第二章 : Hydrostatics and Atmospheric Stability Ben Jong-Dao Jou Autumn 2010
Part I: Hydrostatics 1. Gravity 2. Geopotential: The concept of geopotential is used in measurement of heights in the atmosphere by combining g and z into a single variable 3. The hydrostatic relation 4. The barometric or hypsometric formula 5. Pressure-height evaluation (standard pressure layers) 6. Approximate calculations of depth (thickness) 7. Extrapolation of upper-level pressures and heights from indications at the surface 8. Standard atmospheres 9. The D method of pressure-height evaluation (D: altimeter correction) 10. Pressure reduction to reference levels below the surface (using barometric formula and the variation of lapse rate with temperature is not the same for all stations but is a function of geographic location) 11. Height reductions to pressure references 12. Pressure tendency 13. Height tendencies at constant pressure 14. Space variations of pressure
Gravity force An object of unit mass, at rest of the surface of earth is subject to a centripetal acceleration directed toward the axis of rotation of the earth given by Ω 2 R 1, where R 1 is the position vector from the axis of rotation to the object and Ω=7.292x10-5 (2 /sidereal day= 23h 56min 4s= 86164s) rad s -1 is the angular speed of rotation of the earth. (Ω 2 R 1 = 0.034ms -2 ) The resultant vector, apparent gravity g = g + Ω 2 R 1 indicating an angle of ~0.1 0 to the true direction of gravity (in general taken as constant g =9.81ms -2 ) The local vertical is assumed to be aligned with the resultant gravity vector. g is NOT directed toward the center of the earth, but is perpendicular to a geopotential surface Gravity can be represented in terms of the gradient of a potential function which is just the geopotential, = -g
The hydrostatic equation For the absence of air motion, dp/dz= - g, the gravity force is exactly balanced by the vertical component of the pressure gradient force. Integrating the equation from a height z to the top of the atmosphere p(z) = z g dz The pressure at any point is simply equal to the weight of the unit cross section column of air overlying the point. The mean sea level pressure p(0) =1013.25 hpa is simply the average weight per square meter of the local atmospheric column. Geopotential and geometric height, d = g dz = - dp = - (RT/p) dp Note: = 0 z g dz, the geopotential at height z, is just work required to raise a unit mass to height z from mean sea level.
Station pressure 測站氣壓 sea-level pressure 海平面氣壓 isobars 等壓線 surface maps 地面天氣圖 (a) 四個不同海平面高度的城市之測站氣壓 (station pressure),(b) 這四個城市的海平面氣壓 (sea level pressure), (c) 以 4 mb 為間距的等壓線 (isobars) 分佈 1mb ~ 8m
Virtual temperature 虛溫 T v T {1 e/p [1 - ] } Rather than use a gas constant for moist air (Rv), the exact value of which would depend on the amount of water vapor in the air (which varies considerably), it is convenient to retain the gas constant for dry air (Rd) and use a fictitious temperature (called the virtual temperature) in the ideal gas equation. R d = 287.0 J K-1kg-1 e is specific pressure of water vapor = R d /Rv = Mw/M d (molecular weight) = 0.622 p = p d + e (Dalton law) P = R d T v d = g dz = - dp = - (R d T v /p) dp
重力位高度和測高公式 Geopotential height, Z (z)/g 0, where g 0 =9.80665 ms -2 the global average of gravity at mean sea level. Hypsometric equation, { (z 2 ) - (z 1 )}/g 0 = Z 2 - Z 1 = R d /g 0 [ p2 p1 T v dp/p] The variation of geopotential wrt pressure depends only on virtual temperature. Z T = Z 2 - Z 1, thickness ( 厚度 )
Thickness field, mean temperature, and scale height Z T = Z 2 - Z 1 = R d /g 0 [ p2 p1 T v dp/p] The thickness of the atmospheric layer between the pressure surfaces p2 and p1. Defining a layer mean temperature <Tv> <Tv>= [ p2 p1 T v d lnp] / [ p2 p1 d lnp], and a layer mean scale height H =R<Tv>/g 0 we have Z T = H ln (p 1 /p 2 ) Thus, the thickness of a layer bounded by isobaric surfaces is proportional to the mean virtual temperature of the layer. In an isothermal atmosphere of temperature T, the geopotential height is proportional to the nature logarithm of pressure normalized by the surface pressure, Z = - H ln (p/p 0 ), where p 0 is the pressure at Z=0. Thus in an isothermal atmosphere, the pressure decreases exponentially with geopotential height by a factor of e -1 per scale height, p(z) = p(0) e -Z/H
