Poincaré wave

The Poincaré wave , named after Henri Poincaré (1854–1912), like the Kelvin wave, is a flat water wave in a rotating frame of reference.

properties

The repulsive force of the Poincaré wave depends on its frequency . While in the case of high-frequency Poincaré waves, gravity acts as a restoring force, in the case of low-frequency Poincaré waves it is the Coriolis force . Due to this frequency dependence of the repulsive forces, the Poincaré wave is in principle dispersive, i.e. its frequencies are a non-linear function of its wavelengths , in which the acceleration of gravity , the Coriolis parameter and the dimensions of the body of water in which it spreads are parameters. Poincaré waves exist in an unlimited ocean with finite water depth. In a bordered ocean, the Poincaré waves are modified by the condition that no mass transport (integrated currents from the sea floor to the surface) can exist perpendicular to the coast. In bordered seas, in addition to the Poincaré wave, there are Kelvin waves that propagate along this border in a strip of finite width. In the case of a channel, there are two boundaries, which increases the number of boundary conditions in the mathematical treatment. The equator can also serve as a boundary for the Poincaré wave due to the singularity there in the Coriolis parameter (this is not defined at the equator or is equal to zero); one then speaks of equatorial Poincaré waves. The properties of Poincaré waves depend on the ratio of the wavelength to the Rossby radius , which results in the two borderline cases of short and long Poincaré waves:

• Short Poincaré waves are only weakly dependent on the Coriolis force because they have high frequencies and are therefore hardly dispersive.

• Long Poincaré waves are strongly influenced by the Coriolis force and are highly dispersive.

A dispersive wave is a wave in which the waveform changes over time . The Rossby radius is a length scale that is determined by the ratio of the speed of the gravity wave (without rotation ) to the Coriolis parameter . ${\displaystyle R}$ ${\displaystyle c}$ ${\displaystyle f}$

Rossby radius: ${\displaystyle R={\frac {c}{f}}\,}$

The Rossby radius is the benchmark for short and long Poincaré waves and provides information about the width of the bank zone in which the non-dispersive Kelvin wave propagates.

Occurrences and observations

Poincaré waves occur in all oceans on coastlines, in large bays and at the equator, as well as in large lakes, e.g. B. in Lake Michigan . Gustafson and Kullenberg made the first observation and measurement with flow measuring devices in 1936 in the Baltic Sea.

Mathematical description

Open ocean

The dispersion relation , i.e. the relationship between the frequency and the wave number vector , which defines a wave type, is for barotropic Poincaré waves: ${\displaystyle \omega \,}$ ${\displaystyle {\vec {k}}}$

${\displaystyle \omega ^{2}=f^{2}+gHk_{h}^{2}}$

With

${\displaystyle f\,}$: Coriolis parameter

${\displaystyle g\,}$: Gravity acceleration

${\displaystyle H\,}$: Water depth

${\displaystyle c={\sqrt {gH}}}$: Phase velocity of the shallow water wave in the non-rotating frame of reference

${\displaystyle {\vec {k}}_{h}={\vec {i}}k_{x}+{\vec {j}}k_{y}\,}$: Horizontal wave vector

The above dispersion relation of the Poincaré waves shows that it always holds for their frequencies . Poincaré waves with a wavelength short compared to the Rossby radius, ie , are approximately non-dispersive shallow water waves . This designation is justified because of the barotropic Rossby radius on Earth usually much larger than the water depth H is thus the corresponding wavelength is still much larger than the water depth and H is. Short Poincaré waves therefore propagate very quickly throughout the ocean from the area of ​​their excitation while maintaining their original shape. ${\displaystyle \omega \geq f}$ ${\displaystyle {\frac {\sqrt {gH}}{f}}k_{h}\gg 1}$

For long wavelengths, that is, the dispersion relation of the Poincaré waves is approximate . In this limiting case, the frequency is constantly equal to the Coriolis parameter f . Gravitation no longer has any influence and the liquid particles move in the equilibrium of inertial and Coriolis forces approximately in the form of inertial oscillations. For this reason, f is also called the inertia frequency. ${\displaystyle {\frac {\sqrt {gH}}{f}}k_{h}\ll 1}$ ${\displaystyle \omega \approx f}$

The group velocity of the barotropic Poincaré waves is ${\displaystyle {\vec {c}}_{g}}$

