Governing Equations and Scaling in the Tropics M 1 ( ) e R ε er
Tropical v Midlatitude Meteorology Why is the general circulation and synoptic weather systems in the tropics different to the those in the midlatitudes? Most of the ideas in this lecture come from Charney, J. G., 1963. A note on the large-scale motions in the tropics. J. Atmos. Sci., 20, 607 609.
Tropical v Midlatitude Meteorology
Newton s 2nd Law Horizontal component Du h Dt p + fk u h = h ρ 0 Vertical component 0 = 1 ρ 0 p z g
Equation Scales Du h Dt U 2 L Scale Analysis p + fk u h = h f U ρ 0 δp ρ 0 L Du h Dt ~ U fk u h fl = R o The Rossby number is a measure of the relative importance of rotation.
Scale Analysis Equation Du h Dt p + fk u h = h ρ 0 Scales U 2 L f U δp ρ 0 L Mid-latitude scaling: R 0 << 1 δp ~ ρ ο ful Low-latitude scaling: R 0 ~ 1 δp ~ ρ ο ful~ ρ ο U 2
Geostrophic Balance Newton s 2nd Law Du h Dt p + f k u h = h ρ 0 Let Ro 0 then f k u h = h p ρ 0 This equation holds approximately in synoptic scale motions in the atmosphere and oceans, except possibly in the tropics. Geostrophic wind u h = 1 f k h p ρ 0
Midlatitude Scaling for Pressure Changes Pressure height scale 1 = 1 dp 0 H s p 0 dz H s = p 0 gρ 0 Mid-latitude scaling: R 0 << 1 δp ~ ρ ο ful Then δp p 0 = δp = ful = ρ 0 gh s gh s F2 R 0 R o = U fl Rossby number, F = U Froude number gh s
Low Latitude Scaling for Pressure Changes Low-latitude scaling: R 0 ~ 1 δp ~ ρ ο ful~ ρ ο U 2 δp p 0 = δp = ful ~ F2 ρ 0 gh s gh s R 0 Alternatively δp p 0 = δp = U 2 ~ F 2 ρ 0 gh s gh s These expressions are the same since R0 ~ 1.
Thermodynamic Fluctuations It can be shown that all thermodynamic fluctuation scale the same way δp p 0 ~ δρ ρ 0 ~ δθ θ 0 ~ F2 R 0
Midlatitude Fluctuations Scaling Typically δp ~ δρ ~ δθ ~ F2 p 0 ρ 0 θ 0 R 0 g 10 ms 2, Hs ~ 10 4 m Mid-latitude values: U ~ 10 ms 1, f ~ 10 4 s 1, Ro = 0.1 and F 2 = 10 3 δp ~ δρ ~ δθ ~ 10 2 p 0 ρ 0 θ 0 For geostrophic motion, thermodynamic fluctuations are small.
Low Latitude Fluctuations Scaling δp p 0 ~ δρ ρ 0 ~ δθ θ 0 ~ F2 R 0 Low-latitude values: U ~ 10 ms 1, f ~ 10 5 s 1, Ro = 1 and F 2 = 10 3 δp ~ δρ ~ δθ ~ 10 3 p 0 ρ 0 θ 0 In the tropics, thermodynamic fluctuations are an order of magnitude smaller than in mid-latitudes. The adjustment to a pressure gradient imbalance is less constrained by rotation in the tropics.
First Law of Thermodynamics Dlnθ Dt Q = c p T or lnθ t + u h lnθ + w lnθ z = Q c p T If diabatic processes such as radiative heating and cooling can be neglected, and provided that condensation or evaporation does not occur, the potential temperature θ of an air parcel is conserved.
Reductio Ad Absurdum Adiabatic Scaling in the Tropics Adiabatic Scaling lnθ t U L + u h lnθ + w lnθ z δθ ~ W dθ 0 θ 0 θ 0 dz = 0 where where N 2 ~ g θ 0 dθ 0 dz R i ~ N 2 H s 2 U 2 which is valid for Ro ~ 1 or smaller.
