Earth:Topographic Rossby waves
Topographic Rossby waves are geophysical waves that form due to bottom irregularities. For ocean dynamics, the bottom irregularities are on the ocean floor such as the mid-ocean ridge. For atmospheric dynamics, the other primary branch of geophysical fluid dynamics, the bottom irregularities are found on land, for example in the form of mountains. Topographic Rossby waves are one of two types of geophysical waves named after the meteorologist Carl-Gustaf Rossby. The other type of Rossby waves are called planetary Rossby waves and have a different physical origin. Planetary Rossby waves form due to the changing Coriolis parameter over the earth. Rossby waves are quasi-geostrophic, dispersive waves. This means that not only the Coriolis force and the pressure-gradient force influence the flow, as in geostrophic flow, but also inertia.
Physical derivation
This section describes the mathematically simplest situation where topographic Rossby waves form: a uniform bottom slope.
Shallow water equations
A coordinate system is defined with x in eastward direction, y in northward direction and z as the distance from the earth's surface. The coordinates are measured from a certain reference coordinate on the earth's surface with a reference latitude
where
is the velocity in the x direction, or zonal velocity | |
is the velocity in the y direction, or meridional velocity | |
is the local and instantaneous fluid layer thickness | |
is the height deviation of the fluid from its mean height | |
is the acceleration due to gravity | |
is the Coriolis parameter at the reference coordinate with |
In the equation above, friction (viscous drag and kinematic viscosity) is neglected. Furthermore, a constant Coriolis parameter is assumed ("f-plane approximation"). The first and the second equation of the shallow water equations are respectively called the zonal and meridional momentum equations, and the third equation is the continuity equation. The shallow water equations assume a homogeneous and barotropic fluid.
Linearization
For simplicity, the system is limited by means of a weak and uniform bottom slope that is aligned with the y-axis, which in turn enables a better comparison to the results with planetary Rossby waves. The mean layer thickness
where
Utilizing this expression in the continuity equation of the shallow water equations yields
The set of equations is made linear to obtain a set of equations that is easier to solve analytically. This is done by assuming a Rossby number Ro (= advection / Coriolis force), which is much smaller than the temporal Rossby number RoT (= inertia / Coriolis force). Furthermore, the length scale of
Quasi-geostrophic approximation
Next, the quasi-geostrophic approximation Ro, RoT
where
Neglecting terms where small component terms (
Substituting the components of the ageostrophic velocity in the continuity equation the following result is obtained:
in which R, the Rossby radius of deformation, is defined as
Dispersion relation
Taking for
with
If there is no bottom slope (
The maximum frequency of the topographic Rossby waves is
which is attained for
Phase speed
The phase speed of the waves along the isobaths (lines of equal depth, here the x-direction) is
which means that on the northern hemisphere the waves propagate with the shallow side at their right and on the southern hemisphere with the shallow side at their left. The equation of
which is the speed of very long waves (
which means that
from which it can be seen that
Analogy between topographic and planetary Rossby waves
Planetary and topographic Rossby waves are the same in the sense that, if the term
with
with
In the derivations above it was assumed that
so
where a Taylor expansion was used on the denominator and the dots indicate higher order terms. Only keeping the largest terms and neglecting the rest, the following result is obtained:
Consequently, the analogy that appears in potential vorticity is that
The analogy between planetary and topographic Rossby waves is exploited in laboratory experiments that study geophysical flows to include the beta effect which is the change of the Coriolis parameter over the earth. The water vessels used in those experiments are far too small for the Coriolis parameter to vary significantly. The beta effect can be mimicked to a certain degree in these experiments by using a tank with a sloping bottom. The substitution of the beta effect by a sloping bottom is only valid for a gentle slope, slow fluid motions and in the absence of stratification.[1]
Conceptual explanation
As shown in the last section, Rossby waves are formed because potential vorticity must be conserved. When the surface has a slope, the thickness of the fluid layer
Measurements of topographic Rossby waves on earth
From 1 January 1965 till 1 January 1968, The Buoy Project at the Woods Hole Oceanographic Institution dropped buoys on the western side of the Northern Atlantic to measure the velocities. The data has several gaps because some of the buoys went missing. Still they managed to measure topographic Rossby waves at 500 meters depth.[2] Several other research projects have confirmed that there are indeed topographic Rossby waves in the Northern Atlantic.[3][4][5]
In 1988, barotropic planetary Rossby waves were found in the Northwest Pacific basin.[6] Further research done in 2017 concluded that the Rossby waves are no planetary Rossby waves, but topographic Rossby waves.[7]
In 2021, research in the South China Sea confirmed that topographic Rossby waves exist.[8][9]
In 2016, research in the East Mediterranean showed that topographic Rossby Waves are generated south of Crete due to lateral shifts of a mesoscale circulation structure over the sloping bottom at 4000 m (https://doi.org/10.1016/j.dsr2.2019.07.008).
