Physics of the Atmosphere I


 Roxanne Veronica Stafford
 1 years ago
 Views:
Transcription
1 Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 heidelberg.de
2 Last week The conservation of mass implies the continuity equation: ρ + ( ρ v) = 0 t Thermal wind occurs due to horizontal temperature gradients Direct thermal wind is a small scale process (Coriolis force can be neglected) and leads to a circulation, with wind blowing from cold to warm surfaces Geostropic thermal wind occurs on large scales (e.g., latitudinal temperature gradient) and causes a geostrophic wind pattern. The wind speed increases linearly with altitude. Universität Heidelberg Institut für Umweltphysik
3 Contents Introduction Literature  Vertical structure of the atmosphere Adiabatic processes  Vertical stability Atmospheric radiation: Absorption, scattering, emission Atmospheric radiation: The energy budget of the atmosphere Atmospheric dynamics: NavierStokes equation Atmospheric dynamics: Continuity equation, thermal wind Atmospheric dynamics: Vorticity Atmospheric dynamics: The planetary boundary layer Atmospheric circulation: Global circulation patterns, planetary waves Atmospheric circulation: The ENSO phenomenon Diffusion and turbulence: Molecular diffusion, basics of turbulence Diffusion and turbulence: Theorem of Taylor, correlated fluctuations Diffusion and turbulence: Diffusion of scalar tracers Nearsurface dynamics: Wind profile, influence of surface friction
4 Outline for Today The concept of vorticity: Circulation The Definitions of Atmospheric Vorticity Potential Vorticity (PV) Examples Rigid Rotation Flow on curved trajectory Shear flow Observations of PV
5 The Concept of Vorticity Apart from the conservation of energy and momentum, the angular momentum is a conserved quantity in a dynamical system In atmospheric physics, the conservation of angular momentum is expressed as the conservation of the vortex strength of the wind vector field Different definitions of the vortex strength exist
6 Definition of Vorticity The curl of the wind vector field is only important in the horizontal since the vertical extent of the atmosphere is very small The relative vorticity (relative vortex strength) is defined as the zcomponent of the curl of the wind vector field: with: v y = z x ( v) ζ = = ( rot v) ( v) v x y z z
7 Vorticity and Circulation The curl of a vector field perpendicular to a given surface A with normal vector n is related to the circulation Z of the vector field (i.e., the closed path integral along the border S of A), Z = v ds n A via Z(A) d v = lim = vds A da ( ) n A 0 S(A) A S
8 Vorticity and Circulation Circulation in Cartesian coordinates: ( ) ( ) ( ) ( ) dz = v x y dx + v y x + dx dy v x y + dy dx v y x dy z ζ y y + dy x y x + dx x
9 Vorticity and Circulation ( ) ( ) ( ) ( ) dz = v y dx + v x + dx dy v y + dy dx v x dy x y x y v y v = ( ) ( ) + ( ) + ( ) + x y x v x y dx v y x dy v y x dx dy v x y dy dx v x v y dx dy y x = With da = dx dy the above equation becomes: ζ dz v v da x y y = = Note that these findings are a special case of the Stokes theorem: Sign of ζ: Positive for counterclockwise rotation Negative for clockwise rotation S vds = v da ( ) A x
10 Vorticity of a Rigid Rotator Velocity at distance r from the axis: v = ω Circulation: r 2 Z = vds = 2π rv = 2πr ω Thus, the vorticity is r dz Z 2πr ω ζ = = = = 2 2 da π r π r 2 2ω The vorticity of a rigid rotator is twice its angular velocity Example: High pressure system, R = 500 km, v=10 ms 1 ω = v/r 10/ = s 1 ζ = 2 ω = s 1
