On internal waves. Effects of the horizontal component of the earth's rotation and of a uniform current

1970 ◽  
Vol 23 (1) ◽  
pp. 16-23 ◽  
Author(s):  
Bernard Saint-Guily
Keyword(s):  
Internal Waves ◽  
Horizontal Component ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Uniform Current

scholarly journals Influence of the horizontal component of Earth’s rotation on wind turbine wakes

2018 ◽  
Vol 1037 ◽  
pp. 072003 ◽  
Author(s):  
Michael F. Howland ◽  
Aditya S. Ghate ◽  
Sanjiva K. Lele
Keyword(s):  
Wind Turbine ◽  
Horizontal Component ◽  
Earth’S Rotation ◽  
Earth's Rotation

Flows in a rotating spherical shell: the equatorial case

1994 ◽  
Vol 276 ◽  
pp. 233-260 ◽  
Author(s):  
A. Colin de Verdière ◽  
R. Schopp
Keyword(s):  
Angular Momentum ◽  
Rotation Axis ◽  
Dynamic Pressure ◽  
Vertical Component ◽  
Low Frequency ◽  
Zonal Flow ◽  
Horizontal Component ◽  
Meridional Plane ◽  
Earth’S Rotation ◽  
Earth's Rotation

It is well known that the widely used powerful geostrophic equations that single out the vertical component of the Earth's rotation cease to be valid near the equator. Through a vorticity and an angular momentum analysis on the sphere, we show that if the flow varies on a horizontal scale L smaller than (Ha)1/2 (where H is a vertical scale of motion and a the Earth's radius), then equatorial dynamics must include the effect of the horizontal component of the Earth's rotation. In equatorial regions, where the horizontal plane aligns with the Earth's rotation axis, latitudinal variations of planetary angular momentum over such scales become small and approach the magnitude of its radial variations proscribing, therefore, vertical displacements to be freed from rotational constraints. When the zonal flow is strong compared to the meridional one, we show that the zonal component of the vorticity equation becomes (2Ω.Δ)u1 = g/ρ0)(∂ρ/a∂θ). This equation, where θ is latitude, expresses a balance between the buoyancy torque and the twisting of the full Earth's vorticity by the zonal flow u1. This generalization of the mid-latitude thermal wind relation to the equatorial case shows that u1 may be obtained up to a constant by integrating the ‘observed’ density field along the Earth's rotation axis and not along gravity as in common mid-latitude practice. The simplicity of this result valid in the finite-amplitude regime is not shared however by the other velocity components.Vorticity and momentum equations appropriate to low frequency and predominantly zonal flows are given on the equatorial β-plane. These equatorial results and the mid-latitude geostrophic approximation are shown to stem from an exact generalized relation that relates the variation of dynamic pressure along absolute vortex lines to the buoyancy field. The usual hydrostatic equation follows when the aspect ratio δ = H/L is such that tan θ/δ is much larger than one. Within a boundary-layer region of width (Ha)1/2 and centred at the equator, the analysis shows that the usually neglected Coriolis terms associated with the horizontal component of the Earth's rotation must be kept.Finally, some solutions of zonally homogeneous steady equatorial inertial jets are presented in which the Earth's vorticity is easily turned upside down by the shear flow and the correct angular momentum ‘Ωr2cos2(θ)+u1rCos(θ)’ contour lines close in the vertical–meridional plane.

scholarly journals The Stability of Short Symmetric Internal Waves on Sloping Fronts: Beyond the Traditional Approximation

2012 ◽  
Vol 42 (3) ◽  
pp. 459-475 ◽  
Author(s):  
Alain Colin de Verdière
Keyword(s):  
Kinetic Energy ◽  
Internal Waves ◽  
Relative Vorticity ◽  
Rotation Vector ◽  
Vertical Shear ◽  
Earth’S Rotation ◽  
Horizontal Shear ◽  
Mean Flow ◽  
Earth's Rotation ◽  
The Mean

Abstract The interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth’s rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth’s rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2Ω/N as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean–atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.

Zonal and tesseral harmonic perturbations of an artificial satellite

1966 ◽  
Vol 25 ◽  
pp. 323-325 ◽  
Author(s):  
B. Garfinkel
Keyword(s):  
Angular Velocity ◽  
Spherical Harmonics ◽  
Artificial Satellite ◽  
Compact Form ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Secular Variations ◽  
Tesseral Harmonics

The paper extends the known solution of the Main Problem to include the effects of the higher spherical harmonics of the geopotential. The von Zeipel method is used to calculate the secular variations of orderJmand the long-periodic variations of ordersJm/J2andnJm,λ/ω. HereJmandJm,λare the coefficients of the zonal and the tesseral harmonics respectively, withJm,0=Jm, andωis the angular velocity of the Earth's rotation. With the aid of the theory of spherical harmonics the results are expressed in a most compact form.

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