DOCSLIB.ORG

Applications of the Basic Equations Chapter 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Applications of the Basic Equations Chapter 3 Applications of the Basic Equations Chapter 3 Paul A. Ullrich [email protected] Part 3: The Thermal Wind Question: Is there a relationship between wind speed and temperature? It turns out that there is a close link between vertical wind shear (vertical gradients of horizontal wind speed) and layer thickness, which is governed by temperature. Paul Ullrich Applications of the Basic Equations March 2014 300 hPa Wind Speed Question: Where are the strongest temperature gradients? Paul Ullrich Applications of the Basic Equations March 2014 850 hPa Temperature Wind speeds appear to be largest where temperature gradients in lower layers are strongest. Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind Defnition: The thermal wind is a vector difference between the geostrophic wind at an upper level and a lower level. Figure: Thickness of layers related to temperature, causes a tilt in the pressure surfaces. Change in magnitude of horizontal gradient of pressure then leads to vertical wind shear. Cold Warm Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind Emphasis: The thermal wind is not a real wind, but a vector difference. The thermal wind vector points such that cold air is to the left and warm air is to the right, parallel to isotherms (in the northern hemisphere). Cold air is to the right and warm air is to the left in the southern hemisphere. Cold Warm Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Wind In Pressure Coordinates 1 @p Recall: The geostrophic wind is the component u = of the real wind which is governed by geostrophic g −f⇢ @y balance. On constant height surfaces, it is defned 1 @p to satisfy v =+ g f⇢ @x In pressure coordinates, this relationship takes on the analogous form: 1 @Φ u = g −f @y 1 @Φ v =+ g f @x Paul Ullrich Applications of the Basic Equations March 2014 1 @Φ 1 @Φ Geostrophic u = v = g −f @y g f @x Wind ✓ ◆p ✓ ◆p Differentiate with respect to p @u 1 @ @Φ @v 1 @ @Φ g = g = @p −f @y @p @p f @x @p @Φ @z 1 RT Hydrostatic = g = = On constant p Relationship @p @p −⇢ − p surfaces @ug R @T @v R @T Thermal wind = g = relationship: @p pf @y @p −pf @x ✓ ◆p ✓ ◆p Paul Ullrich Applications of the Basic Equations March 2014 @ug R @T @v R @T Thermal wind = g = relationship: @p pf @y @p −pf @x ✓ ◆p ✓ ◆p This relationship links the horizontal temperature gradient with the vertical wind gradient (vertical shear). Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind @u R @T Thermal wind g = relationship: @p pf @y ✓ ◆p @v R @T g = The thermal wind itself @p −pf @x ✓ ◆p is a vector difference. @ug R @T @vg R @T Rewrite = = @(log p) f @y @(log p) − f @x ✓ ◆p ✓ ◆p R @ T p Integrate u = u (p ) u (p )= h i log 1 T g 2 − g 1 − f @y p ✓ ◆p ✓ 2 ◆ R @ T p Thermal Wind v = v (p ) v (p )= h i log 1 T g 2 − g 1 f @x p ✓ ◆p ✓ 2 ◆ Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind R @ T p u = u (p ) u (p )= h i log 1 T g 2 − g 1 − f @y p ✓ ◆p ✓ 2 ◆ R @ T p v = v (p ) v (p )= h i log 1 T g 2 − g 1 f @x p ✓ ◆p ✓ 2 ◆ Example: Thermal wind vT between 500 hPa and 1000 hPa ug (p2 = 500 hPa) uT thermal wind ug (p1 = 1000 hPa) Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind Alternate form of thermal wind, written in terms of geopotential height (obtained from hypsometric equation): R p 1 u = d k T log 1 = k (Φ Φ ) T f ⇥rph i p f ⇥rp 2 − 1 ✓ 2 ◆ 1 @ 1 @ u = (Φ Φ ) v = (Φ Φ ) T −f @y 2 − 1 T f @y 2 − 1 Index “1” indicates lower level, “2” indicates upper level. Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind R @ T p u u (p ) u (p )= h i log 1 T ⇡ g 2 − g 1 − f @y p ✓ ◆p ✓ 2 ◆ R @ T p v v (p ) v (p )= h i log 1 T ⇡ g 2 − g 1 f @x p ✓ ◆p ✓ 2 ◆ Note that thermal wind always points parallel to lines of constant layer temperature: R p @ T @ T @ T @ T u T = log 1 h i h i + h i h i =0 T · rh i f p − @y @x @x @y ✓ 2 ◆ Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind Thermal wind always points parallel to lines of constant temperature (and lines of constant layer thickness). Defnition: Veering winds are Cold defned by clockwise rotation Thermal wind of the geostrophic wind with Lower level height and are associated with geo. wind Upper level warm air advection. geo. wind Warm Defnition: Backing winds are Cold Upper level defned by counter-clockwise geo. wind Lower level rotation of the geostrophic wind geo. wind with height and are associated with cold air advection. Thermal wind Warm Paul Ullrich Applications of the Basic Equations March 2014 The thermal wind determines the relationship between meridional temperature gradients and zonal winds. Question: Given zonal mean temperature below, where are zonal jets? @T Wintertime DJF Summertime @y ? y Paul Ullrich Applications of the Basic Equations March 2014 The thermal wind determines the relationship between meridional temperature gradients and zonal winds. Question: Given zonal mean temperature below, where are zonal jets? @T Wintertime DJF Summertime @y ? - - - - + - + y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @p pf @y ✓ ◆p @T Wintertime DJF Summertime @y ? - - - + @u g ? @p - - - + + + - - + - y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @ug @ug @p pf @y ✓ ◆p @z ⇠@p @T Wintertime DJF Summertime @y ? - + - - @u g ? @z - + - - + - - + + + y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @ug @ug @p pf @y ✓ ◆p @z ⇠@p Wintertime DJF Summertime y Paul Ullrich Applications of the Basic Equations March 2014 .
Load more

Recommended publications
  • The California Low-Level Coastal Jet and Nearshore Stratocumulus
    Calhoun: The NPS Institutional Archive Faculty and Researcher Publications Faculty and Researcher Publications 2001 The California Low-Level Coastal Jet and Nearshore Stratocumulus Kalogiros, John http://hdl.handle.net/10945/34433 1.1 THE CALIFORNIA LOW-LEVEL COASTAL JET AND NEARSHORE STRATOCUMULUS John Kalogiros and Qing Wang* Naval Postgraduate School, Monterey, California 1. INTRODUCTION* and the warm air above land (thermal wind) and the frictional effect within the atmospheric This observational study focus on the boundary layer (Zemba and Friehe 1987, Burk and interaction between the coastal wind field and the Thompson 1996). The horizontal temperature evolution of the coastal stratocumulus clouds. The gradient follows the orientation of the coastline data were collected during an experiment in the (317° on the average in central California). Thus, summer of 1999 near the California coast with the the thermal wind has a northern and an eastern Twin Otter research aircraft operated by the component and local maximum of both Center for Interdisciplinary Remote Piloted Aircraft components of the wind is expected close to the Study (CIRPAS) at the Naval Postgraduate School top of the boundary layer. Topographical features (NPS). The wind components were measured with may intensify this wind jet at significant convex a DGPS/radome system, which provides a very bends of the coastline like Cape Mendocino, Pt accurate (0.1 ms-1) estimate of the wind (Kalogiros Arena and Pt Sur. At such changes of the and Wang 2001). The instrumentation of the coastline geometry combined with mountain aircraft included fast sensors for the measurement barriers (channeling effect) the northerly flow can of air temperature, humidity, shortwave and become supercritical.
