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Virtual Temperature Thickness Sea Level Pressure Potential Temperature Equivalent Potential Temperature

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Virtual Temperature Thickness Sea Level Pressure Potential Temperature Equivalent Potential Temperature Important Quantities that we will use often in class to examine Atmospheric Processes: Virtual Temperature Thickness Sea Level Pressure Potential Temperature Equivalent Potential Temperature (See P. 57-64 and 195-217 of Bluestein, Vol. I for a detailed discussion) Review of Basic Thermodynamic Equations Ideal gas law relates pressure, density, and temperature for an ideal gas (air is P = pressure considered an ideal gas) ρ = air density R = gas constant P = !RT T = Temperature "P Z = height Hydrostatic equation: = #g$ "z • Upward-directed pressure gradient force per unit mass is balanced by the downward directed force of gravity per unit mass. • Pressure at a given height is given by the weight of the column above per unit horizontal area. ! • Pressure decreases with height, the extent to which an air parcel is compressed also decreases with height. • When applied, you are making the hydrostatic approximation. When is this approximation not valid? • Using these equations, many useful quantities and relations can be derived. Virtual Temperature: The temperature that a parcel of dry air would have if it were at the same pressure and had the same density as moist air. P = pressure = dry air density Derivation: ρd ρv = vapor density Start with ideal gas law for moist air: ρ = air density R= gas constant Rv = vapor gas constant P = !RT (461.5 J kg-1 °-1) Rd = dry air gas constant (287 J kg-1 °-1) P = !d RdT + !v RvT T = Temperature (K) Now treat moist air as if it were dry by introducing the virtual temperature Tv P = (!d Rd + !v Rv )T = (!d + !v )RdTv = !RdTv What is the relationship between the temperature, T And the virtual temperature Tv? ("d Rd + "vRv )T = ("d + "v )Rd Tv M Write: M = mass of air " = Md = mass of dry air V Mv = mass of vapor V = volume " M M % " M M % ! $ d R + v R 'T = $ d + v ' R T # V d V v & # V V & d v ! Multiply and divide second term by Rd " % " % ! Md Mv Rd Rv Md Mv $ Rd + 'T = $ + ' Rd Tv # V V Rd & # V V & Cancel Rd and rearrange " Md Mv Rv % $ + ' ! V V Rd Tv = $ 'T $ Md Mv ' $ + ' # V V & ! " Md Mv Rv % $ + ' From last page: V V Rd Tv = $ 'T $ Md Mv ' $ + ' # V V & Cancel V, and divide top and bottom terms by Md: " Mv Rv % ! $ 1+ ' Md Rd Tv = $ 'T $ Mv ' $ 1+ ' # Md & Introduce mixing ratio: rv = Mv/Md and let ε = Rd/Rv # # 1& & % 1+ rv% ( ( ! $ "' Tv = % (T % 1+ rv ( % ( $ ' ! # # 1& & % 1+ rv% ( ( $ "' From last page: Tv = % (T % 1+ rv ( % ( $ ' -1 Approximate (1+rv) = 1- rv and use 1/ε = 1.61 ! 2 Tv = (1" rv )(1+1.61rv )T = (1" rv +1.61rv "1.61rv )T 2 Neglect r v term T = 1+ 0.61r T ! v ( v ) ! T = 1+ 0.61r T Implications: v ( v ) Moist air has a higher “effective” or “virtual” temperature than dry air. This means that moist air is less dense than dry air. ! Simple explanation: In moist air, water molecules (Molecular weight = 18 g mol-1) -1 replace air molecules (N2 MW = 28 g mol , O2 MW = 32 g mol-1), so volume of air has less mass than equivalent volume of dry air. Why do we use Virtual Temperature? Stability depends on the relative density between an air parcel and its environment. In a dry atmosphere, density differences are determined by comparing the temperature of a parcel and its environment. In a moist atmosphere, we must compare the virtual temperature of a parcel and its environment. From Bluestein Vol II p. 285 Note that the temperature is the same, but the virtual temperature is different. Air west of the dry line is drier, and therefore denser than air east of the dry line. The Hypsometric Equation Ideal gas law: P = !RdTv (1) "P Hydrostatic equation: = #g$ (2) g = gravity "z R T dP Put (1) into (2) and rearrange: d = "dz (3) g P p1 z1 Rd dP Put in integral form, using limits: # T = " # dz p1 z1 ! g p P z 0 0 T R p1 dP z1 T d T = " dz Assume T is mean in layer: g # P # p0 z0 p0 z0 $ ' Rd p1 Integrate: T ln& ) = "(z1 " z0 ) g % p0 ( $ ' Rd p1 z1 = z0 " T ln& ) g % p0 ( Rearrange to get the $ ' Rd p0 hypsometric equation: z1 = z0 + T ln& ) (4) g % p1 ( A really useful equation! ! Atmospheric pressure varies exponentially with altitude, but very slowly on a horizontal plane – as a result, a map of surface (station) pressure looks like a map of altitude above sea level. observed surface pressure variation of pressure over central North America with altitude Station pressures are measured at locations worldwide Analysis of horizontal pressure fields, which are responsible