11 Vorticity of a curved trajectory General case of a curved trajectory Radius of curvature as fu. of ds: dϕ v(r) r r + dr Radius of curvature is always perpendicular to the velocity: dr v = 0 Circulation around infinitesimal area da = r dr dφ: dz = v(r)r dϕ + v(r + dr) (r + dr)dϕ v = v(r)r dϕ + v(r) + dr (r + dr)dϕ r v = v(r)dr dϕ + r dr dϕ + O dr r ( 2 )
12 Vorticity of a Curved Trajectory Dividing by the infinitesimal area da = r dr dφ yields: v v(r)dr dϕ + r dr dϕ dz r v(r) v ζ = = = + da rdrdϕ r r Note that in case of a rigid rotator with dv v = ω r = ω dr this yields ζ = ω + ω = 2ω which is the same result as previously inferred
13 Vorticity of shear winds We assume flow (wind) in xdirection, see figure below (analogous considerations can be made for y direction). Thus v y = 0 and consequently also: v y = 0 x Thus the vorticity is v v v ζ = ( v) = = z x y y y x x y A straight line connecting different air parcels in ydirection will rotate due to wind shear x
14 The Absolute Vorticity The dynamics of the atmosphere is described in a rotating coordinate system The Earth is a rigid rotator with angular velocity Ω Local vorticity in zdirection at latitude φ is given by the Coriolis parameter f = 2 Ω sin φ (see previously discussed vorticity of a rigid rotator) Thus the absolute vorticity η of the wind field is the sum of the relative vorticity ζ (measured relative to the terrestrial coordinate system) and the Coriolis parameter f: η = ζ + f v + 2Ω sinϕ ( ) z
15 The Vorticity Equation To infer a continuity equation for the absolute vorticity, we start with the NavierStokes equation The gravitational term is zero since only horizontal motion is considered (v z = 0),. Furthermore, the external forces (pressure gradient, friction and others) are summarised as f ext : v ( v ) ρ + v = f ext + 2ρ v Ω t external forces Coriolis force The advectional term can also be written as: ( ) 1 2 v v = v v ( v) 2 Thus, using the Coriolis parameter f = 2 Ω sin φ, we have: v v 1 ( ) x v v v f v y = fext t 2 ρ
16 The Vorticity Equation Now we calculate the zcomponent of the curl of this equation. The different terms yield: 1. v ( v) ζ η = = = t t z t t z In the last step, we have used η = ζ + f and f/ t = 0 2. v = 0 ( 2 ) since the curl of a gradient field is zero
17 ( ) 3. The Vorticity Equation 0 v v = v 0 z ζ { } v y ζ = v x ζ 0 = x z ( vx ζ ) ( vy ζ ) x y = ζ vx ζ z y v ζ v ζ vy x x y y x ζ ( ) ( ) = Hv v Hζ with H = y 0
18 The Vorticity Equation 4. = y v y f v x f x 0 z ( vx f) ( vy f) v x f v y f = f vx f vy x x y y = f v v f ( ) ( ) H H
19 The Vorticity Equation Putting these terms together, we have: η 1 + ζ v + v ζ + f v + v f = f t ρ and finally, with η = ζ + f, η 1 + η H v + v Hη = fext t ρ ( ) ( ) ( ) ( ) ( ) H H H H ext ( ) ( ) ( ) η 1 + ( ) ( ) H η v = fext t ρ z This is a continuity equation for the absolute vorticity, known as the vorticity equation Source term for the vorticity is the curl of the external force field Note that pressure gradients do not induce vorticity since the curl of the pressure force f p = p is zero or z z
20 The Potential Vorticity The conservation of vorticity expressed by the vorticity equation is only valid if there is no vertical movement of air. The vorticity is not conserved if lifting of air (i.e., v z 0) occurs. By combining the vorticity equation with the conservation of mass a new quantity is derived: The Potential Vorticity (PV) For this quantity a more general conservation law can be derived, which is directly applicable to the atmosphere The conservation law for the potential vorticity is valid for a barotropic atmosphere, i.e. if the isolines of pressure and temperature are parallel Two equivalent definitions for the potential vorticity after Ertel and Rossby exist (with the former being more popular in atmospheric sciences)