    [Show full text]
  • Using the Weak Temperature Gradient Approximation to Understand Self-Aggregation 1
    Multiple Equilibria in a Cloud Resolving Model: Using the Weak Temperature Gradient Approximation to Understand Self-Aggregation 1 Sharon Sessions, Satomi Sugaya, David Raymond, Adam Sobel New Mexico Tech SWAP 2011 nmtlogo 1This work is supported by the National Science Foundation Sessions (NMT) Multiple Equilibria in a CRM May2011 1/21 Self-Aggregation Bretherton et al (2005) precipitation 1 WVP = g rt dp R nmtlogo Day 1 Day 50 Sessions (NMT) Multiple Equilibria in a CRM May2011 2/21 Possible insight from limited domain WTG simulations Bretherton et al (2005) Day 1 Day 50 nmtlogo Sessions (NMT) Multiple Equilibria in a CRM May2011 3/21 Weak Temperature Gradients in the Tropics Charney 1963 scaling analysis Extratropics (small Rossby number) F δθ ∼ r θ Ro Tropics (Rossby number not small) δθ ∼ Fr θ 2 −3 Froude number: Fr = U /gH ∼ 10 , measure of stratification −5 Rossby number: Ro = U/fL ∼ 10 /f , characterizes rotation nmtlogo Sessions (NMT) Multiple Equilibria in a CRM May2011 4/21 Weak Temperature Gradients in the Tropics Charney 1963 scaling analysis Extratropics (small Rossby number) F δθ ∼ r θ Ro Tropics (Rossby number not small) δθ ∼ Fr θ 2 −3 Froude number: Fr = U /gH ∼ 10 , measure of stratification −5 Rossby number: Ro = U/fL ∼ 10 /f , characterizes rotation In the tropics, gravity waves rapidly redistribute buoyancy anomalies nmtlogo Sessions (NMT) Multiple Equilibria in a CRM May2011 4/21 Weak Temperature Gradient (WTG) Approximation Sobel & Bretherton 2000 Single column model (SCM) representing one grid cell in a global circulation
    [Show full text]
  • NWS Unified Surface Analysis Manual
    Unified Surface Analysis Manual Weather Prediction Center Ocean Prediction Center National Hurricane Center Honolulu Forecast Office November 21, 2013 Table of Contents Chapter 1: Surface Analysis – Its History at the Analysis Centers…………….3 Chapter 2: Datasets available for creation of the Unified Analysis………...…..5 Chapter 3: The Unified Surface Analysis and related features.……….……….19 Chapter 4: Creation/Merging of the Unified Surface Analysis………….……..24 Chapter 5: Bibliography………………………………………………….…….30 Appendix A: Unified Graphics Legend showing Ocean Center symbols.….…33 2 Chapter 1: Surface Analysis – Its History at the Analysis Centers 1. INTRODUCTION Since 1942, surface analyses produced by several different offices within the U.S. Weather Bureau (USWB) and the National Oceanic and Atmospheric Administration’s (NOAA’s) National Weather Service (NWS) were generally based on the Norwegian Cyclone Model (Bjerknes 1919) over land, and in recent decades, the Shapiro-Keyser Model over the mid-latitudes of the ocean. The graphic below shows a typical evolution according to both models of cyclone development. Conceptual models of cyclone evolution showing lower-tropospheric (e.g., 850-hPa) geopotential height and fronts (top), and lower-tropospheric potential temperature (bottom). (a) Norwegian cyclone model: (I) incipient frontal cyclone, (II) and (III) narrowing warm sector, (IV) occlusion; (b) Shapiro–Keyser cyclone model: (I) incipient frontal cyclone, (II) frontal fracture, (III) frontal T-bone and bent-back front, (IV) frontal T-bone and warm seclusion. Panel (b) is adapted from Shapiro and Keyser (1990) , their FIG. 10.27 ) to enhance the zonal elongation of the cyclone and fronts and to reflect the continued existence of the frontal T-bone in stage IV.
    [Show full text]
  • Thermal Conductivity of Materials
    • DECEMBER 2019 Thermal Conductivity of Materials Thermal Conductivity in Heat Transfer – Lesson 2 What Is Thermal Conductivity? Based on our life experiences, we know that some materials (like metals) conduct heat at a much faster rate than other materials (like glass). Why is this? As engineers, how do we quantify this? • Thermal conductivity of a material is a measure of its intrinsic ability to conduct heat. Metals conduct heat much faster to our hands, Plastic is a bad conductor of heat, so we can touch an which is why we use oven mitts when taking a pie iron without burning our hands. out of the oven. 2 What Is Thermal Conductivity? Hot Cold • Let’s recall Fourier’s law from the previous lesson: 푞 = −푘∇푇 Heat flow direction where 푞 is the heat flux, ∇푇 is the temperature gradient and 푘 is the thermal conductivity. • If we have a unit temperature gradient across the material, i.e.,∇푇 = 1, then 푘 = −푞. The negative sign indicates that heat flows in the direction of the negative gradient of temperature, i.e., from higher temperature to lower temperature. Thus, thermal conductivity of a material can be defined as the heat flux transmitted through a material due to a unit temperature gradient under steady-state conditions. It is a material property (independent of the geometry of the object in which conduction is occurring). 3 Measuring Thermal Conductivity • In the International System of Units (SI system), thermal conductivity is measured in watts per -1 -1 meter-Kelvin (W m K ). Unit 푞 W ∙ m−1 푞 = −푘∇푇 푘 = − (W ∙ m−1 ∙ K−1) ∇푇 K • In imperial units, thermal conductivity is measured in British thermal unit per hour-feet-degree- Fahrenheit (BTU h-1ft-1 oF-1).
