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# Atmospheric Dynamics 1 MET 4305

Florida Tech

GPA 3.7

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This 5 page Class Notes was uploaded by Ida Auer on Monday October 12, 2015. The Class Notes belongs to MET 4305 at Florida Institute of Technology taught by Steven Lazarus in Fall. Since its upload, it has received 26 views. For similar materials see /class/221689/met-4305-florida-institute-of-technology in Meteorology at Florida Institute of Technology.

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Date Created: 10/12/15

Introduction Outline The fundamental forces Pressure gradient Gravitational Viscosity N oninertial 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 ie one that is at rest with respect to the 6fixed 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 nonrotating reference frames resulting from coordinate transformations only We will retum 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 ie accelerate uid ow In an inertial nonrotating frame we have I 2 mil where F the sum of forces acting on an object in 2 mass of object a acceleration of object We could also write Newton s 2nd law as F m where 7 the velocity of the object and a E 2 ll quot7 7 Where f is a position vector position of object In classical mechanics the object is typically a rigid solid body In continuum Mechanics eg Atmospheric Dynamics the object is a tiny infinitesimal chunk o uid liquid or gas For meteorology we usually refer to this chunk as a uid element or air parcel We do NOT consider the microscopic aspects of the uid 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 uid element by an outside uid 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 cansbottles aerosols etc Start with a really really really small box infinitesimally small with dimensions 8x 8y and 82 in the x y and 2 directions respectively 5X box volume 8V 8x8y82 What is the net pressure force acting on this uid 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 xcomponent of the pressure force thus we consider only two faces of our cube those that are perpindicular to the xaxis where point 0 is at the box center denoted by x0yozo and F Axis the force on face A in the x direction and FBXis the force on face B in the x direction Also x x x x A o B o 2 2 Pressure at O is given as p0 What is the pressure on face A A for this in nitesimal uid element USE Taylor Series expansion pA 2 p0 g p xA x0 Higher order terms HOT x 1CIJ nvZn p0 a px0 x0 HOT 3x 2 valid at center of box HOT vanish for tiny box 3 ampc 2 p0 valid in the limit as 5x gt 0 3x 2 We recall that pressure is defined as the forcearea thus the force exerted on face A due to the pressure is 3 5x F Ax Po ax 2 jdydz and similarly for face B we have a dc F3 pgayaz p0 al jayaz x 2 Thus the NET force acting on the box due to the pressure is given as FXEFAXFBX apdx apdx 57 52 P0 ax Zj P0 ax Zj a pcc z x SO net pressure force acting on our uid 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 p8x5y82 Thus we have 13p 2 and for the y and 2 components we have x iap F 1 1 and a p or in vector form E V p 82 m s IS s ls i3P pay m p p paxi 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 PF

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