## Microsoft word - cnrs-21_johnson.doc

*Proceedings of the HYDRALAB III Joint User Meeting, Hannover, February 2010 *
**NEAR-CRITICAL OROGRAPHIC FORCING OF STRATIFIED FLOW **
E.R.Johnson(1), J.G.Esler(2), A. Paci(3), S. Cazin(4), E. Cid(4), O. Eiff(4) & L. Lacaze(4)
(1) Department of Mathematics, UCL, U.K., E-mail: e.johnson@ucl.ac.uk
(2) Department of Mathematics, UCL, U.K., E-mail: gavin@math.ucl.ac.uk
(3) CNRM-GAME (URA1357 METEO-FRANCE and CNRS), Toulouse, France, E-mail:
(4) IMFT, Toulouse, France, E-mail: lacaze@imft.fr
Analytical studies show that the dynamics of the flow of two fluid layers of almost equal depths and densities differs significantly from those where the densities are similar but the depths differ. In particular, the waves that appear behind isolated three-dimensional orography in near-critical, but slightly subcritical, flow are long compared to the obstacle dimensions and stretch laterally across the flow with little change of form. Results are presented to show that these aspects of the wake can be observed experimentally. The experiments reported here have been done in the CNRM-GAME (Toulouse) stratified water flume, using an optical measurement technique developed at IMFT.

**1. BACKGROUND **
Two inter-related aspects of flow over orography have proved difficult to model simply -- the structure of the flow-field and the pressure drag exerted on the oncoming flow – yet these are of great interest. Rising and falling air affects cloud cover and rainfall distribution, and drag is of particular importance to oceanographers, climate modellers and researchers involved in the development of numerical weather-prediction models, because of the need to parameterize the drag exerted by orography with spatial scales below the model grid scale.
One class of flows in which analytical progress can be made is the class of flows where the
vertical stratification is such that internal wave energy is trapped and cannot propagate away vertically. The disturbance due to orography then spreads horizontally and the flow can be modelled as the flow of a single shallow layer of fluid. Subcritical flow (with Froude number

*F < *1, where

* F* is the ratio of the undisturbed flow speed to the gravity-wave speed.) over three-dimensional (i.e. height

*h=h(x, y)*) obstacles has been investigated experimentally (e.g. by Lamb & Britter 1984), largely because of its importance to the meteorological and oceanographic communities as a simple model of observed flows around islands, mountains, capes and sea-mounts (e.g. Schar & Smith 1993a). In the shallow-water limit (or 'non-dispersive' limit, as the phase speed of generated gravity waves is independent of wavenumber) the flow behaviour is typically characterized using two parameters --the Froude number

*F* and the non-dimensional obstacle height

*M* measured as the ratio of height to layer depth --. In terms of these parameters, non-dispersive shallow-water models give qualitative, and sometimes reasonable quantitative, predictions for various observed phenomena, including vortex-shedding periods (Schar & Smith 1993b) and wake lengths (Smith & Smith 1995).
An important caveat in the interpretation of the Jiang & Smith (2000) transcritical (

*F ~ *1

*) *results,
however, concerns the validity of the mass- and momentum-conserving hydraulic jumps that characterize the shallow-water solutions. Whilst these hydraulic jumps may sometimes be a good physical model for the situation at breaking waves (Mei 1989; Baines 1995), they are an appropriate model only for those physical situations where a regularizing dissipation dominates over dispersive effects on scales typical of the jump width. In many geophysical and laboratory situations it is dispersion that dominates. For example, there are numerous observations of multiple solitary waves, which are a distinctively dispersive phenomenon, in the flow upstream of obstacles, both in atmospheric flows ahead of islands (Li et al. 2004; Badgley, Miloy & Childs 1969; Burk & Haack 1999) and in laboratory experiments (Maxworthy, Dhieres & Didelle 1984; Johnson et al. 2006). In order to describe such dispersive phenomena, the third important physical parameter in the problem, the ratio

*d* of the layer depth and the obstacle width, must be treated as non-zero. One approach to

*Proceedings of the HYDRALAB III Joint User Meeting, Hannover, February 2010 *
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. (2002b) 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) describe orographically forced atmospheric waves in terms of solutions to the forced KP equation. However, applications to the atmosphere, oceans and experiments tend to be for larger amplitudes and layered flows where, as noted above for one-dimensional motion, it is useful to include cubic nonlinearity, giving here the extended KP (eKP) equation. This project 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. Figure 1, from Johnson & Vilenski (2005) shows the interface displacement for marginally subcritical flow two-layer equal-depth flow over a short ridge. The lee wave train is composed of waves with wavelengths more than five times the downstream obstacle length (scaled to order unity here) and, equally, extending across the flow distances of order five times the ridge length (also scaled to order unity).
The interface displacement for marginally subcritical
two-equal-layer flow over a short ridge.