    [Show full text]
  • Atmospheric Stability Atmospheric Lapse Rate
    ATMOSPHERIC STABILITY ATMOSPHERIC LAPSE RATE The atmospheric lapse rate ( ) refers to the change of an atmospheric variable with a change of altitude, the variable being temperature unless specified otherwise (such as pressure, density or humidity). While usually applied to Earth's atmosphere, the concept of lapse rate can be extended to atmospheres (if any) that exist on other planets. Lapse rates are usually expressed as the amount of temperature change associated with a specified amount of altitude change, such as 9.8 °Kelvin (K) per kilometer, 0.0098 °K per meter or the equivalent 5.4 °F per 1000 feet. If the atmospheric air cools with increasing altitude, the lapse rate may be expressed as a negative number. If the air heats with increasing altitude, the lapse rate may be expressed as a positive number. Understanding of lapse rates is important in micro-scale air pollution dispersion analysis, as well as urban noise pollution modeling, forest fire-fighting and certain aviation applications. The lapse rate is most often denoted by the Greek capital letter Gamma ( or Γ ) but not always. For example, the U.S. Standard Atmosphere uses L to denote lapse rates. A few others use the Greek lower case letter gamma ( ). Types of lapse rates There are three types of lapse rates that are used to express the rate of temperature change with a change in altitude, namely the dry adiabatic lapse rate, the wet adiabatic lapse rate and the environmental lapse rate. Dry adiabatic lapse rate Since the atmospheric pressure decreases with altitude, the volume of an air parcel expands as it rises.
    [Show full text]
  • A Generalization of the Thermal Wind Equation to Arbitrary Horizontal Flow CAPT
    A Generalization of the Thermal Wind Equation to Arbitrary Horizontal Flow CAPT. GEORGE E. FORSYTHE, A.C. Hq., AAF Weather Service, Asheville, N. C. INTRODUCTION N THE COURSE of his European upper-air analysis for the Army Air Forces, Major R. C. I Bundgaard found that the shear of the observed wind field was frequently not parallel to the isotherms, even when allowance was made for errors in measuring the wind and temperature. Deviations of as much as 30 degrees were occasionally found. Since the ther- mal wind relation was found to be very useful on the occasions where it did give the direction and spacing of the isotherms, an attempt was made to give qualitative rules for correcting the observed wind-shear vector, to make it agree more closely with the shear of the geostrophic wind (and hence to make it lie along the isotherms). These rules were only partially correct and could not be made quantitative; no general rules were available. Bellamy, in a recent paper,1 has presented a thermal wind formula for the gradient wind, using for thermodynamic parameters pressure altitude and specific virtual temperature anom- aly. Bellamy's discussion is inadequate, however, in that it fails to demonstrate or account for the difference in direction between the shear of the geostrophic wind and the shear of the gradient wind. The purpose of the present note is to derive a formula for the shear of the actual wind, assuming horizontal flow of the air in the absence of frictional forces, and to show how the direction and spacing of the virtual-temperature isotherms can be obtained from this shear.
    [Show full text]
  • Thermal Equilibrium of the Atmosphere with a Convective Adjustment
    jULY 1964 SYUKURO MANABE AND ROBERT F. STRICKLER 361 Thermal Equilibrium of the Atmosphere with a Convective Adjustment SYUKURO MANABE AND ROBERT F. STRICKLER General Circulation Research Laboratory, U.S. Weather Bureau, Washington, D. C. (Manuscript received 19 December 1963, in revised form 13 April 1964) ABSTRACT The states of thermal equilibrium (incorporating an adjustment of super-adiabatic stratification) as well as that of pure radiative equilibrium of the atmosphere are computed as the asymptotic steady state ap­ proached in an initial value problem. Recent measurements of absorptivities obtained for a wide range of pressure are used, and the scheme of computation is sufficiently general to include the effect of several layers of clouds. The atmosphere in thermal equilibrium has an isothermal lower stratosphere and an inversion in the upper stratosphere which are features observed in middle latitudes. The role of various gaseous absorbers (i.e., water vapor, carbon dioxide, and ozone), as well as the role of the clouds, is investigated by computing thermal equilibrium with and without one or two of these elements. The existence of ozone has very little effect on the equilibrium temperature of the earth's surface but a very important effect on the temperature throughout the stratosphere; the absorption of solar radiation by ozone in the upper and middle strato­ sphere, in addition to maintaining the warm temperature in that region, appears also to be necessary for the maintenance of the isothermal layer or slight inversion just above the tropopause. The thermal equilib­ rium state in the absence of solar insolation is computed by setting the temperature of the earth's surface at the observed polar value.
