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Generation of internal waves by ocean tides

Dr Stephen Griffiths (SoM), Prof Onno Bokhove (SoM)

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Tides are the oceanic response to well-known gravitational forcing by the Sun and Moon. These hugely energetic oscillations influence various parts of the Earth system, such as the history of the lunar orbit, the transport of heat by the large-scale ocean circulation, and ice sheets flowing into the polar oceans [1, 2, 3].

As tidal currents flow over the bottom topography of the ocean, they generate internal gravity waves of tidal frequency in the density-stratified ocean interior. Although almost invisible from the ocean surface, these internal tides play an important role in a range of coastal and open-ocean dynamics [4]. Of particular interest is the amount of energy converted from the (surface) tide to internal tides (along with the magnitude of the implied internal tide drag on the depth-averaged flow), and how this energy is transported within the ocean interior [1, 5].

Figure 1: The internal tide in a numerical model. Left: bathymetry of the domain, in the NW Pacific. Centre: a snapshot of the mode 1 internal tide. Right: a snapshot of the mode 2 internal tide. The arrows indicate the sense of propagation of the internal tide.

Figure 1 shows results from a numerical model of internal tide generation in the Pacific Ocean. Waves are generated at regions of topographic change, such as mid-ocean ridges and continental slopes, and are weakly damped as they propagate away. Shown is the bottom pressure perturbation, which is a measure of how much the density contours are displaced upwards on average in the overlying water column; there would be accom­panying sea-surface displacements of a few centimetres.

Figure 2 shows a side view (or vertical transect) of internal tide generation in a simple two-dimensional model. Above a relatively shallow slope, the internal tide looks wavelike. However, above a relatively steep slope, the internal tide takes the form of a beam, moving away from the steepest part of the slope towards the shallow water.


Figure 2: The internal tide in a simple linear two-dimensional model. The red colours represent warmer (lighter) water, and the blue colours represent cooler (heavier) water. Left: internal tide generation at a relatively shallow slope. Right: internal tide generation at a relatively steep slope. In both cases, the wave propagate away from the slope.


In this PhD project, simple mathematical models of internal tide generation and propagation will be developed in idealised three-dimensional geometries, building on previous two-dimensional studies (e.g., [6, 7]). This involves making some combination of simplifying assumptions about the motion (e.g., linear, weakly nonlinear, time-periodic), density stratification (e.g., uniform, piecewise constant), and sea-floor topography (e.g., small‑amplitude, stepped), so that the nonlinear partial differential equations of motion can be reduced to a more tractable system. Solutions can then be sought analytically or via simple numerical computations, allowing a wide range of parameter space to be explored. This gives a better understanding of the energy transfer from (surface) tides to internal tides in different forcing scenarios, which is known to have a non-trivial dependence upon topographic width and height.

A particular focus would be on extending the solutions and methods of [7], which accounted for arbitrarily large topographic variations but only in two dimensions, to three-dimensional settings. The main targets are to

  • Understand internal tides generated by a three-dimensional surface tide (in particular, the Kelvin wave) above a two-dimensional topography (representing a continental slope). The Kelvin wave is the dominant form of the surface tide in many coastal zones, and the implied energy transfers are significant on a global scale.
  • Understand the role of topographic undulations in the along-shore direction, i.e., for a three-dimensional tide above a fully three-dimensional topography. It is known from observations that canyons (for exam­ple) lead to enhanced internal tide generation (and dissipation).

Potential for high impact outcomes

Internal tide generation at continental slopes (by Kelvin waves, and other tidal modes) is thought to be a major source of tidal dissipation, and thus of energy for vertical mixing and drag on the surface tide. However, observational estimates of the global energy transfer at continental slopes are imprecise, so additional models (or constraints) are of significant interest.


The work will be performed in the Astrophysical and Geophysical Fluid Dynamics research group within the Department of Applied Mathematics. This is one of the leading research groups of its type in the country, with expertise in mathematical and numerical modelling of fluid dynamics in settings varying from the Earth (core, ocean and atmosphere), to our solar system (Solar and planetary interiors), and beyond. Group members are expected to take part in weekly discussion meetings, seminars in fluids dynamics, and seminars in applied mathematics.

In addition to learning analytical, computational and writing skills during their research, our PhD students are also expected to enrol in taught graduate level modules. These might be taught at Leeds (e.g., Advanced Geophysical Fluid Dynamics, Advanced Modern Numerical Methods) or taken as part of the UK-wide MAGIC distributed learning network (e.g., Nonlinear Waves, Numerical Methods in Python).

Student profile

This PhD project would be well suited to somebody with a background in applied mathematics or theoreti­cal physics. Highly desirable would be expertise in fluid dynamics, waves, asymptotic methods, and partial differential equations. Experience with computer programming, or the willingness to learn, is essential.


[1]    Wunsch (2000): Moon, tides and climate. Nature, 405, 743–744.

[2]    Arbic, B. K. D. R. MacAyeal, J. X. Mitrovica & G. A. Milne (2004): Ocean tides and Heinrich events, Nature, 432, 460.

[3]    Gudmundsson, G. H., (2006): Fortnightly variations in the flow velocity of Rutford Ice Stream, West Antarctica, Nature, 444, 1063–1064.

[4]    Garrett, C., and E. Kunze, (2007): Internal tide generation in the deep ocean. Ann. Rev. Fluid Mech. 39, 57–87.

[5]    Egbert, G. D. & R. D. Ray, (2003), Semi-diurnal and diurnal tidal dissipation from TOPEX/Poseidon altimetry, Geophys. Res. Lett., 30, 1907.

[6]    Nycander, J., (2006): Tidal generation of internal waves from a periodic array of steep ridges. J. Fluid Mech., 567, 415–432.

[7]    Griffiths, S. D., and R. H. J. Grimshaw, (2007): Internal tide generation at the continental shelf modelled using a modal decomposition: two-dimensional results. J. Phys. Oceanogr. 37, 428–451.

Related undergraduate subjects:

  • Applied mathematics
  • Mathematics
  • Oceanography
  • Physics