21 Rossby s Potential Vorticity Rossby s potential vorticity is a quantity, related to an air column of finite vertical extent z = z 2 z 1 We assume that the air parcel is bounded at its bottom and top by surfaces of defined pressure or temperature. p = p(z 2 ) p(z 1 ) is the pressure difference between top and bottom of the air column Integrating the continuity equation, z 2, p 2 ρ + ( ρ v) = 0 z 2, p 2 t which expresses the conservation of mass, from z 1 to z 2 yields (after several z 1, p 1 rearrangements, see Roedel): 1 d p z 1, p 1 H v = p dt This equation relates temporal changes in vertical pressure differences to the horizontal divergence of the wind field
22 Rossby s Potential Vorticity Now we combine the continuity equation in the form 1 d p H v = p dt with the vorticity equation dη 1 dη + η Hv = 0 or Hv = dt η dt to 1 d p 1 dη η d p 1 dη + = 0 or + = 0 p dt η dt p dt p dt ( ) 2 This is equivalent to dz dt with Rossby s potential vorticity quantity = 0 or Z const. r r = Z R η = p being a conserved
23 Ertel s Potential Vorticity Assuming the upper and lower boundary layers of the airparcel having the potental temperatures Θ 1 and Θ 2, respectively. The Difference Θ is conserved during adiabatic ascent or descent. dθ Θ = Θ 2 Θ 1 = ( z 2 z 1) = const dz Neglecting the change in density due to different actual temperatures, we can express the altitude difference as pressure difference: p z z = inserting z z g ρ ( ) in the above Eq. yields: g ρ p = const inserting this expression of p in Eq. for ZR : dθ dz η dθ η dθ ZE = = const. or Z * E = = const ' ρ dz ρ dp dp= g ρ dz
24 Some Properties of (Ertel( Ertel s) ) PV η dθ η dθ ZE = = const. or Z * E = = const ' ρ dz ρ dp dp= g ρ dz Dimension: [ Z ] Frequently: 10 E s K K m = = Kg m Kg s 3 m 2 K m = 1PVU (PVUnit) Kg s
25 Z E Example: : Change of Wind Direction in Flows Across Obstacles η dθ = = const. ρ dz Change of wind direction in a flow across an obstacle (hill) due to the conservation of PV. Here a constant coriolis parameter is assumed. Note that the direction of deflection is independent of the direction of flow. Adapted from: W. Roedel, 1992, p. 107.
26 Adiabatic Flow Over a Mountain Range Z E η dθ = = ρ dz const. Holton (1992) uniform zonal flow initial lifting of Θ 0 +dθ layer stretching of Θ 0 +dθ layer horizontal spread of vertical displacement at top of column development of leewave due to changes in f
27 PV and Potential Temperature Gradients Stratosphere, dθ/dz 20 K/km Troposphere, dθ/dz 5 K/km Stratosphere: dθ/dz higher, density lower than troposphere
28 Comparison of Absolute and Potential Vorticity at an Altitude of ca. 8 km. ETA32 Map for Nov 6, 2001, 10:00
29 PV in the Arctic Polar Vortex
30 PV in the Arctic Polar Vortex
31 Summary The conservation of vorticity (curl of the wind field) expresses the conservation of angular momentum The absolute vorticity (sum of relative vorticity and coriolis parameter) also considers the rotation of the Earth The vorticity follows a continuity equation (the vorticity equation) with the curl of the external force field as source term The potential vorticity (PV) is an important concept in atmospheric dynamics. It connects the continuity equation with the vorticity equation and is also valid for vertical movements PV is a conserved quantity in a barotropic atmosphere