    [Show full text]
  • Methods for Diagnosing Regions of Conditional Symmetric Instability
    Methods for Diagnosing Regions of Conditional Symmetric Instability James T. Moore Cooperative Institute for Precipitation Systems Saint Louis University Dept. of Earth & Atmospheric Sciences Ted Funk Science and Operations Officer WFO Louisville, KY Spectrum of Instabilities Which Can Result in Enhanced Precipitation • Inertial Instability • Potential Symmetric Instability • Conditional Symmetric Instability • Weak Symmetric Stability • Elevated Convective Instability 1 Inertial Instability Inertial instability is the horizontal analog to gravitational instability; i.e., if a parcel is displaced horizontally from its geostrophically balanced base state, will it return to its original position or will it accelerate further from that position? Inertially unstable regions are diagnosed where: .g + f < 0 ; absolute geostrophic vorticity < 0 OR if we define Mg = vg + fx = absolute geostrophic momentum, then inertially unstable regions are diagnosed where: MMg/ Mx = Mvg/ Mx + f < 0 ; since .g = Mvg/ Mx (NOTE: vg = geostrophic wind normal to the thermal gradient) x = a fixed point; for wind increasing with height, Mg surfaces become more horizontal (i.e., Mg increases with height) Inertial Instability (cont.) Inertial stability is weak or unstable typically in two regions (Blanchard et al. 1998, MWR): .g + f = (V/Rs - MV/ Mn) + f ; in natural coordinates where V/Rs = curvature term and MV/ Mn = shear term • Equatorward of a westerly wind maximum (jet streak) where the anticyclonic relative geostrophic vorticity is large (to offset the Coriolis
    [Show full text]
  • Synoptic Meteorology
    Lecture Notes on Synoptic Meteorology For Integrated Meteorological Training Course By Dr. Prakash Khare Scientist E India Meteorological Department Meteorological Training Institute Pashan,Pune-8 186 IMTC SYLLABUS OF SYNOPTIC METEOROLOGY (FOR DIRECT RECRUITED S.A’S OF IMD) Theory (25 Periods) ❖ Scales of weather systems; Network of Observatories; Surface, upper air; special observations (satellite, radar, aircraft etc.); analysis of fields of meteorological elements on synoptic charts; Vertical time / cross sections and their analysis. ❖ Wind and pressure analysis: Isobars on level surface and contours on constant pressure surface. Isotherms, thickness field; examples of geostrophic, gradient and thermal winds: slope of pressure system, streamline and Isotachs analysis. ❖ Western disturbance and its structure and associated weather, Waves in mid-latitude westerlies. ❖ Thunderstorm and severe local storm, synoptic conditions favourable for thunderstorm, concepts of triggering mechanism, conditional instability; Norwesters, dust storm, hail storm. Squall, tornado, microburst/cloudburst, landslide. ❖ Indian summer monsoon; S.W. Monsoon onset: semi permanent systems, Active and break monsoon, Monsoon depressions: MTC; Offshore troughs/vortices. Influence of extra tropical troughs and typhoons in northwest Pacific; withdrawal of S.W. Monsoon, Northeast monsoon, ❖ Tropical Cyclone: Life cycle, vertical and horizontal structure of TC, Its movement and intensification. Weather associated with TC. Easterly wave and its structure and associated weather. ❖ Jet Streams – WMO definition of Jet stream, different jet streams around the globe, Jet streams and weather ❖ Meso-scale meteorology, sea and land breezes, mountain/valley winds, mountain wave. ❖ Short range weather forecasting (Elementary ideas only); persistence, climatology and steering methods, movement and development of synoptic scale systems; Analogue techniques- prediction of individual weather elements, visibility, surface and upper level winds, convective phenomena.
