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Introduction Outline: The fundamental forces Pressure gradient Gravitational Viscosity Noninertial frame Centrifugal force Coriolis Forces: If there is a NET force acting on an inertial system –the system will experience acceleration. Conversely, if there are observed accelerations, in an inertial system, then a NET force is acting upon the system. Why do I emphasize ‘NET’? What do I mean by inertial? Newtonian laws hold only in an inertial reference frame, i.e. one that is at rest with respect to the ‘fixed stars’. This means that the coordinate system cannot be rotating if Newtonian laws are to apply. A rotating reference frame gives rise to additional ‘apparent’ forces that will appear in our equations. Note however, that these additional terms can and do appear in non-rotating reference frames – resulting from coordinate transformations only. We will return to these concepts as part of our discussion of the Coriolis ‘force’ and a rotating reference frame. However, for now we consider only the fundamental forces that impact (i.e., accelerate) fluid flow. In an inertial (non-rotating) frame, we have: r r F = ma , where r F = the sum of forces acting on an object m = mass of object, a = acceleration of object We could also write Newton’s 2nd law as: r r r r r DV DV , where V = the velocity of the object, and a ≡ F =m Dt Dt r r D D2 r = mV = 2 (mr ), where r is a position vector (position of object) Dt Dt ( ) In classical mechanics, the object is typically a rigid solid body In continuum Mechanics (e.g., Atmospheric Dynamics), the object is a tiny (infinitesimal) chunk o’ fluid (liquid or gas). For meteorology, we usually refer to this chunk as a ‘fluid element’ or ‘air parcel’. We do NOT consider the microscopic aspects of the fluid in continuum mechanics – only the macroscopic. We will be applying Newton’s 2nd law to the atmosphere – the final result won’t look anything like what we started with! Fluid forces are broken down into two different types of forces” “body” force: Force on an object that is proportional to its mass, acting from a distance. The objects interacting are not in physical contact with each other, yet are able to exert a push or pull despite a physical separation (e.g. gravity, electrical, magnetic). “surface” force: Also known as a ‘contact’ force as it involves objects that are physically interacting with one another. The Force on an object that is proportional to the area of an object. Force exerted on a surface fluid element by an outside fluid. (e.g., pressure gradient, frictional, tensional, air resistance forces, spring force, applied forces etc.) Pressure Gradient Force (PGF): Many examples of this in our every day lives (tire pressure, soda cans/bottles, aerosols, etc.) Start with a really really really small box (infinitesimally small) with dimensions δx, δy, and δz in the x, y, and z directions respectively. δz z y x δx δy box volume =δV = δxδyδz What is the net pressure force acting on this fluid element? Pressure is a compressive force and acts perpindicular to surface element (compare with another surface force ‘stress’ which acts parallel to the surface). Let’s first consider the x-component of the pressure force, thus we consider only two faces of our cube – those that are perpindicular to the x-axis. O FAx FBx z Face B y Face A δx XB XA x where, point ‘O’ is at the box center denoted by (xo,yo,zo), and FAx is the force on face A in the ‘-x’ direction, and FBx is the force on face B in the ‘+x’direction. Also, δx δx x A = x0 + , x B = x0 − 2 2 Pressure at ‘O’ is given as p0. What is the pressure on face A (pA) for this infinitesimal fluid element? USE Taylor Series expansion! p A = p0 + ∂p ( x A − x0 ) + Higher order terms (HOT) ∂x x0 y0 , z0 δx ∂p x0 + − x0 + HOT ∂x 2 valid at center of box HOT vanish for tiny box ∂p δx = p0 + (' =' valid in the limit as δx → 0) ∂x 2 = p0 + We recall that pressure is defined as the force/area, thus the ‘force’ exerted on face A due to the pressure is ∂p δx FAx = − p0 + δyδz , and ∂x 2 similarly for face B we have: ∂p δx FBx = p Bδyδz = p0 − δyδz ∂x 2 Thus, the NET force acting on the box due to the pressure is given as Fx ≡ FAx + FBx ∂p δx ∂p δx = − p0 + δyδz + p0 − δyδz ∂x 2 ∂x 2 ∂p = − δxδyδz ∂x SO, net pressure force acting on our fluid element is proportional to the gradient of the pressure – hence we call it the pressure gradient force (PGF). We’re not quite done however, as the equations of motion are in terms of per unit mass. The mass of the box is m = ρδxδyδz. Thus, we have Fx 1 ∂p , and for the y and z components, we have =− m ρ ∂x r Fy Fz 1 ∂p 1 ∂p F 1 1 ∂p , and , or in vector form = − ∇p = − =− =− m m ρ ∂y m ρ ∂z ρ ρ ∂xi The PGF acts in direction opposite of the gradient in p. This makes sense because the net force is ‘down gradient’. Therefore, the PGF is in toward the center of a low pressure system, and out away from the center of a high pressure system. L PGF ∇p H