3 Vorticity, Circulation and Potential Vorticity.
3 Vorticity, Circulation and Potential Vorticity. 3.1 Definitions Vorticity is a measure of the local spin of a fluid element given by ω = v (1) So, if the flow is two dimensional the vorticity will be
More informationCirculation Bjerknes Circulation Theorem Vorticity Potential Vorticity Conservation of Potential Vorticity. ESS227 Prof. JinYi Yu
Lecture 4: Circulation and Vorticity Circulation Bjerknes Circulation Theorem Vorticity Potential Vorticity Conservation of Potential Vorticity Measurement of Rotation Circulation and vorticity are the
More informationATM 316: Dynamic Meteorology I Final Review, December 2014
ATM 316: Dynamic Meteorology I Final Review, December 2014 Scalars and Vectors Scalar: magnitude, without reference to coordinate system Vector: magnitude + direction, with reference to coordinate system
More informationLecture 4: Pressure and Wind
Lecture 4: Pressure and Wind Pressure, Measurement, Distribution Forces Affect Wind Geostrophic Balance Winds in Upper Atmosphere NearSurface Winds Hydrostatic Balance (why the sky isn t falling!) Thermal
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationRotation, Rolling, Torque, Angular Momentum
Halliday, Resnick & Walker Chapter 10 & 11 Rotation, Rolling, Torque, Angular Momentum Physics 1A PHYS1121 Professor Michael Burton Rotation 101 Rotational Variables! The motion of rotation! The same
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationPressure, Forces and Motion
Pressure, Forces and Motion Readings A&B: Ch. 4 (p. 93114) CD Tutorials: Pressure Gradients, Coriolis, Forces & Winds Topics 1. Review: What is Pressure? 2. Horizontal Pressure Gradients 3. Depicting
More informationTaylor Columns. Martha W. Buckley c May 24, 2004
Taylor Columns Martha W. Buckley c May 24, 2004 Abstract. In this experiment we explore the effects of rotation on the dynamics of fluids. Using a rapidly rotating tank we demonstrate that an attempt to
More informationChapter 4 Rotating Coordinate Systems and the Equations of Motion
Chapter 4 Rotating Coordinate Systems and the Equations of Motion 1. Rates of change of vectors We have derived the Navier Stokes equations in an inertial (non accelerating frame of reference) for which
More informationOcean Circulation: review
Joe LaCasce Section for Meteorology and Oceanography December 2, 2014 Outline Physical characteristics Observed circulation Geostrophic, hydrostatic and thermal wind balances Winddriven circulation Buoyancydriven
More informationPhysics 1A Lecture 10C
Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. Oprah Winfrey Static Equilibrium
More informationOne Atmospheric Pressure. Measurement of Atmos. Pressure. Units of Atmospheric Pressure. Chapter 4: Pressure and Wind
Chapter 4: Pressure and Wind Pressure, Measurement, Distribution Hydrostatic Balance Pressure Gradient and Coriolis Force Geostrophic Balance Upper and NearSurface Winds One Atmospheric Pressure (from
More informationz from Eqn. (3.12.4b) below and the hydrostatic approximation, which is the following approximation to the vertical momentum equation (B9),
Geostrophic, hydrostatic and thermal wind equations The geostrophic approximation to the horizontal momentum equations (Eqn. (B9) below) equates the Coriolis term to the horizontal pressure gradient z
More information4.1 Momentum equation of the neutral atmosphere
Chapter 4 Dynamics of the neutral atmosphere 4.1 Momentum equation of the neutral atmosphere Since we are going to discuss the motion of the atmosphere of a rotating planet, it is convenient to express
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More informationAngular Momentum Problems Challenge Problems
Angular Momentum Problems Challenge Problems Problem 1: Toy Locomotive A toy locomotive of mass m L runs on a horizontal circular track of radius R and total mass m T. The track forms the rim of an otherwise
More informationConservation of Mass The Continuity Equation
Conservation of Mass The Continuity Equation The equations of motion describe the conservation of momentum in the atmosphere. We now turn our attention to another conservation principle, the conservation
More information( ) ( ) 1. Let F = ( 1yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F. F = 1yz x. B. curl F. i j k. = 6xi 8yj+ 2zk F = z 1yz 3xz 9xy
. Let F = ( yz)i + ( 3xz) j + ( 9xy)k. Compute the following: A. div F F = yz x B. curl F + ( 3xz) y + ( 9xy) = 0 + 0 + 0 = 0 z F = i j k x y z yz 3xz 9xy = 6xi 8yj+ 2zk C. div curl F F = 6x x + ( 8y)
More informationLecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)
Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemblemean equations of fluid motion/transport? Force balance in a quasisteady turbulent boundary
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationESS55: EARTH S ATMOSPHERE / Homework #4 / (due 5/1/2014)
ESS55: EARTH S ATMOSPHERE / Homework #4 / (due 5/1/2014) Name Student ID: version: (1) (21) (41) (2) (22) (42) (3) (23) (43) (4) (24) (44) (5) (25) (45) (6) (26) (46) (7) (27) (47) (8) (28) (48) (9) (29)
More informationOutline. Jet Streams I (without the math) Definitions. Polar Jet Stream. Polar Jet Stream
Jet Streams I (without the math) Outline A few definitions, types, etc Thermal wind and the polar jet Ageostrophic circulations, entrance and exit regions Coupled Jets Definitions Jet Stream An intense,
More informationThe Hydrostatic Equation
The Hydrostatic Equation Air pressure at any height in the atmosphere is due to the force per unit area exerted by the weight of all of the air lying above that height. Consequently, atmospheric pressure
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13.
Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the
More informationChapter 4 Atmospheric Pressure and Wind
Chapter 4 Atmospheric Pressure and Wind Understanding Weather and Climate Aguado and Burt Pressure Pressure amount of force exerted per unit of surface area. Pressure always decreases vertically with height
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationCircular Motion. We will deal with this in more detail in the Chapter on rotation!
Circular Motion I. Circular Motion and Polar Coordinates A. Consider the motion of ball on a circle from point A to point B as shown below. We could describe the path of the ball in Cartesian coordinates
More informationLecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows
Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3. 1 Basics: equations of continuum mechanics  balance equations for mass and momentum  balance equations for the energy and the chemical
More informationRELATIVE MOTION ANALYSIS: VELOCITY
RELATIVE MOTION ANALYSIS: VELOCITY Today s Objectives: Students will be able to: 1. Describe the velocity of a rigid body in terms of translation and rotation components. 2. Perform a relativemotion velocity
More informationME19b. SOLUTIONS. Feb. 11, 2010. Due Feb. 18
ME19b. SOLTIONS. Feb. 11, 21. Due Feb. 18 PROBLEM B14 Consider the long thin racing boats used in competitive rowing events. Assume that the major component of resistance to motion is the skin friction
More informationElectromagnetism  Lecture 2. Electric Fields
Electromagnetism  Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More informationCBE 6333, R. Levicky 1. Potential Flow
CBE 6333, R. Levicky Part I. Theoretical Background. Potential Flow Potential Flow. Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous
More informationProblem Set 9 Angular Momentum Solution
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 801 Fall 01 Problem 1 Sedna Problem Set 9 Angular Momentum Solution 90377 Sedna is a large transneptunian object, which as of 01 was
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationMthSc 206 Summer1 13 Goddard
16.1 Vector Fields A vector field is a function that assigns to each point a vector. A vector field might represent the wind velocity at each point. For example, F(x, y) = y i+x j represents spinning around
More informationATMS 310 Jet Streams
ATMS 310 Jet Streams Jet Streams A jet stream is an intense (30+ m/s in upper troposphere, 15+ m/s lower troposphere), narrow (width at least ½ order magnitude less than the length) horizontal current
More informationChapter 28 Fluid Dynamics
Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example
More informationUniversity of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P. Pebler
University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P Pebler 1 Purcell 66 A round wire of radius r o carries a current I distributed uniformly
More informationIntroduction to basic principles of fluid mechanics
2.016 Hydrodynamics Prof. A.H. Techet Introduction to basic principles of fluid mechanics I. Flow Descriptions 1. Lagrangian (following the particle): In rigid body mechanics the motion of a body is described
More informationScalars, Vectors and Tensors
Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector
More informationChapter 6 Circular Motion
Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example
More informationSynoptic Meteorology I: Thermal Wind Balance. 9, 14 October 2014
Deriving the Thermal Wind Relationship Synoptic Meteorology I: Thermal Wind Balance 9, 14 October 214 Recall from our most recent lecture that the geostrophic relationship applicable on isobaric surfaces
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationGravity waves on water
Phys374, Spring 2006, Prof. Ted Jacobson Department of Physics, University of Maryland Gravity waves on water Waves on the surface of water can arise from the restoring force of gravity or of surface tension,
More informationLecture L5  Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5  Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationChapter 27 Magnetic Field and Magnetic Forces
Chapter 27 Magnetic Field and Magnetic Forces  Magnetism  Magnetic Field  Magnetic Field Lines and Magnetic Flux  Motion of Charged Particles in a Magnetic Field  Applications of Motion of Charged
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationEkman Transport Ekman Pumping
