently large initial data evolves into table-top waves. To explicitly compare unsteady eKdV solutions with the equivalent steady feKP solutions requires choosing the correct initial condition for the unsteady integrations. It is not clear how to do this once significant wavefields appear in the flow field, except by using the centreline interface displacement from the steady two-dimensional problem as in JV and Vilenski & Johnson (2004). Comparisons of numerical integrations (using the methods of §4) of the unsteady eKdV and the steady feKP, not presented here, show that supercritical two-dimensional flow with small and moderate forcing is well described by the related unsteady one-dimensional solutions, interpreted in terms of the waves described by Grimshaw *et al*. (2002*a*), and thus purely supercritical flow is not treated further here.

## 4. Cross-stream ridges

As in JV and Vilenski & Johnson (2004) it is most straightforward here to obtain steady numerical solutions by integrating an initial value problem. Equation (2.6) is integrated using the linearly implicit spectral method of Fornberg & Driscoll (1999) which treats carefully the inherent stiffness due to high wavenumbers in the equations for the Fourier components. Equation (2.6) is integrated forward in time subject to the conditions (2.7) until the flow pattern is steady within a large domain centred on the obstacle. The time development of the flow is similar to that described in JV and Vilenski & Johnson (2004) and so attention is confined below to description of the eventual asymptotically steady state. For simplicity and smoothness the obstacle in the integrations is chosen to be the stretched Gaussian(4.1)where *w*_{x} and *w*_{y} are typical horizontal length scales (in the *x* and *y* directions when is *ϑ*=0) and *ϑ* is the angle between the obstacle's line of symmetry and the crossflow direction (so it is sufficient to consider *w*_{y}≥*w*_{x} with 0≤*ϑ*≤*π*/2). Without loss of generality *d* is taken as unity and, except where explicitly noted otherwise, the obstacle height is taken as *M*=2, so that nonlinear effects are important, the dispersion coefficient as *c*=0.1, sufficiently small so separate flow features do not merge, and *q*=0.3 with *a*=0 to isolate cubic nonlinearity. This section describes first the flow forced by a finite length ridge perpendicular to the oncoming flow with *w*_{x}=1, *w*_{y}=6 and *ϑ*=0 in (4.1).

### (a) Departures from criticality: the detuning parameter *b*

Steady supercritical (*b*>0) flows (not shown here) with *a*≠0 closely resemble the supercritical one-layer flows of JV with bow waves upstream of the obstacle, and a dispersive wavetrain downstream. For supercritical flows with *a* identically zero the absence of isolated solitary wave solutions of (3.2) means that large amplitude bow waves are absent from the flow. Strong localized forcing at the obstacle drives a weak wavefield very similar to the linear dispersive wavefield of much weaker forcing. Subcritical flows (*b*<0) differed markedly from the one-layer flows of JV. In all cases they rapidly become steady in a large region about the obstacle and near criticality finite-amplitude cross-stream leaps appear. Figure 2 shows an interfacial wave with table-top streamwise profile appearing in steady flow immediately behind the obstacle as the detuning parameter *b* decreases from zero. In critical flow (*b*=0) there is a rapid transition from subcritical to supercritical flow as the interface plunges on the lee side of the ridge (as expected from one-dimensional shallow-water theory) followed by an oscillatory recovery to undisturbed flow. For slightly subcritical (*b*=−1/16) flow the leading wave of the wake, centred far downstream at *x*∼14 increases in amplitude, becoming table-topped with leading-edge and trailing-edge leaps. Figure 3 shows that the surface elevation of this table-top wave varies only slowly away from the centreline, giving almost one-dimensional flow over the ridge. For more subcritical flow (*b*=−1/8 in figure 2) the table-top wave remains with the wake amplitude initially growing with increasing detuning. Two-dimensional effects begin to appear by *b*=−1/4 (not shown here) and figure 4 shows that by *b*=−1 the leading leap in the wake is strongly distorted by waves propagating towards the centreline from the ridge ends even for this wide ridge. After these waves collide they form the edges of the usual dispersive ‘ship-wave’ wake in the lee. The table-top standing wave remains visible but is confined closer to the axis and has shorter dispersive waves superposed on it.

