Skip to main content Accessibility help

Wave–vortex decomposition of one-dimensional ship-track data

  • Oliver Bühler (a1), Jörn Callies (a2) and Raffaele Ferrari (a3)


We present a simple two-step method by which one-dimensional spectra of horizontal velocity and buoyancy measured along a ship track can be decomposed into a wave component consisting of inertia–gravity waves and a vortex component consisting of a horizontal flow in geostrophic balance. The method requires certain assumptions for the data regarding stationarity, homogeneity, and horizontal isotropy. In the first step an exact Helmholtz decomposition of the horizontal velocity spectra into rotational and divergent components is performed and in the second step an energy equipartition property of hydrostatic inertia–gravity waves is exploited that allows a diagnosis of the wave energy spectrum solely from the observed horizontal velocities. The observed buoyancy spectrum can then be used to compute the residual vortex energy spectrum. Further wave–vortex decompositions of the observed fields are possible if additional information about the frequency content of the waves is available. We illustrate the method on two recent oceanic data sets from the North Pacific and the Gulf Stream. Notably, both steps in our new method might be of broader use in the theoretical and observational study of atmosphere and ocean fluid dynamics.


Corresponding author

Email address for correspondence:


Hide All
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Callies, J. & Ferrari, R. 2013 Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1–200 km). J. Phys. Oceanogr. 43 (11), 24562474.
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 10871095.
Ferrari, R. & Rudnick, D. L. 2000 Thermohaline variability in the upper ocean. J. Geophys. Res. 105 (C7), 1685716883.
Ferrari, R. & Wunsch, C. 2010 The distribution of eddy kinetic and potential energies in the global ocean. Tellus A 62 (2), 92108.
Lindborg, E. 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388, 259288.
Lindborg, E. 2007 Horizontal wavenumber spectra of vertical vorticity and horizontal divergence in the upper troposphere and lower stratosphere. J. Atmos. Sci. 64 (3), 10171025.
Müller, P., Lien, R.-C. & Williams, R. 1988 Estimates of potential vorticity at small scales in the ocean. J. Phys. Oceanogr. 18 (3), 401416.
Munk, W. H. 1981 Internal waves and small-scale processes. In Evolution of Physical Oceanography (ed. Warren, B. A. & Wunsch, C.), chap. 9, pp. 264291. The MIT Press.
Nastrom, G. D. & Gage, K. S. 1985 A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42 (9), 950960.
Riley, J. J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65 (7), 24162424.
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145168.
Tulloch, R. & Smith, K. S. 2006 A theory for the atmospheric energy spectrum: depth-limited temperature anomalies at the tropopause. Proc. Natl Acad. Sci. USA 103 (40), 1469014694.
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281308.
Wang, D.-P., Flagg, C. N., Donohue, K. & Rossby, H. T. 2010 Wavenumber spectrum in the Gulf Stream from shipboard ADCP observations and comparison with altimetry measurements. J. Phys. Oceanogr. 40 (4), 840844.
Wortham, C., Callies, J. & Scharffenberg, M. 2014 Asymmetries between wavenumber spectra of along- and across-track velocity from tandem-mission altimetry. J. Phys. Oceanogr. 44 (4), 11511160.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Wave–vortex decomposition of one-dimensional ship-track data

  • Oliver Bühler (a1), Jörn Callies (a2) and Raffaele Ferrari (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.