Lecture 10: : Ocean Circulation Ekman Transport Ekman Pumping WindDriven Circulation Basic Ocean Structures Upper Ocean (~100 m) Warm up by sunlight! Shallow, warm upper layer where light is abundant
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationOcean Processes I Oceanography Department University of Cape Town South Africa
Ocean Processes I Isabelle.Ansorge@uct.ac.za Oceanography Department University of Cape Town South Africa 1 9/17/2009 Lecturer in Oceanography at UCT About me! BSc in England University of Plymouth MSc
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationPhysics 2212 GH Quiz #4 Solutions Spring 2015
Physics 1 GH Quiz #4 Solutions Spring 15 Fundamental Charge e = 1.6 1 19 C Mass of an Electron m e = 9.19 1 31 kg Coulomb constant K = 8.988 1 9 N m /C Vacuum Permittivity ϵ = 8.854 1 1 C /N m Earth s
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311  Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationThe Shallow Water Equations
Copyright 2006, David A. Randall Revised Thu, 6 Jul 06, 5:2:38 The Shallow Water Equations David A. Randall Department of Atmospheric Science Colorado State University, Fort Collins, Colorado 80523. A
More information52. The Del Operator: Divergence and Curl
52. The Del Operator: Divergence and Curl Let F(x, y, z) = M(x, y, z), N(x, y, z), P(x, y, z) be a vector field in R 3. The del operator is represented by the symbol, and is written = x, y, z, or = x,
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More information39th International Physics Olympiad  Hanoi  Vietnam  2008. Theoretical Problem No. 3
CHANGE OF AIR TEMPERATURE WITH ALTITUDE, ATMOSPHERIC STABILITY AND AIR POLLUTION Vertical motion of air governs many atmospheric processes, such as the formation of clouds and precipitation and the dispersal
More information3 1.11 10 5 mdeg 1 = 4.00 4.33 3.33 10 5. 2.5 10 5 s 1 2
Chapter 7 7.7 At a certain location along the ITCZ, the surface wind at 10 Nisblowing from the east northeast (ENE) from a compass angle of 60 at a speed of 8ms 1 and the wind at 7 N is blowing from the
More information(Should use spherical coords for Earth, but ideas are similar)
1 Streamfunction and Vorticity Can decompose 2D vector field U into a potential Φ(φ, θ) and a streamfunction Ψ(φ, θ): Cartesian components: U = ẑ Ψ + Φ. (1) U = Ψ y + Φ x V = Ψ x + Φ y. (2a) (2b) (Should
More informationFundamental Theorems of Vector Calculus
Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use
More informationFLUID DYNAMICS. Intrinsic properties of fluids. Fluids behavior under various conditions
FLUID DYNAMICS Intrinsic properties of fluids Fluids behavior under various conditions Methods by which we can manipulate and utilize the fluids to produce desired results TYPES OF FLUID FLOW Laminar or
More informationChapter 3: Weather Map. Weather Maps. The Station Model. Weather Map on 7/7/2005 4/29/2011
Chapter 3: Weather Map Weather Maps Many variables are needed to described weather conditions. Local weathers are affected by weather pattern. We need to see all the numbers describing weathers at many
More informationThen the second equation becomes ³ j
Magnetic vector potential When we derived the scalar electric potential we started with the relation r E = 0 to conclude that E could be written as the gradient of a scalar potential. That won t work for
More informationRELATIVE MOTION ANALYSIS: VELOCITY
RELATIVE MOTION ANALYSIS: VELOCITY Today s Objectives: Students will be able to: 1. Describe the velocity of a rigid body in terms of translation and rotation components. 2. Perform a relativemotion velocity
More informationarxiv:1111.4354v2 [physics.accph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.accph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationEXAMPLE: Water Flow in a Pipe
EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along
More informationExamples of magnetic field calculations and applications. 1 Example of a magnetic moment calculation
Examples of magnetic field calculations and applications Lecture 12 1 Example of a magnetic moment calculation We consider the vector potential and magnetic field due to the magnetic moment created by
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationNUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES
Vol. XX 2012 No. 4 28 34 J. ŠIMIČEK O. HUBOVÁ NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES Jozef ŠIMIČEK email: jozef.simicek@stuba.sk Research field: Statics and Dynamics Fluids mechanics
More informationThe purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.
