* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Coriolis Force - Atmosphere Physics
Derivations of the Lorentz transformations wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Hunting oscillation wikipedia , lookup
Analytical mechanics wikipedia , lookup
Lift (force) wikipedia , lookup
Routhian mechanics wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Mechanics of planar particle motion wikipedia , lookup
Inertial frame of reference wikipedia , lookup
Flow conditioning wikipedia , lookup
Reynolds number wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Work (physics) wikipedia , lookup
Seismometer wikipedia , lookup
Fictitious force wikipedia , lookup
Equations of motion wikipedia , lookup
Centrifugal force wikipedia , lookup
Classical central-force problem wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Fluid dynamics wikipedia , lookup
PHYS-575/CSI-655 Introduction to Atmospheric Physics and Chemistry Lecture Notes 7: Atmospheric Dynamics 1. Kinematics of Large-Scale Horizontal Flow 2. Dynamics of Horizontal Flow 3. Primitive Equations 4. The Atmospheric General Circulation 5/7/2017 1 Announcements: April 4, 2010 April 4: Homework #4 Due No more homework assignments! April 4 – Finish Clouds (Chapter 6), Begin Dynamics (7) April 11 – Dynamics (Chapter 7), Intro to Climate (10) April 18 – Climate (10); Review for exam April 25 – Exam #2 May 2 – Climate – continued; Last Day of Classes May 16 (4:30-7:20pm)– Term Paper Presentations 5/7/2017 2 Term Paper Format The term paper must follow standard guides for research papers, and have the following sections: Title Abstract Introduction & background Body of paper - with a significant number (10-15) references to primary literature and/or review articles. This may include discussion of scientific theories, observations, and/or methods. Conclusions Figures are important in the body of the paper. References to sources The paper must be typed, double spaced, and have ~ 15-25 pages of text, not including figures, and at least 3 figures (may have more, include captions). Please number all pages. 5/7/2017 3 Term Paper Presentations Wednesday, May 16, 2010 (4:30-7:20pm) Overview & Summary of Term Paper 5 minutes time limit No more than 5 slides in PowerPoint file. No math derivations, 1-2 key equations ok Summary figures Outline Motivation Summary Please email presentation to me by 10pm on Sunday, 5/15 5/7/2017 4 Exam #2, Monday April 25 Note Change of Date!!!!!!! Closed Book/Notes Covers Text Chapter 5 through Chapter 7. Last approximately 1-1.5 hours Total of 100 points ~30-35 questions You are responsible for all the material from the text, lecture notes, and lecture discussion. Some sketches required. You need to know key equations and physical significance. Always define all terms. 5/7/2017 5 Spatial and Time Scales of Atmospheric Flow Hadley Circulation Tornado http://www.ux1.eiu.edu/~cfjps/1400/FIG07_014B.jpg Hurricane Katrina 5/7/2017 6 The General Circulation: Details Effects of rotation Geostrophy Atmospheric waves Boundary layers, friction and stresses Turbulence and mixing The planetary boundary layer Instabilities and wave breaking 5/7/2017 7 Scales of Atmospheric Motions Type of Motion Molecular motion Micro-scale flow Surface Layer Small scale eddies Dust devils Tornado Small clouds Thunderstorm Hurricane Weather front Planetary wave 5/7/2017 Horiz. Spatial Scale 0.1 micron 1 cm 10 cm 1m 10 m 100 m 1 km 10-100 km 1000 km 1000 km 10,000 km Time Scale 10-9 s 0.1 s 0.1 s 1s 10 s 10 s 1 min 10-100 min 1 hour 10 hours 3-5 days Small and Mesoscale atmospheric flow Synoptic scale motions 8 Synoptic Scale Motions Horizontal Scale ~ 100’s of km Vertical Scale ~ depth of troposphere Timescales ~ hours to days Motions on these scales are directly and strongly influenced by the Earth’s rotation. They are in hydrostatic balance. The vertical component of the velocity ~ 1000 times smaller than the horizontal component. This scale of motion is dominant in controlling the transfer of energy and momentum in the Earth’s atmosphere. 