• Internal waves

Focusing on wave attractors

Field observations

Shelf-edge front

Dead-water

inertial waves

• Tides

Current observations

Coastal resonance

Topographic filtering

Rectified flows

• Ocean circulation

Thermohaline circulation

• Vortices

Meddies

• Miscellaneous

Surface-area ellipsoid

New pressure sensory feeding mechanism in birds

Diffusion in strained flow

Anna Rabitti: (ongoing) - The Equatorial boundary layer.

Hazewinkel, J.: 2010 - Attractors in stratified fluids - Utrecht University.

Swart, A. N.: 2007 - Internal waves and the Poincaré equation - Utrecht University.

( download as PDF-file )

Lam, F.P.-A.: 2007 - Ocean and laboratory observations on waves over topography - Utrecht University. ( download as PDF-file )

Terra, G.M.: 2005 - Nonlinear tidal resonance - Utrecht University.

( download as PDF-file )

Manders, A.M.M.: 2003 - Internal wave patterns in enclosed density-stratified and rotating fluids - Utrecht University. ( download as PDF-file 

Schrier, G. van der: 2000 - Aspects of the thermohaline circulation in a simple model - Utrecht University.

Gerkema, Th.: 1994 - Nonlinear dispersive internal tides: generation models for a rotating ocean - Utrecht University

Haren, H. van: 1990 - Observations on the structure of currents at tidal and sub-tidal frequencies in the central North Sea - Utrecht University.

Sjoerd Groeskamp: 2010 – Thesis: Solitary Internal Waves in Marsdiep Tidal Channel, Using ADCP measurements - Utrecht University. ( download as PDF-file )

Jordy de Boer: 2010 – Thesis: Dynamics of a tidal estuary - KTH Stockholm.

( download as PDF-file )

Selvi Makarim: 2009 - M2 Internal tide in Mozambique Channel - Utrecht University.

Chrysanthi Tsimitri: 2007 - Thesis: Perturbed Internal Wave Attractors – Utrecht University. ( download as PDF-file )

Pieter van Breevoort: 2007 - Thesis: Experiments on internal wave attractors - Utrecht University. ( download as PDF-file )

Sander Ganzevles: 2007 -   Zwemmen in gelaagd water - VU Amsterdam.

Fons van Nuland: 2007 -   Zwemmen in gelaagd water - VU Amsterdam.

Huussen, T:. 2006 - Internal wave analysis of recent LOCO measurements in the Mozambique channel – University of Amsterdam.

Kopecz, S. : 2006 - Internal waves in a tilted square – Kassel University.

Verwer, E.: 2005 - Rossby– Inertiaalgolfparadox -   Utrecht University.

Wijngaards, E.F.: 2003 - Thesis: Experiments on internal wave attractors - Univ. Twente. ( download als PDF-file )

Berg, W.J. van den: 2003 - Laboratory experiments with tidal Helmholtz resonators - Utrecht University

Ypma, G.: 2002 - Internal waves: solitons in the ocean - Utrecht University.

Swart, A.: 2001 - A finite element method for internal gravity waves - Utrecht University.

Rienstra, M.: 2001 - Experimental and theoretical research on inertial waves in enclosed basins. - Utrecht University.

Sollie, H. 2001 - A 3D finite element model of internal waves - Univ. Twente.

Koenderink, F.: 1999 - Chaotic behaviour of quasi-periodically driven Helmholtz oscillators - Utrecht University.

Pörtzgen, N.: 1998 - Dataverwerking van metingen aan inertiaalgolven - Tech. Univ. Delft.

Öllers, M.: 1998 Towards a validation of a low-order ocean model for the thermohaline circulation with two Ocean General Circulation Models. - Utrecht University.

Lam, F.P.A.: 1992 - Getij-analyse en interpretatie van stroommetergegevens uit de IJslandzee in de nabijheid van een shelfrand - Utrecht University.

Bokhove, O. 1990 - Nonlinear Poincaré waves - Tech. Univ. Delft.

Hulscher, S. 1990 - Hogere getij-harmonischen in de Noordzee - Utrecht University.