For even slightly more subcritical flows the table-top wave effectively disappears. Figure 5 for *b*=−1.5 shows the strong convergence of waves from the ridge ends destroys the table-top wave. The dispersive wavetrain extends no further from the axis than the ridge itself and near the axis forms a train of alternate upward and downward leaps. By *b*=−2 (not shown here) the flow is sufficiently subcritical that leaps do not form. The wavetrain is a simple subcritical wavetrain confined behind the highest central section of the ridge.

These large changes in the interface field cause dramatic changes in the drag on the ridge. Figure 6 shows the normalized drag *C*_{x}/*M*^{2} as a function of *b*, where the non-dimensional form drag is given by (JV)(4.2)

The drag attains a clear maximum near *b*=−1 when the transition above the ridge is steepest. In the present flow where cubic nonlinearity is important the drag increases by a factor of 12 as the flow moves from critical to *b*=−1 differing significantly from transcritical flow dominated by quadratic nonlinearity in JV where changes of drag with detuning are small near critical and there is no sharp maximum. It appears that near *b*=−1 the table-top wave is so positioned as to increase the steepness of the transition above the ridge. Further into the subcritical regime the drag drops towards the small linear value, negligible compared to the resonant value.

### (b) Quadratic nonlinearity

For small quadratic nonlinearity, i.e. |*a*| small, the structures noted above for *a*=0 survive. However the cross-stream extent of the table-top waves decreases rapidly with increasing |*a*|. Figure 7 shows the interface elevation for the same flow as figure 3 but for *a*=−0.5. The leaps curve and weaken with distance from the axis and the wake no longer has the simple streamwise profile of a table-top wave. Moreover, the flow at large time (*t*=50 here) remains weakly unsteady as, in the absence of cubic nonlinearity, the purely quadratic flow would lie in the unsteady modulated wavetrain regime of figure 7 of JV.

### (c) The cubic term

Figure 8 shows the interface elevation along the centreline *y*=0 for *b*=−1/8 at three values of the cubic coefficient *q*. The patterns for *q*=0.1 and *q*=0.5 have very similar shapes with however the amplitude of the leaps significantly smaller at *q*=0.5. This agrees with borecube which requires the height of a leap to decrease as *q* increases. Further increases in *q* lead to lower leaps. The table-top wave is also small for *q*=0.3 but this appears to be related to a weaker, more confined interface deviation over the ridge. The qualitative effect of variations in *q* is small compared to variations in detuning.

### (d) The orientation of the ridge

Figure 9 shows the effect of changes in the ridge's orientation to the oncoming flow. The table-top wave previously present behind the cross-stream ridge has disappeared by *ϑ*=*π*/4 to be replaced by a series of lee-wave crests parallel to the ridge but lying only in *y*<0. Further downstream these crests disperse giving a spreading pattern centred about *y*∼−18. By *ϑ*=*π*/2 the ridge is aligned with the flow causing on the centreline only a smooth, slow transition from subcritical to supercritical flow above the obstacle and then an undular bore recovery. The disturbance decays rapidly away from the centreline.

## 5. Axisymmetric obstacles

Figure 10 shows that as the cross-stream length *w*_{y} of the ridge decreases, the normalized drag also decreases, although more weakly with a reduction from *w*_{y}=6 to *w*_{y}=1 reducing the drag by less than half. The interface elevations in figure 11 shows why this weak variation might be expected. Figure 11*a* shows the interface for a cross-stream ridge with *w*_{y}=6 with first a depression behind the ridge and then the table-top wave in the wake. Figure 11*b* shows the same pattern for *w*_{y}=1. Remarkably, although the wide depression behind the obstacle is absent (as expected), the downstream wavefield of this axisymmetric (in the current scaling) obstacle contains a table-top standing wave as in flows over ridges. It is this close resemblance of the radiated wavefields that leads to the similar wave drags even when the ridge length varies. As noted earlier, although axisymmetric obstacles in the current scaling correspond to obstacles elongated across the flow in physical variables, the experiments reported in Johnson *et al*. (in press) show that solutions to weakly two-dimensional equations capture the general behaviour of transcritical flow over obstacles with horizontal aspect ratios of order unity. Figure 11*c*,*d* demonstrates the essential nonlinearity of the flow by showing the interface displacements of the linear flows corresponding to the nonlinear flows of figure 11*a*,*b*. The linear solutions miss entirely the flat depressed region behind the ridge and the table-top wave in both wakes. It is the absence of theses features that appears to lead to the lower drags noted in figure 6 in linear subcritical flows (provided, of course, that *b* is not so close to zero that the linear flow is strongly resonant).