5/7/2017 9 1. Kinematics of Large-Scale Horizontal Flow Kinematics deals with properties of flows that can be diagnosed (but not necessarily predicted) without recourse to the equations of motion. Streamlines: lines whose orientation is such that they are everywhere parallel to the horizontal velocity vector V. Isotachs: contours of constant scaler wind speed V. 5/7/2017 10 Natural Coordinates Natural Coordinates: a pair of axes (s, n) where s is the arc length directed downstream along the local streamline, and n is the distance directed normal to the streamline and toward the left. The direction of flow is denoted by the angle ψ, which is defined relative to a reference direction. At any point in the flow, the scaler wind speed: ds V dt dn 0 dt 5/7/2017 11 Properties of Flows Shear is the rate of change of velocity in direction transverse to the direction of flow. Curvature is the rate of change of direction of flow. dV Shear dn d Curvature V ds Shear and curvature are labeled as cyclonic (anticyclonic) and have a sign in the same sense as to cause an object in the flow to rotate in the same (opposite) sense as the Earth’s rotation when looking down on the N pole. Cyclonic means counterclockwise in the N hemisphere and clockwise in the S hemisphere. 5/7/2017 12 Kinematical properties of horizontal flow that can be defined at any point in the flow (all have units s-1). 5/7/2017 13 Vorticity and Divergence Vorticity and divergence are scaler quantities that can be defined not only in natural coordinates (s,n), but also in Cartesian Coordinates (x,y), for a horizontal wind vector V. Vorticity is the sum of the shear and curvature. ξ=2ω where ω is the rate of spin of an imaginary object moving with the flow. Divergence is the sum of the diffluence and the stretching, but is more easily intuited as outflux per unit volume.. 5/7/2017 14 Idealized Flows Sheared flow without curvature Solid body rotation with cyclonic shear and cyclonic curvature but without divergence. Radial flow with velocity directly proportional to radius, but without curvature or shear. Hyperbolic flow that has both shear and curvature but no vorticity or divergence. 5/7/2017 15 Deformation is the sum of the Confluence and Stretching terms Even simple horizontal flow can rapidly distort a field of passive tracers. This gives rise to a variety of complex transport effects such as eddy diffusion (also known as eddy mixing). 5/7/2017 16 Frontal Zones Deformation can sharpen preexisting horizontal gradients creating features known as frontal zones. Frontal zones can be stationary, but more often are in motion relative as as consequence of the flow field. 5/7/2017 17 Streamlines are horizontal trajectories only if the flow is steady 5/7/2017 18 Atmospheric Dynamics: The Basics 1) Forces: Pressure gradient Friction Gravity Electromagnetic - ∂P/∂x - ν ∂2U∂z2 -ρg -eExB 2) Reference Frames: Inertial Non-Inertial Newtonian Dynamics Apparent Forces (Coriolis) 3) Time Tendency: Fixed observer Observer “riding” motion Material Deriviative Lagrangian Deriviative 4) Conservation Laws: Mass Momentum (force equation) Energy m linear + angular 1/2mv2 + ρCpT + ρgz 5) Scaling of the Equations of Motion: Geostrophic Balance Turbulence 5/7/2017 Rossby Number Reynold’s Number 19 2. Dynamics of Horizontal Flow Real Forces: are the fundamental forces, e.g. - Gravitation - Electricity & Magnetism - Friction - Pressure gradient Apparent Forces: arise due to the acceleration of the reference frame. - Centrifugal force - Corriolis force 5/7/2017 20 Real vs. Apparent Forces In an inertial (non-accelerating) reference frame Newton’s Laws of Motion can be directly applied to a parcel of gas in order to determine its time tendency (acceleration). Euler’s Equation: m Dv/Dt = - dP/dx + ρg + other forces (1-dimensional, x-direction) Dv/Dt is the material or advective deriviative in an inertial reference frame. It can be related to the Lagrangian deriviative (“riding” the parcel) via D/Dt = - ∂/∂t + v ∂/∂x, where v is the vector velocity. 5/7/2017 21 Real vs. Apparent Forces In an accelerating reference frame (planet rotating with angular velocity ω), the acceleration produces an apparent force to the fixed observer. This is called the Coriolis Force. The acceleration due to this “fictitious” force is given by: Dv/Dt = -2 ω x v The Coriolis Force is always perpendicular to the direction of motion and thus cannot do work on the fluid. 5/7/2017 22 Gravity vs. Gravitation; Or Which Way is Down? Gravitation: The force between two objects due to their mass. On the surface of the Earth gravitation is denoted by the vector g* directed toward the center of the Earth. However, the Earth is rotating with angular acceleration Ω = 2π rad day-1 = 7.292 x 10-5 s-1 which produces a radial force = Ω2R, where R is the radial vector from the axis of rotation. Gravity: g = g* + Ω2R 5/7/2017 23 Focault Pendulum The Focault Pendulum is an example of simple harmonic motion in inertial space. The Focault Pendulum swings back and forth in inertial space while the Earth rotates “underneath” it. From the point of view of someone watching the motion on the surface of the Earth, it appears that the pendulum rotates. This is inertial motion. 