${\displaystyle {\vec {c}}_{g}={\vec {i}}{\frac {\partial \omega }{\partial k_{x}}}+{\vec {j}}{\frac {\partial \omega }{\partial k_{y}}}=c^{2}{\frac {\vec {k}}{\omega }}}$

and thus assumes the maximum value c in the limit case of short waves. It tends towards zero when the wavelengths go towards infinity. These properties of the group velocity of Poincaré waves determine the behavior of the propagation of an initial localized disturbance in such a way that a wavefront spreads dispersion-free circularly around an initially localized disturbance and long waves with , i.e. , remain in the center of the disturbance . In this case, the distance between the propagating fronts must be much larger than the Rossby radius. ${\displaystyle c_{g}\approx 0}$ ${\displaystyle \omega \approx f}$

Poincaré waves in an infinite channel

The two-sided boundaries in a channel that is infinitely long in the x-direction have the consequence that the velocity component perpendicular to the channel axis (here v in the y-direction) must always be at y = 0 and at y = B v = 0. This is always the case if v is proportional to a set of sine waves perpendicular to the channel axis with discrete wave numbers, namely for n = 1, 2, .... The individual sine waves with the discrete wave numbers are called modes . ${\displaystyle k_{yn}={\frac {n\pi }{B}}}$

Dispersion diagram of the Poincaré and Kelvin wave in a rotating channel, the width of a Rossby radius B = R is.

Channel in the rotated system:

Fig. 1: Infinitely long channel (x-direction), with width B (y-direction). Rotation with angular velocity ${\displaystyle {\frac {f}{2}}\,}$

The dispersion relation of the Poincaré waves in the infinite channel is obtained from the one for the unlimited ocean by replacing the continuous wave number component with the discrete wave number component . It then reads: ${\displaystyle k_{y}}$ ${\displaystyle k_{yn}}$

${\displaystyle \omega ^{2}=f^{2}+c^{2}\left({\frac {n^{2}\pi ^{2}}{B^{2}}}+k_{x}^{2}\right)}$, with n = 1, 2, 3, ...

From the above dispersion relation it follows that the minimum frequency of a Poincaré wave in the infinite channel is for . For very large channel widths B >> R , the minimum frequency approaches the inertia frequency f . For narrow channel widths B << R , the minimum frequency is approximately which represents the largest seiches period for natural oscillations transverse to the channel axis. ${\displaystyle \omega _{\min }^{2}=f^{2}+c^{2}{\frac {\pi ^{2}}{B^{2}}}}$ ${\displaystyle k_{x}\rightarrow 0}$ ${\displaystyle \omega _{\min }={\frac {\pi c}{B}}}$

Dispersion diagram of the Poincaré and Kelvin wave in a rotating channel, the width 10 Rossby radii B = 10 · R is.

The group velocity for Poincaré waves in the channel has only one component along the channel axis, which is given by

${\displaystyle {\vec {c}}_{g}={\vec {i}}{\frac {\partial \omega }{\partial k_{x}}}={\vec {i}}c^{2}{\frac {k_{x}}{\omega }}}$

The limit values ​​of the group speed of the Poincaré waves in the channel are analogous to those in the open ocean for and for . From a local disturbance in the channel, non-dispersive wave fronts propagate in all directions with the maximum group speed of a shallow water wave in the non-rotating reference system. If parts of the front reach the walls of the canal, the waves are reflected there several times and a discrete spectrum of waves with vanishing group speed and minimal frequencies is formed between the wave fronts propagating towards both ends of the canal . ${\displaystyle c_{g}=0}$ ${\displaystyle k_{x}\rightarrow 0}$ ${\displaystyle c_{g}=c}$ ${\displaystyle k_{x}\rightarrow \infty }$ ${\displaystyle \omega _{\min }}$

In a rotating channel, in addition to Poincaré waves, Kelvin waves are a possible form of movement. They automatically meet the boundary condition on the banks of the canal, as they have no velocity component perpendicular to the canal axis. Your dispersive relationship is . Kelvin waves are non-dispersive shallow water waves which, in contrast to Poincaré waves, only propagate within a bank zone with the width of the Rossby radius. This shore zone is also called a coastal waveguide . The direction of propagation of the Kelvin waves in the coastal waveguide is always such that, looking in the direction of propagation, the shore on the northern hemisphere (southern hemisphere) is on the right (left). Kelvin waves are always excited when pressure gradients occur parallel to the shore within the coastal waveguide. Since the Kelvin waves behave like non-dispersive shallow water waves, the initial form of their excitation is retained during the propagation in the coastal waveguide. ${\displaystyle \omega ={\sqrt {gH}}k_{x}}$