Bachelor Students:

Beckebanze, F.: 2011 Experiments and theory on freak waves - Utrecht University

Gregorian, P.: 2011 Synthetic seismograms for a spherically symmetric, non-rotating, elastic and isotropic Earth model - Utrecht University

Wright, J.: 2010 - University College London.

Witte J.: 2007 - Opwaarts gedreven schijf - Delft Tech. University.

Kruijts, M.: 2006 - Metingen aan roterende stromingen - Twente University.

Technical High school Students:

 Smit, N.S.: 2005 – Drukmetingen aan roterende stromingen - TH Rijswijk.

When the sidewalls of a container slope steeper than the characteristics, these waves are focused upon reflection, but are reflected back into the interior. However, in enclosed domains focusing reflections dominate over defocusing ones, so that internal waves tend to be steered towards certain periodic orbits (wave attractors), where viscous and nonlinear effects act to absorb these. An example of the streamfunction field of standing internal waves in such a configuration is shown at the right. The following two movies present experimental results (see one of accompanying papers) obtained in the fluid dynamics Laboratory of Dr. J. Sommeria, who was then at the Ecole Normale et Superieure de Lyon. It presents a side-view of a uniformly-stratified fluid, which is visualized by means of fluorescent dye, injected in alternating horizontal layers, upon using a vertical laser sheet. The fluid has a sloping wall at the right, and the table on which the tank sits is oscillated vertically. By the parametric excitation mechanism internal waves are generated, that become visible 5 minutes after the start of the oscillation. See the internal wave experiment.

These waves appear to be localized on a limit cycle: a wave attractor (here: the obliquely oriented, rectangular shaped object). Two distinct phases can be discerned, which is perhaps better appreciated by subtracting the initial, horizontal dye lines, shown here. 1: An initial phase, in which the localized oscillations are all in phase. In this growing phase, the wave thus appears to be standing. (This can be appreciated by noting that there are certain instances at which in the latter movie the whole picture is black). And, 2 a quasi-stationary final state, in which there appears to be a continuous propagation of phase (and, hence, energy) and the waves thus, paradoxically are of propagating type. This phase propagation is seen from the (black) nodal lines that 'cross over' in a direction perpendicular to the edges of the rectangular box-shaped attractor. Given that internal wave energy propagation is perpendicular to its phase propagation direction (having opposite vertical components) the sense in which each of these phase lines propagates is consistent with a energy propagating around (and into) the attractor in a clockwise direction, consistent with the sense dictated by the focusing at the sloping side.

Publications:

Maas, L.R.M., Benielli, D., Sommeria, J., Lam, F.-P. A. (1997)

Observation of an internal wave attractor in a confined stably-stratified fluid.   Nature, 388: 557-561. ( Download PDF-file )

Maas, L.R.M. (1995)

Focusing of internal waves and the absence of eigenmodes. 'Aha Huliko' a 1995 conference proceedings, eds: P.M. Müller and D. Henderson. ( Download PDF-file )

Maas, L.R.M., Lam, F.-P. A. (1995)

Geometric focusing of internal waves. Journal of Fluid Mechanics, 300: 1-41 ( Download PDF-file )

Unpublished webscript:

Leo Maas, Uwe Harlander, Astrid Manders & Arno Swart (2003)

Stratified fluid experiments in an annulus with sloping inner wall under slight modulation of the rotation speed. ( Download PDF-file )

Links to related sites:

• Thierry Dauxois

• Bruce Sutherland

Internal tide amplitude (left; m/s) and phase (right, degrees) in vertical cross-sections obtained from Bay of Biscay field observations (top) and a numerical model (bottom).The observed area in the top panels corresponds to that inside the dashed rectangle in the bottom panels. Horizontal and vertical distances are given in kilometers (see Lam et al 2004, and Gerkema et al 2004)

The observed area in the top panels corresponds to that inside the dashed rectangle in the bottom panels. Horizontal and vertical distances are given in kilometers (see Lam et al 2004, and Gerkema et al 2004)

Dead-water is a nautical term referring to a drastic decrease of a ship’s speed compared to its normal speed that arises when traversing through water that is density-stratified. The decrease in speed was reported to be up to a factor 5, giving the ship the appearance of having been brought to a full stop.

