Wave-induced mean flows in rotating shallow water with uniform potential vorticity. (English) Zbl 1419.86016
Summary: Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.
MSC:
| 86A05 | Hydrology, hydrography, oceanography |
| 76U05 | General theory of rotating fluids |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
References:
| [1] | Alford, M. H.; Mackinnon, J. A.; Simmons, H. L.; Nash, J. D., Near-inertial internal gravity waves in the ocean, Ann. Rev. Marine Sci., 8, 95-123, (2016) |
| [2] | Andrews, D. G.; Holton, J. R.; Leovy, C. B., Middle Atmosphere Dynamics, (1987), Academic Press |
| [3] | Andrews, D. G.; Mcintyre, M. E., An exact theory of nonlinear waves on a Lagrangian-mean flow, J. Fluid Mech., 89, 609-646, (1978) · Zbl 0426.76025 |
| [4] | Babin, A.; Mahalov, A.; Nicolaenko, B., Global splitting and regularity of rotating shallow-water equations, Eur. J. Mech. B, 16, 5, 725-754, (1997) · Zbl 0889.76007 |
| [5] | Van Den Bremer, T. S.; Sutherland, B. R., The mean flow and long waves induced by two-dimensional internal gravity wavepackets, Phys. Fluids, 26, 10, (2014) |
| [6] | Bretherton, F. P., On the mean motion induced by internal gravity waves, J. Fluid Mech., 36, 785-803, (1969) · Zbl 0175.52805 |
| [7] | Bühler, O., A shallow-water model that prevents nonlinear steepening of gravity waves, J. Atmos. Sci., 55, 2884-2891, (1998) |
| [8] | Bühler, O., On the vorticity transport due to dissipating or breaking waves in shallow-water flow, J. Fluid Mech., 407, 235-263, (2000) · Zbl 0986.76009 |
| [9] | Bühler, O., Waves and Mean Flows, (2014), Cambridge University Press · Zbl 1286.86002 |
| [10] | Bühler, O.; Holmes-Cerfon, M., Particle dispersion by random waves in rotating shallow water, J. Fluid Mech., 638, 5-26, (2009) · Zbl 1183.76878 |
| [11] | Bühler, O.; Jacobson, T. E., Wave-driven currents and vortex dynamics on barred beaches, J. Fluid Mech., 449, 313-339, (2001) · Zbl 1015.76012 |
| [12] | Bühler, O.; Mcintyre, M. E., On non-dissipative wave – mean interactions in the atmosphere or oceans, J. Fluid Mech., 354, 301-343, (1998) · Zbl 0922.76281 |
| [13] | Bühler, O.; Mcintyre, M. E., Remote recoil: a new wave – mean interaction effect, J. Fluid Mech., 492, 207-230, (2003) · Zbl 1063.76628 |
| [14] | Eliassen, A.; Palm, E., On the transfer of energy in stationary mountain waves, Geofys Publ., 22, 3, 1-23, (1961) |
| [15] | Garrett, C.; Kunze, E., Internal tide generation in the deep ocean, Annu. Rev. Fluid Mech., 39, 57-87, (2007) · Zbl 1296.76026 |
| [16] | Lighthill, J., Waves in Fluids, (1978), Cambridge University Press · Zbl 0375.76001 |
| [17] | Majda, A. J.; Embid, P., Averaging over fast gravity waves for geophysical flows with unbalanced initial data, Theoret. Comput. Fluid Dynam., 11, 3, 155-169, (1998) · Zbl 0923.76339 |
| [18] | Mcintyre, M. E., A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves, J. Fluid Mech., 189, 235-242, (1988) · Zbl 0643.76014 |
| [19] | Plumb, R. A., The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation, J. Atmos. Sci., 34, 1847-1858, (1977) |
| [20] | Thomas, J.; Smith, K. S.; Bühler, O., Near-inertial wave dispersion by geostrophic flows, J. Fluid Mech., 817, 406-438, (2017) · Zbl 1387.86024 |
| [21] | Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, (2006), Cambridge University Press · Zbl 1374.86002 |
| [22] | Wagner, G. L.; Young, W. R., Available potential vorticity and wave-averaged quasi-geostrophic flow, J. Fluid Mech., 785, 401-424, (2015) · Zbl 1381.86018 |
| [23] | Wagner, G. L.; Young, W. R., A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic, J. Fluid Mech., 802, 806-837, (2016) · Zbl 1456.76036 |
| [24] | Xie, J.-H.; Vanneste, J., A generalised-lagrangian-mean model of the interactions between near-inertial waves and mean flow, J. Fluid Mech., 774, 143-169, (2015) · Zbl 1328.76055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