5/7/2017 24 The Coriolis Force Local Gravity: g = g* + Ω2R An object on the surface of the Earth will experience an outward directed force (away from the axis of rotation) due to the Earth’s rotation of magnitude Ω2R. An object moving with velocity V in the plane perpendicular to the axis of rotation experiences an additional apparent force that is known as the Coriolis Force of magnitude - 2Ω x V In spherical coordinates the Coriolis Force = - f k x V where f = 2Ω sinφ is the Coriolis parameter, and k is the local unit vector in the vertical. 5/7/2017 25 The Coriolis Force In spherical coordinates the Coriolis Force = - f k x V The Coriolis Force is a deflecting force, always acting perpendicular to the direction of motion. Thus the Coriolis force cannot do work on the parcel/object. The magnitude of the Coriolis force is = – f V 5/7/2017 26 The Pressure Gradient Force The vertical force balance is known as hydrostatic equilibrium: 1 dp g dz The vertical force per unit mass: In general the total pressure force: P p Horizontal components of the pressure gradient force: 1 p Px x 5/7/2017 1 p Py y 27 The Horizontal Equations of Motion East-West direction (x, u positive toward East): du 1 p fv Fx dt x North-South direction (y, v positive toward North): dv 1 p fu Fy dt y where Fx,y is the external force (e.g. friction) 5/7/2017 28 Geostrophy and the Rossby Number du 1 p fv Fx E-W: dt x dv 1 p fu Fy N-S: dt y Outside of the Boundary Layer (above ~1km altitude) the friction components are insignificant. If the acceleration terms are small, then those too may also be ignored. To determine if du/dt & dv/dt can be ignored, compare their magnitude to the Coriolis Force for a typical velocity (U), lengthscale (L) and timescale (T ~ L/U) dU/dt ~ U/T ~ U2/L; fu ~ f U The non-dimensional Rossby Number: (dU/dt)/fu ~ U/fL is a measure of the relative magnitude of the acceleration to Coriolis terms. Ro = U/fL 5/7/2017 29 The Horizontal Equations of Motion Above ~1km altitude (outside the boundary layer) Fx,y =0: 1 p fv x 1 p fu y Note that this force balance is diagnostic, not prognostic, i.e., there is no tendency or time evolution. This is called the Geostrophic Balance. The wind velocity that exactly solves these equations is known as the Geostrophic Wind. 1 p ug f y 5/7/2017 1 p vg f x 30 Geostrophy Above ~1km altitude (outside the boundary layer) Fx,y =0: 1 p 1 p fv fu x y For Synoptic Scale Motions: L ~ 1000 KM ~ 106 m U ~ 10 ms-1 f ~ 7 x 10-5 s-1 Ro = U/fL ~ 1/7 ~14% The smallness of the Rossby number is a measure of the accuracy of the Geostrophic Approximation. At mid-latitudes the geostrophic equations are generally accurate to about 10%. 5/7/2017 31 Geostrophic Equations of Motion The smallness of the Rossby number is a measure of the accuracy of the Geostrophic Approximation. At mid-latitudes the geostrophic equations are accurate to about 10% 1 p fv x 1 p fu y These equations of motion are diagnostic equations, i.e., they can be used to infer the velocity field if the pressure variation is known, or they can be used to determine the pressure field if the velocity field is known. These equations cannot be used to determine the time evolution of either the pressure or temperature fields. In order to determine the time evolution, the du/dt term is needed. 5/7/2017 32 Schematic View of Geostrophic Balance To first approximation, the horizontal balance of forces in the Earth’s atmosphere is between the pressure gradient force and the Coriolis force. The approximation that they are in perfect balance is known as the Geostrophic Wind. It is typically accurate to of order 10% at mid latitudes, away from the equator, for synoptic scale motions. 5/7/2017 33 The Coriolis Force and Deflection of Flow Pressure gradients, usually due to temperature differences, cause the air to flow. Once set in motion, the Coriolis Force deflects the force to the right in the northern hemisphere and to the left in the southern hemisphere. This is an apparent force which reflects the tendency of the air to move in an inertial reference frame (“fixed to the distant stars”). 