References
- ↑ Jump up to: 1.0 1.1 1.2 Cushman-Roisin, Benoit; Beckers, Jean-Marie (2011), Introduction, International Geophysics, 101, Elsevier, pp. 3–39, doi:10.1016/b978-0-12-088759-0.00001-8, ISBN 9780120887590, http://dx.doi.org/10.1016/b978-0-12-088759-0.00001-8, retrieved 2022-03-17
- ↑ Jump up to: 2.0 2.1 Thompson, Rory (January 1971). "Topographic Rossby waves at a site north of the Gulf Stream". Deep Sea Research and Oceanographic Abstracts 18 (1): 1–19. doi:10.1016/0011-7471(71)90011-8. ISSN 0011-7471. Bibcode: 1971DSRA...18....1T. http://dx.doi.org/10.1016/0011-7471(71)90011-8.
- ↑ Louis, John P.; Petrie, Brian D.; Smith, Peter C. (January 1982). <0047:ootrwo>2.0.co;2 "Observations of Topographic Rossby Waves on the Continental Margin off Nova Scotia". Journal of Physical Oceanography 12 (1): 47–55. doi:10.1175/1520-0485(1982)012<0047:ootrwo>2.0.co;2. ISSN 0022-3670. Bibcode: 1982JPO....12...47L. http://dx.doi.org/10.1175/1520-0485(1982)012<0047:ootrwo>2.0.co;2.
- ↑ Oey, L-Y.; Lee, H-C. (December 2002). "Deep Eddy Energy and Topographic Rossby Waves in the Gulf of Mexico". Journal of Physical Oceanography 32 (12): 3499–3527. doi:10.1175/1520-0485(2002)032<3499:deeatr>2.0.co;2. ISSN 0022-3670. Bibcode: 2002JPO....32.3499O.
- ↑ Hamilton, Peter (July 2009). "Topographic Rossby waves in the Gulf of Mexico". Progress in Oceanography 82 (1): 1–31. doi:10.1016/j.pocean.2009.04.019. ISSN 0079-6611. Bibcode: 2009PrOce..82....1H. http://dx.doi.org/10.1016/j.pocean.2009.04.019.
- ↑ Schmitz, William J. (March 1988). <0459:eotefi>2.0.co;2 "Exploration of the Eddy Field in the Midlatitude North Pacific". Journal of Physical Oceanography 18 (3): 459–468. doi:10.1175/1520-0485(1988)018<0459:eotefi>2.0.co;2. ISSN 0022-3670. Bibcode: 1988JPO....18..459S. http://dx.doi.org/10.1175/1520-0485(1988)018<0459:eotefi>2.0.co;2.
- ↑ Miyamoto, Masatoshi; Oka, Eitarou; Yanagimoto, Daigo; Fujio, Shinzou; Mizuta, Genta; Imawaki, Shiro; Kurogi, Masao; Hasumi, Hiroyasu (May 2017). "Characteristics and mechanism of deep mesoscale variability south of the Kuroshio Extension". Deep Sea Research Part I: Oceanographic Research Papers 123: 110–117. doi:10.1016/j.dsr.2017.04.003. ISSN 0967-0637. Bibcode: 2017DSRI..123..110M. http://dx.doi.org/10.1016/j.dsr.2017.04.003.
- ↑ QUAN, QI; CAI, ZHONGYA; JIN, GUANGZHEN; LIU, ZHIQIANG (2021-03-22). "Topographic Rossby Waves in the Abyssal South China Sea". Journal of Physical Oceanography 51 (6): 1795. doi:10.1175/jpo-d-20-0187.1. ISSN 0022-3670. Bibcode: 2021JPO....51.1795Q.
- ↑ Wang, Qiang; Zeng, Lili; Shu, Yeqiang; Li, Jian; Chen, Ju; He, Yunkai; Yao, Jinglong; Wang, Dongxiao et al. (October 2019). "Energetic Topographic Rossby Waves in the Northern South China Sea". Journal of Physical Oceanography 49 (10): 2697–2714. doi:10.1175/jpo-d-18-0247.1. ISSN 0022-3670. Bibcode: 2019JPO....49.2697W. http://dx.doi.org/10.1175/jpo-d-18-0247.1.
Original source: https://en.wikipedia.org/wiki/Topographic Rossby waves.
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