If one lets the width of the channel approach infinity when viewed from one of the banks, the branch of the Kelvin wave belonging to the infinitely distant bank disappears. The Poincaré modes all converge to a dispersion curve with the minimum frequency , so that the dispersion curve of the unlimited ocean arises with respect to the Poincaré waves. In contrast to the open ocean, in the one-sided limited ocean there is, in addition to the Poincaré waves, a Kelvin wave that only propagates in the coastal waveguide in the direction permitted. ${\displaystyle \omega _{\min }=f}$

Baroclinic Poincaré waves

In stratified water there are baroclinic Poincaré waves as well as barotropic waves. The simplest form of stratification of the sea consists of a thin, near-surface layer of density , which extends from the sea surface z = 0 to the depth z = - h . Below that, down to the sea floor at depth z = - H, there is a second layer with greater density . The density jump layer, which separates the two bodies of water in the depth z = - h , can carry out wave-like movements around its rest position, which are called internal waves . In the case of a wavelength that is large compared to the water depth, the long internal waves propagate without dispersion with the phase velocity . Then there is the barotropic Rossby radius defined by the barotropic phase velocity c , which is approximately 2000 km in the ocean. In addition, there is the baroclinic Rossby radius defined by the baroclinic phase velocity , which is orders of magnitude smaller than the average density of the water, namely only 10 km to 100, due to the generally thin cover layer and in particular due to the low density difference between the two water layers km is. For this reason, oceans and marginal seas can be viewed as an infinitely wide channel with regard to internal Poincaré waves. The internal Poincaré waves with vanishing group velocity then all have minimal frequencies that are only slightly larger than f and appear like a single spectral peak in measured flow spectra with low frequency resolution. The small size of the internal Rossby_Radius also means that the coastal waveguide, in which the baroclinic Kelvin wave propagates, is much more bundled, since the baroclinic Rossby radius is 1 to 2 orders of magnitude smaller than the barotropic one. ${\displaystyle \rho _{1}}$ ${\displaystyle \rho _{2}}$ ${\displaystyle c_{i}^{2}={\frac {\rho _{2}-\rho _{1}}{\rho _{2}}}g{\frac {h\left(H-h\right)}{H}}}$ ${\displaystyle c_{i}}$ ${\displaystyle R_{i}}$

Poincaré waves in the equatorial waveguide

The so-called equatorial Poincaré waves, which are limited by the equatorial Rossby radius instead of the canal banks, can be described in the same way as the canal with fixed borders. In this case, the boundary condition for the meridional component of the velocity is based on the assumption that it approaches zero outside the equatorial waveguide.

The meridional velocities of the various Poincaré modes n can then be described in the following form: ${\displaystyle V_{n}}$

${\displaystyle V_{n}=V_{n0}\exp \left({\frac {-y^{2}}{2R^{2}}}\right)H_{n}\left({\frac {y}{R}}\right)\,}$

With

${\displaystyle H_{n}\left({\frac {y}{R}}\right)}$: Hermitian polynomial

${\displaystyle R^{2}={\frac {c}{\beta }}\,}$: Rossby radius at the equator, describes the meridional extension of the equatorial waveguide

${\displaystyle \beta \,}$: Gradient of the Coriolis parameter at the equator .

${\displaystyle y\,}$: meridional distance from the equator, positive to the north

In the equatorial waveguide, in addition to Poincaré waves, equatorial Kelvin waves are also a possible form of movement. They are characterized by the fact that their meridional velocity component vanishes and their zonal velocity component is proportional . The Kelvin wave always propagates to the east in the equatorial waveguide and has the phase and group velocity c of the shallow water wave in the non-rotating reference system. The baroclinic Rossby radius at the equator is approximately 300 km. ${\displaystyle U_{K}(y)\approx \exp \left({\frac {-y^{2}}{2R^{2}}}\right)}$

Web links

• Glossary of Meteorology
• A Model of Ocean Waves
• Animations of Poincaré waves

swell

• A. Gill, Atmosphere-Ocean Dynamics. International Geophysics, 1982. ISBN 0-12-283522-0
• T. Gustafson and B. Kullenberg, (1936). Investigation of inertial currents in the Baltic Sea. Sven. Hydrogr. Biol. Come on. Skr., Hydrogr. No 13
• J. Pedlosky, Geophysical fluid dynamics. Springer, 1998. ISBN 0-387-96387-1