Inertial (gyroscopic) waves in homogeneous rotating fluids:

Homogeneous, rotating fluids support waves that are restored by the Coriolis force. These waves of frequency less than twice the rotation rate are similar to internal gravity waves in uniformly-stratified, non-rotating fluids in being also transverse, and in again being constrained in propagation direction, but now relative to the rotation axis. In particular, the waves will focus when reflecting from a sloping boundary onto wave attractors. However, because particle motion is circular instead of rectilinear, as for the internal gravity waves [which mathematically translates into a requirement that the pressure field satisfies oblique-derivative (Robin) boundary conditions], these waves generate spatially complicated patterns, even when just reflecting from vertical boundaries. See for instance the patterns of inertial wave energy distribution in a horizontal cube, shown in a plan view (horizontal mid-plane) in the left-hand corner of these web pages. Both stratified as well as rotating fluid systems (and also their oceanographically relevant combination) can accommodate waves of arbitrary frequency, thus denying the existence of eigenmodes. The spatial structure of these waves is determined by the Poincare equation, which is hyperbolic in spatial coordinates, and which is responsible for the unusual behaviour of these waves. See the spatial pattern of the observed current magnitudes (low/high presented as blue/red colors) and the redicted attractors (solid line) or standing mode (dashed line, upper right figure). These figures present side views taken from Astrid Manders' thesis, and the first reference below.

Some short movies reveal the time-dependent inertial wave.

A mean flow is generated above the location where the waves approach an attractor. This can be discerned in the following movie (in Microsoft .WMV format) taken from above:

• Movie for Dial-up connection: 38Kbps (160*120 / ~710 kB)
• Movie for ISDN connection: 48 Kbps (160*120 / ~875 kB)
• Movie for Broadband connection: 150 Kbps (320*240 / ~2.5 MB)

 In the lower/upper half of this frame the bottom is flat/ sloping upwards. It displays a steady, cyclonic dye displacement, presumably driven by the inertial waves that are forced by the modulation of the rotation (which itself is responsible for the observed periodic part of the motion).

The inertial waves are geometrically focused by the sloping bottom. The mean flow that results from the angular momentum mixing that ensues is particularly strong over the middle part of the sloping region (3/4 upwards from the lower boundary of this frame) where the theoretical attractor reflects from the slope (and focusing occurs).

Publications:

Manders, A.M.M., Maas, L.R.M. (2004)

On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary. Fluid Dynamics Research 35,1-21. ( Download PDF-file )

Manders, A.M.M., Maas, L.R.M. (2003)

Observations of inertial waves in a rectangular basin with one sloping boundary. Journal of Fluid Mechanics 493, 59-88. ( Download PDF-file )

Maas, L.R.M. (2003)

On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dynamics Research, 33, 373-401.( Download PDF-file )

Maas, L.R.M. (2001)

Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids   Journal of Fluid Mechanics, 437: 13-28.( Download PDF-file )

User Report:

Coriolis turntable experiments - Mean flow generation by geometrically focused gyroscopic waves

Links to related sites:

• Joel Sommeria at rotating lab Coriolis/LEGI, Grenoble

• Michel Rieutord

• Boris Dintrans

Observations of (oscillatory) tidal currents reveal the coexistence of two counterrotating Ekman layers. Only the thicker (anticyclonic) one affects the shear over a thermocline further up from the bottom. Lagrangian (drifter) compared to Eulerian (moored) current measurements reveal an increase of velocity gradients upon a decrease in separation scale.

Here are observed amplitude W±and phase q± of the anticlockwise, or cyclonic (+) and clockwise, or anticyclonic (-) current components as a function of depth z, divided by water depth H.

In the presence of a stratified thermocline the transfer of only the thicker anticyclonic current component is affected.

Publications:

Maas, L.R.M., Haren, J.J.M. van (1987)

Observations on the vertical structure of tidal and inertial currents in the central North Sea. Journal of Marine Research, 45: 293-318.

Maas, L.R.M. (1989)

A comparison of Eulerian and Lagrangian current measurements. Deutsches Hydrographisches Zeitschrift, 42: 111-132.

