Buoyancy Frequency Calculator

Displace a water parcel upward in a stably stratified column and it finds itself denser than its new surroundings, so gravity pulls it back; it overshoots, and oscillates. The angular frequency of that oscillation is the buoyancy or Brunt-Vaisala frequency, N, defined by N squared = -(g / rho0) * d(rho)/dz with z measured upward. Positive N squared means stable layering, zero means neutral, and negative means dense water sits above light water and the column will overturn convectively. Because N squared is proportional to the density gradient rather than the density itself, quite small differences matter: a change of 2.5 kg per cubic meter over 100 m gives N squared near 2.4e-4 per second squared, N near 0.0155 rad/s, and a buoyancy period around 7 minutes. Typical ocean values span three orders of magnitude in N squared. The surface mixed layer is nearly homogeneous with N close to zero. The seasonal and permanent pycnocline, usually between 50 and 300 m, holds the maximum, with N of roughly 5 to 20 cycles per hour. Below about 1000 m stratification weakens to 1 to 3 cycles per hour, and abyssal water is weaker still. N sets the upper frequency bound for internal gravity waves, which can exist only between the local inertial frequency and N, and it enters the Richardson number Ri = N squared divided by the square of the vertical shear; below the critical value of 0.25 Kelvin-Helmholtz billows form and mixing follows. The practical caveats are that N must be computed from potential density referenced to a suitable pressure rather than in-situ density, otherwise the adiabatic compressibility of seawater produces spurious stratification, and that finite-difference gradients from raw CTD casts are noisy and normally smoothed before differentiation. The Richardson number reported here assumes a nominal shear and is illustrative only.

The frequency carries two names because two scientists arrived at it independently in the early twentieth century while working on atmospheric stability. David Brunt, a Welsh meteorologist who became professor at Imperial College London and later president of the Royal Meteorological Society, published the result in 1927 in the Quarterly Journal of the Royal Meteorological Society in work on the period of simple vertical oscillations in the atmosphere. Vilho Vaisala, the Finnish physicist who founded the instrument company that still bears his name and who developed an early practical radiosonde, derived the same oscillation frequency in the mid-1920s in the course of his atmospheric sounding work. In some literature the underlying physics is credited further back to Karl Schwarzschild and to the nineteenth-century convective stability criterion associated with William Thomson and with Karl Schwarzschild's 1906 radiative-convective analysis. The transfer of the concept from meteorology to oceanography followed the growth of quantitative physical oceanography in the mid-twentieth century, particularly through the work of Carl-Gustaf Rossby's students and of Walter Munk at Scripps. Munk and Christopher Garrett's 1972 and 1975 papers describing a universal internal wave spectrum scaled by N made the buoyancy frequency the natural vertical coordinate for interpreting ocean variability, and the WKB stretching that follows from it remains standard. Adrian Gill's 1982 textbook Atmosphere-Ocean Dynamics consolidated the treatment for both fluids. Routine computation became possible only with the CTD, which replaced Nansen bottles and reversing thermometers during the 1960s and 1970s, and global coverage arrived with the Argo float array, which began deployment in 2000 and now supplies the profiles used to track decadal strengthening of upper ocean stratification.