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Geophysical Fluid Flows
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
There are two main fluids found on our Planet Earth, namely water and air. Their appearances on Earth’s surface make our planet habitable for living organisms. Air and water control both climate and weather on Earth’s surface, through atmospheric and oceanic flows. The former one relates to meteorology and the latter one relates to oceanography. With growing concerns on climate change, sea-level rise, drought, flooding, and more extreme weather occurrences (such as the increasing intensity of hurricanes, tropical storms, and typhoons), the understanding and prediction of geophysical fluid flows become more important. This chapter focuses on the fundamental fluid dynamics used in modeling geophysical flows. As shown in Chau (2018), fluid flows are governed by Navier-Stokes equations. For geophysical flows, we need to incorporate the rotational effects of the Earth. On the global scale, Rossby waves, or planetary waves, are upper atmospheric cold fronts that influence the mid-latitude weather in the Northern Hemisphere in winter. Heat transfer in the atmosphere from the equatorial areas to the polar areas occurs often through violent weather systems in the form of hurricanes or tropical cyclones. Such extreme weather has been making profound impacts on all kinds of human activities, including explorations, trading, travels, and fisheries. Mathematical treatment of the Navier-Stokes equations under various simplified conditions leads to analytical solutions, providing meaningful insights for weather prediction and forecast. Meaningful analytical models for hurricanes or tropical cyclones are still being developed. In this chapter, after reviewing the governing equations for geophysical flows, we discuss the prediction of storm surges considering the inverse barometer effect, moving center of low pressure, and wind-induced effects. Ekman transport is then discussed in explaining the orthogonal drifting of icebergs with respect to the wind. Various vortex models are discussed in detail and their implications on the wind speed of tornadoes. Although these solutions for tornadoes are on much smaller scales than that of hurricanes, they do provide insight to the structures of hurricanes.
Nonlinear Dynamics of the Oceanic Flow
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
S.V. Prants, M.V. Budyansky, M.Yu. Uleysky
Motion of 2D-incompressible fluid on the rotating Earth is governed by the equation for conserving potential vorticity (∂/∂t+v→⋅∇→)Π=0. In the quasi-geostrophic approximation, one gets Π=∇2Ψ+βy where β is the Coriolis parameter. The x-axis is chosen along the zonal flow, fromthe west to the east and y—along the gradient from the south to the north. Barotropic perturbations of zonal flows produce planetary Rossby waves propagating to the west and producing an essential impact on transport and mixing in the ocean and the atmosphere. It is possible to find in a linear approximation an exact solution for the stream function obeying the equation for conserving potential vorticity [9] and consisting of steady zonal flow with the velocity profile of a Bickley jet and two propagating Rossby waves with the amplitudes A1 and A2 and wave numbers n1 = mN1 and n2 = mN2, where m ≠ 1 is the greatest common divisor and N1/N2 is an irreducible fraction. The normalization, introduced in References 28 and 29, is very convenient because it reduces the number of control parameters to the main ones N1 and N2. The stream function in the frame moving with the phase velocity of the first wave is Ψ(x,y,t)=−tanhy+A1sech2ycos(N1x)+A2sech2ycos(N2x+ω2t)+C2y.
Thermal versus mechanical topography: an experimental investigation in a rotating baroclinic annulus
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
The concept of thermal topography may also be in some manner related to the possible origins of current “extreme weather” conditions associated with climate change. Studies such as those of Francis and Vavrus (2012) and Liu et al. (2012), for example, conclude that a reduction in Arctic sea-ice from global warming has led to significant changes to the atmospheric circulation. Francis and Vavrus suggest that the resulting smaller land-sea temperature difference has led to a slower eastward progression of Rossby waves due to weakened zonal winds, and increased wave amplitudes. Liu et al. note how this links to the impacts of blocking, and thus propose that a lack of sea-ice may lead to more frequent blocked states, in turn leading to cold surges and perhaps at least partially accounting for the “recent cold and snowy winters”.
Rossby waves in the ocean covered by compressed ice
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
Y. A. Stepanyants, I. V. Sturova
Nowadays, there is a great interest to flexural-gravity waves in the oceans covered by a compressed ice (Das et al.2018a,b,c). However, one more type of fluid motion in the ocean which can be also potentially influenced by ice cover, the Rossby waves, was not studied yet to our best knowledge, whereas the influence of ice cover on two other types of planetary waves, the Kelvin and Poincare waves, was studied recently in Muzylev and Tsybaneva (2019). Barotropic Rossby waves in the ocean covered by ice can occur in the winter periods in the high and even middle latitudes in the Northern Hemisphere and in the Southern Ocean adjoining Antarctica. A similar problem can be topical for the oceans (not necessarily containing water but some other liquids, e.g. methane) on other planets of the Solar System (e.g. Europa, Jupiter's moon, or Titan, Saturn's moon) and even on exoplanets. Rossby waves per se are the well-known atmospheric and oceanic motions in a rotating planet (Pedlosky 1982, Brekhovskikh and Goncharov 1994, Chelton and Schlax 1996). Therefore we see the aim of this publication in the closing of the gap in the theoretical fluid mechanics in application to the Earth' oceans and oceans on other planets.
Trajectory analysis of the propagation of Rossby waves on the Earth’s δ-surface
Published in Atmospheric and Oceanic Science Letters, 2018
Jian SONG, Hai-Le XUE, Chao-Jiu DA
Rossby waves are a fundamental wave motion in large-scale atmospheric and oceanic dynamics, affecting weather and climate. The propagation of Rossby waves is related to the variation in the Coriolis force with latitude, the so-called β-effect. There have been many studies on the subject of Rossby waves. Plazman (1968) and Hoskings, McIntyre, and Robertson (1985) explained the Rossby wave propagation mechanism. Holton and Hakim (2012) showed the mechanism of Rossby wave propagation in a barotropic atmosphere. Cai and Huang (2013) used the mechanical-Coriolis oscillation mechanism to show how the β-effect and topographic-effect induce Rossby waves. Song and Yang (2009, 2010) and Song et al. (2009, 2017) have studied solitary Rossby waves in barotropic fluids and stratified fluids using the Wentzel-Kramers-Brillouin theory. In the barotropic and baroclinic model, Schneider (2015) analyzed the mechanism for the westward propagation of Rossby waves on the β-plane. Yang (1987, 1988) introduced the δ-surface approximation of the Earth’s surface by considering the second derivative of the Coriolis parameter with respect to latitude (the variation of β with the latitudinal δ-effect), revealing Rossby wave packet structural vacillation to exist in some basic currents or topographies on the Earth’s δ-surface.