260 171 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this
More informationThompson/Ocean 420/Winter 2005 Inertial Oscillations 1
Thompson/Ocean 420/Winter 2005 Inertial Oscillations 1 Inertial Oscillations Imagine shooting a hockey puck across an icecovered (i.e., frictionless) surface. As the hockey moves at speed u in a given
More informationGradient, Divergence and Curl in Curvilinear Coordinates
Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
More informationDimensional Analysis
Dimensional Analysis An Important Example from Fluid Mechanics: Viscous Shear Forces V d t / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Ƭ = F/A = μ V/d More generally, the viscous
More informationExample of Inversion Layer
The Vertical Structure of the Atmosphere stratified by temperature (and density) Space Shuttle sunset Note: Scattering of visible light (density + wavelength) Troposphere = progressive cooling, 75% mass,
More informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More informationFaraday s Law of Induction
Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction...1010.1.1 Magnetic Flux...103 10.1. Lenz s Law...105 10. Motional EMF...107 10.3 Induced Electric Field...1010 10.4 Generators...101
More informationGlobal winds: Earth s General Circulation. Please read Ahrens Chapter 11, up to page 299
Global winds: Earth s General Circulation Please read Ahrens Chapter 11, up to page 299 The circulations of the atmosphere and oceans are ultimately driven by. solar heating. Recall: Incoming radiation
More informationIntroduction to COMSOL. The NavierStokes Equations
Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following
More information3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field
3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationChapter 8 Steady Incompressible Flow in Pressure Conduits
Chapter 8 Steady Incompressible Flow in Pressure Conduits Outline 8.1 Laminar Flow and turbulent flow Reynolds Experiment 8.2 Reynolds number 8.3 Hydraulic Radius 8.4 Friction Head Loss in Conduits of
More informationD Alembert s principle and applications
Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationV. Water Vapour in Air
V. Water Vapour in Air V. Water Vapour in Air So far we have indicated the presence of water vapour in the air through the vapour pressure e that it exerts. V. Water Vapour in Air So far we have indicated
More informationRELATIVE MOTION ANALYSIS: ACCELERATION
RELATIVE MOTION ANALYSIS: ACCELERATION Today s Objectives: Students will be able to: 1. Resolve the acceleration of a point on a body into components of translation and rotation. 2. Determine the acceleration
More information1 The basic equations of fluid dynamics
1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which
More informationSolutions to Practice Problems for Test 4
olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationMesoscale Meteorology
Mesoscale Meteorology METR 4433 Spring 2015 3.5 Nocturnal LowLevel Jet The broadest definition of a lowlevel jet (LLJ) is simply any lowertropospheric maximum in the vertical profile of the horizontal
More informationSynoptic Meteorology II: Isentropic Analysis. 31 March 2 April 2015
Synoptic Meteorology II: Isentropic Analysis 31 March 2 April 2015 Readings: Chapter 3 of Midlatitude Synoptic Meteorology. Introduction Before we can appropriately introduce and describe the concept of
More informationMagnetism Conceptual Questions. Name: Class: Date:
Name: Class: Date: Magnetism 22.1 Conceptual Questions 1) A proton, moving north, enters a magnetic field. Because of this field, the proton curves downward. We may conclude that the magnetic field must
More informationChapter 6 MOMENTUM ANALYSIS OF FLOW SYSTEMS
Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGrawHill, 2010 Chapter 6 MOMENTUM ANALYSIS OF FLOW SYSTEMS Lecture slides by Hasan Hacışevki Copyright The
More informationChapter 7. Fundamental Theorems: Vorticity and Circulation
hapter 7 Fundamental Theorems: Vorticity and irculation 7.1 Vorticity and the equations of motion. In principle, the equations of motion we have painstakingly derived in the first 6 chapters are sufficient
More informationReview of Vector Calculus
Review of Vector Calculus S. H. (Harvey) Lam January 8th, 2004 5:40PM version Abstract This review is simply a summary of the major concepts, definitions, theorems and identities that we will be using
More information