Thickness Z = Z 2 - Z 1 = H ln (p1/p2) = R d /g 0 <T v > ln (p1/p2) P1 = 1000 hpa, p2 = 500 hpa Z = Z 500mb - Z 1000mb = R d /g 0 <T v > ln (1000/500) = 20.3 <Tv> m In the tropics, <Tv>= +15 C (288K) = 5846m In the pole, <Tv>= -40 C (233K) = 4730m Warm core low Cold core low
Warm core tropical cyclone
Warm core low Tropical cyclone 11
Upper level cold core low
Surface map analysis 60-180E, 50N: (hpa) 1025/1020/1018/1016/1010/1016/ 1018/1019/1017/1012/1017/1010/1003 天氣現象和天氣系統的綜合分析
Pressure (scalar) field analysis 地面氣壓 ( 純量 ) 場分析 Terminology: isobars, height contours, radius of curvature, positively curved (R>0), negatively curved (R<0), ridge axis (ridge line), {tilted} trough axis (trough line), inverted trough, a high pressure center or high, a low pressure center or low, low is usually more intense than high because of inertial instability, shape (circle, elliptic, or wave-like), centers do not need to be closed isobar, wave train, a saddle point (common in the horse latitudes) is called a col. Subjective analysis and objective analysis: Objective analysis techniques including Cressman weighting function, influence of radius, Barnes weighting function, the first guess field, optimal interpolation scheme (Gandin and Eddy 1960), and the variational alalysis technique (Sasaki 1950). Propagating (move zonally or meridionally or dugging or lifting out); intensity: intensifying, weakening, deepening, building, filling Isallobars (lines of constant pressure tendency), isallohypses (lines of constant height tendency) Issues related to local analysis: Wake depression, Semi-diurnal pressure oscillation, Station pressure and sea level pressure, hypsometric equation, Altimeter setting Issues related to tropical analysis: Tropical depression, tropical storm, typhoon, cyclone, tropical cyclone, hurricane, Monsoon trough, monsoon gyre
The surface pressure field (kinematics) By kinematics, we mean a description of the motion of a particular field without regard to how it came about or how it will evolve. Isobars (surface map) and height contours (upper air maps) Radius of curvature (R) is measured from the center, radially outward to the isobars. Low pressure center (L), the isobars are positively curved (R p >0) -convex, and for high pressure center (H), the isobars are negatively curved (R p <0) -concave. The ridge axis (ridge line) and the trough axis (trough line) The tilting of the trough line is related to the sign of the eddy transport of westerly momentum in the meridional direction by the trough and ridge. Troughs in easterly flow are often called inverted trough. ( 倒槽 )
The intensity of the ridge/high (trough/low) The intensity of the ridge (trough) is given by - 2 p/ x 2 and is determined by the gradient of the pressure gradient itself across the ridge (trough) axis. A high (low) pressure center or high (low) is a local maximum (minimum) in the pressure field, z p =0 and 2 z p <0 ( 2 z p >0). The intensity of a high (low) is thus determined by the Laplacian of the pressure field at the center of the high (low), and not by the maximum (minimum) value of the pressure. However, intense highs (lows) usually have relatively high (low) values of central pressure. Low pressure centers are usually more intense than high pressure centers (due to inertial instability). Lows and highs may assume shapes that are intermediate between perfect circles and wavelike troughs and ridges. (heart- or elliptical shapes are two examples)
Center pressure? Centers of high and low pressure do not necessary have closed contours drawn around them. it is also highly likely that the actual central pressure of the highs and lows are not measured by the stations. The uncertainty depends upon coverage and the intensity of the centers (over land ~1-2 hpa and over ocean ~2 hpa or more).