5/7/2017 34 Friction The role of friction in the atmosphere is to produce a “drag” on atmospheric motion. However, the magnitude of the frictional drag is generally very difficult to quantify. The reason is that the drag force is due to a variety of physical processes, all of which transfer momentum between the surface and the free atmosphere. The Frictional Force per unit mass: 1 F z where τ represents the vertical component of the shear stress (rate of vertical exchange of horizontal momentum) due to the presence of smaller, unresolved scales of motion. The shear stress at the surface of the Earth is: τs = - ρ CD Vs V CD = drag coefficient, Vs = scaler velocity, V = vector velocity 5/7/2017 35 Effects of Friction The Coriolis Force and Pressure Gradient Force are always perpendicular to the direction of flow. However, the Frictional Force is always opposite to the direction of flow. 5/7/2017 36 Effects of Friction on Flow Direction The effect of friction is to slow down the flow. The resulting balance of forces leads to a cross-isobar drift, generally from high pressure to low pressure. Thus friction is a primary means of generating flow which is not parallel to the isobars. 5/7/2017 37 The Gradient Wind For rapidly rotating flow outside of the surface friction layer, the angular rotation produces a centripetal acceleration. The three-way balance of forces in Gradient Flow (Pressure gradient, Coriolis, and Centripetal) leads to a modification of the force balance equations so that: du 1 p u2 fu dt dy R where R is the radius of curvature of the flow streamlines. For steady flow, du/dt = 0, and we can use the definition of geostrophic wind to give an equation that can be solved algebraically for the flow velocity u. 5/7/2017 u2 0 u g fu R 38 Horizontal Balanced Flow For rapidly rotating steady flow: du 1 p u2 fu 0 dt dy R where R is the radius of curvature of the flow streamlines. For rapid rotation the pressure gradient force is balanced by the centripetal term. The relative magnitude of the centripetal to the Coriolis term is: Rossby No. ~ (U2/R)/fU ~ U/fR Thus the largeness of the Rossby # is a measure of the accuracy of the Cyclostropyhic Approximation. 5/7/2017 39 The Three-Way Balance in the Gradient Wind 5/7/2017 40 Thermal Wind Equation A horizontal gradient in temperature produces a vertical gradient in the horizontal velocity field. Hydrostatic equilibrium: dP = -ρ g dz = - (P/RT) dz So the difference in altitude between two pressure levels differing by dP is -dz = (RT/P) dP So increasing T (in the horizontal) implies increasing dz in the vertical. 5/7/2017 41 Isotherms and Geopotential The Thermal Wind Equation implies a diagnostic relationship between the temperature structure and the wind structure. 5/7/2017 42 Vorticity (Angular Momentum) Conservation 5/7/2017 43 3. Primitive Equations 5/7/2017 44 Cross-Isobar Flow 5/7/2017 45 Wave Propagation 5/7/2017 46 4. Development of the General Circulation Non-rotating planet: motion is mainly in the meridional plane. Rotating planet: Motion is highly 3-dimensional 5/7/2017 47 Steady Circulation Heating: Tropical latent heat release IR heating from ground Cooling: Adiabatic expansion IR cooling 5/7/2017 48 The Atmospheric General Circulation 5/7/2017 49 The Atmosphere as a Heat Engine 5/7/2017 50 Numerical Weather Prediction 5/7/2017 51 Ensemble Forecasts 5/7/2017 52 Weather Models 5/7/2017 53 Surface flow due to the Coriolis Effect 5/7/2017 54 Vertical Flow Consequences of the Coriolis Effect 5/7/2017 55 The Hadley Cell 5/7/2017 56 Jupiter’s Winds 5/7/2017 57 Jupiter’s Great Red Spot 5/7/2017 58 Condensation Flows: Saturation Vapor Pressure is the Driving Force Sulfur Dioxide flows from the day side to night side. 5/7/2017 Seasonal sublimation of the polar caps produce pressure gradients and global scale winds. 59 Questions for Discussion (1) Is Geostrophic Flow a universal property of a planetary atmosphere? (2) How large can atmospheric cyclones (e.g. hurricanes) become in a planetary atmosphere? (3) In what way would the general circulation be different without surface friction? (4) If the Coriolis Force does no work on a parcel of air, then how does the air accelerate? (5) Which is more important for driving atmospheric winds: Solar heating (direct and latent) and Earth’s rotation? 5/7/2017 60 Rotation Experiment In a rotating reference frame, the only force is the Coriolis Force 5/7/2017 61 “Rotating” Trajectories 5/7/2017 62