*Proceedings of the HYDRALAB III Joint User Meeting, Hannover, February 2010 *
**EXPERIMENTAL RESULTS **
Experiments were carried out in the CNRM-GAME stratified water flume (Toulouse, France). The flume was used in a towing tank configuration, with net dimensions Ht × Wt × Lt = 1 m × 3 m × 22 m. This unique facility is indeed very pertinent for such a study, in particular due to its ability to generate density stratified flow at high Reynolds numbers with low confinement effect.
Two different axi-symmetric obstacles with a gaussian shape, a diameter of about 100cm, and a
height of 8.0cm and 12.5cm were towed upside-down at uniform velocity in a two density layer fluid. The layers were 15cm deep with a density difference of 0.06. The towing speed varied from about 5cm/s to more than 30cm/s, leading to a Froude number range of 0.2-1.6. Interface was seeded with particles and its displacement measured using a non intrusive technique developed at IMFT based on a stereoscopic optical system. Drag measurements were also done using a technique developped at CNRM-GAME.
Figure 2 shows the two-dimensional interface pattern just after the obstace left the field of view,
and Figure 3 shows profiles of the interface behind the obstacle in the same experiment (Froude number is 1.08). The lee wave train has a wavelength of order 2.5m, approximately five times the obstacle radius. The cross-stream structure shows that interface profile along the centerline persists for some tenth of centimeters away in the cross-flow direction.
The two-dimensional interface displacement pattern in the tank for an experiment
with Fr=1.08, just after the obstacle left the field of view. X-axis is along the flow
(and flume main axis, center of the obstacle is close to X~130cm, below the figure Y-axis).
Y-axis is perpendicular to the flow (center of the obstacle and flume axis are close to Y~135cm).
Flow is going from the bottom to the top.

*Proceedings of the HYDRALAB III Joint User Meeting, Hannover, February 2010 *
The interface displacements a behind the towed obstacle plotted in a frame moving
with the obstacle in the same experiment than Figure 2 (Fr=1.08). This figure, similar to Figure 1
(but for a different Froude number), is computed from about 40 two-dimensional interface fields
including the one shown Figure 2. Z is the interface displacement expressed in cm from the
measurement disposal, i.e. it increases when the interface is moving away from the obstacle.
Y is the distance from the flume (and obstacle) axis along a line perpendicular to the flow. X is the
distance along the flume axis from the center of the obstacle (located on the left side, at X=0cm).

**ACKNOWLEDGMENT **
This work has been supported by European Community's Sixth Framework Programme through
the grant to the budget of the Integrated Infrastructure Initiative HYDRALAB III within the Transnational Access Activities, Contract no. 022441. We thank B. Beaudoin, B. Bourdelles, J.-C. Canonici, M. Morera, F. Murguet, S. Lassus-Pigat, and H. Schaffner of the CNRM-GAME (URA1357, METEO-FRANCE and CNRS) fluid mechanics laboratory, as well as Y. Dossmann, M. Foucherot, N. Tonnelier and B. Sudrie, for their kind support during the experiments.

**REFERENCES **
Esler J.G., Rump O.J., Johnson E.R. (2005): Steady rotating flows over a ridge. Physics of Fluids
Esler J.G., Rump O.J., Johnson E.R. (2007): Transcritical rotating flow over topography. Journal of
Johnson, E.R., and G.G. Vilenski (2004): Flow Patterns and Drag in Near-Critical Flow over Isolated
Orography.

*J. Atmos. Sci.*,

**61**, 2909–2918.

Johnson E.R. and G.G. Vilenski (2005): Two-dimensional leaps in near-critical flow over isolated
orography. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 461(2064): 3747.
Johnson E.R., Esler J.G, Rump O., Sommeria J., Vilenski G. (2006): Orographically generated
nonlinear waves in rotating and non-rotating two-layer flow. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 462(2065): 3.

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