Coriolis force, atmospheric motion

Horizontal motion of the atmosphere, or wind, is a response of the air to the forces that are present. These include the force due to the pressure gradient, the Coriolis force associated with the rotation of the Earth, and frictional forces acting to retard any motion. If the acceleration of the air mass and frictional effects are small, the horizontal velocity is described by the following expression ... 

The Coriolis force becomes particularly important when fluid motion occurs at large scales, as in vast lakes such as Lake Michigan (Fig. 2-6 b), or in atmospheric circulation (Fig. 4-12). Figure 4-12 explains the origin of the Coriolis force in terms of the radial and tangential components of the velocity of a fluid parcel in a rotating mass of fluid. 

The atmospheric boundary layer (ABL) is that portion of the atmosphere where surface drag due to the motion of the air relative to the ground modifies synoptic-scale motions caused by horizontal pressure gradients, Coriolis forces, and buoyancy. The depth of the ABL is highly variable (50 to 2000 m), but it generally increases with proximity to the equator, with wind speed, and as the earth surface roughness, but it decreases... 

The distribution of most chemical species in the middle atmosphere results from the influences of both dynamical and chemical processes. In particular, when the rates of formation and destruction of a chemical compound are comparable to the rate at which it is affected by dynamical processes, then transport plays a major role in determining the constituent distribution. In an environment like the Earth s atmosphere, air motions, and hence transport of chemical species, are strongly constrained by density stratification (gravitational force) which resists vertical fluid displacements, and the Earth s rotation (Coriolis force) which is a barrier against meridional displacements. Geophysical fluid dynamics describes how atmospheric motions are produced within these constraints. 

From the standpoint of air motion, the atmosphere can be segmented vertically into two layers. Extending from the ground up to about 1000 m is the planetary boundary layer, the zone in which the effect of the surface is felt and in which the wind speed and direction are governed by horizontal pressure gradients, shear stresses, and Coriolis forces. Above the planetary boundary layer is the geostrophic layer, in which only horizontal pressure gradients and Coriolis forces influence the flow. 

In the forced convection layer, wind shear plays a dominant role, and the Monin-Obukhov similarity l pothe-sis applies. To develop their similarity theory Monin and Obirkhov idealized the field of motion that is frequently used in the atmosphere near the ground. It was assumed that all statistical properties of the temperature and velocity fields are homogenous and do not vary with time. The steady mean motion was considered to be unidirectional at all heights in the x-direction. Second-order terms in the equations of the field were considered to be negligible. The scale of motion was considered to be small enough to omit the Coriolis Force, and the radiative heat inflow was neglected. In the surface layer the turbulent fluxes are approximately constant at their smface values. 

A sound wave is manifested as one kind of the atmospheric normal modes, known as the acoustic mode, and is caused by the compressibility of air. There are two more kinds One is called the gravity-inertia mode, which is caused by a combinations of the restitutive force of gravity against thermally stable atmospheric stratification and the Coriolis force due to the earth s rotation. The other kind is called the rotational or planetary mode, which is caused by the meridional variation of the Coriolis force. The importance of the latter kind of normal mode as a prototype of upper tropospheric large-scale disturbances was clarified by C. -G. Rossby and his collaborators a little over one decade prior to the dawn of the numerical prediction era (see Section I). In retrospect, the very natrrre of this discovery was hidden in complicated calcnlations for the normal modes of the global atmospheric model. The mathematical analysis was initiated by the French mathematician Marquis de Laplace (1749-1827), and the complete solntions became clear only with the aid of electronic compnters. It is remarkable that Rossby was able to capture the essence of this important type of wave motion, now referred to as the Rossby wave, from a simple hydrodynamic principle of the conservation of the absolute vorticity that is expressed by the sum of the vertical component of the relative vorticity and the planetary vorticify /. 

To motivate the geostrophic approximation, we invoke a scale analysis approach. Assume frictional forces can be neglected, and the atmospheric motions have a characteristic horizontal length-scale, L, and velocity scale, U. Recalling the definition of the advective derivative operator D/Dt, we find that the magnitude of the acceleration terms, Dw/Dt and Du/Dt in Eqs. (9.2.20) and (9.2.21), is 17 /L, provided the magnitude of the time scale of the motion is greater than or equal to the advective time, L/U. The terms proportional to tan 0 are of order U /a, and the Coriolis terms are of order /(/. If L < a, then the ratio of each of the terms to the Coriolis term is less than or comparable to the Rossby number, defined as... 

In the vector equation of motion, the form assumed by the vertical component when all Coriolis, earth-curvature, frictional, and vertical-acceleration terms are con- sidered negligible compared with those involving the vertical pressure force and the force of gravity. The error in applying the hydrostatic equation to the atmosphere for cyclonic-scale motions is less than 0.01%. In extreme situations, the strong vertical accelerations in thunderstorms and mountain waves can be 1% of gravity, hypsithermal period... 

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