    [Show full text]
  • ATM 316 - the “Thermal Wind” Fall, 2016 – Fovell
    ATM 316 - The \thermal wind" Fall, 2016 { Fovell Recap and isobaric coordinates We have seen that for the synoptic time and space scales, the three leading terms in the horizontal equations of motion are du 1 @p − fv = − (1) dt ρ @x dv 1 @p + fu = − ; (2) dt ρ @y where f = 2Ω sin φ. The two largest terms are the Coriolis and pressure gradient forces (PGF) which combined represent geostrophic balance. We can define geostrophic winds ug and vg that exactly satisfy geostrophic balance, as 1 @p −fv ≡ − (3) g ρ @x 1 @p fu ≡ − ; (4) g ρ @y and thus we can also write du = f(v − v ) (5) dt g dv = −f(u − u ): (6) dt g In other words, on the synoptic scale, accelerations result from departures from geostrophic balance. z R p z Q δ δx p+δp x Figure 1: Isobaric coordinates. It is convenient to shift into an isobaric coordinate system, replacing height z by pressure p. Remember that pressure gradients on constant height surfaces are height gradients on constant pressure surfaces (such as the 500 mb chart). In Fig. 1, the points Q and R reside on the same isobar p. Ignoring the y direction for simplicity, then p = p(x; z) and the chain rule says @p @p δp = δx + δz: @x @z 1 Here, since the points reside on the same isobar, δp = 0. Therefore, we can rearrange the remainder to find δz @p = − @x : δx @p @z Using the hydrostatic equation on the denominator of the RHS, and cleaning up the notation, we find 1 @p @z − = −g : (7) ρ @x @x In other words, we have related the PGF (per unit mass) on constant height surfaces to a \height gradient force" (again per unit mass) on constant pressure surfaces.
    [Show full text]
  • Thickness and Thermal Wind
    ESCI 241 – Meteorology Lesson 12 – Geopotential, Thickness, and Thermal Wind Dr. DeCaria GEOPOTENTIAL l The acceleration due to gravity is not constant. It varies from place to place, with the largest variation due to latitude. o What we call gravity is actually the combination of the gravitational acceleration and the centrifugal acceleration due to the rotation of the Earth. o Gravity at the North Pole is approximately 9.83 m/s2, while at the Equator it is about 9.78 m/s2. l Though small, the variation in gravity must be accounted for. We do this via the concept of geopotential. l A surface of constant geopotential represents a surface along which all objects of the same mass have the same potential energy (the potential energy is just mF). l If gravity were constant, a geopotential surface would also have the same altitude everywhere. Since gravity is not constant, a geopotential surface will have varying altitude. l Geopotential is defined as z F º ò gdz, (1) 0 or in differential form as dF = gdz. (2) l Geopotential height is defined as F 1 z Z º = ò gdz (3) g0 g0 0 2 where g0 is a constant called standard gravity, and has a value of 9.80665 m/s . l If the change in gravity with height is ignored, geopotential height and geometric height are related via g Z = z. (4) g 0 o If the local gravity is stronger than standard gravity, then Z > z. o If the local gravity is weaker than standard gravity, then Z < z.
    [Show full text]
  • Temperature Gradient
    Temperature Gradient Derivation of dT=dr In Modern Physics it is shown that the pressure of a photon gas in thermodynamic equilib- rium (the radiation pressure PR) is given by the equation 1 P = aT 4; (1) R 3 with a the radiation constant, which is related to the Stefan-Boltzman constant, σ, and the speed of light in vacuum, c, via the equation 4σ a = = 7:565731 × 10−16 J m−3 K−4: (2) c Hence, the radiation pressure gradient is dP 4 dT R = aT 3 : (3) dr 3 dr Referring to our our discussion of opacity, we can write dF = −hκiρF; (4) dr where F = F (r) is the net outward flux (integrated over all wavelengths) of photons at a distance r from the center of the star, and hκi some properly taken average value of the opacity. The flux F is related to the luminosity through the equation L(r) F (r) = : (5) 4πr2 Considering that pressure is force per unit area, and force is a rate of change of linear momentum, and photons carry linear momentum equal to their energy divided by c, PR and F obey the equation 1 P = F (r): (6) R c Differentiation with respect to r gives dP 1 dF R = : (7) dr c dr Combining equations (3), (4), (5) and (7) then gives 1 L(r) 4 dT − hκiρ = aT 3 ; (8) c 4πr2 3 dr { 2 { which finally leads to dT 3hκiρ 1 1 = − L(r): (9) dr 16πac T 3 r2 Equation (9) gives the local (i.e.
    [Show full text]