Reductio Ad Absurdum Adiabatic Scaling in the Tropics Scaling W D ~ U L 1 R 0 R i Typical values for synoptic scale systems: U ~ 10 ms 1, L ~ 1000 km, Hs ~ 10 km, N ~ 10 2 s 1, Ri ~ 10 2 W 10 3 Ro ms 1 In the tropics, Ro ~ 1 W ~ 1 mm s -1, which unrealistically small. In the midlatitudes, Ro << 1 W ~ 1 cm s -1, which about right.
Diabatic Processes Our conclusion is that, unlike the midlatitudes, it is essential to consider diabatic processes in the tropics (weak temperature gradient approximation). Consider first the diabatic contribution in regions away from active convection. Then the net diabatic heating is associated primarily with radiative cooling to space alone. Active Convection Radiative Cooling
Heating Budget Units are percent of incoming solar radiation. The solar fluxes are shown on the left-hand side, and the long wave (IR) fluxes are on the right-hand side. The Earth-atmosphere system continuously absorbs radiation from the Sun and emits radiation to space. Over periods longer than several days the incoming and outgoing radiation fluxes are almost equal.
Heating Budget SW LW Convection 100 31 9 60 Atmosphere 46 115 100 7 + 24 net radiative cooling = (100-31 - 46) + (115-100 - 60-9) = -31 net convective heating = (7 + 24) = 31
Distribution of Incoming Solar Radiation The incoming solar radiation of S = 1360 Wm -2 (the solar constant) intercepted by the Earth (πa 2 1360 W) is distributed, when averaged over a day or longer, over an area 4πa 2.
Outgoing Terrestrial Radiation Radiative cooling ΔQ = (cooling as a fraction of SW) (SW averaged over the surface) = 0.31 (0.25 1360) Wm 2. Change in temperature ΔT in unit time (first law of thermodynamics) ΔQ = cp Ma ΔT where Ma = (mean surface pressure) /g = 1.013 10 4 kg is the mass of a column of atmosphere 1 m 2 in cross-section. Hence
Latitudinal Variation in Cooling Actually, the rate of cooling varies with latitude. From the surface to 150 mb (approx. 85% of the atmosphere s mass): ΔT = 1.2 K/day from 0-30 lat ΔT = 0.88 K/day from 30-60 lat ΔT = 0.57 K/day from 60-90 lat
Subsidence from Radiative Cooling Radiative cooling at the rate Q c p = -1.2 K/day: w lnθ z = Q c p T W g N 2 T Q c p 10 10 4 300 1.2 24 3600 0.5 cm / s Slow subsidence over much of the tropics. The vertical velocities associated with radiative cooling are larger than those arising from synoptic scale adiabatic motions.
Diabatic Processes Consider first the diabatic contribution in regions in regions of active convection. Active Convection Radiative Cooling
Latent Heating SW LW Convection 100 31 9 60 Atmosphere 46 115 100 7 + 24 Three quarters (24/31) of the radiative cooling is balanced by latent heating. For 0-30 latitude, the warming rate is about 24/31 x 1.2 = 0.9 K/day. Tropical weather systems cover about 20% of the tropics. A warming rate Q c p 5 x 0.9 = 4.5 K/day in weather systems.
Ascent Associated with Latent Heating Latent heating at the rate Q c p = 4.5 K/day: w lnθ z = Q c p T W g N 2 T Q c p 10 10 4 300 4.5 24 3600 1.5 cm / s Note that the effective N is smaller in regions of moist convection implying that the estimate for w is conservative.
Area Occupied by Precipitation z These simple calculations can be used to obtain an estimate for the horizontal area occupied by precipitating disturbances. From mass conservation, the ratio of the area of ascent to descent must be inversely proportional to the ratio of the corresponding vertical velocities. Adown/Aup = wup/wdown = 1.5/ -0.5 = 3.
Area Occupied by Precipitation z The fraction of total area covered by convection is Aup/(Aup + Adown) = 1/(1 + 3) = 1/4. Allowing for a smaller N in convective regions will decrease the estimate of 1/4 to something closer to our estimate of 1/5.