for the earth’s winds and are critical to analysis of weather systems, requires that station pressures be converted to a common level, which, by convention, is mean sea level. Application of the Hypsometric Equation #1: Reduction to Sea Level Pressure Ideal gas law: P = !RdTv (1) "P Hydrostatic equation: = #g$ (2) g = gravity "z dP ! g Put (1) into (2) and rearrange: = dz (3) P RdTv P Integrate left side from SL dP 0 " g ! = dz (4) sea level pressure (PSL) ! P ! RT PSTA zSTA v to station pressure (PSTN) and right side from z = 0 g (5) ln(PSL )= ln(PSTN )+ zSTN to station altitude (ZSTN). RdTv Reduction of station pressure to sea level pressure: Ideal gas law: P = !RdTv (1) #P Hydrostatic equation: = "g! (2) #z dP "g Put (1) into (2) and rearrange: = dz (3) P Rd Tv P Integrate left side from SL dP 0 " g = dz (4) sea level pressure PSL to station ! P ! RT PSTA zSTA v pressure PSTN and right side from ! g z = 0 to station altitude ZSTN. (5) ln(PSL )= ln(PSTN )+ zSTN RdTv BUT WHAT IS Tv? WE HAVE TO ASSUME A FICTICIOUS ATMOSPHERE THAT IS BELOW GROUND!!! National Weather Service Procedure to estimate Tv 1. Assume Tv = T 2. Assume a mean surface temperature = average of current temperature and temperature 12 hours earlier to eliminate diurnal effects. 3. Assume a lapse rate between the station and sea level of o 6.5 C/km to determine TSL. 4. Determine average T and then PSL. In practice, PSL is determined using a table of “R” values, where R is the ratio of station pressure to sea level pressure, and the table contains station pressures and average temperatures. Table contains a “plateau correction” to try to compensate for variations in annual mean sea level pressures calculated for nearby stations. A weakness of the method: Imagine if a very cold, but very shallow airmass (2 km thick) moved across the high plains of eastern Colorado (elevation ~5000 ft). Observed MSLP may rise 6 mb with the passage of this cold air mass. Note, however, that much of this pressure rise will be the result of the integration of the cold temperatures below the ground to sea level, not the added pressure from the weight of the cold shallow airmass. If this same airmass moved over a surface station near sea level (with the same ambient conditions), the rise in observed MSLP with the passage of this air mass will be significantly less; but the change in station pressure should be the same! Can you think of why? Net result: Pressure analysis at sea level altitude Application of Hypsometric Equation #2 p1 z1 T The Thickness Equation T p z Rearranging the Hypsometric equation: 0 0 # & Rd p0 h = z1 " z0 = T ln% ( Thickness equation g $ p1 ' This says that the thickness of the layer p0 to p1 is directly proportional to the mean temperature of the layer. ! We are ignoring one factor, however (think about Tv). What? Now we can calculate how much layer mean temperature change occurs for a given thickness change and vice versa. Also, we can calculate the thickness for key atmospheric parameters related to temperature (e.g. 0°C layer thickness threshold for snow). What are potential pitfalls of this method? Potential Temperature Temperature a parcel of air would have if it were brought dry adiabatically to a pressure of 1000 mb. “Dry adiabatically” Implies no exchange of mass or energy with the environment, and no condensation or evaporation occurring within the air parcel. Potential temperature equation derived from 1st law of thermodynamics: S = Entropy Cp = Specific Heat at constant pressure dS = C pd lnT ! Rd d ln P (1005 J kg-1 °-1) Rd = dry air gas constant For an adiabatic process, dS = 0 C pd lnT ! Rd d ln P = 0 C pd lnT ! Rd d ln P = 0 Integrate equation from an arbitrary temperature and pressure To the potential temperature θ and a pressure of 1000 mb. 1000 " R dlnT = d dlnP #T # Cp P # " & R # 1000& ln% ( = d ln% ( $ T ' Cp $ P ' ! Rd # 1000& C p " = T% ( ! $ P ' ! Significance of Potential Temperature in Synoptic Analysis 1. θ increases with height* and can therefore serve as a vertical coordinate. (* θ can decrease with height locally near strong heat sources (e.g. a hot parking lot), but these can be ignored in synoptic analyses). Significance of Potential Temperature in Synoptic Analysis 2. θ is conserved for air parcels moving about in the atmosphere provided: a) there is no condensation or evaporation within the parcel b) there is no mixing of the parcel with air of dissimilar properties c) there is no radiational heating or cooling of the parcel. 3. In the absence of condensation or evaporation, and radiational heating or cooling: AIR MOVES ALONG SURFACES OF CONSTANT POTENTIAL TEMPERATURE.
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