### (a) Obstacle height

Figure 12 gives interface displacements for obstacles of height *M*=2 and *M*=4. Comparing figure 12*a*,*b* shows that the higher obstacle produces higher, more nonlinear waves with sharper, more skewed fronts. The details for *M*=4 in figure 12*c*,*d* show a curved leap extending several obstacle widths away from the downward transition above the obstacle. The flow recovers in a steep finite-amplitude wave that at its extremes marks the beginning of the dispersive wavetrain. The curved leap in *y*<0 is of similar length and orientation as the ridge in figure 9*c*,*d* and the dispersive wavetrain in *y*<0 in figure 12*b* resembles that in the lee of the ridge in figure 9*d*. The pattern near the centreline here is however of the typical form noted for weakly subcritical flow with a table-top wave extending almost unchanged for several obstacle widths across the flow before it is disrupted by the dispersive wavefield.

### (b) Dispersion

Figure 13 shows the interface height along the centreline *y*=0 as a function of *x* for different strengths of the dispersion *c*. Dispersion has little effect on the transition above the obstacle demonstrating that the transition is basically hydraulic (*c*=0). Neither does dispersion affect the height of the table-top wave in the wake. Comparing the profiles for *c*=0.1 and *c*=0.3, however, shows that, in accord with (3.1), increasing *c* widens the leaps forming the leading and trailing edges of the table-top wave. The profile for *c*=0.01 shows that if *c* becomes too small, for fixed *M* and *q*, the balance between dispersion and nonlinearity in the table-top wave is broken and dispersive waves appear superposed on the nonlinear wave. The drag is determined predominantly by the height of the transition above the obstacle and figure 14 shows that the relative insensitivity of this height to changes in *c* means that even a 30-fold increase in *c* has little effect on the drag.

## 6. Discussion

The distinctive feature of the steady flows found numerically here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a table-top wave extending almost one-dimensionally for many obstacles widths across the flow. The integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges by *b*<−1. The wave appears after the subcritical flow has passed through a transition to supercritical over the obstacle and its leading and trailing edges are dissipationless leaps standing in supercritical flow as described propagating in one dimension by Melville & Helfrich (1987). It is noted in §3 that for weak forcing two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable. Thus the wide cross-stream extent of the table-top wave appears to derive from its presence in a supercritical region of the flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow, described in detail by Grimshaw *et al*. (2002*a*). The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag, an effect absent when cubic nonlinearity absent (JV). Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

The extended KP, as the KP itself, applies to geometries where the cross-stream, *y*, scale is large compared to the streamwise, *x*, scale. As noted in §2 the ratio of *x* to *y* scale is of order the square root of the fractional departure of the oncoming flow speed from the interfacial wave speed and this departure is taken to be small here. This has two consequences. Firstly, axisymmetric orography in the scaled coordinates here corresponds to cross-stream extended orography in unscaled coordinates. Secondly, the slow cross-stream variation of the table-top wave is even slower in unscaled coordinates. Karpman (1975) demonstrates that the KP (and similarly the eKP here) describe the farfield (in *y*) behaviour for near-critical flows in quite a general sense and thus, for two-layer flows with stratifications and depth ratios such that quadratic nonlinearity is negligible and cubic nonlinearity dominant, cross-stream extending table-top waves appear likely to be a generic feature of slightly subcritical flow even for narrow orography concentrated near the *y*-axis.

## Acknowledgments

This work was funded by the UK Natural Environment Research Council under grant NER/A/S/2000/01323. We are indebted to referees on an earlier version of this paper for their constructive comments.

## Footnotes

- Received September 27, 2004.
- Accepted June 10, 2005.

- © 2005 The Royal Society