Implied Rainfall ΔT = 0.9 K day -1 Mass of column Ma = (mean surface pressure)/g=1.013 10 4 kg m -2 Latent heating ΔQ = Lv ΔMw = cp Ma ΔT where ΔMw is the mass of condensed water kg m -2 and L = 2.5 10 6 J/kg is the latent heat of condensation ΔMw = cp Ma ΔT/Lv = 3.7 kg rainfall = ΔMw/density of water = 3.7/10 3 4 mm day -1 averaged over the tropics rainfall 5 x 4 mm day -1 = 20 mm day -1 in weather systems
Annual Average Rainfall Rate The annual average precipitation (mm day 1) between 1979 and 2009.
Summary Fractional changes in the thermodynamics quantities scale as δp ~ δρ ~ δθ ~ F2 p 0 ρ 0 θ 0 R 0 In the midlatitudes F 2 /Ro ~ 10-2. In the tropics F 2 /Ro ~ 10-3. Hence, the thermodynamics fields are an order of magnitude flatter in the tropics.
Summary For adiabatic motion, vertical motion scale as W 10 3 Ro ms 1 In the midlatitudes, W ~ 1 cm s -1, which is realistic. Hence, diabatic processes are not essential in describing the midlatitude circulation. In the tropics, W ~ 1 mm s -1, which is unrealistically small. Hence, diabatic processes are essential in describing the tropical circulation. Temperature gradients in the tropics are too weak for horizontal advection to balance vertical advection.
Summary For diabatic motion, vertical motion scale as w lnθ z In regions of radiative cooling, W ~ -1.5 cm s -1. In regions of latent heating, W ~ 4.5 cm s -1. Continuity implies that subsidence occupies about 3/4 of the tropics and ascent occupies about 1/4. The implied rain rates are around 4 mm/day when averaged over the whole tropics and 20 mm/day when averaged over tropical weather systems. = Q c p T
Extended Material Implications for Scaling Vertical component of the unapproximated vorticity equation t + u h ζ + ζ u h + w ζ z + k w u h h z + u h f + f u h = k ρ p T ρ 2 Compare the scales of each term with the scale for the first term for Ro << 1 and Ro ~ 1.
t + u h ζ + ζ u h + w ζ z + k hw u h z + u h f + f u h = k A B C D ρ p T ρ 2 E Term A B C D E General 1 LW DU L 2 U 2Ω a cosφ LW DU 1 R 0 2 F 2 R 0 Ro << 1 1 1 R 0 R i L 2 U 2Ω a cosφ 1 R 2 0 R i F 2 R 0 2 Ro 1 1 1 R i L 2 U 2Ω a cosφ 1 R i F 2 Note: the adiabatic scaling has been used for w.
t + u h ζ + ζ u h + w ζ z + k hw u h z + u h f + f u h = k A B C D ρ p T ρ 2 E Typical values: Ri = 10 2, F 2 = 10 3 Term A B C D E Ro << 1 1 10-1 1 1 10-1 Ro ~ 1 1 10-2 1 10-2 10-3 Middle latitudes Ro << 1 Low latitudes Ro 1 t + u h ζ + f t + u h ζ + f ( ) + f u h = 0 ( ) = 0
Extended Material Barotropic or Baroclinic? Middle latitudes: Ro << 1 Or, using continuity, Vertical gradients of vertical mass flux can generate absolute vorticity. Low latitudes: Ro 1 t + u h There is no generation of absolute vorticity in the absence of diabatic processes. Air parcels are confined to a particular level, where they move conserving their absolute vorticity. t + u h t + u h ζ + f ( ) + f u h = 0 ζ + f ( ) = f ρ ζ + f ( ) = 0 ( ρw) z
Extended Material Implications for Diabatic Scaling and the Weak Temperature Approximation A balanced theory for motions in the deep tropics. Assume that θ/ t and uh θ are much less than w( θ/ z) so that w θ z = Q θ c p T = S θ Vorticity equation t + u h ( ( ) = ζ + f ) ζ + f ρ ( = ζ + f ) ρ ( ρw) z z ρs θ θ z
Extended Material Implications for Diabatic Scaling and the Weak Temperature Approximation t + u h ( ( ) = ζ + f ) ζ + f ρ z ρs θ θ z Were there no diabatic heating (Sθ = 0), the RHS would be zero the absolute vorticity is simply advected by the horizontal wind. Heating produces horizontal divergence. Div > 0 ζ + f decreases, Div < 0 ζ + f increases