A sequence of highs and lows along the flow is called a wavetrain, and each cycle of alternating troughs and ridges is referred to as a wave. A simple analytical expression for a onedimensional wavetrain is the pressure field is given as: P(x, t) = A sin ( [2 /L] {x-ct} ) + p* A is the amplitude of the wave, L is the wavelength in the x-direction, c is the phase speed in the x-direction, p* is the average pressure 2 c/l is the angular frequency L/c = T = 2 / is the period = 2 /L is the wave number.
Waves in the atmosphere: diurnal cycle and semidiurnal cycle
Part II: Static stability 1. The concept of hydrostatic stability 2. Stability of unsaturated and saturated air 3. Degree of stability (LCL, LFC, CCL, instability index) 4. Conditional instability and the parcel method of convection 5. Objective use of the parcel method of convection 6. Entrainment in the parcel method of convection 7. Layer stability: unsaturated case 8. Layer stability: saturated case (convectively or potentially unstable, θ E ) 9. Layer stability: Air becomes saturated during lifting 10. Effect of mixing on vertical lapse rates (MCL) 11. Additional processes affecting stability (radiation, surface process, vertical shear) 12. The role of stability in the modification of air columns (heat, moisture, momentum) 13. Formation of temperature inversions {lower stratosphere, free atmosphere, and ground inversions}; (differential horizontal temperature advection, differential vertical motions, differential radiation, turbulence, radiation cooling, contact cooling) 14. Local temperature changes due to adiabatic vertical motions 15. Relation of stability and vertical motion to weather (air mass meteorology) 16. Remarks on the nature of the vertical motions 17. Brief survey of space and time variations of stability
溼度 Humidity 乾空氣 vs 濕空氣 ( 空氣塊 air parcel 的概念 ) 乾空氣分子量 (m d )= 氮氣 + 氧氣 + 微量氣體 ~28.96 g/mole = m v /m d ~ 0.622 溼度 : 描述空氣中水氣的含量 空氣密度 : 單位體積內空氣質量,1.275 kgm -3 乾空氣加水氣 ρ= M(=M d +M w ) /V 水氣密度 ( 絕對溼度 ): 單位體積內的水氣質量,ρ w = M w /V 比濕 : 氣塊內水氣質量和空氣質量比值, q = M w /(M d +M w ) 混合比 : 氣塊內水氣質量和乾空氣質量比值, = M w /M d 水氣壓 : 氣塊內水氣所佔有的分壓, p = p d + e 相對溼度 : 實際水氣壓 / 飽和水氣壓 RH = (e/e s ) x 100% 飽和 : 描述氣塊水氣含量的一種狀態, 此時氣塊表面的蒸發率等於凝結率 飽和 (saturation): 未飽和 (RH<100) 與過飽和 (RH>100)
Moist air: its vapor content Temperature and dew point temperature Vapor pressure (saturation vapor pressure) Vapor density Mixing ratio Specific humidity Relative humidity Virtual temperature Ways of reaching saturation Dew point temperature Td (frost point temperature wrt ice) Convective temperature Tc (isentropic condensation temperature) Wet-bulb temperature Tw (Td Tw) Equivalent temperature Te Pseudo-adiabatic process Adiabatic wet-bulb temperature Tsw Wet-bulb potential temperature w Equivalent temperature Te (adiabatic definition) Equivalent potential temperature e Wet equivalent potential temperature q
1999, 8, 29 0000Z 台北冰雹之板橋探空 右為斜溫圖, 上為風徑圖 CAPE ~ 1405.3 m²s² 0 溶解層高度 ~ 4842 m Ric = 109
Ps surface pressure, Po=1011.0mb, Tair = 26.6C, Td = 23.9C Lifting condensation level (LCL), 972 hpa Convective condensational level (cloud base of cumulus cloud) CCL= 893 hpa; Convective temperature (Tc) Mixing condensation level (MCL)= Level of free convection (LFC) = 752 hpa Equivalent level (EL) = 156 hpa CAPE: convective available potential energy CIN: convective inhibition Bulk Richardson number (Ri) Total total index Showater index K index Sweat index Lifted index
不穩 ( 定 ) 度 : Instability
絕對不穩定和條件性不穩定
Various Inversion Heights Inversion height: ~800m at 00Z ~600m at 06Z
Simulations of Radar Ray Paths with Various Inversion Heights between 600 m and 3000m: lapse rate T: 7.5 C/km Td: 1.7 C/km 19.2 36.7