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MET 61 MET 61 Introduction to Meteorology - Lecture 4 “Heat in the atmosphere” Dr. Eugene Cordero San Jose State University W&H: Chap 3, Pg 74-84 Ahrens: Chapter 4 Class Outline: • Sensible and latent heat • First law of thermodynamics 1 MET 61 Introduction to Meteorology Thermodynamic Diagram MET 61 Green Dry Adiabats Red Moist Adiabats Yellow Saturation Mixing Ratio 2 MET 61 Introduction to Meteorology Sensible heat MET 61 • Is related to the energy exchange that can be measured by a temperature change: Q H C p T m air Cp=specific heat at constant pressure Cp=Cpd(1+0.84w); w=water vapor mixing ratio MET 61 Introduction to Meteorology 4 Latent Heat MET 61 • Latent heat is the energy exchanged during a phase change (I.e. liquid to vapor etc.) • Heat exchanged without a temperature change. Q E L m water 6 MET 61 Introduction to Meteorology Latent Heat MET 61 • The heat energy required to change water from one state to another (e.g. water from a vapor to a solid). • Latent heat of evaporation: Heat lost by environment • Latent heat of condensation Heat given to environment 8 MET 61 Introduction to Meteorology MET 61 9 MET 61 Introduction to Meteorology MET 61 10 MET 61 Introduction to Meteorology MET 61 11 MET 61 Introduction to Meteorology 1st law of thermodynamics MET 61 • Relates the change in internal energy with the heat added and the work done by the body du dq dw U, q and w are defined as per unit mass du- internal energy (proportional to motion of molecules) dq- increment of heat added dw- work done by the body 13 MET 61 Introduction to Meteorology 1st law of thermo MET 61 • Work: related to a force acting on an object to cause a displacement. • Basically, the temperature of a parcel changes when heat is added (dq) or when work is done (dw) • Alternate form: dq c p dT dp 15 MET 61 Introduction to Meteorology Water Vapor MET 61 The amount of water vapor present in the air can be expressed in a variety of ways: Mixing Ratio: Where mv is the mass of water vapor; md is the mass of dry air: Units for w are typically given in: (g of water vapor/ kg of air) Typical values; 1-5 g/kg midlatitudes and up to 20 g/kg in the tropics If there is no condensation or evaporation, then the mixing ratio of an air parcel is a conserved quantity. Note: symbol r is also commonly used for the mixing ratio 17 MET 61 Introduction to Meteorology Saturation Vapor Pressure MET 61 The saturation vapor pressure is defined as the maximum amount of water vapor necessary to keep moist air in equilibrium with a surface of pure water or ice. Consider a box containing air and water. If the box is initially dry, the water will evaporate and the water vapor pressure in the air will increase. Eventually, an equilibrium will be reached where evaporation and condensation are equal. At this point, the air is ‘saturated’ and the vapor pressure, e=es. The saturation vapor pressure: es=es(T) 18 MET 61 Introduction to Meteorology Saturation Vapor Pressure MET 61 The Clausius-Clapeyron equation describes the relationship between the saturation vapor pressure and temperature. You may see the derivation of this expression later in the course or further courses. For now, the relationship is: Where e0=6.11 hPa, T0 = 273°K and L represents the latent heat of vaporization (Lv== 2.453 × 106 J/kg) or the latent heat of deposition (Ld=2.8 x 106 J/kg). 22 MET 61 Introduction to Meteorology The Clausius-Clapeyron equation MET 61 Example problem: What is the saturation vapor pressure when the temperature is 30° C? Answer: Convert temperature to Kelvins, 30° C = 303 K ln(Es/6.11) = (2.453×106 J/kg/461 J/kg)(1/273 - 1/303) Es = (e1.92)(6.11) = 42.1 hPa 24 MET 61 Introduction to Meteorology Saturation Mixing Ratio MET 61 The saturation mixing ratio, with respect to water is defined as the ratio of the mass of saturated water vapor, mvs, to the mass of dry air, md . We can express the saturation mixing ratio in terms of pressure as: Where vs' is the partial density of water vapor required to saturate air and d' is the partial density of dry air. 27 MET 61 Introduction to Meteorology Saturation Mixing Ratio MET 61 The above equation can be further simplified as: For typical values of temperatures in the Earth’s atmosphere, p>>es, and thus Thus, ws=ws(T,P) 29 MET 61 Introduction to Meteorology Relative Humidity MET 61 The relative humidity is generally the ratio of the amount of water vapor in the air compared to the maximum amount of water the air can hold. actual vapor pressure RH saturation vapor pressure water vapo r content RH water vapo r capacity RH is a function of temperature: 31 MET 61 Introduction to Meteorology MET 61 32 MET 61 Introduction to Meteorology Dew Point MET 61 The dew point temperature, Td, is the temperature air must be cooled at constant pressure for the air to become saturated. Or, this is the temperature where the actual mixing ratio and the saturation mixing ratio are equal. Dewpoint is a good measure of the amount of water in the air. Td ~ 20°C uncomfortable Td > 24 °C very uncomfortable! The Clausius-Clapeyron can be used to determine the dew point. If you know the saturation vapor pressure, es, and Td=T and solve for Td, giving: 34 MET 61 Introduction to Meteorology Lifting Condensation Level MET 61 The lifting condensation level (LCL) defines the level where a moist parcel lifted adiabatically will become saturated. Because dry air must be lifted further to reach the LCL than moist air, the LCL height serves as another measure of the amount of water vapor in the air. During the lifting of a parcel, one often assumes that: - the mixing ratio, w, of the parcel is conserved. - the potential temperature of the parcel is conserved. At the LCL, the mixing ratio, w and the saturation mixing ratio, ws are equal. One often uses a pseudoadiabatic chart to find the LCL. However, there is a relationship: LCL=125m (T-Td) in ºC 36 MET 61 Introduction to Meteorology Thermodynamic Diagram MET 61 Green Dry Adiabats Red Moist Adiabats Yellow Saturation Mixing Ratio 37 MET 61 Introduction to Meteorology MET 61 Activity 3 (Due Feb 14th) 1. Show that for adiabatic motions, increases in temperatures are accompanied by decreases in geopotential. 2. Derive an expression for the dry adiabatic lapse rate. 3. Plot out the vertical distribution of potential temperature between the surface and 10hPa. 4. Exercise 3.42 5. Exercise 3.46 6. Exercise 3.47 38 MET 61 Introduction to Meteorology Questions MET 61 1. Calculate the density of water vapor which exerts a pressure of 9 mb at 20°C. 2. Determine the virtual temperature of moist air at 30 °C which has a mixing ratio of 20 g/kg. 3. Air at 1000hPa and 18 °C has a mixing ratio of 6g/kg. What is the relative humidity and dew point? 4. In (3), determine the LCL using the given relationship between dew point and temperature. 39 MET 61 Introduction to Meteorology