Tides: Coastal resonance

Coastal tides may either be choked (as in Lagoon type estuaries) or resonantly amplified. These two, contrasting type of responses might actually coexist as two stable equilibria when the resonance horn (ratio of reesponse over forcing amplitude versus frequency) is bent ove. This happens due to nonlinear effects (such as simply due to a sloping bottom). In that case, an irregular response may result when a perturbation is kicking the response from one to the other equilibrium and back.

Publications:

Terra, G. M., Berg, W.J. van den, Maas L.R.M. (2003)

Experimental verification of Lorentz' linearization procedure of quadratic bottom friction. Submitted to Physics of Fluids. ( Download PDF-file )

Terra, G. M., Doelman, A., Maas, L.R.M. (2003)

A weakly nonlinear approach to coastal resonance. Submitted to Journal of Fluid Mechanics. ( Download PDF-file )

Doelman, A., Koenderink, F.A., Maas, L.R.M. (2002)

Quasi-periodically forced nonlinear Helmholtz oscillators.   Physica D, 164: 1-27. ( Download PDF-file )

Maas, L.R.M., Doelman, A. (2002)

Chaotic tides.     Journal of Physical Oceanography, 32: 870-890. ( Download PDF-file )

Maas, L.R.M. (1998)

On an oscillator equation for tides in almost enclosed basins of non-uniform depth. Physics of Estuaries and Coastal Seas, Dronkers, J. & M. Scheffers eds., A.A. Balkema Rotterdam: 127-132.

Maas, L.R.M. (1997)

On the nonlinear Helmholtz response of almost-enclosed tidal basins with a sloping bottom.   Journal of Fluid Mechanics, 349: 361-380. ( Download PDF-file )

Links to related sites:

• Arjen Doelman

Tides: Topographic filtering

A topographic irregularity (particularly a continental shelf or ocean ridge) may act as a spectral (often low-pass) filter for incident long waves, which is evident through its presence in the potential of a Schrödinger equation. For some circumstances, however, the potential is reflectionless, and the shelf region behind a slope becomes vulnerable to incident long waves satisfying the corresponding criteria.

For a typical shelf-slope topography (upper pannel) the localized nature of the scattering potential of the Schrödinger equation into which the problem can be cast is shown below.

Publications:

Maas, L.R.M. (1996)

Topographic filtering and reflectionless transmission of long waves. Journal of Physical Oceanography, 27: 195-202.

Tides: Rectified flows

Internal tides are both generated as well as advected by barotropic tides. The combined process acts to generate not only overtides, but also a mean flow. In a homogeneous sea, this barotropic, tidally-rectified flow extends over at most two tidal excursions beyond the slope in bottom topography. In the Northern Hemisphere it is directed such that it has the shallow side at its right-hand side. Stratification suppresses the vorticity stretching mechanism, responsible for the rectification, further away from the bottom. Apart from a frictional reduction close to the sea bed, this thus predicts a bottom enhancement of the tidally rectified flow.

Publications:

Lam, F.-P.A., Maas, L.R.M., Gerkema, T. (2003)

Spatial structure of tidal and residual currents as observed over the shelf break in the Bay of Biscay. Submitted to Deep-Sea Research I. ( Download PDF-file )

Maas, L.R.M., Zimmerman, J.T.F. (1989)

Tide-topography interactions in a stratified shelf sea II. Bottom-trapped internal tides and baroclinic residual currents. Geophysical and Astrophysical Fluid Dynamics, 45: 37-69.

Maas, L.R.M., Zimmerman, J.T.F. (1989)

Tide-topography interactions in a stratified shelf sea I. Basic equations for quasi-nonlinear internal tides. Geophysical and Astrophysical Fluid Dynamics, 45: 1-35.

Maas, L.R.M., Zimmerman, J.T.F., Temme, N.M. (1987)

On the exact shape of the horizontal profile of a topographically rectified tidal flow.   Geophysical and Astrophysical Fluid Dynamics, 58: 105-129.

Maas, L.R.M. (1987)

Tide-topography interactions in a stratified shelf sea. Ph.D. Thesis. University of Utrecht: 241 pp.

Ou, H.W., Maas, L. (1986)

Tidal-induced buoyancy flux and